Mode-stirred reverberating chamber autocorrelation ... - IEEE Xplore

IET Science, Measurement & Technology Special Issue on Recent Developments on the Use of Reverberation Chambers for Testing Wireless Systems

Mode-stirred reverberating chamber autocorrelation function: model, multifrequency measurements and applications

ISSN 1751-8822 Received on 29th November 2014 Revised on 13th March 2015 Accepted on 18th March 2015 doi: 10.1049/iet-smt.2014.0350 www.ietdl.org

Antonio Sorrentino ✉, Angelo Gifuni, Giuseppe Ferrara, Maurizio Migliaccio Dipartimento di Ingegneria, Università degli Studi di Napoli Parthenope, Centro Direzionale Isola C 4, 80143 Napoli, Italy ✉ E-mail: [email protected]

Abstract: In this study, a simple model for the autocorrelation function (ACF) within a reverberating chamber (RC) is developed and verified. In particular, in a well-stirred RC when n mechanical stirrers operate, the ACF of the total field is given by the product of n ACFs obtained when the n stirrers are singularly operated. Hence, a parameterisation of the ACF is accomplished. Accordingly, it is possible to parameterise the coherence time, T C, of the RC emulated wireless propagation channels. Experiments accomplished at the RC of the Università degli Studi di Napoli Parthenope, formerly Istituto Universitario Navale, confirm the soundness of the proposed model and its operational effectiveness.

1

Introduction

The reverberating chamber (RC) is a metallic room, in which different stirring techniques continuously change the boundary conditions randomising the electromagnetic field. The most used technique is mechanical tuning, which consists of one or more metallic stirrers rotating within the chamber [1] but other techniques such as frequency, polarisation, platform and source stirring techniques are also employed [2–4]. The goal of the above-mentioned stirring techniques is to give rise to an electromagnetic environment, that is, on average uniform and isotropic. Initially, the RC has been used as a high-field amplitude test facility for electromagnetic interference and electromagnetic compatibility [5]. According to the well-known first-order statistics of the electromagnetic field, several applications using the RC have been developed. In particular, the use of an RC as a reliable, repeatable and controllable environment for emulating wireless propagation channels and testing the performance of wireless systems has become increasingly popular in the past few years. Applications have been developed that use the RC as a free-field test-site of wireless systems for both wireless channels and devices [6–21]. Although first-order statistics of the electromagnetic field are well-known, the RC modelling is an open problem in the RC community especially in wireless applications where studies on the rate of change of RC geometry and hence on the second-order statistics of the electromagnetic field are in progress. Second-order statistics are concerned with the field’s rate of change rather than the field itself. In particular, the autocorrelation function (ACF) of the received field expresses the correlation between a field at a given time and its value at some delay ‘t’ later. If the ACF is evaluated, by fixing a threshold for the ACF, it is possible to evaluate the coherence time, TC, of the propagation channel emulated within the chamber [13]. It must be noted that the above-mentioned application is of importance to study the propagation problems of modern communication systems [22, 23]. Second-order statistics have been most commonly specified in the frequency domain by means of the spectrum of the received field. As matter of fact, according to the mechanical stirrers, the Doppler effect within an RC has been investigated but no formula for analytically modelling the received spectrum has been provided [24, 25]. Recently, a first model was developed in [26], where an

IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 5, pp. 547–554 & The Institution of Engineering and Technology 2015

empirical formula on the spectrum of the RC electromagnetic field is accomplished for the first time. By proper setting the shape parameter of the empirical model, a desired spectrum of the total received field can be obtained [26]. Nevertheless, in [26] no explanation is given about how several stirrers are combined to provide a desired spectrum and hence a proper field correlation. Although the spectral analysis is a useful tool to characterise the received field in an RC, for real-life applications and in particular wireless applications, the characterisation of an RC field in the time domain by means of the ACF method is with a direct impact. It directly gives a measure of the time dispersion of the emulated propagation channels, that is, the maximum interval over which the emulated propagation channel gains are relatively constant. Hence, it directly provides the TC [13]. In practice, in a well-randomised RC when n different continuous mechanical stirrers are operated (but the rationale can be extended also at other continuous stirring techniques), the total field is given by the product of the n contributions relative to the stirrers operating within the chamber. Hence, the ACF of the total field is given by the product of the n ACF obtained when the stirrers are individually operated. This means that, in a well-stirred RC, by parameterising the stirrer actions, it is theoretically possible to obtain every kind of electromagnetic field with a desired ACF. Following this rationale, in this paper, based on a large set of measurements conducted at the RC of the Università degli Studi di Napoli Parthenope, formerly Istituto Universitario Navale (IUN), experimental verification of theoretical ACF of the total electromagnetic field, is achieved. Measurements have been performed at different working frequencies and different working stirrer configurations. Then, according to [13], TC of different wireless propagation channels emulated within the RC have been evaluated. It is important to note that a sort of parameterisation of the IUN RC has been first accomplished in [26] where the spectrum of the total electromagnetic field is achieved. Such spectrum is obtained by means of the average periodogram approach that reduces the variance of the estimate of the final periodogram by averaging together several independent periodograms [27]. By increasing the number of periodograms for a fixed record length of data decreases not only the variance of the final periodogram but also the spectral resolution [27], that is, a meaningful parameter for the spectrum characteristics estimation. The smaller is the spectral resolution the worst is the spectrum estimation and hence the worst is the RC parameterisation.

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Moreover, the technique proposed in [26] analyses the spectrum of the electromagnetic field obtained when all the stirrers are simultaneously operated within the chamber. In this paper, a direct evaluation of the ACF of the field is accomplished to have a full temporal and hence spectral resolution for the evaluation of the RC parameterisation. The most important novelty of this paper is that the estimate of the ACF of the total field obtained when n stirrers are operated within the RC is achieved by using the n ACFs estimates of the fields obtained when the stirrers are individually operated. Summarising, the technique proposed in this paper outlines two key points: † on one side, in a well-randomised RC where n mechanical stirrers are operating, the ACF of the total field can be evaluated as the product of n ACFs obtained when the n stirrers are individually operated. Then, by comparing time- and frequency-domain [26] results obtained with two different techniques, it is possible to express the ACF in terms of a shape parameter and † on the other side, since the TC is a function of the ACF, by parameterising the ACF allows expressing TC in terms of the shape parameter, hence it allows parameterising the TC of the propagation channels emulated within the RC. TC values obtained from measurements in RC have been applied for measuring the modulation error ratio (MER) on signals with different digital modulations transmitted through different emulated propagation channels to analyse the quality of the transmission [10]. Experimental results confirm the soundness of the proposed model and the effectiveness of the approach from an operational viewpoint.

statistically independent, is given by [28] RPP (t) = RWW (t) · RQQ (t)

Let us apply (5) to the RC system, that is, a linear system having an own impulse response h(t, t) [24], in which the input deterministic electromagnetic field is randomised by means of stirring techniques. When a mechanical stirrer is operated, the RC can be approximated as a linear stochastic filter, in which the rate of change of chamber geometry is function of the stirrer velocities [10, 13]. The envelope of the stirred field (stochastic process) ET(t) is more or less correlated in function of the velocity of stirrers employed. Without loss of generality, one can suppose that two stirrers operate within an RC, let us call such stirrers S1 and S2. The RC filter output depends on the operating stirrers within the chamber in a way that if E(t) is the electromagnetic field injected within the chamber, E1(t) and E2(t) are the outputs corresponding to only S1 and only S2 operating within the chamber, respectively. It must be noted that E1(t) and E2(t) are given by the elementary contributions corresponding to the number of independent positions, N1,ind and N2,ind obtained when only S1 and only S2 operating within the chamber, respectively. It is important to note that E1(t) and E2(t) are two independent stochastic processes when one stirrer, that is, only S1 or only S2 operate within the RC. When S1 and S2 stirrers operate at the same time within the RC, S1 and S2 continue performing in independent way. In such a case, the total number of independent positions, NT,ind, is given by the product of the number of independent positions obtained with only S1 and the one obtained with only S2 operated [29] NT , ind = N1, ind · N2, ind

2 ACFs of individually and simultaneously operated stirrers As it is known, in a wireless propagation channel the received field is given by a superposition of replicas of the transmitting field each one with own amplitude, frequency, phase and time delay. Then, at receiving antenna, a time series of random variable, that is, w(ti), is present. Hence, a random process is present at receiving antenna. In other words, if we focus on the ensemble of values taken at an arbitrary collection of n fixed time instants, t1, t2, …, tn for arbitrary n, we are dealing with a stochastic process. The most widely used random process models have a special structure that allows computation of such a statistical specification. In particular, when we deal with linear systems, we analyse the stochastic processes by considering only the first and the second moment processes, that is, [28]

mW (ti ) = E[W (ti )] = mW

(1)

RWW (t) = E[W (t + t)W (t)] = E[W (t)W (t − t)]

(2)

where μW(·) and RWW(·) are the mean and the ACF of the stochastic process W(ti), t is the delay time, that is, the difference time between two consecutive instants, ti and tj. Equation (1) states that the statistical mean of the W(·) process is not time dependent. When a sum of two stochastic processes, that is, P(t) = W(t) + Q(t), is present, the ACF, RPP(t) is given by RPP (t) = RWW (t) + RWQ (t) + RQW (t) + RQQ (t)

(3)

(6)

Note that (6) is a theoretical upper limit for the number of independent positions within an RC. Such limit can be practically achieved when the RC is large in terms of wavelength and when the mechanical stirring employed within the chamber is really efficient (note that the term ‘efficient’ is used for indicating a received field totally decorrelated within the RC when the antennas are not faced each other, also when a single stirrer is individually operated within the RC). Suppose that (6) holds on. When S1 and S2 are operated together, that is, at the same time within the chamber, the total field obtained, that is, ET(t), can be written as ET (t) = E1 (t) · E2 (t)

(7)

It must be noted that (7) is satisfied if the following condition is verified ET2 (t) ET2 (t)

E12 (t) · E22 (t) = 2 ET2 (t)

(8)

Hence, E1(t) and E2(t) are two statistically independent processes either when S1 and S2 operate alone or when they operate concurrently within the chamber. In Fig. 1 a schematisation of the RC system when S1 and S2 are operated one at time and together, is shown. Following this rationale and according to (5), the ACF of two independent processes is given by RET ET (t) = E[E1 (t)E2 (t)E1 (t + t)E2 (t + t)]

that is, reduced to RPP (t) = RWW (t) + RQQ (t)

(4)

if the two processes are statistically independent. In this last case, when a product of two stochastic processes, that is, P(t) = W(t)·Q (t), is present, the ACF of P(·), RPP(t), when W(t) and Q(t) are

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(5)

= E[E1 (t)E1 (t + t)E2 (t)E2 (t + t)] = RE1 E1 (t) RE2 E2 (t)

(9)

where the E before the square brackets is the statistical average symbol. Equation (9) states that the ACF of ET(t) is given by the product of the single ACF of the two processes generating ET(t). IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 5, pp. 547–554 & The Institution of Engineering and Technology 2015

Fig. 1

Sketch of the RC system when two stirrers, S1 and S2, are operated

a Singularly b At the same time

Equations (7)–(9) can be generalised to n stirrers operating within the RC. This model is verified in Section 4 on a large set of RC data.

3

Coherence time

Second-order fading statistics give information about how rapidly the field level changes between different levels. This distribution is most commonly specified by the spectrum of the field. In general, the time variation of the channel which arises from the transmitting and the receiving antennas motion or from the motion of the objects through the propagation path with fixed antennas causes a Doppler

Fig. 3 Inner of the IUN RC used for measurements

shift in the frequency of the received field. Since several propagation paths between the transmitting and the receiving antennas are in place and each of them is related to an elementary field contribution, several elementary fields are present at receiving side, each one with an own phase and frequency [23]. Therefore a Doppler effect on the receiving side is obtained. Accordingly, a Doppler spread phenomenon on the spectrum of the received field is achieved. Doppler spread phenomenon causes the spreading of the bandwidth of the received field with respect to its transmitted bandwidth. The overall spectral width is called Doppler bandwidth. The maximum value of frequency present in the spectrum of the received field is called Doppler spread, that is, fm. It is a measure

Fig. 2 Simulated (continuous line) and theoretical (dashed line) ACF for different TC a 2.232 ms b 0.089 ms

IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 5, pp. 547–554

& The Institution of Engineering and Technology 2015

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of the spectral broadening caused by the rate of change of the channel and is defined as the range of the frequencies over which the received spectrum is essentially non-zero, that is Sd ( fm ) . 0

(10)

where Sd( fm) is the Doppler spectrum of the electromagnetic field. The variations of the field bandwidth, caused by the Doppler spread phenomenon, are strictly related, by means of a Fourier Transform, to the ACF of the complex envelope of the received field, that is, REE(t). The ACF of the received field defines how each time variant environment changes and hence how the field decorrelates over the time. It is important in the study of wireless propagation channels since it is directly related to the TC of the channel. Accordingly, TC and fm are strictly related each other. TC is one of the most important parameter that describes the varying nature of the wireless propagation channels. It is the time-domain dual of Doppler spread and it is a statistical measure of the time duration over which the channel response is essentially invariant. TC is the time duration over which the received field has a strong amplitude correlation [22]. The received field is a strong amplitude correlation when its ACF remains close to unity for a certain time interval. In other words, REE(t) ≅ 1 for TC ≃ 1/fm where the fm is the Doppler spread. It must be noted that in a modern communication systems if the symbol period is quantitatively

greater than TC, then the channel will change during the symbol period thus causing distortion at the receiving antenna. TC is evaluated by the ACF of the received field by means of a fixed threshold. In the literature, several values for the ACF threshold are present [23]. In this paper, according to Sorrentino et al. [13], a threshold value equal to 1/√2, that is, 0.707, is chosen. Hence, TC value is defined by [13] TC =

9 16p fm

(11)

In Fig. 2, two coupled theoretical and simulated ACFs for a TC equal to 2.232 and 0.089 ms, that is, for fm equal to 80.5 and 2012 Hz, are shown. The simulated ACF has been obtained by employing the sum-of-sinusoids model given in [30]. The black curves are relative to the theoretical ACF and the grey curves to the simulated ones for two different TC. The simulation results and the theoretically calculated results show a good agreement. Note that with the same input parameters, the ACF rapidly drops as the coherence time decreases.

4

Experimental results

In this section, a meaningful set of experimental results are shown. Experiments are conducted at RC of the IUN, which has been

Fig. 4 ACF obtained from measurements at IUN RC Working frequency is 10 GHz a S1–S2–S3 b S2–S3 c S1–S2 d S1–S3 stirrers operated within the RC Marker shows the coherence time evaluated in the corresponding case

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Table 1 TC and β values for different couples of stirrers employed within the IUN RC RC configuration

Frequency, GHz

Shape parameter values

TC, ms

MSE

S1–S2–S3 operated S2–S3 operated S1–S2 operated S1–S3 operated S1–S2–S3 operated S2–S3 operated S1–S2 operated S1–S3 operated

10

45 × 10−4

0.28

1.42 × 10−4

10 10 10 1.8

50 × 10−4 72 × 10−4 100 × 10−4 30 × 10−4

0.31 0.44 0.53 0.45

2.43 × 10−4 3.05 × 10−4 4.87 × 10−4 8.71 × 10−4

1.8 1.8 1.8

33 × 10−4 46 × 10−4 49 × 10−4

0.49 0.7 0.74

9.57 × 10−4 1.27 × 10−3 1.7 × 10−3

successfully employed as emulator of real-life wireless propagation channels [10, 11, 13, 16–19]. The IUN RC is an 8 m3 metallic chamber wherein three mechanical stirrers are present. The first one (S1), placed on the left of the entrance door, has a rectangular shape of 1.84 m × 0.45 m; the second stirrer (S2) and the third stirrer (S3) have a Greek-cross shape. The S2 has bars of 1.84 m × 0.25 m; it is placed in front of the entrance door. The S3 stirrer has bars of 1.2 m × 0.18 m and it is placed in the ceiling. S1, S2 and S3 stirrers can rotate in continuous mode with maximum speeds of 190, 390 and 320 rate per minute, respectively. Two EMC Test System (ETS)-Lindgren double ridged waveguide horn in the 1–18 GHz frequency range are

employed. Moving a distance of 0.3 m from walls and stirrers, the working volume (WV) considered exhibits an area of about 2 m2 and a volume of about 3 m3 [15]. The square shape of the WV is well suited to the channel conditions emulated in the RC. Fig. 3 shows a photograph of the IUN RC interior used for measurement configurations. The measurements can be divided into two main sets: in the first one, a monochromatic signal is transmitted in the chamber. The scattering coefficient S21 is measured by using the Agilent Technologies vector network analyser. For every measurement, a data set of 16 001 samples is acquired in 3.66 s, that is, at sample time of 0.229 ms. Measurements are performed at frequencies from 1.8 to 10 GHz in order to be independent from the used wavelength: to save space and to have a direct comparison with the measurements in [26] only measurements at 10 and 1.8 GHz are presented and discussed. In the second set of measurements, in order to apply the experimental results given by a monochromatic signal to a modern communication systems, a modulated phase shift keying (PSK) and a quadrature amplitude modulation (QAM) signals used into the Universal Mobile Telecommunication Systems (UMTSs) modulation are transmitted through the different propagation channels emulated within the RC. Such modulations, based on low-bit-rate (PSK) and high-speed packet data (QAM) are employed to emulate within the IUN RC the real propagation conditions on UMTS systems [31, 32]. Hence, an MER is measured. A National Instrument controller (NI PXI 1042Q) is used to generate and to receive the digital systems employed. In all the measurements, the same antenna configuration is

Fig. 5 Same of Fig. 4. The working frequency is 1.8 GHz a S1-S2-S3 b S2-S3 c S1-S2 d S1-S3 stirrers operated within the RC


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Fig. 6 PSK constellations from the emulated wireless propagation channels for TC equal to a 0.45 b 0.49 c 0.7 d 0.74 ms

employed, see Fig. 3. The transmitting and the receiving antenna are co-polarised and are placed about 1 m above the floor. In the first set of measurements, three stirrers are operated together inside the chamber. The working frequency is 10 GHz. In Fig. 4a, the correlation of the output RC electromagnetic field is shown. The blue dotted one is relative to the acquired data when the three stirrers are together operated, that is, at the same time. The red curve is relative to the combined ACF, that is, obtained by data acquired in three different moments. In particular, measurements are performed when only S1, only S2 and only S3 is operated, and then combined (multiplied) together to provide the autocorrelation of the total field, according to the theoretical model proposed in Section 2. A good agreement between the two curves is shown providing that the IUN RC is a well-stirred chamber and that the stirring process in the RC is a multiplicative process in which the ACF of the total electromagnetic field is given by the product of the single ACF evaluated when S1, S2 and S3 stirrers are individually operated. The third curve, that is, the black dotted one, is the empirical ACF. Such ACF is obtained as the inverse Fourier transform of the empirical spectrum of the RC field given in [26]. It is important to note that the inverse Fourier transform is numerically accomplished. Once again, a good agreement is shown. Hence, the shape parameter technique of the empirical model given in [26] can be applied to the ACF obtained from measurements. The minimum square error (MSE) is evaluated, see Table 1. Note that the MSE evaluated for the ACF method are

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smaller with respect to the corresponding ones given in [26] witnessing that an improved accuracy in evaluating the second-order statistics of the electromagnetic field within an RC is achieved. Furthermore, the TC of the wireless propagation channel emulated within an RC is evaluated for each emulated case, see Fig. 4. Each TC was parameterised by means of the shape parameter of the corresponding ACF. TC in the case of three stirrers operated within the RC is equal to 0.28 ms. Such a TC value is associated to a shape parameter value equal to 45 × 10−4. The values of TC and of the shape parameter corresponding to the ACF evaluated in the emulated propagation channels are listed in Table 1. In Figs. 4b–d, the autocorrelation of the output RC field is shown for two stirrers operating together (S2S3, S1S2 and S1S3, respectively) within the chamber. Once again, the agreement among the three curves is good. It must be noted that in Fig. 4d the shape parameter given in [26] has a value greater than the one given by the corresponding ACF measurements, see Table 1. The MSE of such a case is equal to 0.05 × 10−2 much smaller than 0.78 confirming that a better accuracy in evaluating the second-order statistics of the electromagnetic field in an RC is accomplished with the ACF method. In Fig. 5, the autocorrelation of the RC field is shown for a working frequency of 1.8 GHz. In each figure, the agreement among the three curves (measured, combined and empirical ACF) is again good showing the effectiveness of the employed method by an operational point of view. The TC and the shape parameter IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 5, pp. 547–554 & The Institution of Engineering and Technology 2015

Fig. 7 QAM constellations from the emulated wireless propagation channels for TC equal to a 0.45 b 0.49 c 0.7 d 0.74 ms

values are listed in Table 1. Moreover in this case, the MSE values of the comparison between the measured and empirical ACFs are smaller with respect to the ones given in [26] witnessing that a better accuracy is accomplished. Last but not least, the coherence time values evaluated from the monochromatic field are applied to the PSK and QAM modulations in order to evaluate the quality of transmission in the emulated wireless propagation channels. The PSK is mainly used for signalling logical/physical channels where a low-bit-rate is required. The QAM is adopted for high-speed packet data services, such as the voice service, which are deeply affected by the lack of coherence because of the multipath amplitude incoherence. In both cases, because of the NI PXI 1042 Q, the channel bandwidth is 1

MHz. In Fig. 6 and 7, the quadrature-PSK (QPSK) and the 16-QAM constellations for the different propagation channels emulated within the IUN RC are shown. Although a visual analysis cannot discriminate the transmission quality of the QPSK and QAM modulations through several propagation channels, a quantitative analysis shows that the smaller is the TC emulated in RC the smaller MER applies, see Table 2. Note that differently to global system of mobile cases presented in [13], the QPSK modulation is more robust to the RC environments, that is, it is more robust to the most hostile wireless propagation channels emulated within the chamber, when all stirrers are operated together. On the other hand, the QAM signal is substantially affected by the lack of coherence because of multipath environments emulated in RC. The MER values of the QAM modulation are negative indicating that a very poor quality of transmission applies. All MER values are listed in Table 2.

Table 2 MER values for the PSK and QAM through the propagation channel emulated within the IUN RC RC configuration

Frequency, GHz

TC, ms

MER–PSK, dB

MER–QAM, dB

S1–S2–S3 operated S2–S3 operated S1–S2 operated S1–S3 operated

1.8

0.45

2.3

−1.49

1.8 1.8 1.8

0.49 0.7 0.74

2.45 3.26 4.05

−0.65 −0.16 −0.13


5

Conclusions

In this paper, a simple model for the ACF is proposed and verified on IUN RC measurements. In summary, it can be observed that † in a well-randomised RC where n stirrers operate in independent way, the ACF is obtained by the product of the single ACFs achieved

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by measuring the field in n single measurements done with only one stirrer operating; † by comparing the results obtained here with the corresponding one given in [26] a parameterisation of the ACF is obtained. Moreover, a better accuracy in the shape parameter evaluation with respect to [26] is applied; and † following the rationale of the previous points, that is, by parameterising the ACF, it is possible to parameterise the TC. This is of fundamental importance in the emulation of the wireless propagation channels since allow to reproduce the time variant characteristics of the real wireless propagation channel by simply setting the shape parameter. The experiments undertaken over a large set of measurements confirm that the proposed technique is operationally effective and attractive.

6

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