radio wave propagation fundamentals

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A.S. SAAKIAN

RADIO WAVE PROPAGATION FUNDAMENTALS

ARTECH HOUSE - 2011

To my wife Arous, and my sons, David and Mark

RADIO WAVE PROPAGATION FUNDAMENTALS A.S.Saakian Table of Contents Preface Chapter 1. Introduction 1.1. Historical overview 1.2. Classification of radio waves by frequency bands 1.3. The earth’s atmosphere and structure 1.4. Classification of radio waves by propagation mechanisms 1.5. Interferences in RF transmission links References Problems Chapter 2. Basics of electromagnetic waves theory 2.1. Electromagnetic process 2.1.1. Maxwell’s equations of electrodynamics 2.1.2. Boundary conditions of electrodynamics 2.1.3. Time-harmonic electromagnetic process. Classification of media by conductivity 2.2. Free propagation of uniform plane radio waves 2.2.1. Uniform plane wave in lossless medium 2.2.2. Uniform plane wave in lossy medium 2.2.2.a. Low-loss dielectric medium 2.2.2.b. High-loss conducting medium 2.3. Polarization of the radio waves 2.4. Reflection and refraction of plane radio wave from the boundary of two media 2.4.1. Normal incidence on a plane boundary of two media 2.4.2. Oblique incidence of vertically polarized radio wave 2.4.2.a. Radio wave incident from sparse medium onto the border with dense medium

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2.4.2.b. Radio wave incident from dense medium onto the border with sparse medium 2.4.3. Oblique incidence of horizontally polarized radio wave 2.4.3.a. Radio wave incident from sparse medium onto the border with dense medium 2.4.3.b. Radio wave incident from dense medium onto the border with sparse medium 2.4.4. Reflection of the radio wave with arbitrary polarization 2.4.5. Power reflection and transmission 2.4.6. Reflection of the radio wave from the boundary of non-ideal dielectric medium 2.5. Radiation from infinitesimal electric current source. Spherical waves 2.6. Spatial area significant for radio waves propagation 2.6.1. Principle of Huygens-Kirchhoff 2.6.2. Fresnel zones 2.6.3. Knife-edge diffraction 2.6.4. Practical applications of the Fresnel zones concept 2.6.4.a. Ring-shaped antenna directors 2.6.4.b. Ring-segment diffractors as passive repeaters for radio-relay links 2.6.4.b. Effective area of the radio wave reflection from the flat boundary References Problems Appendix-1 Appendix-2 Appendix-3 Appendix-4 Chapter 3. Basics of antennas for RF radio links 3.1. Basic parameters of antennas 3.1.1. Radiation pattern and directivity 3.1.2. Radiation resistance and loss resistance. Antenna gain and efficiency factor 3.1.3. Antenna effective length and effective area of the aperture 3.2. General relations in radio wave propagation theory

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References Problems Chapter 4. Impact of the earth surface on propagation of ground waves 4.1. Propagation between antennas elevated above the earth surface. Ray trace approach 4.1.1. Flat earth approximation case study 4.1.2. Propagation over the spherical earth surface 4.1.2.1. Evaluation of the distance to reflection point 4.1.2.2. Divergence of energy of the radio wave while reflecting from the convex earth surface 4.1.3. Specifics of the propagation over the rough and hilly terrains 4.1.4. Optimal path clearance and choice of the antennas elevations 4.1.5. Propagation prediction models in urban, suburban and rural areas 4.1.5.1. Empirical models 4.1.5.1a. The Okumura-Hata model 4.1.5.1b. Other empirical models 4.1.5.2. Physical models 4.1.5.2a. Non-LOS (Line-Of-Sight) paths 4.1.5.2b. LOS paths 4.2. Propagation between ground-based antennas over the flat earth 4.2.1. Antennas over the infinite, perfect ground plane 4.2.2. Leontovich approximate boundary conditions and structure of radio waves near the earth’s surface 4.2.3. Propagation over the real homogeneous flat earth 4.2.4. Propagation along the real inhomogeneous flat earth. Coastal refraction 4.3. Asymptotic diffraction theory of propagation over the spherical earth surface 4.3.1. Basic concepts 4.3.2. Propagation between ground-based antennas 4.3.3. Propagation between elevated antennas 4.3.4. Specifics of propagation estimates in penumbra zone References Problems Appendix-5

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Appendix-6 Chapter 5. Atmospheric effects in radio waves propagation 5.1. Dielectric permittivity and conductivity of the ionized gas 5.2. Regular refraction of the radio waves in atmosphere 5.3. Standard atmosphere and tropospheric refraction 5.4. Reflection and refraction of the sky waves in ionosphere 5.5. The impact of earth’s magnetic field on propagation of the radio waves in ionosphere 5.5.1. Longitudinal propagation of the radio wave 5.5.2. Transverse propagation of the radio wave 5.5.3. Propagation of the radio wave arbitrary oriented relative to the earth’s magnetic field 5.5.4. Reflection and refraction of radio waves in magneto-active ionosphere 5.6. Over-the-horizon propagation of the radio waves by tropospheric scatterings mechanism. Secondary tropospheric radio links 5.6.1. Analytical approaches in description of the random tropospheric scatterings 5.6.2. Physical interpretation of tropospheric scatterings 5.6.3. Effective scattering cross-section of the turbulent troposphere 5.6.4. Statistical models of tropospheric turbulences 5.6.4.1. Gaussian model 5.6.4.2. Kolmogorov-Obukhov model 5.6.5. Propagation factor on secondary tropospheric radio links 5.6.6. The specifics of the secondary tropospheric radio links performance 5.6.6.1. Antennas gain effect on link performance 5.6.6.2. Signal level fluctuations at the receiving point (fading) 5.6.6.3. Limitations to signal transmission bandwidth 5.7. Attenuation of the radio waves in atmosphere 5.7.1. Attenuations in troposphere 5.7.2. Attenuations in ionosphere References Problems Appendix-7

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Chapter 6. Receiving of the radio waves: Basic outlines 6.1. Multiplicative interferences (signal fades) 6.1.1. Fluctuation processes and stability of radio links 6.1.2. Fast fading statistical distributions 6.1.2a. Two-Ray random interference 6.1.2b. Random interference of the large number of independent wavelets 6.1.2c. Further generalization of the fast fading statistics 6.1.3. Slow fading statistical distribution 6.1.3a. Normal (Gaussian) distribution of the random variable 6.1.3b. Lognormal distribution of the random variable 6.1.4. Combined distribution of fast and slow fades. Signal stability in long-term observations 6.2. Additive interferences (noises) 6.2.1. Internal noises of one-, and two-port networks. Noise figure 6.2.2. Noise figure and noise temperature of the cascaded two-port networks 6.2.3. Noise figure of the passive two-port networks 6.2.4. Antenna noise temperature 6.2.5. Receiver sensitivity and signal threshold definition 6.2.6. Environmental (external) noise 6.2.6a. Atmospheric noise 6.2.6b. Thermal noise of the earth’s surface 6.5.6c. Cosmic noise 6.3. Methods of improvement of the radio waves reception performance 6.3.1. Noise-suppressing modems in analog CW-systems 6.3.2. Use of spread-spectrum discrete signals 6.3.3. Diversity reception technique References Problems List of Symbols List of Abbreviations Index

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PREFACE The main goal of this text is to satisfy the growing demand in propagation study materials that may be used by a proper audience of students and specialists. The core materials of the proposed text are developed based on lecture notes that have been offered by author for the graduate students at Patuxent Graduate Center of Florida Institute of Technology. The materials included into the text are extended beyond the needs of the single-semester course and may be used for continuous self-study. Main objective of this text is to support senior level undergraduate and graduate EE-students by introduction of the basic principles of electromagnetic waves propagation of radio frequencies (RF) in real conditions relevant, but not limited to communications and radar systems. It is also to emphasize the primary role of the antenna-to-antenna propagation path in overall performance of those systems. Some of the practicing engineers who need quick references to the basics of propagation mechanisms and principles of the engineering estimates and designs may use this text in their everyday routine. It may be useful not just for the students and specialists in the area of radar and communication technologies, but also for the students, scientists, and engineers of the adjacent areas of science and engineering / technology such as antenna engineering, astrophysics, geomagnetism, aeronomy, etc.

Chapter-1 is an introductory chapter, which outlines the definitions and classifications that are commonly used and are adopted by international organizations such as IEEE and ITU. A brief survey of the structure of the earth's atmosphere is considered in this chapter in support of atmospheric propagation phenomena that are covered in chapter-5.

Chapter-2 covers the basics of electromagnetic waves theory with the emphasis on those, which are specific for the RF propagation, such as polarization of radio waves, their reflections and transmission at the interface of two mediums, as well as diffraction on the knife-edge obstacles. As a subject of the special interest the Fresnel's zones are analyzed based on Huygens-Kirchhoff's principles of electromagnetics. This is to clarify the concept of ray forming, i.e. the concept of the spatial area actively involved in canalization of energy of the radio waves. Some engineering applications are presented in this chapter to demonstrate the variety of application of that concept.

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Chapter-3 is a brief journey to the basic antenna parameters that is needed to evaluate and analyze antenna-to-antenna propagation path. The end-chapter section presents the main relations, such as Friis formula, link budget equation etc., as well as introduces propagation factor, which is a most important measure of the impact of the real conditions on propagation of the radio waves. Those main relations are of importance to evaluate the radar equation, as well as for the radio links power budget analysis in communication systems.

Chapter-4 is devoted to propagation of the ground radio waves, i.e. the waves that propagate in vicinity of the Earth's surface, being affected by that interface. It appears to be methodically reasonable not to involve any atmospheric effects into consideration within the scope of this chapter, in order not to confuse the reader by mixing ground effects and atmospheric effects. The methods of propagation factor calculations are first considered here for the flat-earth approximation, based on Leontovich’s boundary conditions with further extension for the cases of convexity of the Earth’s ground. Propagation features over the inhomogeneous paths are presented within Mandelshtam’s “take-off-landing” concept that is qualitative, rather that quantitative: it allows reader better understand the behavior of the waves that propagate along “mixed” type paths. Some quantitative estimates are given to support the engineering estimates. The coastal refraction effect is analyzed as a particular case along with numerical estimates discussed. As a particular case of the propagation in mixed, rough terrains the propagation in urban, suburban and rural areas is considered in subsection 4.1.5. Last section of the chapter is concluded by introduction of the asymptotic diffraction theory (V.A. Fock) – theory of diffraction of radio waves over the spherical Earth’s surface. Engineering approaches for the practical applications are demonstrated by using sample examples.

Chapter-5 is dedicated to the effects of the atmosphere on propagation of the radio waves. Smooth refraction in troposphere and reflections from the ionospheric layers are analyzed in conjunction with the regular inhomogeneties of the refraction index in those atmospheric regions. Scattering of the radio waves of UHF and higher frequency bands from the random variations of the tropospheric refraction index (from air turbulences) are considered here by using the principles of statistical radio-physics. The results are brought to the level of engineering applications and design of the over-the-horizon troposcatter communication links.

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Despite the troposcatter radio-links are not widely used nowadays, though the understanding of the physical mechanisms of the scatterings in troposphere may become a background for the understanding of scattering phenomenon in general, so there’s no need to discuss them oneby-one for other scatterings such as from ionospheric meteor trails, raindrops, etc. The chapter is concluded by analysis of absorptions in atmosphere. Both, absorptions in tropospheric gases and hydrometeors, as well as in ionospheric layers are introduced to support signal attenuation estimates on variety of RF radio links.

Chapter-6, the last chapter, that is devoted to the reception of radio waves, which is of the high importance for the radar and communications systems design. Two types of interferences, namely multiplicative (fading) and additive (noise) are analyzed in conjunction with the signal-tonoise ratio (SNR) and communication stability. Statistical distributions of fast and slow fades are considered, as well as a combined distribution that is to predict a long-term stability of the communication system's performance. This analysis results in engineering method for power margin calculation, which is to insure that the objective of the communication stability is met. Two components of the additive noise are considered: internal noise (receiver noise) and external noise (environmental noise) that is received from the surrounding areas. The sensitivity of receiver is discussed in order to define a threshold of the received signal level. The chapter is concluded by the section that outlines the basic methods of improvement of the radio waves reception performance: (1) the use of noise-resistive signals, such as analog FM, (2) the use of the spread-spectrum signals, (3) the diversity reception technique. The main goal of this chapter is to build the “bridge” between RF-system structure and the propagation conditions and mechanisms, i.e. to show a direct “coupling” between them.

The scope of the book includes a wide variety of aspects of radio-physics; therefore the proposed text may not pretend to cover all details of the subject, but rather to encourage a creative approach amongst the students. A background in mathematics and electromagnetics that is required in engineering and physics curriculums is assumed. Some of the unique mathematical techniques and evaluations are either incorporated within proper chapters, or presented separately in appendixes. Appendix-1 provides some useful mathematical relations as well as notations adopted for the complex quantities.

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The numerous of the calculation examples are to support better understanding of the core materials of the text. The end chapter problems may become highly supportive for the study process. The problems solution manual is available for teachers / instructors from Artech House to support the teaching process. This text may be used in senior elective or entry-level graduate courses.

The list of symbols contains only generalized notations for the quantities (e.g., f for frequency) in base or derived units. Subscripts are used within the text to denote specific application of the particular notation (e.g., f c for critical frequency of the ionospheric plasma in Hz); for the multiple and fractional quantities within the text the units are indicated in symbols by the proper subscripts (e.g., f c ,

MHz

for the ionospheric plasma frequency in MHz).

First, second, third, etc. derivatives are notated with prime, double prime and triple prime respectively. Complex numbers and complex vectors are notated with the dot above the symbol.

Now, I wish to express my deepest gratitude to my reviewers, especially to Dr. D.K. Barton whose valuable comments and recommendations helped improve the text significantly. I’d like to thank Prof. V.E. Arustamyan from State Engineering University of Armenia for his revision and constructive criticism of very first draft of the manuscript in Armenian, as well as Ms. Svetlana Avanian for help with typing. I’d also like to express my appreciation to Ms. Catherine Wood for her help with grammatical and syntax corrections of most of the chapters. I’m also thankful to my colleague, Mr. Frederick Werrell for his review of chapter-3, as well as to my students for critical remarks during and after the classroom test-studies. My appreciation is forwarded to Mr. Norm Chlosta, Director of Patuxent Graduate Center of Florida Institute of Technology for his support and for providing me the opportunities to test the course with smart audience of graduate students. Finally I’d like to express my thankfulness to my wife Arous: without her understanding, patience, and support this work would not be possible.

Artem Saakian December, 2010

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Chapter 1 INTRODUCTION

1.1 HISTORICAL OVERVIEW Utilization of radio waves for communication purposes debuted in 1895, when the Italian-born American engineer G.M. Marconi and the Russian physicist A.S. Popov independently introduced the first wireless transmission of telegraph signals through the earth’s atmosphere. This initial effort was particularly notable, as it was conducted without the support of the then traditional wire guiding line first used by Marconi in 1844 for his wire telegraphy efforts. Instead, a spark-gap was implemented as a transmitting source for the electromagnetic radiation, and a coherer was utilized as a reception device. It is difficult to overestimate the importance of that invention for human society, and for the advancements it heralded for wireless communications. However, that invention might not have been possible without prior theoretical hypotheses of the existence of free propagating electromagnetic waves; which were made in 1864 by the Scottish mathematician and theoretical physicist J. C. Maxwell. Maxwell’s greatest merit was a theoretical prediction of the displacement currents in dielectrics and vacuums, which had generalized the concept of current continuity in Ampere's law [1]. The fundamental equations of electromagnetism (known as Maxwell's equations) were later updated to achieve complete and symmetric form by the introduction of magnetic currents. The introduction of displacement electric and magnetic currents in dielectrics, and in free space, made possible the comprehension of the nature of electromagnetic waves capable of propagating for long distances independent of any physical guidance such as wires, waveguides, etc. Those electromagnetic waves are identified as radio waves, when relative to scientific and commercial applications. Experimental verification of the existence of electromagnetic waves was achieved by H. Hertz in the 1880's, when he demonstrated a "propagation" of the spark from a transmitting Leyden jar to the terminals of remote receiving antenna. The revolutionary role of Maxwell's equations, which are based on experimental investigations of M. Faraday and A.M. Ampere, is hard to overemphasize for the immense progress they have allowed science, engineering, and associated technologies.

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Soon after Marconi's and Popov's experiments with the transmission of telegraph signals over the distance of several miles in 1895, Marconi was able to greatly extend the range of propagation from the UK to Canada, over the Atlantic Ocean, in December 1901. This accomplishment was made possible by the use of a sinusoidal carrier, a resonant LC-filter at the receiver's input, and a vertical grounded radiator (antenna), which thereby demonstrated the advantage of vertical polarization for the frequency ranges of tens and hundreds of kilohertz. The immense success of Marconi’s long-range signal transmission motivated engineers and research scientists to find a reasonable explanation for existence of propagation mechanisms [2]. Ionospheric propagation is a mechanism that was initially assumed to be overall predominant. It was introduced in 1902 after A.E. Kennelly, in the U.S., and O.Heaviside, in the U.K., postulated the existence of the ionized region in the upper atmosphere, which seemed to reflect the radio waves and thereby support their long range propagation. The second mechanism of propagation was based on the assumption of the existence of surface waves, which may occur only at the interface of two mediums, such as the boundary of atmosphere and earth's ground. A detailed theoretical analysis of surface propagation waves was conducted by the German physicist A. Sommerfeld in 1909, an analysis that was based on Maxwell's equations. A specific approach for the solution near the flat boundary of two ideal mediums was achieved, and was repeated ten years later by H. Weyl in a more explicit form. It was shown, that two propagating modes are existent: (1) A regular TEM mode of a spherical phase front that is specific to free space propagation; and for this mode the field strength is in inverse proportion to propagation distance; (2) A surface mode of a cylindrical phase front, which is tightly bound to the interface of two mediums, and which may exist only in the vicinity of that interface, whose field strength is in inverse proportion to the square root of the propagation distance. For moderate and large distances the second component becomes predominant. In the late 1930s the conceptualization of surface waves, relevant to the real earth's ground constants, was further developed by J. Zenneck and K.A. Norton. The third mechanism of propagation was based on the assumption of the diffraction of radio waves around the earth surface. These assumptions were proposed by several mathematicians and scientists prior to and during World War I, and were first presented by G.N. Watson in 1919 in the form of estimates of the field at the receiving

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point for the ideal conductive earth surface, as an attempt of direct solution of Maxwell's equations. Later, in 1937, Van-Der-Pol and Bremmer adopted Watson's approach to demonstrate arbitrary ground constants. Based on those developments, C.R. Burrows created an engineering approach that allowed representation of the results in a form that was more convenient for the applications. Directly following World War-II, in 1945-46, Soviet physicists V.A. Fock and M.A. Leontovich introduced a Maxwell's equations solution in parabolic form relevant to the diffraction problem. The complete theory was developed by taking into account the properties of the earth ground in a wide frequency range. It was broadly used, until quite recently, for the over-the-horizon propagation analysis (diffracted field analysis) of radio waves associated with frequencies up to tens of megahertz. Both surface wave and diffracted wave approaches result in a single conclusion: The lower a radio wave's frequency, the more favorable the propagation conditions are. Prior to and after World War-II, the need to further increase the volume of transmitted information within a single communication link, as well as the need to advance radar system performance motivated an increase in carrier frequencies. Thereby higher and higher frequencies were employed for the support of communication and radar systems designs, along with newly developed antenna radiating systems and advanced Radio Frequency (RF) components, together providing increased performance and effectiveness. Radio waves implementing frequencies of hundreds of MHz and up were considered to be employed. For those higher frequency ranges, propagation analysis is based most substantially on geometric optical approaches. The first empirical expression for the intensity of the current induced in a remote receiving antenna for distances within Line-of-Sight (LOS) was developed by L.W. Austin in 1911, in the U.S. That expression was fairly close to the formula developed analytically, in the former USSR, by B.A. Vvedensky in the late 1920s. Vvedensky’s formula was based on a ray-tracing (geometrical optics) approach. For higher frequency bands the atmospheric effects became considerable, and had to be taken into account. Thus, after World War II a significant amount of attention was focused on issues such as attenuation in atmospheric gases, as well as refraction, reflection, and scatter of radio waves in the lower and upper atmosphere. One of those effects, namely the tropospheric scatter of microwaves by atmospheric turbulence, was discovered in the late 1940s and was later theoretically validated by H.G. Booker and W.E. Gordon in 1950. Some limited military

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and commercial applications of the tropospheric scattering effect take place currently, in nominated troposcatter radio-links within the U.S. and several European countries. A new era of ionospheric propagation research and investigation initiated after the launch of the first satellite ("Sputnik"), by the USSR in 1957, to the earth's orbit. This culminated in our current large network of specialized ionospheric ground-based radars, sounding stations (ionosondes), and satellite systems worldwide which allow obtainment of a complete set of data for long-term predictions of the status of ionospheric layers. Those predictions are widely used for the radio links design and for deployment in HF and higher frequency bands, which are directly affected by the ionosphere. In modern times, the radio wave propagation theory and applications still remain as subjects of extremely high interest to the science and engineering technological world, and are in consistently in further development and expansion via numerous worldwide programs.

1.2 CLASSIFICATION OF RADIO WAVES BY FREQUENCY BANDS Radio Wave is defined by the Institute of Electrical and Electronics Engineers (IEEE) as “An electromagnetic wave of radio frequency” [3]. Each particular radio frequency belongs to the radio spectrum, which is a wide range of frequencies from several Hertz up to 3 THz. The entire radio spectrum is divided into frequency bands, shown in Table 1.1, that are based on decimal division. This standard classification is accepted by the International Telecommunications Union (ITU), which is comprised of 189 member-countries. A selection of applications pertinent to the radio waves used in engineering, science and technological efforts includes, but is not limited to: 

Wireless communication systems, including satellite communication systems and wireless local area networks (WLAN)



Radar systems



Telemetry, radio-remote control, and radio-navigation

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Adhering to ITU-R 1 recommendations, the range for each frequency band extends from 0.3x10N to 3x10N Hz 2, where N is a band number given in the first column of Table 1.1. There are many subdivisions within each band, reliant upon allocations to services and the inhabitant worldwide regions [4, 5]. Terminology noted in the last two columns of Table 1.1 is commonly used, but not officially accepted. A subdivision of Microwave Bands, shown in Table 1.2, is widely used in radar and satellite applications [6]. Both, the upper and the lower limits of the radio spectrum are outlined conventionally, and rely entirely on progressions made in science and technology. For instance, until the mid-1930’s radio-communications designs were based on technologies that allowed only utilization of frequencies lower than 100 MHz, as that represented the upper limit of radio frequencies at that time. In the 1930s and 1940s, overarching progress in the design of a new generation of the radar systems invoked a claim of involving higher frequencies. Invention of multiple new types of devices, including magnetrons, klystrons, traveling-wave tubes, and other technologies, allowed the expansion of the upper limit of radio frequencies to approximately 10 GHz and higher. In the mid 50-s to early 60-s when the new generation of quantum electronic devices, such as MASERs, were developed by C.H. Townes, (USA), N.G. Basov, and A.M. Prokhorov (USSR), they were based on achievements in quantum radiospectroscopy. Further broadening of the upper limit of radio frequencies became possible not just by including CMW, MMW and SMMW, but by also instituting coherent optical waves to develop Lasers. Implementation of these “new” types of devices made possible the amplification and generation of coherent radiations even in an optical domain, and thereby traditional radio technologies and principles became applicable to the optical frequency domain as well. Optical waves that are used for wireless information transmission and processing are sometimes called optical radio waves [7]. Table 1.3 shows the classification of optical waves by frequency bands, which is considered as a non-official classification, and is widely used by specialists in different areas of engineering and science. In tandem with other items contributing to expansion of the lower limit of the radio spectrum, there are also rationales of expansion based on the needs of global military communication services and navigation, as well as scientific research categories such as geophysics, atmosphere science, and radio astronomy.

1 2

The Radio Communications section of the ITU Conventionally the upper limit is included into the band, and the lower limit is excluded.

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Name by ITU-R

Extremely Low

-

Frequency Very Low

4

Frequency Low

5

Frequency Medium

6

Frequency High

7

Frequency Very High

8

Frequency Ultra High

10

11

12

Microwaves *

9

Frequency Super High Frequency Extremely High Frequency -

Frequency Range, Hz

< 3x10

VLF

(3 to 30)x10

MF

HF

VHF

5

3

(30 to

(0.3 to 3)x10

(30 to

SHF

(3 to 30)x10

2

3

9

9

Kilometer Waves

-

Waves Decameter

-

Waves Meter Waves

Centimeter

-1

-2

-3

10 to 10

DMW

Waves*

10 to 10

10 to 10

MW

Decimeter

-1

-4

-

Hectometer

10 to 1 -2

-

Waves

10 to 10

-3

300)x10

3)x10

4

1 to 10

6

300)x10

12

3

-

Miriameter

5

2

10

(0.3 to

4

10 to 10

10 to 10

6

(0.3 to 3)x10

-

6

(3 to 30) x

9

-

10 to 10

3

300)x10

(30 to

Name

in Meters > 10

UHF

EHF

Descriptive

length

3

ELF

LF

Wave-

Acronym

Frequency Band

Acronym

Band

Number, N

Table 1.1

CMW

Waves* Millimeter Waves

MMW

* Sub-Millimeter Waves *

SMMW

Table 1.2 Band Name

L

S

C

X

Ku

Frequency Range, GHz

1-2

2-4

4-8

8 – 12

12 – 18

Band Name

K

Ka

V

W

mm band

Frequency Range, GHz

18 – 27

27– 40

40 – 75

75 - 110

110 - 300

10

Table 1.3 Wavelength in Optical Band name

Frequency Range,

Atmosphere,

in Hz

in meters -5

-4

Far Infrared (IR) Band

2x10 to 10

Medium IR Band

1.5x10 to 2x10

Near IR Band

7x10 to 1.5x10

Visible Light

4x10 to 7x10

Ultraviolet Rays

10 to 4x10

-6

-7

-7

(3 to 15)x10 -5

(15 to 200)x10

-6

2x10 to 4.3x10

-7

-8

12

-7

12

14

14

14

4.3x10

14

7.5x10

14

to 7.3x10

16

to 3x10

1.3 THE EARTH’S ATMOSPHERE AND STRUCTURE The real earth’s atmosphere has a complex structure, which may significantly impact radio wave propagation, causing such effects as smooth refraction, scatter and energy absorption of the radio wave. Variations of the electro-magnetic parameters of the atmospheric air are highly dependent on its gaseous composition, pressure, humidity and ionization. The vertical profile 1 of distribution of the main composites of atmospheric air is shown in Figure 1.1.

1

The graph of the elevation dependence of any parameter of the atmosphere is called a "vertical

profile" of that particular parameter. Another example of vertical profiles is free electrons' distribution, which is shown in Figure 1.2, or elevation dependence of the tropospheric air refraction index (refractivity) which is shown in Figure 5.4.

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Figure 1.1 Diagram of the gaseous composition of atmospheric air (by percentage).



As one may notice from the diagram, for heights of up to approximately 90 km, the gaseous composition of the atmosphere is homogeneous as a result of the continuous mixture caused by ascending, descending and horizontal air streams that permanently exist in that area. At that range of height, the atmospheric air is composed of about 78 percent molecular nitrogen and about 20 to 21 percent molecular oxygen, despite the fact that they have different molecular weights. The remainder is a mix of carbon dioxide, argon, ozone, and other gases. This atmospheric area of homogeneous distribution of gases is conventionally divided into two regions, the troposphere and the stratosphere. The troposphere is the lowest portion of the earth's atmosphere. It contains approximately 80 percent of the atmosphere's mass, and 99 percent of its water vapor and aerosols. The average ceiling of the troposphere is approximately 17 km in its middle latitudes. It is deeper in tropical regions (up to 20 km more), and is shallower near the earth’s poles (about 7 km in summer, and an indistinct measurement in the winter). The remaining portion of the homogeneous atmospheric area is known as stratosphere. One of the specific features of the troposphere, which distinguishes these two regions, is a rapid decrease of the concentration of water vapors reliant on elevation. In fact, the humidity level is highly dependent on weather conditions. The main characteristics of the troposphere are air pressure,

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usually measured in millibars, the absolute temperature, measured in kelvins, and absolute humidity, also measured in millibars. Based on numerous observations and collaborative measurements carried out worldwide in 1925 the International Commission for Aeronavigation introduced the “International Standard Atmosphere”, which was later renamed, and is called now the Standard Troposphere. It represents a hypothetical troposphere with characteristics that are averaged from the measurements to portray all locations and seasonal influences. Those characteristics are noted below [7]: •

A sea-level air pressure of 1013 millibars



A constant vertical gradient of the air pressure of negative 120 millibars per kilometer



A sea-level temperature of 290 K



A constant vertical gradient of the air temperature equal negative 5.5 K per kilometer



Relative humidity of 60 percent, which is assumed to remain elevationindependent.

The average vertical profile of the temperature is:

T (h ) = T0 − 5.5 ⋅ h .

(1.1)

Here T0 = 290 K and elevation h in kilometers. (1.1) may be understood based on the following rationale: the tropospheric air is transparent to solar thermal radiation; thereby it does not cumulate the heat directly from solar radiation. The bulk of that thermal radiation penetrates through the troposphere freely and reaches the earth's surface, where it is absorbed. The air layers adjacent to the earth’s surface become heated due to heat transfer and air convection processes. The higher the elevation, the less is the effect of these processes, resulting in linear decay of the temperature given by expression (1.1). A similar linear vertical profile is specific for averaged tropospheric air pressure 1. As noted later in chapter 5, from the viewpoint of atmospheric propagation problems analysis, the most important parameter is the dielectric permittivity of the atmospheric 1

As noticed from numerous observations, for the altitudes higher than 10 km the linearity of the

vertical profile of mean temperature and air pressure becomes significantly destroyed. However, the atmospheric air at those high altitudes is extremely sparse; therefore those distortions do not play a significant role in propagation mechanisms specifically on radar and communication paths.

13

air. It closely relates to the refraction index, defined as: n =

ε . The mean value of

tropospheric air refraction index, being tightened to atmospheric air characteristics, appears as a smooth, linearly decaying function of elevation (see section 5.3). At the same time a large number of globally noted experiments and measurements, undertaken over many decades, have demonstrated existence of seasonal and random fluctuations of atmospheric air in all atmospheric regions. In the troposphere the main mechanism of their generation is stipulated by the horizontal and vertical movements of air masses. Under proper conditions those movements become turbulent, i.e. the air masses of different refraction indexes are mixed randomly in space and time resulting in random space-time fluctuations of the refraction index. These turbulent movements may be observed in the visible region of the spectrum of electromagnetic radiation in multiple ways to include the twinkling of stars, the wavering appearance of objects seen over the earth's surface that is heated by sun, and the conversion trails left by the exhaust gases of aircraft jet engines. The same processes take place in radio frequency bands, and all of these examples demonstrate that the air in the troposphere is present in a random, erratic flow. The stratosphere does not alter the propagation of radio waves significantly, as it is the atmospheric region containing fairly constant gaseous, whose composition is of a very low density. The Ionosphere is the upper part of the earth’s atmosphere which extends from 60 kilometers upwards. At these elevations the atmospheric air becomes ionized, i.e. the neutral atoms and molecules split into positively charged ions and free electrons. This state of matter is called plasma. Regarding the latest data obtained from the ionospheric research, the upper border of the ionosphere is above 20,000 km. Ionization of atmospheric air is caused by intensive radiation flow, emanating from the outer space that is sometimes referred to as cosmic rays. Cosmic rays are the intensive flow of a variety of elementary particles and photons 1 composed of a wide range of energies. A major contribution to the total intensity of those radiation comes from the sun. Physics courses will teach that in order to ionize the gas cloud, (i.e. to tear off an electron from

1

Acting as “the envelopes” of electromagnetic waves, photons are often considered as particles,

due to some properties that are specific to the elementary particles. For instance, these particles are able to eject the electrons from the atom's boundary by bombarding them, or bouncing them from each other like the balls in billiard game.

14

an atom or a molecule) a quantum of energy greater than a work function, We is to be applied. This amount of energy must be expended to break the bond between electron and atom (or electron and molecule), relative to each particular type of atom (or molecule), and may be acquired from the composite parts of cosmic ray, i.e. from this radiation that comes from outer space. Forms of this radiation may include: •

Elementary particles (protons, neutrons, electrons etc.), or



Photons of electromagnetic character, such as ultraviolet, X-rays and

γ − radiation. Hence, two types of ionization are to be distinguished: (1) Strike-Ionization, caused by particles, and (2) Photo-Ionization, caused by photons. For photo-ionization the energy carried by photon must be greater than, or equal to the work function, i.e.

h f ≥ We ,

(1.3)

where: h = 6.626 ⋅ 10 − 34 J ⋅ s is a Planck's constant, and f is the frequency of the photon. The maximum wavelength (threshold wavelength) of the radiation, which is able to cause the ionization, may be found from (1.3) as

λ ≤ λ max = where c = 3 ⋅ 10

8

ch , We, min

(1.4)

m / s is the speed of light in free space1, and We, min represents the

minimal energy in joules that a single particle or photon in a cosmic ray may have. That portion of energy may also be expressed in electron-volts (eV), if the relation

1 J = 1.6 ⋅ 10 −19 eV is applied. Among the composite gases of atmospheric air, nitrogen −18 J . Thus the oxide has lowest value of the work function, We = We , min = 1.48 ⋅10

maximum wavelength of radiation that is able to ionize this gas is found from (1.4) as

λ max = 0.134 µ . From (1.4) one also may realize that only the ultraviolet radiation, as well as the radiation of the shorter wavelengths such as X-rays and γ − radiation, are able to cause the ionization of atmospheric air. From numerous observations and

1

Note that the ionization process itself does not depend on the intensity of radiation, but instead

on the wavelength of the ionizing radiation.

15

measurements it has been concluded, that the photo-ionization in the real atmosphere is caused by radiation in the range of wavelengths from 0.03 to 0.14 µm. The ionization degree of atmospheric air may be expressed by the number of free electrons per unit volume (predominating per cubic centimeter), which is known as the concentration of electrons (or plasma concentration), N e . An experimental graph of the vertical profile of ionospheric plasma concentration is shown in Figure 1.2.

Figure 1.2 Vertical profile of plasma concentration in the real ionosphere

The ionization of atmospheric air starts from the height of about 60 km on upwards. It is shown analytically [7] that for the hypothetical homogeneous gaseous composition of the atmosphere and for the exponential model of the vertical profile of air pressure, the single-layered vertical profile of the plasma concentration will be obtained when the ionizing radiation is monochromatic. However, in real conditions for a complex structure of air composition, as well as for the complex mixture of ionizing radiation (multi-particle and multi-photon cosmic rays), the vertical profile of the ionospheric plasma concentration becomes multilayered (stratified) as shown in Figure 1.2. As may be noted from that figure, four layers exist during the day: D-layer (60 to 90 km), E-layer (90 to 120 km), F1-layer (180 to 230 km), and F2-layer (230 km and up). A sporadic ES-layer of fairly high plasma concentration may appear and disappear randomly. Starting from elevations of about 400 km and higher, there is no stratification, but only a smooth decrease of the concentration of ionospheric plasma.

16

Aside from ionization, there are also recombination processes, where randomly roaming free electrons may collide with positively charged atoms and molecules, resulting in the recovery of neutral particles. It thereby becomes plain, that the rate of the recombination process is as great as the number of free electrons and positive charged particles, i.e. as great as the corresponding plasma concentration. Consider a limited spatial volume of initially neutral atmospheric air is being ionized. Then, after ionization is complete, ( i.e. plasma is generated) the considered volume will remain neutral overall, due to the number of generated negative free electrons remaining equivalent to the number of positively charged atoms, or to the molecules with an equal amount of total charge. Initially, when ionizing radiation is applied to neutral gas, the rate of generation of charged particles is nearly constant, causing an increase of plasma concentration. When plasma concentration increases, after a certain period of time a balance between ionization and dissociation will be achieved, so far as the rate of recombination is proportional to the ionospheric plasma concentration N e . Thus, the plasma's concentration stabilizes after the transition period ends. Typically the ionization and recombination processes are in balance at noon and at midnight; whereas during sunrise (or sunset), when the radiation coming from the sun appears (or disappears) this balance is destroyed, and smooth changes of the ionization concentration (increasing in the morning hours, and decreasing in the evening hours) may be observed. This phenomenon results in the disappearance of two layers during night-time hours: the Dlayer and the F1-layer. Only two of the overall layers, namely the E-layer and F2-layer, will remain during night-time hours, with a plasma concentration that is much less than it is during the day (Figure 1.2). Finally, it must be noted that the profile of ionospheric plasma concentration shown in that figure is simply a graph of the averaged values of N e . In reality, there are random fluctuations of N e around each point of the graph, similar to those occurring in a refraction coefficient of troposphere. There are several factors, which cause the fluctuations of N e : o

Random fluctuations of intensity of the ionizing radiation emanating from outer space

o

Turbulent movements, caused by horizontal and vertical drafts of the ionospheric plasma

17

o

Fast invasion of micro-meteors and cosmic dust, acting as an additional source of the ionization, causing highly ionized and randomly distributed prolonged paths of ionizations

o

Magneto-hydrodynamic waves originated by the influence of the earth's magnetic field in the presence of mobile masses of ionized air.

These random fluctuations of ionospheric plasma concentration result in random scatterings (not reflections) of t radio waves, which are most intensively observed in HF and VHF bands.

1.4 CLASSIFICATION OF RADIO WAVES BY PROPAGATION MECHANISMS Two types of radio waves propagation are: (1) guided propagation, and (2) free (unguided) propagation. Free (unguided) propagation of radio waves occurs between corresponding antennas in the earth’s atmosphere, under-water, or in free space 1; in contrast to guided propagation which occurs in man-made guiding systems, such as wire-lines, coaxial cables, waveguides, and optical fibers. However, only free propagating radio waves are subjects for detailed consideration in this textbook. The following terms are introduced for classification of radio waves by propagation mechanisms between transmitting and receiving antennas: 1. A direct radio wave (or simply “direct wave”) is a radio wave that propagates from a transmitting to a receiving point over “an unobstructed ray path” [3], i.e. over the trajectory that is either a straight line, or close to one. One example of a direct radio wave is a radio wave that propagates via an earth-to-space (uplink), space-to-space, or space-to-earth (downlink) path of a satellite communication system (see Figure 1.3a).

1

Free space is defined as "Space that is free of obstructions and that is characterized by the

constitutive parameters of a vacuum" [3].

18

Figure 1.3. A). Direct radio wave, b). Reflected radio wave, c). Scattered radio wave, d). Diffracted radio wave

A reflected radio wave (or a reflected wave) is a wave that travels to the receiving point via a reflection from a boundary of two media, where the boundary of a size that is much larger than a wavelength and is relatively close to the flat surface 1. The examples portray radio waves traveling to the receiving point via reflections are: the reflections from the earth's surface or via structures such as landscapes, metallic bodies placed into orbits, etc. Near ideal reflection occurs via the ionized layers in ionosphere, when the radio wave of low frequencies (up to 30 MHz) propagates between corresponding points A and B, as shown in Figure1.3b. 3. A scattered (or secondary) radio wave is one that appears when the scatterings take place during propagation. Scatterings may be observed when the radio wave stochastically reflects from a rough, random surface with the average size of the roughness comparable or less than the wavelength, or during propagation of the radio wave through a medium which contains randomly shaped and/or space-time-distributed irregularities. Typically these volumetric scatterings are observable when the dimensions of scatterers (or random globules) are comparable or less than the wavelength itself. 1

The IEEE standard definition [3] is as follows: “For two media, separated by a planar interface,

that part of the incident wave that is returned to the first medium.”

19

Each globule then plays the role of a secondary (virtual) radiator of the random origin. A superposition of the multitude of secondary waves that arrive to the receiving point B produce the resultant field. Radio waves scattered from the small-scale irregularities of the refractive index of tropospheric air may be considered as an example of a secondary radio wave (Figure 1.3c). These random irregularities exist in the lower portion of the atmosphere, or the troposphere (even in clear atmospheric air), as turbulences caused by the horizontal and vertical movements of atmospheric air masses. The random volumes of the irregularities of atmospheric air are able to scatter the microwaves effectively within wide range of angles. That effect is the main mechanism of long-range propagation of DMW and CMW, which are able to propagate over the horizon for distances of many hundreds of kilometers. This phenomenon is known as far (over-thehorizon) tropospheric scatter propagation of microwaves, or as troposcatter propagation. The scatter propagation caused by the irregularities of ionization in the ionosphere is another example of propagation by the mechanism of secondary radio waves. Those irregularities are mainly generated by small-scale particles (micro-meteors) and dust coming from outer space. These particles contain highly ionized footprints with an average length of several meters. Therefore the phenomenon of scatter propagation through the irregularities of the ionosphere takes place mainly with radio waves within the VHF frequency band. Note, that both micro meteors and space dust are present in the ionosphere permanently, so these types of secondary waves may be observed all day long, regardless of the season. 4. A diffracted radio wave (or simply a diffracted wave) is defined as “an electromagnetic wave that has been modified by an obstacle or spatial inhomogeneity in the medium by means other than a reflection or refraction” 1 [3]. As known from the college physics course, any material body placed across a propagation path may be considered as an obstacle only if its linear dimensions are comparable or greater than the wavelength. Otherwise the wave will spill over that material body (i.e. will diffract on it) and will easily arrive at the observation point placed behind the obstacle. For a rough estimate of propagation distance one may take into account that the diffraction will take place when

h ≤ λ , where h is shown in Figure 1.3d. The following approximate geometrical 1

Later we will use the term “refraction” to identify the bending of the propagation path in the

stratified troposphere, or ionosphere, and the term “refracted wave” to identify the wave that penetrates from one medium into the second through their interface.

20

relations may be written based on an expansion of the cos Θ into Taylor's series for small Θ angles. In fact, for real conditions the propagation distances are much smaller than the earth’s radius: a = 6370 km. Thus

  1 Θ2 Θ h = a − a cos ≈ a 1 − 1 − 2   2 4 where

 Θ 2 a  = , 8 

Θ = r/a

(1.5) (1.6)

represents a geo-central angle between corresponding points A and B, and r indicates a curvilinear distance (arc) between those points. Taking into account the formula h ≤ λ , as well as (1.5) and (1.6) one may define a maximum distance of the propagation of a diffracted wave as

r≈

8aλ ,

(1.7)

or, by expressing r in kilometers and λ in meters, we may obtain

rkilometers ≈ 7

λ meters .

(1.8)

The expression (1.8) portrays a rough estimate of the limits of propagation distances through diffraction mechanism, for radio waves of differing frequency bands. From expression (1.8) it may be noted that the greater the wavelength and the longer the propagation distance, the easier the diffraction can occur. This mechanism creates more favorable conditions for the propagation of the radio waves at frequencies less than 30 kHz, at which the propagation distances may reach up to 1000 km. On the other hand, for higher frequencies such as HF, the diffraction mechanism of propagation may not be considered as an essential propagation mechanism as the maximum distances of HF propagation, caused by the diffraction, become almost equal or even less than the LineOf-Sight (LOS) 1 distance, while the real observations portray the distances as much greater than as noted from (1.8). The set of terms, “direct”, “reflected”, “scattered”, and “diffracted” relate to the mechanisms specifying how the radio wave arrives the observation (receiving) point.

1

“Line-of-Sight” is a term that is common to the propagation paths in frequency ranges VHF and

higher. The higher a frequency, the closer propagation properties are to those of the optical waves.

21

Now we present another set of terms, which allow the classification of radio waves based on spatial area that the propagation paths are traveling through: 1. A sky wave (or ionospheric wave) is “…a radio wave that propagates obliquely toward, and is then returned from the ionosphere”. [3].This type of radio wave is localized in the spatial region between the ionosphere and the earth’s surface, and is shown in Figure 1.3b. It is thereby evident that it may also be called a reflected wave, if we intend to specify the mechanism of propagation. Note, that the long range propagation distances of HF radio waves are stimulated by the mechanism of subsequent reflections of sky waves from the ionosphere and the earth's surface. These results in propagation distances of thousands of kilometers (see Figure 1.4).

Figure 1.4. Illustration of the ionospheric propagation mechanism of the HF radio waves

2. A ground wave is a radio wave that propagates “From a source in the vicinity of the surface of the earth, i.e. a wave that would exist in the vicinity of the earth's surface in the absence of the ionosphere” [3]. Two ground wave modes may be considered as independently existing: 

A surface wave is a non-TEM mode that propagates along the earth’s surface and is guided by the air-ground boundary; this type of wave is specific to radio waves generated by “so called” low-elevated antennas

1

Antenna elevation above the earth’s surface that is close to, or less than a wavelength

22

1



A space wave is a superposition of direct and ground-reflected TEMwaves in the vicinity of the earth’s surface; this type of wave is specific to radio waves generated by so called high-elevated antennas 1, mainly in frequency ranges of VHF and higher.

Per above definitions, one may conclude that the contribution of each component into the ground wave depends strictly on the radiating antenna height above the earth's ground surface. For an antenna with an elevation of several or more wavelengths above ground level (high-elevated antenna) the space wave component of the ground wave is predominant. Otherwise, for a low-elevated antenna, i.e. for an antenna with an elevation that is comparable or less than the wavelength, the surface wave component will become predominant. These issues will be discussed further in chapter 4.

1.5 INTERFERENCES IN RF TRANSMISSION LINKS The quality of information transmission, via a radio transmission link between corresponding points, as well as the quality of radar performance, is highly impacted by the presence of disturbances to the desired signal’s reception path. The term interference is commonly used in communications engineering practices to manifest those disturbances of a desired signal. Meanwhile, in physics and in electromagnetic theory the same term is used to convey the superposition of the considered electromagnetic wave intermingling with other electromagnetic wave(s): either of different origin, or of the same origin, but arriving at the observation point from different propagation directions. In this section we will amplify the first meaning of the term interference. However, the second meaning will be examined in the following chapters. Interferences typically destructively impact the content of information received, vs. the information actually transmitted. Considered as destructive random processes, which

1

Antenna elevation above the earth’s surface that is greater than several wavelengths. In order

to be considered as a high-elevated antenna, the feeding line must be non-radiating.

23

occur in the receiving mode, these interferences may come into view in two different forms: •

They may be in the form of random fluctuations of the parameters of the desired signal. For instance, when a monochromatic signal 1 passes through the propagation path, then both the amplitude and the phase of the signal will randomly fluctuate. The rate of these random fluctuations is much less than the rate of the oscillations of the signal’ carrier; as typically the quasi-period of random fluctuations is hundreds of milliseconds and up. Thereby, the fluctuations may be simply interpreted as multiplications of the amplitude of the signal by slow random variable(s). This is similar to passing the signal through a linear two-port network with a randomly fluctuating transmission coefficient, where the input signal and voltage (or current) is multiplied by the transformation coefficient of a random character to determine the output. Hence, this type of interference is referred to as multiplicative interference, or simply signal’s fading. This form of interference is originated in the propagation medium, i.e. along the propagation path.



Another form of interference appears in both the propagation medium and in conjunction with the receiver (including a receiving antenna). It occurs independently and simultaneously with the desired signal, superimposing (overlaying) to the desired signal, and is therefore known as additive interference, or in simple terms: noise. In contrast with fading, this type of interference may be specific to an extremely wide spectrum, affecting all applicable RF frequency bands

Based on the above definitions, the output signal of the receiver may be written in time domain as

~ s (t ) = κ (t ) ⋅ s (t ) + n(t ) .

(1.9)

Here s ( t ) notes the desired signal, κ (t ) is the multiplicative interference, and n (t ) represents the additive interference. Both, κ (t ) and n (t ) indicate random processes.

1

To present a strict approach, the quasi-monochromatic signals are to be discussed, simply

because any information carrying, modulated signal is never purely monochromatic. However, in practice, RF signals that are transmitted through propagation paths are narrow-banded in most cases, thus they may be considered as monochromatic for analytical purposes.

24

To present an example of multiplicative interference, the fading of the voice volume of an HF-broadcast audio signal may be considered. Another example is the “deep fades” encountered with a receiving antenna output signal, which is indicated by secondary troposcatter associated with over-the-horizon microwave radio links. Two types of multiplicative interference, the slow and fast fading of signal level 1, are presented in the classification chart shown in Figure 1.5. Additionally, a display of seasonal variations of a received signal is included (conventionally), to indicate a multiplicative interference. The uniform hum of an HF-broadcasting receiver’s audio output may be considered as an example of additive interference(s) which permanently exists, regardless of the existence of a desired signal within a broadcasting channel. In Figure 1.5 below, the reader will find clarifications for the classification of additive interference types (noise(s)). Beneath Figure 1.5, interference / noise types are defined in greater detail.

Figure 1.5 The classification of interferences associated with RF links

1

Occasionally the seasonal variations of signal levels may also be considered as an example of

multiplicative interference. However, they may be excluded if the pre-known character of those variations is accounted for.

25

1. Receiver noise (or internal noise): A noise type that appears in different areas of the pre-detection (linear) section of receivers including antennas, feeders, RF and IF amplifiers, filters, etc. The nature of inner noise is stipulated by several physical phenomenons to include: 

Random thermal movements of free electrons in resistors, conducting wires, antennas, and associated elements: The long-term averaged summation of vector velocities for electrons is zero. Therefore, if no outer forces are applied, there is no draft of the electron cloud within the circuit. For short-time periods, the chaotic motions of electrons result in “jumps” (short pulses) of current/voltage; and the overlay of a large number of these short and overlapping pulses emanates as a steady hum, which is known as thermal noise. The spectrum of thermal noise covers the entire RF range, and the higher the temperature, the more intensive are the movements are and bigger the power spectral density of thermal noise. However, thermal noise may exist even without a current flowing through the associated element



The discrete character of associated particles, i.e. electrons or vacancies, when flowing through an active electronic component, such as a diode, transistor, running wave tube, etc. may cause a random sequence of splashes within the current contained in the circuits. This phenomenon is known as shot effect. However, it will not occur if the current is sufficiently large, as the natural averaging effect of a larger current will flatten off the fluctuations of the current. It is only for small currents flowing through a component that the shot effect becomes significant. This is particularly specific to the first conducive stage of RF signal amplifiers (typically placed right at the receiving antenna output, i.e. before the feed line associated with receiver).

2. External noise: This noise type usually penetrates into the receiver from the propagation medium. The three types of external noises are described as follows: •

Man-made noise: Examples of man-made noise include the noise generated by the ignition system of automobiles, powerful electric motors, power transmission lines, and high power distribution equipment as well as by other residential and commercial systems. In these cases, electromagnetic energy will be radiated into our frequency bands of interest

26



Atmospheric noise, which is generated by two sources, and therefore may appear within two types:  Lightening discharges within the troposphere, and  A steady noise background, generated by collisions between atoms and molecules within the tropospheric air layer. The spectral density of the first type of atmospheric noise will typically be concentrated in the lower part of the RF spectrum; i.e. in a frequency range extending up to 30 MHz. For higher frequencies (i.e. VHF, UHF and higher) the intensities of this first type of noise may be ignored, as they are no comparable to the effects of the second type of noise. The second type of noise is characterized by “noise spectral power density”, which increases in direct proportion to the square of the frequency, and therefore, reigns significantly within the microwave frequency bands.



Cosmic noise signifies the RF radiation emanating along the observation direction border(s) of outer space. The intensity of cosmic noise is reliant upon the location of where the receiving antenna is directed / aimed. Note that the ionosphere is impenetrable by radio waves with frequencies of less than 30 MHz, so this type of noise is applicable only to radio links at operating frequencies higher than VHF; most typically of those whose receiving antenna(s) is directed skywards ( i.e. for satellite downlinks). The RF thermal radiations of the earth's surface may also be considered as cosmic relative to these radio links. Numerous radio-astrophysical observations have indicated that the most intense cosmic noise comes either from the sun, or the center of our associated galaxy i.e. the Milky Way, and/or from several other constellations within the universe.

Additional details about interferences, and their estimates, may be found in chapter 6.

REFERENCES [1] Maxwell, J.C., “A Dynamical Theory of the Electromagnetic Field,” Proc. Roy. Soc., London, Vol. 13, 1864, pp. 531-536 [2] Burrows, C.R., “The History of Radio Wave Propagation Up to the End of WW-I,” Proceedings of the IRE, Vol. 50, 1962

27

[3] IEEE Standard Definitions of Terms for Radio Wave Propagation. Std 211-1997 [4] Withers, D., Radio Spectrum Management, IEE Telecommunication Series 45, 1999 [5] Derek, M.K., Ah Yo, and Emrick, R., “Frequency Bands for Military and Commercial Applications,” Ch.2 in Antenna Engineering Handbook, 4-th Ed. McGraw-Hill Co., 2007 [6] IEEE Standard 521-2002 (Revision of IEEE Standard 521-1984), IEEE Standard Letter Designations for Radar-Frequency Bands, 2002 [7] Dolukhanov, M. P., Propagation of Radio Waves, Moscow, USSR: Mir Publishers, 1971

PROBLEMS P1.1 Create a chronological timeline and place all the historical events described in section 1.1 into it. Use that time-line to indicate other events you have found from your own educational and/or professional career to archive the history of radio wave propagation theory and practice. P1.2. Convert the following free-space wavelengths into frequencies and oscillation periods utilizing scientific notations and proper engineering prefixes if applicable: 300 km, 1m, 0.5 µm. Determine which frequency bands they are associated with: Answer Wavelength

Frequency

Periods of Oscillation

300 km = 3 ⋅ 10 5 m

1000 Hz = 1 kHz (ELF)

10 −3 s = 1 ms

1m

3 ⋅ 10 8 Hz = 300 MHz (VHF)

3.33 ⋅ 10 −9 s = 3.33 ns

0.5 µ = 5 ⋅ 10 −7 m

6 ⋅ 1014 Hz = 600 THz (visible light)

1.67 ⋅ 10 −15 s = 1.67 fs

28

P1.3. Use expression (1.4) to calculate the ionizing radiation wavelengths ( λ max thresholds) for the following component-gases of the atmospheric air, when the ionization work functions We are given in electron-volts (eV):  Nitrogen oxide – 9.25  Atomic oxygen – 13.61  Molecular hydrogen – 15.42  Atomic hydrogen – 13.6 Answer: 0.134µ, 0.09131 µ, 0.0806 µ, 0.09135 µ P1.4. Based on the answer from Problem P.1.3, and the information from Table 1.3, assess whether or not ultraviolet radiation is able to ionize molecular hydrogen. (The answer is YES or NO; provide the numerical validation of your answer.) P1.5. Determine the correct information to include, and populate the table below based on the material addressed in expression (1.8). Show the results for the wavelength limits provided for each frequency band. The following statement, "The greater the length of the radio wave, the bigger is the propagation range" may be concluded from this table. Would that statement prove true or false for all propagation cases? Explain your answer. Table P1.1 Maximum propagation Band

Wavelength, λ in meters

distances, rkm of radio waves, in kilometers

3

4

LF

10 - 10

MF

102 - 103

HF

10 - 102

P1.6. What are the advantages of the increase in carrying frequencies supporting communication systems and radars? Additionally, how may that increase change radio link designs? Explain in your own words. P1.7. While listening to an AM broadcast (particularly in the HF-band), how may one identify the presence of multiplicative interference (fading), and additive interference (noise), and distinguish them from one another? Explain your answer in your own words.

29

CHAPTER 2. BASICS OF ELECTROMAGNETIC WAVES THEORY 2.1 ELECTROMAGNETIC PROCESS

2.1.1 MAXSWELL'S EQUATIONS OF ELECTRODYNAMICS

1

Dynamic electromagnetic process is considered a unity of two processes, identified as time-varying electric and magnetic. Analytically, the unification of these processes is expressed by a system of Maxwell's equations, as follows:

∇×H =

∂D + J tot ∂t

(Ampere's law)

(2.1)

(Faraday's law)

(2.2)

∇ ⋅ D = ρ tot

(Gauss' law)

(2.3)

∇⋅B = 0

(Law of Continuity of Magnetic Field Lines) (2.4)

∇×E = −

∂B ∂t

In this system of equations, E and H represent electric and magnetic field strengths respectively, whereas D and B represent electric and magnetic induction vectors (or electric and magnetic flux density vectors) for particular points of space. These vectors are further coupled by constitutive parameters of medium ε , ε 0 , µ , and µ 0 .

D = ε 0 ε E , and B = µ 0 µ H .

(2.5)

Here absolute dielectric permittivity and absolute magnetic permeability of free space (vacuum) are

1

The term "Electrodynamics" is utilized here to describe the area of "Electromagnetism" relative

to time-varying electromagnetic processes, in contrast to "Electrostatics" and "Magnetostatics" which are to represent to areas of study relevant to time-constant electric and magnetic fields, respectively.

30

ε0 =

1 10 −9 F/m, and µ 0 = 4π ⋅ 10 −7 H/m 36π

(2.6)

respectively, whereas ε and µ signify relative dielectric permittivity (dielectric constant) and relative magnetic permeability (magnetic constant) specific to the particular medium. All media herein are categorized based on ε and µ as follows: 1). Constant and parametric media presenting time dependence 2). Homogeneous and inhomogeneous media presenting spatial dependence 3). Specific character such as isotropic (for scalar ε and/or µ ), and anisotropic (for tensor ε and/or µ ), media 4). Linear, if ε and/or µ are field intensity independent, and non-linear, otherwise. If the relations in equations (2.5) are taken into account, then equations (2.1) and (2.2) may be rewritten as:

∇ × H = ε 0ε

∂E + J tot , ∂t

∇ × E = −µ 0 µ

(2.1a)

∂H . ∂t

(2.2a)

Per expressions (2.1a), (2.2a), and (2.3), one may realize that electric and magnetic field vectors E and H interrelate and are coupled to the volumetric total conducting electric current J tot and volumetric total electric charge ρ tot . They both ( J tot and ρ tot ) are defined by the presence of the same electric charges (electrons, ions) that are able to move freely within the considered spatial area 1. Therefore, it is not surprising that they are coupled by the law of current continuity:

∇⋅J = −

∂ρ tot . ∂t

(2.7)

The movements of free electric charges may be stipulated either by external force, or by internal electromagnetic field, once it's generated so far by the externally forced movements of charges. Thus, the total conducting current and charge may be presented as sums:

J tot = J Ext + J

(2.8)

ρ tot = ρ Ext + ρ

(2.9)

1

Here defined in contrast to the bonded charges, such as those bonded to the crystalline lattice of solid matter.

31

Here J Ext and ρ Ext represent the components of the current and charge that are stipulated by external source(s), and J and ρ induced by the electromagnetic field within the medium that contains free moving charges. It’s well known, that Coulomb's force, applied to the charged particle, is in direct proportion to the amount of charge and the electric field intensity. Therefore current J is expected to be in direct proportion to the electric field as well:

J =σ E

(Ohm's Law in differential form)

(2.10)

The coefficient σ is called a conductivity of the medium, and is proportional to the concentration of free charges (i.e. proportional to volumetric density of free electric charge) in considering point of space. If equations (2.8) and (2.10) are substituted, then (2.1) may be rewritten as:

∇ × H = ε 0ε

∂E + σ E + J Ext . ∂t

(2.1b)

The physical meaning of the first term, in right hand side, translates to a volumetric current density that exists in space, regardless of the existence of free charges, i.e. it is stipulated in dielectric medium (or vacuum) by the time variations of the electric field. This term is conventionally known as a volumetric displacement electric current, or displacement electric current volumetric density, J dis . By analogy, the right hand side term in equation (2.2) is known as a displacement magnetic current volumetric density. The importance of these two displacement currents is difficult to overestimate. Indeed, only these currents are accountable for keeping the electromagnetic process running, as they permit continuous energy exchange between electric and magnetic fields within the united electromagnetic process. The balance of energy interchange within the spatial area containing the electromagnetic process may be assessed as follows: First we multiply both sides of equations (2.1b) and (2.2) by E and H respectively, and subtract equation (2.1b) from equation (2.2). Next, we will refer to identity (A1.2.4.2), given in Appendix-1, which may be applied to the left hand side to result in the following equation:

H ⋅ (∇ × E ) − E ⋅ (∇ × H ) = ∇ ⋅ (E × H ) Now use the transforms that are applied to the right hand side:

32

(2.11)

− µ0 µ H

∂H ∂E − ε 0ε E − σ E ⋅ E − E ⋅ J Ext = ∂t ∂t

∂  µ 0 µ H 2 ε 0ε E 2 = −  + 2 2 ∂t 

  − σ E 2 − E ⋅ J Ext . 

(2.12)

Here, the expression in parenthesis represents the energy per unit volume (volume density of energy) cumulated by electric and magnetic fields respectively. The second term represents the power loss per unit volume, due to finite conductivity of the medium. The higher the conductivity of medium, the greater is the rate of collisions of chargecarrying free particles within the crystalline lattice, and therefore the rate of transformation of energy of the electric field into heat is increased. The last term in equation (2.12) represents solely the power of the external source implemented into the electromagnetic field. Now we may integrate both, right and left hand sides, presented by the equations (2.11) and (2.12) respectively within the volume V that contains an external source J Ext .

∫ [∇ ⋅ (E × H )] dV =...

V

... = −

 ε 0ε E 2 ∂  µ0 µ H 2 ∫  − ∫ σ E 2 dV − ∫ E ⋅ J Ext dV . + dV dV ∫   ∂ t V 2 2 V V  V



(2.13)



Here Wm = (1 / 2) µ 0 µ H 2 dV and We = (1 / 2) ε 0 ε E 2 dV denote total energies V

V



cumulated within volume V by magnetic and electric fields, PL = σ E 2 dV denotes the V

total thermal loss of power of the electromagnetic field within volume V, and

PExt = − ∫ E ⋅ J Ext dV -- the total power given to the electromagnetic field by the external V

source 1. The identity (A1.2.2.2), given in Appendix-1 that is known as Gauss theorem, may be applied to the left hand side of equation (2.13), namely as:

∫ [∇ ⋅ (E × H )] dV = ∫ (E × H ) d S ,

V

(2.14)

S

1

The negative sign indicates a power that is inserted into the electromagnetic field, in contrast to the positive power that is subtracted from the field.

33

Here the volume integral is replaced by the integration along the closed surface S, which surrounds volume V. Note that d S illustrates a vector surface element directed outward of the volume, normally to the surface at the considering point. Therefore, equation (2.13) may finally be rewritten as:

PExt =

∂ (Wm + We ) + PL + ∫ (E × H ) ⋅ d S . ∂t S

(2.15)

This is a mathematical formulation of the balance of energy of the electromagnetic field known as Poynting theorem: the amount of power given to the electromagnetic field by the external power source within a limited spatial area: •

Is partially consumed to increase the energy stored by electric and magnetic fields in that spatial area



Is partially dissipated within the volume as a thermal power loss



Partially flows away from the volume as a radiated power

The surface integral in the right hand side of equation (2.15) allows introduction of a vector of power flow density known as Poynting vector, which shows the amount of power passed through the unit surface that is placed orthogonal 1 to the direction of flow in a particular point of space.

Π = E×H ,

(2.16)

It may be seen from (2.16) that the unit for the Poynting vector, i.e. for the unit for the power flow density (magnitude of vector Π ) is: (V/m)·(A/m)=W/m 2. Figure 2.1 shows the transformations between time-varying electric and magnetic fields, within the electromagnetic process, as it follows from Maxwell's Equations.

1

If the unit surface is not orthogonal to vector Π from (2.16), then a scalar product is appropriate.

Namely in equation (2.15), the arbitrary oriented elementary surface dS is represented by the surface element vector

dS , so the scalar product Π ⋅ dS represents an infinitesimal amount of

power that flows through that element.

34

Figure 2.1. Sketch of the main constituents of the electromagnetic process

The external source of electric current J Ext generates the initial magnetic field H . The time variant magnetic field, at any arbitrary point A, appears as a source of displacement electric current µ 0 µ

∂H , which forces the generation of the electric field E . ∂t

Consequently, a time varying electric field E , at the point B, appears as a source of displacement magnetic current ε 0 ε

∂E that initiates the secondary magnetic field H ′ , ∂t

and so on. This process may remain infinitely long in time, if there are no losses in considering spatial area, i.e. if the conductivity of the medium is equal to zero.

2.1.2. BOUNDARY CONDITIONS OF ELECTRODYNAMICS Maxwell's Equations noted in (2.1) – (2.4) represent differential equations in partial derivatives. In a spatial area free of sources, the electromagnetic field is to be considered standalone, as an independently existing form of matter. The first two Maxwell’s equations are utilized here to describe the interrelations between electric and magnetic fields, and may be rewritten as:

35

∇ × H = ε 0ε

∂E +J, ∂t

(2.17)

∂H , ∂t

(2.18)

∇ × E = −µ 0 µ

Here J depicts a conducting current from Ohm's Law, defined by equation (2.10). (2.17) - (2.18) is a system of two first order linear equations with two unknowns being depicted as E and H . As evident in collegiate mathematics, a general solution of the system contains arbitrary (undefined) integration constants, creating a multi-valued (ambiguous) solution. For real conditions, when configuration of boundaries between mediums is known, the initial conditions may be set up to calculate those integration constants. In electrodynamics applications, the initial conditions are conventionally called boundary conditions, because they represent the act of "bonding" (restraining) the values of electric and magnetic fields to those boundaries. In other words, these boundary conditions allow transformation of a general solution of Maxwell's equations into a particular solution that is specific for the given configuration of the boundaries in which the electromagnetic field is being defined. That particular solution is known as a boundary value problem. For more consistency we will now consider those conditions in detail. First, we take volume integrals of equations (2.3) and (2.4) within a volume V:

∫ (∇ ⋅ D )dV = ∫ D ⋅ dS = ∫ ρ

V

S

tot

dV ,

(2.20)

V

∫ (∇ ⋅ B )dV = ∫ B ⋅ dS = 0 ,

V

(2.21)

S

Within these equations, the divergence theorem is utilized (see (A.2.2.2) in Appendix A). Here S denotes a closed surface representing the boundary of the volume V. The vector surface element d S is always directed outbound to volume V. This volume encompasses a part of the boundary between two mediums via constitutive parameters ( ε 1 , µ1 , σ 1 ) and ( ε 2 , µ 2 , σ 2 ) as shown in Figure 2.2.

36

Figure 2.2. Integration area in (2.20) and (2.21) integrals Additionally, n indicates a unit vector normal to boundary of mediums and directed from medium-2 towards medium-1. As D and B vectors are specific to any particular point of space, then we may choose to minimize volume V enough to allow uniformity of the field distribution within that volume as well as on its boundary. Taking that fact into account we may write the following expression

D ⋅ d S = ( D1 ⋅ n ) d S = Dn1 d S

(2.20a)

for the top base of cylinder, and

D ⋅ d S = −( D2 ⋅ n ) d S = − Dn2 d S

(2.20b)

for the bottom base of cylinder, wherein D1 , Dn1 and D2 , Dn 2 represent the electric field inductions vectors and their normal components of the first and second media respectively. These vectors and their components are considered to be constant along the top and bottom surfaces ∆S, as mentioned above. Hence, the left hand side of equation (2.20) indicating the D -vector flow through the closed surface may be rewritten as:

∫ D ⋅ dS = [( D

1

− D2 ) ⋅ n ] ∆S = ( Dn1 − Dn 2 ) ∆S + Ξ D .

(2.22)

S

Here the first term in the right hand side shows a D -vector flow through both bases of the cylinder, and Ξ D represents a flow through the side surface of the cylinder. Hence it is evident, that if we shrink the cylinder vertically towards the boundary of media, i.e. take ∆h → 0 , then Ξ D will disappear.

37

The right hand side of expression (2.20) represents a total charge enclosed within the volume V. If ρ tot is considered as a volumetric charge density, then obviously for

∆h → 0 the right hand side of expression (2.20) will also disappear. However, for a wide range of electromagnetic problems, free charge is allocated within a tiny layer directly atop the boundary surface of two media. With that, it is appropriate to consider a surface charge ρ S that is distributed in a minute layer of infinitesimal thickness per unit surface atop the boundary 1. Thus, the right hand side of equation (2.20) may be represented as

ρ S ∆S , and finally that expression may be transformed into the following equation:

( D1 − D2 ) ⋅ n = Dn1 − Dn 2 = ρ S .

(2.23)

Similarly expression (2.21) may be transformed into the following:

( B1 − B2 ) ⋅ n = Bn1 − Bn 2 = 0 .

(2.24)

Equations (2.23) and (2.24) represent boundary conditions for the components of electric and magnetic fields that are normal to the boundary of two mediums. As one may conclude, the normal component of the magnetic field flux density consistently remains continuous across the boundary between mediums, whereas the normal components of the electric field flux density may have a discontinuity if a free surface charge exists on the boundary, i.e. exists within infinitesimal layer that surrounds the boundary of two media. Now, the boundary conditions may be evaluated for the tangential components of the electric and magnetic fields if (2.17) and (2.18) are integrated within a surface S (ABCD) residing across the boundary and orthogonal to them as shown in Figure 2.3.

1

Such charges do not exist in nature, thus this solution simply represents a mathematical

abstraction.

38

Figure 2.3. Integration area in surface integrals (2.25) and (2.26)

∫ (∇ × H )⋅ d S = ∫ ε ε

∂E ⋅d S + ∫ J ⋅d S , ∂t S

∫ (∇ × E )⋅ d S = −∫ µ

µ

0

S

S

S

0

S

∂H ⋅d S . ∂t

(2.25)

(2.26)

Here d S = s0 d S displays a surface vector-element that is directed orthogonal to the surface ABCD, with s 0 as a unit vector orthogonal to that surface. Now we apply the Stoke’s theorem (see (A1.2.2.1) from the Appendix-1) to the left hand side of the equation (2.25):

∫ (∇ × H )⋅ d S = S

∫H ⋅d S .

(2.27)

Contour ABCD

It is apparent from Figure 2.3 that the integration path along the rectangular contour ABCD may be expressed as:

∫H ⋅d S = H

1

⋅ AB + H 2 ⋅ CD + Int ,

(2.28)

Contour ABCD

Where Int represents a line-integral along sides BC and DA of height ∆h . That integral will disappear for the vanishing height of the rectangle, i.e. for ∆h → 0 . If the following three unit cross-orthogonal vectors s 0 , n , and l 0 are positioned as it is shown in Figure 2.3, then the vectors AB and CD may be replaced by AB = l 0 ⋅ ∆L = (s 0 × n ) ⋅ ∆L , and

CD = − l 0 ∆L = − (s 0 × n ) ∆L . Then, expression (2.28) may be rewritten after simple transformations as:

39

lim ∆h→0

∫ H ⋅ d S = [ H ⋅ (s 1

0

× n ) − H 2 ⋅ (s 0 × n )] ∆L =

Contour ABCD

= [n × ( H 1 − H 2 )] ⋅ ( s 0 ∆L) .

(2.29)

The first term in the right hand side of (2.25) will vanish to zero when ∆h → 0 , as well as will a second term, except in the case when the conducting current J has a surface instead of spatial, distribution (i.e. flows into a layer of infinitesimal thickness along the boundary of mediums as shown in Figure 2.3). Hence if J S represents a surface conducting current, then the second term in the right hand side of formula (2.25) may be replaced by the flow of J S through the line segment MN as shown:

∫J

S

⋅ ( s 0 ⋅ dL) = ( J S ⋅ s 0 ) ∆L .

(2.30)

MN

Making (2.29) and (2.30) equal will result in:

[n × ( H 1 − H 2 )] = J S ,

(2.31)

or, in scalar form:

H 1τ − H 2τ = J S .

(2.31a)

The above procedure may be repeated for expression (2.26), and will result in:

[n × ( E1 − E 2 )] = 0 ,

(2.32)

or, in scalar form:

E1τ − E 2τ = 0 .

(2.32a)

Physical meaning of equations (2.31) and (2.32) is defined as follows: Tangential components of the electric and magnetic field strengths remain continuous across the boundary of two mediums, unless there exists surface current flowing on the boundary that results in a jump of the tangential component of the magnetic field strength. That jump is equal to the value of that surface current density. In some of applications in antennas and microwave theory a boundary with perfect electric conductor (PEC) is of interest, and is identified for its infinite conductivity, i.e. for

σ = ∞ . Based on expression (2.1a), for the spatial areas free of sources ( J ext = 0) , it becomes evident that the electric field must be assumed to be equal to zero due to the fact that the induced conducting current σ E may not be infinitely large. The absence of the electric field results in the absence of the magnetic field, thereby indicating an absence of the entire electromagnetic process within the PEC medium. Hence, the

40

boundary conditions (2.23) (2.24), (2.31), and (2.32) may be rewritten for the PEC boundary via the following formulations:

E ⋅ n = En =

ρS . ε 0ε

(2.33)

H ⋅n = Hn = 0 .

(2.34)

n × H = J S , or

Hτ = J S

(2.35)

n×E =0 ,

Eτ = 0

(2.36)

or

From formulas (2.33) through (2.36) one may conclude, that the tangential component of the electric field and the normal component of the magnetic field on the boundary of PEC are always equal to zero, i.e. the electric field-lines are always perpendicular to the PEC boundary, whereas the magnetic field-lines are always tangential to it.

---------------------------------------------------------------------------Example 2.1 The slope angle of electric field lines changes when passing from one medium into another as shown in Figure E.2.1. Find the angle α 2 if ε 1 = 5 , ε 2 = 1 , angle α 1 = 60 0 , and there are no surface charges distributed along the boundary of those media.

Figure E.2.1. Sketch of the electric field lines on the border of two media

Solution

Eτ 1

Eτ 2



tan α 1 =



Based on (2.32a) E1τ = E 2τ . Thus, E n 1 tan α 1 = E n 2 tan α 2 .

E n1

,

tan α 2 =

En 2

.

41



Based on (2.23) for ρ S = 0 we may assume: ε 1 E n 1 = ε 2 E n 2 . By substitution into the previous expression we may derive finally:

ε2

 1 tan α 1  = tan −1  5  ε1 

α 2 = tan −1 

 3  = 11.30 . 

(Answer)

-------------------------------------------------------------------------------

2.1.3. TIME-HARMONIC ELECTROMAGNETIC PROCESS. CLASSIFICATION OF MEDIA BY CONDUCTIVITY

If time variations of electric and magnetic fields in dynamic electromagnetic processes are assumed to be harmonic (sinusoidal), which is of predominant interest in science and technology, then significant simplifications in Maxwell's Equations may be achieved by applying a complex variables analysis. If, for example, time-harmonic oscillations of the electric field are presented analytically as E cos(ω t + ϕ ) , then transformation into complex form E exp[ i (ω t + ϕ )] allows representation as follows 1:

E cos(ω t + ϕ ) = Re {E exp[i (ω t + ϕ ]} = Re {E exp[iω t ] exp[iϕ ]}.

(2.37)

The most attractive feature of complex analysis is the fact that time derivatives (or time integrals) may simply be replaced by multiplication (or division) by the factor iω , which transforms differential equation into algebraic. Indeed

∂ {E exp[i (ω t + ϕ )]} = i ω {E exp[i (ω t + ϕ )]} . ∂t

(2.38)

Even a time-harmonic multiplier, exp( i ω t ) may be cancelled out of Maxwell's equations. Maxwell's equations are written now not for the time-varying electric and magnetic fields but for their vector-phasors, such as:

E = x 0 E X exp(iϕ X ) + y 0 EY exp(iϕ Y ) + z 0 E Z exp(iϕ Z ) , where E X , EY , E Z and ϕ X ,

ϕ Y , ϕ Z represent a coordinate-dependent amplitudes and initial phases of oscillation of

1

Here the Euler's formula

exp(iα ) = cos α + isin α is used, where i =

42

−1 .

the electric field vector in Cartesian coordinates. A phasor for the magnetic field vector or any other scalar or vector variable may be introduced similarly. Taking the above statements into account, Maxwell's equations for the source-free spatial region may be expressed in complex form as follows:

∇ × H = iω ε 0 ε E + σ E ,

(2.39)

∇ × E = − i ω µ 0 µ H ,

(2.40)

ρ ∇ ⋅ E = tot

(2.41)

∇ ⋅ H = 0

(2.42)

ε ε0

Then expression (2.39) may be transformed to 1:

 σ  E . ∇ × H = J tot = J dis + J = iω ε 0 ε E + σ E = iωε 0  ε − i ωε 0  

(2.43)

Here the total volumetric current density J tot is shown as a sum of displacement, J dis and conducting, J current densities. The same expression (2.43) for the lossless medium ( σ = 0 ) can be rewritten as:

∇ × H = J dis = iω ε 0 ε E .

(2.44)

If equation (2.43) is compared with equation (2.44), then a relative complex permittivity of the considering medium (or material) may formally be introduced as:

ε = ε − i

σ = ε − i 60λ 0σ . ωε 0

(2.45)

In (2.45) the following substitution is used: ω = 2π c / λ0 , where λ0 represents the wavelength in free space, and c = 3 ⋅ 10 8 m/s represents the speed of light in free space.

1

Here and below we’ll assume the absence of the magnetic losses, which is specific for the

propagation problems. Moreover, for most of the cases of RF propagation in earth’s atmosphere, along the earth’s surface, and in space the propagation medium(s) are assumed to be nonmagnetic, i.e. µ = 1.

43

For further analysis we’ll assume both, ε and σ scalar quantities, i.e. the media under consideration being isotropic 1. Now, without limitations to generality, consider for simplicity a single-component electric field (e.g. a field that is directed along x-axis) of zero initial phase shift. Then in (2.43) we may replace E by scalar E . Thus for the lossy medium expression (2.43) may be rewritten for the same component in phasor form as

J tot = iω ε 0 ε E = iωε 0 (ε − i 60λ 0σ )E = J dis − i J

(2.43a)

As one may notice from (2.43a) the total volumetric current density is a complex quantity with displacement current as a real part, and conducting current as its imaginary part. The ratio between imaginary and real parts of the complex dielectric constant (2.45) is the same as the ratio of conducting and displacement current densities. This ratio may be used to specify the rate of losses in considering medium or material. In complex plane it represents tangent of the slope angle δ ε , identified as loss angle (Figure 2.4).

tan δ ε =

60 λ 0 σ J σ . = = J dis ω ε 0 ε ε

(2.46)

Figure 2.4. Conducting, displacement, and total current densities in a complex plane Based on the values of tan δ ε , classification of media by conductivities is introduced conventionally as follows: o 1

tan δ ε < 10 −1 – for dielectrics

Anisotropic properties of ionospheric plasma interacting with earth’s magnetic field and its

impact on radio waves propagation is considered shortly in chapter 5. All other propagation mechanisms that are of interest for communications and radar applications relate to propagation in isotropic media only.

44

o

10 −1 ≤ tan δ ε ≤ 10 - for semiconductors

o

tan δ ε > 10 – for conductor

From (2.46) one may notice, that the value of tanδε is dependent on frequency ω. The same medium may exhibit different behavior in different frequency ranges. For several media listed in Table 2.1, frequency dependences are presented in Figure 2.5. As one may notice from the figure, marble and mica may be considered as ideal dielectrics for the entire radio frequency (RF) spectrum. Quartz, paraffin, glass, atmospheric air, polyethylene, (etc.) may also be considered as ideal dielectrics for the entire radiospectrum. Nearly all metals, including copper, iron, aluminum, mercury, and others behave as ideal conductors in the entire RF spectrum up to nearly 1010 Hz. Several media, such as wet and dry soil, sea and fresh water, ice, etc. may display variant properties in different frequency bands. For example, a dry soil behaves as a conductor in the LF frequency band, as a semiconductor in MF and HF frequency bands, and as a lossy dielectric in higher frequency bands.

Figure 2.5. Frequency dependence of tanδε for variant media (see Table 2.1 for details by reference numbers)

45

Reference No

(from figure 2.5)

Table 2.1

Relative Medium/material

Conductivity,

dielectric

in S/m

permittivity

Applicable frequency range (Hz)

1

Sea water

~ 80

1–5

0 – 108

2

Wet soil

10 - 30

3.10 - 3 – 3.10 - 2

0 – 108

3

Fresh water

~ 80

10 - 3 – 2.4.10 - 2

0 – 108

4

Dry soil

3-6

1.1.10 - 5 – 2.10 - 3

0 – 108

5

Marble

~8

10 - 7 – 10 - 9

103 - 108

6

Mica

~7

10 - 11 – 10 - 15

103 - 108

---------------------------------------------------------------------------------------------------------Example 2.2 Estimate the frequency ranges for sea water ( ε = 80, σ = 1 S / m ) to be considered as conductor, semiconductor, and dielectric. Solution •

First transform (2.46) to obtain the frequency:

f =

60 σ c ε ⋅ tan δ ε

(E-2.2.1)

Here c = 3 ⋅ 10 8 m / s is speed of light in free space •

The results of calculations based on expression (E-2.2.1) are shown in the table below.

tan δ ε < 10 −1

f > 2.25 GHz

Dielectric

10 −1 < tan δ ε < 10

22.5 MHz ≤ f ≤ 2.25 GHz

Semiconductor

tan δ ε > 10

f < 22.5 MHz

Conductor

------------------------------------------------------------------------------------------------------

46

2.2. FREE PROPAGATION OF UNIFORM PLANE RADIO WAVES As an electromagnetic process, we'll analyze the propagation of the radio waves based on Maxwell's equations that are specifically conditioned to free propagation in the media such as earth's atmosphere, outer space, water of seas and ponds, and the earth's ground. Consider a time-harmonic field 1 that is initially generated by an external source. The first two of Maxwell's equations in a region, which do not contain a source (a source-free region) may be expressed in complex form as:

∇ × H = iωε 0 ε E ,

(2.47)

∇ × E = −iωµ 0 µ H .

(2.48)

To solve this system of two linear differential equations in partial derivatives we exclude one of the unknowns ( E or H ) to bring this system to one equation of one unknown. With that in mind we will take a curl from both sides of (2.48), and then substitute ∇ × H from (2.47):

∇ × ∇ × E = −iωµ 0 µ ⋅ ∇ × H = k 2 E ,

(2.49)

k = ω

(2.50)

where:

ε 0 ε µ 0 µ = β − iα

represents a complex propagation constant with a physical meaning that we will define later in this text. If the equality (A1.2.3.1) from Appendix-1 is applied to the left hand side of (2.49), and (2.3) is taken into account, that is ∇ ⋅ E = 0 , for the medium free of electric charges, then (2.49) may be rewritten as

∇ 2 E + k 2 E = 0 .

1

(2.51)

Known also as monochromatic. The term comes from optic, and originated from the Greek

phrase "single colored". "Single colored" radiation in optics means a radiation of a single frequency. The same meaning is adopted for the RF spectrum.

47

Here ∇ 2 indicates a Laplacian operator applied to E -vector. In Cartesian coordinates,

∇ 2 is expressed as:

∇2 =

∂2 ∂2 ∂2 , + + ∂x2 ∂ y2 ∂z2

(2.52)

Thus for the vector phasor E ( x, y , z ) = x 0 E x ( x, y, z ) + y 0 E y ( x, y, z ) + z 0 E z ( x, y, z ) the Laplacian is

∇ 2 E = x0 ∇ 2 E X + y 0 ∇ 2 E Y + z 0 ∇ 2 E Z ,

(2.53)

where x0 , y 0 , and z 0 are the unit vectors along X, Y, and Z coordinates. Equation (2.51) displays a particular form of well known wave equation called Helmholtz equation that is a wave equation written for the time-harmonic process. The function, given below by (2.54), may be considered as one of the simplest solutions of Helmholtz equation:

E = x0 E m X exp [i (ω t − k z )] .

(2.54)

In this expression, E m X is a complex amplitude (phasor) that includes the real amplitude iϕ E m X and the initial phase shift ϕ 0 , i.e. E m X = E m X e 0 . It’s easy to show by

substitution, that (2.54) satisfies equation (2.51). Regarding formula (2.54), the electric field contains a harmonic, time and Z-dependence only, with an amplitude value of E m X . In other words all planes parallel to the XOY are the planes of constant values of the field for the fixed time instance. As electric and magnetic fields are coupled to each-other in every point of space, it is reasonable to expect, for field components of both, electric and magnetic fields are portrayed as:

∂E ∂E ∂H ∂H = = = = 0. ∂x ∂y ∂x ∂y

(2.55)

The value of the magnetic field coupled to solution (2.54) may be found by substituting the field into (2.47). In Cartesian coordinates, a curl in the left hand side of (2.47) is conveniently presented in form of determinant, i.e. (2.47) may be presented as:

x0 ∂ ∇ × H = ∂x H X

y0 ∂ ∂y H Y

z0 ∂ = iωε 0 ε ⋅ E . ∂z H Z

48

(2.56)

Taking into account (2.55), and observing the fact that the right hand side of (2.56) does not contain X- and Y-components of E -field, (2.56) may be rewritten as:



dH Y = iωε 0 ε E m X exp[ i (ω t − k z )] . dz

(2.57)

Here the partial derivative is replaced by the regular derivative due to single-coordinate dependence. If we integrate (2.57), then the result will be:

H = y 0 H m Y exp[ i (ω t − k z )] ,

(2.58)

ω ε 0 ε  H m Y = Em X . k

(2.59)

where

Both vector-phasors E and H contain the total phase

ϕ = ωt − β z ,

(2.60)

where β is a real part of the complex number defined by (2.50). Note that it affects a phase and is called phase coefficient, whereas the imaginary part, α affects the amplitude and is called attenuation coefficient, or attenuation constant (see subsection 2.2.2 for details). Surfaces represented by ϕ = const in space, namely the surfaces of constant values of the phase, are called wave fronts. Consider the value of ϕ that remains unchanged, i.e. we’ll try to find a velocity of movement of the wave front along Z-axis. Then any increase in time ∆t is to be related to the change in coordinate ∆ z (Figure 2.6), so thus:

ϕ 1 = ω t − β z = ϕ 2 = ω (t + ∆t ) − β ( z + ∆ z ) = const .

(2.61)

Figure 2.6. Movement of the plane wave front. Only a square fragment of the Infinity wave front is represented here.

49

From this equation (2.61) it is logical to determine the phase velocity, which is the velocity of the movement of the plane wave front along the Z-Axis:

v=

∆z ω = . ∆t β

(2.62)

If, in particular, ∆z represents a distance between two adjacent maximums of the wave pattern, i.e. ∆ z = λ , then the time interval that is needed to move from one maximum to the next is ∆t = T = 1 / f = 2π / ω , where T is a period of the oscillations, f is the linear frequency, and ω is angular frequency. From (2.62) the phase coefficient may be expressed in terms of the wavelength in considering medium as

β=



λ

.

(2.62a)

Based on above considerations one may conclude that the electromagnetic process described by the solution in formulas (2.54) and (2.58) represents a wave that propagates along Z-Axis with the constant velocity v and contains plane infinite wave fronts, supporting the uniform distribution of the electric and magnetic fields along those planes. For the considering particular case of propagation along Z-direction, wave fronts are flat surfaces parallel to the X0Y plane. These types of waves are known as uniform plane waves because of the constant distribution of the amplitudes and phases of electric and magnetic fields along those planes. They do not exist in nature in their pure form and are introduced simply as a mathematical abstraction to simplify understanding of the physical concepts and make easier the formal mathematical evaluations. In reality, the distances, covered by the radio waves, are much greater than the geometric size of source. It will be shown later, that for those long distances this abstraction is a good foundation for the radio wave propagation theory, and is capable of outlining all propagation problems qualitatively and quantitatively. From equation (2.59) one may realize that if a radio wave contains only an X-component of the electric field, i.e. that is directed along positive X-axis then the magnetic field may have only a Y-component that is directed along the positive Y-axis. From the same equation it may then be confirmed, that for the propagation of time-harmonic radio wave in lossless mediums ( σ = 0 ) both, ε and k become real numbers, therefore the Poynting vector (2.16) also becomes real if the cross product is defined as:

~  = E × H , Π

(2.63)

50

~

i.e. if the phasor H represents a conjugate of the complex vector H . In that particular case of a lossless medium, the Poynting vector becomes real vector, whereas for any

 is a complex vector. Disposition of E and H vectors satisfies the lossy medium Π right-hand rule as shown in Figure 2.7.

Figure 2.7. Disposition of vectors E , H and Π in a free space according the right hand law

An expression similar to (2.63) may be written for the amplitude phasor of the Poynting vector as:

~  = E × H Π m m m.

(2.64)

For practical engineering applications, it is more convenient to introduce a scalar that is derived from the complex Poynting vector as an RMS value of the instant power flow density, i.e. power flow averaged within a period of oscillation:

Π=

(

)

~ 1 Re E m × H m . 2

(2.65)

The factor 1/2 is due to the ratio between amplitude and RMS values of any timeharmonic oscillations. It’s analogous to the relation between voltage amplitude, current amplitude, and the power consumption within an electric circuit element. As one may note from equation (2.59) if the parameters of the propagation medium ε and µ are constant in time and space, then the ratio between electric and magnetic fields is time- and space-independent. That ratio may be used to identify a property of the propagation medium called intrinsic impedance of medium and may be defined from (2.59), if (2.50) is taken into account, as well as µ = 1 is assumed:

51

E W = mX = H

µ0 120 π 377 , Ohm, = ≈ ε 0 ε ε ε

mY

The values for ε 0 and µ

0

(2.66)

are utilized from (2.6).

2.2.1. UNIFORM PLANE WAVE IN LOSSLESS MEDIUM In expression (2.50) for parameter k the value of relative magnetic permeability µ may be assumed to be µ = 1 , due to the non-magnetic types of mediums the propagation theory deals with, such as salt and sweet water, ground soil, and atmospheric air. Another assumption is σ = 0 , due to lossless propagation conditions in the atmospheric air. Under those two conditions k becomes a real number, i.e. k = β . If this value of k is substituted into (2.62) then the result for propagation phase velocity will become

v=

1

ε 0ε µ 0

=

c

ε

,

(2.67)

where

c=

1

ε 0 µ0

=

1 (10 / 36π F / m) ⋅ (4π ⋅ 10 −9

−7

= 3 ⋅ 10 8 m/s

(2.68)

H / m)

represents speed of light in vacuum. Expression (2.67) demonstrates that propagation phase velocity in any real medium is always less than in free space. An exception is the case of propagation within ionospheric plasma medium, which will be discussed later in chapter 5. Recall that the wavelength is a distance that is covered by the radio wave within one period of oscillation, i.e. within time interval T = 1 / f = 2π / ω , as follows:

λ = v ⋅T =

1 2π c λ 0 1 c . = = ε f ε ω ε

(2.69)

Here:

λ0 =

c 2π c = ω f

(2.70)

52

denotes a wavelength of the radio wave that propagates in free space, i.e. in vacuum 1. Now consider the propagation constant k defined by (2.50). For this particular case of propagation in lossless medium:

k =ω

ε 0 µ0 ⋅ ε =

ω

ε =

c



λ0

ε =



λ

rad/m.

(2.71)

Parameter k is usually referred to as a wave number. To be more specific we have to note that in physics the wave number is defined as a reciprocal of the wavelength, 1 / λ , showing the number of the wavelengths on a unit distance along the propagation path whereas the quantity 2π / λ is referred to as angular wave number. However most of the authors in electromagnetics are using the term wave number to specify 2π / λ , so we'll do all the way in this text. To summarize (2.71) we emphasize that the higher the oscillation frequency, the greater the wave number is. As mentioned earlier, in the case of the absence of losses, σ = 0 regarding (2.45), (2.50), and (2.66) three complex quantities, ε , k , and W become real numbers with the following values:

k=β =

W =



λ0 / ε

120 π

ε



=

377

ε



λ

rad/m

Ohms.

(2.72)

(2.73)

Thus, as is indicated in (2.59), vectors E and H , being cross-perpendicular to each other, are not phase-shifted, i.e. maximums and minimums of these vectors are allocated in the same points of space. As follows from (2.54) for this case of lossless medium ( k is a real number) the amplitude of the wave is Z-independent, i.e. it remains unchanged along the entire propagation path, i.e. along Z-Axis. If the conjugate of H mY is substituted from (2.66) and into (2.65), then the effective (RMS) power flow may be found as:

1

Do not confuse the term “free propagation” with “propagation in free space”. “Free propagation”

is referred to unguided propagation that may occur in any medium such as air, water, soil, etc., whereas “propagation in free space" means propagation in ideal conditions (vacuum) that is free of any material substance.

53

~ Em 1  Π = Re  E m × 2  W 

 E 2 = m .  2 W

(2.74)

This expression allows making a conclusion that the Poynting vector also becomes real, time- and space-independent. The distribution of the electric and magnetic fields along the propagation path is shown in Figure 2.8.

Figure 2.8. Structure of the radio wave in lossless medium

2.2.2. UNIFORM PLANE WAVE IN LOSSY MEDIUM

In this case the complex relative permittivity both, intrinsic impedance and wave number are complex, and the expressions (2.45) and (2.66) may be rewritten in following forms:

ε = ε − i

σ = ε ′ − i ⋅ ε ′′ , ωε 0

where ε ′′ =

σ ωε0

W = W ⋅ e iΦW .

(2.75) (2.76)

In this case the expression (2.50) is complex and is rewritten here:

k = β − iα .

(2.77)

For the plane uniform radio wave that propagates along Z-direction, the expression (2.54) may be presented in the following form, after (2.77) is substituted into it:  E = x0 E m X e i (ω t −k z ) = x0 E m X e −α z ⋅ e i (ω t − β z ) .

(2.78)

Similarly, based on (2.58), the expression for the magnetic field may be presented as:  H = y 0 H m Y e i (ω t − k z ) = y 0 H m Y e −α z ⋅ e i (ω t − β z ) ,

54

(2.79)

Where:

H m Y =

E m X E m X −i ΦW = e . W W

(2.80)

It is obvious from equations (2.78) and (2.79) that the amplitudes of electric and magnetic components of the radio wave decay exponentially as e

−α z

. Coefficient α 

introduced in (2.50) as attenuation coefficient and β, the phase coefficient, which is a part of the total phase ω t − β z . The expression (2.80) demonstrates the fact that there is an initial (permanent) phase shift, Φ W between electric and magnetic fields oscillations. This phase shift may be transformed into the distance shift ∆ z = Φ W / β between electric and magnetic waves as shown in Figure 2.9.

Figure 2.9. Structure of the uniform plane wave in lossy (semi conducting) medium

The decrease of the intensity of radio wave during propagation may also become evident if the expression for effective (RMS) value of Poynting vector is recalled. That expression may be derived if (2.78) and the complex conjugate of H m Y from (2.80) are used along with (2.65).

 1   E m X iΦW Π = Re E m X e 2  W

 E m X 2 − 2α z = cos Φ W . e  2 W 

(2.81)

From equation (2.81) it is evident, that in case of propagation in lossy medium the average power density flow not just decays exponentially along the propagation path, but also its initial value becomes smaller than that for propagation in lossless medium, by the value of the so called power factor, which is equal to cos Φ W .

55

The important parameters of propagation, namely attenuation coefficient α and phase coefficient β , are considered as composite parts of the complex propagation constant

k . If in equation (2.50) we assume µ = 1 , then taking into account (2.75) it will results in

κ = β − iα = ω ε 0 µ 0 ε = ... =



ω

... =

ε exp − i arctan 

c





δε



2

ε  cos

λ0

ω

ε −i

c

σ = ... ωε 0

σ  2π = ωε 0 ε  λ 0

− i ⋅ sin

ε exp(− iδ ε ) = ... .

δε 

. 2 

(2.82)

Here

 σ ε +   ωε 0

ε =

2

2

 ε  = cos δ ε 

(2.83)

is the magnitude of complex dielectric permittivity, expressed by its real part, ε and by angle of losses, δ ε given by (2.46) and shown in Figure 2.4. Thus (2.82) may be rewritten as:

β − iα =

δ δ    cos ε − i ⋅ sin ε  . cos δ ε  2 2 



ε

λ0

(2.84)

Thereby, the attenuation and phase coefficients may be derived by taking real and imaginary parts in both sides equal to each other:

α= β=



ε

λ0

cos δ ε



ε

λ

0

cos δ ε

sin

δε

cos

2

δε 2

,

(2.84a)

.

(2.84b)

Here λ 0 is a wavelength in free space. Expressions (2.84a) and (2.84b) may be simplified for following two extreme cases.

56

2.2.2.a. LOW-LOSS DIELECTRIC MEDIUM In this case tan δ ε 1 (or δ ε ≈

cos

δε 2

≈ sin

δε 2



1

π 2

), therefore:

ε cos δ ε

=

ε ≈

σ , and ωε 0

. Thus:

2

α ≈β ≈

σ ω µ0 2

.

(2.87)

In all other cases of semi-conducting mediums, which are more specific for the real propagation conditions, only the original (2.84a) and (2.84b) formulas are applicable. The attenuation constant is specified by a Neper-per-meter (Np/m) unit. The physical meaning of that unit may become clear if (2.78) and (2.79) are recalled. From any of these equations one may conclude that if α = 1 Np/m, then the amplitude of the radio wave, which passes a distance of one meter, will decrease e = 2.71.... 1 times. Indeed, from the equation (2.78) the attenuation constant is found as

α Np / m =

1 E m X ( z = 0) ln z Em X ( z )

(2.88)

Then, for the ratio of the amplitudes equal e = 2.71.... , and for the propagation distance of z = 1 m, the value of the right hand side in (2.88) is equal to unity. Another unit, decibel-per-meter (dB/m), which is widely used in engineering applications, is defined as

1

This is the base of the natural logarithm.

57

1 z

 E m X ( z = 0)    E ( z)  m X  

α dB / m = ⋅ 20 log 

(2.88a)

and is more commonly annotated. The relation between those two units is

α dB / m = 8.68 ⋅ α Nep / m .

(2.89)

Now the term attenuation of the radio wave may be introduced as:

A = α z ( AdB = α dB / m ⋅ z )

(2.90)

that displays a total decrease of the radio wave amplitude along the entire homogeneous propagation path of the length z . In inhomogeneous mediums, where ε and/or σ are varying in space from point to point, α also becomes variable. Suppose the entire inhomogeneous propagation track consists on several discrete segments of homogeneous paths as shown in Figure 2.10. Then it’s apparent, that after passing the entire z distance the amplitude of the radio wave will become

E m ( z ) = E m ⋅ e −α1 ⋅∆z1 ⋅ e −α 2 ⋅∆z2 ⋅ ⋅ ⋅ ⋅ ⋅ e −α n ⋅∆zn = E m ⋅ e



∑ α k ⋅∆zk k

(2.91)

for the total distance of

z = ∑ ∆z k .

(2.92)

k

Figure 2.10. Radio wave propagation pattern in a discretely inhomogeneous medium If α is a continuous function of the distance, it is apparent that (2.91) may be transformed into integral form as follows: − α ( z ′ )⋅dz ′ . E m (z ) = E m0 ⋅ e ∫

(2.93)

Here E m 0 is the initial amplitude. Thus, the expression for total attenuation along the entire propagation path may be rewritten in an integral from as

58

z

A = ∫ α ( z ′) ⋅ dz ′ .

(2.94)

0

----------------------------------------------------------------------------------------------Example 2.3 Calculate attenuation coefficient and the phase coefficient for the radio wave propagating in dry soil ( ε = 4 , and σ = 2 ⋅ 10 − 4 S/m). Calculations are to be performed for the following three frequencies: f 1 = 6 kHz, f 2 = 600 kHz, f 3 = 6 GHz: Solution •

For the frequency f 1 = 6 kHz the wavelength in free space is λ 01 = 50000 m. Then regarding (2.46):

tan δ ε =

60 λ 01 σ

ε

= 150 >> 1 .

(E-2.3.1)

Hence, expression (2.87) is applicable for both, the attenuation coefficient and phase coefficient, i.e.

α≈

σ ω µ0

2 ⋅ 10 −4 ⋅ 2π ⋅ 6 ⋅ 10 3 ⋅ 4π ⋅ 10 −7 = 2.177 ⋅ 10 −3 Np/m 2

=

2

β ≈ 2.177 ⋅ 10 −3 rad/m •

For the frequency f 2 = 600 kHz the wavelength in free space is λ 02 = 500 m.

tan δ ε =

Then

60 λ 0 2 σ

ε

= 1.5 ( δ ε = 0.9828 rad)

(E-2.3.2)

Hence, neither of the approximate approaches given in sections 2.2.2.a and 2.2.2b is applicable. The general expressions (2.84a) and (2.84b) must be used for the attenuation coefficient and phase coefficient respectively.

α= β= •



ε

λ 02

cos δ ε



ε

λ 02

cos δ ε

⋅ sin

δε

⋅ cos

2

δε 2

=

2π 500

4  0.9828  sin   = 0.0159 Np/m cos 0.9828  2 

=

2π 500

4  0.9828  cos  = 0.03 rad/m cos 0.9828  2 

For the frequency f 2 = 6 GHz the wavelength in free space is λ 03 = 0.05 m. Then

tan δ ε =

60 λ 0 3 σ

ε

= 1.5 ⋅ 10 − 4 ϕ > 0 . Hence the reflection phase Φ Γ E , vert jumps from 1800 to 00 at ϕ 0 . The patterns of the angular dependencies of the magnitude and phase of reflection coefficient for this particular case are shown in Figure 2.19 (dotted lines).

71

2.4.2.b. RADIO WAVE INCIDENT FROM DENSE MEDIUM ONTO THE BORDER WITH SPARSE MEDIUM (ε 1 > ε 2 ) If ϕ = 0 in this particular case, then the reflection coefficient is positive, therefore the reflection phase is equal to zero. The increase of ϕ from zero will result in same total reflection phenomenon at the same Brewster's angle ϕ 0 , defined by (2.120), as with the

 E vert previous case. When the angle of incidence passes through the value of ϕ 0 , then Γ turns the sign from positive to negative, i.e. the reflection coefficient jumps from zero to 1800. Further increase of the angle of incidence ( ϕ > ϕ 0 ) will result in another phenomenon that occurs at so called critical angle, ϕ = ϕ cr , when the expression under the square root in (2.119) becomes equal to zero. Then ϕ cr is defined as

sin ϕ cr =

ε2 . ε1

(2.121)

For the values of angle of incidence ϕ > ϕ cr the expression under the square root sign becomes negative, thus the numerator and denominator of (2.119) become complex

 vert will conjugate relative to each-other. Therefore, for all ϕ > ϕ cr the magnitude of Γ remain equal to unity, whereas the reflection phase Φ Γ E , vert will smoothly decrease from 1800 to zero degrees as shown in Figure 2.20 (dotted lines). Hence ϕ cr is called the total reflection angle.

72

2.4.3. OBLIQUE INCIDENCE OF HORIZONTALLY POLARIZED RADIO WAVE In this case electric field vectors of all of three rays lie in the same horizontal plane as shown in Figure 2.18.

Figure 2.18. Positions of vectors for the horizontally polarized electromagnetic wave of oblique incidence to the flat reflection boundary

The boundary conditions (2.110) and (2.111), may be rewritten in the following form:

H in cos ϕ − H 1 cos ϕ = H 2 cosψ ,

(2.122)

E in + E 1 = E 2 .

(2.123)

Simplifications similar to those provided for the vertically-polarized incident waves are applicable to (2.122) and (2.123) as well: •

ψ may be expressed in terms of ϕ using Snell's law (2.109)



Replace magnetic field strengths by electric, using (2.66)



After combining (2.122) and (2.123) the ratio (2.112) for electric field reflection coefficient may be found as

Γ E horiz =

ε 1 cos ϕ − ε 2 − ε 1 sin 2 ϕ ε 1 cos ϕ + ε 2 − ε 1 sin 2 ϕ

.

(2.124)

From (2.124) it may be seen that unlike the previous case of the vertical polarization, the horizontally polarized radio wave can never satisfy the total refraction condition, i.e. the

73

horizontally polarized radio wave will never be totally refracted (penetrate) into the second medium. Below we consider two particular cases of reflection of the horizontally polarized waves.

2.4.3.a. RADIO WAVE INCIDENT FROM SPARSE MEDIUM ONTO THE BORDER WITH DENSE MEDIUM (ε 1 < ε 2 )  E horiz represents a real and negative fraction, For all angles of incidence ϕ the value of Γ

 E horiz which urges to minus one, when ϕ becomes close to 90o, i.e. the magnitude of Γ smoothly increases, while the reflection phase remains constant and equal to minus 1800 for all values of angle of incidence, ϕ as shown in Figure 2.19 (solid lines).

2.4.3.b. RADIO WAVE INCIDENT FROM DENSE MEDIUM ONTO THE BORDER WITH SPARSE MEDIUM (ε 1 > ε 2 ) In this case the total reflection occurs at the same condition (2.121). Hence, for the

 E horiz remains a angles of incidence between zero and ϕ cr , the reflection coefficient Γ positive fraction and tends to one when ϕ becomes close to ϕ cr . The magnitude of

Γ E horiz tends smoothly to one, while the reflection phase remains equal to zero. For the  E horiz remains equal to one, while values of ϕ between ϕ cr and 90o the magnitude of Γ the value of reflection phase increases smoothly from zero to 1800 as shown in Figure 2.20 (solid lines). Note, that both expressions (2.119) and (2.124) will convert to (2.113) regardless of sense of polarization if the normal incidence ( ϕ = 0 ) is considered.

74

Figure 2.19. Magnitude and angle of the electric field reflection coefficients for the boundary of two ideal dielectrics with parameters

ε 1 = 1, ε 2 = 3 .

Figure 2.20. Magnitude and angle of the electric field reflection coefficients for the boundary of two ideal dielectrics with parameters

ε 1 = 3, ε 2 = 1 . 75

2.4.4. REFLECTION OF THE RADIO WAVE WITH ARBITRARY POLARIZATION As was mentioned earlier in this text, any arbitrary polarized radio wave may be represented as a superposition of two linearly-polarized radio waves: vertically polarized and horizontally polarized. Each of those two radio waves satisfies their own reflection properties. Therefore, after reflection at the same angle, the superimposition of those may result in a radio wave that demonstrates the new polarization properties. If, for instance, the initial linearly polarized radio wave has a arbitrary polarization angle with the vertical XOZ-plane (neither zero, nor 90 0), then, before the reflection, it may be decomposed into two linearly polarized waves (vertically and horizontally polarized) with properly chosen amplitude-phase relations between them. If after reflection the phase shift between components becomes Φ vert − Φ horiz = π / 2 , then it may result either in circular or elliptic polarization reliant on the relationship between their amplitudes. This phenomenon is utilized in optics to construct the polarization converter. Additionally, the separation of cross-polarized components may also be achieved. It is used in optics to separate the cross polarized components of the non-polarized radiation. To accomplish that, the transparent slab is placed with the proper positioning of the surface, relative to the direction of primary ray-trace (see Figure 2.21).

Figure 2.21 Spatial decomposition of the non-polarized (or arbitrary polarized) optical radiation onto two cross-polarized components.

76

The angle of incidence is chosen to be equivalent to Brewster angle ϕ 0 , thus the vertically-polarized component penetrates completely through the slab, while the horizontally-polarized component reflects from the slab's surface alone.

2.4.5. POWER REFLECTION AND TRANSMISSION

Electric field reflection coefficients for the vertically and horizontally polarized radio waves incident to the border of two dielectric mediums are given by (2.119) and (2.124). Magnetic field reflection coefficients may be derived similarly, based on the systems of equations represented by (2.117) – (2.118), and (2.122) – (2.123). If in those equations electric fields are replaced by magnetic fields using W1 / 2 = W0 /

ε 1 / 2 (see (2.73)), and

then further evaluate (2.112a) for both polarizations, then it may be shown simply that

Γ H = −Γ E .

(2.125)

For the transmission coefficients the definitions are based on the following expressions for the electric and magnetic fields respectively:

E −i⋅Φ T E = 2 = T E ⋅ e T E .  Ein

(2.126)

H −i⋅Φ T H = 2 = T H ⋅ e T H .  H in

(2.126a)

 E , Γ H , T E , and T H are provided in Table 2.2. Evaluation results for Γ To best evaluate reflection and transmission of the power of radio wave, we have to keep in mind that the power flow must be considered across the boundary in a direction normal to the border. In other words, the normal components of the incident, reflected, and transmitted (refracted) waves ( Π in

Norm

, Π refl

2.22, must be taken into account.

77

Norm

, and Π trans

Norm

) shown in Figure

Figure 2.22 Power flow directions for the oblique incidence of the radio wave onto the border of dielectric media

Power flow density at the border of loss-less media may be defined from (2.81). So for the origin of the coordinate system that is placed on the border ( z = 0) the power flow is defined as Π = E 2 / 2W , and an expression for the power balance right at the interface may be written as

Π N in − Π N refl = Π N trans .

(2.127)

Then (2.127) is rewritten as: 2

2

2

Ein E E cos ϕ − 1 cos ϕ = 2 cosψ , 2 W1 2 W1 2 W2

(2.128)

where W1 and W2 indicate the intrinsic impedances of first and second media respectively. Now we may define the power reflection and transmission coefficients as:

Π N refl E Γ = N = 1 2 = Γ E Π in Ein 2

P

E Π N trans T = = 22 N Π in Ein 2

P

2

,

(2.129)

ε 2 cosψ = TE ε 1 cos ϕ

78

2

ε 2 − ε 1 sin 2 ϕ ε 1 cos ϕ

.

(2.130)

Hence, taking into account (2.129) and (2.130), the expression (2.128) may be rewritten as:

1 − ΓP = T P .

(2.131)

Note that (2.131) represents the power balance that may be easily verified by direct substitution of Γ E and T E for either vertical or horizontal polarizations into (2.129), (2.130), and further into (2.131). Table 2.2 summarizes analytical expressions that relate to the cases above. Table 2.2

Electric Magnetic

Vertical polarization

coefficient Transmission

Electric

T E vert =

2 ε 1 ε 2 cos ϕ

ε 2 cos ϕ + ε 1 ε 2 − ε 1 sin 2 ϕ

Reflection

ε cos ϕ − ε 1 ε 2 − ε 1 sin 2 ϕ Γ H vert = 2 ε 2 cos ϕ + ε 1 ε 2 − ε 1 sin 2 ϕ

coefficient Transmission

Reflection

Magnetic

2 Γ E vert = − ε 2 cos ϕ − ε 1 ε 2 − ε 1 sin ϕ ε 2 cos ϕ + ε 1 ε 2 − ε 1 sin 2 ϕ

coefficient

coefficient

Horizontal polarization

For Electric and Magnetic Fields

Reflection

coefficient Transmission

T H vert =

2 ε 2 cos ϕ

ε 2 cos ϕ + ε 1 ε 2 − ε 1 sin 2 ϕ

Γ E horiz =

T E horiz =

ε 1 cos ϕ − ε 2 − ε 1 sin 2 ϕ ε 1 cos ϕ + ε 2 − ε 1 sin 2 ϕ 2 ε 1 cos ϕ

coefficient

ε 1 cos ϕ + ε 2 − ε 1 sin 2 ϕ

Reflection

ε cos ϕ − ε 2 − ε 1 sin 2 ϕ Γ E horiz = − 1 ε 1 cos ϕ + ε 2 − ε 1 sin 2 ϕ

coefficient Transmission coefficient

For Power Reflection and

Reflection

Transmission

coefficient

for either vertical or horizontal

Transmission

polarizations

coefficient

79

T H horiz =

2 ε 2 cos ϕ

ε 1 cos ϕ + ε 2 − ε 1 sin 2 ϕ Γ P = Γ E

T P = T E

2

2

ε 2 − ε 1 sin 2 ϕ ε 1 cos ϕ

2.4.6. REFLECTION OF THE RADIO WAVE FROM THE BOUNDARY OF NON-IDEAL DIELECTRIC MEDIUM

This case is specific for radio waves propagation conditions, as for the RF frequencies the soil, fresh and sea water, as well as the ionosphere behave as semiconductors; therefore in (2.119) and (2.124) either both, ε1 and , ε2 or at least one of them must be considered as a complex number, defined by (2.45). It may be seen, that for non-ideal dielectric boundary a phenomenon such as total reflection or total refraction will not purely appear, due to the presence of the power losses. An example of the angular dependence of the reflection coefficient for the real conditions is shown in Figure 2.23, obtained from (2.119) by using the MATLAB subroutine.

Figure 2.23. The angular dependence of magnitude and phase of reflection coefficient for the reflection of vertically polarized radio wave from soil with the following parameters: ε = 3 , σ = 0.01 , λ = 5 m .

80

2.5. RADIATION FROM INFINITESIMAL ELECTRIC CURRENT SOURCE. SPHERICAL WAVES

For the spatial area that contains time-harmonic source, the first two Maxwell's equations may be written as

∇ × H = iωε 0 ε ⋅ E + J Ext ,

(2.132)

∇ × E = −iωµ 0 µ ⋅ H ,

(2.133)

where electric current density, J Ext is induced by the external source, and is distributed within a limited volume V. Regarding Poynting theorem (2.15) the energy of the source is spent not only to increase the energy stored by electric and magnetic fields, and dissipated in form of a loss within the area V, but also generates an outgoing electromagnetic radiation. This radiation may exist if last integral in (2.15) is non-zero 1. Solution of the systems (2.132) – (2.133) for the radiated field may not be obtained directly, meaning if we try to turn the two first order linear differential equations for two unknowns into a single differential equation of the second order for one unknown, then generically it becomes unsolvable. The main idea of the commonly used indirect approach for obtaining the result is to introduce “so called” auxiliary magnetic potentials, i.e. vector potential A , and scalar potential Φ 2. Introduction of these auxiliary functions allows obtainment solutions for the fair number of the radiation problem. As an example, the radiation from the linear electric current of the infinitesimal size is considered in Appendix-3. Results of the solution are presented by expressions (A3.22) and (A3.25).

1

The integral

∫ (E × H ) d S 

~

may become equal to zero because of boundary conditions on the

S

closed surface S. If, for instance, the volume V is surrounded by the PEC, then regarding (2.36) the tangential component of the electric field becomes equal to zero, which results in zero value of the above integral. 2

Not to be confused with phase shift. For the phase shift notation we always use

subscript.

81

Φ along with a

If the time-harmonic multiplier exp (iω t ) is included into those analytical results, then (A3.22) and (A3.25) may be rewritten as follows: •

For the magnetic field

I l  exp[i (ω t − k r )] H = ik sin θ ⋅ ϕ 0 r 4π •

(2.134)

For the electric field:

exp[i (ω t − k r )] Il sin θ E = θ 0 iω µ 0 µ 4π r

(2.135)

In those two above expressions we assume that the electric current filament, I with uniform current distribution is placed symmetrically at the origin of the spherical coordinates, and directed along Z-Axis as shown in Figure 2.24.

Figure 2.24. Hertzian dipole in spherical coordinates The length of the current filament is assumed to be infinitesimal, i.e. l > 1 ) and first Fresnel zone, ∆ (n) = ( Rn +1 − Rn ) / R1 in the vicinity of the radiating source (

r01 1 Ψ > 45 0 . If tan Ψ = 1 Ψ = 45 0 then τ = ± 45 0 . In order to obtain ε , first we find the product ± a b by multiplying (A2.13) by (A2.15), and (A2.14) by (A2.16), and adding the results.

± a b = (cosτ + M cos δ sin τ )(− M sin δ cosτ ) + ... ... + M sin δ sin τ (− sin τ + M cos δ cosτ ) .

(A2.26)

After simplifications the following expression may be obtained:

± a b = − M sin δ .

(A2.27)

Now we use (A2.9) to evaluate the ratio 1 .

2ab ± 2 =± a + b2

b a = 2 tan ε e = sin 2ε e b 2 1 + tan 2 ε e 1+ 2 a 2

(A2.28)

On the other hand we may modify the same expression by using (A2.17) and (A2.27)

±

1

2 M sin δ 2 tan Ψ sin δ 2ab =− =− = − sin 2Ψ sin δ . 2 2 1 + tan 2 Ψ a +b 1+ M 2

Trigonometric relation

2 sin x cos x 2 tan x = = sin 2 x is used. 2 1 + tan x sin 2 x + cos 2 x

120

(A2.29)

Combination of (A2.28) and (A2.29) results in

sin 2ε = − sin 2Ψ sin δ .

(A2.30)

Expressions (A2.22) and (A2.30) allow obtaining the direct relations between initial parameters of the linear cross-polarized components ( M = tan Ψ and δ ) and parameters of polarization ( τ and ε ). Axial ratio, AR may further be calculated from (A2.10). To find the inverse relations one may obtain cos δ from (A2.30), substitute into (A2.22), and solve for cos 2Ψ . The result is

cos 2Ψ = cos 2τ cos 2 ε

(A2.31)

which may be used to obtain Ψ . If (A2.30) is divided by (A2.22), then

tan δ = −

sin 2 ε 1 . cos 2 Ψ tan 2τ

(A2.32)

The final expression may be developed if cos 2Ψ is substituted from (A2.31). Then the result

tan δ = −

tan 2 ε sin 2τ

(A2.33)

allows finding the phase shift δ between initial LP-components of the elliptically polarized wave. -------------------------------------------------------------------------------------------------Example A2.1. Find polarization parameters of the radio wave, i.e. sense of polarization, angle of ellipticity, ε , axial ratio, AR, and tilt angle τ of the polarization ellipse, if parameters of cross-polarized components (vertical and horizontal) are: •

Amplitude of V-pol component: E1m = 3 mV/m



Amplitude of H-pol component: E 2 m = 8 mV/m



Phase shift between components: δ = −60 0 . Solution



M = tan Ψ = 8 / 3 = 2.67 ,

thus Ψ = 69.44 0 ,

121

2 Ψ = 138.89 0



We’ll use (A2.25) because of the condition M = tan Ψ > 1

1 2

τ = tan −1[tan(2Ψ ) cos δ ] + 90 0 = ... ... = •

1 tan −1[tan 138.89 0 cos(−60 0 )] + 90 0 = 78.210 2

(Answer)

Now using expression (A2.30) to obtain ε

(

)

sin 2 ε = − sin 2 Ψ sin δ = − sin 138.89 0 ⋅ sin − 60 0 = 0.5694 , therefore ε = •

Axial ratio

1 −1 sin (0.5694 ) = 17.35 0 . 2

(

)

AR = tan −1 (ε ) = tan −1 17.35 0 = +3.2 (10 dB).

(Answer)

(Answer)

The sense of polarization is RHEP for positive ε (see Figure A2.5).

Figure A2.5. Polarization ellipse sketch for the calculated parameters in Example A-2.1

----------------------------------------------------------------------------------------------Example A2.2. Find parameter M = tan Ψ (the ratio of the amplitudes of initial linear cross-polarized components) of the radio wave and the phase shift δ , if the following parameters of polarization ellipse are provided: angle of ellipticity, ε = −15 0 (LHEP), and tilt angle,

τ = 75 0 . Solution

122



Using the expression (A2.31)

(

)

cos 2Ψ = cos 2τ ⋅ cos 2ε = cos 2 ⋅ 75 0 ⋅ cos (2 ⋅ (−15) ) = −0.75 , thus Ψ = 69.3 0 , M = tan Ψ = 2.65 . •

(Answer)

Using the expression (A2.33)

( (

) )

tan 2 ε tan − 30 0 =− = 1.1547 , thus δ = 49.10 . tan δ = − 0 sin 2τ sin 150

(Answer)

APPENDIX-3 A3.1. HELMHOLTZ EQUATION FOR VECTOR POTENTIAL Recall the Maxwell’s equation (2.4) that testifies the principle of continuity of the magnetic field lines. Rewrite the equation in form

∇ ⋅ H = 0.

(A3.1)

It proves that magnetic fields are always solenoidal. Therefore any magnetic field may be represented as a curl of the other vector field, i.e.

H =

1

µ0 µ

∇× A.

(A3.2)

Here A is called vector potential. If we substitute (A3.2) into (2.2), then the Faraday's law for time-harmonic process may be rewritten as

(

)

∇ × E + i ω A = 0 .

(A3.3)

From (A3.3) one may conclude that because the curl of the vector field E + i ω A is equal to zero, then the field E + i ω A is purely potential. It is well known from the Electromagnetics course, that vector gradient of any scalar field Φ is always purely potential. Thus vector E + i ω A may be represented as a gradient of another scalar field

Φ called scalar potential i.e.

E + i ω A = −∇Φ .

(A3.4)

123

Now we substitute (A3.2) into the expression for the Ampere's law (2.1), written for the lossless spatial area that contains a time-harmonic current source J Ext .

1 ∇ × H = ∇ × ∇ × A = iω ε 0 ε E + J Ext .

(A3.5)

µ0 µ

In this expression a double cross product (double curl) may be modified by using the identity (A1.2.3.1) from Appendix -1.

∇ × ∇ × A = ∇ (∇ ⋅ A ) − ∇ 2 A .

(A3.6)

Now we substitute (A3.6) and E from (A3.4) into (A3.5).

∇ 2 A + ω 2 ε 0 ε µ 0 µ A − ∇(iω ε 0 ε µ 0 µ Φ + ∇ ⋅ A ) = − µ 0 µ J Ext .

(A3.7)

In order to simplify (A3.7) we may apply Lorentz condition or otherwise called calibration condition that allows make both, vector and scalar potentials interrelate to each-other. Regarding Lorentz calibration condition

iω ε 0ε µ 0 µ Φ = − ∇ ⋅ A .

(A3.8)



Indeed, both potentials, A and Φ has been initially introduced independently. Now an additional "calibration" request (A3.8) allows making significant simplification of (A3.7) without loosing the generality of further analysis. Under that condition the equation (A3.7) may be rewritten in form of Helmholtz equation as

∇ 2 A + k 2 A = − µ 0 µ J Ext 1,

(A3.7a)

where k is same complex number as that defined by (2.50) called propagation constant. Here we're considering the propagation in lossless medium, i.e. we'll assume k = k = β as being a real number. In Cartesian coordinates the linear operator ∇ 2 may be applied to each component of

{

}

the vector A = A X , A Y , A Z , thus (A3.7a) may be decomposed into the system of three linear independent partial differential equations for each scalar components as follows:

1

∇ 2 A X + β 2 A X = − µ 0 µ J Ext , X ,

(A3.8a)

∇ 2 A Y + β 2 A Y = − µ 0 µ J Ext , Y ,

(A3.8b)

(A3.7a) looks similar to (2.51). The difference is in right hand side. That's because (A3.7a),

inhomogeneous Helmholtz equation, is written for the source containing spatial region in contrast to (2.51), homogeneous Helmholtz equation, that is written for free space propagation medium.

124

∇ 2 A Z + β 2 A Z = − µ 0 µ J Ext , Z ,

(A3.8c)

which may be solved separately. Unfortunately similar decomposition is impossible in other coordinate systems, such as cylindrical or spherical curvilinear coordinates. The three equations, (A3.8a) – (A3.8c) are identical in form, therefore it's reasonable to expect the same approach in their solutions. For the infinitesimal point source located at the origin the right hand side of those equations become zero except of the point of the origin. For the source-free spatial region a general form that is common for all three expressions may be expressed in form of Helmholtz’s homogeneous equation

∇ 2ϑ + β 2ϑ = 0 ,

(A3.9)

where ϑ is one of the components of the A vector. This problem is spherically symmetric, hence it’s meaningful to expand the ∇ 2 operator in spherical coordinates based on (A1.2.4.13) from Appendix-1. The physical interpretation of this case of the scalar point source allows assuming that the solution will have only the radial dependence, thus in ∇ 2 derivatives by θ and ϕ are taken equal to zero. Then (A3.9) becomes single-coordinate ordinary differential equation and may be presented as

1 d  2 dϑ  r  + βϑ = 0 . r 2 dr  dr 

(A3.10)

It may be shown by substitution, that the following general solution satisfies (A3.10):

ϑ = C

exp(± iβ r ) . r

(A3.11)

Here C is an arbitrary integration constant. If we include time dependence into (A3.11), then it will represent two waves of the phases similar to that given by (2.60), while the amplitudes are decaying inverse proportional to the radial distance. One may conclude, that the solution with the negative sign represents the wave that propagates from origin to infinity, whereas the solution with the positive sign represents the wave that propagates from infinity to the origin. Hence the positive sign may be eliminated as meaningless. The integration constant may be defined based on specific conditions in the origin, where the point source is located. Hence (A3.11) is to be substituted into the inhomogeneous Helmholtz equation

1 d  2 dϑ  r  + βϑ = −δ (r ) r 2 dr  dr 

(A3.12)

125

that contains − δ (r ) function representing a point source at the origin 1. If a volume integral is taken from both sides of (A3.12), and letting r → 0 the integration constant may be found as C = (4π ) −1 . This procedure is nothing, but the application of the initial conditions at the point of origin, which allows turning a general solution (A3.11) into the particular solution of the following form:

ϑ =

exp(− iβ r ) , 4π r

(A3.13)

which satisfies the initial conditions. Expression (A3.13) represents a scalar spherical wave that propagates from the point of origin radially in all directions. Indeed, if timeharmonic multiplier exp(iω t ) is included into (A3.13), then

ϑ (r , t ) =

exp[i (ω t − β r )] 4π r

(A3.13a)

is a harmonic wave process with the constant phase ϕ = ω t − β r that moves along radial direction with the velocity of v = ω / β (compare with (2.60) – (2.62)), and with the amplitude that decays, being inverse proportional to the distance. In other words for the fixed time moment t = const the constant values of the phase ϕ = const will be associated with r = const distances from the origin. The relation r = const is the equation of the sphere in spherical coordinate system, thus the constant values of the phase, ϕ = const (wave fronts) are represented as source-centered spherical surfaces. Hence the scalar point source radiates a spherical waves radially propagating away from the point source. Now suppose that the scalar inhomogeneous Helmholtz equation (A3.8a) contains the source that has a weighted distribution J Ext , X within a finite volume V, i.e. instead of the point source, we consider the source of the finite size. Based on principle of linear superposition we may assume, that the solution will be represented as a sum of the numerous waves coming from the multiple point sources located within the volume V. In other words the solution (A3.13a) for that case may be written as

1

δ (r ) is a Dirac's delta function. It’s equal to zero everywhere except of the point of origin. For

that point δ (r ) tends to infinitely, so thus the overall integral along the entire area of coordinates

r = (0, ∞) is equal to unity.

126

exp[i (ω t − β r ′′)] A X (r , t ) = ∫∫∫ µ 0 µ J Ext , X (r ′, t ) dV , 4π r ′′ V

(A3.14)

where the distances r , r ′ and r ′′ are shown in Figure A3.1.

Figure A3.1. Definition of distances in (A3.14) integral

Expressions similar to (A3.14) may be written for Y- and Z-components of the magnetic



vector potential. Hence the total magnetic vector potential A may be written as a vector:

exp[i (ω t − β r ′′)] A (r , t ) = ∫∫∫ µ 0 µ J Ext (r ′, t ) dV . 4π r ′′ V

(A3.15)

A3.2. RADIATION FROM THE ELECTRIC CURRENT POINT SOURCE Consider a piece of infinitely thin wire of the length l with an electric current I directed along Z-axis as shown in Figure A3.2. If the current is time-harmonic, then it is considered as a source that radiates radio waves. For the current uniformly distributed within the region − l / 2 < z < l / 2 , and for the wire length l > λ ) is considered. Thus (A2.25) may approximately be rewritten as

(

)

I l  2 exp − ikr sin θ . k ∇H × ϕ 0 = −θ 0 4π r

(A3.26)

Here, the cross product r0 × ϕ 0 is replaced by − θ 0 based on disposition of those unit vectors (see Figure A1.1b from Appendix-1). Now we can substitute ∇ × ( H ⋅ ϕ 0 ) by (A3.26) in expression (A3.23), then the final result for the electric field phasor will become

exp(−ik r ) Il sin θ ⋅ θ 0 , E = iωµ 0 µ 4π r

(A3.27)

where k is defined as ω ε 0ε µ 0 µ . It may be seen from the last expression that the maximum value of the electric field appears at θ = 90 0 , i.e. in direction perpendicular to dipole’s axis. For non-magnetic medium ( µ = 1 ) the magnitude of that electric field may be written as

Em =

I lωµ 0 60π I l . = λr 4π r

(A3.28)

For some applications it is more convenient to express (A3.27) in terms of the dipole moment. The Hertzian dipole is his original form is shown in Figure A3.3, where the spheres on both ends of the dipole are to cumulate charges that are moving due to current I .

Figure A3.3. Hertzian dipole

130

The general definition for the vector of dipole moment is

p = ql ,

(A3.29)

where q is the charge that cumulates on both ends of the dipole shoulders, and l is a vector of the length of the dipole, and direction along its axes. For the time-harmonic source we may write

dq dp l = Il . = iω p = dt dt

(A3.30)

Now if we substitute the magnitude of (A3.30) into (A3.27) it will result in

k 2 p exp(−ik r ) sin θ . E = θ 0 4π ε 0ε r

(A3.31)

Note: minus sign is omitted in the last expression.

APPENDIX-4

FRESNEL'S INTEGRALS Expression (2.153) and (2.154) for C and S surface integrals may be modified as based on trigonometric identities (A1.1.1) and (A1.1.2) from Appendix A-1.

π C = ∫∫ cos  u 2 + v 2 2 S′

(

∞ π  π  cos  u 2 du ⋅ ∫ cos  v 2 dv − −∞ u0 2  2 

) du dv = ∫ 



∞ ∞ π  π  − ∫ sin  u 2 du ⋅ ∫ sin  v 2  dv , −∞ u0 2  2 

π S = ∫∫ sin  u 2 + v 2 2 S′

(

∞ π  π  sin  u 2 du ⋅ ∫ cos  v 2 dv + −∞ u0 2  2 

) du dv = ∫ 

(A4.1)



∞ ∞ π  π  + ∫ cos  u 2 du ⋅ ∫ sin  v 2  dv . − ∞ u0 2  2 

As mentioned in section 2.6.3 the limits for variables in (A4.1) and (A4.2) are

u 0 < u < ∞ , and − ∞ < v < ∞ . 131

(A4.2)

Now consider the integral ∞

 π  I = ∫ exp i t 2  dt .  2  0

(A4.3)

To evaluate (A4.3) the following change of the variable is applied:

π

ζ 2 = −i t=

Then

2

1+ i

t2 .

(A4.4)

ζ.

π

(A4.5)

If (A4.5) is substituted into (A4.3), then

I=

1+ i

π





0

(

)

exp − ζ 2 dζ .

(A4.6)

The integral in this expression is well known Gaussian error-integral, which is equal to

π / 2 . With that in mind if we recall (A4.6), we may obtain I=

1 (1 + i ) . 2

(A4.7)

On the other hand (A4.3) may be rewritten as ∞





π  π   π  I = ∫ exp i t 2  dt = ∫ [cos t 2  + i ∫ sin  t 2  ] dt =C F (∞ ) + i S F (∞ ) . 2  0 2   2  0 0

(A4.8)

From the comparison of (A4.7) and (A4.8) we may define ∞ ∞ 1 π  π  C F ( ∞ ) = ∫ cos  t 2  dt = S F (∞ ) = ∫ sin  t 2  dt = . 0 0 2 2  2 

(A4.9)

Based on those values, the expressions (A4.1) and (A4.2) may be transformed to ∞ ∞ π  π  C ( u 0 ) = ∫ cos  u 2  du − ∫ sin  u 2  du , u0 u0 2  2 

(A4.10)

∞ ∞ π  π  S ( u 0 ) = ∫ cos  u 2  du + ∫ sin  u 2  du , u0 u 0 2  2 

(A4.11)

where the integrals in (A4.1) and (A4.2) with the integration limits (−∞, ∞) are twice larger, than those with the limits (0, ∞) . Then (A4.10) may be rewritten as

C ( u0 ) =

1 u0 1 u0  π  π  − ∫ cos  u 2  du − + ∫ sin  u 2  du = 2 0 2 0 2  2 

= S F ( u0 ) − C F ( u0 )

(A4.12)

Similar transforms may be applied to (A4.11). Then it may be rewritten as

132

S ( u 0 ) = 1 − [ S F ( u 0 ) + C F ( u 0 )] .

(A4.13)

In (A4.12) and (A4.13) we used the following notations

and

u0 π  C F ( u 0 ) = ∫ cos  t 2  dt , 0 2 

(A4.14)

u0 π  S F ( u 0 ) = ∫ sin  t 2  dt . 0 2 

(A4.15)

(A4.14) and (A4.15) are called Fresnel's cosine and Fresnel's sine respectively. Originally this two functions of u 0 had been presented in form of so called Cornu spiral that is shown in Figure A4.1. It's easy to see from (A4.14) and (A4.15) that both, C F ( u 0 ) and S F ( u 0 ) are odd functions, i.e.

C F (− u 0 ) = − C F ( u 0 ) , and S F (− u 0 ) = − S F ( u 0 ) .

Figure A4.1. Cornu spiral

133

(A4.16)

Chapter 3. BASICS OF ANTENNAS FOR RF RADIO LINKS

3.1. BASIC PARAMETERS OF ANTENNAS Antenna is defined by IEEE as “…part of the transmitting or receiving system that is designed to radiate or receive electromagnetic waves” [1]. There are a variety of antennas designed for a wide range of applications, which are being characterized by the common set of parameters. Those are as follows: •

Radiation pattern/diagram



Directivity and gain



Input impedance



Radiation resistance



Effective area of aperture / Effective antenna-length



Polarization



Noise temperature

Regarding to the principle of reciprocity in electromagnetics the same antenna may be used in both regimes, namely in transmitting (radiating) and receiving. The basic parameters are the same in both regimes besides of the noise temperature, which is specific just for receiving mode of the antennas and discussed in Chapter 6.

3.1.1. RADIATION PATTERN AND DIRECTIVITY Assume a hypothetical isotropic radiating antenna of infinitely small size is placed in point A of space. Then the parameters of radiated field of harmonic excitation will only be distant-dependent. The isotropic antenna is just a useful abstraction in antennas and propagation theory. For non-isotropic antennas the radiated electromagnetic field strength at the observation point B depends not just on the distance between communicating A and B points, but also on antenna's spatial orientation relative to

134

orientation of the straight line between them. In other words, if the radiating antenna is placed in the origin of the spherical coordinate system (Figure 3.1a), then the field strength at B point will depend not just on radial distance r but on azimuth angle ϕ and zenith angle θ . For large distances 1, when antenna's relative dimensions are ignorable, and the antenna is considered as a point in space, this statement may be expressed in the following form:

K  (θ , ϕ ) , E B (r , θ , ϕ ) = Ψ r

(3.1)

i.e. electric field is inverse proportional to distance. For the particular type of the radiating element such as infinitesimal line current (Hertzian dipole) electric field is given by (2.135), so from the comparison with (3.1) one may define a constant of proportionality

K=

Il  (θ , ϕ ) = sin θ . The phase exponent, exp[i (ω t − kr )] is omitted iω µ 0 µ , and Ψ 4π

in (3.1) for simplicity.

 (ϕ , θ ) is a function that presents dependence on the angular coordinates. In general, Ψ  (ϕ , θ ) is usually more complicated than that for Hertzian dipole and the expression for Ψ becomes a complex function of angular coordinates.

Figure 3.1. a). Spherical coordinate system: θ – zenith angle, ϕ – azimuth angle, γ – angle of elevation. b). Surface element dS in spherical coordinates (dΩ – elementary solid angle). 1

Observation point is assumed to be allocated in so called “far zone”, i.e. when the distance

r ≥ 2 D 2 / λ . Here D is a maximum dimension of the antenna.

135

For the lossless propagation medium the power flow density (magnitude of Poynting vector) in any point of space may be defined by substitution of (3.1) into (2.74), i.e.

Π (r , ϕ ,θ ) =

2 1 K2  Ψ (ϕ , θ ) . 2 2W r

(3.2)

Now recall, that at the observation point for any arbitrary direction (ϕ , θ ) the portion of the total radiated power that flows through the surface element dS = r d Ω , which is 2

placed across the observation direction, is

dPΣ = Π dS =

2 K2  Ψ (ϕ , θ ) dΩ , 2W

(3.3)

where dΩ = sin θ dϕ dθ is solid angle subtended by the surface element dS from the origin at the observation point as shown in Figure 3.1b. The factor that represents the 2

 (ϕ ,θ ) ≤ 1 . Based on (3.3) the term angular dependence is a limited function, i.e. Ψ radiation intensity may be introduced now as

I Σ (ϕ ,θ ) =

dPΣ K 2  = Ψ (ϕ ,θ ) dΩ 2W

2

 (ϕ ,θ ) 2 , = I Σ , max Ψ

(3.4)

where I Σ , max = K 2 /( 2W ) is the maximum value of the radiation intensity. It may be seen from (3.4) that the radiation intensity, I Σ (ϕ , θ ) has physical meaning of the power that is radiated within the unit solid angle (within the angle of one steradian) along the observation direction. On the other hand, from (3.4), the function

 (ϕ , θ ) 2 = I Σ (ϕ , θ ) , Ψ I Σ , max

(3.5)

is solid geometrical figure that represents the angular distribution of the normalized radiation intensity. In antenna theory and practice its known as a power radiation pattern

 (ϕ , θ ) is called amplitude radiation of the antenna. The real part of complex quantity Ψ pattern, Ψ (ϕ , θ ) (or just simply radiation pattern), whereas the imaginary part of it,

arg [Ψ (ϕ , θ )] is called phase pattern. All three, power pattern, amplitude pattern, and phase pattern are solid figures, which may be presented either in spherical or in Cartesian (rectangular) coordinates. An example of three-dimensional (3D) amplitude radiation pattern is shown in Figure 3.2a for the vertically placed electric current point source (elementary electric dipole) that is analyzed in Appendix-3. As may be noticed

136

from (A3.25) the electric field has a Z-axial symmetry, thus the radiation intensity is only

θ -dependent, which, in this case, is expressed in a simple form: Ψ ( θ ) = sin θ . In spherical coordinates Ψ ( θ ) is toroidally shaped figure as shown in Figure 3.2a. It must be noted that in engineering practice the use of spatial 3D radiation patterns is not always convenient. The use of two-dimensional (2D) cuts of that solid figure is more preferable in most of applications. They may be derived from Ψ (ϕ , θ ) by keeping either

θ = const, which results in so called conical cuts 1, or by keeping ϕ = const, which results in vertical cuts. Examples of those cuts are given in Figures 3.2b and 3.2c for the same electric current point source. Directional properties of the antenna may be identified not only by radiation pattern, but also by a numerical quantity called directivity, (marked D). It is a quantity that shows how many times the total radiated power of the hypothetical isotropic antenna placed at the transmission point must be increased in order to achieve the same field intensity at the receiving point if antenna of directed radiation were be placed in the same transmission point under the same conditions. In other words, if the antenna with directed radiation is placed in point of origin and the radiated field is observed at point B, then the observer, who assumes the radiating antenna to be isotropic, will attribute to it the total hypothetical radiated power of

′ PΣ = PΣ ,hypothetical = D ⋅ PΣ .

(3.6)

Here PΣ is the total power radiated by the real antenna.

1

Horizontal cut for

θ = 90 0

is a particular case of it.

137

Figure 3.2. Examples of amplitude radiation patterns of the vertically placed symmetric electric dipole: a). Three-dimensional pattern, b). Vertical cut, c). Horizontal cut In order to express directivity, D in terms of radiation pattern, the expression (3.6) may be modified as follows:

′ PΣ D(ϕ 0 ,θ 0 ) = = PΣ

4π I Σ (ϕ 0 ,θ 0 ) 2π

π

∫ dϕ ∫ I 0

Σ

=

(ϕ ,θ ) sin θ dϕ dθ

0

4π [Ψ (ϕ 0 ,θ 0 )]2 2π

π

∫ dϕ ∫ [Ψ (ϕ ,θ )] 0

2

,

(3.7)

sin θ dϕ dθ

0

where ϕ 0 and θ 0 are angular coordinates of the observation point, D(ϕ 0 , θ 0 ) is the value of directivity for the direction from antenna towards the observation point, ϕ and

θ are independent variables. Total power PΣ in denominator of (3.7) is found as an integral of (3.3) taken for the entire space. The expression for the elementary solid angle is dΩ = sin θ dϕ dθ . In the numerator we assume, that the radiation intensity along any given direction (ϕ 0 , θ 0 ) for the isotropic hypothetical antenna, I Σ (ϕ 0 , θ 0 ) is constantly



spread all around the full solid angle 4π . Thus for the total power PΣ the integration is simply replaced by the product 4π I Σ (ϕ 0 ,θ 0 ) . If we take into account that the average radiation intensity is defined as the actual radiated power PΣ divided by 4π , i.e.

138

I Σ , ave =

1 4π



π

0

0

∫ dϕ ∫ I Σ (ϕ ,θ ) sin θ dϕ dθ

(3.8)

then another definition for directivity may be formulated from (3.7) as

D(ϕ 0 , θ 0 ) =

I Σ (ϕ 0 , θ 0 ) I Σ , ave

,

(3.9)

namely, the directivity shows how much more the antenna’s radiation intensity in any particular direction (ϕ 0 , θ 0 ) is greater than antenna’s average radiation intensity. In some cases in engineering practices the term radiation pattern refers to the angular dependence of directivity and is presented as D (ϕ , θ ) function. When the term “radiation pattern” is used for the function D (ϕ , θ ) , then the term “directivity” is used to identify the

(

)

maximum value of that function, Dmax = D ϕ max , θ max . Depending on the type of antenna, its frequency range, working conditions, etc. the directivity may vary in a wide range: from 1.5 to 106 (unitless), and even more.

 (ϕ , θ ) | 2 , Ψ (ϕ , θ ) , and arg [Ψ  (ϕ , θ )] the function D(ϕ , θ ) is also a 3D-solid Similar to | Ψ figure and, as said before, for the practical applications its sometimes useful to refer to both, conical (elevation) and vertical (azimuthal) cuts as follows:

Dcon (ϕ ) = D ( ϕ , θ = const ) , Dvert (θ ) = D ( ϕ = 0, θ ) .

(3.10)

If antenna has a predominant direction of radiation, then another parameter is applicable to specify the directivity indirectly: that is the radiation beam width (or antenna’s angular aperture). This parameter is defined as a range of angular coordinates that surrounds the direction of maximum radiation. For those angles the value of D remains either •

greater than zero, which defines a first null beam width, FNBW = 2∆ϕ 0 , or



greater than 0.5 Dmax , which defines a half power beam width, HPBW = 2∆ϕ 0.5 .

An example of sketch for the angular apertures 2∆ϕ 0.5 and 2∆ϕ 0 is shown on Figure 3.3. This approach is equally applicable for both, conical and vertical 2D cuts.

139

Figure 3.3. Example of 2D cut of the antenna radiation pattern in polar coordinates: 2 ∆ϕ 0.5 is FNBW, and 2 ∆ϕ 0 is HPBW

Several lobes may occur in the radiation pattern of the directive antenna as shown in Figure 3.3. The biggest that associates to the most intensive radiation direction is called the main lobe, others are called side lobes, or back lobe. For some applications the real radiation pattern (the example is shown in Figure 3.4a) is idealized, i.e. replaced with the hypothetic pattern, where the total radiated power is concentrated within the main lobe of the uniform intensity Dmax = D (ϕ max ,θ max ) = const, as shown in Figure 3.4b.

Figure 3.4. a). Real 3D radiation pattern of the antenna, b). Equivalent replacement (idealized) radiation pattern in form of 3D spherical sector

140

Recall that denominator of (3.7) is equal to total power radiated by the real antenna. The replacement by the idealized pattern assumes that the same power is radiated within the solid angle Ω , where the directivity remains constant and equal Dmax . From normalization condition (3.5) we may have Ψmax = 1 , thus for the radiation solid angle 2

Ω the following expression may be written for denominator of (3.7). 2π

π

2 ∫ dϕ ∫ [Ψ (ϕ ,θ )] sin θ dϕ dθ = Ψmax Ω = Ω 2

0

(3.11)

0

It's easy to see, that in right hand side of (3.7) the numerator becomes equal 4π . Then the expression (3.7) may be rewritten as

Dmax = D(ϕ max , θ max ) =

4π , Ω

(3.12)

This formula defines the directivity as a ratio of the whole solid angle 4π and the solid beam width Ω . It confirms the fact that the narrower the antenna’s main beam Ω is, the greater the directivity is. If the main beam of the antenna is single-lobed solid figure, then the solid angle Ω may be approximately calculated as a product Ω ≈ (2∆ϕ 0.5 ) ⋅ (2∆θ 0.5 ) of both 2∆ϕ 0.5 , and

2∆θ 0.5 , if they’re small enough, so the expression (3.12) becomes [2,3]

Dmax ≈

4π 41253 . = (2∆ϕ 0.5 ) rad ⋅ (2∆θ 0.5 ) rad (2∆ϕ 0.5 ) deg ⋅ (2∆θ 0.5 ) deg

(3.13)

Note that If the radiation pattern is ϕ -independent, i.e. it has a shape of body of rotation about Z-axes (see example in Figure 3.2), then antenna is called omnidirectional 1. -----------------------------------------------------------------------------------------------------Example 3.1 Find the directivity of the electric current point source (Hertzian dipole). Solution •

The amplitude radiation pattern may be defined from the expression for the electric field (A3.27) given in Appendix-3.

Ψ (ϕ , θ ) = sin θ •

1

(E3.1.1)

The solid angle of the main lobe is defined by (3.11).

Do not confuse with isotropic, which has a radiation pattern independent on both,

141

ϕ

and

θ.



π



π

0

0

0

0

Ω = ∫ dϕ ∫ [Ψ (ϕ , θ )]2 sin θ dϕ dθ = ∫ dϕ ∫ sin 3 dθ = •

8π . 3

(E3.1.2)

The directivity may be found from (3.12), i.e.

Dmax =

4π = 1.5 (1.76 dB) Ω

(Answer)

------------------------------------------------------------------------------------------------

3.1.2. RADIATION RESISTANCE AND LOSS RESISTANCE. ANTENNA GAIN AND EFFICIENCY FACTOR Generally speaking, an antenna may be considered as a complex load for the feeding line (feeder) that has a complex impedance called the input impedance of the antenna (see Figure 3.5). It is defined as

Z ant = Rant + i X ant = (Rant , 0 + Rant ,Σ ) + i X ant ,

(3.14)

where Rant , 0 is the one of the components of real part of the input impedance representing thermal losses from metallic parts of the antenna, and X ant is a reactive component of the input impedance.

Figure 3.5. Equivalent schematic representation of real antenna

Another component of the antenna's input impedance that is specific for the radiating systems is radiation resistance, Rant , Σ . The physical meaning of Rant , Σ is a virtual

142

resistance, which would dissipate the radiated power PΣ irreversibly, if antenna is replaced by the equivalent load at the feed’s terminal. In other words, the power PΣ radiated by the antenna into free space is represented as thermal loss on resistor R ant,Σ . If antenna with the input impedance Z ant = Rant + i X ant is connected to a feed line with characteristic impedance Zf , then from circuit theory, the maximum power transfers from the feed line to the load (antenna) will be achieved if

X ant = 0 Rant = Z f

  

(3.15)

The first of these two conditions means that antenna is tuned to feeding line (adjusted), while the second condition means that antenna is matched with the feeding line. Even if both conditions are satisfied, just part of the total power PTx from the transmitter, guided by feeding line towards the antenna’s input, is utilized as a radiated power PΣ . Another part of it will be wasted as an irreversible thermal loss P0 in metallic body of the antenna. In other words, P0 is dissipated on equivalent resistor Rant , 0 as heat. Taking into account, that the active power is in direct proportion to the resistance, the expression for antenna efficiency is given as

η ant =

Rant , Σ PΣ PΣ , = = PTx P0 + PΣ Rant , 0 + Rant , Σ

(3.16)

which shows what part of the total power PTx, that inputs antenna from the feeder, is radiated into free space. The meaning of antenna efficiency may be generalized by including several types of losses other than just thermal losses that occur in antenna. In addition to thermal losses, those are: •

Losses caused by the mismatch between the antenna input impedance and the feeding line



Polarization losses (for the receiving antennas only), caused by the mismatch between antenna polarization and polarization of the radio wave received



Losses caused by the non-efficient use of the antenna’s radiating aperture (for the aperture antennas only), such as non-uniform distribution of the amplitude and phase of the near field along the aperture



Spillover losses of the energy for the aperture illuminating radiation

143

Now if PΣ is found from (3.16) as

PΣ = η ant ⋅ PTx

(3.17)

and substituted into (3.6), then another important parameter, antenna gain may be introduced as

G = η ant

′ PΣ . ⋅D = PTx

(3.18)

From (3.18) the physical meaning of antenna gain may be stated as follows: it shows how many times the power delivered to the input of hypothetical isotropic ideal (lossless) antenna must be greater than the power delivered to the input of real antenna under consideration to get the same field strength at the observation point in the same conditions. In other words, G shows what the gain of power is when the real antenna with directive radiation is used instead of hypothetic ideal isotropic antenna in the same conditions.

3.1.3. ANTENNA EFFECTIVE LENGTH AND EFFECTIVE AREA OF THE APERTURE For practical applications its sometimes useful to introduce two closely related antenna parameters such as effective length and effective area of the aperture. Note that these parameters may have a real physical meaning just for specific types of antennas. In order to introduce the antenna effective length, consider a particular case of lossless, twin-wire open-end transmission line with the harmonic voltage source at the input (Figure 3.6a). If we move the open endpoints of the wires away from each-other as shown in Figure 3.6b, then the electric field lines will spread in the surrounding area. Time variations of the electric field will produce a time-varying magnetic field and vice versa, so this continuous electromagnetic process will result in generation of the radio waves that freely propagate away from the point of origin. This is a way how to come up with the simplest type of antenna, namely a dipole that is widely used for many applications. The distribution of currents along the wires in dipole is shown in Figure 3.6c with dashed line. To define the effective length l eff of this antenna, the area under the current distribution curve I (l ) is replaced by the rectangular (shaded region in Figure 3.6c) with

144

the same value of maximum current I max and the same area, so the height of this rectangular, l eff represents the effective length of antenna 1. Mathematically it may be expressed as

l eff =

1 I max

l/2

∫ I ( x)dx ,

(3.19)

−l / 2

where x is the integration variable, and the origin of coordinates is assumed to be at the center-point of dipole. It is obvious, that one may get different values of the ratio l eff / l for different current distributions I ( x ) along the wire. It also may be concluded from the geometrical picture that the more evenly the current distribution is, the closer this ratio is to unity. Some calculations using (3.19) are recommended to reader in problem P3.4.

Figure 3.6. a). The distribution of current I (l ) and voltage U (l ) along twin-wire open-end line, b). Radiation illustration of the symmetric electric dipole, c). Definition of the effective radiating length of the dipole, d). Equivalent schematic diagram of the dipole as a symmetric two-port network. 1

General definition of the antenna effective length may be found in [2] or [3].

145

For a wire antenna with any current distribution the amplitude value of electric field strength may be defined analogous to that for the ideal (Hertzian) dipole given by expression (A3.28) of Appendix-3, i.e.

E max =

60π I max l eff

, V/m,

λr

(3.20)

Recall the wave impedance of the free space, W0 = 120π Ohms, then if (3.20) is substituted into (2.74) the following expression may be obtained for the power flow density at the considering distance r from the antenna.

 I max ⋅ l eff Π = 30π   λr

2

  . 

(3.21)



If it is assumed that the total power is radiated isotropically, i.e. PΣ = I max ⋅ Rant , Σ , then 2

power flow density for the distance r and for any direction may be defined as

Π Isotr

2 ′ I max Rant , Σ PΣ . = = 4π r 2 4π r 2

(3.22)

Then based on definition (3.7) the directivity may be found as the ratio of (3.21) and



(3.22), which is the same as the ratio PΣ / PΣ :

 I max l eff 30π   λr D= 2 I max RΣ 4π r 2

  

2

.

(3.23)

Finally, (3.23) may be solved for the effective antenna length as

l eff =

λ π

D Rant , Σ 120

.

(3.24)

It is easy to show, that instead of (3.24) the expression

l eff =

λ π

G Rant 120

(3.25)

may be used, where Rant is an input impedance of antenna in optimal regime (tuned and matched). Generally speaking the expressions (3.24) and (3.25) may be applicable to any type of antenna, and not just to dipoles or even not only to wire antennas. In other

146

words l eff may be considered as a universal parameter and may be used to define the RMS of electromotive force inducted by radio wave in receiving antenna

emf = E ⋅ l eff ,

1

(3.26)

where E is the RMS value of electric field strength of the radio wave in close vicinity of receiving antenna. For the analysis of reception of the radio waves, the antenna is replaced by the Thevenin’s equivalent voltage source at the receiver's input with the equivalent parameters that are emf voltage and Rant as the inner resistance of the source (see Figure 3.7).

Figure 3.7. Equivalent replacement of the receiving antenna by Thevenin’s voltage source. Rant is the antenna input impedance, emf is the electromotive force induced by the radio wave, Z input is the input impedance of the receiver

As it is known from circuits theory, maximum power is transferred from source to load (i.e. maximum efficiency of the antenna-to-input is achieved), if Z input = Rant . Then the power that inputted to the receiver is equal to

1

2

This expression is true only if antenna polarization is the same as polarization of the received

radio wave, i.e. if there is no polarization loss. Generally effective length is introduced as a vector with the magnitude defined by (3.26) and direction the same as polarization. Hence (3.26) must be replaced by the dot product of two vectors. 2

When

Z ant = Rant + iX ant and Z input = Rinput + iX input are complex, then optimum conditions

means both, matching

Rant = Rinput and tuning X ant = − X input are satisfied.

147

2

E 2 l eff e2 . = PRx = 4 Rant 4 Rant

(3.27)

An alternate to the effective length of antenna is the antenna’s effective are of the aperture. Recall, that the magnitude of Pointing vector at the observation point is the power flow density, which is expressed by (2.74), so at this point in free space 2

E E2 E2 , Π= m = = W 120 π 2W where E = E m /

(3.28)

2 is the RMS value of the electric field. The effective area of the

antenna's aperture across the propagation direction, S eff is defined as a parameter that allows converting power flow density of the radio wave into received signal power, and backwards in transmitting antennas. Thus the total power absorbed by receiving antenna may be found as

PRx = Π ⋅ S eff

E2 = ⋅ S eff . 120 π

(3.29)

If (3.25) and (3.27) are substituted into (3.29), then the effective area of the aperture (effective aperture) may be found as

S eff

G λ2 . = 4π

(3.30)

Only for several types of antennas the term effective area of the aperture has a physical meaning. For the antennas, such as reflector type antennas, horn antennas, lens antennas, etc. where radio wave radiation/reception occurs from the geometrical aperture, the term antenna's effective aperture relates to the antenna's geometrical aperture, i.e. to the surface, where electromagnetic field is distributed during radiation or reception. For those types of antennas the effective aperture S eff relates to real geometrical aperture S as

S eff = ν ⋅ S ,

(3.31)

where ν is a aperture utilization factor called aperture efficiency. It shows what part of the geometrical aperture is being used effectively for radiation / reception. It may be realized, that ν is always less than one, and depends on field distribution of radio wave along geometrical aperture: the closer to a uniform distribution the field is across the

148

aperture, closer the value of ν is to unity. In practical applications the value of ν may vary approximately from 0.5 to 0.8.

3.2. GENERAL RELATIONS IN RADIO WAVE PROPAGATION THEORY Suppose a hypothetical isotropic transmitting antenna is placed at a point A in free space. Recall the expression (3.22) for the power flow density at reception point B at the distance r

Π Isotr

′ PΣ = 4π r 2

(3.32)

that is found as a surface density of the power uniformly distributed along the spherical surface of the radius r. Now we use (3.28) for that isotropic radiator in free space and combine with (3.32). Then the effective field strength at the receiving point B for this ideal case may be defined as

E Isotr =

30 PΣ′ r

.

(3.33)

For the same free space, when a real transmitting antenna with the gain GT and with



the input power PT is used, then PΣ must be replaced by the product PT GT in (3.32) and (3.33). Then field induced in ideal conditions (free space) by the real antenna at the observation point B may be written as

E0 =

30 PT GT r

,

(3.34)

and the power flow density is

Π0 =

PT GT . 4π r 2

(3.35)

At the observation point, the power generated by the incoming radio wave at the output of the receiving antenna may be found if Π 0 is multiplied by the receiving antenna effective aperture (3.30), i.e.

149

PR , 0 =

PT GT G R λ2

(4π r )2

,

(3.36)

where G R is the gain of receiving antenna. As mentioned, both expression (3.34) and (3.36) are derived for so called reference condition, i.e. for ideal propagation path, that is known as a reference path 1, when the propagation in free space (vacuum) is considered. In order to take into account the real conditions, when the propagation is affected by a number of factors, such as attenuations and scatterings in atmosphere, the impact of the lossy earth’s ground, etc, the propagation factor, F is introduced as a multiplier for E 0 . Propagation factor is officially defined as follows: “For the time-harmonic wave propagation from one point to another, the ratio of the complex electric field strength at the second point to that value which would exist at the second point if propagation took place in a vacuum” [4]. It is apparent that in general, the propagation factor is a complex quantity, i.e. it affects both, the amplitude and phase of the electric field strength at the reception point. It is apparent also, that for the received power on real propagation path the value PR , 0 must be multiplied by the square of propagation factor, F 2 . On the other hand it is to be noted that the real natural propagation conditions are randomly variable, especially if we take into account variations of the atmospheric refractive index, variations of attenuations in atmospheric precipitations and in ionosphere, etc., which results in random variations of the propagation factor around its median value. Thus the evaluations of the propagation factor are expected to be done in statistical sense. It may be concluded from the above definition, that for the real propagation conditions the reference values E 0 and PR , 0 must be replaced by the following 2:

30 PT GT

E = E0 F =

PR = PR , 0 F = 2

and

1 2

r

F,

PT GT G R λ 2

(4π r )

2

(3.37)

F2.

Here and below reference conditions will be denoted by “0” subscript. The effectiveness of feeding lines are not considered in (3.37) and (3.38). They may be

emerged into the values of and

ηF,R

(3.38)

′ ′ GT and G R as GT = GT ⋅ η F , T and G R = G R ⋅ η F , R , where η F , T

are transmission coefficients of the feeders on transmitting and receiving sites

respectively.

150

Expression (3.38) is referred to as Friis transmission formula. Both, (3.37) and (3.38) are general expressions for engineering calculations in propagation problems. The overall goal of the radio wave propagation theory is to define the value of F, by taking into account all factors, affecting propagation between transmitting and receiving antennas such as the impact of the earth’s ground and atmospheric effects, except of the free space losses that are included so far in (3.34) and (3.36). Equation (3.37) is more conveniently used for the frequency bands lower than HF, where wire-type antennas are mostly used, and therefore the EMF induced by radio wave at the terminals of the receiving antenna: in this case EMF is calculated by (3.26). For other cases, when aperture-type antennas are in use (usually for frequency bands VHF and higher) the expression (3.38) is more preferable. This expression may be written in logarithmic units (decibels) in the following form:

PR , dB = PT , dB + GT , dB + G R , dB − L0, dB − L A, dB ,

(3.38a)

where PR , dB is the power received (called power level of the received radio wave), PT , dB is transmitted power, G R , dB and GT , dB are the gains for receiving and transmitting antennas respectively,

 4π r  L 0, dB = 20 log  , (dB)  λ 

(3.39)

is the free space loss or the reference loss, and is common for any propagation path. The last term in right hand side of (3.38a)

1 L A, dB = 20 log   , (dB) F

(3.40)

is the propagation factor in logarithmic form that physically expresses the additional losses relative to the reference conditions. Formula (3.38a) is a logarithmic form of Friis formula, is sometimes referred to as an equation of the RF communication link budget, which is widely used for the engineering estimates in RF communication systems design. For the practical applications a term total propagation loss is introduced as a ratio between transmitted and received powers. It may be defined based on (3.38) as

P 1 1  4π r  ⋅ 2, = T =  ⋅ PR  λ  GT G R F 2

Ltot

151

(3.41)

which shows how many times the received power will decrease while propagating from transmitting antenna's input to receiving antenna's output. It may be seen from (3.41), that the total loss on propagation path may be reduced by proper choices of GT and G R antenna gains. However, for the generalized approaches, the antennas' properties must be excluded from (3.41). Having that in mind the more convenient form of the overall propagation loss between corresponding points caused only by the propagation path may be defined as

P EIRPT  4π r  1 , L = T GT G R = =  PR ( PR / G R )  λ  F 2 2

(3.42)

where a new useful term is introduced here as

EIRPT = PT GT

(3.43)

called effective isotropic radiated power (EIRP) that shows how much power must be radiated by a hypothetical isotropic antenna, placed in the same transmitting point, in order to achieve the same field strength at the reception point. Analogous to EIRPT in (3.42) a physical meaning of denominator, PR / G R is the power at the output of the hypothetical isotropic receiving antenna, considered as predicted effective isotropic received power. Note, that this term is not widely used in engineering practices. Friis formula as well as the expression for communication link budget is useful tool for the communication radio links engineering analysis and assessments. The propagation phenomena impact considerably not just communication, but also the RADAR 1 systems performance. In traditional radars the radiated signal in form of a series of short pulses with a low duty cycle 2 is used. For the target ranging, i.e. for the measurement of the distance from radar station to target a time delay between sounding (direct) pulse and its echo replica, t R is defined and then is further transformed into the value of range by using the following formula [5 - 6]

r=

ct R , 2

(3.44)

where c is the velocity of electromagnetic waves in free space.

1

Abbreviation of the “RADio Detection And Ranging”.

2

Pulse duty cycle is a ratio between pulse “active” state and period of repetition, i.e. is the

duration of the pulse relative to period of its sequence.

152

For finding the direction of target the antennas with a sharp “pencil” shape beam, or vertically positioned “fan” shape beams are used. Both, ranging and direction finding may be considerably affected by the parasitic reflections from the earth’s surface (Figure 3.8), as well as by refractions and absorptions in the atmosphere. The free space propagation losses, as well as the disturbances caused by the earth and the atmosphere relative to the reference path may be taken into account in the radar equation while estimating the power budget for the radar links, similarly to what Friis formula allows for the communication links.

Figure 3.8. Signal propagation pattern on radar targeting track. r – target range (real distance), d – horizontal distance

In order to obtain the radar equation, we first introduce a term radar cross section (RCS) that is used as a “…measure of reflective strength of a radar target…” [4]. The following explanations are to clarify this statement: •

Consider Π 0 is power flow density of the radio wave incident on a target with specified direction of incidence



For any direction of the scattered radio wave a hypothetic flat PEC surface of the area σ RCS is used, so that the total power scattered along that direction may be found as a product Π 0σ RCS



In other words RCS is the area of hypothetical PEC surface, σ RCS that is used for the equivalent replacement of target, and creates the same mirror-reflected power in direction of observation, as the power of the wave that is scattered from the real target

153



Apparently RCS is highly dependent on signal frequency, shape of the target, its orientation, and materials used

Based on above explanations, the power flow density of the echo signal that comes back from target to the radar station point of location may be found if Π 0σ RCS is divided on the area of spherical surface with the radius r, i.e.

Π 0, radar , received =

PT GT 1 . σ RCS 2 4π r 4π r 2

(3.45)

In order to obtain the amount of total power of the echo signal that is received by the radar antenna, this value must be multiplied by the effective area of the antenna, 1 which is defined by expression (3.30). Hence the power received in reference condition is

PR , 0 = Π 0, radar , received S eff

PT G 1 Gλ2 PT G 2 λ 2 σ RCS . = = σ RCS 4π r 2 4π r 2 4π (4π ) 3 r 4

(3.46)

This expression does not count the impacts caused by the real propagation conditions, such as earth’s and atmosphere’s impacts, and is valid for free space only. As in case of Friis formula all propagation phenomena may be included into consideration by involving the relative loss factor, so thus (3.46) may be replaced by

PR = PR , 0 F 4 =

PT G 2 λ 2 σ RCS F 4 . 3 4 (4π ) r

(3.47)

In contrast to (3.38), where F 2 is implemented to count for the losses in real condition on communication links, in (3.47) the propagation factor is counted to the power of four because the signal is affected twice: for direct propagation, and then for the backwards propagation of the echo. The radar range equation may then be found from (3.47) by solving it for the distance, i.e. (see also [5])

 P G 2λ 2  r =  T 3 σ RCS F 4   (4π ) PR 

1/ 4

.

(3.48)

If the sensitivity of the radar receiver is denoted as PR , min 2, then the maximum detectable range of the radar is

1

In most widely used so called "mono-static" radars the same antenna is used for sounding and

reception of the echoes, i.e. 2

GT = G R = G is assumed.

PR , min is the lowest signal power the receiver is able to “sense”.

154

rmax

  P G 2λ 2 σ RCS F 4  = T 3   (4π ) PR , min

1/ 4

.

(3.48a)

The values of F may be estimated based on the materials provided in chapters 4 and 5, and some estimates for PR , min may be found in Chapter 6.

REFERENCES [1] IEEE Std 145-1983. IEEE Standard Definitions of Terms for Antennas [2] Stutzman, W.L., Thiele, G.A. Antenna Theory and Design. John Wiley & Sons, Inc. 1998. [3] Balanis, C.A. Antenna Theory. Analysis and Design. John Wiley & Sons, Inc. 2005. [4] IEEE 100. The Authoritative Dictionary of IEEE Standard Terms. Seventh Edition. IEEE Press 2000 [5] Collin, R.E., Antennas and Radio Wave Propagation, NY: McGraw-Hill Book Co.,1985 [6] Skolnik, M.I., Introduction to RADAR systems, 3-rd edition, NY: McGraw-Hill Co. 2001.

PROBLEMS P3.1. Derive the expression (3.25) based on expression (3.24). P3.2. Confirm the expression (3.13). P3.3. Based on results of problem P2.24 calculate how many times the diameter of parabolic reflector antenna must be increased in order to achieve the same gain increase, as in case of using ring-type director. What the advantages and disadvantages of those two methods are for different applications. Explain. Answer: 7 P3.4. Use (3.19) to calculate the effective length l eff for the following wire antennas: a) Ideal dipole (Hertzian dipole) with uniform current distribution: I ( x) = I max



b) Short dipole ( l 1 km .

If (4.78) is expressed in non-logarithmic (linear) form, then one may note that

B / 10 equates to the same meaning as parameter n in (4.75), being fairly close to the value B / 10 ≈ 4 , which is specific for the ground wave propagating over the flat earth. The base station antenna elevation must be defined relative to the average ground level within the distances of 3 – 10 km from the base station along the propagation path. Therefore hb may vary slightly when mobile station moves.

191

4.1.5.1.b OTHER EMPIRICAL MODELS The COST 231-Hata 1 model allows extension of the Okumura-Hata model to the frequency band 1500 – 2000 MHz, applicable for small and medium cities [7, 8].

where:

LdB = F + B log r0, km − E + G ,

(4.83)

F = 46.3 + 33.9 log f MHz − 13..82 log hb .

(4.84)

B is defined by (4.77), E is defined from Table 4.2 for small and medium cities, and

 0 dB G=  3 dB

for medium − sized cities and suburban areas for uburban (metropoli tan) areas

(4.85)

Parameter G here is not to be confused with the antenna gain. Other restrictions to this method are the same as for the Okumura-Hata method.

The Lee model [10] is expressed as a power of distance:

LdB = 10 n log r0, km − 20 log hb , eff − M − 10 log hm + 29 ,

(4.86)

where parameters M and n are obtained from the measurements, and are displayed in Table 4.3. Here, in this method, a new term, base antenna effective height hb , eff is introduced as follows: a line that is tangent to the landscape profile at the mobile antenna location is extended towards the vertical line at the base station antenna; hb , eff is found as a distance from the base antenna positioning point to that intersection as shown in Figure 4.12.

Table 4.3

1

Propagation Area

n

M

Free space

2

- 45

Ideal flat earth

4

-45

Open (rural) area

4.35

- 49

Suburban

3.84

- 61.7

COST stands for Committee on Science and Technology. This committee programs and coordinates

European Community collaborative studies in the areas of science and technology.

192

Urban (averaged)

3.96

- 73.75

Figure 4.12 Demonstration of the base antenna effective heights in the Lee model.

----------------------------------------------------------------------------------------------Example 4.6 Use the Okumura-Hata model to obtain the spatial distribution of the median values of the total power loss at 800 MHz around the base station of the cellular phone network, located in an urban area of a large city, utilizing the initial data given below: •

Base station antenna average height: hb = 30 m



Average height of mobile phones above the ground level: hm = 3 m. Solution

Based on (4.76), (4.77), and formulas provided in Table 4.2 proper parameters are calculated as follows: •

A = 69.55 + 26.16 log 800 − 13.82 log 30 = 125.1 dB



B = 44.9 − 6.55 log 30 = 35.22 dB



E = 3.2 ( log (11.75 ⋅ 3)) − 4.97 = 2.69 dB 2

Results of calculations of the total loss LdB from (4.78) are shown in Table E-4.6 below. Table E-4.6 Distance from the base station, r0, km

1

1.5

2

2.5

3

3.5

Total power loss, dB

122.41

128.6

133

136.4

139.2

141.57

193

4.1.5.2. PHYSICAL MODELS Amongst the numerous types of physical models created during last several decades, the Walfish-Ikegami Model is the one most widely recommended by ITU-R. This method is based upon the combination of two: one proposed by Walfish and Bertoni [9], and the other proposed by Ikegami, et al [10]. This combined methodology allows the prediction of overall losses (in dB) between corresponding points. It is based on fairly precise analytical counts of the propagation phenomenon. However, some instituted corrections may permit better matching of predicted losses with those obtained from experimentally observed data. Note that a complete description of propagation physical models and mechanisms may be found in [4], [5] and [8].

4.1.5.2a. Non-LOS (LINE-OF-SIGHT) PATHS In this case the total loss (3.42) is defined as a combination of three composite terms:

LdB = 10 log [ EIRPT /( PR / G R )] = L 0 + Lmsd + Lrts ,

(4.87)

where L0 represents a free space loss (reference path loss) defined by (3.39) for the ideal conditions and isotropic antenna. Lmsd acts as the “multiple screen diffraction loss”, i.e. the total loss due to diffraction from multiple roof tops along the propagation path (see Figure 4.13)), and Lrst represents the loss caused by diffraction from the roof of the last building down to street level.

194

Figure 4.13 Sketch of the simplified urban/suburban terrain and propagation path for the Walfish-Ikegami "flat edge" model.

Note that a simplified "flat edge" approach for the buildings is employed in this model, i.e. all buildings are considered of the same height (average value of the roof top heights) and equally spaced. On Figure 4.13 w denotes the width of a street, which is usually assumed to be equal to the wall-to-wall distance between buildings, thus w = b / 2 , where b is the average distance between buildings, and ϕ (in degrees) is the street orientation relative to the direct propagation path.

Lrst term introduced in (4.87) is dependent on ϕ angle. Lrts = −16.9 + 10 log f MHz + 10 log  − 10 + 0.354 ϕ ,  where L (ϕ ) =  2.5 + 0.075 (ϕ − 35), 4.0 − 0.114 (ϕ − 55 0 ), 

(hroof − hm ) 2 w

+ L (ϕ ) ,

0 0 < ϕ < 35 0 35 0 < ϕ < 55 0 55 0 < ϕ < 90 0

(4.88)

(4.89)

Multiple screen diffraction loss: Lmsd is defined as

Lmsd = Lbsh + k a + k d log r0 + k f log f MHz − 9 log w with the following values for the Parameter k f :

195

(4.90)

   f MHz − 4 + 0.7  925 − 1,   kf =  f   − 4 + 1.5  MHz − 1,    925

suburban areas (4.91)

urban areas

The other parameters in (4.90) are dependent on positioning of the antenna of the base station relative to the average roof tops level as follows: Antenna of the base station is above the average roof tops level ( hb > hroof ).

Lbsh = −18 log [1 + (hb − hroof )]

(4.92a)

k a = 54 ,

(4.93a)

k d = 18 ,

(4.94a)

Antenna of the base station is below the average roof tops level ( hb < hroof ).

Lbsh = 0

(4.92b)

  54 − 0.8 (hb − hroof )  ka =   (hb − hroof ) ⋅ r0 54 − 0.8 0.5 

k d = 18 − 15

hb − hroof hroof

r0 ≥ 0.5 km ,

(4.93b)

r0 < 0.5 km

,

(4.94b)

Note that for the LOS condition between corresponding points ( ϕ = 0 0 ) formula (4.87) is simplified to L ≈ L0 , which means in this case the term Lmsd + Lrts ≤ 0 , and must be excluded from consideration. A more accurate approach is given by the Walfish-Ikegami model in (4.96) below. It must be noted here, that this considering model is restricted to the following limits: 800 < f MHz < 2000 , 4 m ≤ hb ≤ 50 m , 1 m ≤ hm ≤ 3 m , 0.2 km ≤ r0 ≤ 5 km . -----------------------------------------------------------------------------------------------------Example 4.7 Use the Walfish-Ikegami model to define the spatial distribution of the median values of the total power loss around the base station of the cellular phone network located in the urban

196

area of a large city at 800 MHz, with the initial data given in Example 4.6 if the average height of the roof tops is hroof = 20 m, and streets with the width of w = 30 m, are positioned orthogonal to propagation path ( ϕ = 90 0 ). Compare the results with those given in Table E-4.6. Solution •

Combination of (4.88) and (4.89) is as follows:

Lrts = −16.9 + 10 log f MHz + 10 log

(hroof − hm ) 2 w

+ [4 − 0.114 (ϕ − 55)] .

(E4.7.1)

After proper substitutions into (E4.7.1) the result is Lrts = 21.98 dB •

For this case of the base antenna above the average roof tops ( hb > hroof ) the combination of (4.90) with (4.92a), (4.93a), and (4.94a) is

Lmsd = −18 log[1 + (hb − hroof )] + 54 + 18 log r0 + .....  f  ..... + − 4 + 1.5  MHz − 1 log f MHz − 9 log w = 9.76 + 18 log r0  925  

(E4.7.2)



Calculation results with formula (E4.7.2) are given in Table E-4.7



Total power loss is found from (4.87), and is presented in the table below



The differences between two considered prediction models are shown in the last row of the same table. Table E-4.7

Distance from the base station,

r0, km

1

1.5

2

2.5

3

3.5

Free space loss, L 0 dB (formula (3.39))

90.5

94

96.5

98.5

100

101.4

Lmsd , dB

9.76

12.93

15.18

16.92

18.35

19.55

122.24

128.9

133.66

137.4

140.33

141.9

-0.17

+ 0.3

+ 0.66

+1

+1.13

+ 0.33

Total power loss, dB ( from Walfish-Ikegami model) Difference between Walfish-Ikegami and Okumura-Hata predicted data, dB (compared with data from Table E4.6)

197

4.1.5.2b. LOS PATHS In the presence of the LOS between communication end points, when streets are radially directed from mobile station towards the base station ( ϕ = 0 0 ) 1, a two-ray interference approach that is presented in sections 4.1 and 4.2 for flat-earth approximation may be applied. Generally speaking, that approach is applicable if at least the first Fresnel's zone is clear. In other words, based on (2.179) the width of the street must exceed the maximum cross-sectional dimension of the first Fresnel's zone in order for LOS approach to be applied, i.e.

w ≥ R1 / 3 =

1 2 3

λ r0 = 0.3 λ r0 .

(4.95)

Here we assume r10 = r20 = r0 / 2 . For most practical combinations of w , f , and r0 the expression (4.95) is valid. Indeed, in the worse case of the lowest usable frequency

f = 100 MHz, and a large enough distance of r0 = 50 km, the inequality (4.95) becomes w ≥ 116 m, which is practically reasonable. The Walfish-Ikegami model provides an expression the total loss for LOS in the following form that is adjusted to experimental results [4]:

LdB = 42.6 + 26 log r0 + 20 log f MHz .

(4.96)

However the ITU-R Recommendation P.1411 provides the following set of formulas that allows estimation of the upper and lower borders for variations of the total loss, and may be used to check the accuracy of the range for the total loss calculated from (4.96):

LdB , lower

LdB , upper

1

   = Lbp +    

r 20 log  0 r  bp r 40 log  0 r  bp

   = Lbp + 20 +    

       

r 25 log  0 r  bp r 40 log  0 r  bp

r0 ≤ rbp (4.97)

r0 > rbp        

Sometimes refers to as street canyons

198

r0 ≤ rbp (4.98)

r0 > rbp

Here rbp is so called breaking point distance, which is simply the distance of the first maximum of the field defined by (4.19), i.e.

rbp = r1, max =

4 hb hm

λ

.

(4.99)

Then

 8π h b h m   Lbp = 20 log  2  λ 

(4.100)

is the total loss at the breaking point distance. Expression (4.100) is developed based on the fact that at the breaking point distance r0 = rbp = r1, max the unitless (linear) value of the free space loss becomes equal to

L 0 = 4π r0 / λ = 16π h b h m / λ 2

(4.101)

after we substituted (4.99) into (4.100). The difference between expressions (4.100) and (4.101), as noted in Section 4.2, is relative to the real field demonstrating twice the strength of the reference field in free space, when constructive interference occurs between direct and reflected waves. ----------------------------------------------------------------------------------------------Example 4.8 Based on the general approaches given in Sections 4.1 and 4.2, calculate predictable losses if a LOS condition exists between corresponding points ( ϕ = 0 0 ) for the initial data provided in Example 4.6. Compare with results from the Walfish-Ikegami Model, and verify that they satisfy the upper and lower boundaries given by (4.97) and (4.98). Solution Table E-4.8 (Example 4.8 calculation results)

ITU-R

outlined by

Boundaries

Distance from the base station, r0, km

1

1.5

2

2.5

3

3.5

LLOS , lower , dB

84.84

91.88

96.88

100.25

103.92

106.6

LLOS , upper , dB

104.84

111.88

116.88

120.25

123.92

126.6

90.5

94

96.5

98.5

100

101.4

Free space (reference) loss, L 0 dB – expression (3.39)

199

Total power loss in dB for two-ray interference (sections 4.1 and 4.2) Total power loss, dB (Walfish-Ikegami model, formula (4.96))

96.51

98.58

99.25

99.56

99.72

99.82

100.66

105.24

108.49

111

113.07

114.81

From Table E-4.8, one may confirm the fact that the Walfish-Ikegami model (4.96) for LOS condition provides field predictions for urban propagation that are closest to the mid-values outlined by ITU-R boundaries. -----------------------------------------------------------------------

4.2. PROPAGATION BETWEEN GROUND-BASED ANTENNAS OVER THE FLAT EARTH

Recall that in the case of elevated antennas of horizontal polarization a general expression for the propagation factor is (4.14), and an approximate formula for the field strength at the reception point over the “flat earth” is (4.30). From those expressions we may conclude that the decrease of antenna heights, h 1 and h 2 , or increase of the wavelength will result in the decrease of the field strength. It tend to zero much faster than it does in free space. As mentioned in sections 4.1 and 4.2, in the case of a horizontally-polarized transmitting antenna, the resultant field tends to zero due to cancellation between direct and reflected waves of opposite phases, (i.e. the smaller is h 1 , the closer is the phase shift between those waves to 1800). Indeed, the smaller the antennas heights are, and the closer the angle of incidence to 900, then the closer the phase shift ∆Φ to zero, thus the overall phase shift between direct and reflected waves becomes equal to 1800 as shown in Figure 4.3. In the case of a vertically polarized transmitting antenna. lowering the antenna heights does not result in complete vanishing of the total field. However, in reality if the antenna heights (either h 1 or h 2 , or both) are vanishing, the intensity of the field in the vicinity of air-to-ground interface may differ significantly from the field intensity predicted by two-ray

200

interference 1 as demonstrated in sections 4.1 and 4.2. In reality both, structure of the field and the propagation mechanisms of the radio wave near that interface, are significantly distinct from what is predicted by the ray-tracing approach. Below, a surface wave propagation mechanism that is specific to ground-based antennas is considered. We will use the term ground-based not only to the antennas that are placed on the earth’s surface, but also near it, i.e. we will assume that the antenna elevation heights over the ground are less or comparable to the wavelength. Generally speaking, we will consider not the absolute heights of the antenna elevations, but rather their relative values: h1 / λ and h2 / λ are to be considered as key terms for distinctions between elevated and ground-based antennas, and also to distinguish the propagation mechanisms associated with them. From this viewpoint the antennas are considered as elevated when both of those ratios are greater than several unities. Alternatively, the antennas are considered as ground-based, if one (or both) of those ratios is (are) less than a unity or even close to zero. It is evident that for real antenna heights (from several meters up to hundreds of meters) antennas are considered mainly as ground-based in the frequency range is limited to less than approximately 100 MHz. In other words, the physical concepts described in this section are reasonably applicable for relatively low frequencies such as VHF, HF and lower.

4.2.1. ANTENNAS OVER THE INFINITE, PERFECT GROUND PLANE As a reference (ideal) case for real propagation conditions, we now consider a vertical ground-based electric line current that is placed on infinite ground plane constructed of the perfect electric conductor (PEC) as shown in Figure 4.14a. We will assume the length of the line current, l small enough, i.e. l / λ > 1 ,

whereas for the propagation in the atmospheric air it is assumed as

(4.107)

ε1 = 1 . As

demonstrated below, condition (4.107) allows definition of relations between components of the electromagnetic field in the area above the ground by using the electromagnetic parameters of the ground, without “interweaving” fields into upper and lower mediums, i.e. without taking into account the values of electromagnetic field components below the airto-Ground interface. Recall section 2.4, and consider the radio wave that falls onto the boundary of two mediums, as shown in Figure 2.15 of Section 2.4. As illustrated in that section, the angles of incident, ϕ and refraction, ψ are bounded by the Snell's Law that may be presented here in general form for the lossy mediums as:

205

ε1 sin ϕ =

ε2 sinψ ,

(4.108)

where ε1 and ε2 represent the magnitudes of the dielectric constants of the upper and lower mediums respectively. Then, from (4.107) and (4.108) it becomes evident that the angle of refraction ψ is fairly close to zero, regardless of what the angle ϕ is, even when it becomes close to its maximum value, ϕ ≈ 90 o , (i.e. when radio wave propagation occurs along the horizontal axis). Thereby, part of the radio wave that penetrates into the second medium (refracted wave) will propagate in the direction close to perpendicular to the interface. As shown in Figure 4.16a, a vertically polarized, horizontally propagating wave that is initially generated by the vertical monopole above the earth’s ground will induce a downwards propagating plane wave within the ground. The electric and magnetic fields are tangential to the interface, and relate to each other via the intrinsic impedance of the ground expressed by formula (2.66).

E Y 2 = W 2 H X 2 =

120π

H X 2 .

ε 2 − i (60λσ )

(4.109)

Now we rewrite the boundary conditions (2.31a) and (2.32a) for tangential components of electric and magnetic fields as

E Y 1 = E Y 2 H X 1 = H X 2

1

 . 

(4.110)

Regarding (4.110) E Y 2 and H X 2 in (4.109), they may be replaced by E Y 1 and H X 1 . Then (4.109) may be rewritten as:

E Y 1 = W 2 H X 1 =

120π

ε 2 − i (60λσ )

H X 1 .

(4.111)

On the other hand, if (4.111) is taken into account, for the total electric field above the ground we may write

E 1 = W1 H X 1 = 120π H X 1 =

ε2 E Y 1 ,

(4.112)

and 2 2 2 2 E Z 1 = E 1 − E Y 1 = (ε2 − 1) E Y 1 .

1

(4.113)

Note that the condition for the magnetic field in (4.110) does not include J S , as it is shown in

(2.31a), as the presence of the conducive currents in the earth's ground are taken into account by introducing the complex form for the magnetic fields.

206

Figure 4.16 a) Disposition of the field vectors on the air-earth ground interface. b). Power flux density (Pointing vector) and its components

Thus, referring to (4.107), expression (4.113) may be rewritten in approximate from as

E E Y 1 = W0 H X 1 ≈ Z 1 . ε2

(4.114)

This expression is one of the forms of the Leontovich approximate boundary condition. Another form of Leontovich approximate boundary condition may be derived by following [22], based on the Gauss Law (2.3), which may be rewritten for both, upper and lower mediums as

∂E X 1 ∂E Y 1 ∂E Z 1 ∂E X 2 ∂E Y 2 ∂E Z 2 + + = + + = 0. ∂x ∂y ∂z ∂x ∂y ∂z

(4.115)

The volume charge, ρ tot , which is responsible for carrying conductive currents, is already included in the expression for complex dielectric constant indirectly, therefore it is omitted here. Below we will make the same assumption, namely that within the second medium the refracted plane wave is propagating vertically, along the negative Z-axis. Hence, the field components that are tangential to the interface remain continuous when crossing the airto-ground border, i.e.

∂E X 1 ∂E Y 1 ∂E X 2 ∂E Y 2 . + ≈ + ∂x ∂y ∂x ∂y Then, taking (4.116) into account from (4.115) we may conclude:

207

(4.116)

∂E Z 1 ∂E Z 2 . ≈ ∂z ∂z

(4.117)

Despite the E Z 2 component of the electric field below the interface is vanishingly small, however its propagation along the negative Z-axis may be expressed analytically as

E Z 2 = E Z 2, max exp (ik2 Z ) ,

(4.118)

∂E Z 2 / ∂ z = ik2 E Z 2 ,

(4.119)

Therefore:

and (4.117) may be rewritten as

∂E Z 1 ≈ ik2 E Z 2 = ik 0 ε2 E Z 2 , ∂z

(4.120)

where k 0 is a wave number in free space. Now we will utilize the boundary condition (2.23), which in our case of ε1 = 1 is presented as

E Z 1 = ε2 E Z 2 .

(4.121)

If we use this expression to replace E Z 2 by E Z 1 in (4.120), then we will ultimately end up with

ik ∂E Z 1 ≈ 0 E Z 1 , ∂z ε2

(4.122)

which is another form of the Leontovich approximate boundary condition. Compared to exact boundary conditions, the advantage of expressions (4.114) and (4.122) for the approximate conditions at the air-to-ground boundary is obviously clear: they allow expressing the relations of the field components in the air through the parameters of the medium (ground) beneath the interface. It allows achievement of the solution for the ground wave propagation problems without referring to the field in the second medium. In the next section we will consider analysis of the propagating field strength, namely the propagation factor that calculates for the losses obtained along the propagation path. Here we will consider the structure of a radio wave that propagates along the earth's surface and within its close vicinity. Assume a vertical monopole is used to generate a primary electric field E Z 1 at the transmission point. In the presence of the lossy ground a tangential component E Y 1 will

208

occur that closely relates to E Z 1 through expression (4.114). E Y 1 literally represents the voltage drop on unit length of a real, non-perfect interface along the propagation path, which does not occur while propagation along the PEC ground. The occurrence of E Y 1 component results in the tilting of the resultant electric field vector by angle χ , as shown in

~  = E × H Figure 4.16a. Hence the Poynting vector Π will also be tilted by the same angle towards the ground (Figure 4.16b), resulting in the splitting of the total power flow into two portions: one relates to the propagation along the interface, and the other that is directed normally downwards, perpendicular to the interface, and relates to the penetration into the ground, thereby identifying the power losses along the propagation path. Due to the complex character of the dielectric constant ε2 there is a phase shift between field components E Y 1 and E Z 1 that may be found from (4.114). Based on the complex forms (2.45) and (2.46), that phase shift may be expressed as

1 2

δ ground = δ ε 2 =

 60 λ 0 σ 1 tan − 1  2  ε2

  , 

(4.123)

where δ ε 2 indicates the argument of the complex dielectric constant of the earth’s ground. As may be seen from (4.123), in the low-frequency range, when λ 0 is large enough,

δ ground ≈ 45 0 , whereas for the high frequencies, when λ 0 is small, δ ground ≈ 0 0 . It becomes obvious that for the given amplitude-phase relations between E Y 1 and E Z 1 components, the total field E 1 will become elliptically polarized. Unlike the elliptically polarized waves considered in section 2.3, where the plane of polarization ellipse is positioned perpendicular to the propagation direction, here we have the case where the plane of the polarization ellipse is vertically coincident to the propagation direction, as shown in Figure 4.17. Figure 2.5 shows that for frequencies above 100 MHz both, the fresh water, as well as ground soil are considered as dielectrics, thus δ ε 2 ≈ δ ground ≈ 0 0 . Hence, regarding section 2.3, the elliptic polarization will become linear with the tilt angle

χ = tan −1 ( EY 1 / E Z 1 ) = tan −1 (1 /

ε2 ) > 1 ) the wavelength becomes much smaller than in the upper hemisphere, (i.e. λ 2 = λ 1 /

ε > 1 , then the combination of (4.137) with (4.132) and (4.133) will result in:

60λσ π  1  Φ F (∞ ) ≈ arg  = arg[ i (ε − i 60λσ )] = − tan −1 = 2 ε  2 x  =

π

 1  − tan −1  . 2  tan δ 

(4.145)

For the most conducive type of the earth surface, such as sea water, we have

tan δ >> 1 , and therefore Φ F (∞ ) ≈

π 2

. For the land-area such as dry soil that is

close to dielectric, we have tan δ 1 , then z n , q ≈ z n , q =∞ + . q z n

(4.159)

226

As follows from the right-hand-side of (4.159), if q > 30 , then for practical calculations the value of z n may be chosen approximately the same as for q = ∞ , i.e. z n , q ≈ z n , ∞ . Based on data provided in Table 2.1 (see chapter 2), we may confirm by simple calculations, that for any type of real earth's ground the statement q horiz = ∞ is always true for the horizontally polarized wave of any RF frequency.

(

Magnitude z n = z n = z n exp − i π

n, root order

3

)*

q=∞

q=0

1

2.338

1.019

2

4.088

3.248

3

5.521

4.820

For n = 4, 5, 6, …(and higher)

≈ [1.5 ( n − 0.25)π ]

2/3

Table 4.4

≈ [1.5 ( n − 0.75)π ]

2/3

* Note: the complex number z n is defined by taking a product of the numerical value from proper column, and exponential multiplier

exp ( i π / 3) . For example, z 2, ∞ = 4.088 exp ( i π / 3) .

On the other hand, if q < 0.5 , then we may choose the value of z n , q ≈ z n , 0 , which directly relates to q = 0 . It is thereby evident from (4.155) and (4.158) that this case may occur for vertically polarized waves only. Figure 4.27 demonstrates the dependence of parameter q on frequency for two polarizations, and for radio waves propagating along different types of earth surfaces.

227

Figure 4.27. Frequency dependence of parameter q for propagation along sea water ( ε = 80 , σ = 3 S / m ), wet soil ( ε = 30 , σ = 0.03 S / m ), and dry soil ( ε = 3 ,

σ = 10 −5 S / m ) As one may conclude from the above graphs, for horizontally polarized radio waves the values of z n may always be chosen from the second column of Table 4.4 regardless of the frequency and type of the earth's ground along the propagation path, i.e. z n = z n , q =∞ . For the vertically polarized radio wave some caution is needed, i.e. •

The assumption z n = z n , q =∞ is applicable for frequencies higher than 500 MHz



In the case of sea water and wet soil, the assumption z n = z n , q =0 (last column of Table 4.4) is applicable for frequencies less than 50 kHz,



For intermediate frequencies expressions (4.158) and (4.159) must be used for

z n estimates Within the illuminated zone-1 (see Figure 4.26) the series (4.153) converges very slowly; hence the calculation of the propagation factor becomes either highly time/effort consuming, or even impossible. Therefore, it is meaningless to use it instead of the interference formulas given in sections 4.1 to 4.3. However, in the shadow-zone it converges fairly fast, thus any term higher than the first one may easily be ignored and a "single-term" (sometimes called “single-mode”) formula may be adopted as follows:

F =2 π X

exp (i X z1 ) w ( z1 − y1 ) w ( z1 − y 2 ) . w ( z1 ) w ( z1 ) z1 − q 2

(4.160)

This expression represents a main achievement of Fock's approach known as the outcome of the asymptotic diffraction theory (ADT).

228

Expression (4.160) contains three separate terms. First term,

U (x ) = 2 π X

exp ( i X z1 ) z1 − q 2

(4.161)

we’ll call a attenuation factor. It depends only on normalized distance X . The second and third terms are symmetric and may be written in general form (4.162) given below. They depend only on the transmitting and receiving antennas' elevations y1 and y 2 respectively, i.e.

V ( y1, 2 ) =

w (z1 − y1, 2 ) w ( z1 )

.

(4.162)

Those terms are called height-gain function. So, in general, for the given value of q, the “single-term” propagation factor may be expressed as a product:

F ( X , y1 , y 2 , q ) = U ( X , q ) V ( y1 ) V ( y 2 ) .

(4.163)

The fact that both height-gain functions are counted in (4.163) symmetrically comes from the principle of reciprocity in electromagnetic, and confirms the correctness of all statements of the Fock's asymptotic diffraction theory.

4.3.2. PROPAGATION BETWEEN GROUND-BASED ANTENNAS As noted previously, the term "ground-based antenna" signifies y1, 2 f 02 , where f 01 and f 02 are derived from (5.60) by taking it equal to zero: 2

f 01 =

f  fH  2  + fc − H ,  2  2 

f 02 =

f  fH  2  + fc + H  2  2 

(5.62)

2

(5.63)

Those expressions are the same as (5.55) and (5.56). Some properties of transverse-propagation in magneto-active plasma are as follows: 1. While propagation in magneto-active plasma, an extraordinary ray generates an additional electric field component E z along the propagation direction, which, regarding some estimates provided in [2-4]), is 900 phase shifted relative to E y , and is much smaller by magnitude, than the initial component E y . If the extraordinary ray is the only one that propagates in considered ionospheric plasma-medium, then the superposition of those two electric field components, E y and E z will result in elliptic polarization of the wave, with polarization ellipse lying in Y0Z-plane. 2. Similar to the previous case of longitudinal propagation, in this case of transverse propagation magneto-active plasma exposes different properties ( ε ord and ε ext ) for ordinary and extraordinary rays. Therefore each of those two components of linearly-polarized radio waves will propagate with different velocities, so thus at the end of propagation distance an extra phase shift between them will occur:

∆Φ =

ω c

r

(

)

ε ord − ε ext .

(5.64)

The total field strength, as a superposition of those two components

E = x0 E x + y 0 E y

(5.65)

may have different polarization for different values of ∆Φ (see section 2.4). While moving along the medium one polarization type may smoothly be transformed into another, depending on distance covered. It must also be noted that the sense of polarization may become even more complicated if E z component is taken into account in (5.65).

295

3. When the frequency of the radio wave becomes close to resonant,

f ≈ f ∞ , then similarly to

the case of longitudinal propagation one of the components of the arbitrary polarized wave, namely extraordinary component, will experience an intensive attenuation and may disappear almost completely, being absorbed by the medium. In other words the magneto-active plasmamedium may be considered as a polarization filter at that particular resonant frequency.

5.5.3. PROPAGATION OF THE RADIO WAVE ARBITRARY ORIENTED RELATIVE TO THE EARTH’S MAGNETIC FIELD Consider the general case when the Pointing vector of the radio wave has an arbitrary γ angle with earth’s magnetic field H 0 . Let’s decompose the magnetic field vector into longitudinal

H L = H 0 cos γ

(5.66)

and transversal

H T = H 0 sin γ

(5.67)

components as it's shown in Figure 5.18.

Figure 5.18. Disposition of earth’s magnetic field into transversal and longitudinal components for an arbitrary direction of propagation in magneto-active ionospheric plasma

For the previous case of transverse propagation the radio wave splits into two linearly-polarized waves: ordinary ( E x ) and extraordinary ( E y ). Without restrictions to the generality we may assume, that the magnetic field H 0 lies in XOZ-plane, thus for the linearly polarized plane wave

296

propagating along Z-axes the E x -component is concurrent with the direction of H T and E y is perpendicular to it. Thus E x occurs as an ordinary component of the radio wave, and E y , as an extraordinary component. Regarding to longitudinal H L -field the primary wave may be decomposed into RHCP ( E+ ) and LHCP ( E − ) circularly polarized components. Superposition of one of the linear components ( E x , for instance) with one of the circular components ( E − , for instance) will result in left-hand-elliptically-polarized wave E1 , propagating along 0Z-axes 1 and with the main axes of the polarization ellipse coincide with H T . Similarly, the superposition of

E y with the other circular component E + will result in right-hand-elliptically-polarized wave, E 2 with the main axes of the polarization ellipse orthogonal to H T , and with same propagation direction (see figure 5.19).

Figure 5.19. The pattern of two elliptically-polarized waves, generated in magneto-active plasma by primary wave. In this general case the earth’s magnetic field is arbitrarily oriented relative to the direction of propagation of the primary radio wave

Electric field vector of each elliptically-polarized wave may be represented as

E1 = E ord + E −  . E 2 = E ext + E + 

1

(5.68)

It's assumed that in general the magnitudes of components are not equal to each-other.

297

For this general case the dielectric constant of magneto-active plasma is defined by the following formula [4 - 7]

2 fc

ε 1, 2 = 1 − 2f2−

fT

2

2

 fT ±  1 − f c 2 / f 4

1 − fc / f 2

2

(

)

2 2

+4 f

2

 2 fL  

1/ 2

.

(5.69)

Here ε 1 and "+" sign relate to propagation of E1 - wave, while ε 2 and "-" sign relate to propagation of E 2 - wave. Two other notations in (5.69) are

and

fT =

e µ0 HT , 2 π me

(5.70)

fL =

e µ0 HL . 2 π me

(5.71)

Expression (5.69) may be reduced considerably in two extreme cases:

Case-1:

fT

4

2  f 1 − c  f2 

   

2

>> 4 f 2 f L

2

(5.72)

which is satisfied under the condition f ≈ f c , which usually takes place in HF- or MF-bands. Then (5.69) may be simplified down to (5.59) and (5.60), so thus the propagation conditions become the same as for transverse propagation, regardless to the value of γ angle. In other words the ray will split into two cross-orthogonal linearly-polarized rays, which propagate independently (ordinary and extraordinary, instead of two elliptically polarized components with cross-orthogonal polarization ellipses). Therefore this case is often called quasi-transverse propagation. Case-2:

fT

4

2  f 1 − c  f2 

   

2

> f c , which is typical for the frequencies UHF and higher. Then (5.69) may simplified down to (5.53) and (5.54), so thus the propagation conditions become the same as for longitudinal propagation, regardless to the value of γ angle; the ray

298

will split into two circularly-polarized components, which propagate independently. This case is called quasi-longitudinal propagation. Note finally, that after transformations of (5.69) the values f L or f T must be kept the same as from (5.70) and (5.71) respectively, instead of f H .

5.5.4. REFLECTION AND REFRACTION OF THE RADIO WAVES IN MAGNETO-ACTIVE IONOSPHERE The presence of earth's magnetic field modifies the patterns of reflection and refraction of the radio waves in ionosphere, because of dependence of parameters of the medium on propagation direction and polarization of the radio wave. This is a typical case of propagation in an anisotropic medium. Consider the oblique incidence of the radio wave on the boundary of magneto-active ionosphere. The incident wave splits into two independently propagating waves, as it was discussed above (see Figure 5.20).

Figure 5.20. Split of the radio wave in magneto-active ionosphere: a). reflection pattern for the extraordinary (1), and ordinary (2) rays, b). refraction pattern for LHCP (1), and RHCP (2) rays.

The reflections from the ionosphere are generally used for HF long-distance propagation (HF long-range radio links), so thus the radio wave frequency is usually close to f c . Based on (5.72) for those links the reflection may be considered for quasi-transverse propagation condition. At the input of the ionospheric layer the incident wave, radiated from point A, splits into two independent linearly polarized cross-orthogonal waves: ordinary and extraordinary. The reflection condition, (5.37) for the ordinary ray may be rewritten here as

299

f ≤ f max = sec ϕ

80.8 ⋅ N e , ordinary ,

(5.74)

where N e , ordinary is the ionospheric plasma concentration needed to satisfy reflection condition for the ordinary ray. Now rewrite the reflection condition (5.37) for the extraordinary ray, which may appear hypothetically in two cases:

f ≤ f 0,1 sec ϕ and

(5.75)

f ≤ f 0, 2 sec ϕ ,

(5.76)

where f 0,1 and f 0, 2 are defined by (5.62) and (5.63) respectively. Due to the relation f 0,1 < f c < f 0, 2 , the expression (5.76) is easier to satisfy, so we may ignore condition (5.75). Then the MUF for the extraordinary ray may be found from (5.76) and (5.63) as 2 f max = sec ϕ  80.8 N e , extraord + ( f H / 2 ) + f H / 2 .  

(5.77)

The same frequency and the same angle of incidence are considered for both, ordinary and extraordinary rays. Hence taking (5.74) and (5.77) equal to each-other, the following expression may be written here:

80.8 N e , ordinary =

80.8 N e, extraord + ( f H / 2 ) + f H / 2 . 2

(5.78)

The expression (5.78) shows clearly the following relation between ionospheric plasma concentration needed to satisfy the reflection conditions for the ordinary and extraordinary rays:

N e , extraord


h 1 . It must be noted, that the absorption rate, as well as dispersion rate for the extraordinary ray is much higher than that for ordinary ray. That's the reason of using for HF communication lines the radio waves of polarization, which produces the ordinary ray rather than extraordinary. The structure of geomagnetic field is such, that for the low and medium latitudes the earth's magnetic field-lines are close to horizontal, and directed from north to south. That's the reason the ordinary ray has a horizontal polarization, therefore the horizontally polarized waves are at the most preference for the HF communication lines design. At the higher latitudes, namely for the earth’s polar zones the HF-communications are rarely used due to the high absorption rate (abnormal absorptions) in those areas. Considering the refraction of the ray path in ionosphere (e.g. on earth-to-space, or space-toearth links) it must be emphasized, that for the frequencies higher than VHF, the radio wave penetrates through the ionosphere, being shifted from his initial propagation direction due to refraction as shown in Figure 5.20b. From the general expression (5.69) for dielectric constant of magneto-active ionospheric plasma it is clear, that the impact of medium will be as low, as greater is the difference between the currier frequency of radio wave and the cutoff frequency (plasma frequency), i.e.

f >> f c .

(5.80)

Under this condition: 1. The dielectric constant become close to one, so thus the ray path will become close to the straight line

301

2. Ionospheric absorptions will decrease because regarding (5.9b) the conductivity of the medium will decrease 3. The polarization disturbances and signal dispersions will also decrease The condition (5.80) is to be met while designing the earth-to-space radio links as much as possible. As mentioned above the frequencies VHF and higher are preferably to be used for earth-to-space or space-to-earth propagation paths, which relates to the quasi-longitudinal propagation (see condition (5.73)). If the linearly polarized radio wave propagates vertically trough the ionosphere then the Faraday's rotation may be expected. The random rotations in polarization plane of LP signal may results in so called polarization losses at the reception point due to polarization random mismatches between Tx and Rx antennas. If the propagation track is sloped, then most likely just one of the circularly polarized (CP) components of the linearly polarized (LP) radio wave will arrive the destination due to different paths for the CP components, as shown in Figure 5.20b. Hence the usage of LP radio wave may result in loss of 50% energy (3 dB). To avoid those losses a CP radio wave is considered as more preferable for the satellite radio links such as communication, navigation, broadcast, etc.

5.6. OVER-THE-HORIZON PROPAGATION OF THE RADIO WAVES BY TROPOSPHERIC SCATTERINGS MECHANISM. SECONDARY TROPOSPHERIC RADIOLINKS Although the asymptotic diffraction theory (ADT) examined in section 4.3 appears to be a powerful tool for radio wave propagation predictions in shadow region behind the horizon line, however the only way of counting the impact of the atmosphere within ADT concepts is the introduction of the earth’s equivalent radius that takes into account only a smooth standard atmospheric refraction. ADT prediction results are in fairly good agreement with measurements in relatively low frequency bands up to HF. For the VHF frequencies and up, ADT predicts the abrupt decrease of the field intensity for over-the-horizon distances, i.e. in diffraction zone, whereas numerous observations performed after WW-II has shown much higher field intensities than those predicted by ADT. Figure 5.22 demonstrates qualitatively the dependence of the field

302

intensity on wide range of distances that are predicted by ADT, as well as observed in real conditions at the frequencies VHF and higher: in fact, higher is the frequency, shaper is predicted field decay in shadow zone.

Figure 5.22. Distance-dependence of field strength for the VHF/Microwave propagation (a qualitative pattern). R0 – horizon distance, 1 – theoretical prediction from ADT 2 – real measurements (averaged curve), 3 – reference path (free-space)

In the mid 30-s several pioneering measurements have been carried by G. Marconi and other investigators, which have shown a considerable difference (up to several hundreds dB-s) compared to diffraction theory predictions. One of the features of the received field is very deep fast fading of the signal level at the reception point. Form mid 40-s to mid 50-s the formal explanation as well as fairly good quantitative description of this phenomenon has been given by H. Booker and W. Gordon [11] with the later updates by V. Tatarskii and S. Rytov [8]. The basic explanation of the nature of fairly intensive field at the receiving point for over-the-horizon distances is as follows: •

In presence of real atmosphere the existence of the field in shadow area is a result of scattering of the primary radio wave from random irregularities of the tropospheric air known as turbulences



The facts such as twinkling stars and the wavering appearance of the objects observed over the earth’s surface heated by sun testify that the atmospheric air is in random, erratic movements that exists permanently, regardless on weather condition, geographic region, and season

303



Statistically those irregularities appear as spherical volumes (globules, eddies) with the refraction index slightly different from the surroundings; those globules exist permanently in troposphere, even in “clear air” condition



Typical dimensions of those globules that are most intensively involved in scattering of the UHF and higher band frequencies vary from several millimeters to hundreds of centimeters, hence maximum effectiveness of those re-radiations must be expected in a range of the wavelengths comparable to the range of those dimensions, which typically belong to microwave frequency bands



Although a scattering from one single globe may never be “sensed”, however the superposition of the huge amount of those scattered fields, called random ensemble or Rayleigh ensemble of waves, causes in fairly “sensible” field strength at the observation point



Number of those scattering globules is unpredictably large, and is defined by the common volume between transmitting and receiving antenna beams; that common volume shown in Figure 5.33 is located at the heights from about 1 to 8 km above the surface



Deep fading of the received signal, which is specific for the troposcatter propagation, is a result of random interference of the scattered (elementary) fields at the reception point, i.e. is a result of constructive and destructive interference of those elementary fields due to random phase shifts between them.

5.6.1. ANALYTICAL APPROACHES IN DESCRIPTION OF THE RANDOM TROPOSPHERIC SCATTERINGS Assume that the dielectric constant ε of the troposphere is a variable that is randomly fluctuations in time and space. The time variations of ε are much slower, than the rate of oscillations of propagating radio wave. Thus it may be assumed, that within several periods of oscillations of the radio wave the random spatial distribution ε remains unchanged, i.e. the "randomness" occurs only as a function of coordinates. In other words only the random spatial fluctuations around the spatial average, ε may be accounted to simplify the analysis, i.e.

304

ε (r ) = ε + ∆ε (r ) ,

(5.81)

where ∆ε is a randomly fluctuated part of dielectric constant, and r = xˆ 0 x + yˆ 0 y + zˆ 0 z is the radial vector-coordinate of the observation point. This time-independent model of so called frozen turbulences is quite convenient for analytical evaluations instead of time-dynamic approach, and as follows from the numerous of experiments, the final results are close enough to those observed. For further simplifications we may assume ε = 1 , which is acceptable for the tropospheric scatterings analysis. Recall (2.5), which constitutes relation between vectors of the electric field strength ( E ) and its induction ( D ):

D = ε 0ε E = ε 0 (1 + ∆ε ) E = ε 0 E + P ,

(5.82)

where P , by its classical definitions, is microscopic vector of polarization of the medium at particular point of space within a unit volume. From (5.82)

P = ε 0 ∆ε E .

(5.82a)

If referred to Figure 5.23, one can realize that time-harmonic polarization P of the medium at point C is forced by electric field E incident from transmission point A. In other words the field, scattered from point C (secondary radiation) may hypothetically be presented as an primary field radiated by the dipole with the moment of polarization

d p = ε 0 ∆ε EC dVsc .

(5.82b)

Here d p is a resultant dipole moment from all polarized particles (atoms, molecules) induced by the incident electric field EC within elementary volume dVsc . The following vector quantities are introduced to support further analysis (see Figure 5.23): •

k 0 the wave-vector of the incident wave, which is directed orthogonal to incident wavefront; its magnitude is equal 2π / λ 0



r1 is the radius-vector of the elementary volume dVsc . It is directed from the transmitting point A to the point where the elementary volume dVsc is located (see Figure 5.23).

305

Figure 5.23. Positioning of vectors for tropospheric scattering analysis. (Origin of the coordinate system is located at A point)

Taking into account the above introduced quantities, the expression (4.1) for the amplitude of the incident field at point C may be rewritten in modified form as

E C =

60 PTx GTx r1

[

]

exp i (ω t − k 0 ⋅ r1 ) .

(5.83)

For the elementary dipole with the dipole moment d p , placed in point C, the field strength in reception point B at the distance r2 from point C in complex scalar form is defined by (A3.31) from Appendix-3, which may here be rewritten as

dE B =

k0

2

4πε 0ε r2

[

]

d p sin θ exp − i k sc ⋅ r2 ,

(5.84)

where k sc is the wave-vector of the scattered wave, shown in Figure 5.23. Note that the multiplier sin θ in (5.84) is the normalized radiation pattern of the elementary dipole as expected from its physical meaning. In our case the dipole axis is collinear to the exciting field

E C , thus θ must be counted form that axes (see Figure 5.23). After combining (5.82b), (5.83) and (5.84) the following result may be obtained for the scalar complex field strength at the observation point

dE B =

k 0 ∆ε (r1 ) 4π ε 2

60 PT GT r1 r2

[

]

sin θ exp i (ω t − k 0 ⋅ r1 − k sc ⋅ r2 ) dVsc .

(5.85)

Here is assumed, that the vector of the scattered field strength will keep the same direction as the original vector E C . As one can see from (5.85) the field dE B is proportional to ξ ε = ∆ε / ε .

306

Hence both, the amplitude and phase of dE B will randomly fluctuate as the position of the scattering point C varies within the scattering volume Vsc . In order to define the power flow density of the scattered radio wave at the receiving point (the average magnitude of Pointing vector), consider scattered field strengths dE B′ and dE B′′ arriving





at the observation point B from elementary scattering volumes dVsc and dVsc positioned at point C’ and C” respectively as shown in Figure 5.24.

Figure 5.24. Superposition of two scattered rays at the reception point from different volumetric elements of the scattering volume

Taking into account (2.25) the total average power flow intensity at the receiving point B may by expressed in the following form of the volumetric integral:

Π B , ave =

 1 1 ′ ~ ″  Re  ∫ (dE B dE B ) , 2 Vsc W0 

(5.86)

~ ″

where W0 = 120π Ohm is the wave impedance of the free space, dE B is a complex conjugate



~ ″

of dE B . Using (5.85) we may substitute both, dE B′ and dE B into (5.86):

Π B , ave

4 k0 60 PT GT 1  ′ sin θ ″  ∆ε ′ (r1 ) ⋅....... θ sin = Re  ∫∫  ε  2  Vsc (4π )2 W0 r1 r2 r1 + ρ r2 − ρ  

 ∆ε ′′(r1 + ρ ) ′ ′ ″ ″ ′ ″ ..... exp − i k 0 ⋅ r1 − i k sc ⋅ r2 + i k 0 ⋅ (r1 + ρ ) + i k sc ⋅ (r2 − ρ ) dVsc dVsc  ,      ε   

(5.87)

where the vector-coordinates, shown in Figure 5.24 are related to each-other as follows:

″ ′ ″ ″ ′ ″ r1 = r1 , r1 = r1 + ρ = r1 + ρ , r2 = r2 , r2 = r2 − ρ = r2 − ρ .

307

(5.88)

The heavy expression (5.87) may considerably be simplified for the real conditions, when both antenna beams, Tx and Rx, are extremely narrow, so thus the distances r1 and r2 are much greater, than the linear dimensions of the common scattering volume Vsc . This volume has an elevation above the earth's surface much less, than the horizontal distances. Hence

′ ″ k0 ≈ k0 ,

′ ″ k sc ≈ k sc ,

θ′ ≈θ″,

r1 + ρ ≈ r1 ,

r2 − ρ ≈ r2 .

(5.89)

For conditions (5.89), as well as W0 = 120π Ohms, the expression (5.87) may be simplified down to (see also the footnote on the bottom of this page)

k 0 PT GT sin 2 θ 4

Π B , ave ≈

(4π )

3

2

r1 r2

2

∫ exp [ − i (k

sc

Vsc

  ∆ε ′ (r1 ) ∆ε ′′ (r1 + ρ ) − k 0 )⋅ ρ  ∫ dVsc  dVsc ε  Vsc ε

]

(5.90)

Evidently the integral in braces in (5.90) is a the spatial autocorrelation function of the random scalar field, ξ ε (r ) = ∆ε (r ) / ε , which we denote here as

ψ ε (r1 , ρ ) =

1 ξ ε ′ (r1 + ρ ) ξ ε ″ (r1 ) dVsc . ∫ Vsc Vsc

(5.91)

In general, ψ ε depends on r1 , as well as on the distance ρ of two point within the common volume Vsc . Fortunately the fluctuations of tropospheric refraction index are statistically homogeneous, i.e. they are independent on position r1 , and are only dependent on distance ρ between two points. Thus (5.91) may be written as

ψ ε = ψ ε (ρ ) ,

(5.92)

and (5.90) is represented as

k 0 PT GT Vsc sin 2 θ 4

Π B , ave ≈ where

(4π )3 r12 r2 2

∫ ψ ε (ρ ) ⋅ exp [− i q ⋅ ρ ]dV

sc

,

(5.93)

Vsc

q = k sc − k 0

(5.94)

is called a scattering vector or vector of spatial wave number. ψ ε (ρ ) is considered as one of deterministic characteristics of the random field ∆ε (r ) . Then the integral in (5.93) may be expressed in terms of volumetric Fourier-spectrum of ψ ε (ρ ) field 1:

1

As seen later the volumetric function ψ s ( ρ ) is symmetric, hence (5.95) becomes a real function.

Therefore when transforming (5.87) into (5.90) the sign Re was omitted.

308

S ε (q ) =

ψ ε (ρ ) exp (− i q ⋅ ρ ) dVsc ,

1

(2π )3 Vsc∫

(5.95)

with the inverse transform:

ψ ε (ρ ) = ∫ S ε (q ) exp (i q ⋅ ρ ) dVsc .

(5.96)

Vsc

The physical meaning of the (5.96) transform is that the autocorrelation function ψ ε (ρ ) is represented as superposition of the continuum of deterministic plane waves S ε (q ) exp ( i q ⋅ ρ ) each of which has an amplitude of S ε ( q ) as a function of the spatial frequency q (scattering vector). Note that in the theory of random processes (theory of time-domain random functions) the transform (5.95) is known as Wiener-Khinchin transform, whereas in theory of random fields (theory of space-domain random functions) it's known as Shannon-Whittaker transform. If (5.95) is substituted into (5.93), then the result is

k 0 PT GT sin 2 θ S ε ( q ) 4

Π B , ave ≈

2

8 r1 r2

2

Vsc .

(5.97)

Finally we use the replacement θ = π / 2 + θ S that is illustrated in Figure 5.23. Then (5.97) will be rewritten as

k 0 PT GT cos 2 θ sc S ε ( q ) 4

Π B , ave ≈

2

8 r1 r2

2

Vsc .

(5.97a)

Here θ sc is a scattering angle (see Figure 5.23).

5.6.2. PHYSICAL INTERPRETATION OF TROPOSPHERIC SCATTERINGS As shown above, spatial distribution of the random field ξ ε (r ) may be expanded into the superposition of continuum of elementary plane waves. To be more specific, note that the expansion is applied not to the random field ξ ε (r ) itself, but to one of its deterministic characteristic such as autocorrelation function ψ ε (ρ ) . It contains the main statistical parameters, such as mean square deviation, ξ ε2 = (∆ ε / ε ) and the average dimension of 2

random irregularities Lε, i.e. the average sizes of the globules of turbulences (see Figure 5.25).

309

Figure 5.25. a). Flat projections of the autocorrelation function of the random scalar field, b). Spectrum of the spatial harmonics of autocorrelation function of the random field ∆ε .

One of the properties of Shannon-Whittaker transform is similar to property of the Fourier transform 1, namely the narrower is the initial function ψ ε (ρ ) , the wider the spectrum becomes, and vise-versa. There's a following approximate relation between the width of the main lobe of the correlation function (the average dimension of the irregularities), Lε and the width of spatial spectrum of irregularities: q max ~ 1 / Lε . To understand the mechanism of how the random medium impacts the scattering process, let’s consider a single S ε (q ) component of 3D spectrum of irregularities. If, for instance, q is directed vertically down, then 3D shape of that component of the spectrum of autocorrelation function may be represented as a stratified medium with the harmonically changing intensity of

ε in vertical direction (Figure 5.26) with the amplitude Sε and the spatial period of Λ = 2π / q . Loci of the maximums of spatial densities are the parallel planes, separated by Λ , as shown with dashed lines on Figure 5.26.

1

As it was mentioned above the Shannon-Whittaker transform is the same as a Fourier transform, which

is applied to spatial correlation function ψ ε (ρ ) , i.e. allows finding the Fourier spectrum of the spatial harmonics of the given function

ψ ε (ρ ) . 310

Figure 5.26. Scattering of the radio wave from single spatial harmonic of the turbulent fluctuation of dielectric constant in troposphere

As seen from Figure 5.26 all rays reflected towards the receiver will superimpose at the receiving point with the same phase (constructive interference) if the phase shift between two adjacent rays is equal to 2π (or is multiple of 2π ). If we take into account that the difference in distances between two adjacent rays is equal 2 BC , then from the Figure 5.26 it’s easy to derive geometrically

k 0 (2 BC ) =

θ  2π   2Λ sin sc  = 2π . λ0  2 

(5.98)

Then the spatial period of the considering harmonic may be found from (5.98) as

Λ=

λ . 2 sin (θ sc / 2)

(5.99)

The same expression may be found from the same Figure 5.26 based upon disposition of three vectors, k 0 , k sc and q . Indeed, taking into account the equal magnitudes

k 0 = k sc = and

q =



(5.100)

λ

2π Λ

(5.101)

311

the following relation may be developed form triangle ABC:

k sc − k 0 = 2 k 0 sin

θ 2

= q .

(5.102)

After substitution (5.100) and (5.101) into (5.102) the expression (5.99) may be verified, which is known in optics as a Bragg's diffraction condition with the vector form of (5.94). It allows finding relations between λ , Λ and θ sc , i.e. the relations between directions of incident and diffracted waves, if the diffraction takes place on one of the spatial harmonics of 3D correlation function of random field of the dielectric constant. Now if the expressions (5.97) or (5.97a) are recalled, then one may notice that the average power flow density at the receiving point is proportional to the amplitude of spatial harmonic. Hence the intensity of scattered field, Π B , ave will vanish for scattering angles greater than θ max when S ε becomes very small (see Figure 5.25). The relation between θ max and q max is found from (5.102) as

  λ0 q max  .   4π

θ max = 2 sin −1 

(5.103)

As mentioned above, the upper limit of spatial spectrum of ξ ε fluctuations, namely q max is in inverse proportion to the average dimension of irregularities ( q max ∝ 1 / Lε ), therefore greater the average dimensions of irregularities, narrower is the spatial spectrum, thus narrower is the angular aperture of the scattered rays as demonstrated in Figure 5.27.

312

Figure 5.27. Patterns of scatterings of the radio waves on random irregularities of the tropospheric turbulences and their vector diagrams ( 2θ max -- angular aperture of scatterings). a). scatterings on small-scale irregularities, b). scatterings on large-scale irregularities

5.6.3. EFFECTIVE SCATTERING CROSS-SECTION OF THE TURBULENT TROPOSPHERE For the engineering applications it's useful to introduce the effective scattering cross-section (ESCS) of the troposphere. Consider a unit volume (e.g. 1 m3) that surrounds point C located within the scattering volume Vsc . Now we replace hypothetically the unit volume of scatterers that surround point C by a flat PEC surface σ sc , called ESCS, which produces at the receiving point B the same amount of the received power as it is produced by the primary irregularities located within that unit volume. Similarly to (3.35) we may obtain the incident power flow density at the scattering point C produced by the transmitter. That is

ΠC =

PT GT 4π r1

2

,

(5.104)

313

where r1 is the direct distance from transmitting point A to the scattering point C. Then, regarding the definition for ESCS, the total power flow of the wave scattered by the unit volume that surrounds point C may be found as

PC = Π C σ sc =

PT GT 4π r1

2

σ sc .

(5.105)

The power flow density at the receiving point B is

′ ΠB =

PC 4π r2

2

=

PT GT σ sc

(4π ) 2 r12 r2 2

,

(5.106)

where P C is substituted from (5.105), and r2 is the direct distance from scattering point C to receiving point B. Finally the total power flow density at B point, induced by the entire scattering volume may be defined if we multiply (5.106) by Vsc , i.e.

1

PG σ V ′ Π B = Π B Vsc = T T 2 sc2 2sc . (4π ) r1 r2

(5.107)

If (5.107) is compared with (5.97a), then SSCS may be found in the following form:

σ sc = 2π 2 k 0 4 cos 2 θ sc S ε (q ) .

(5.108)

Expression (5.108) demonstrates σ sc (θ sc ) dependence not just due to the presence of the

cos 2 θ sc factor, but mostly because of the presence of the spectral component S ε (q ) , which is highly dependent on q vector, namely on both, on its magnitude and orientation θ sc . It's easy to realize, that the maximum value of ESCS will appear at θ sc = 0 , i.e. σ sc , max = σ sc (0 ) . The graph of the normalized function

Σ sc (θ ) =

σ sc (θ sc ) σ sc , max

(5.109)

is called a scattering diagram, and is considered below for particular troposcatter models.

1

For higher accuracy the product

σ scVsc

must be replaced by the volume integral

∫σ

sc

dVsc . However

VSC

σ sc is assumed to be uniformly distributed within scattering volume Vsc , which results in simple product σ scVsc instead of the volume integral. for the practical engineering applications

314

5.6.4. STATISTICAL MODELS OF TROPOSPHERIC TURBULENCES 5.6.4.1. GAUSSIAN MODEL Most commonly the autocorrelation function ψ ε (ρ ) in problems that relate to statistical radiophysics is presented in form of Gaussian function. It is assumed, that this function doesn't depend on direction of the spatial coordinate ρ but only on its magnitude. This kind of statistically homogeneous random medium is called statistically isotropic random field.

 ρ2  , ψ ε (ρ ) = ξ ε exp  − 2  2 L ε   2

(5.110)

where Lε is the average radius of the irregularities of the tropospheric air that are spherically shaped by statistical means, and

 ∆ε     ε 

ξ ε2 =

2

(5.111)

is standard deviation of fluctuations of the relative dielectric permittivity of the clear tropospheric air that usually belongs to the range from 1.5 ⋅10

−7

to 3.3 ⋅10

−6

. The lower limit corresponds to

the winter season, and the upper limit, to the summer season. It is independent on the elevations above the earth’s surface (at least for the elevations of up to 5 km). Spatial spectrum (5.95) of statistically isotropic field presented particularly by (5.110) may be introduced in the following form after transformations provided in Appendix-7.

S ε (q ) =

1 2π 2 q



∫ ψ ε (ρ ) sin (qρ ) ρ dρ .

(5.112)

0

If (5.110) is substituted into (5.112), then the expression for the spatial spectrum may be derived as [10, p.495]

 q 2 Lε 2  ξ ε2 Lε 3  . S ε (q ) = exp  − (2π ) 3 / 2  2 

(5.113)

Now substitute (5.113) into (5.108) and (5.109) and take into account (5.102) to determine final expressions for SSCS and for the scattering diagram respectively:

σ sc (θ sc ) =

θ   4 3 2 2 k 0 ξ ε2 Lε cos 2 θ sc exp − 2k 0 Lε sin 2 sc  , 2 2  

π

315

(5.114)

θ   2 2 Σ sc (θ sc ) = cos 2 θ sc exp  − 2k 0 Lε sin 2 sc  . 2  

(5.115)

The 2D-scattering diagrams that are expressed in (5.115) analytic form in polar coordinates are illustrated in Figure 5.28 for different Lε / λ 0 ratios obtained by using MATLAB routine.

Figure 5.28. Scattering diagrams of the turbulent troposphere plotted in polar coordinates, for the Gaussian model of turbulences (verifies the concept demonstrated in Figure 5.27)

5.6.4.2. KOLMOGOROV-OBUKHOV MODEL Although the previously considered Gaussian model (5.110) has various practical applications, however, in case of tropospheric turbulences it is applied notionally, and is not based on real aerodynamic processes related to generation of the tropospheric clear-air inhomogenieties of the dielectric constant (eddies). More accurate quantitative results may be obtained, if scattering

316

phenomenon analysis is linked to the model, which is developed based on the real aerohydrodynamic processes of generation of those inhomogenieties. Regarding to the principles of hydrodynamics, there're two types of movement of any gaseous or liquid masses: •

Laminar – a particular type of streamline flow where the gas (or fluid) in thin, parallel layers tends to maintain uniform velocity with the constant magnitudes and directions (Figure 5.29a);



Turbulent – when, at certain condition, a laminar flow of gas (liquid) turns into stochastically distributed eddies, i.e. when the magnitudes and directions of the composite atoms and molecules turn to be randomly distributed in time and space (see Figure 5.29b).

Figure 5.29. Movements of tropospheric air masses; a) Laminar, b). Turbulent

The type of movement of gas or liquid is usually specified by so called Reynolds number:

Re =

ϑ v l flow , ν~

(5.116)

Here ϑ is a density, v is the magnitude of velocity vector v , l flow is the cross-sectional dimension of the flow, and ν~ is the viscosity of the medium. Each gas or liquid has the specific critical value of Reynolds number, Re cr . So if for particular medium Re > Re cr then the movement of gas or liquid becomes turbulent, otherwise it is laminar. The values of variables in (5.116) for the real troposphere are such that the movement of the air masses is almost always turbulent, which results in permanent presence of the fluctuations of its dielectric constant.

317

Figure 5.30 demonstrates the spectrum of spatial harmonics of the tropospheric dielectric constant fluctuations as a function of the “mechanical” spatial wave numbers. As seen from the figure, the spectrum is divided into three ranges: 1. Range of formation of the eddies (globules) of largest sizes, L0 called the outer scale of irregularities, which may vary in the range from 100 to 1000 m, and even larger 2. Inertial range. Kinetic energy of the movements remains unchanged. Eddies become smaller in size gradually, hence with gradually increasing velocity of spinning of the air masses inside each eddy. The range of sizes is l 0 < l < L0 ( 2π / L0 < q < 2π / l 0 ), where

l 0 is its lower limit called the inner scale of irregularities, which varies in range of dimensions approximately from 1 centimeters to 1 millimeter 3. Dissipation range, where the lower limit l 0 of the eddy’s size is reached, thus the maximum spinning velocity results in destruction of the eddies followed by the dissipation of their kinetic energy, i.e. irreversible transformation of that energy into heat while disappearance of the globules

Figure 5.30. Spatial spectrum of random fluctuations of dielectric constant of the troposphere. 1 – range of eddies formation 2 – energy conservation range 3 – energy dissipation range

Regarding to theoretical investigations by Kolmogorov and Obukhov the spatial spectrum of correlation function for random fluctuations of dielectric constant of the troposphere in the area of inertial transforms is expressed in the following analytical form [8]

318

Sε (q ) = 0.033 Cε q 2



11 3

,

(5.117)

2

In (5.117) Cε here is in m-2/3 is called a structural constant that is dependent on season of the 2

year and on elevation of the scattering point. Sometimes the values of Cε are given in m-2/3 2

units. To transform Cε from one unit to the other recall that (1cm)-2/3 = (0.01m)-2/3 = 2

= 21.54(1m)-2/3, which means the value of Cε in cm-2/3 must be multiplied by 21.54 to get it in m-2/3 units. Seasonal variations of the structural constant near the earth's surface are in the range from 10 −14 cm-2/3 (summer) to 10 −16 cm-2/3 (winter). 2

The average vertical profile of Cε may be expressed as

h Cε ( h ) = Cε ( h0 )    h0  2

where Cε

2

2

−α

,

(5.118)

( h ) is defined for the elevation of h 0

0

= 30 m. α is a constant, which has the

following values, defined empirically: α = 2 / 3 for the winter season, and α = 4 / 3 for the summer season (see Figure 5.31).

2

Figure 5.31. Vertical profiles of the structural constant Cε for the turbulent troposphere Note, that expressions (5.108) and (5.109) become meaningless if θ sc → 0 , because (5.107) is valid only in inertial range shown in Figure 5.30, i.e. is valid for the spatial harmonics q starting with their minimum value of

319

q = 2 k 0 sin

θ sc 2

≥ qmin =

2π L0

(5.119)

up to the maximum q max . Indeed, for the scattering angle θ sc = 0 the Kolmogorov-Obukhov model violates the physical picture: it results in infinite scattering intensity, while limited value is expected. This is the shortcoming of this model. On the other hand if that is compared with the Gaussian model then one may realize that the Kolmogorov-Obukhov's model considers an existence of multi-scale irregularities, which is typical for the troposphere, while the Gaussian model considers only a mono-scale irregularities, with the dimensions, dispersed (spread) around its mean value, Lε given in (5.110). Despite this fact, the use of Gaussian model is still applicable if the angles θ sc are close to zero. In MMW, and optical bands, when the values of scattering vector become large enough, i.e., when the interaction between incident waves with the turbulent medium takes place in dissipation range of the spectrum, the following relation

q = 2 k 0 sin

θ sc 2

=



λ

sin

θ sc 2



2π = qmax , l0

(5.120)

dictates the usage of an additional exponential multiplier to be inserted into (5.117), which allows to take into account an abrupt decrease of S ε (q ) seen from Figure 5.28:

 q 2 S ε (q ) = 0.033 Cε q −11 / 3 exp  −  q max

  . 

(5.121)

For the small values of the magnitude of scattering vector, i.e. in range of formation of the tropospheric turbulences, where

q = 2 k 0 sin

θ sc 2



2π = qmin , L0

(5.122)

the data obtained from the Kolmogorov-Obukhov model considerably distinguishes from real observed values of S ε (q ) . Carman [8] introduced later update of the model by transforming (5.121) into his final form, applicable for the practical applications:

 q 2 2 S ε (q ) = 0.033 Cε (q 2 + q min ) −11 / 6 exp  −  q max

320

  . 

(5.123)

It was shown theoretically by Kolmogorov and Obukhov that the outer scale of atmospheric turbulences, L0 is linked to the mean-square value of fluctuations dielectric constant of the troposphere ξ ε2 and to the structural constant Cε by the following expression [8] 2

Cε =

2 ξ ε2

2

L0

2/3

.

(5.124)

If the decaying tendency of the vertical profile Cε (h ) is taken into account from (5.118) or from 2

Figure 5.31, then it may be realized, that the higher is the elevation, the greater the outer scale of the turbulences is, i.e. greater the dimensions of the new-born eddies are. Now we substitute (5.123) into (5.108) and (5.109) to define ESCS and scattering diagram in analytical form for this model:

σ sc (θ ) = 3

4

2

π 0.033 k 0 Cε L 0 4 (2π ) 2

11 / 3

  L θ 2 cos θ sc 1 +  2 0 sin sc  2   λ0 

  L θ 0 Σ sc (θ sc ) ≈ cos 2 θ sc 1 +  2 sin sc 2   λ 0 

   

2

   

−11 / 6

   

2

   

 l0 θ exp − 2 sin sc  2 λ0 

−11 / 6

 l0 θ exp − 2 sin sc  λ0 2 

  , (5.125)  

 .  

(5.126)

Examples of scattering diagrams calculated based on (5.126) for the Kolmogorov-Obukhov model of the tropospheric turbulences is shown in Figure 5.32.

Figure 5.32. Scattering diagrams of the turbulent troposphere plotted in Cartesian coordinates, based on (5.128) model for the fixed values of the inner ( l 0 ) and outer ( L 0 ) scales, and for different wavelengths (verifies the concept demonstrated in Figure 5.27)

321

5.6.5. PROPAGATION FACTOR ON SECONDARY TROPOSPHERIC RADIO LINKS In order to obtain the total power received at the point B the power flow density Π B form (5.107) must be multiplied by antenna's effective aperture form (3.30), i.e.

PTx GTx G Rx λ 0 σ sc (θ sc )Vsc 2

PB =

.

(4π r1 r2 )2 4π

(5.127)

If this expression is compared with (3.38), then the propagation factor may be found as

F=

σ sc (θ sc )Vsc , 4π

r r1 r2

(5.128)

where r1 and r2 are the distances from scattering point C and transmitting, A and receiving, B points respectively, and r is the shortest distance between transmitting and receiving points; Expression (5.128) may be rewritten for the total horizontal distance R between transmission and reception points if assumed r1 ≈ r 2 ≈ R / 2 :

F=

2 R π

σ scVsc .

(5.128a)

The scattering volume Vsc is defined as intersection of main beams of the transmitting and receiving antennas, as shown in Figure 5.33. For simplicity we'll assume both diagrams symmetric relative to the main radiation direction, i.e. half power beam widths (HPBW) are equal in both, E- and H-planes for each antenna:

γ 1E = γ 1H = γ 1 ,

(5.129)

γ 2E = γ 2H = γ 2 .

(5.130)

Then by definition (3.13) given in chapter 3 antenna gains may be expressed in terms of beam widths as

GTx ≈



γ1

2

and

G Rx ≈



γ 22

.

(5.131)

For maximum efficiency of the secondary tropospheric radio links the angles of elevation of both antennas are chosen close to zero, i.e. the main beams of both antennas are directed tangential to the earth’s surface, i.e. directed horizontally. That allows having a common scattering volume

Vsc to be as close to earth’s ground as possible: the reason is because the lower the elevation 322

of the scattering volume is, then smaller is the scattering angle, thus higher the intensity of scatterings are. From Figure 5.33b

VS = Area MNLP ⋅ MM ′ ,

(5.132)

Figure 5.33. The definition of the scattering volume of troposphere. a). The intercept of the radiation diagrams of Tx and Rx antennas, b). The detailed sketch of the scattering volume. (note that the vertical scale is highly exaggerated for clarity)

where

Area MNLP =

b d , and sin θ

d = MM ′ .

(5.133)

If C is the center point of the scattering volume, then from Figure 5.33a the following approximate expressions may be written down:

d ≈ AC ⋅ γ 1 = r1 γ 1 b ≈ BC ⋅ γ 2 = r2 γ 2

  

(5.134)

It’s easy to realize that if the elevations of antennas main beams are close to horizontal then the scattering angle θ sc is the same as the geo-central angle. For the real conditions the following approximation may be applied:

sin θ sc ≈ θ sc ≈

R , ae

(5.135)

where ae = 8500 km is the earth's equivalent radius, and R is the horizontal distance between corresponding points. Another assumption is r1 ≈ r 2 ≈ R / 2 . If γ 1 and γ 2 are found from (5.131), then, by combining (5.132) - (5.135), we find

323

Vsc ≈

R 2 ae (4π )3 / 2 . 8 GT G R

(5.136)

Yet was assumed GT > G R , because the width d of the volume VS was defined based on the beam width of the antenna with higher gain (transmitting antenna in this case). It's easy to show, that for GT < G R denominator in (5.126) will become G R GT . In real conditions the mean-geometric of those two values may be used instead:

GTx G Rx G Rx GTx = GTx

3/ 4

G Rx

3/ 4

.

(5.137)

Thus (5.136) may be rewritten as

Vsc ≈

R 2 ae (4π )3 / 2 . 8 GTx 3 / 4 G Rx 3 / 4

(5.138)

This expression may easily be transformed to

( R γ ) 3 R 3 (4π ) 3 / 2 , ≈ Vsc ≈ 8θ sc 8θ sc G 3 / 2

(5.138a)

where (5.135) is taken into account as well as the similarity between corresponding antennas, i.e. G = GTx ≈ G Rx ≈ 4π / γ 2 . If (5.138) is substituted into (5.128a), then the following formula may be found for the power propagation factor

F 2 ≈ 4 π ae where

∆G = GTx

3/ 4

σ sc (θ sc )

G Rx

∆G 3/ 4

,

(5.139)

.

(5.140)

Now before we refer to σ sc (θ sc ) to make expression (5.139) applicable for practical estimates we have to take into account the following statements: •

The path length of the troposcatter links is practically limited from 200 km (for relatively wideband systems with the bandwidth up to several MHz) to 1000 km (for narrowband systems with the bandwidth no more tens of kHz)



The carrying frequencies are limited from 200 MHz (due to antenna size limitations) to 5 GHz (due to increasing of the atmospheric absorptions)

It’s easy to estimate that for the given distances the geo-central angle θ sc = R / ae along the great circle is limited to 0.0235 < θ sc < 0.1177 radians. Then the magnitude of scattering vector from expression (5.102), q = q = 2 k 0 sin θ sc / 2 will be ranged within 0.0986 m-1 < q < 12.3 m-1

324

limits. If those limits are compared with the limits ( q min ≤ 0.063 m-1, q max ≥ 628 m-1) of the energy conservation range (range-2 on Figure 5.30), then easy to realize that only those irregularities are responsible for the scattering, which are located in range-2 only. Therefore for further evaluation of (5.139) for the power propagation factor the original version (5.117) of the spectrum of turbulences may be employed, without counting on Carman’s corrections made to adjust both ends of the spectrum. After combining (5.108), (5.117), and (5.139) we obtain

F2 ≈ ... =

4 π ae 2π 2 k 04 cos 2 θ sc 0.033 Cε2 q −11 / 3 = ... ∆G

θ  39.26 ⋅10 6 4  k 0 cos 2 θ sc Cε2  2k 0 sin sc  ∆G 2  

−11 / 3

.

(5.141)

The following notations are used in (5.141): •

∆G is defined by (5.142) as unitless



k 0 is a free space wave number in 1/m



ae = 8.5 ⋅10 6 m is equivalent earth’s radius



θ sc is a scattering angle defined by (5.135)



Cε2 is a structural constant for atmospheric turbulences in m-2/3. ------------------------------------------------------------------------------------------------Example 5.2

Estimate the propagation factor on the tropospheric radio link for the frequency 1 GHz (free space wavelength, λ 0 = 30 cm, wave number, k 0 = 2π / λ = 20.944 , 1/m), and horizontal distance of R = 300 km between transmitting and receiving antennas with the gains

GTx = G Rx = 50 dB ( 10 5 unitless). The value of the structural constant in vicinity of the earth surface, at the height h 0 = 30 m, may approximately be taken: Cε = 10 −14 cm-2/3 for summer 2

season and Cε = 10 −16 cm-2/3 for winter season (see Figure 5.31). Find the power received, if 2

the radiated power is PTx = 2 kW. Solution •

Scattering angle is calculated taking it equal to geo-central angle,

θ = r / aeq = 300 / 8500 = 0.0353 rad , •

The height of scattering volume is found from triangle AOC on Figure E5.1:

325

Figure E5.1. Definition of the scattering volume elevation above the ground

    1 1 hC = ae  − 1 = 8500  − 1 ≈ 5.3 km ,  cos(θ / 2)   cos (0.0353 / 2 )  •

We define the structural constant at the elevation hC from (5.118) (or roughly estimated from Figure 5.31): 2

 Cε (5.3 km) = 10-16(5300/30)-2/3 = 3.18x10-18 cm-2/3 = 6.85x10-17 m-2/3 (for winter season) 2

 Cε (5.3 km) = 10-14(5300/30)-4/3 = 10-17 cm-2/3 = 2.15x10-16 m-2/3 (for summer season)  Note: here we ignore seasonal variations of the smooth tropospheric refraction that may result in variations of the common volume elevation above the ground; standard tropospheric refraction is assumed here •

Power propagation factor is calculated from (5.143):  for winter seasons

F2 =

39.26 ⋅10 6 0.0353   20.944 4 cos 2 0.0353 ⋅ 6.85 ⋅10 −17  2 ⋅ 20.944 ⋅ sin  5 3/ 2 2  (10 ) 

... = 4.95 ⋅10 −11 (-103 dB)

−11 / 3

= ...

(Answer)

 for summer seasons

39.26 ⋅10 6 0.0353   F = 20.944 4 cos 2 0.0353 ⋅ 2.15 ⋅10 −16  2 ⋅ 20.944 ⋅ sin  5 3/ 2 2  (10 ) 

−11 / 3

2

... = 1.55 ⋅10 −10 (-98.1 dB) •

(Answer)

Power received on reference propagation path (free space) is found from (3.36):

326

= ...

PRx , 0 = •

PTx GTx G Rx λ 20

(4π R )2

=

2000 ⋅10 5 ⋅10 5 ⋅ 0.32 = 0.1267 W (126.7 mW) (4π ⋅ 300000) 2

Power received on real propagation path is found as PR = PR , 0 ⋅ F 2 : 

for winter seasons, PRx = 6.27 ⋅10 −12 W (-112 dBW)

(Answer)



for summer seasons, PRx = 1.96 ⋅10 −11 W (-107 dBW)

(Answer)

--------------------------------------------------------------------------------------------

The above considerations of the propagation factor and power received on troposcatter radio links allow make only the rough assessments of their median values. Unfortunately more accurate calculations may not be performed due to lack of complete and precise set of data about the parameters of the atmospheric turbulences: we do mean the data of structural constant global distributions, and its spatial and time variations. For more accurate results the reader may be referred to ITU-R document [12], which provides an empirical calculation method based on numerous of observation on existing troposcatter radio links, but it is not based on analysis of physical mechanisms of propagation.

5.6.6. THE SPECIFICS OF THE SECONDARY TROPOSPHERIC RADIO LINKS PERFORMANCE 5.6.6.1. ANTENNAS GAIN EFFECT ON LINK PERFORMANCE As seen form (5.141) the propagation factor on secondary tropospheric radio links is highly dependent on antenna gains GTx and G Rx . The propagation factor is getting lower by ∆G when either GTx or G Rx (or both) increase unlike other types of radio links, where propagation factor is independent on antennas. In other words the higher antennas performance, the greater the propagation factor is for all other conditions remaining. The cause of this phenomenon is quite clear: the increase of the antenna gains will result in decrease of the beam widths γ 1 and γ 2 , thus, regarding (5.138) and (5.138a) will result in decrease of the scattering volume VS as demonstrated in Figure 5.33. If (5.140) is expressed in dB-s

327

∆GdB =

3 (GTx, dB + GRx, dB ) . 4

(5.142)

then, as follows from the plot of (5.142) shown in Figure 5.34, the loss of the antennas' gains (in dB-s) as a function of total antenna gain is linearly increasing with the slope of 3/4.

Figure 5.34. Antennas' total gain losses on secondary tropospheric radio-link In reality ∆G becomes considerable when the total gain exceeds 65 -- 70 dB, as may be seen from Figure 5.34. Another reasonable explanation of this phenomenon is as follows: due to turbulences, scattered field is a subject of intensive amplitude and phase fluctuations at the reception point along the antenna’s aperture; correlation distance of those fluctuations across the propagation path is comparable to the antennas dimensions; if the antenna’s dimension is less than that correlation distance, then within the antenna’s aperture received field is correlated, i.e. its structure becomes close to the structure of plane wave; if the antenna’s dimension is greater than correlation distance, then uncorrelated fluctuation along the receiving aperture (especially phase fluctuations) result in destructive integration of the field values, hence resulting in destruction of the antenna gain; it is clear that when transmitting and receiving antennas are not identical, then antenna with larger dimension suffers more than that of smaller dimension. Total antennas gain of GTx + G Rx = 90 --100 dB seems to be the limit, which is meaningless to exceed; further investments are not paid off. Hence the value for each antenna gain is usually taken not larger than 45--50 dB.

328

5.6.6.2. SIGNAL LEVEL FLUCTUATIONS AT THE RECEIVING POINT (FADING) One of the specifics of troposcatter radio link is in presence of intensive random fluctuations of the signal level at the receiving point, which may be classified into three independent categories: •

Fast fading,



Slow fading,



Seasonal variations.

As pointed above, the cause of fast fading is a random interference of the multitude of secondary waves, interfering at the receiving point 1. On this type of radio-links the multitude of secondary, interfering partial waves may be considered as almost ideal Rayleigh ensemble, thus the fast fading is considerably deep, and have the Rayleigh-type probability distribution of the field strength (see chapter 6 for details). Slow fading on troposcatter radio-links is described by lognormal probability distribution with fair accuracy (see subsection 6.1.3.b for more details). Standard deviation of the median signal level, σ y depends on propagation path length and season of the year as shown in Figure 5.35.

Figure 5.35. Dependence of the standard deviation of slow fading of the received field level on distance for troposcatter radio link

To understand the reason for the decease in intensity in slow fades distance-dependence recall that greater the distance is, higher is the elevation of the scattering volume VS , hence more

1

Detailed analysis of the random interference (multipath interference) is given in Chapter 6, along with

the references for terms and definitions.

329

stable the atmospheric conditions are. Those conditions are less dependent on seasonal variations at high elevations, rather than at low elevations. The seasonal dependence of those variations may become clear if taken into account the fact, that during winter season the earth's surface is heated less, therefore less is the vertical gradient of the atmospheric refractivity, dN / dh ; regarding (5.25) the curvature of ray trace due to regular smooth tropospheric refraction is less, hence higher is position of the scattering volume VS . Higher is the elevation of scattering volume, less is the impact of the seasonal variations. In spite the intensity of slow fades is less during the winter season, however the short-term median signal at the reception point is always less than in summer. Therefore it is common practice to design such radio-links for the winter season that guaranties the performance all year long. On the other hand in terms of stability of the signal level at the receiving point, the worst conditions arise during the summer season, because of increasing of deepness of fading (both, fast and slow). Therefore the estimates for the stability of those radio links is carried out preferably for the summer seasons, based on approaches given in chapter 6, and the values of σ y obtained from Figure 5.35. One of the specific features of troposcatter communication systems is the use of diversity receiving principles that allows struggling against very deep fading. Note that the above approaches allow estimating the long-term median received power only. If the sensitivity of the receiver is taken equal to that median power level, then apparently the designer may not expect the overall performance quality (steadiness) of the system greater than 50% by definition. To reach the required performance quality, the increase of the radiated power, i.e. the imbedding of power margin is not the solution for such systems because of the limitations in power budgets. The only means of achieving the required performance steadiness is a combination of the power margin with diversity reception techniques that are widely used in troposcatter communication lines. Those techniques are considered in subsection 6.3.3 in details.

5.6.6.3. LIMITATIONS TO SIGNAL TRANSMISSION BANDWIDTH Consider an amplitude-modulated signal with the carrier frequency f 0 and simple doublesideband sinusoidal modulation. The waveform has a periodical shape with the amplitude envelope of the sinusoidal form. If Fmod is modulating frequency, then the frequency bandwidth is ∆ f = 2Fmod , hence the spectrum is located in frequency range from f min = f 0 − Fmod to

f max = f 0 + Fmod . 330

To analyze transmission of this signal through the troposcatter propagation path we refer to Figure 5.33 in order to define the phase shifts between spectral components f min and f max . Two extreme rays scattered from the highest and lowest points of the scattering volume VS , ANB and APB have the difference in distances, which may roughly be estimated as

∆ r = APB − ANB ≈ R γ θ =

R2 γ , ae

(5.143)

for γ 1 = γ 2 = γ and equal distances r1 ≈ r2 = R from transmission/reception point to the center of the common volume Vsc along the great circle. Expression (5.143) may be rewritten, taking into account (5.131):

R2 ∆r ≈ ae

4π / G ,

(5.144)

where ae = 8500 km is the earth’s equivalent radius. The phase shift between two rays passed through different extreme paths for the lowest spectral harmonic is

∆ϕ min =

2π ( f 0 − Fmod ) 2π f min ⋅∆r = ⋅∆r , c c

(5.145)

whereas for the highest spectral harmonic it is equal to

∆ϕ max =

2π ( f 0 + Fmod ) 2π f min ⋅∆r = ⋅∆r , c c

(5.146)

where c = 3⋅10 8 m/s is a speed of light in free space. As noted in subsection 4.1.3 the phase distortions of the signal may be neglected if the maximum difference in phase shifts between above extreme rays and extreme spectral harmonics is less than π / 2 , i.e.

δϕ = ∆ϕ max − ∆ϕ min = 2

2 π Fmod π ∆r ≤ . 2 c

(5.147)

Then the limit for signal transmission bandwidth may be defined by substitution of (5.144) into (5.147)

∆ f = 2 Fmod ≤

c ae c ≈ 4∆ r 4 R2

G . 4π

(5.148)

An example below demonstrates, in particular, how the signal transmission frequency band may depend on troposcatter propagation path length.

331

---------------------------------------------------------------------------------------------Example 5.3 Calculate and plot the distance dependence of the maximum signal transmission bandwidth for the range of distances from 200 to 1000 km. Assume identical antennas of the gain G = 47 dB (G = 50000) for both transmitting and receiving stations. Solution Based on (5.142) the calculation results are presented in Table E5.3 below and plotted in Figure E5.3. Table E5.3 R, km ∆f, kHz

100

200

4021.2 1005.3

300

400

500

600

700

800

900

1000

446.8

251.3

160.8

111.7

82.1

62.8

49.6

40.2

Figure E5.3. Example of the distance dependence of maximum bandwidth for troposcatter signal transmission

---------------------------------------------------------------------------------------------

From the considered example it may be seen that the bandwidth of the secondary-tropospheric transmitting system is strictly limited and cannot exceed several megahertz. From (5.148) it may be concluded, that some improvement of the bandwidth can be achieved by increasing of the antennas gains. However the gain limitations are to be applies. Unfortunately the restrictions do not allow transmission of the video signals or high speed digital data.

332

5.7. ATTENUATION OF THE RADIO WAVES IN THE ATMOSPHERE 5.7.1. ATTENUATIONS IN TROPOSPHERE Propagation of the radio wave through the atmospheric air is accompanied by the losses due to transfer of the energy of the wave into the thermal movements of atoms and molecules of the composites: gases and hydrometeors. Those composites are considered to be uniformly distributed along the propagation path, so thus the attenuation coefficient in (2.90) is counted as a sum

α = αg +αh ,

(5.149)

where α g and α h are the attenuations per unit distance (attenuation coefficients 1) in atmospheric gases, and hydrometeors respectively. The most significant contributions into gaseous component of the attenuation coefficient, namely α g occurs in molecular oxygen (O2), called “dry air specific attenuation”, and in water vapors (H2O), i.e. the total attenuation coefficient due to losses in atmospheric gases in dB/km is

α g ≈ αO + α H O . 2

(5.150)

2

Frequency dependencies of α O2 and α H 2O are shown on graphs in Figure 5.36; it’s seen that both terms in (5.150) are highly frequency-sensitive, called therefore frequency-selective absorptions [13].

1

Cited in [13] as specific attenuation.

333

Figure 5.36. Attenuation coefficient of sea-level atmosphere due to molecular oxygen and water vapour for standard atmosphere: air pressure, 1013 mbar, temperature, 15 0C, water vapour density, 7.5 g/m3 (for α H 2O curve only).

From the given figure one may realize, that the attenuation of energy of radio waves becomes considerable for the frequencies higher than 5 GHz. For the frequencies less than 5 GHz those attenuations can be neglected for any type of radio links including terrestrial, as well as earth-tospace and space-to-earth. Considering the frequency range of f ≥ 5 GHz, the value of α g increases with the significant jumps of attenuations caused by mechanical resonances in molecular structures of O2 and H2O as seen from Figure 5.36. Apparently the use of those frequencies is meaningless. From the same figure it may be seen that there're also so called transparency windows with relatively low attenuations; only those windows are acceptable for the radio links design. Particularly one of the windows is located in a frequency range between 22 GHz and 60 GHz with the minimum value of specific attenuation factor of approximately

α g ≈ 0.15 dB/km at f = 35 GHz. Total attenuation (losses in dB) on terrestrial radio links due to absorptions in atmospheric gases may easily be found as

Lg = α g R , dB

(5.151)

334

where R is a horizontal distance between corresponding antennas in kilometers. Note that for the slant propagation paths, such as those for earth-to-space, space-to-earth, as well as for the VHF, UHF, and microwave communication links between ground-based and airborne stations expression (5.151) is not valid any more. The reason is uneven vertical distribution of the atmospheric air temperature, pressure, and density of the water vapours. Last two factors have most significant impact on the values of α O2 and α H 2O . For the standard atmospheric conditions they both decrease exponentially. To obtain the total attenuation on slant propagation path the product (5.151) is to be replaced by the (2.94) integral. If ∆ h is the difference in altitudes of the communicating antennas, the equivalent height ∆ heq < ∆ h is introduced [13] for both, dry air attenuations, ∆ heq , O2 and water vapours attenuations, ∆ heq , H 2O . It must be emphasized that the air pressure decreases much slower than water vapour density, therefore ∆ heq , H 2O < ∆ heq , O2 . Then for the zenith angle θ of the slant propagation path the value of the total attenuation may be calculated based on secant law, i.e.

(

)

Lg = α O2 ∆ heq , O2 + α H 2O ∆ heq , H 2O secθ .

(5.152)

The values of α O2 and α H 2O are taken at the height of the lowest antenna. Figure 5.36 may be used only for the rough graphical estimates of α g . For precise (line-by-line calculation method), and approximate analytical calculations of α g , as well as calculation of ∆ heq , O2 and ∆ heq , H 2O the ITU document [13] may be used. The second term in (5.149), namely the attenuation coefficient due to hydrometeors, has two composite parts:

α h = α prec + α fog ,

(5.153)

where α prec is a component, caused by precipitations (rain, snow, hail), and α fog is a component, caused by clouds and fog. Note, that each hydrometeor may be considered as a small particle (raindrops, ice-balls, or snowflakes) with the semi-conducting properties. The propagating radio wave excites a displacement and/or conducting currents in that particle. Regarding (2.44) the intensity of displacement currents is in direct proportion to dielectric constant of propagation medium. The density of these currents within each raindrop is significant because the dielectric constant of water is 80 times greater than in air. On the other hand they are proportional to the frequency; hence they become heavier at the higher frequency. From the same expression (2.44) one may realize that the displacement currents do

335

not result in dissipation of energy of the radio wave into heat within the particle, because they’re shifted by 900 relative to electric field. However they result in decay of energy due to scatterings similar to scatterings on tropospheric turbulences: the incident radio wave turns each particle (raindrop, snowflake) into the elementary radiator with wide radiation pattern, thus the energy of directed radiation transforms into the radiation widely spread into surrounding space. This scattering mechanism may have a significant impact on loss of energy especially if the wavelength is comparable to the size of particles. Those attenuations may be neglected for the frequencies less than 1 GHz, whereas they become considerable in microwave bands. An additional attenuation in hydrometeors is caused by conducting currents, which turn irreversibly part of the energy of radio wave into heat that dissipates within the particle. The frequency-dependencies of composite parts of α h are shown in Figure 5.37 [3]

Figure 5.37. a). Frequency dependence of the attenuation coefficient, α h due to hydrometeors (rain and fog): (1) Drizzling rain (0.25 mm/hr); (2) Light rain (1mm/hr); (3) Moderate rain (4 mm/hr); (4) Heavy rain (15 mm/hr); (5) Light fog (0.03 g/m3 – 600 m visibility); (6) Medium fog (0.3 g/m3 – 120 m visibility); (7) Dense fog (2.3 g/m3 – 30 m visibility). b). Propagation path range within precipitation area for earth-to-space communication link

Graphs on Figure 5.37 may be used for rough, approximate references only. For precise calculations the reader may be referred to [14,15]. Note that for the dry snow and hail α prec

336

becomes almost 25 times less than that for the same intensity of the rain at the same frequency. This statement may be taken into account when similar calculations are performed for snow and hail based on the data from Figure 5.37a. However, the wet snow results in almost the same

α prec as a rain of the same intensity. Total loss due to precipitations on earth-to-space communication path of length ∆ prec depends on the zenith angle θ (see Figure 5.37b) as follows:

L prec = α prec ∆ prec =α prec

hclouds . cosθ

(5.154)

More complete data including geographical and seasonal statistical distributions may be found in [14].

-------------------------------------------------------------------------------------------Example 5.4 Calculate approximate diameter of parabolic dish antenna onboard geostationary satellite for the space-to-earth communication link for the initial data outlined below. Assume receiving station positioned in under-the-satellite point ( θ = 0 ). •

The wavelength

λ 0 = 7.5 cm ( f = 4 GHz),



Power radiated

PT = 30 W (14.8 dB/W),



Minimum power received

PR = 1.6 ⋅ 10 −11 W = 16 pW (-108 dB/W),



Receiving antenna gain

50 dB ,



The distance

r = 36000 km , (distance to geostationary orbit)



Intensity of the rain

15 mm/hr ,



Visibility in foggy condition

120 m,



Height of the precipitation ceiling

1 km



Losses in antenna feeding lines, and total losses in ionosphere ignored



Consider for simplicity the equivalent path lengths:

∆ heq , O2 = 7 km, and ∆ heq , H 2O = 1.5 km Solution 1. Specific attenuation (attenuation coefficient) is found from Figure 5.36 approximately as: α O2 = 0.007 dB/km, α H 2O = 0.001 dB/km

337

2. Total attenuation loss, due to atmospheric gases found from (5.152):

Lg = (0.007 ⋅ 7 + 0.001 ⋅1.5) sec 0 = 0.0505 dB 3. attenuation coefficient due to rain and fog defined from Figure 5.37a: for rain

α prec ≈ 0.015 dB/km

for fog

α fog ≈ 0.02 dB/km

4. As a worst-case scenario we consider the same path length for both, rain and fog. Then the total loss due to attenuations in hydrometeors:

Lh ≈ ( 0.015 + 0.02) ⋅1 / cos 0 = 0.035 dB 5. Total losses due to attenuations in troposphere (dry air and hydrometeors):

LTrop = L g + L h = 0.0505+0.035 ≈ 0.1 dB 6. From the expression (3.38a) the satellite-based transmitting antennas gain:

GT = PR , dB − PT , dB − G R + L0 + Ladd = = −108 − 14.8 − 50 + 0.1 + 195.6 = 22.9 dB ( ≈ 195 unitless) , 2

2  4π r   = 10 log  4π ⋅ 36,000,000  = 195.6 dB is a free-space where L0 = 10 log   λ0  0.075    

loss 7. Now we refer to (3.30) and (3.31) and take ν ≈ 0.6 . Then the geometrical aperture of the satellite transmitting antenna may be calculated as:

S=

GT λ 0 4π ν

2

=

π D2 195.6 ⋅ 0.075 2 , ≈ 0.146 m2 = 4 ⋅ 3.14 ⋅ 0.6 4

where D is the diameter of the satellite transmitting antenna 8. Finally, form the previous expression

D=

4S

π

=

4 ⋅ 0.146 = 0.43 m 3.14

(Answer)

-----------------------------------------------------------------------------------------

5.7.2. ATTENUATIONS IN IONOSPHERE Ionosphere is considered as a low-loss dielectric with the attenuation coefficient defined by (2.85) for tan δ > ξ 2 . Then (5.9) may be rewritten as

σ≈

2.8 ξ N elec / cm3

(2π )

2

f Hz

⋅10 −2 ≈ 7.1

2

ξ N elec / cm f MHz

2

3

⋅10 −16 , S/m,

(5.157)

where N elec / cm3 is a number of free electrons per cubic centimeter, and ξ is a number of free electrons collisions per second. The substitution of (5.157) into (5.156) will result in

α ≈ 1.34

ξ N elec / cm f MHz

3

2

10 −13 , Np/m.

(5.158)

For the simple estimates we may assume the values ξ and N e remaining constant within the absorbing layer. Particularly for the E-layer the number of collisions is taken form Table 5.1 as

ξ ≈ 10 5 1/sec, then (5.149) can be expressed as α E ≈ 1.66 where

f c E , MHz =

f c E , MHz f MHz

2

2

⋅10 −4 , Np/m ,

80.0 N elec / cm3 ⋅10 −3

(5.159) (5.160)

is the plasma frequency (otherwise called critical or Langmuir frequency) of the ionospheric Elayer in MHz. Because the value of α E is taken approximately constant within the absorbing layer, then the total losses in that particular layer can be estimated based on (2.90) (instead of (2.94)). The distance z, covered by the radio wave within the absorbing layer may be estimated based on geometric sketch given in figure 5.38a.

339

Figure 5.38. The reflection pattern of the radio wave from F2 Ionospheric layer in presence of the absorbing layers D, E and F1 layers

z = ∆ hE sec ϕ E ,

(5.161)

where ∆hE is a thickness of the E-layer, and ϕ E is the angle of radio wave incidence on Elayer. By substitutions of (5.150) and (5.152) as well as the average thickness of the E-layer

∆ hE = 30 km into (2.90), a formula for total attenuation of the radio wave in E-layer may be presented as

AE =

AE



f MHz

2

,

(5.162)

′ 2 AE ≈ 5 f c , E , MHz sec ϕ E

where

(5.163)

is the attenuation for the carrier frequency of 1 MHz (unit is: Np·MHz2). The expressions (5.162) – (5.163) may be generalized for the other absorbing layers such as Dand F1-layer. To be more accurate it must be recalled, that those expressions have been developed without taking into account the following factors for simplicity: •

Within each layer the value of α doesn't remain constant due to variations of ξ (h ) and

N e (h ) (see section 1.3). It results in changes of α from point-to-point along the ray path, thus (2.90) may not provide enough accuracy; (2.94) must be used instead •

The earth's magnetic field has a proper impact on the value of attenuation, therefore it must be taken into account as well

The exact evaluations provide the following expression for the total attenuation, consolidating all three absorbing layers

340

AΣ =

(f





+ fL )

2

,

(5.164)

where

′ ′ ′ ′ 2 AΣ = AD + AE + AF 1 = f c , E , MHz (3 sec ϕ D + 2.5 sec ϕ E + 0.4 sec ϕ F 1 )

(5.165)

is for the day-time propagation, and f L is defined from (5.52), (5.66) and (5.71) as

f L = f H cos γ ,

(5.166)

where f H ≈ 1.4 MHz, and γ is the angle between the direction of propagation and the geomagnetic meridian (see Figure 5.18), which is almost the same as the direction of the earth's







magnetic field, especially for the low and middle latitudes.. The fact that AD , AE and AF 1 are defined by only one parameter, f c , E , MHz is because the critical frequencies of all three ionospheric layers, f c , D , f c , E and f c , F 1 are dependent on just one single factor, namely on the solar activity. In order to simplify the engineering approaches it's reasonable to use just one of those three parameters, e.g. f c , E , keeping in mind that all three attenuations are bounded



(

)

originally with just one source. The family of curves for the total attenuation AΣ f c , E as a function of the length of a single-hop propagation distance (horizontal distance along the earth’s surface), R h is presented in Figure 5.39.

Figure 5.39. Total nonreflecting attenuation of the radio wave of the frequency 1 MHz in ionosphere as a function of the E-layer’s plasma frequency f c E , MHz

341



As seen form (5.165) the increase of single-hop distance results in increase of AΣ due to increase of the angles of incident ϕ D , ϕ E and ϕ F 1 . Besides the nonreflecting attenuation, some attenuation takes place while reflection form F2 layer, which may be calculated by using the following formula

AF 2 = BF 2 f MHz , 2

(5.167)

where BF 2 may be defined from Figure 5.40 for the given distance and effective height of F2 layer, heff shown in Figure 5.38b.

Figure 5.40. Attenuation of the radio wave in reflecting, F2 ionospheric layer for the frequency of 1 MHz as a function of single-hop distance, R h and the effective reflection height, heff

The decaying shape of the family of attenuation curves on Figure 5.40 is due to the increase of the angle of incidence, ϕ F 2 when the horizontal distance Rh is increased, and as a result lowering the altitude of the reflection point C ′ , shown in Figure 5.38b. Lower is the height of the reflection point, smaller the distance MC ′N will be covered by the propagation path, less attenuation will occur. The height of the vertex C ′ of the track heff depends on the ratio of the

342

carrier frequency and the critical frequency of the reflecting layer, namely f / f c , F 2 , of ionospheric MUF-predictions; otherwise it may be taken roughly heff ≈ 300 to 350 km. Those advance monthly predictions have been provided in the US for several decades by CRPL (Central Radio Propagation Laboratory) of the National Bureau of Standards (currently NIST, National Institute of Standards and Technology). Similar governmental services exist in other countries. Ionospheric predictions are in strong correlation with the index of solar activity measured in so called Wolf number-s (also known as Zürich number-s), which is highly predictable for many years in advance, taking into account the 11-year solar activity cycle. The total attenuations from all layers, nonreflecting and reflecting, is defined for the day-time as

A = AΣ + AF 2 .

(5.168)

Note finally that sometimes the reflections from the layers lower than F2 , such as E or F1 , are possible (see Figure 5.11). The specific approaches must be applied in each particular case. For instance if reflection takes place from the E-layer, then the total attenuation may be found as follows

A≈

where

(f

BE ≈

AD



+ fL )

2

4 f c , E , MHz

+ BE f

f f + fL

(5.169)

cos 2 ϕ E

(5.170)

is a semi-empiric expression, which represents the reflecting attenuation at the frequency of 1 MHz. The first term represents the non-reflective absorption from D-layer only, whereas the second term represents the reflective absorption within E-layer. The median value of the field strength at the receiving point for the multi-hop propagation path may be found from the following empirical expression:

E B = E0

~ 1 + Γ ~ n −1 Γ exp (− A) , 2

(5.171)

where • E 0 is the free space field strength (3.34) for the radio wave, which covers a distance counted along the earth's surface between radiation, A and reception, B points • n is a number of hopes • A is total attenuation in ionospheric layers, in nepers (Np), including both, reflective and non-reflective absorptions. They’re defined either by (5.168) or (5.169)

343

(

~

)

• The multiplier 1 + Γ / 2 takes into account the impact of the earth surface at the

~

transmitting and receiving points for Γ as the average reflection coefficient 1 at those points. For the practical calculations it may be taken approximately equal to 0.8. Finally it is to be mentioned that the approaches considered in this subsection are mostly applicable to HF communication lines assessments/designs. Those lines are still in use by military and commercial users despite high competitiveness of the satellite communication (satcom) systems; however HF communication systems are still used as substitutes or back-ups for those highly developed satcom systems.

------------------------------------------------------------------------------------------------------Example 5.5 Determine which ionospheric layer reflects the radio wave on a single-hop HF communication line ( n = 1 ) at the frequency f = 28 MHz, and estimate the RMS field strength at the reception point if the horizontal propagation distance is R = 1800 km. The angle between the direction of propagation and geomagnetic meridian is γ = 45 degrees. Power radiated by transmitter is

PT = 200 W with the antenna gain of GT = 20 dB (100 unitless). For ionospheric layers parameters assume the following average values taken from Table 5.1 for a day-time: •

D-layer - N e = 5 ⋅10 3 electrons/cm3, hmax = 75 km (non-reflecting layer)



E-layer - N e = 2.8 ⋅10 5 electrons/cm3, hmax = 120 km



F1-layer - N e = 3.25 ⋅10 5 electrons/cm3, hmax = 210 km



F2-layer - N e = 2 ⋅10 6 electrons/cm3, hmax = 315 km Solution

1. Check the reflection conditions for each, E F1 and F2 layer: (a) find the reflection angles from (5.43), (b) calculate plasma frequencies from (5.10a), (c) calculate MUF-s for the given distance from (5.37), (d) check if the condition f < MUF is satisfied • For E-layer – (a) ϕ = 1.369 rad (78.44 degrees), (b) f c = 4.76 MHz, (c) MUF-E-1800=23.7 MHz, (d) f > MUF (this is non-reflecting layer) • For F1-layer – (a) ϕ = 1.275 rad (73.04degrees), (b) f c = 5.12 MHz, 1

~ R is not to be confused with the horizontal distance R .

344

(c) MUF-E-1800=17.56 MHz, (d) f > MUF (this is non-reflecting layer) • For F2-layer – (a) ϕ = 1.1715 rad (67.12 degrees), (b) f c = 12.71 MHz, (c) MUF-E-1800=32.7 MHz, (d) f < MUF (this is reflecting layer) 2. Find the angle of incidence on absorbing D- E- and F1-layer, considering F2 as a reflecting layer. For the geo-central angle between communicating points along the great circle θ = R / a = 1800 / 6370 = 0.2826 rad (16.19 degrees),

ψ = π − (ϕ + θ / 2) = 1.8288 rad (104.78 degrees, see Figure E5.5).

Figure E5.5. Reflecting F2 and absorbing layers of ionosphere: habs is the height of the absorbing layer, ϕ abs is the angle of incidence of the radio wave onto absorbing layer In triangle ADO, DO= a + h Abs . Then based on the law of sines

 a  sin ϕ Abs sinψ , hence ϕ Abs = sin −1  sinψ  = a + hAbs a  a + h Abs 

(E5.5.1)

Angles of incidence for proper layers are calculated from (E5.5.1) as follows: • D-layer - ϕ D = 1.27186 rad (72.872 degrees) • E-layer - ϕ E = 1.25 rad (71.63 degrees) • F1-layer - ϕ F 1 = 1.21 rad (69.4 degrees) 3. Calculating f L from expression (5.166): f L = f H cos γ = 1.4 cos 45 0 = 0.99 MHz



4. AΣ for non-reflecting layers is defined from(5.165)

′ AΣ = 4.76 2 (3 / cos1.272 + 2.5 / cos1.25 + 0.4 / cos1.21) = 436.22 345

5. Total non-reflecting absorption in D- E- and F1-layers is defined from (5.164):

AΣ =

(f





+ fL )

2

=

436.22 = 0.52 Np (4.51 dB) (28 + 0.99) 2

6. The reflective absorption in F2 layer is defined by (5.167) for BF 2 = 7 ⋅10 − 4 obtained from Figure 5.40 approximately is: AF 2 = BF 2 f MHz = 7 ⋅10 − 4 ⋅ 28 2 = 0.55 Np (4.76 dB) 2

7. The total ionospheric absorptions from (5.168) are A = AΣ + AF 2 = 0.52 + 0.55 = 1.07 Np (9.29 dB) 8. RMS value of the electric field strength in free space (reference conditions) is defined by expression (3.34) as: E0 =

30 PT GT R

=

30 ⋅ 200 ⋅100 = 4.4 ⋅10 − 4 V/m = 0.44 mV/m 3 1800 ⋅10

9. The final RMS value of the electric field strength is defined by (5.171) as:

~ 1 + Γ ~ n −1 1 + 0.8 E = E0 Γ exp (− A) = 0.44 ⋅ ⋅ exp (−1.07) = 0.136 mV/m 2 2 ~ for Γ = 0.8 , and n = 1 .

(Answer)

---------------------------------------------------------------------------------------------------

REFERENCES [1] Levis, C. A., Johnson, J.T., and Teixeira, F.L., Radiowave propagation: physics and applications, John Wiley & Sons, Inc., 2010 [2] Staelin, D.H., Morgenthaler, A.W., and Kong, J.A., Electromagnetic waves, NJ: Prentice-Hall, Inc., 1994. [3] Dolukhanov, M. P., Propagation of Radio Waves, Moscow, USSR: Mir Publishers, 1971 [4] Rawer, K., Wave Propagation in the Ionosphere. Theory and Applications, Vol. 5, Springer Verlag,1993 [5] Al'pert, Ya. L., Radio Wave Propagation and Ionosphere, Vol. I, The Ionosphere, NY: Consultants Bureau, 1974 [6] Budden, K.G., The Propagation of Radio Waves. The Theory of Radio Waves of Low Power in the Ionosphere and Magnetosphere, NY: Cambridge University Press, 1988. [7] Davies, K., Ionospheric Radio Propagation, NBS, Washington, DC, 1965.

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[8] Rytov, S.M., Kravtsov, Yu.A., and Tatarskii, V.I., Principles of Statistical Radiophysics, Springer Verlag, 1989 [9] Collin, R.E., Antennas and Radio Wave Propagation, NY: McGraw-Hill Book Co.,1985 [10] Gradshtein, I.S., Ryzhik, I.M., Tables of Integrals, Series, and Products, Seventh Ed., Elsevier, Inc., 2007 [11] Booker, H.G., and Gordon, W.E., “Radio Scattering in the Troposphere,” Proc. IRE, Vol. 38, April 1950, pp. 401-421 [12] ITU-R Recommendation P.617-1, “Propagation Prediction Techniques and Data Required for the Design of Trans-Horizon Radio-Relay Systems,” International Telecommunication Union, 1992 [13] ITU-R Recommendation P.676-7, “Attenuation by atmospheric gases,” International Telecommunication Union, 2007 [14] Crane, R.K., Electromagnetic Wave Propagation through Rain, Wiley, 1996 [15] ITU-R Recommendation P.838-3, “Specific attenuation model for rain for use in prediction methods,” International Telecommunication Union, 2003

PROBLEMS P5.1. LOS communication radio link has a transmitting antenna of horizontal polarization with the gain of 15 dB. It is placed at the height of h1 = 40 m. Transmitter’s output power is PT = 5 W at the frequency f = 3 GHz. The receiving antenna of the height h2 = 30 m is placed at the distance of 20 km. Calculate the values of field strength of the received signal (in dB/ per mV/m) and plot its dependence on gradient of refractivity of the troposphere within the following range:

− 0.157 < dN / dh < 0 . Assume the magnitude of reflection coefficient equal unity for the flatEarth approximation. Explain the behavior of the plot. Note: Use (4.69) to calculate propagation factor. Replace H (0) by H (0) + ∆H to take into account the tropospheric refraction, with ∆H from (5.29). P5.2. A local TV station broadcasts on UHF channel-77 at the frequency of 850 MHz. Transmitted signal has a horizontal polarization with omnidirectional EIRP = 35 dBW. Calculate

347

and compare field strengths at the receiving point without (E1) and with (E2) impact of the standard (normal) tropospheric refraction for the distance of 20 km and antenna heights of h1 = 50 m (transmitting) and receiving, h2 = 10 m. Consider the ideal reflection from the earth’s surface for simplicity. Note: Use the approaches given in subsection 4.1.2. To take the atmospheric refraction into account replace the real earth’s radius by equivalent. Answer: E1 = 7.06 mV/m, E2 = 8.39 mV/m

P5.3. Confirm the values of equivalent earth’s radius ae and radius of ray curvature A provided in Table 5.2 for the given values of the gradient of refractivity index dN / dh . P5.4. Reconnaissance radar detected a target at the range of r 0 = 40 km and the elevation of

h = 5 km. Radar is calibrated for the standard atmosphere. However, the current condition of the troposphere is such that ae = 2.5 a (relation between equivalent and real earth’s radii). Calculate the error in elevation measurement ∆ h by assuming the measured range remaining unchanged. Answer: ∆ h = 43 m P5.5. Estimate the values of scattering angles 2 θ max for the radio wave propagating in the troposphere with the average size of the turbulent inhomogenieties Lε ≈ 15 cm for the following frequencies: 0.5 GHz, 5 GHz, 50 GHz. Hint:

Assume q max ∝

1 for the rough estimate. Lε Answer: 74.40, 7.30, 0.720

P5.6. Repeat calculations from the Example 5.2 for the frequency 3 GHz and compare the results. P5.7. Calculate the frequency bandwidth of the tropo-scatter radio-link of the length R = 300 km if transmitting and receiving antennas are identical with the beam width of the radiation pattern

γ = 2.5O. Does the regular tropospheric refraction impact the bandwidth? Explain. Answer: 162.4 kHz

348

P5.8. Estimate the decrease of the level of the signal received from the TV broadcasting satellite. The drop of the signal level (in dB) is due to rain of intensity 50 mm/hr, and the fog with visibility of 100 m compared to “clear weather” condition. Frequency – 12 GHz. Assume the rain and fog ceiling equal 0.8 km. The zenith angle of arrival of the radio wave from satellite to reception point is 45O. Answer: 2.34 dB (decreased 1.714 times) P5.9. Calculate the angle of Faraday rotation for the radio wave propagating in magneto-active ionosphere along the geomagnetic field line for the distance of 20 km if the carrier frequency is 30 MHz and the concentration of free electrons (ionospheric plasma concentration) is 106 el/cm3. Answer: 145.4 degrees

P5.10. Calculate MUF and angle of incidence onto F2-layer for HF broadcast link of the length 6000 km, N e , max = 10 6 electrons/cm3, and hmax = 300 km. You may use the diagram on Figure 5.10 to verify your analytical estimate. Note: 1. Assume that the propagation mode is based on the least number of ionospheric reflections. 2. Ignore the impact of the earth’s magnetic field Answer: MUF=29.5 MHz ϕ = 72.270

P5.11. The HF communication link of the path length 2600 km uses reflections from F2 layer of the ionosphere. The daily variations of electrons concentration, obtained from ionospheric observations, are shown in Figure P5.11 provided below. Design the schedule of changes of the carrier frequency, f carrier = 0.8 ⋅ MUF − F2 − 2600 , i.e. fill out the Table P5.11. Show your calculations.

349

Figure P5.11

Table P5.11 Observation time From

To

1:00

7:00

7:00

17:00

17:00

1:00

Nel per cm3

MUF0 = fc

MUF -

MUF -

4000

2600

Notes: 1. Assume the height of F2 layer hmax = 300 km. 2.

Ignore the impact of earth’s magnetic field.

350

fcarrier

APPENDIX-7 VOLUMETRIC SPECTRUM FOR AUTOCORRELATION FUNCTION OF STATISTICALLY HOMOGENEOUS AND ISOTROPIC RANDOM FIELD For the statistically isotropic random field ∆ε (r ) the volumetric autocorrelation function depends only on the magnitude of the distance between two correlating points, ρ = r2 − r1 i.e.

ψ ε ( r1 , r2 ) = ψ ε (ρ ) = ψ ε (ρ ) .

(A7.1)

Assume the function ψ ε (ρ ) is given in spherical coordinates ( ρ , ζ , ϕ ) . First we define the elementary volume dV in spherical coordinates as (see Figure A7.1b below)

dV = ρ 2 sin ζ ⋅ dρ ⋅ dϕ ⋅ dζ .

(A7.2)

Figure A7.1. a). Positions of vectors q and ρ ; b). The pattern of elementary volume dV in spherical coordinates.

Vector q of the spatial frequency is conventionally taken directed along negative Z-axes as shown in Figure A7.1a. Then the scalar product q ⋅ ρ is needed for (5.95):

q ⋅ ρ = q ⋅ ρ ⋅ cos (π − ζ ) = − q ⋅ ρ ⋅ cos ζ .

(A7.3)

Now substitute (A7.2) and (A7.3) into (5.95), taking into account statistical isotropy (A7.1).

351

S ε (q ) = S ε (q ) =

1

(2π )

3

⋅ ∫ψ ε (ρ ) ⋅ exp (i q ρ cos ζ ) ⋅ ρ 2 sin ζ ⋅ dρ ⋅ dϕ ⋅ dζ .

(A7.4)

V

Variables in (A7.4) may be separated and volumetric integral may be written as

S ε (q) =

1

(2π )3

∞ 2π π  ⋅ ∫ dϕ ⋅∫ψ ε (ρ ) ⋅  ∫ exp ( i q ρ ⋅ cos ζ ) ⋅ sin ζ ⋅ dζ  ⋅ ρ 2 dρ . 0 0 0 

(A7.5)

Now consider the brackets in last expression: π

π

0

0

∫ exp ( i qρ ⋅ cos ζ ) ⋅ sin ζ ⋅ dζ = −∫ exp (i qρ ⋅ cos ζ ) ⋅ d (cos ζ ) =

=−

1 ⋅ exp(iqρ ⋅ cos ζ ) i qρ

π 0

=

2 exp (i q ρ ) − exp (− i qρ ) 2 ⋅ = ⋅ sin (q ρ ) . 2i qρ qρ

(A7.6)





If dϕ = 2π is substituted into (A7.5) along with (A7.6), then final expression for the spatial 0

spectrum of the isotropic, homogeneous field ψ ε (ρ ) may be obtained as

S ε (q ) =

1 2π 2 q



⋅ ∫ψ ε (ρ ) ⋅ sin (qρ ) ⋅ ρ dρ , 0

which is ready to be used with different ψ ε (ρ ) models.

352

(A7.7)

Chapter 6.

RECEIVING OF THE RADIO WAVES: BASIC OUTLINES

6.1. MULTIPLICATIVE INTERFERENCES (SIGNAL FADES) 6.1.1. FLUCTUATION PROCESSES AND STABILITY OF RADIO LINKS The filed strength of the radio wave at the reception point in the real conditions is counted based on propagation factor F in (3.37), which is introduced in section 3.2 to accommodate the reference path (ideal propagation track) to the real conditions. In other words the propagation factor allows taking into account the impact of the earth's surface and the atmospheric effects. The electro-magnetic parameters of the earth's surface and the atmospheric propagation medium are not constant, but randomly fluctuating in time and space, therefore generally the magnitude and initial phase of the complex propagation factor F , must be considered as random variables in time domain at any particular receiving point. Those random timevariations/fluctuations may be expressed as

F = F ( t )exp [ i Φ F ( t )] ,

(6.1)

where F ( t ) is a random magnitude of the propagation factor, and Φ F (t ) is its random phase. Based on (3.37) for the RMS (effective) value of the electric field strength, the expression for complex time-harmonic field strength, may be written in the following form

E (t ) = E m, 0 F exp(iω t ) = E m, 0 exp {i [ω t + Φ E ( t ) ] }F ,

(6.2)

E m , 0 = E m , 0 exp[i Φ E (t )]

(6.3)

where

is the amplitude phasor of the signal in free space with the magnitude of

E m, 0 = E0

2=

60 PT GT r

,

(6.4)

and E 0 is the effective (RMS) value of the reference field strength of the radio wave (compare with (3.37)). The randomly fluctuating amplitude of the field strength may be defined from (6.1) and (6.2) as a product

Em ( t ) = Em, 0 F ( t ) .

(6.5)

353

From (6.5) one may notice that in real condition the field amplitude at the reception point randomly fluctuates because of random variations of F (t ) . The value of amplitude E m , 0 for the ideal propagation conditions is multiplied by a random factor that represents the interferences caused by fluctuations on real propagation path. Therefore this type of interference is called multiplicative interference or fading. In other words fading is considered as fluctuation of the wanted signal caused by random changes of the signal propagation conditions 1. Methods of the probability theory and statistics are employed for the quantitative description of fading statistics. For numerical estimates the probability of any value E of the continuous random variable E ′ is formally defined as a ratio of (1) the sum of the time periods, ∆ Tn , when

E ′ faded below E ( E ′ ≤ E ) to (2) the total observation time tTot (see Figure 6.1), i.e.

P( E ) =

∑ ∆T

n

n

tTot

=

∆T1 + ∆T2 + ∆T3 + ∆T4 . tTot

(6.6)

Figure 6.1. Random variations of effective field strength of the signal at the receiving point in presence of multiplicative interference (fading).

1

Taking into account (6.1) and (6.2), the random total phase of the radio wave at the reception point is

Φ E ( t ) + Φ F (t ) . For the AM systems the initial phase Φ E ( t ) = Φ 0 is constant, i.e. doesn’t carry information . In FM and PM systems component

Φ E (t ) carries the information, thus at the output of demodulator the

Φ F (t ) appears as an additive interference (noise) (see section 6.2). Fortunately the rate of

fluctuations of

Φ F (t ) is usually much slower, than the rate of time-variations of Φ E (t ) due to the

informative signal. Hence the power spectrum of this additive noise is limited to less than 10 Hz and may be easily cleared off with the high-pass filter.

354

As one may see form Figure 6.1, the greater the value of E, the larger the probability P (E ) is. This functional dependence is called the cumulative distribution function (CDF). Another important statistical measure of the random variable is probability density function (PDF). Suppose ∆P(E ) is the probability of the field strength E ranged between E and

E + ∆E , i.e. in expression (6.6) the quantity

∑ ∆T

n

is defined not as the cumulative time

n

interval, when the random field strength remains below the value E , but the cumulative time interval, when it remains within E and E + ∆E limits. Then PDF is defined as:

∆P dP = = P ′(E ) , ∆E →0 ∆E dE

w ( E ) = lim

(6.7)

i.e. it is the probability of the random variable E ′ (t ) remained within the unit interval that surrounds the particular value E . From (6.7) it's easy to see that the relation between CDF and PDF is E

P( E ) = ∫ w ( E ) dE .

(6.8)

0

Per definition, the cumulative probability, CDF may not exceed the unity, therefore w ( E ) must satisfy the normalization condition, i.e. ∞

∫ w ( E ) dE = 1 .

(6.9)

0

One of the useful statistical measures of the random E ′ (t ) process that is widely used in the radio communications and radar engineering applications is the median value, E med defined from the following equation

P( E med ) = 0.5 .

(6.10)

As follows from (6.10) E med has a meaning of deterministic parameter of the random variable

E ′ (t ) that represents the value of E being exceeded with the probability of 0.5, i.e. being exceeded within 50% of the total observation time. Another important parameter of the random variable is the mean value E , called average value or statistical expectation defined from the following averaging integral

E =

1 tTot

tTot

∫ E ′ (t ) dt

(6.10a)

0

355

The definitions (6.10) and (6.10a) are considerably different, so thus the quantities E and

E med are different in general. However, for the statistics those describe fading of the radio waves, the values E and E med are fairly close to each other, so the difference may often be ignored. It’s easy to realize that neither E , nor E med allow specify the depth of fades. It is customary to obtain the fade depth based on deciles E0.1 and E0.9 , which show the levels of observing signal (usually in dB) that are exceeded within 10% and 90% of total observation time respectively 1. In other words from the practical point as a measure of the fades depth the use of the difference

E0.1 − E0.9 (in dB) in RF propagation is more convenient than standard deviation that is widely used in other probability and statistics applications. In engineering practice the term channel reliability is introduced as a measure of quality of the service: it is defined as percentage of the total serving time while the performance of the communication system is not less than its minimum (threshold) 2; this threshold is usually specified by the industry standards. Therefore, in practical applications another statistical measure, namely complementary cumulative distribution function (CCDF) is introduced similar to that given in subsection 4.2.3 for the complementary error function. CCDF is more commonly used in communications and radars, rather than cumulative distribution function: CCDF indicates the percentage of total observation time while random variable E ′ is greater or equal than considered fixed value E . Hence, taking into account (6.8) and (6.9), the expression for CCDF may be written as ∞

CCDF = W ( E ) = 1 − P( E ) = ∫ w( E )dE .

(6.11)

E

Within time intervals ∆t1 , ∆t 2 , and ∆t 3 of the total observation time, t Tot , shown in Figure 6.2 the fluctuating field E may fade down the minimum allowable value E min , which causes communication outage (break). E min is the lowest electric field, which receiver is able to “sense” in presence of additive noises in terms of its ability to extract the modulating signal from the

E med = E0.5 thus E med may be considered as a decile.

1

Note that

2

In some references [16], the term channel reliability =

outage (break) time,

100 ⋅ (1 − t 0 / t tot ) = t a / t tot is used: t 0 is the total

t tot is total observation time, and t a is total availability time. 356

carrier. In order to estimate channel reliability one may refer to Figure 6.2, where the shortterm 1 pattern of the field strength at the receiving point is presented.

Figure 6.2. Sketch of the complementary cumulative distribution function W(E) of the fading field of the radio wave

From the above definition the quantitative value of channel reliability may be found based upon CCDF as (see Figure 6.2)

W0 = W ( E min ) .

(6.12)

It is equal to probability of the fluctuating field at the reception point being above its minimum value, i.e. it is equal W (E ) for the value of E = E min . It may be realized, that higher is the margin between E med and E min , greater is the channel reliability W0 . In other words for any particular radio link the greater is ratio E med / E min , the higher is channel reliability. This ratio is called fade margin that in most cases is expressed in dB-s, as follows: fade margin = E med , dB − E min, dB . On the other hand the probability of communication break is apparently equal to

T0 = 1 − W0 .

1

(6.13)

Short-term is introduced to emphasize that during the total observation time-period

tTot the median

value of electric field remains constant, i.e. this random process remains stationary. Therefore we call it interval of stationarity, HF-links

t Stat . For different types of radio links t Stat has different values. For instance for

t Stat is equal approximately several to tens of minutes. 357

Generally all above statements are applicable not just to randomly fluctuating E (t ) , but also to Signal-to-Noise Ratio (SNR), which actually behaves as a random variable with the same statistical properties as for E (t ) , if assumed the noise intensity unchanged within the observation time interval. Channel reliability standards for most of the service types are defined in terms of SNR rather than in terms of E (t ) , i.e. the quantities such as SNRmed , SNRmin , fade margin for SNR are commonly used in radar and communications engineering. If, instead of short-term observation, the long-term observations (hours, days, months, etc.) are conducted, then in predominant propagation cases the median field strength, as well as other statistical parameters, do not remain constant, but also are changing randomly as shown in Figure 6.3. This means that generally the fading is a complex, non-stationary random process, i.e. the process that have non-stable statistical parameters, namely average, median, standard deviation, etc., which are also randomly fluctuating. It was noticed from numerous observations, that in real conditions any long-time interval may be divided into a several short-time intervals so that the statistical parameters within each of those short-time intervals may be considered as being almost constant, i.e. each of those short-time intervals may practically be considered as intervals of stationarity, t stat (see Figure 6.3).

Figure 6.3. Random variations of the effective field strength at the receiving point for long term observation interval

From the above consideration one may conclude, that two types of fading persist in radio links: fast and slow. Fast fading appears as random variations within the stationarity intervals, whereas slow fading is considered as long-term, slow variations of the median value of the field strength, which usually appears together with variations of other statistical parameters, such as standard deviation and mean value.

358

The natures of fast and slow fades are totally different and independent. The cause of fast fades relates to multi-path propagation. A typical example is tropospheric scattered propagation considered in section 5.6. For multi-path propagation different independent rays approach the reception point by passing different paths with random propagation distances (∆ r ) , causing so called random interference. Those fluctuations of ∆ r are the result of fluctuating positions of scattering irregularities of the tropospheric "clear air" on tropo-scatter radio links (Figure 6.4a), or by random changes of the reflection points and reflection heights on HF-ionospheric radiolinks (Figure 6.4b). Hence the phase shifts between different rays ∆Φ = (2π / λ ) ∆ r becomes a random variable. It’s "sensitive" to both: fluctuations of paths differences ∆ r , and the wavelength. Smaller is the wavelength λ , the larger the fluctuations of phase shifts are, and therefore higher is the fast fading rate.

Figure 6.4. The multi-path propagation mechanisms on tropospheric (a) and ionospheric (b) radio links

When arriving the destination point B with random phase shifts within wide range [0, 2π] different rays are superimposing differently. The extreme cases are: Equal-phase superposition causing constructive interference, and opposite-phase superposition causing destructive interference. The random interference is typical for most of RF links. The greater the number of superimposing rays, the deeper the fading is, and more considerable is its impact on communication stability. Slow fading of the RW is caused by random changes of attenuations along the radio propagation path, which are considered as a result of slow changes in the electromagnetic parameters of earth's surface and atmosphere, due to the random variations of temperature, humidity, air pressure, ionization, etc. From the above one may realize, that fast and slow fading are two independent random processes acting simultaneously within antenna-medium-antenna propagation path. Therefore

359

statistical distributions are expected to be composite functions, combining fast and slow fade statistics. This issue will be considered in section 6.1.4.

6.1.2. FAST FADING STATISTICAL DISTRIBUTIONS Here we will start with simple case of two-ray random interference, continue to the case of multiray superposition of the independent wavelets (Rayleigh ensemble), and then further consider more complex cases in subsection 6.1.2d to provide generalized approaches.

6.1.2.a. Two-Ray Random Interference Consider a random interference of two rays, i.e. a superposition of two sine waves with the constant, equal amplitudes E max , same spatial orientation of electric field and random phase shift, Φ (t ) with evenly distributed probability within the range [ 0 ÷ π ]. The sum of electric field collinear vectors of those two waves may be replaced by a sum of two complex scalars, i.e.

E (t ) = E max exp (iω t ) + E max exp [i (ω t − Φ )] =

= E max exp ( iω t ) ⋅ [1 + cos Φ − i sin Φ ] .

(6.14)

From (6.14) the effective (RMS) field strength value may be found as

E = E1

(1 + cos Φ )2 + sin 2 Φ = 2 E1 cos Φ 2

where E1 = E max /

,

(6.15)

2 is an effective field strength of one of the rays.

Now we differentiate both sides of (6.15) 1 as follows:

dE = E1 ⋅ sin

Φ dΦ , 2

(6.16)

From (6.15) we define sin (Φ / 2) , substitute into (6.16) and divide both sides by π :

1

π

dE

dΦ =

π

E2 E1 − 4

.

(6.17)

2

For evenly distributed probability density of the random phase Φ shown in Figure 6.5, the left hand side of is a probability of random variable Φ being located in the range [Φ, Φ + dΦ ] : in

1

Minus sign is ignored, because it doesn’t effect the final result, as it may be seen later.

360

general form it may be expressed as w (Φ ) d Φ , where w (Φ ) = 1 / π is a probability density for this particular case 1.

Figure 6.5. Probability density distribution of the resultant phase for randomly superimposing two independent rays

Hence the right hand side of (6.17) may be considered as a probability of the resultant randomly fluctuating E (t ) field being within the range [ E , E + dE ] , and may be written in the form of w (E ) d E . Therefore the following expression may be found for the PDDF, w (E ) from (6.17):

w (E ) =

1

π

E1 − E 2 / 4 2

.

(6.18)

Based on (6.18) the CPDF may be found, taking into account the fact that the upper limit in (6.8) is equal 2 E1 , i.e. may not exceed the double value of the effective field E1 of the single wave.

W (E ) =

2 E1

1

∫π

E1 − E / 4 2

E

dE = 1 −

2

 E   . sin −1  π  2 E1  2

(6.19)

The definition (6.10) for the median value of random variable E may now be applied to (6.19) as

W (E med ) = 1 −

E sin −1  med π  2 E1 2

  = 0.5 , 

(6.20)

then from (6.20) the value of E med may be expressed in terms of E1 :

E med =

1

Note, that

2 E1 .

(6.21)

w (Φ ) distribution is considered only in {0, π }, because the values of E are repeated in rest

of the range, i.e. within

{π , 2π }. 361

The reason for that replacement is that the median value of the electric field at the reception point is measurable, whereas the effective value of the electric field of only one of the components of the total field E1 is really hard to separate in practical reception procedure. Using (6.21), the value of E1 in (6.19) may be replaced by E med as follows:

W (E ) = 1 −

 sin −1   π  2

E 2 E med

  .  

(6.22)

The final expression (6.22) for the CPDF of the random variable E (t ) is shown graphically in Figure 6.6 as a function of the ratio E / E med . Note that due to the condition 0 ≤ E ≤ 2 E1 the argument remains in the range 0 ≤ E / E med ≤

2.

Figure 6.6. Cumulative Probability Distribution Functions for different scenarios of multi-path random interference

6.1.2.b. Random interference of the large number of independent wavelets In most applications a large number of partial, independent waves of equal amplitudes and random phases, evenly distributed in [0, 2π ] range, are interfering at the reception point. The intensity of each wavelet is vanishing. However the number of those partial wavelets tends to infinity resulting in a finite overall power. This group of partial rays, called Rayleigh ensemble, is a theoretical model that closely presents the real conditions that are specific for scatterings from

362

tropospheric turbulences or scatterings from the ionospheric irregularities due to micro-meteor ionized trails 1 (see Figure 6.4a).

Figure 6.7. a). Partial waves ( E n and E k ) presentation in complex plane b). Probability density distribution of the components of the resultant E - magnitude of Rayleigh ensemble

Each wavelet may be represented in the complex plane as a phasor with the constant amplitude and random phase as shown in Figure 6.7a. As pointed above, the number of independent partial waves is extremely large, i.e. N → ∞ , with vanishing intensities of each wave. Then each of them may be decomposed into real and imaginary parts, either positive or negative, depending on random initial phase shift Φ of the single partial wavelet. Note, we silently assume that all those wavelets propagate parallel to each-other, resulting in collinear electric fields. Hence one may replace a vector sum by algebraic summation in complex plane, as it’s been done in previous cases of random interference, i.e. N

N

N

n =1

n =1

n =1

E = ∑ E n = ∑ Re E n + i ⋅ ∑ Im E n = X + i Y .

1

(6.23)

Rayleigh model of the random interference is equally applicable to diffuse scatterings (such as

scatterings from the tropospheric turbulences) and to discrete scatterings (such as scatterings from the micro-meteor trail-paths in ionosphere)

363

Regarding to the central limit theorem, if N is large enough, then both terms in right hand side of (6.23) have a Gaussian probability distribution, regardless to what the distribution of each N

component of the sum

∑ E

n =1

n

is, i.e.

w (X ) =

 X2 exp  − 2 2π σ  2σ

  , 

(6.24)

w (Y ) =

 Y2 exp  − 2 2π σ  2σ

  , 

(6.25)

1

1

where σ is a standard deviation of X and Y random variables that are the resultant components of the total vector E 1 in complex plane. In order to evaluate the probability distribution of the E = E magnitude of the electric field, let’s find first the probability for the tip of vector E to be located in the area d S that surrounds point A with the coordinates ( X , Y ) as shown in Figure 6.7b. The joint probability density function, w ( X , Y ) of the events X ∈ [ X , X + dX ] , and Y ∈ [Y , Y + dY ] , which appear simultaneously, may be counted as a product of the proper distribution functions, due to independency of X and Y random variables, i.e.

w ( X , Y ) = w ( X ) ⋅ w (Y ) =

 X 2 +Y 2  − exp 2π σ 2 2σ 2  1

It may be seen from Figure 6.7b that E 2 = E

2

  . 

(6.26)

= X 2 + Y 2 , therefore (6.26) may be considered

also as the joint probability density of the magnitude and phase of E = E exp ( iΦ ) phasor:

w ( X , Y ) = w (E , Φ ) =

 E2  − exp 2 2π σ 2  2σ 1

  . 

(6.27)

Now, based on probability density distribution function w (E , Φ ) we may find the probability for the tip of vector E to be located in the area d S = E dE dΦ (see Figure 6.7b).

d W [E ∈ (E , E + dE ), Φ ∈ (Φ, Φ + dΦ )] =

1

Note that the standard deviation

σ

is taken the same for both, X and Y because it’s invariant to the

rotations of complex coordinates.

364

= w (E , Φ ) d S =

 E2 exp  − 2 2π σ  2σ 1

  ⋅ E dE dΦ . 

(6.28)

In order to find the probability distribution of the magnitude E , the phase Φ to be excluded from (6.28). In other words (6.28) is to be integrated for Φ form 0 to 2π , which allows finding the probability for the tip of E to be located within the ring E ∈ (E , E + ∆E ) shown in Figure 6.7b. 2π

dW [E ∈ (E , E + dE )] = ∫ dW [E ∈ (E , E + dE ), Φ ∈ (Φ, Φ + dΦ )]dΦ = 0



=

∫ 0

 E2 exp  − 2 2π σ 2  2σ 1

  E2 E  E dE dΦ = 2 exp  − 2 σ   2σ

  dE . 

(6.29)

From (6.29) the probability density distribution function for the Rayleigh ensemble, or so called differential Rayleigh distribution function, may be found as

w (E ) =

 E2 dW E = 2 exp  − 2 dE σ  2σ

  . 

(6.30)

The graph of (6.30) is shown in Figure 6.8a. Cumulative Rayleigh distribution function may be obtained by substitution of (6.30) into (6.11). ∞



 y2  − exp 2 2  2σ Eσ

W (E ) = ∫ w ( y ) dy = ∫ E

1

  E2  y dy = exp  − 2   2σ

  . 

(6.31)

Similarly to the previous case the parameter σ is likely be replaced by the median value, E med . By definition

 E med 2  . W (E med ) = 0.5 = exp  − 2   2σ 

(6.32)

Then σ may be found by solving (6.32):

2σ 2 =

2

2

E med E = med . ln 2 0.693

(6.33)

365

Figure 6.8. Rayleigh distribution: a).Probability density function. b). Cumulative probability function

Final expression for CPDF as a function of E / E med may be found if (6.33) is substituted into (6.30):

  E W (E ) = exp − 0.693    E med

  

2

  . 

(6.34)

The graph of probability density distribution function (6.34) is presented in figure 6.6a in linear scale and cumulative probability distribution function is shown in Figure 6.8b in so called Rayleighian scale, which ends up with the straight line, as shown.

6.1.2.c. Further Generalization of the Fast Fading Statistics In his pure form the Rayleigh distribution is not quite common for fast fading statistical description in practical applications in radars and communications practices. Its main restriction is the lack of flexibility that is in needed to support a variety of propagation mechanisms, hence the variety of statistical properties, particularly the fading depths and shapes of probability distribution functions. Variety of types of radio links and propagation mechanisms leads to the need of more complex analytical expressions for fading statistics that are able to: 1. Specifics of the physical processes in radio waves propagation 2. Adjust to statistical properties of the particular propagation path of interest Further updates of the fast fading statistics that involve more detailed propagation mechanisms are outlined in this subsection without going inside the mathematical evaluations procedures.

366

One of the advantages of those statistics is the ability to adjust the probability distribution functions to 1). Rice-distribution (Generalized Rayleigh-distribution) appears as a result of superposition of Rayleigh ensemble with the strong monochromatic (deterministic) component of E 0 field strength (see Figure 6.9). This approach has been originally presented in [4], where the probability density distribution function was evaluated in the following form (see graph in Figure 6.10a)

 E 2 + E0 2   E E0   ⋅ I0  2  . w ( E ) = 2 exp  − 2   σ  σ 2 σ   E

(6.35)

Here I 0 is the modified Bessel function of the first kind, zero order. If E 0 = 0 , then the distribution reduces to a regular Rayleigh distribution. The adjustment parameter β = E 0 / σ is to adopt (6.35) to the real conditions for the given ratio between deterministic component E 0 and total intensity of Rayleigh ensemble. This distribution is specific for the terrestrial LOS or Earth-to-Space / Space-to-Earth communication links at UHF and higher frequencies. CPDF is calculated numerically, and is shown in Figure 6.10b in Rayleighian scale.

Figure 6.9. Superposition of a strong, stand-alone monochromatic component with the continuum of single-scattered 1 uncorrelated wavelets (Rayleigh ensemble).

1

Superposition of the continuum of multi-scattered waves with intensive monochromatic component is

mostly observable on atmospheric-optical communication links rather than on RF links. Multi scattering of the partial waves results in log-normal probability distribution (see [7] for references).

367

Figure 6.10. Rice-distribution: a). Probability density distribution, b). CPDF

2). m-distribution of Nakagami has been originally presented in [5]. It is one of the most generalized descriptions of the random interference for the superimposing wavelets of various amplitude and phases distributions. The probability density distribution function is presented by

 mE 2  2m m E 2 m−1 exp − 2  , wm ( E ) = Γ ( m) σ 2m  σ 

(6.36)

where m and σ are the adjustment parameters. This expression is applicable to any type of radio links as a fast fading statistic (see graph on Figure 6.11).

Figure 6.11. Probability density of m-distribution Nakagami

Based on (6.8), and (6.36) the CPDF for m-distribution may be defined as

368

Wm ( E ) =

B=

where





2 mm 1   m z 2 m −1 exp  − 2 z 2  dz = t m −1 exp (− t ) dt , 2m ∫ ∫ ( ) Γ m σ Γ(m )σ E   B

mE2

σ2

(6.37)

.

(6.38)

The integral in (6.37) may be expressed in closed from through the incomplete gamma-function [1] as follows:

Wm ( E ) = 1 −

 

 E2  σ . Γ (m )

γ  m,

m

2

(6.39)

It may be shown, that the second order mathematical moment of the random variable E (t ) is defined from (6.39) as

E2 = σ 2 ,

(6.40)

and it is constant and independent on m. Median values may be found from Wm (E med ) = 0.5 , by using (6.39) in the following form 2   E γ  m, m med2  σ  

Γ (m )

= 0.5 .

(6.41)

The ratio E med / σ 2 is m-dependent. If we denote 2

p (m ) = E med / σ 2 , 2

(6.42)

then values of p (m ) may be calculated: the results are given in Table 6.1. Table 6.1

m

0.5

0.75

1.0

1.5

2.0

3.0

4.0

5.0

6.0

p ( m ) 0.456 0.606 0.693 0.789 0.839 0.891 0.918 0.922 0.945 p (m ) values may also be found from the graph shown in Figure 6.12. Taking into account (6.42) the expression (6.39) may be rewritten as

Wm ( x ) = 1 − where

γ (m, m ⋅ q (m ) ⋅ x 2 ) Γ (m )

x=

,

(6.43)

E E med

(6.44)

369

is E-level, relative to median value. The sketch of CPDF for m-distribution is shown in Figure 6.12b in Rayleighian scale, so thus for m = 1 (for Rayleigh distribution) the graph becomes a straight line.

Figure 6.12. a). Graph of the p (m) parameter, b). CPDF of m-distribution

3). n-distribution (Quasi-Rayleigh distribution) is a statistical model that was verified based on numerous observations of fast fading on HF radio links, and has been originally presented in [6]. The adjusting parameter n is introduced to modify Rayleigh distribution and make it adjustable to the observed fade statistics on real propagation paths. The idea relates to diversity reception in auto-selection mode that results in (6.150). Similar expression is adopted here for CPDF of Quasi-Rayleigh distribution as follows:

   E Wn ( E ) = 1 −  1 − exp − 0.693 ⋅ q (n) ⋅    E med 

where

q (n ) =

2

E med 2σ 2

  

2

n

   ,  

    1  = ln   1  1 − exp  n ln 0.5    

(6.45)

(6.46)

is n-dependent parameter, so thus Wn (E ) becomes adjustable. The calculated values of q (n ) are given in Table 6.2 and are presented graphically in Figure 6.13. CPDF of the n-distribution (quazi-Rayleigh distribution) is shown in Figure 6.14 with Rayleigh-scale for horizontal axis.

370

Table 6.2

n

0.25

0.5

1

2

3

q(n)

0.0645

0.2877

0.6931

1.2279

1.5784

Figure 6.13. Quasi-Rayleigh distribution: a) Graph of the q (n) parameter, b). CPDF

6.1.3. SLOW FADING STATISTICAL DISTRIBUTION As noted above the cause of slow fading (slow random variations of the short-term median field at the reception point) are the random changes in attenuations along the propagation path. Those attenuations depend on frequencies and propagation path configurations'. They may arise in different atmospheric layers and along Earth's surface, being impacted by random changes in temperature, humidity, pressure, ionization intensity or other physical conditions. Numerous observations on various radio links exhibited the logarithmic-normal (lognormal) probability distribution of slow fades.

6.1.3.a. Normal (Gaussian) distribution of the random variable. First consider the normally distributed (Gaussian distribution) random variable z. This type of random variable is described by the following probability density distribution function:

w (z ) =

 ( z − z )2 exp  − 2 2π σ z  2σ z 1

 ,  

(6.47)

371

The graph is shown in figure 6.14, where z is a mean value of random variable z (statistical expectation), and σ z =

( z − z )2

is its standard deviation.

Figure 6.14. Normal (Gaussian) probability density distribution function for z = 1 Based on (6.10) for this case the value of zmed may be found by solving the equation below. ∞



z med

 ( z − z )2  exp − dz = 0.5 2  2π σ z  2σ z  1

(6.48)

The result is

z med = z

(6.49)

because of the symmetry of the function (6.47) about z = z line 1.

6.1.3.b. Lognormal distribution of the random variable. Term "lognormal" means the normal (Gaussian) distribution of the logarithm of random variable E, i.e.

z = ln E .

1

Otherwise, for non-symmetric distributions,

(6.50)

z med ≠ z . 372

For the practical applications it's more convenient to rewrite (6.47) as a function of E-variable, taking into account (6.50):

dW ′ [z ] = w (z ) d z = w [z (E )]⋅

dz ⋅ d E = d W (E ) . dE

(6.51)

Probability density distribution function may be found from (6.51) as (Figure 6.15)

(

 ln E − ln E exp − 2 2σ z 2π σ z 

dz w (E ) = w [z (E )] ⋅ = dE E

1

)  . 2



(6.52)

Figure 6.15. Lognormal probability density distribution (simplified version of (6.56))

Based on (6.49) and (6.50) the expressions for statistical expectation and standard deviation may be rewritten as

ln E = ln Emed ,

σz = where

(z − z )2 y = ln

(6.53a)



 E  ln  Emed

  = 

y2 = σ y ,

(6.53b)

E , [Np] Emed

(6.54)

is a relative level of field strength expressed in nepers. Cumulative (integral) distribution function for this case may be evaluated if (6.52) is substituted into (6.11) as follows: ∞



E

E

W (E ) = ∫ w (E ) dE = ∫

(

 ln E − ln E exp − 2 2σ z 2π σ z  1

)  dE . 2

 E

If replacements (6.53) and (6.54) are applied to (6.55), then the result is

373

(6.55)

 y′2 exp ∫  − 2 σ 2 2π σ y y y 

W (y) =

  d y′ ,  



1

(6.56)

i.e. the slow fading of the relative level (in dB or Np) of short-term medians of the electric field strength is normally distributed. In order to “standardize” this expression the replacement

x=

y

(6.57)

2σ y 1

is used, which results in



1 exp(− x ′ ) dx ′ = [1 − erf ( x )] , ∫ 2 π

W ( y ) = W (x ) =

1

2

(6.58)

x

where

exp (− x′ ) dx′ = −erf (− x ) π ∫

erf ( x ) =

2

x

2

(6.59)

0

is so called error function, given in numerous of math references, e.g. [1]. Examples of lognormal cumulative distributions are shown in Figure 6.16, where nonlinear scale was adopted for the horizontal axes (so called "Gaussian" scale), which allows to sketch the graphs as straight lines.

Figure 6.16. Lognormal cumulative probability distribution function

The graphs on Figure 6.16 are drawn based on the following procedure:

1

The normalization

1

π



∫ exp (− x′ )dx′ = 1 is used to evaluate (6.62). 2

−∞

374

2 , i.e. y = σ y , then from (6.58) W = 0.16 (16 %) , and



if x = 1 /



if x = −1 /



note also the fact, that for any σ y graph will cross the point (0, 50%), which

2 , i.e. y = − σ y , then from (6.58) W = 0.84 (84 %) .

comes from the definition of the median value. Hence in Gaussian coordinate system the any graph of CPDF is a straight line that goes through the points:

(− σ

y

, 84) , (0, 50) , and (σ y , 84) .

Log-normal distribution is appropriate as a fast fading statistics for atmospheric-optical communication lines only. That’s because of multi-scatter propagation of each partial wave coming to the reception point along with the strong “deterministic” component [7], as it’s shown in Figure 6.17.

Figure 6.17. Mechanism of the superposition of strong “deterministic” component with an ensemble of partial, multi-scattered waves In (6.53) and (6.56) σ y is an independent parameter, which specifies the deepness of fading and may be used to adopt the graph to real data observed. Note again, the analytical expressions, given in subsection 6.1.2d, describing fast fading also include that kind of "adopting" parameter, e.g. parameter E 0 /( 2 σ ) in Rice-distribution, parameter m in Nakagami-distribution, or parameter n in quasi-Rayleigh distribution.

375

6.1.4. COMBINED DISTRIBUTION OF FAST AND SLOW FADES. SIGNAL STABILITY IN LONG-TERM OBSERVATIONS Fast and slow fades always act simultaneously, i.e. within the short-term stationarity intervals of observations the median value of the field strength remains almost constant, whereas during long-term observation periods the random variations of the median E med ( t ) become observable and are considered as slow fading (see Figure 6.3). The evaluation provided below is based on assumption, that the fast fading may be counted by introduction of the normalized random variable κ fast (t ) with the median value equal to one. Thus within the interval of stationarity (short term observation) the field strength at the reception point may be written in following form:

E (t ) = κ fast (t ) ⋅ E med .

(6.60)

If observation period is prolonged, becoming greater than the interval of stationarity, then in order to count slow random variations of the median field E med ( t ) we may introduce another variable κ slow (t ) , similar to that for the fast fading, so the fast and slow fades may be taken into account simultaneously by the following expression:

E (t ) = κ fast (t ) ⋅ κ slow (t ) ⋅ E med , slow ,

(6.61)

where E med , slow is an overall median value of the field strength for a long-term observation period, i.e. the median of the slow fading. Now we introduce the level of signal as a unitless quantity referenced to the long-term median

E med , slow :

Z (t ) = ln

E (t ) E med , slow

.

(6.62)

Then taking the logarithm from both sides, the expression (6.61) may be rewritten as

Z (t ) = X (t ) + Y (t ) ,

(6.63)

X (t ) = ln κ slow (t ) Y (t ) = ln κ fast (t )

(6.64)

where

 . 

In (6.64) X (t ) and Y (t ) represent slow and fast fading components in Np, respectively. They easily may be converted into dB by using the relation 1 Np = 8.69 dB.

376

From (6.63) one may notice, that the randomly fluctuating signal level Z ( t ) is displayed in form of the sum of two random variables, X ( t ) and Y ( t ) that are representing slow and fast fades respectively. It's known from the probability theory, that if w ( X , Y ) is the joint probability density distribution function of X and Y random variables, then in order to figure out what is the joint probability of the combination ( X , Y ) . Based on (6.63) one may realize that in XOY-plane only S-region, shown in Figure 6.18 is to be considered as the area of existence for X and Y variables.

Figure 6.18. Area S of integration in (6.69) Then having w ( X , Y ) as a joint differential probability distribution function, the integral PDF may be written

W = ∫∫ w ( X , Y ) dX dY .

(6.65)

S

As known from the probability theory course, for the independent X and Y random variables the joint differential probability density distribution function may be simplified to the product

w ( X , Y ) = wslow ( X ) ⋅ w fast (Y ) .

(6.66)

Hence, taking into account (6.63), the integration limits in (6.65) may be set as follows:

W (Z ) =

+∞

+∞

−∞

Z−X

∫ wslow ( X )

∫ w (Y ) dY dX = fast

377

+∞

+∞

−∞

Z −Y

= ∫ w fast (Y )

∫ w ( X ) dX dY .

(6.67)

slow

The convolution integral (6.67) in probability theory is called composition of probabilities, and a special symbol, " ⊗ " is used to express (6.67) in compact presentation form. Taking into account (6.11), we may rewrite the convolutions (6.67) as

W (Z ) = Wslow ( X ) ⊗ W fast (Y ) = ...

... =

+∞

+∞

−∞

−∞

∫ wslow ( X ) ⋅W fast (Z − X )dX = ∫ w fast (Y ) ⋅Wslow (Z − Y )dY .

(6.68)

The expression (6.67) for the cumulative probability distribution function in form of double integral may be transformed to the single-integral form, written for the probability density distribution functions of both, fast and slow fades. I.e. (6.67) may be rewritten as follows

w (Z ) =



1



∫ w ( X ) ⋅ w (Z − X ) dX = ∫ w (Y ) ⋅ w (Z − Y ) dY . slow

fast

fast

−∞

(6.68a)

slow

−∞

It’s may be shown, that if one of the functions, w fast (Y ) or wslow ( X ) is even (symmetric about vertical axis), then (6.68a) may be written as

w (Z ) =





∫ w ( X ) ⋅ w (Z + X ) dX = ∫ w (Y ) ⋅ w (Z + Y ) dY , slow

fast

fast

−∞

(6.68b)

slow

−∞

which means, that in this care of symmetry both expressions, (6.68a) and (6.68b) may be applied to either sum of two random variables, Z = X + Y , or to their difference, Z = X − Y . Note: (6.68a) type convolutions may easier be calculated based upon the fact that the Fourier transforms of both functions, w fast (Y ) and

wslow ( X ) are just to be multiplied.

--------------------------------------------------------------------------------------------------------Example 6.1 Consider a combined distribution of fast and slow fades if both are log-normally distributed with σ 1 and σ 2 , standard deviations respectively. Note that this is typical case for the atmospheric optical communication links (not for RF links). Find the joint probability distribution function as well as its standard deviation. ∞ 1

Expressions



∫ w ( X ) dX = W (Z − X ) , or ∫ w (Y ) dY = W (Z − X ) are employed here. slow

Z −Y

slow

fast

Z−X

378

fast

Solution The expressions probability distribution differential functions for the logarithmic variables of fast and slow fades are presented the in the first column of Table E6.1, with proper Fourier transforms in the second column. The expression for the Fourier transform of the joint distribution density is a product:

F (ξ ) = F1 (ξ ) ⋅ F2 (ξ ) =

(

)

 ξ2  1 σ 12 + σ 2 2  , exp − 2π  2 

(E6.1)

As seen from (E6.1) the inverse transform is also expected to be a log-normal, and may be easily obtained from [1] with same form as w1 ( X ) , or w 2 (Y ) , and with the standard deviation

σ 2 = σ 1 2 + σ 2 2 , dB

(E6.2)

Table E6.1. Probability density distributions

Fourier cosine transforms [1, p.1127]

w1 ( X ) =

 X2 exp − 2 2π σ 1  2σ 1

   

F1 (ξ ) =

1

 ξ 2σ 1 2 exp − 2 2π 

   

w 2 (X ) =

 X2 exp −  2σ 2 2π σ 2 2 

   

F2 (ξ ) =

1

 ξ 2σ 2 2 exp − 2 2π 

   

1

1

----------------------------------------------------------------------------------------------------------------

The expressions (6.68) and (6.68a) allow determining the long-term probability distribution by combining lognormal distribution for slow fades, and one of the distribution models given in section 6.1.2 for fast fades. Unfortunately for RF links any combinations between fast and slow fading models may not be evaluated in closed form and, therefore, only numerical evaluations are applicable. The graphs of several combined distributions are shown in Figures 6.19 – 6.21.

379

Figure 6.19. Combination of lognormal and m-distributions

Figure 6.20. Combination of lognormal and n-distributions

380

Figure 6.21. Combination of lognormal and Rayleigh distributions

To generalize the above statements one may consider three, four and more independently acting fading mechanisms instead of two: the problem is to obtain the resultant distribution function if the cumulative distribution functions (W1, W 2, … W N) due to different independent mechanisms of fades are known. The following approach may be proposed, based on (6.68): first W 1 must be composed with W 2, then the result with W 3 and so on , i.e.

W = { [ (W1 ⊗W2 ) ⊗ W3 ]........ ⊗W N } .

(6.69)

The approaches presented in this subsection may be considered as a background for the assessments of the RF link stability.

--------------------------------------------------------------------------------------Example 6.2 Assume, E ( t ) = Eth = 4.3 µV/m is a minimum value of the electric field the receiver is able to “sense” (threshold value) for the conditions described in Example 5.5. Fast fading of the signal has Nakagami statistics with parameter m = 1.5, and slow fading is log-normally distributed with parameter σ = 8 dB. Estimate the stability of communication link. Solution If the output power of the transmitter is chosen to induce the long-term median electric field at the reception point equal to threshold, E med , slow = Eth then apparently from (6.66) follows Z = 0 , which, regarding to Figure 6.19 results in 50% of stability. For the considered case

381

E med , slow = 136 µV/m, therefore Z = 20 log( E / E med , slow ) = − 30 dB. From Figure 6.19 for the given parameter m = 1.5 we may determine approximately: W = 99.9 % (Answer) As seen, the power margin of 30 dB results in fairly high communication link stability

----------------------------------------------------------------------------------------

Now consider the case, when the power margin is such that received signal stability is high enough, like it is demonstrated in Example 6.2. It means only a very deep fades may result in signal outage, when it drops down the threshold level. The reason is a significant difference between threshold and long-term median fields. Now rewrite (6.68) for the combined probability distribution function:

W(Z)=

+∞

∫ w ( X ) ⋅W (Z − X )dX . slow

(6.70)

fast

−∞

As seen, in case of deep fades, i.e. for large negative values of Z the cumulative distribution

W fast (Z − X ) is fairly close to unity (100%) in wide range of variations of the argument X , where the contribution of wslow ( X ) is significant, the value of W fast may approximately be considered as a constant and taken out of the integral. Then based on (6.67), i.e. Z − X = Y , the expression (6.70) may be rewritten as

W (Z ) ≈ Wslow ( X ) ⋅ W fast (Y )

(6.70a)

in other words the convolution (6.70) may be replaced by the product (6.70a) of CPDF-s of slow and fast fades, with acceptable accuracy. The same approach may be applied to the case when more than two independent fading mechanisms remain in force. From (6.11) that shows the relation between IPDF and CPDF we may obtain:

P = 1−W .

(6.71)

P represents the probability of dropping down the fixed value of the signal (e.g. down the threshold value). Based on (6.70a) CPDF for two independent fades may be rewritten as

Wresult ≈ W1 ⋅ W2 = (1 − P1 ) ⋅ (1 − P2 ) = 1 − (P1 + P2 ) + P1 ⋅ P2 = 1 − Presult , hence

Presult = P1 + P2 − P1 ⋅ P2 .

(6.72) (6.73)

The last term is the probability of simultaneous appearance of two drop-downs, caused by two different independent mechanisms. Under the assumption P1 1 is a modulation index; α = 2 ⋅ (1 + m FM + Wideband frequency modulation (FM)

m FM ) = ∆f / F ;

∆f dev is a maximum deviation of carrying

3 2 m FM α 4

frequency; F is an initial, modulating frequency; ∆f is width of FM signal spectrum

Figure 6.34. Demonstration of SNR threshold for FM signal (both axes are in logarithmic scale). Note: if the input SNR is less than threshold then FM becomes unacceptable

Significant improvement in SNR makes FM systems widely used for broadcasting, and for VHF and microwave communication applications. That improvement is achieved for expense of the signal bandwidth, 2 ⋅ ∆f max = α ⋅ F which becomes much wider than that for linear types of modulation such as AM, DSB-SC and SSB. If SNRDet , out for detector’s output, as well as the allowable width of signal spectrum is pre-assigned (either by existing standard, or it is userdefined) then SNRDet , in for detector’s input becomes linked to those values. That allows defining receiver’s sensitivity, based on developments given in 6.2.5.

409

The modems such as AM, DSB-SC, SSB, and FM are just examples of how detection of the signal may affect both, Tx-Rx structure design, and link budget calculations. Complete analysis of all types of analog and digital modems is behind the scope of this text, and may be subject for additional reading.

6.3.2. USE OF SPREAD-SPECTRUM DISCRETE SIGNALS For the special class of discrete signals called spread-spectrum signals a significant improvement in SNR may be achieved when a matched-filtering reception 1 is used. The idea of the matched-filtering was first introduced by D.O. North in 1943 [22], and further evaluated in late 40-s and early 50-s [23]. Below is a brief, heuristic outline of the basics of the matchedfiltering. Consider a pulse signal s (t ) of duration T and of the arbitrary waveform that is applied to the input of the linear two-port network resulting in output response s out (t ) of duration Tout . The goal of the matched filter (MF) 2 is to “squeeze” (compress) the signal from input to output with maximum possible compression as shown in Figure 6.35. Then the envelope of the output

Figure 6.35. Signal waveforms at input/output of the matched filter

pulse presumably is to be sin(t ) / t -shaped with duration

Tout ≈ 1 / ∆f ,

(6.118)

where ∆f is a bandwidth of the input signal. For the proper reason ∆f remains the same from input to output of MF. Considering MF is lossless network, the total energy of the signal remains unchanged from input to output, i.e. 1

Optimum filtering or correlation detection terms are also in use.

2

Not to be confused with abbreviation for Medium Frequency (see Chapter 1)

410



E =

2 2 ∫ s (t )dt = T ⋅ s =

−∞



∫s

2 out

(t )dt = Tout ⋅ s out , 2

(6.119)

−∞

2

where s 2 and s out are the values of mean power 1 of input and output signals respectively. Then, taking into account (6.118), the following relation between input and output RMS voltages may be written:

sout , rms s rms

=

sout s

2

2

=

T = Tout

T ∆f =

B,

(6.120)

where the time-bandwidth product B = T ∆f is called pulse compression ratio when having a deal with discrete signal. It shows how many times the pulse duration decreases from input to output of the matched filter, and as a result, how many times increases the peak power of the input signal (not the energy of the signal). As shown below in presence of white Gaussian uncorrelated noise, which equally affects both, input and output signals the ratio (6.120) is nothing but the ratio of SNR from input to output, i.e. it is a SNR gain for this type of transform. The values of that gain may become significant, especially for the radar applications, where the value of B may achieve 1000, and even more. In order to develop a structure of the matched (optimal) receiver that maximizes SNR consider a signal s (t ) of the complex spectrum

S ( jω ) = S (ω ) ⋅ exp[ jΦ S (ω )]

(6.121)

that passes through MF along with the Gaussian white noise n(t ) of the spectral density N 0 . In (6.121) S (ω ) is the amplitude and Φ S (ω ) is a phase spectrums of the input signal respectively. Let MF has a transmission function in complex from:

H ( jω ) = H (ω ) ⋅ exp[ jΦ H (ω )] ,

(6.122)

where H (ω ) and Φ H (ω ) are amplitude and phase transmission coefficients of MF. If one needs to superimpose all spectral components of signal at the output with the same phase (constructive superposition) at any time moment t 0 , then the signal’s phase variations vs.

1

Actually

2

s 2 and s out are the mean squares of the voltages within the pulse durations. They turn to

become mean powers if the voltage drops are considered on unit resistances of 1 Ohms. This assumption does not put any restrictions on further considerations

411

frequency Φ S (ω ) must be compensated by MF in order to obtain the phase spectrum of output signal in the form of (−ω t 0 ) . Hence Φ S , out = Φ S (ω ) + Φ H (ω ) = −ω t 0 , therefore

Φ H (ω ) = −Φ S (ω ) − ω t 0 .

(6.123)

Indeed, the output response of MF will appear at the time moment of dΦ S , out / dω = − t 0 , i.e. it is shifted from start-point of the input signal by t 0 . As pointed, the condition (6.123) allows maximizing the output signal by superimposing all spectral components at the time moment t 0 . To minimize the noise intensity we may assume that MF is “matched” to amplitude spectrum of signal, i.e.

H (ω ) = a ⋅ S (ω ) ,

(6.124)

where a is a real constant. By combining (6.123) and (6.124) the transmission function of MF may be expressed as a complex conjugate Sˆ ( jω ) of the spectrum of primary signal, i.e.

H ( jω ) = a ⋅ S (− jω ) ⋅ exp(− jω t 0 ) = a ⋅ Sˆ ( jω ) ⋅ exp(− jω t 0 ) .

(6.125)

For further evaluations the following statements from the theory of Fourier transforms must be recalled: •

The Fourier transform of convolution of two functions is a product of the Fourier transforms of proper functions, i.e. ∞

∫ s(t ′) ⋅ h(t − t ′) ⋅ dt ′ ⇔ S ( jω ) ⋅ H ( jω )

(6.126)

−∞



If in expression (6.126) the replacement H ( jω ) = Sˆ ( jω ) is applied, then

S ( jω ) ⋅ Sˆ ( jω ) ⇔



∫ s(t ′) ⋅ s(t ′ − t ) ⋅ dt ′

= ψ (t )

(6.127)

−∞

it becomes an autocorrelation function for s (t ) . •

Time shift property:

exp(− jω t 0 ) ⋅ S ( jω ) ⇔ s (t − t 0 ) .

(6.128)

Based on (6.126) - (6.128) the expression for output signal may be written as

S out ( jω ) = S ( jω ) ⋅ H ( jω ) ⇔ ∞

⇔ s out (t ) = a ⋅ ∫ s (t ′) ⋅ s[t ′ − (t − t 0 )] ⋅ dt ′ = a ⋅ψ (t − t 0 ) , −∞

412

(6.129)

where ψ (t − t 0 ) is the autocorrelation function of the input signal 1. It may be seen, that for the time moment t = t 0 the expression (6.129) becomes ∞

s out (t 0 ) = aψ (0) = a ∫ s 2 (t ′) dt ′ = a E ,

(6.130)

−∞

i.e. the maximum value of the output signal at the time moment t = t 0 is directly proportional to the total energy of input signal (see Figure 6.36).

Figure 6.36. Optimum reception of spread-spectrum signal: a). Block-diagram of the signal correlation processing in matched (optimal) receiver, b). Input signal, c). Output signal

Finally let’s derive the SNR gain based on above analyses. For the input SNR, the signal power may be defined from (6.119) as PS , in = E / T = s 2 . The input white Gaussian noise spectral density N 0 is evenly distributed in frequency range ∆f = f max − f min along the entire spectrum of the signal, thus the input noise power is PN , in = N 0 ∆f . Then taking into account B = T ∆f we may write

SNRin =

1

s2 = E / ( N 0 B) . N 0 ∆f

(6.131)

(6.129) is known as Wiener-Khinchin theorem, which states, that Fourier transform of the signal power

spectrum,

S 2 (ω ) = S ( jω ) ⋅ Sˆ ( jω ) is equal to autocorrelation function of the signal.

413

For the output SNR the noise power spectral density may be defined if N 0 is multiplied by power transmission function of the MF from (6.124): N out (ω ) = N 0 a 2 ⋅ S 2 (ω ) . Then the total output noise power may be defined by integrating the output noise power density within the effective frequency band:

PN , out =

1 2π

f max

∫ N out (ω ) dω =

f min

1 2π

f max

∫N

0

a 2 S 2 (ω ) dω =N 0 a 2 E.

(6.132)

f min

Taking into account (6.130) and (6.132) the output SNR may be represented as

SNRout =

PS , out PN , out

= E / N0 .

(6.133)

Hence, the SNR gain is found as

χ=

SNRout = B. SNRin

(6.134)

An important conclusion from (6.134): matched (optimal) reception of the spread-spectrum signals allows achieving significant gain in SNR by increasing the base of signal; thus the proper gain will allow detecting the signals, which are completely buried within the noise. The following two types of signals with the large pulse compression ratio B are widely used in modern radars and communication systems: •

“Colored” pulses, with linear frequency modulation (FM) waveform (Figure 6.37a ); those signals are sometimes called "chirp" signals



Phase-coded pulses, with several jumps of carrier’s phase between 0 and 1800 (Figure 6.37b) within signal’s duration

Figure 6.37. Examples of spread-spectrum signals: a). “Colored” pulse of linear FM waveform (chirp pulse), b). Four-element phase-coded pulse

414

In those two types of signals the energy is “spread” along a wide frequency range. The reader may obtain the details about practical utilization of the compression for those types of signals in [23 - 24].

6.3.3. DIVERSITY RECEPTION TECHNIQUE An alternative for improvement of the reception reliability is diversity reception technique (DRT). The use of DRT allows decreasing of the fast fade depth by altering its statistics. In other words the same communication stability may be achieved for less power fade margin for particular propagation conditions compared to that for non-DRT reception. The idea of DRT is based on the use of two or more copies of the same signal (Signal-1, Signal2, etc.) sent to the reception point through different statistically-independent tracks, followed by merging those signals in receiver to decrease fading depth. Those independent tracks are called diversity branches. The independent (uncorrelated) copies of the signal are merged in receiver by using different techniques to achieve less communication breaks. Statistically independent copies of the signal have different fluctuation patterns, so thus if one of them falls down the desired SNR-level, the others most likely will remain above that level, so the proper combination of independent copies of signal may result in decrease of probability of outages, and therefore in increase of transmission channel reliability, as illustrated in Figure 6.38.

Figure 6.38. Illustration of the fade depth decrease while combining of two statistically independent copies of signal by using DRT technique

415

The methods of achieving statistically-independent diversity branches are: •

Space diversity, when different spatially separated receiving antennas are used: the separation between antennas (that is called diversity base, L – see figure 6.39a) must be greater than the spatial correlation distance, which is usually taken as L ≥ (5 ÷10 ) λ



Polarization diversity, when two orthogonally polarized radio waves of the same frequency and propagation path are used to carry different copies of the same signal; this type of diversity branches are fairly effective for HF radio links



Angle diversity, when different angles of arrivals of the radio waves are used to form the independent copies of the signal; for instance if a dish antenna is used, then multiple diversity branches may be formed by using multiple feeds, shifted off the focal point perpendicular to the axial line of dish reflector



Frequency diversity, when different carrying frequencies are used: the separation between carrying frequencies must be large enough in order the copies to become uncorrelated



Time diversity, when the signal is repeated two or more times with the time shift that is greater, than fading correlation interval in time domain; the gain in reception quality is achieved by sufficient decrease in transmission rate



Rake technique. When the pulse-signal arrives the reception point being carried by the radio waves propagating along different paths, then results in time-series of the same single pulse at the reception point. Those series are the result of different time delays between different propagation paths; shift-summation (discrete convolution) of those series may allow overlaying those pulses at a specific time moment, and therefore improve SNR at the receiver’s output.

Last two of listed approaches are used mostly for digital data transmissions links, when the discrete signals are transmitted; those two approaches are considered in details in [15]. Simplified block-diagrams of DRT-systems that allow realizing those approaches are shown in Figure 6.39.

416

Figure 6.39. Simplified block-diagrams of basic DRT-s (two-fold DRT cases): a). Space diversity, b). Polarization diversity, c). Angle diversity, d). Frequency-diversity

An important issue is how to combine the copies of signal at the reception point in order to achieve maximum efficiency in SNR, or maximum communications stability. Consider two separate diversity branches with u1 (t ) and u 2 (t ) signal voltages mixed with the additive noise voltages u N1 (t ) and u N 2 (t ) in each channel, i.e.

1 − st branch 2 − nd branch

u1 (t ) + u N 1 (t ),   u 2 (t ) + u N 2 (t ) . 

(6.135)

Assume, that the mixes, coming from diversity branches, are combined by adding to each-other with weighting coefficients a1 and a 2 respectively, i.e. the resultant output voltage is

U out (t ) = u out (t ) + u N , out (t ) = a1 [u1 (t ) + u N 1 (t )] + a 2 [u 2 (t ) + u N 2 (t )] .

(6.136)

The power carried by the total wanted signal at the output is in direct proportion to the following mean-square

417

u out ( t ) = [a1 ⋅ u1 (t ) + a 2 ⋅ u 2 (t )] 2

2

(6.137)

whereas the power carried by total unwanted noise at the output is in direct proportion to

u N , out (t ) = a1 ⋅ u N 1 2

2

2

(t ) + a2 2 ⋅ u N 2 2 (t ) .

(6.138)

Here u N 1 (t ) and u N 2 (t ) are the mean-square noise voltages in channel-1 and channel-2 2

2

respectively. The difference in presentations (6.137) and (6.138) comes from the fact, that u1 (t ) and u 2 (t ) are strongly correlated in contrast to u N1 (t ) and u N 2 (t ) that are uncorrelated processes therefore their mean product becomes equal to zero. Then for the output SNR the following expression may be written:

SNRout =

u out

2

u N , out

2

=

[a1 ⋅ u1 (t ) + a2 ⋅ u 2 (t )] 2 . 2 2 2 2 a1 ⋅ u N 1 (t ) + a 2 ⋅ u N 2 (t )

(6.139)

Now we assign signal-to-noise ratios

u1 (t )

u N 1 (t ) 2

u 2 (t ) 2

2

SNR1 =

SNR 2 =

, and

(6.140)

u N 2 (t ) 2

to each channel before we combine them. Then, taking into account (6.140), the expression (6.139) may be rewritten as

a1 ⋅ SNR1 ⋅ u N 1 + 2a1 a 2 ⋅ u1 (t ) ⋅ u 2 (t ) + a 2 ⋅ SNR2 ⋅ u N 2 2

SNRout =

2

2

a1 ⋅ u N 1 + a 2 ⋅ u N 2 2

2

2

2

If a1 ⋅ SNR2 ⋅ u N 1 − 2a1 a 2 ⋅ u1 (t ) ⋅ u 2 (t ) + a 2 ⋅ SNR1 ⋅ u N 2 2

2

2

2

2

.

(6.141)

is added and subtracted in numerator,

then after proper simplifications the expression (6.141) may be rewritten as

SNRout

  a1 ⋅ = SNR1 + SNR2 − 

2

2  SNR1 ⋅ u N 2   . 2 2 2 2 a1 ⋅ u N 1 + a 2 ⋅ u N 2

SNR2 ⋅ u N 1 − a 2 ⋅ 2

(6.142)

In (6.142) the replacement

u1 (t ) ⋅ u 2 (t ) =

u1 ⋅ u 2 = 2

2

SNR1 ⋅ u N 1 ⋅ SNR2 ⋅ u N 2 , 2

2

(6.143)

is used based on fact that both signals, u1 (t ) and u 2 (t ) are exact scaled copies of each-other. An important conclusion can be made from (6.142), namely, maximum SNRout after combining

SNRout

max

= SNR1 + SNR2

(6.144)

may be achieved if the expression in parenthesis in (6.142) is equal to zero, i.e. if

418

a1

u N1

2

=

SNR1

a2

uN 2

2

= C = const .

SNR2

(6.145)

If (6.140) is taken into account, then (6.145) may be rewritten as

a1 ⋅ u N 1 u1

2

=

a2 ⋅ u N 2

2

u2

2

=C.

(6.146)

2

From (6.146) the unknown weighting coefficients a1, a2 … an may be found for the general case of n-folded diversity as follows

a1 = C ⋅

u1 u N1

2

2

, a2 = C ⋅

u2 uN 2

2

2

, ………. a n = C ⋅

un uN n

2

2

.

(6.147)

From (6.144) it may be concluded that maximum SNR (maximum SNR combining technique 1) may achieved if the receiver transfer function of each diversity branch is adjusted individually to keep the ratio of wanted signal (voltage or current) and unwanted noise power the same and equal to each-other before summation. One of the appropriate block-diagrams based on this algorithm is shown in Figure 6.40a.

Figure 6.40. Examples of two-folded DRT block-diagrams: (a) for maximum SNR combining, (b) for equal gain combining, (c) for switching (signal auto-selection). 1

This principle is called also weighted summation technique.

419

Each diversity branch has two AGC (automatic gain control) loops: one of them to control the signal voltage level (numerator in (6.147)), another one is to control the noise power level (denominator in (6.147)) so thus the ratio remains constant. In order to make it applicable, a pilot-tone is mixed up with the wanted signal at the transmission. This block-diagram is used in most advanced systems, which allows obtaining of the maximum SNR. The simplified version of DRT called equal gain combining is represented in Figure 6.40b. In this case all weighting coefficients are taken equal to each-other. Finally the selection combining is presented in figure.6.40c. In this case the diversity branch that has greatest value of the SNR is switched (connected) to detector’s input. This principle is sometimes called auto-selection DRT. As a particular case we’ll consider an analytical expression for output fast fades statistics of the auto-selection DRT to confirm the decrease of deepness of fading (see Figure 6.38 for the qualitative demonstration). Consider the case of n-folded diversity, when signal-1, signal-2, … signal-n from diversity branches are due to independently fading fields E1, E2, … En; each of those fields assumed to have the same cumulative probability distribution function W (E ) . Then integral probability distribution function is P(E ) = 1 − W (E ) . Because all branch signals are statistically independent, then the probability of combined signal being below the E-value at the selector’s output is the product of probabilities of individual branch signals, i.e.

Tn (E ) = T n (E ) = [1 − W (E ) ] . n

(6.148)

Thus the probability of E-value being exceeded by all diversity branch fields simultaneously (cumulative probability distribution function) becomes

Wn (E ) = 1 − Tn (E ) = 1 − [1 − W (E )] . n

(6.149)

If Rayleigh distribution is considered for W (E ) as most common CDF, that describes most deep fades of single-branch signal, then in (6.153) it may be replaced by (6.35), so n

  E 2   . Wn (E ) = 1 −  1 − exp  − 2  2 σ   

(6.150)

Expression (6.150) is referred in section 6.1.2.d as quasi-Rayleigh distribution, which provides statistical description of the non-Rayleigh type fades with variable depth. Expression (6.45) is nothing but a modified version of (6.150). The difference between those two expressions is that the parameter n in quasi-Rayleigh distribution may vary within the range ( 0, ∞ ) continuously, whereas parameter n in (6.150) may only belong to the discrete sequence of natural numbers.

420

Recall the argument of the expression (6.45), and insert the noise power PN into numerator and denominator:

 E   E med

2

 E 2 / PN E2 SNR  = γ = , = = 2 2 E med E med / PN SNRmed 

(6.151)

where the ratios are replaced by proper SNR-s taking into account that signal power is in direct proportion with proper electric field strength. Then expression (6.45) may be rewritten as

Wn (γ ) = 1 − {1 − exp[− 0.693 ⋅ q (n) ⋅ γ ]} , n

(6.152)

where γ is SNR relative to its median per expression (6.151). The values of q (n) may be taken from Table 6.2 or from Figure 6.13. As mentioned above, the expression (6.152) is only applicable to selection combining along with the graph in Figure 6.13b. In two other considered cases, namely maximum SNR combining and equal gain combining, the CDF, Wn (γ ) is expressed in terms of χ 2 -distribution function

1

(chi-square distribution) that is given in [15] in details. Graphs shown in figure 6.21 [17] may be conveniently used for the estimates of the gain in fade margin (i.e. how much the fade margin may be decreased in comparison to non-diverse reception) as a function of number of diversity folds n and for different types of DRT combining schemes.

Figure 6.41. SNR gain for n-folded diversity reception for following DRT combing schemes: 1 – Maximum SNR combining, 2 – Equal gain combining, 3 – Auto-selection

1

Do not confuse with

χ

that represents the SNR gain

421

REFERENCES 1. Gradshtein, I.S., Ryzhik, I.M. Tables of Integrals, Series, and Products. Seventh Ed., Elsevier, Inc., 2007 2. Vaughan, R., Andersen, J.B. Channels, Propagation and Antennas for Mobile Communications. IEEE, 2003. 3. Dolukhanov, M.P., Propagation of Radio Waves. Translated from the Russian by Boris Kuznetsov. Moskow, Mir Publishers, 1971 4. Rice, S. O. Mathematical Analysis of Random Noise. Bell System Technical Journal 24 (1945), pp. 46–156 5. Nakagami, M. "The m-Distribution, a general formula of intensity of rapid fading". In W. G. Hoffman, editor, Statistical Methods in Radio Wave Propagation: Proceedings of a Symposium held at the University of California, pp 3-36. Permagon Press, 1960. 6. Долуханов, М.П., Саакян, А.С., Энтина, Н.Н. Квазирелеевские замирания на коротких волнах. Известия АН Арм.ССР, серия XXVIII, Апрель, 1975. (Dolukhanov, M.P., Saakian, A.S., Entina, N.N., “Quasi-Rayleighan Faded on HF”, Proceedings of the Academy of Sciences of the Rep of Armenia, Series XXVII, April, 1975, in Russian). 7. Rytov, S.M., Kravtsov, Ya.A., Tatarskii, V.I. Principles of statistical radio-physics. Springer-Verlag, 1987. 8. CCIR. Report 322, “World Distribution and Characteristics of Atmospheric Radio Noise”, Xth Plenary Assembly, Geneva, 1963. 9. Spaulding, A.D., Washburn, J.S. Atmospheric Radio Noise: Worldwide Levels and Other Characteristics. NTIA Report 85-173, US Dept of Commerce, 1985. 10. ITU-R Recommendation P.372-7, “Radio Noise,” International Telecommunication Union, 2001 11. Smith, A.A. Radio Frequency Principles and Applications. IEEE Press, 1998 12. Uman, M.A. Understanding Lightning. Bek Technical Publications, Carnegie, PA, 1971 13. Freeman, R.L., Reference Manual for Telecommunications Engineering. V.1, WileyInterscience Publication, 2002. 14. Kraus, J.D., Radio Astronomy. McGraw-Hill Book Co. 1966. 15. Schwartz, M., Bennett, W.R., Stein, S. Communication systems and techniques. IEEE Press, NY, 1996.

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16. Weik, M.H. Communications Standard Dictionary. Van Nostrand Reinold Co., N.Y., 1983 17. Pratt, T., Bostian, C.W., Satellite Communications, John Wiley & Sons, 1986 18. Balanis, C.A., Antenna Theory, Analysis and Design, John Wiley & Sons, 2005 19. Stutzman, W.L., Thiele, G.A., Antenna Theory and Design, John Wiley & Sons, 1998 20. Ziemer, R.E., Tranter, W.H. Principles of communications. Houghton Mifflin Co., Boston, 1990. 21. Carlson A.B. Communication systems. McGraw-Hill Publishing Co., 1986 22. North, D.O., An Analysis of the Factors Which Determine Signal/Noise Discrimination in Pulsed-carrier Systems, RCA Tech.Rept. PTR-6C, June, 1943 (reprinted, Proc IEEE 51, No. 7, Jul 1963, 1016-1027; reprinted: Detection and estimation, (S. S. Haykin, ed.), Halstad Press, 1976, 10-21) 23. Cook, C.E, Bernfeld, M., Radar Signals. An Introduction to Theory and Application, Academic Press, 1967 (reprinted, Artech House, 1993) 24. Skolnik, M.I. Introduction to Radar Systems. 3-rd edition, McGraw-Hill Co., 2002 25. O’Flinn, M., Moriarty, E. Linear Systems. John Wiley & Sons, NY, 1987.

PROBLEMS P6.1. Use (6.30) to confirm it satisfies normalization condition (6.9). Express median value

E med of the Rayleigh distribution in terms of parameter σ . Answer: E med = 1.177 σ P6.2. Find the depth of the Rayleigh fading relative to its median value (E0.1/Emed)|dB – (E0.9/Emed)|dB and assess the result by comparing with that found from the graph on Figure 6.8a. Answer: 13.4 dB P6.3. The stability of HF communication radio-link is affected by log-normally distributed slow fading with the standard deviation of σ = 8 dB, and by Rayleighian fast fading. The power margin on transmitting side is to compensate both fades and to achieve communication stability of 99%. Find the value of the power margin by using two approaches: 1) As a sum of margins assessed for slow and fast fades separately from Figures 6.16 and 6.12b (m = 1) respectively, and 2) From the combined distribution graphs shown in Figure 6.19 (m = 1) Which one is larger? Why? Explain in your own words.

423

Answer:

1). 20 dB + 18.5 dB = 38.5 dB 2). 25 dB

P6.4. For the probability density distribution function (PDDF) of two-ray random interference that is given by (6.18) find the mean value of fluctuating electric field E expressed in terms of

E1 and compare with median value given by (6.21). x2

Hint: For any random variable x mean value is defined as x =

∫ x w ( x) dx ; for the considering

x1

case x1 = 0 , x 2 = 2 E1 . Answer: E =

4E1

π

P6.5. A three-stage passive filter has a transformation coefficients of the stages, K1 , K 2 , and

K 3 respectively. Based on (6.91) and (6.92) show that the overall noise parameters for the filter may be represented as N = 1 /( K1 K 2 K 3 ) , and T = T0 ( N − 1) .

P6.6. In Example 6.3 for the case-1 scenario calculate the minimum value of the gain for LNA if predefined (required) value of the system noise temperature for the RF-unit is T = 800K ? Answer: 11.47 dB P6.7. Confirm (6.142) by modifying (6.141).

P6.8. Estimate the increase in maximum detectable radar range if sounding monochrome pulse signal (case-1) is replaced by linear-FM spread-spectrum signal with time-bandwidth product equals B = T ∆f = 50 , and optimal filtering is applied (case-2). Transmitter’s EIRP and the system’s bandwidth are assumed to remain unchanged. Hint: use expressions (3.48a), (6.104a), and (6.134). Take amplitude modulation depth for monochrome pulse equal 100% (mAM = 1). Answer: increased 2.94 times P6.9. For the sufficient reception quality of the satellite TV broadcast FM-signal a required signal-to-noise ratio at the detector’s output is SNRDet, out = 26.2 dB (416.9 unitless) 1. Standard TV video signal spectrum is limited to the maximum frequency of F = 4.5 MHz. System’s RF

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bandwidth is ∆f = 30 MHz (same as the width of FM signal spectrum). System noise temperature is Tsys = 145K. Based on this data make the following estimates: •

Modulation index m FM and detector’s SNR gain χ (Table 6.6)



Minimum input signal power PRx , min required to meet the value of χ (6.104a)



Compare PRx , min with the power received in real conditions, PRx if signal power density at the reception point (magnitude of Poynting vector) is -110 dBW/m2 ( Π = 10 pW/m2). A dish antenna of diameter d = 0.6 m with the aperture efficiency ν = 0.67 is used at the receiver’s input.



Find the power margin ∆PRx = PRx / PRx , min in dB

What changes will take place if the modulation index m FM is increased? Explain. 1

Note: In real systems an additional 18.8 dB improvement in SNRDet, out is

achieved due to technical modifications that are not discussed here; so the resultant SNRDet, out is 26.2+18.8 = 45 dB. For details refer to Pratt,T., Bostian,C.W., Jeremy,E.A., Satellite Communications, John Wiley & Sons, 2003 Answer: m FM = 4.44 , χ = 20 dB, PRx , min = -126 dBW, PRx = -117.2 dBW, ∆PRx = 8.8 dBW

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LIST OF SYMBOLS AND ABBREVIATIONS A

Diffraction loss Parameter in Okumura-Hata propagation model









A , A′

Attenuations in Ionosphere (e.g., AΣ , AΣ , AD , AD , AE , AE , AF 1 , AF 1 , AF 2 )

A

Magnetic vector potential, Wb/m

ADT

Asymptotic diffraction theory

AM

Amplitude modulation

a

Major axis of the ellipse Earth’s average geometric radius (6370 km)

B

Time-bandwidth product for spread-spectrum signal Parameter in Okumura-Hata propagation model

B

Vector of the magnetic field induction, or magnetic flux density, T

BF 2

Specific attenuation in ionospheric reflecting layer F2 (at 1 MHz)

b

Minor axis of the ellipse

C

Parameter in Okumura-Hata propagation model

CF

Fresnel cosine integral



Structural constant of the tropospheric turbulences

CPDF

Cumulative probability distribution function

CW

Continuous wave

c

Speed of light in vacuum ( 3 ⋅ 10 8 m/s)

D

Antenna directivity Parameter in Okumura-Hata propagation model

D

Vector of the electric field induction (Electric flux density), C/m2

Ddiv

Divergence factor

E

Parameter in Okumura-Hata propagation model

E med

Median electric field strength, V/m

E

Average electric field strength, V/m

E

Vector of the electric field strength, V/m

EIRP

Effective isotropic radiated power, W 426

ELF

Extremely Low Frequency (< 3 kHz)

EHF

Extremely High Frequency (30 – 300 GHz)

e

Electron’s charge, 1.6 ⋅10

erf

Error function

erfc

Complementary error function

emf

Electro-motive force

F

Complex propagation factor

F

Force, N

F

Modulating frequency, Hz

FA

Atmospheric noise factor (relative brightness temperature)

FNBW

Fist null beam width of the antenna radiation pattern, rad (or degrees)

FM

Frequency modulation

f

Carrier frequency, Hz

∆f

RF signal bandwidth, Hz

−19

C

Receiver’s passband, Hz

G

Antenna gain

H (0)

LOS path clearance without atmospheric effect, m

H (N )

Total path clearance on LOS link in presence of the atmospheric refraction, m

H

Vector of the magnetic field strength, A/m

Hh

Natural unit of heights, m

HF

High Frequency (3 – 30 MHz)

HPBW

Half power beam width of the antenna radiation pattern, rad (or deg)

h

Height, elevation, m Planck's constant = 6.626 ⋅ 10 − 34 J ⋅ s

I

Current, A



Radiation intensity, W/sr

IEEE

Institute of Electrical and Electronics Engineers

IF

Intermediate Frequency

IPDF

Integral probability distribution function

ITU-R

International Telecommunication Union – Radio Communication Sector

i=

Imaginary unit

−1

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J

Vector of electric current spatial density, A/m2

JS

Vector of conducting current surface density, A/m

K

Transmission coefficient of TPN

k

Relative distance of the reflection point on LOS microwave radio link

k

Propagation constant (complex), 1/m

kB

Boltzman’s constant [ 1.38 ⋅10 −23 W /( K ⋅ Hz ) ]

L

Total propagation path loss

LF

Propagation path loss in real conditions

L0

Free space (reference) propagation loss Outer scale of tropospheric turbulences

Lmsd

Multiple screen diffraction loss in urban area, dB

Lrst

Natural unit of distance in ADT approach, m

Lr

Roof-to-street diffraction loss, dB

LF

Low Frequency (30 – 300 kHz)

LHCP

Left Hand Circular Polarization

LOS

Line-of-Sight

l eff

Antenna effective length, m

l0

Inner scale of tropospheric turbulences

M

Parameter in Lee propagation model Large parameter in ADT approach

MF

Medium Frequency (0.3 – 3 MHz) Matched filter

MUF

Maximum usable frequency

m

Parameter in m-distribution of Nakagami

me

Mass of the electron ( 9.1 ⋅10

N

Refractivity

−31

kg)

Noise figure

Ne

Plasma concentration, 1/m3

NT

Noise power spectral density, W/Hz

n

Refraction index 428

Parameter in Lee propagation model Parameter in Quasi-Rayleigh distribution (n-distribution)

n

Unit vector normal to surface

P

Power, W

PM

Phase modulation

PN

Noise power, W

PS

Signal power, W

P(E )

IPDF for the electric field fades

P

Polarization vector of the unit volume, C/m2



Radiated power, W

PTx

Power transmitted (applied to antenna’s input), W

PRx

Power received (from the antenna’s output), W

PDDF

Probability density distribution function

PEC

Perfect Electric Conductor

p (m)

Parameter in m-distribution of Nakagami

q

Earth’s ground parameter in ADT approach

q

Scattering vector

q (n)

Parameter in n-distribution

R

Horizontal distance between corresponding antennas along the earth surface, m Radius, m

Rant

Real part of the antenna input impedance, Ohm



Antenna radiation resistance, Ohm

RHCP

Right Hand Circular Polarization

RMS

Root mean square value of any variable

RF

Radio Frequency

r

Direct distance between two points, m

S

Area of the surface, m2

S eff

Area of the antenna effective aperture, m2

SF

Fresnel sine integral



Spatial spectrum of the autocorrelation function for dielectric permittivity

429

SHF

Super High Frequency (3 – 30 GHz)

SNR

Signal-to-noise ratio

s

Distance scale, 1/m

T

Temperature (including noise temperature), K Tesla (Unit for the magnetic field induction/ flux density)

T E

Electric field transmission coefficient (complex)

T H

Magnetic field transmission coefficient (complex)

TP

Power transmission coefficient

TEM

Transverse Electromagnetic

TPN

Two-port network

t

Time, s

U (x)

Attenuation factor in ADT approach

UHF

Ultra High Frequency (0.3 – 3 GHz)

V

Volume, m3

V ( y)

Height-gain function in ADT approach

VLF

Very Low Frequency (3 – 30 kHz)

VHF

Very High Frequency (30 – 300 MHz)

v

Velocity of the radio wave, m/s

W (E )

CPDF for the electric field fades

W

Intrinsic impedance (complex), Ohm

We

Ionization energy, J Spatial density of the energy of electric field, J/m3

Wm

Spatial density of the energy of magnetic field, J/m3

WLAN

Wireless local area network

w (t )

Airy function

w (E )

PDDF of the electric field fades

X

Normalized distance in ADT approach

X ant

Imaginary part of the antenna input impedance, Ohm

x

Cartesian coordinate Relative electric field strength (

x

x = E / E med )

Numerical distance in W&VdP method for ground wave propagation 430

x0

Unit vector in Cartesian system

y

Normalized height in ADT approach Cartesian coordinate

y0

Unit vector in Cartesian system

Z ant

Antenna input impedance, Ohm

z

Cartesian coordinate

z0

Unit vector in Cartesian system

α

Attenuation coefficient, Np/m (or dB/m) Angle, rad

β

Phase coefficient of the radio wave, rad/m Angle, rad Parameter in generalized Rayleigh distribution function (Rice distribution)

Γ E

Electric field reflection coefficient

Γ H

Magnetic field reflection coefficient

ΓP

Power reflection coefficient

γ

Slant angle (elevation angle)



The allowable average height of the surface roughness

δε

Loss angle of the dielectric medium, rad

ε

Complex relative dielectric permittivity

ε0

Absolute dielectric permittivity of free space, (1 / 36π ) ⋅10 − 9 F/m

ξ

Number of collisions in ionospheric plasma, 1/s

ξε

Relative fluctuation of the dielectric permittivity ( ξ ε = ∆ε / ε )

η ant

Antenna efficiency

Θ

Geocentric angle, rad

θ

Zenith angle in spherical coordinates, rad (or dergree)

κ fast

Fast fading factor

κ slow

Slow fading factor

λ

Wavelength, m

µ

Relative magnetic permeability

431

µ0

Absolute magnetic permeability of free space (vacuum), H/m

ν

Antenna aperture efficiency

Π

Pointing vector (power flow spatial density of the radio wave), W/m2

ρ

Electric charge volumetric density, C/m3

σ

Conductivity of the medium, S/m Standard deviation of the normally distributed random variable

σ sc

Scattering cross-section of the turbulent troposphere (per unit volume), 1/m

σ RCS

Scattering cross-section of the target in radar applications, m2

Φ

Phase of the radio wave, rad Scalar magnetic potential, V

ΦF

Phase of propagation factor, rad

ΦΓ

Reflection phase, rad

ϕ

Azimuth angle in spherical coordinates Angle of incidence

χ

Detector’s SNR gain

Ψ

Amplitude radiation pattern Angle of transmission of the radio wave on the interface between two media

ψε

Spatial autocorrelation function of the random dielectric permittivity



Solid angle, sr

ω

Angular frequency, rad/s



Nabla vector operator

∇2

Laplacian Operator

432