Statistical Uncertainty Budget in a Reverberation Chamber

Transcription

1 ADVANCED ELECTROMAGNETICS SYMPOSIUM, AES 2012, APRIL 2012, PARIS FRANCE Statistical Uncertainty Budget in a Reverberation Chamber Philippe Besnier, Christophe Lemoine, Abdou Khadir Fall Université Européenne de Bretagne, INSA, IETR, UMR CNRS 6164, 20 avenue des buttes de Coësmes F RENNES, France *corresponding author, philippe.besnier@insa-rennes.fr Abstract This communication deals with the a priori estimation of the statistical uncertainty budget in a reverberation chamber. Under the assumption of an ideal random field, given a sample of N independent measurements, the theoretical performance of a stirring process is defined from the starting point of the acquisition identically and independent measurement data. However, in a real reverberation chamber, there is no proof for independence of individual measurements although we may check for correlation within data. A direct method may consist in quantifying the empirical uncertainty of the statistical first moment. According to the central limit theorem, this follows a gaussian probability density function. In fact, given an appropriate estimation of the standard deviation of the estimation of the moment of the parent distribution, thus uncertainty budget may be evaluated. This paper discusses some experiment results that deals with either frequency stirring or mechanical stirring procedures. These measurements tends to prove that frequency stirring is an easier way to provide as many independent measurements as anticipated from correlation analysis between consecutive realizations (frequency steps). However, mechanical stirring may not operate with the same efficiency. 1. Introduction Reverberation chambers are nowadays used for many applications ranging from EMC measurements (immunity testing) to Radio Frequencies applications. A reverberation chamber operates as an oversized electromagnetic cavity where some modifications of the boundary ce onditions and/or excitation are able to drastically modify the field distribution in a quasi-random way [1][2]. Finally, a reverberation chamber measurement is based on the post-processing of a set of currents, voltages measured in one or different places in the working volume of the chamber. Postprocessing techniques consist in providing a statistic of this sample which is mainly obtained through a selected stirring procedure. The stirring process must be good enough to provide a stochastic sequence of measured values. Thus, in principle, if one gets a sample of size N iid made up of statistically independent measurements the theoretical uncertainty budget may be determined. As far as the first moment estimation is concerned, it follows the central limit theorem [3]. As predicted from the plane wave spectrum theory, the underlying probability density functions (pdfs), at least in a well-oversized cavity, is either gaussian or that of a combination of normal (Gaussian) independent variables depending on the physical parameter under investigation. Numerous experimental results confirm this prediction of the plane wave spectrum theory. For example, a rectangular projection of the electric field follows a Rayleigh distribution. Therefore, the estimation of the first moment of this projection would be given by : Ê p Norm (µ, σ Niid ) (1) where µ and σ are, respectively, the moment and the standard deviation of the underlying Rayleigh distribution, in this case related to each other as σ 0, 52µ and p Norm is the pdf of a Gaussian distribution. Therefore, in a well oversized reverberation chamber the statistical uncertainty budget would be only dependent on the size sample, N iid. This sample may be obtained through any stirring process (mechanical stirrer, frequency hopping or receiver/transmitter stirring) or a combination of these. The common method for choosing an adequate stirring process is to check that measurements are uncorrelated. In practice, given a reasonable estimation of the correlation among individual measurements, one checks that correlation is not detected [4]. However, even a very residual correlation has been proved to break the rule given by (1) [5]. Last but not least, in case we could claim that correlation observed through a given correlation operator does not exist, it is only a necessary condition for independence, but it does not prove independence. In the case of a mechanical stirrer, we may suspect that a correlation operator would not be sensitive to some types of correlation that may appear between non adjacent positions of the stirrer [6]. The limited density of propagating modes may support this hypothesis. This paper discusses two correlation models and then introduces a discussion about some experimental results aiming at determining the uncertainty budget of the ensemble average estimation. 2. Correlation models and effective sample size Analysis of the correlation between individual realizations of a given sample is not by far en easy process. The relationship between correlation and the various physical parameters of a reverberation chamber is not well known. Various hypothesis may be considered. The stirring process

2 2 suggests that there are at least two types of correlations. A well known factor for correlation is due to an unsufficient modification of the boundary conditions of the electromagnetic field between consecutive stirrer positions, frequencies of operation, or movement of a transmitter/receiver in the RC. Correlation for these various stirring processes may be studied through a model of correlation that suppposes that equal changes induce equal correlations. It s obviously a rough approximation that may be not always consistent for example with any mechanical stirrer geometry and shape. Once the consecutive correlation between realizations is made negligible, the question of correlation is bound to the fact that the random properties of the RC are related to the number of modes involved in the stirring process. Then, a set of realizations yield to partially correlated data. This suggests a simple hypothesis of a residual and maybe uniform or average correlation. The following subsections provide a quick analysis of these two correlation models, i.e., the uniform residual correlation and the persistent correlation between consecutive realizations. Note that these models do not exclude each other Variance of the average estimation of independent data We start with X = (X 1, X 2,..., X n 1, X N ) a sample of size N individual measurements in a reverberation chamber. In the following development, one is interested to estimate the variance of ˆµ w hich is the estimation of the first moment. The variance of this estimator is indeed an indication of the statistical uncertainty budget in the reverberation chamber according to (1 ). The variance of this set of data is given by : N N var( X i ) = var(x i ) + 2 N i 1<j N cov(x i, X j ) (2) If data are supposed independent, they are therefore uncorrelated. Thus, the variance of the estimation of the first moment, ˆµ is given by : var(ˆµ) = var( 1 N N X i ) = 1 N 2 Nσ2 = σ2 N where σ 2 is the variance of the random variable X. This is the best performance one might expect from a set of N measurements is a reverberation chamber Variance of the average estimation. Uniformly correlated data In case of uniformly correlated data, we define the autocorrelation function ρ u as : (3) ρ u = cov(x i,x j ) σ 2 i, j i j (4) In that case, the variance of the estimation of the first moment becomes : var(ˆµ) = σ2 N (1 + (N 1)ρ u) (5) Thus, for a large value of N, i.e. N 1 ρ u, one gets : var(ˆµ) σ 2 ϱ u (6) Even a low residual correlation causes a limit in the reduction of the uncertainty budget of the estimation of ˆµ. This result was mentionned in a recent communication [5] as a possible cause of unexpected uncertainties in reverberation chamber measurements depending on the stirring process under investigation Variance of the average estimation. Persistent correlation between consecutive realizations This correlation model applies only if the sample is collected and ordered in such a way that the consecutive realizations are performed with small changes in the boundary conditions in the reverberation chamber. Its means the collection consists of a set of consecutive mechanical stirrer positions or consecutive frequencies (electronic stirring). This may applied to detect the maximum performance of the stirring process and provides an indication of the minimum change of angle or frequency that has to be made, at a given central frequency of operation of the chamber. It suggests also that the correlation between consecutive data is somewhat uniform within the sample. Suppose that the correlation is reduced to the first order autoregression model below : X i = ρ c X i 1 + ɛ t (7) In this expression ρ c is the correlation coefficient between two consecutive data and ɛ t is an independant and identically distributed random variable called the innovation. This innovation follows the following stationarity condition and therefore, var(x i ) = σ 2 still holds. var(ɛ t ) = (1 ρ 2 c)σ 2 (8) From equation (7)The covariance cov(x i, X i 1 ) is given by : cov(x i, X i 1 ) = cov(ρ c X i 1 + ɛ t, X i 1 ) = ϱ c σ 2 (9) Moreover, the first order autoregression model also yield to the following relation between data at a distance p from each other : cov(x i, X i p ) = ϱ p cσ 2 (10) From equation (2) and given the relationships (9) and (10) we obtain : var( n N 1 X i ) = Nσ 2 + 2Nσ 2 p=1 ϱ p c (11)

3 3 If we suppose ϱ 1 we find : var( ˆµ) = σ2 (1 + ρ c ) (12) N (1 ρ c ) We can eventually define an effective sample size such as : N ESS = N 1 ρ c (13) 1 + ρ c This relationship of the effective sample size in equation (13) was previously established on the only basis of the properties of the autoregression model hypothesis and was also proposed for the second order autoregression model for higher values of correlation [4]. 3. Statistical uncertainty evaluation from measurements. From the previous developments a main question arises. Is it possible to check for the effective sample size of the sample that was measured given a stirring process? As already mentionned, a measurement in a reverberation chamber is the result of an estimation, that of the first moment for example. And this estimation behaves as in equation (1). Given of sufficient number of M samples of size N iid (either its real size in the optimum case or its equivalent size) the estimation of the standard deviation ˆσ µ among the M estimation of ˆµ should result to a value close to σ / N iid. Therefore, we would find : ˆN iid = σ2 ( σ ˆ µ ) 2 (14) In principle, it s therefore possible to get an empirical estimation of the effective sample size from a sufficienlty accurate estimation of standard deviation of the moment estimation (ensemble average). It requires however that the number M of samples reach at least a hundreed to get a rough estimation and more than a thousand to get an estimation of ˆN iid within a few percents. An observation of the cdf obtained from a hundreed estimation of ˆµ enables to get a picture of the result as well. The following experiments aim at providing some examples of results obtained for two types of stirring methods. The first experiment investigates the efficiency of the frequency strirring method while the second experiment investigates the efficiency of the mechanical stirring method. Both experiments were carried out within the IETR reverberation chamber (8.7 m x 3.7m x 2.9 m) at a central frequency of operation of 700 MHz, well above the lowest usable frequency of the chamber. A exponential distribution for the magnitude for the measured power at the terminal of a receiving antenna Measurements in frequency stirring mode Principle The frequency stirring procedure is applied over a 20 MHz bandwidth, using N equally spaced frequencies of operation. Therefore, we would like to check that the set of N realizations (or its equivalent sample size N ESS given for example by equation (13) ) forming a sample of these measurements at these different frequencies of excitation corresponds to i.i.d.realizations. An individual measurement consists of the power received at the terminal of a receiving log-periodic antenna. The power received at an antenna follows a exponential distribution function for which σ = µ. To get a sufficient amount of samples we make use of M=100 hundreed positions of the stirrer and P=15 receiving antenna positions to estimate σ ˆ µ such as : with, ˆσ µ,p = 1 M and m=1 ˆσ µ = 1 P P ˆσ µ,p (15) p=1 ( M 1 M µ 2 p,m, M µ p,m = 1 N m=1 ) 2 µ p,m (16) N X p,m,n (17) n=1 ˆσ µ,p corresponds to a standard deviation estimated for the antenna position p. This standard deviation is calculated within a sample of M data (stirrer positions) : µ p,m. Each of these µ p,m represents the ensemble average at the antenna position p and stirrer position m of a set of measurements X p,m,1,..., X p,m,n,..., X p,m,n at N equally spaced frequencies within the 20 MHz bandwidth Example of results and analysis As an exemple, we choose first of all N = 100 stirring frequencies. The step frequency between two measurements is therefore about 200 khz. A preliminary analysis of the realizations suggested that the persistent correlation between consecutive realizations is inexistent (i.e. ρ c is assumed to be null). The effective sample size within this frequency band was also estimated to be 100. We focus first on an arbitraty position of the antenna, say p = 1. Fig. 1 shows the cumulative distribution function of the ensemble averages performed for the M=100 different stirrer positions for this antenna position. The normalized standard deviation to the mean of this sample is calculated to be Applying equation (14) for an exponential distribution leads to estimate N iid = 84. Fig.1 compares the theoretical distributions for N iid = 84 and N iid = 100 with the experimental distribution. We see a slight departure from the N=100 curve in this case. Looking at another arbitrary position 2 of the antenna, we get a different picture from Fig.2 The corresponding estimation of N iid corresponds to 67 and the departure from N=100 is now much sensitive. These variations may have two different causes at least. First of all, from an antenna position to another, the random

4 4 Antenna position N iid from ˆσ µ,p Antenna position ˆσ µ N iid from ˆσ µ,p TABLE 1 Estimation of the size Niid from the estimation of the standard deviation of the ensembe average for the different positions of the receiving antenna. N=100 stirring frequencies Antenna position N iid from ˆσ µ,p Antenna position ˆσ µ N iid from ˆσ µ,p FIGURE 1 Distribution of the ensemble average sample over M=100 stirrer positions for a frequency stirring performed over N=100 frequencies as observed for antenna position p=1. TABLE 2 Estimation of the size Niid from the estimation of the standard deviation of the ensembe average for the different positions of the receiving antenna. N=200 stirring frequencies nature of the electromagnetic field may be slightly different. However, such a factor may not contribute at all, if we consider that electronic stiring provides a very large amount of natural modes within the chamber. Another factor to be considered is the quality of the standard deviation estimation. Tab. 1 gathers the results of all the 15 antenna positions. The average over the 15 positions ˆσ µ yields to the final estimation of N iid = 84. According to this final estimation, it seems that the sample of 100 stirring frequencies thus gives a slightly reduced sample size. Therefore, the sequence of equally spaced frequencies may not offer the optimum choice. Indeed, the effective sample size was estimated with a much larger sample of N=400 frequencies within the same 20 MHz frequency band, using a second order autoregressive model. FIGURE 2 Distribution of the ensemble average sample in the same conditions as in Fig.1 for antenna position p=2. In order to check this hypothesis, we choose to double the number of frequencies, N=200. In this situation correlation between consecutive frequencies occurs. The same procedure is then applied and results are reported in Tab. 2. The ensemble of different estimated size for the sample, from an antenna position to another still varies to some extent since the estimation of variance suffers from the still limited size of the samples. However, values are significantly higher than in the case of N=100 stirring frequencies. And the average over the 15 positions ˆσ µ yields to the final estimation of N iid = 104. This is now close to the effective sample size as determined from the model of persistent correlation between consecutive frequencies. Further analysis shows that the numbers of table 2 are not significantly modified when choosing higher values for N. As far as frequency stirring operation is concerned, it can be concluded that the effective sample size as determined from this correlation model is consistent enough with measurements.

5 5 Rotation of the mode-stirrer Number of independent stirrer locations N 7 N 8 N 6 N 5 N 1 N 2 N 4 N 3 Angular section covered by the stirrer FIGURE 3 Subdivision of the stirrer in 8 sectors and corresponding independent N i realizations estimated from experiments 3.2. Mechanical stirring mode As far as mechanical stirring is concerned the same set of data is used. Frequencies and stirrer positions play each other role. N is now the number of stirrer positions, and the purpose of the analysis is to check that the stirrer provides an effective sample size as close as possible to the one calculated from equation (13). The estimation of the effective sample size was N ESS = 130. Here we call N the number of independent samples given by the central limit theorem, among the N = 400 locations of the mode-stirrer. We have estimated using autoregressive models that, at 700 MHz, there are 130 uncorrelated positions of the mode stirrer over 360. Estimation of ˆσ µ requires a different approach, the standard deviation of the estimated average over the stirrer positions has to be performed on the antenna positions first. Then the normalized average standard deviation is averaged over all frequencies. After these operations, we find that the equivalent size of the sample equals 80. Therefore, N iid = 80 equivalent independent realizations instead of the expected size N ESS = 130 corresponds to a significant shift of the standard deviation to mean ratio. Therefore, for mechanical stirring, having uncorrelated consecutive realizations may not necessarily imply that these realizations are strictly independent. In order to deepen this assumption, we propose to make the same analysis, but dividing the complete rotation of the mode stirrer into several angular subsections as shown in Fig. 3. The procedure is then repeated independently for each of these 8 sectors. Tab. 3 provides the estimation of the equivalent sample size for each one. Two conclusions may be established. First of all, the hypothesis of a somewhat uniform spreading of the independant positions over the entire rotation of the stirrer seems to be a reasonable assumption. This conclusion holds only for this specific stirrer and may also be expected since the chamber is well overmoded. However, the most important observation to be made, is that the sum of each sector of the stirrer provides a higher total (114), than the effective sample size provided by analysis of the full rotation of the stirrer (80). This result in- Sector of the stirrer Estimation of the sample size Total 114 TABLE 3 Size of the equivalent iid sample for each each of 8 sections of the stirrer dicates that a residual correlation may exist among stirrer positions. If such a residual correlation follows a uniform correlation model, equation (6) states that the effective sample size reaches a floor value. Analysing 1/8th revolution of the stirrer provides a far less number of independent realizations not yet hidden by the residual correlation. These results seem therefore to be well in line with some recent works [5]. A further simple analysis may confort this idea. Supposing that the effective sample size for the complete stirrer position is the one provided by a residual correlation according to equation (6). Let s recall that our estimation for this effective sample size was N ESS = 130. This sample size may be considered as a set of measurement without consecutive correlation. We may therefore estimate this residual correlation from the difference between the theoretical sample size N th = 130 and the estimated sample size N est = 80 using equation (5) : N est = N th 1 + (N th 1)ϱ u (18) This yields to ρ u = Giving a 1/8th sector of the stirrer rotation, the new theoretical value for N th is 130/8. Applying a second time equation (18), the new estimated N est for one sector of the stirrer becomes 15. This value is found to be coherent with the results of Table3. 4. Conclusions The statistical uncertainty budget must be controled in day to day operation of reverberation chambers. The moment estimation is a key parameter for a lot of ensemble average based measurements such as EMI measurements or antenna measurements (efficiency, total radiated power...). Starting from a presupposed maximum available uncorrelated data, one may look for the maximum performance in terms of statistical uncertainty. This communication presented some possible ways of assessing this performance from a set of many measurements. The empirical uncertainty thus observed is analysed with respect to the theoretical sample size as determined from correlation analysis between individual realizations.

6 6 The above results seem to acknowledge that performances of a mechanical stirring and of a frequency stirring in a reverberation chamber may appear as different. In particular, the mechanical stirrer may be not efficient enough to provide as many as independent measurements as expected from an estimation of the effective sample size, if it is calculated from the observation of correlation between consecutive realizations. On the contrary, the performance of a frequency stirring is in line with what is expected from the determination of the effective sample size calculated in the same conditions. A possible explanation of this result is obviously the completely different mode of operation of the chamber. Frequency stirring operation is able to generate numerous propagation modes in a well overmoded chamber and modifies drastically the fields in the chamber. There will be unlikely a stationnary wave not affected by this stirring process. The mechanical stirrer is may be not able to do so, due to possible symetries in his shape and/or the limited number of modes that are sufficiently involved by the stirrer. References [1] D.A.Hill, Electromagnectic fields in cavities. Deterministic an statistical theories,ieee press, Series on Electromagnetic Wave Theory,Wiley, [2] P. Besnier, B. Démoulin, Electromagnectic reverberation chambers, ISTE/Wyley&Sons, August 2011 [3] A. Papoulis, Probability, Random Variables and Stochastic Processes,New York : Mac Graw Hill 4th edition, [4] C. Lemoine, P. Besnier, M. Drissi, Estimating the effective sample size to select independent measurements in reverberation chamber, IEEE trans. EMC, Vol. 50, No. 2, , 2008 [5] A. Cozza, A skeptic s view of unstirred components, EMC Europe, Proc. int. symp. on, [6] C. Lemoine, P. Besnier, M. Drissi, Evaluation of frequency and mechanical stirring efficiency in a reverberation chamber, EMC Europe, Proc. int. symp. on, 2008.