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antenna system transfer function (ASTF). A network parameter analysis method is presented for transfer func- tion characterization. Transfer functions based on ...

RADIO SCIENCE, VOL. 41, RS1002, doi:10.1029/2005RS003287, 2006

Characterization of ultrawideband antennas using transfer functions Xianming Qing, Zhi Ning Chen, and Michael Yan Wah Chia Institute for Infocomm Research, Singapore

Received 16 May 2005; revised 19 September 2005; accepted 11 October 2005; published 20 January 2006.

[1] Transfer functions are useful parameters for ultrawideband (UWB) antenna

characterization. The response of an antenna system to any excitation can be completely determined in terms of the transfer functions (or the ‘‘impulse response’’ in time domain) of the antennas. In this paper, the transfer functions are defined as a transmitting antenna transfer function, a receiving antenna transfer function, and an antenna system transfer function to characterize UWB antenna systems readily and accurately. The transfer functions are expressed by ABCD and S parameters when the antenna system is considered as a two-port network. As examples, an antipodal Vivaldi antenna and a disc cone antenna are investigated to validate the proposed method. The measured results are further verified by XFDTD simulation. Citation: Qing, X., Z. N. Chen, and M. Y. W. Chia (2006), Characterization of ultrawideband antennas using transfer functions, Radio Sci., 41, RS1002, doi:10.1029/2005RS003287.

1. Introduction [2] Since the 1970s, ultrawideband (UWB) technology has been widely investigated and developed for wireless communications applications [Ross, 1973; Bennett and Ross, 1978; Harmuth, 1981; Carin and Felsen, 1995]. Recently, much effort has been devoted to commercial UWB systems. UWB-based systems offer opportunities for high-resolution radar imaging; rejection of multipath cancellation effect; transmission of high data rate signals; coding for security; and low probability of intercept, especially in multiuser network applications [Win et al., 1997; Cramer et al., 2002; Welborn and McCorkle, 2002; Win and Scholtz, 1998]. For example, UWB-based systems transmit and receive temporally short pulses without carrier or modulated short pulses with carriers. Practically, short pulses mean wide spectra; that is, the bandwidth may exceed at least 25% of the nominal center frequency. The Federal Communications Commission has opened the spectrum from 3.1 to 10.6 GHz, that is, a bandwidth of 7.5 GHz, for unlicensed use with a limited emission level of 41.25 dBm/MHz [Federal Communications Commission (FCC), 2002]. It allows UWB-based systems to span the entire spectra and to use this band like an industrial, scientific, and medical band.

Copyright 2006 by the American Geophysical Union. 0048-6604/06/2005RS003287

[3] UWB-based systems usually require antennas that feature acceptable performance over a wide operating band. This results in special design and measurement considerations for UWB antennas [Chen et al., 2004]. Therefore the classical methods of antenna characterization are not enough to provide a clear portrayal of the UWB antenna’s behaviors. Several new methods, such as radiation energy pattern, peak amplitude pattern, slop pattern, fidelity, transfer function (impulse response in time domain), group delay, and effective height [Harmuth, 1984; Farr and Baum, 1992; Allen et al., 1993; Lamensdorf and Susman, 1994; Atchley et al., 2003] have been used to describe the characteristics of the UWB antennas. [4] This paper describes the method for characterizing UWB antenna systems by using transfer functions, namely, a transmitting antenna transfer function (TATF), a receiving antenna transfer function (RATF), and an antenna system transfer function (ASTF). A network parameter analysis method is presented for transfer function characterization. Transfer functions based on voltage ratio definition are described by ABCD parameters and S parameters when the antenna system is considered as a two-port network. By means of the network parameter method, the transfer functions of an UWB antenna system can be measured in frequency domain accurately and then can be assessed in time domain with the aid of inverse Fourier transform. This methodology is verified by experiments and simulation. Two case studies are carried out to show the advantage of this method.

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Figure 1. Transmitting-receiving antenna equivalent circuits: (a) antenna system, (b) transmitting equivalent circuit, and (c) receiving equivalent circuit.

2. Definition of Transfer Functions [5] For an UWB-based transmitting-receiving link, the received signal cannot be obtained as a product of power terms; it should be done in terms of transfer functions [Mayo et al., 1961]. The radiation field or output response of an antenna system can be conveniently predicted by using transfer functions. Kanda [1980] used the transfer function to investigate the transmitting and receiving characteristics of a resistively loaded linear antenna, a transverse electromagnetic horn, and a conical antenna. Lamensdorf and Susman [1994] classified the transfer function as transmitting antenna transfer function, receiving antenna transfer function, and overall signal transfer function. Recently, Mohammadian et al. [2003] presented vector transfer functions, which take antenna polarization into account, to characterize the transmitting and receiving UWB antenna. We will adopt this definition in this paper. [6] Consider a transmitting-receiving antenna system as shown in Figure 1. The transmitting antenna transfer ~ TA(w, q, j), is defined as the ratio of function (TATF), H the distance normalized electric far field, E(w, q, j) = ejkR , at the spatial test point to the input E(w, q, j, R)/ R signal, Vin(w), at the input of the transmitting antenna. ~ RA(w, The receiving antenna transfer function (RATF), H

q, j), is the ratio of the output signal of the receiving antenna, Vout(w, q, j, R), to the incident field, E(w, q, j, R). The antenna system transfer function (ASTF), H(w, q, j, R), is the ratio of the output signal, Vout(w, q, j, R), to the exciting signal, Vs (w). The definitions are expressed as follows: ~ TA ðw; q; jÞ ¼ HTA ðw; q; jÞRTA ¼ Eðw; q; jÞ H Vin ðwÞ Eðw; q; j; RÞ jkR ¼ Re Vin ðwÞ

ð1Þ

~ RA ðw; q; jÞ ¼ HRA ðw; q; jÞRRA ¼ Vout ðw; q; j; RÞ ð2Þ H Eðw; q; j; RÞ H ðw; q; j; RÞ ¼

Vout ðw; q; j; RÞ ; Vs ðwÞ

ð3Þ

where w = 2pf, f is the operating frequency, k is the free space wave number, (q, j) is the orientation, and R is the distance between the transmitting antenna and receiving antenna; RTA and RRA are unit vectors that indicate the polarization direction of the transmitting antenna and receiving antenna, respectively. ~ TA (w, q, j) and H ~ RA (w, q, j) are [7] The functions H frequency- and orientation-dependent and are determined

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Figure 2. Equivalent ABCD parameters of the antenna system. by an antenna’s inherent properties, such as geometry, material, and operating mode of the antenna. They are more useful in the UWB radar system to analyze the characteristics of the illuminating pulses from the transmitting antenna or the echo from the target [Taylor, 1995]. On the other hand, the function H(w, q, j, R), which describes the overall characteristics of a transmitting-receiving antenna link, is suitable for UWB communication systems characterization. The function H(w, q, j, R) is not only a function of operating frequency and antenna orientation but also a function of distance ~ RA between the antennas. According to the definition, H ~ TA(w, q, j) and (w, q, j) is measured in meters, while H H(w, q, j, R) are dimensionless. [8] From Figure 1, we have Vout ðw; q; j; RÞ ¼

¼

3. Expressions of Transfer Functions by Network Parameters

Zl ðwÞ ~ RA ðw; q; jÞ H Zl ðwÞ þ Zin2 ðwÞ  Eðw; q; j; RÞ Zl ðwÞ ~ RA ðw; q; jÞ H Zl ðwÞ þ Zin2 ðwÞ jkR ~ TA ðw; q; jÞ e H Vin ðwÞ R

Vin ðwÞ ¼

Zin1 ðwÞ Vs ðwÞ: Zin1 ðwÞ þ Zs

ð4Þ ð5Þ

Combining (4) and (5), the antenna system transfer function can be expressed as Zl ðwÞ ~ RA ðw; q; jÞ H Zl ðwÞ þ Zin2 ðwÞ jkR Zin1 ðwÞ ~ TA ðw; q; jÞ e H R Zin1 ðwÞ þ Zs Zl ðwÞ HRA ðw; q; jÞ ¼ Zl ðwÞ þ Zin2 ðwÞ ejkR Zin1 ðwÞ HTA ðw; q; jÞ  R  RTA : Zin1 ðwÞ þ Zs RA R ð6Þ

H ðw; q; j; RÞ ¼

[9] It is known that the transmitting transient response of an UWB antenna is different from the receiving transient response of the same antenna. Schmitt [1960], using the Rayleigh-Carson relationship, and Kanda [1986], using plane wave – scattering theory, showed that an UWB antenna’s transmitting transient response is proportional to the derivative of the receiving transient response. In other words, the ratio of the transmitting antenna transfer function of an UWB antenna to the receiving antenna transfer function of the same antenna is proportional to operating frequency [Taylor, 1995; Scheers et al., 2000]. Therefore we can assume ~ TA ðw; q; jÞ ¼ jw H ~ RA ðw; q; jÞ: H ð7Þ 2pC0

[10] Some works have reported on the transfer function characterization using S parameters. Mohammadian et al. [2003] used S21 of the equivalent network to describe the system transfer function of an UWB transmittingreceiving link. In the paper by Scheers et al. [2000], a normalized transfer function (impulse response) was presented and was characterized by S12 of a transmittingreceiving antenna system. A method to describe the transfer function by using ABCD and S parameters was presented by Qing and Chen [2004]. [11] Referring to the transmitting-receiving antenna system shown in Figure 1, two identical antennas are in the far field of each other at a distance R and are oriented with the same polarization so that the same parts of their radiation patterns point at each other like a mirror between them. This antenna system can be considered as a two-port network and can be presented by the ABCD parameters as shown in Figure 2. [12] According to microwave network theory [Pozar, 1990; Fooks and Zakarevicius, 1990], ABCD parameters describe the relationship of the input/output voltage and current, while S parameters relate the incoming/outgoing traveling waves of a two-port network. The antenna system transfer function, which is defined by voltage

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Figure 3. Transfer function measurement setup. ratio, can be easily described by ABCD parameters. However, ABCD parameters cannot be obtained with high accuracy experimentally because it is very difficult to measure the voltage and current at microwave frequencies. On the contrary, S parameters can be easily measured by a vector network analyzer with very high accuracy. Therefore we derive the antenna system transfer functions by ABCD parameters first and then convert them to S parameter expression according to the microwave network analysis method. [13] Referring to Figure 2, the receiving antenna is connected to a receiver, which is with a load impedance Zl; the relationship between the signal at the input of the transmitting antenna and the voltage and current at the output end can be written as Vin ðwÞ ¼ AVout ðw; q; j; RÞ þ BIout ðw; q; j; RÞ ¼ AVout ðw; q; j; RÞ þ BVout ðw; q; j; RÞ=Zl ¼ ð A þ B=Zl ÞVout ðw; q; j; RÞ:

Combining (6), (7), and (10) and considering jRRA . RTAj = 1, we obtain HTA ðw; q; jÞ ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Zl ðwÞ þ Zin2 ðwÞ jw pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RejkR pffiffiffiffiffiffiffiffiffiffiffiffiffi Zl ðwÞ 2pC0 a Zc1 =Zc2 þ b Zc1 Zc2 =Zl

ð11Þ HRA ðw; q; jÞ ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Zl ðwÞ þ Zin2 ðwÞ 2pC0 jkR pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Re : pffiffiffiffiffiffiffiffiffiffiffiffiffi Zl ðwÞ jw a Zc1 =Zc2 þ b Zc1 Zc2 =Zl

ð12Þ

In most cases, the source impedance and receiver impedance are equal to the characteristic impedance of the transmission lines connecting to them, namely Zs = Zc1 = Zc2 = Zl, and the transfer functions can be expressed by the S parameters as S21 ð13Þ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2S21 jw HTA ðw; q; jÞ ¼ RejkR ð14Þ ð1 þ S11 Þð1  S22 Þ 2pC0 H ðw; q; j; RÞ ¼

ð8Þ

Then Vout ðw; q; j; RÞ 1 ¼ Vin ðwÞ A þ B=Zl 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð9Þ a Zc1 =Zc2 þ b Zc1 Zc2 =Zl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi where a = A Zc1 =Zc2 and b = B/ Zc1 Zc2 are the normalized ABCD parameters and Zc1 and Zc2 are the characteristic impedance of the transmission lines connecting the antennas to the source/receiver. [14] Substituting (5) into (9), we get the system transfer function Vout ðw; q; j; RÞ H ðw; q; j; RÞ ¼ Vs ðwÞ 1 Zin1 ðwÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : pffiffiffiffiffiffiffiffiffiffiffiffiffi Z a Zc1 =Zc2 þ b Zc1 Zc2 =Zl in1 ðwÞ þ Zs ð10Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2S21 2pC0 jkR Re : ð15Þ HRA ðw; q; jÞ ¼ ð1 þ S11 Þð1  S22 Þ jw

As seen in equation (13), the antenna system transfer function, under the above impedance constraint, can be directly described by S21 or S12 of a transmitting-receiving antenna system no matter what the antenna’s impedance match is. On the other hand, the impedance characteristic has impact on transmitting and receiving antenna transfer function.

4. Measurement of Transfer Functions [15] Generally, transfer function can be measured directly in the frequency domain. It can also be obtained by

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Figure 4. Configuration of the measured antenna prototypes: (a) antipodal Vivaldi antenna and (b) disc cone antenna.

Fourier transform from the impulse response, which is measured in the time domain. The measurement in the frequency domain is preferable because much higher accuracy can be achieved. By measuring the S parameters of a transmitting-receiving antenna system, the transfer functions can be obtained with equations (13) – (15). 4.1. Transmitting Antenna Transfer Function and Receiving Antenna Transfer Function Measurement [16 ] The measurement configuration is shown in Figure 3, where two ‘‘identical’’ antennas were placed in an anechoic chamber with sufficient separation to be in the far field of each other. The antennas were oriented with same polarization so that the same parts of their radiation patterns point at each other. The S parameters of the antenna system were measured by an Agilent 8510C vector network analyzer. The frequency range is from 45 MHz to 30 GHz, with a step of 37.5 MHz. [17] Two points should be noted in the measurements. The first is that one must ensure that the measured antennas are ‘‘identical.’’ It is similar to the antenna gain measurement by using two identical antennas. We should check the impedance match and radiation patterns of the antennas and select two prototypes with the closest features as the measured samples. The second is that a phase unwrap should be done before the square root is taken from a complex number in (14) and (15); otherwise, the results will be nonphysical. The origin of the task comes from the fact that only the wrapped phase values can be measured or directly computed. The wrapped phase values are restricted to the interval (p, p). Phase unwrapping is the task of finding the true absolute phase values, and there are different approaches to the problem [Ghiglia and Pritt, 1998; Tribolet, 1977].

4.2. Antenna System Transfer Function Measurement [18] The measurement configuration of the ASTF is the same as the TATF and the RATF measurement. One antenna was used as transmitting and the other as receiving; the two antennas should be with sufficient separation in the far field. Antenna polarization was kept the same in our measurement. The S21 of the transmitting-receiving antenna system can be measured by vector network analyzer, and the ASTF was then calculated by (13).

5. Results and Discussions [19] To validate the proposed method, two types of UWB antennas are designed and measured. First, a

Figure 5. Measured TATF of antipodal Vivaldi antenna (on boresight).

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Figure 6. Measured TATF of disc cone antenna (at q = 90, j = 0). directional antipodal Vivaldi antenna is taken into account. Then, the measurement on an omnidirectional disc cone antenna is carried out. Last, an antipodal Vivaldi antenna– disc cone antenna link is discussed. The configurations of the antipodal Vivaldi antenna and the disc cone antenna prototypes are illustrated in Figure 4. The antipodal Vivaldi antenna is fabricated on a low-loss substrate (RO4003, er = 3.38, tan d = 0.002, and thickness of 0.8128 mm). The return loss, gain, and radiation patterns are verified to ensure that the antenna prototypes are a pair of identical antennas. 5.1. Transmitting Antenna Transfer Function [20] After obtaining S parameters of the antenna system, HTA (w, q, j) can be calculated with equation (14). Figures 5 and 6 show the measured TATF of the antipodal Vivaldi antenna (on boresight) and the disc

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cone antenna (at q = 90, j = 0), respectively. The TATF of the antipodal Vivaldi antenna shown in Figure 5 illustrates flat amplitude and linear phase response over 2.5– 19.0 GHz, which is important for pulse transmitting. Those pulses, which have dominant spectra over the above frequencies, will be little distorted by the antenna. Beyond 19.0 GHz, the amplitude shows more variation, while the phase becomes chaotic, which indicates that the reflections of the signal from the ends of the antenna cause more dispersion at higher frequencies. The TATF of the disc cone antenna is illustrated in Figure 6. Flat amplitude and linear phase response are obtained over the bandwidth from 3.0 to 12.5 GHz, and more variation of amplitude and chaotic phase occurs beyond 12.5 GHz. [21] Referring to Figures 5 and 6, the directional antipodal Vivaldi antenna prototype shows larger amplitude and more phase variation cycles. More phase cycles are reasonable: Because the antipodal Vivaldi antenna is a slow-wave structure [Gazit, 1988], the signal will take more time to go through the antenna before it radiates out. [22] After obtaining the TATF, the radiation fields can easily be calculated by multiplying the specified input signal, Vin (w), the transmitting transfer function, HTA (w, q, j), and the distance factor, ejkR/R. The waveform of radiation field is then obtained by calculating the inverse Fourier transform [Van de Vegte, 2002; Wu and Chen, 2004]. The input signal is a Gaussian monocycle as shown in Figure 7. The result for the antipodal Vivaldi antenna (on boresight, 1 m away) is shown in Figure 8a and is indicated as a measurement result. Similarly, the result for the disc cone antenna is shown in Figure 8b. The waveform is similar to the first-order derivative of the input Gaussian monocycle. The antipodal Vivaldi antenna prototype shows little distortion to the waveform

t 2 Figure 7. Gaussian monocycle in time and frequency domains, V(t) =  e(t/s) (s = 35 ps): s (a) waveform and (b) spectrum. 6 of 10

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t 2 Figure 8. Waveform of the radiation field with a Gaussian monocycle exciting, Vin (t) =  e(t/s) s (s = 35 ps): (a) antipodal Vivaldi antenna (on boresight, 1 m away) and (b) disc cone antenna (at q = 90, j = 0, 1 m away).

of radiated pulse because the transmitting transfer function shows wider bandwidths of flat amplitude and linear phase characteristics. [23] So far, no commercial software can provide the simulation results for transmitting antenna transfer function or receiving antenna transfer function as defined above. Therefore the measured transfer functions cannot be verified directly with simulated results. However, the radiation field of an antenna with a pulse exciting can be easily calculated by some methods, such as finite difference time domain used in an electromagnetic simulator XFDTD [Remcom, Inc., 2003]. The methodology mentioned above can be verified by comparing the radiation field on the basis of the measured transmitting antenna transfer function and the simulation result by the

XFDTD. The results shown in Figure 8 are in very good agreement.

Figure 9. Measured RATF of antipodal Vivaldi antenna (on boresight).

Figure 10. Measured RATF of disc cone antenna (at q = 90, j = 0).

5.2. Receiving Antenna Transfer Function [24] HRA (w, q, j) can be calculated by equations (15) or (7), which is relative to HTA (w, q, j) . Figures 9 and 10 show the measured RATF of the antipodal Vivaldi antenna (on boresight) and the disc cone antenna (at q = 90, j = 0), respectively. Comparison with the TATF and RATF shows different amplitude characteristics. For the antipodal Vivaldi antenna, the amplitude of the RATF linearly decreases from 2.5 to 19.0 GHz. However, the phase characteristic of the RATF is similar to the TATF. The linear phase response can be observed over the above frequencies.

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Figure 11. Waveform of the output signal: (a) antipodal Vivaldi antenna (incident field is on boresight) and (b) disc cone antenna (incident field is at q = 90, j = 0). The RATF of the disc cone antenna shows similar characteristics over 3.0 to 12.5 GHz. [25] The output response can easily be predicted by multiplying the incident field and the RATF. Assuming the incident field has the same waveform of a 35 ps Gaussian monocycle, the waveform of the output response of the antipodal Vivaldi antenna is shown in Figure 11a; the pulse is duplicated with little distortion. Similarly, the results for the disc cone antenna are shown in Figure 11b.

ASTF and output response of a pair of antipodal Vivaldi antennas, a pair of disc cone antennas, and an antipodal Vivaldi antenna-disc cone antenna link are shown in Figures 12, 13, and 14, respectively. The separation between the antennas is 2 m. The response of the antipodal Vivaldi antenna pair shows higher amplitude and less ringing. The simulation results of the ASTF are not available for comparison because the simulation consumes much time and computer memory.

5.3. Antenna System Transfer Function

6. Conclusions

[26] For UWB communications application, antenna system transfer function of a transmitting-receiving UWB antenna link will be more useful. The measured

[27] This paper has presented a method to characterize UWB antenna systems by using transfer functions, namely, a transmitting antenna transfer function, a re-

Figure 12. Measured ASTF and output waveform of a pair of antipodal Vivaldi antennas (exciting: Gaussian monocycle, s = 35 ps): (a) measured ASTF and (b) output response. 8 of 10

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Figure 13. Measured ASTF and output waveform of a pair of disc cone antennas (exciting: Gaussian monocycle, s = 35 ps): (a) measured ASTF and (b) output response.

ceiving antenna transfer function, and an antenna system transfer function. Transmitting antenna transfer function and receiving antenna transfer function are more useful in UWB radar for radiated or received waveform analysis. They are determined by the antenna’s inherent properties, such as geometry, material, and operating mode. Antenna system transfer function characterizes the overall response of an UWB transmitting-receiving link; it takes both the effects of the antennas and distance factor into account and is more meaningful for UWB communications system design. The transfer functions

have been described by ABCD parameters and S parameters when the antenna system is considered as a twoport network. Two types of UWB antenna have been investigated to validate the methodology. The measured results have shown good agreement with XFDTD simulation. In addition, the investigation has shown that the bandwidth of the flat amplitude and linear phase response of the transfer function is very important for pulse transmission. The antipodal Vivaldi antenna, which has wider bandwidth, shows less distortion to the waveform of pulse.

Figure 14. Measured ASTF and output waveform of antipodal Vivaldi antenna – disc cone antenna link (exciting: Gaussian monocycle, s = 35 ps): (a) measured ASTF and (b) output response. 9 of 10

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Z. N. Chen, M. Y. W. Chia, and X. Qing, Institute for Infocomm Research, 20 Science Park Road, 02-21/25 TeleTech Park, Singapore 117674. ([email protected])

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