A Skeptic's View of Unstirred Components Andrea Cozza

Département de Recherche en Électromagnétisme Laboratoire des Signaux et Systèmes (L2S), UMR8506 SUPELEC -Univ Paris-Sud -CNRS 91192 Gif sur Yvette, France Email: andrea.cozza@supelec.fr

Abstract-In the practice and research on reverberation chambers, the concept of unstirred field components have often been invoked as soon as empirical distributions fail to comply with goodness-of-fit tests applied to data collected at a frequency of operation far beyond the lowest usable frequency. The current explanation for this phenomenon is that under certain conditions, the field samples generated by means of a stirring procedure are not characterized by a zero average-value, but actually present a deterministic offset term that is linked to a line-ofsight contribution. In this paper, we prove that this practice is not sound, as it does not acknowledge the fact that even a very low but non-zero residual correlation between the samples is enough to put in jeopardy the validity of the limits imposed by most hypothesis-test statistics, and hence their ability in properly detecting any constant contribution. An alternative approach is here proposed, based on the analysis of the variability of the line-of-sight contribution estimate, capable of accounting for the residual correlation in a reverberation chamber. Experimental results are presented to support the validity of our approach, exposing the critical use of goodness-of-fit tests as currently applied.

Index Terms-Reverberation chambers, Statistics, Hypothesis test, Unstirred components.

I. INTRODUCTION

Reverberation chambers (RCs), when operated at a sufficiently high frequency jointly with a stirring technique, are expected to be characterized by a field spatial distribution following a Gaussian probability law at any point within a volume of space, with equal moments, i.e., stationary in space [START_REF] Kostas | Statistical model for a mode-stirred chamber[END_REF], [START_REF] Corona | Reverberating chambers as sources of stochastic electromagnetic fields[END_REF]. In fact, the practice of RCs is not so close to this ideal depiction, as non-agreements appear at different levels: not only field samples appear not to agree flawlessly with a Gaussian probability distribution, but even when they do, they may happen to present a non-zero average value. This phenomenon has been studied in several atypical (with respect to the usual practice for EMC tests) configurations [START_REF]Reverberating chamber electromagnetic field in presence of an unstirred component[END_REF], [START_REF] Holloway | On the use of reverberation chambers to simulate a Rician radio environment for the testing of wireless devices[END_REF] where a direct line-of-sight (LoS) contribution between the excitation antenna and the field probe is established on purpose, leading to a fixed contribution negligibly affected by the stirring technique.

Clearly, admitting the existence of this LoS term implies that the field statistics will be skewed with respect to the ideal diffuse case. Perhaps because of this state of affairs, in several works presented in the last few years when the conditions expected to lead to an ideal diffuse field are reasonably true, if field samples do not comply with their expected behavior the presence of a LoS term has been invoked [START_REF] Lundén | How to avoid unstirred high frequency components in mode stirred reverberation chambers[END_REF], [START_REF] Primiani | Numerical and experimental investigation of unstirred frequencies in reverberation chambers[END_REF]. In other words, it is considered that if the frequency of operation of the chamber is well above the lowest usable frequency (LUF) [START_REF]Reverberation chamber test methods[END_REF] and that the stirrer respects certain empirical constraints [START_REF] Wellander | Experimental investigation and mathematical modeling of design parameters for efficient stirrers in mode-stirred reverberation chambers[END_REF], the only way to explain the existence of non-ideal field statistics is to invoke the presence of LoS contributions. In practice, this is regarded as due to a non-optimal positioning and orientation of the excitation antenna, allowing the establishment of a direct propagation path between this antenna and the probe one. As this propagation model implies that the LoS contribution is weakly affected by the stirrer (as at least one path do not cross it), the LoS term is referred to as an unstirred component.

We are actually very skeptical of this explanation: it is hard to imagine non-negligible LoS terms appearing when the source antenna is not pointed directly towards the probe one. As it will be shown in the remainder of this paper, there are sound arguments that point to the fact that the methods currently used to detect the presence of unstirred components are not statistically nor physically justified. Our analysis proceeds from an original description of the behaviour of an RC as a multivariate random generator: this leads to a sounder definition of the correlation between samples. Experimental results interpreted through this model show the presence of residual weak correlation levels even in conditions deemed to be ideal. This fact is shown to be sufficient to lead to wrong conclusions on the presence of LoS contributions.

Our statistical description allows to define in a very natural manner an alternative hypothesis test that is expected to lead to a better decision for the existence of unstirred components. Experimental results are shown to comply in a very effective way with the theoretical behavior predicted by our model, thus suggesting that it can be regarded as closer to the real behavior of a real-life RC.

II. REVERBERATION CHAMBERS AS MULTIVARIATE

RANDOM GENERATORS

Reverberation chambers are often regarded as random field generators [START_REF] Corona | Reverberating chambers as sources of stochastic electromagnetic fields[END_REF]. When field probes are used to assess the statistical behavior of any scalar components of the electric field at a given position, the sequence of values generated by means of the chosen stirring technique is usually regarded as a random sequence. In particular, the field is modeled as a univariate random variable x ∈ C, while the N s samples obtained by operating the stirring technique are regarded as random realizations of this random variable. In the context of the present paper, stirring techniques will be assumed to be operated through discrete steps, for the sake of simplicity. The continuous case can be eventually inferred from our derivation.

The random sequence {x i } (corresponding, e.g., to a scalar field component) is typically analyzed by estimating its first two empirical moments μx = x i Ns (1a)

σ2 x = N s N s -1 |x i -μx | 2 Ns (1b)
with y i N the arithmetic average of the N -sample sequence {y i }. The random sequence is further studied by considering its empirical probability distribution function and its autocorrelation function (ACF), defined as

φ(k) = Ns i=1 |x i | 2 -1 Ns i=1 x i x ⋆ i+k , (2)
having chosen its normalized definition. The first sum is to be taken circularly over the entire set of samples, as these usually form a periodic sequence, depending on the stirring technique. Random samples thus generated are expected to comply with a series of constraints dictated by the asymptotic model presented in [START_REF] Hill | Plane wave integral representation for fields in reverberation chambers[END_REF]. If any scalar component of the electric field is taken at turn to be the {x i }, then a sound approach is to apply goodness-of-fit (GoF) tests, intended to check the consistency of the samples with a null-hypothesis that requires them to follow a zero-average normal law. The samples are subsequently regarded as iid random variables, i.e., their ACF is considered to well approximate a Kronecker delta, i.e., to fall to negligible values for k = 0.

This approach has several drawbacks and limitations: first of all, it is applied in a local way, i.e., the results of the test can differ from two different positions within the same chamber. Such an outcome is critical, as it is actually inconsistent with the spatial stationarity of the random properties of the field distribution within an ideal RC [START_REF] Hill | Plane wave integral representation for fields in reverberation chambers[END_REF], [START_REF] Kostas | Statistical model for a mode-stirred chamber[END_REF]. Secondly, the assumption that the field be stationary is physically justified only when moving in space, and not when a stirring technique is applied and the field observed at the same position. Clearly, the iid assumption is the ideal case of reference, but their is no physical reason to expect the field samples generated by, e.g., a mechanical stirrer to be iid. The same remark holds for the modeling of the random sequence as a time series, as suggested in [START_REF] Lemoine | Estimating the effective sample size to select independent measurements in a reverberation chamber[END_REF]: regarding the correlation between two consecutive samples to be independent of the absolute position of the stirrer is clearly not physically justified.

We should add to these remarks a further fact. As the field is probed at different positions, the hypothesis test can yield different and inevitably contrasting conclusions. Should this be regarded as a token of the fact that the random properties of the field are not stationary in space, or rather that the application of GoF methods to a local level is not the right procedure? In fact, the former conclusion is not consistent with the original constraints imposed by the ideal behavior of an RC: as a result, it would be necessary to regard a chamber as nonideal as soon as GoF tests were negative over some locations. As a matter of fact, why should one consider that a sequence with non-zero average value is regarded as unacceptable, while accepting that the field moments be non stationary? Still, to the best of our knowledge, the claim of unstirred components is used as a figure of merit of stirrer efficiency only over one position [START_REF] Lundén | How to avoid unstirred high frequency components in mode stirred reverberation chambers[END_REF], without checking how it fares from a global point of view. In other words, the current approach could lead to the conclusion that a stirrer is effectively generating a proper random sequence at one position and not at another.

Moreover, the ACF cannot be properly estimated from the N s samples provided by the stirring technique at one position in most real-life configurations. Its statistical uncertainty can be assessed by means of the laws recalled in [START_REF] Lemoine | Estimating the effective sample size to select independent measurements in a reverberation chamber[END_REF]: as the correlation gets weaker, the uncertainty can be overwhelmingly higher than the actual value to estimate. As we will show that even weak residual sample correlations play a central role in GoF tests, the relatively high uncertainty affecting its estimate when working on a single position would not allow to draw any useful information.

We would rather suggest to regard an RC as a multivariate random generator. Now the field generated over N p different positions for a given stirrer configuration will be modeled as a random vector, treated as a single random realization X i = (x i,1 , ..., x i,Np ), with X i ∈ C Np×1 . The sample average will now be referred to as

μ = X i Np , (3)
with the arithmetic average applied to each of the N s scalar components x i,j . The sample covariance matrix Ĉ will be given by

Ĉ = N p N p -1 (X i -μ)(X i -μ) H Np ( 4
)
and the sample correlation between two stirrer configurations

ρi,j = C i,j C i,i C j,j (5)
now evaluated over N p positions. This type of approach allows studying the behavior of an RC in a global way, rather than the local one currently used when looking for unstirred components. Having access to a broader view is fundamental in the estimate of the residual correlation affecting the stirring technique, and thus the accuracy of the estimate of eventual unstirred components. As a matter of fact, the stirrer efficiency should be evaluated as its ability to generate uncorrelated samples over an entire range of positions within the chamber, more specifically taken within its test volume. The same goes for the eventual existence of unstirred components.

Another valid representation would have consisted in switching the role of the stirrer realizations and the spatial ones. Although entirely valid, this description would lead to a correlation matrix assessing the correlation between the samples taken at different positions, i.e., the spatial correlation of the field within an RC. As opposed to the modelling of the correlation between stirrer-generated samples, the spatial correlation quantity features a good agreement between experimental data and theoretical models [START_REF] Hill | Spatial correlation function for fields in a reverberation chamber[END_REF]. Hence, it is expected to take relatively low values as the distance between the probe position go beyond the wavelength and changes sign as the distance increases. As a result, its average value in space can be expected to be much lower than the correlation between the samples generated by the stirring technique. This assumption is supported by experimental results presented in Section V. As a consequence, the alternative description of mixing together spatial-and stirring-obtained data should not be regarded as a valid approach when correlation data are being examined, since their correlations are remarkably different, as will be shown in Section V.

III. ON THE DETECTION OF AN OFFSET

Following this discussion, each realization X i = (x i,1 , ..., x i,Np ) will be assumed to have elements following the same family of probability law, but not necessarily with the same statistical moments. No assumption will be invoked on the moments of X in this work: the only assumption will be that the x i,j be normally distributed. Such requirement should actually not be regarded as a constraint, as it is part of the definition of the null hypothesis we are going to detail in the next section.

Let us recall, for the time being, that the assumption of an overmoded cavity implies that the field samples should be indeed normally distributed, with a zero average value, independently from the use of an efficient stirrer, or for that of a stirrer at all [START_REF] Schroeder | Statistical parameters of the frequency response curves of large rooms[END_REF], a well-known fact in acoustics. It is thus natural to consider that the field samples should be normally distributed, even though a LoS component is present: this latter will only affect the arithmetic average Xi = X i Ns [START_REF] Primiani | Numerical and experimental investigation of unstirred frequencies in reverberation chambers[END_REF] of the N s samples generated by the stirrer at the i-th position.

The typical approach to estimate the intensity (and presence) of an eventual LoS contribution is to compute Xi at a given position. It is thus interesting to study its behavior in the framework presented in the previous section. The offset Xi can be regarded as a random variable, whose ensemble average can be shown to be

µ X = E Xi = µ Ns , (7)
with E [y] the ensemble average of the random variable y and µ the ensemble average of the multivariate random variable X. The ensemble average should not be confused with the arithmetic averages typically used as estimates of statistical moments. The arithmetic average in [START_REF]Reverberation chamber test methods[END_REF] is taken over the N s scalar values constituting the vector µ.

The variance of the offset estimator can be shown to be

σ 2 X = E | Xi | 2 = σ2 1 N s + ρ , (8)
where

σ2 = C i,i Ns (9a) ρ = Re ρ i,j Ns(Ns-1) , (9b)
having taken the arithmetic average in (9b) over all of the offdiagonal elements of the covariance matrix C. The quantities σ2 and ρ are the average variance and correlation factor of the field measured within the RC over the N s stirrer realizations and N p positions. Following the assumption of a multivariate normal law, the offset estimator follows a normal law, too, with mean and variance given in [START_REF]Reverberation chamber test methods[END_REF] and [START_REF] Wellander | Experimental investigation and mathematical modeling of design parameters for efficient stirrers in mode-stirred reverberation chambers[END_REF], respectively. It is fundamental to recall that the assumption of normality is not linked to the use of a stirring technique. The normality of the field samples is asymptotically expected in any overmoded cavity, even in a static configuration [START_REF] Schroeder | Statistical parameters of the frequency response curves of large rooms[END_REF]. This fact will play a fundamental role in the next section, when defining an alternative hypothesis test. In particular, the assumption of a normally distributed X does not require to invoke the central limit theorem (CLT).

Getting back to [START_REF] Wellander | Experimental investigation and mathematical modeling of design parameters for efficient stirrers in mode-stirred reverberation chambers[END_REF], the presence of a correlation term ensures that the CLT and therefore the law of large numbers cannot be invoked, as the N s scalar random variables cannot be assumed to be independent. Whence, even if N s → ∞, the uncertainty over the estimate X of the offset can never go to zero. Indeed lim

Ns→∞ σ 2 X = ρσ 2 , (10)
leading to a lower bound in the accuracy of the estimator. This lower bound is far from negligible: for what is usually regarded as a negligible correlation, e.g., ρ = 0.1, the reduction of the original standard deviation in the samples is translated into a mere reduction of about a factor 3 in the standard deviation of the offset estimator. This is far from being a satisfying accuracy gain obtained by collecting a large number of samples. This is where the current approach to detecting unstirred components fail. As a matter of fact, the GoF tests usually applied are based on the assumption of iid samples. Such assumption has profound implications: indeed, it implies that for an increasing number of samples N s , the accuracy of the data-set is increasingly fine. In other words, GoF tests tend to ignore imprecise data, especially those with low probability, as N s keeps sufficiently low. But as N s increases, these statistics tends to become more demanding, leading to rejections of the null hypothesis as soon as small imperfections take the data set apart from the reference one. In the presence of a finite correlation, the statistics of the GoF tests are incapable of acknowledging the fact that further data do not provide new complementary information for the rejection of the null hypothesis.

The problem is that the estimation of the residual correlation is typically based on one single observation point, thus affected by a large statistical uncertainty. Moreover, it is often assumed that a sufficiently low correlation factor implies that the data are "almost" independent [START_REF] Lemoine | Estimating the effective sample size to select independent measurements in a reverberation chamber[END_REF]. Equation ( 8) clearly shows that even the smallest correlation (the real one, not the estimate) rules out the use of the CLT. In other words, the actual value of the offset X cannot be accessed. This residual uncertainty is actually in contradiction with the assumptions at the base of currently used GoF tests. As N s increases, they do expect the accuracy of the estimate to improve. Hence a misconception and misuse of these tools. In a practical way, the CLT and hence most of the GoF tests can be used only as long as

N s ≪ 1 ρ , (11)
so that the usual reduction in the statistical uncertainty predicted by the CLT holds. For larger populations, the results from GoF tests are going to be too conservative, as the actual amount of information provided by the data is less than might be expected for uncorrelated samples.

It could be argued that this limitation can be bypassed by extracting a subset of independent samples from the original samples. In fact, as demonstrated by the very definition of the effective sample size, the amount of information in the two sets, and in particular the uncertainty it involves in the estimation of the ensemble average is the same [START_REF] Lemoine | Estimating the effective sample size to select independent measurements in a reverberation chamber[END_REF]. Hence, this limitation would subsist. Moreover, the concept of independent samples is controversial, as it is not related to a threshold value in the residual correlation, nor to a clearly defined procedure that allows to extract in a precisely defined manner these samples. The actual approach suggested in international standards [START_REF]Reverberation chamber test methods[END_REF] and recent works [START_REF] Lemoine | Estimating the effective sample size to select independent measurements in a reverberation chamber[END_REF] is just to decimate the original population, which does not ensure the independence of the the remaining samples with certainty.

IV. AN ALTERNATIVE HYPOTHESIS TEST

The presence of an offset can now be restated in an alternative manner. Let us consider the null hypothesis of a perfect RC, presenting all of the features required by asymptotic models [START_REF] Hill | Plane wave integral representation for fields in reverberation chambers[END_REF], in particular a zero-average. We are interested in assessing what is the probability of observing a sample X i at a given position presenting an offset Xi , although its actual ensemble average is equal to zero.

Under these hypothesis, a standardized random variable ζ can be introduced

ζ = X σ X = X σ 1 N s + ρ -1 , (12)
following [START_REF] Wellander | Experimental investigation and mathematical modeling of design parameters for efficient stirrers in mode-stirred reverberation chambers[END_REF]. As the null hypothesis consider that the chamber is ideal, the field samples will follow a normal law, and thus ζ too. It is therefore possible to state that the presence of an unstirred component should be considered as statistically significant to a probability α only if the value taken by ζ at one position (single realization) is larger than the quantile q α of a standardized normal distribution. In any other case, the fact that Xi = 0 should not be regarded as due to unstirred components.

It is noteworthy that this definition accounts for the existence of a residual correlation between the samples. Indeed, as the average correlation increases, ζ is scaled down, so that it becomes less likely to decide for the presence of an unstirred component only on the basis of a non-negligible offset. The decision on the actual presence of an unstirred component should be based on a stronger foundation, i.e., by comparing the estimated offset with the uncertainty affecting its estimation. To this effect, a z-test could be applied, since ζ is a standardized normal variable, thus rejecting the null hypothesis if

|ζ| > q 1-α/2 N p , (13)
where q 1-α/2 is a quantile for the standard normal distribution.

It could be argued that the z-test requires independent samples. This is true, but since the samples ζ i are now taken over N p different positions in space their residual correlation can be expected to be far lower than that of a stirring technique, if these positions are sufficiently far apart. This assumption will be experimentally shown to hold in the next section. The strength of this approach lies in the fact that the validity of the proposed model can be checked from an experimental point of view. Indeed, the quantities appearing in (12) can be estimated from measurements: if the experimentally-derived values of ζ do appear to follow a standardized normal distribution, than our modeling approach should be regarded as sound. It is fundamental to notice that the role of ρ has a dramatic impact on the behavior of ζ, even for small values. This will be show in the next section to be the reason why we regard the proposed test as physically and statistically correct.

V. EXPERIMENTAL RESULTS

We carried out measurements in Supelec's RC, with dimensions 3.08 × 1.84 × 2.44 m, with a lower usable frequency of about 550 MHz as defined in [START_REF]Reverberation chamber test methods[END_REF]. We considered the frequency range between 1 and 3 GHz, in order to ensure a negligible probability of incurring into a configuration where the RC would not behave as an overmoded cavity. Moreover, at these frequencies our mechanical stirrer is electrically large and aperiodic, so that it can ensure low correlation between samples if sufficiently large angular steps are considered: the 50 angular samples used provided an estimated correlation below 0.3. A log-periodic antenna was used to excite the field within the chamber, oriented towards a corner in such a way as to have its main lobe scattered towards the stirrer. We can thus regard as unlikely that unstirred components in the sense given in [START_REF]Reverberating chamber electromagnetic field in presence of an unstirred component[END_REF] be present. A horizontal component of the electric field (normal to the sample plane) thus generated was measured by means of an electro-optic field probe (Enprobe EFS-105), capable of preserving the phase of the field while providing a negligible scattering cross-section: it was thus possible to move it freely within the RC without any risk of altering the actual field distribution. Forty positions were considered, distributed over a 5 × 8 (vertical and horizontal, with a step of 37.5 cm and 25 cm, respectively) Cartesian grid dividing into two halves the chamber along its longer dimension.

As proposed in Section II, we treated the spatial samples for a given stirrer position as one single realization of a multivariate random variable. The empirical covariance and correlation matrices were computed for this case and for the complementary one, where the role of the stirrer and spatial samples are switched. This yielded the ρ results presented in Fig. 1: it is indeed proven that the average correlation in much lower among the spatial samples than between the stirrer ones, as anticipated in Section III. As a result the assumptions on the modeling of the ζ random variable hold, so that its definition is statistically sound. The empirical moments of ζ where than computed, and are shown in Fig. 2: as required by our assumptions, ζ presents a unitary variance. This result should not be underestimated, as neglecting the residual correlation ρ would have led to a twofold increase of the variance of ζ, which would have led to a higher rate of rejection of the null hypothesis. The fundamental role played by ρ even for weak values is thus also confirmed.

The consistency of the statistical properties of the field distribution were then tested against the zero-mean assumption expected from their asymptotic modeling. Three approaches were applied: 1) the minimum p-value was considered for Kilmogorov-Smirnov (KS) GoF tests based on the N s -sample population at each of the N p positions in space; 2) the real and imaginary parts of ζ were tested for normality through KS tests over N p samples; 3) |ζ| was tested through KS against a χ 2 distribution. The results presented in Fig. 3 show that indeed, if the presence of "unstirred" components is considered as unacceptable at one position, than our RC does not allow to generate an unbiased field distribution over a region of space. This overreaction of the KS test comes from the fact that the N s samples measured over one single position are hardly independent, and that since the condition in [START_REF] Hill | Spatial correlation function for fields in a reverberation chamber[END_REF] is not satisfied, the effect of the correlation is not negligible. Another important issue is the fact that testing against χ 2 (and in general any function of the modulus of the field components) does not allow assessing whether the real and imaginary parts have identical features. Indeed, the test based on the |ζ| is much less strict than that applied to the real and imaginary parts of ζ. Taking a significance level α = 0.05, leads to the identification of a number of frequencies where the average value of the field generated within the RC can no more be explained through statistical uncertainty alone. An example is given in Fig. 4a, where the scatter plots of the N s = 40 population of ζ i are shown for two close frequencies, 1.744 GHz and 1.754 GHz, where the ζ-test null hypothesis is respectively accepted and rejected. The KS test rejected the null hypothesis of normal distributions respectively in 27 and 37 configurations for these two frequencies: it is hence not possible to assess whether the RC presents major differences at these two frequencies. Indeed, the scatter plots clearly show the huge uncertainty in the estimate of the LoS component, as the confidence margins associated to the nullhypothesis for a single local measurement do not lead to a clear decision scenario. At 1.754 GHz, only 5 samples lie outside the confidence margin for the local test. Conversely, the global test provides a much stricter decision criterium, as the arithmetic averages of the two populations are wide apart from the smaller confidence region. Such global approach indicates without doubts that at 1.754 GHz, the RC is not behaving as expected, with the ζ i clearly biased away from the origin. The proposed approach indeed allows assessing in what scenario the RC is found: at 1.744 GHz our RC presented a zero-mean ζ, well explained by statistical uncertainty alone, as soon as residual correlation is taken into account.

A last result of interest is presented in Fig. 4b, where the quantile-quantile plots of the real part of the ζ i are considered against a standard normal variable (null hypothesis) at the two same frequencies as before. This figure shows that at 1.754 GHz the rejection of the ζ test is not only motivated by the existence of an offset, since the plot is not close to a line, but strongly distorted. In other words, removing its average value does not lead to the test being accepted. This points to the fact that apart in atypical configurations as those considered in [START_REF]Reverberating chamber electromagnetic field in presence of an unstirred component[END_REF], the field distribution is not merely modified by a LoS term, but the field distribution is likely superposed to a stationary wave that is weakly interacting with the stirrer. This would better explain the non constant bias and the statistics distortion at the same time, since the field does not present stationary moments in space. This could also likely be linked to a loss of field uniformity.

VI. CONCLUSIONS

This paper has introduced an alternative approach in the detection of "unstirred" components. Rather than applying the usual local approach, a more general one was employed, based on the modeling of an RC as a multivariate random generator. Studying the statistical properties of the moments of this model, it was shown that weak correlation values should never be neglected, particularly when testing experimental data against theoretical reference ones.

The introduction of the test distribution ζ capable of taking into account this residual correlation has led to a sounder test, whose physical validity has been tested against experimental data, thus pointing to the intrinsical importance of taking into account the residual correlation. Experimental results also proved that the reason of rejection of the null hypothesis cannot be entirely put on the presence of a bias, since the appearance of a profoundly different statistical behavior of the RC points to deeper reasons not well understood.

Future work will be needed in order to establish a proper testing approach on a global scale, rather than on a local one, in particular in order to set a relationship between the number of positions needed and the accuracy of the test.

Fig. 1 .

 1 Fig.1. Residual correlation ρ as defined in[START_REF] Hill | Plane wave integral representation for fields in reverberation chambers[END_REF], computed from experimental data when considering the stirrer (blue) or the positions (red) as random realizations of multivariate samples.

Fig. 2 .

 2 Fig. 2. Empirical moments of ζ estimated from the Ns multivariate samples: (a) real (blue lines) and imaginary (red lines) parts of the sample average (m r ζ , m i ζ ); (b) standard deviations for the real and imaginary parts of ζ (s r ζ , s i ζ ); (c) standard deviation of ζ (s ζ ). Shaded areas represent the expected confidence margin at 95 % probability for the null hypothesis, for a sample population of Np realizations.

Fig. 3 .

 3 Fig. 3. Probabilities of rejection of the null hypothesis, as obtained by means of three different approaches: (a) acceptance of a KS test applied to univariate data over all of the Np positions (minimum of the Np p-values), in blue; (b) null hypothesis of Reζ and Imζ as standard normal variables, in black; (c) null hypothesis of |ζ| behaving as a χ 2 variable. The black dots stand for the frequencies over which the test (b) is rejected with a significance level α = 0.05 for either the real or the imaginary parts.

Fig. 4 .

 4 Fig. 4. The ζ i samples measured at 1.744 GHz (blue) and 1.754 GHz (red): (a) scatter plot of the ζ i samples, with their average values marked by circles. The confidence limits for α = 0.05 are shown for the case of a single local measurement (solid line) and for the global 40-sample population (dashed line); (b) quantile-quantile plots based on a comparison with a standard normal random variable.


Andrea Cozza
email: andrea.cozza@supelec.fr

Florian Monsef

How a Physical Definition of Overmodedness Can Explain Local Statistical Non-Compliancy

Keywords: Statistics, Cavities, Losses, Error analysis

Current understanding of reverberation chambers is firmly funded upon their being operated as overmoded cavities. A proper definition of this condition has been lacking for a long time, while being regarded as automatically satisfied as soon as a large (and unspecified) number of normal modes are set to resonate at the frequency of operation of the chamber. In a recent work, we have introduced in a formal way a proper definition of the overmoded condition, showing how, from a strict mathematical viewpoint, it is not related to any threshold frequency, but rather requirs a statistical framework. In this paper we revisit the most important results from our analysis, and show how our theory can straightforwardly explain the local non-compliancies sometimes observed for field statistics in what are deemed to be overmoded cavities. Partial overlapping between the normal modes of the cavity are shown to lead to an alternative and more physically-based explanation for this phenomenon, without invoking the concept of an ineffective stirring technique, nor the presence of unstirred components. The proposed theory is finally applied for the assessment of the relationship existing between the actual confidence margin based on the ideal properties of a reverberation chamber and those predicted by our model, showing how there is a non-negligible probability of having a large number of samples outside the original confidence margin, i.e., a potentially erroneous estimate of the true variability of a field-related quantity.

I. INTRODUCTION

The way reverberation chambers (RCs) are commonly utilized in EMC radiated tests is based on a strong assumption, that they are capable of providing a diffused field. This property is never directly invoked in RC theory as developed within the EMC community, though it is of fundamental importance in the physics of cavities and the study of room acoustics [START_REF] Schroeder | Statistical parameters of the frequency response curves of large rooms[END_REF]. Historical reasons are likely to blame for this mismatch, as well as the fact that electrical engineers have long preferred to have a more practical and test-driven approach to the understanding of these facilities. Still, the lack of the clear definition of a diffused field has been an invisible obstacle, in our opinion, to the cross-fertilization that should have existed between the EMC community and the acoustics one. Indeed, EMC engineers could benefit from the deeper understanding and modeling tools developed in acoustics. In this respect, we will regard the concepts of overmoded RC and that of diffused field as synonyms, as both refer to a cavity that ensures, asymptotically, to expect a very specific statistical behavior for field-related quantities.

Yet, both fields lacked a proper physics-based model capable of explaining under what conditions a cavity can be considered to allow a diffused field to establish. Acoustics, too, tends to consider the idea of a threshold frequency [START_REF] Schroeder | Statistical parameters of the frequency response curves of large rooms[END_REF], often referred to as Lowest Usable Frequency (LUF) in EMC. In a recent paper [START_REF] Cozza | The role of losses in the definition of the overmoded condition for reverberation chambers and their statistics[END_REF], we have proposed such a model, establishing a direct link between the physics of an RC and the statistical properties of the field it supports. In this paper we recall the most important steps and assumptions that have led to this result, and show how it can explain another phenomenon with an unsatisfying description, that of local statistical noncompliancy of field-related quantities.

Our definition of an ovemoded cavity (and thus of a diffused field) is based on the use of a modal description of the field excited within an RC, coupled to a statistical approach. The interest of this approach is that it allows avoiding making non-physical assumptions on the need for a threshold on the modal density, as apparently assumed on the EMC side, or on the sheer fact that a threshold frequency exists. Of particular importance is the prediction that losses should not be regarded as a limitation or nuisance in the operation of RCs, but that they rather lie at the heart of its functioning. Without losses, an RC would never be capable of supporting a diffused field, independently of the modal density it supports: hence, our regarding the use of the term overmoded as a potentially treacherous habit, as the mere availability of a high modal density is not a sufficient condition. In the following, it will be shown that an overmoded RC is a cavity that supports a large number of overlapped modes, which is not the same as just supporting a large number of modes, as estimated by looking at modal density alone. This fact was unacknowledged in EMC, while well understood in acoustics and, as we will show in this paper, this misinterpretation can be regarded as one of the reasons why the phenomenon of local non-compliancy is still characterized by an unsatisfying explanation.

II. DEFINING THE OVERMODED CONDITION

The common sense given to the concept of overmoded cavity is tightly linked to the expectation of the following properties: 1) field statistics are independent of the position in space where they are tested, at least in a sub-volume of the cavity; 2) the three orthogonal components of the electric (respectively, magnetic) field behave as independent and identically (iid) distributed random variables; 3) they follow a Gaussian distribution law, with zero mean-value. It has been shown that these hypothesis are met whenever the field distribution can be described by means of a continuous random plane-wave spectrum, with specific statistical properties [START_REF] Hill | Plane wave integral representation for fields in reverberation chambers[END_REF]. Although such type of model well approximates the behavior of real-life RCs in their high-frequency range, it is well-known that there might exist some frequencies where it is incapable of predicting a drift from it [START_REF] Primiani | Numerical and experimental investigation of unstirred frequencies in reverberation chambers[END_REF]. These disagreements are often explained through a stirrer inefficiency or a sub-optimal alignment for the sources [START_REF] Lundén | How to avoid unstirred high frequency components in mode stirred reverberation chambers[END_REF], though we are quite skeptical of these explanations [START_REF] Cozza | A skeptic's view of unstirred components[END_REF], whereas in the lower frequency range they are usually just fitted to alternative probability distribution functions [START_REF] Orjubin | Statistical model of an undermoded reverberation chamber[END_REF].

It is, in our opinion, by far more interesting and useful to get back to the basics of the physics of cavities, while taking on a more statistical approach from the beginning. To this end, we will consider a modal expansion for the electric field distribution observed at the position r within an RC at the working frequency f [8]

E(r, f ) = i∈M γ i (r, f )ψ i (f ) ξi (r, f ) , (1)
where the γ i (r, f ) reflect the excitation of each mode, depending on the nature of the source and its position relative to the modal topographies, ψ i (f ) is the bell-shaped frequency response of each mode and ξi (r, f ) is the polarization of the electric field contribution provided by each mode. The sum in (1) is taken over the set M of modes that are effectively excited at the working frequency. A proper justification for this model has been provided in [START_REF] Cozza | The role of losses in the definition of the overmoded condition for reverberation chambers and their statistics[END_REF]. Rather than attempting to apply (1) in a deterministic way, we will regard all of these modal quantities as random ones, with statistical properties dictated (or suggested) by their physical properties. In this respect, the modal weights {γ i } can be expected to behave as centered random variables, as they are directly related to the scalar projection of the equivalent currents representing a source and the modal topographies, which are standing waves, thus characterized by a pseudo-periodic change in their sign over space. No further assumption is needed on the nature of the probability law for the γ i . The functions {ψ i (f )} represent the responses of second-order systems, defined as follows

ψ i (f ) = 1 f 2 i (1 + j/2Q i ) 2 -f 2 , (2)
where f i is the resonance frequency of the i-th mode and Q i its quality factor. As the presence of a stirring technique implies, from an ideal perspective, that the frequencies of resonance of the modes are thoroughly mixed and swapped about the working frequency, it is reasonable to assume that the variables {f i } be uniformly distributed. Moreover, the quality factors {Q i } can be characterized by means of their average (or composite) value, as will be shown later. This latter quantity can be experimentally assessed. Finally, the polarization unit vectors { ξi } will be assumed to be uniformly distributed over 4π steradian, again as a consequence of the assumption of an ideal stirring technique at work. It is important to notice that assuming a perfect stirring is actually sensible, as we are interested in assessing whether it is possible to explain the eventual non-compliancy of an RC even having enforced an ideal performance on the stirring technique.

With this model at our disposal, we now aim at studying the first two statistical moments of the electric energy density

W (r, f ) = ǫ 0 E(r, f ) 2 , (3)
with ǫ 0 the dielectric permittivity of the medium filling the cavity. The electric energy density can be described, in the case of an ideally diffused field, as a random variable distributed as a χ 2 6 law, as a consequence of the features recalled at the beginning of this Section. As opposed to this hypothesis, by computing the first two moments of W (r, f ) and employing the previously introduced assumptions on the statistics of the modal quantities should allow to check under what conditions the ideal behavior is actually met. The procedure for this calculation has been presented in details in [START_REF] Cozza | The role of losses in the definition of the overmoded condition for reverberation chambers and their statistics[END_REF], and leads to the following standardized variance

ς 2 W = σ W µ W 2 = 1 3 + 1 π µ 4 µ 2 2 1 M M , (4)
with M M = mB M the average number of modes overlapping within the average -3 dB bandwidth B M = f /Q centered around the working frequency, for a modal density m(f ) and an average composite quality factor Q. The terms µ n refer to the moments of the modal weights {γ i }

µ n = E [|γ i | n ] , (5)
so that µ 4 /µ 2 2 = 2 for the case of normally distributed modal weights, which is the usual assumption applied for these quantities. Other distribution laws do not have much of an effect on this ratio, so that the final result is weakly dependent on any assumption on the nature of the modal weights.

Comparing (4) to the standardized variance ς 2 χ 2 6 = 1/3 obtained for the ideal diffused field scenario shows what is needed for an RC to be as close as possible to the ideal case, i.e., presenting a large number of overlapped modes M M . Whence, it is clear that there is no reason for assuming that any threshold frequency ensures an RC to support a diffused field, as M M is actually a complex function of frequency, with a non monotonous trend. As a matter of fact, both the composite quality factor Q and the modal density m are nonmonotonous; attention must be paid to the fact that the often used Weyl's approximation is indeed a smooth fitting curve to the real m(f ), which is generally not known. Local variations around this smooth function can be far from negligible, leading to modal depletion or excess: it can therefore be intuitively understood that M M follows a similar trend, whose range of variations is made further wider and unpredictable by the dominant role of the composite quality factor Q.

As a result, the standardized variance ς 2 W can get quite larger than the 1/3 value expected in the ideal case. This value is attained only asymptotically as M M → ∞, a condition that can be thought as well approximated as the working frequency increases, since the modal density will increase (on average) quadratically with the frequency, while the composite quality factor has typically a slower rate of growth. In any case, (4) links the rate of convergence of the statistics of W to the average number of overlapped modes M M : hence, by settling on an acceptable error between the real and ideal ς 2 , we can come up with a threshold on the minimum number of modes that should overlap on each other bandwidth, as a way to ensure this maximum error.

In fact, the results here recalled are just sensible and can be understood by recalling that any component of the electric field is actually made up by a discrete superposition of contributions, with a small number of them dominating the final result. If all of these terms are treated as random variables, ideally independent, the central-limit theorem ensures that their sum, even weighted, can be expected to converge to a normally distributed random variable, with a degree of accuracy that broadly increases with the number of terms involved. In other words, a large number of independent degrees of freedom are needed. Here we can see two physical reasons that can put in jeopardy the normal-distribution hypothesis: 1) there is no way of ensuring that the modes are excited as independent random variables by a single source, even when applying a stirring technique; 2) the number of degrees of freedom is actually directly related to the number of modes effectively excited at the working frequency. The first point is hard to be assessed, as the actual excitation of the modes is intimately related to the modal topographies, which are hardly accessible in reallife configurations. We are convinced that a given amount of residual correlation is always present, and that it can lead to surprising results when neglected [START_REF] Cozza | A skeptic's view of unstirred components[END_REF]. Concerning the second point, the advantage of a modal approach is that it makes visible the intrinsical discrete nature of the degrees of freedom actually needed to describe the electromagnetic field within a cavity. Even more importantly, it allows not getting out of sight of a fundamental phenomenon, i.e., the bandwidth of the frequency response of each mode is strongly dependent on the amount of losses associated to each modal topography. In such conditions, the central-limit theorem cannot be fully invoked, so that disagreements with the ideal-scenario requirements should not come as a surprise.

Since M M (f ) can be reasonably expected to broadly increase with the frequency, (4) predicts that the standardized variance of the energy density should converge to the asymptotic value 1/3 at high frequency. This fact is demonstrated in Fig. 1a, where experimentally determined estimates of ς 2 W are shown over a wide frequency range, as obtained with a 100-step mechanical stirrer in an RC with a volume of 13.8 m 3 . The RC was excited by means of a log-periodic dipole antenna pointed at one corner, while the field samples were measured with an electro-optical probe positioned at the center of the cavity. Fig. 1 confirms that indeed the error ǫ ς 2 = ς 2 W -1/3 broadly decreases with the frequency, while it is far from negligible in the lower frequency range. The thicker line in Fig. 1 has been obtained by applying (4), using a loose estimate for M M . This latter quantity was derived by using the average modal density predicted by Weyl's formula and a smooth majorant of an experimentally-derived estimate of the composite quality factor. This operation was aimed at assessing whether (4) is capable of providing a loose upperbound for the error observed in practice on the standardized variance ς 2 W . Indeed, a good agreement between the prediction of the maximum error yielded by (4) and the experimental results is observed, with an envelop well identified by the former. The case treated in Fig. 1b deals with the introduction of additional losses: in this case too our model predicts in a fairly good way the reduction of the variability of W . Further details about the validity of (4) are provided in [START_REF] Cozza | The role of losses in the definition of the overmoded condition for reverberation chambers and their statistics[END_REF], particularly about the role of losses in the statistical compliance of the energy density in an RC.

III. STATISTICAL COMPLIANCE

In the present work we are rather more interested in utilizing the physical understanding brought by (4), for interpreting why an RC can present a non-compliant behavior at some specific frequencies. As already recalled, the idea of unstirred field components and ineffective stirring techniques are often invoked in such cases. But another possibility is perhaps more likely: the fact that if at a given frequency the modal overlapping (represented by M M ) is not large enough, the number of degrees of freedom effectively involved in the generation of the electromagnetic field is not high enough to lead to a good approximation of a Gaussian law. Hence, a larger variance for the electric energy density, with respect to that expected in the asymptotic modeling. It is noteworthy to recall that this argument is based on the assumption of an ideal stirring technique, thus involving a random excitation of the modes and a random positioning of their resonance frequencies.

In order to assess the soundness of this explanation, we have given a closer look at the field samples measured for an empty cavity at two frequencies where our RC can be regarded as behaving ideally or not; we have chosen 1.438 GHz and 1.441 GHz to represent this phenomenon, with an error ǫ ς 2 about 0 % and 100 %, respectively. These frequencies are almost three times higher than the LUF of our RC, intervening at about 550 MHz, assessed as required in [START_REF]Reverberation chamber test methods[END_REF]. The stirring paddle is about 5 wavelengths wide, and non-periodic, so that hardly any doubts can subsist on its effectiveness. At the same time, the likeliness of the presence of unstirred components has been minimized by orienting the excitation antenna in such a way as to have its main lobe of radiation pointing towards a corner of the RC, as to be reflected over the stirrer and avoid any direct path towards the probe.

The empirical cumulative distribution functions (cdfs) of the real and imaginary parts of the three Cartesian components of the samples are shown in Fig. 2, for the two frequencies chosen above. The first thing to notice is that for f = 1.441 GHz the statistical non-compliancy of our RC cannot be explained by any unstirred component alone. As a matter of fact, it even seems, on a qualitative level, that some field components is actually more biased for f = 1.438 GHz, though in this case the energy density presents a very good agreement with a χ 2 6 law. A closer look suggests that the non-compliancy is likely due to the higher kurtosis featured at 1.441 GHz: 3.7 on average against 3.0 at 1.438 GHz, the latter corresponding to the kurtosis of a Gaussian law. This is clearly visible in the longer transitions from the central part of the cdf towards its ends. Recalling that the field is related to the energy density by a square power, the kurtosis is directly linked to the variance of W , while the field variance is rather related to average value of W . As predicted by (4), in the case of a low modal overlapping, the standardized variance is to increase with respect to the asymptotic case: this is what is observed in these results.

Hence, it seems likely that the non-compliancy is better explained by a local limitation of the RC rather than the stirrer or other phenomena. As a matter of fact, it is hard to imagine why a stirrer and the positioning of the sources should work fine over a majority of frequencies, while leading to a bad performance just over a handful of frequencies. On the other hand, the concept of low modal overlapping can easily explain this disturbing phenomenon, as the combination of high composite quality factor and local modal depletion can indeed appear. Moreover, the fact that the insertion of losses yields an improved performance (see Fig. 1), i.e., a lower incidence of non-compliant frequencies, supports this interpretation, as already pointed out in [START_REF] Cozza | The role of losses in the definition of the overmoded condition for reverberation chambers and their statistics[END_REF].

A higher statistical dispersion has a direct impact on the results of compliancy tests required in international standards [START_REF]Reverberation chamber test methods[END_REF]. As a matter of fact, the upper-bounds defined for the maximum acceptable spatial variability of the maximum field intensity are based on an asymptotic model [START_REF] Ladbury | Evaluation of the NASA Langley Research Center mode-stirred chamber facility[END_REF]. Hence, a higher variability for the field samples implies a higher 3. Theoretical quantile-quantile plot of the asymptotic distribution laws for W in the asymptotic and non-asymptotic case, as based on (4), for a number of average overlapped modes M M = {1, 3, 10, 20, 50}. The dashed line stands for the bisector, which is asymptotically approached as M M increases. The size of the sample population is not needed in this type of plots, as it only intervenes in setting the confidence interval, with no effect on the relationship between the two quantile distributions. Apparent significance level α ′ associated to the one based on asymptotic statistics, α. As the number of overlapped modes M M decreases, the statistical interpretation of the arithmetic mean of experimental samples can be increasingly misleading, due to a higher variance with respect to the samples mean.

probability for them to get closer to the upper-bound. The context of this paper is not suitable for studying the statistical properties of the extreme values of W and the related squared field components. Still, as shown in [START_REF] Hoijer | Maximum power available to stress onto the critical component in the equipment under test when performing a radiated susceptibility test in the reverberation chamber[END_REF], there exists a direct relationship between the average value of the energy density components and their maximum value, as estimated from a finite population of samples. A similar reasoning applies for the energy density W , which is our case of interest in this paper.

It is therefore interesting to assess how the increment of variability predicted by (4) could affect the probability of having samples out of a given confidence margin. This latter could be set, e.g., in order to ensure a minimum accuracy for a measurement or, as it is the case for industrial tests, the reproducibility of a test result.

To this end, let us consider a number N of independent samples W i large enough as to have their arithmetic average W converging towards a normal law, thanks to the centrallimit theorem. We aim at computing how a confidence margin defined for a significance level α under the hypothesis of an asymptotic behavior, translates into another confidence margin whenever M M happens not to be high enough. A good and intuitive way of assessing the effect of a finite M M is to check how the quantiles of the asymptotic case relate to those of the modal description we have introduced. As we are dealing with this problem under the approximation of a normally distributed W (central-limit theorem), the quantiles are given by [START_REF] James | Statistical methods in experimental physics[END_REF]

q p = µ + σ √ 2erf -1 (2p -1) , (6)
where p = P ( W < q p ) is the probability associated to the quantile q p , while µ and σ are the expected mean-value and the standard deviation for W . The central-limit theorem allows to compute a good approximation of the quantiles just by knowing these two moments, for the case of an ideal χ 2 6distributed W i and for the more realistic case provided by [START_REF] Primiani | Numerical and experimental investigation of unstirred frequencies in reverberation chambers[END_REF]. The result of this operation is presented in Fig. 3, where any confidence margin established on the asymptotic case for a given significance level α is shown to lead to an inevitably larger margin with the same significance level as soon as M M is found to be finite. It is interesting to notice that the case M M = 3, often regarded as a good compromise for a diffused field in room acoustics, actually provides a more than twofold increase in the original confidence margin, whereas the case M M = 20 suggested as a tentative threshold on M M in [START_REF] Cozza | The role of losses in the definition of the overmoded condition for reverberation chambers and their statistics[END_REF] settles to an about 50 % larger interval.

The other way round, choosing the same interval margin for the ideal and non-ideal cases, if this interval corresponds to a significance level α for the former case, this will lead to a corresponding significance level α ′ in the latter, related as

α ′ = 1 -erf q 1-α/2 √ 2 M M 18/π + M M . ( 7
)
This function is plotted in Fig. 4, for several values of α. These results provide a direct feeling about the increased probability of incurring into samples falling outside the originally intended confidence margin. The probability α ′ of this event increases not only when M M is relatively low, but also when the significance level α is reduced. Such relationship has a potentially harmful impact, as α is typically reduced in order to improve the significance of the results of a statistical test: when expecting asymptotic results from a realistic RC, this risks leading to a higher rate of rejection of eventual hypothesis tests, rather than the opposite. One straightforward way of reducing this eventuality is to increase the average modal overlapping for the RC, as suggested by (4): the most practical approach is likely to slightly increase the average amount of dissipated power, as long as this procedure is not against minimum field intensity constraints.

IV. CONCLUSIONS

In this paper we have briefly recalled the reasons of our introducing a physically-motivated definition of the overmoded condition for an RC. The use of a modal approach, though inevitably based on simplifying assumptions, has led to pointing out that there is no threshold frequency for an real RC to behave as an ideal one. Local disagreements for the statistics of field-related quantities can appear at any frequency, depending on eventual configurations featuring a weak overlapping of modes. The probability for these events is intuitively expected to decrease for an increasing frequency, and the proposed model indeed supports this idea. Still, having put the assumption of an ideal stirring technique at the heart of our model, while it admits configurations where an RC can present a non-compliant statistics, it is no more reasonable to infer that statistical non-compliancy is a synonyms of nonideal stirring techniques. This fact was further discussed in this paper, showing how the excess statistical dispersion found in practical configurations cannot be explained with respect to the stirring techniques, but is more likely due to an intrinsical local limitation of the RC, particularly its finite modal overlapping at certain frequencies. This fact becomes critical as soon as an ideal RC behavior is taken for granted.

Fig. 1 .

 1 Fig. 1. Estimates of the standardized variance of the energy density W , as assessed from experimental results obtained for : (a) an empty cavity (apart for the excitation antenna) and (b) one loaded with a set of 4 pyramidal absorbers about 30 cm high.

Fig. 2 .

 2 Fig. 2. Empirical cumulative distribution functions for the real and imaginary parts of the three Cartesian components of the electric field, as estimated from 100-sample populations for two frequencies: (a) 1.438 GHz and (b) 1.441 GHz.

  Fig.3. Theoretical quantile-quantile plot of the asymptotic distribution laws for W in the asymptotic and non-asymptotic case, as based on (4), for a number of average overlapped modes M M = {1, 3, 10, 20, 50}. The dashed line stands for the bisector, which is asymptotically approached as M M increases. The size of the sample population is not needed in this type of plots, as it only intervenes in setting the confidence interval, with no effect on the relationship between the two quantile distributions.

  Fig. 4.Apparent significance level α ′ associated to the one based on asymptotic statistics, α. As the number of overlapped modes M M decreases, the statistical interpretation of the arithmetic mean of experimental samples can be increasingly misleading, due to a higher variance with respect to the samples mean.


Andrea Cozza
email: andrea.cozza@supelec.fr

A Statistics-Based Definition of the Overmoded Condition for Reverberation Chambers

Current understanding of reverberation chambers is firmly funded upon their being operated as overmoded cavities. A proper definition of this condition has been lacking for a long time, while being regarded as automatically satisfied as soon as a large (and unspecified) number of normal modes are set to resonate at the frequency of operation of the chamber. In this paper we summarize the most important steps needed to introduce in a formal way a proper definition of the overmoded condition, showing how, from a strict mathematical viewpoint, it is not related to any threshold frequency, but rather requires a statistical framework.

Introduction

Reverberation chambers (RCs) are electrically large closed cavities characterized by very weak dissipation phenomena over its boundaries (walls) and within their volume. Apart as a fundamental tool for experimental investigations in quantum chaos and other fundamental research topics, RCs are widely used as tests facilities in Electromagnetic Compatibility (EMC), mostly in the case of radiated tests. The interested Reader can refer to [START_REF] Hill | Electromagnetic theory of reverberation chambers[END_REF][START_REF]Reverberation chamber test methods[END_REF] as a useful introduction on the way RCs are actually utilized in EMC and their physics. The topic of concern in this paper is rather the fact that their use is based on a strong assumption, that they are capable of providing a diffused field. This property is never directly invoked in RC theory as developed within the EMC community, though it is of fundamental importance in the physics of cavities and the study of room acoustics [START_REF] Schroeder | Statistical parameters of the frequency response curves of large rooms[END_REF]. Historical reasons are likely to blame for this mismatch, as well as the fact that electrical engineers have long preferred to have a more practical and test-driven approach to the understanding of these facilities. Still, the lack of the clear definition of a diffused field has been an invisible obstacle, in our opinion, to the cross-fertilization that should have existed between the EMC community and the acoustics one. Indeed, EMC engineers could benefit from the deeper understanding and modeling tools developed in acoustics. In this respect, we will regard the concepts of overmoded RC and that of diffused field as synonyms, as both refer to a cavity that ensures, asymptotically, to expect a very specific statistical behavior for field-related quantities.

Yet, both fields lacked a proper physics-based model capable of explaining under what conditions a cavity can be considered to allow a diffused field to establish. Acoustics, too, tends to consider the idea of a threshold frequency [START_REF] Schroeder | Statistical parameters of the frequency response curves of large rooms[END_REF], often referred to as Lowest Usable Frequency (LUF) in EMC. In a recent paper [START_REF] Cozza | The role of losses in the definition of the overmoded condition for reverberation chambers and their statistics[END_REF], we have proposed such a model, establishing a direct link between the physics of an RC and the statistical properties of the field it supports. In this paper we recall the most important steps and assumptions that have led to this result, and show how it can explain another phenomenon with an unsatisfying description, that of local statistical non-compliancy of field-related quantities. Our definition of an ovemoded cavity (and thus of a diffused field) is based on the use of a modal description of the field excited within an RC, coupled to a statistical approach. The interest of this approach is that it allows avoiding making non-physical assumptions on the need for a threshold on the modal density, as apparently assumed on the EMC side, or on the sheer fact that a threshold frequency exists. Of particular importance is the prediction that losses should not be regarded as a limitation or nuisance in the operation of RCs, but that they rather lie at the heart of its functioning. Without losses, an RC would never be capable of supporting a diffused field, independently of the modal density it supports: hence, our regarding the use of the term overmoded as a potentially treacherous habit, as the mere availability of a high modal density is not a sufficient condition. In the following, it will be shown that an overmoded RC is a cavity that supports a large number of overlapped modes, which is not the same as just supporting a large number of modes, as estimated by looking at modal density alone. This fact was unacknowledged in EMC, while well understood in acoustics: still, both have always considered that the idea of a threshold frequency be reasonable, a fact that is disproved in this paper.

The overmoded condition as a statistical asymptotic condition

The common sense given to the concept of overmoded cavity is tightly linked to the fulfillment of the following properties: 1) field statistics are independent of the position in space where they are tested, at least in a sub-volume of the cavity; 2) the three orthogonal components of the electric (respectively, magnetic) field behave as independent and identically (iid) distributed random variables; 3) they follow a Gaussian distribution law, with zero mean-value. It has been shown that these hypothesis are met whenever the field distribution can be described by means of a continuous random plane-wave spectrum, with specific statistical properties [START_REF] Hill | Plane wave integral representation for fields in reverberation chambers[END_REF]. Although such type of model well approximates the behavior of real-life RCs in their high-frequency range, it is well-known that there might exist some frequencies where it is incapable of predicting a drift from it [START_REF] Primiani | Numerical and experimental investigation of unstirred frequencies in reverberation chambers[END_REF]. These disagreements are often explained through a stirrer inefficiency or a sub-optimal alignment for the sources [START_REF] Lundén | How to avoid unstirred high frequency components in mode stirred reverberation chambers[END_REF], though we are quite skeptical of these explanations [START_REF] Cozza | A skeptic's view of unstirred components[END_REF], whereas in the lower frequency range they are usually just fitted to alternative probability distribution functions [START_REF] Orjubin | Statistical model of an undermoded reverberation chamber[END_REF].

It is, in our opinion, by far more interesting and useful to get back to the basics of the physics of cavities, while taking on a more statistical approach from the beginning. To this end, we will consider a modal expansion for the electric field distribution observed at the position r within an RC at the working frequency f [10]

E(r, f ) = i∈M γ i (r, f )ψ i (f ) ξi (r, f ) , (1)
where the γ i (r, f ) reflect the excitation of each mode, depending on the nature of the source and its position relative to the modal topographies, ψ i (f ) is the bell-shaped frequency response of each mode and ξi (r, f ) is the polarization of the electric field contribution provided by each mode. The sum in ( 1) is taken over the set M of modes that are effectively excited at the working frequency. A proper justification for this model has been provided in [START_REF] Cozza | The role of losses in the definition of the overmoded condition for reverberation chambers and their statistics[END_REF]. Rather than attempting to apply (1) in a deterministic way, we will regard all of these modal quantities as random ones, with statistical properties dictated (or suggested) by their physical properties.

In this respect, the modal weights {γ i } can be expected to behave as centered random variables, as they are directly related to the scalar projection of the equivalent currents representing a source and the modal topographies, which are standing waves, thus characterized by a pseudo-periodic change in their sign over space. No further assumption is needed on the nature of the probability law for the γ i . The functions {ψ i (f )} represent the responses of second-order systems, defined as follows

ψ i (f ) = 1 f 2 i (1 + j/2Q i ) 2 -f 2 , (2)
where f i is the resonance frequency of the i-th mode and Q i its quality factor. As the presence of a stirring technique implies, from an ideal perspective, that the frequencies of resonance of the modes are thoroughly mixed and swapped about the working frequency, it is reasonable to assume that the variables {f i } be uniformly distributed. Moreover, the quality factors {Q i } can be characterized by means of their average (or composite) value, as will be shown later. This latter quantity can be experimentally assessed. Finally, the polarization unit vectors { ξi } will be assumed to be uniformly distributed over 4π steradian, again as a consequence of the assumption of an ideal stirring technique at work. It is important to notice that assuming a perfect stirring is actually sensible, as we are interested in assessing whether it is possible to explain the eventual non-compliancy of an RC even having enforced an ideal performance on the stirring technique.

With this model at our disposal, we now aim at studying the first two statistical moments of the electric energy density

W (r, f ) = ǫ 0 E(r, f ) 2 , ( 3
)
with ǫ 0 the dielectric permittivity of the medium filling the cavity. The electric energy density can be described, in the case of an ideally diffused field, as a random variable distributed as a χ 2 6 law, as a consequence of the features recalled at the beginning of this Section. As opposed to this hypothesis, by computing the first two moments of W (r, f ) and employing the previously introduced assumptions on the statistics of the modal quantities should allow to check under what conditions the ideal behavior is actually met. The procedure for this calculation has been presented in details in [START_REF] Cozza | The role of losses in the definition of the overmoded condition for reverberation chambers and their statistics[END_REF], and leads to the following standardized variance

ς 2 W = σ W µ W 2 = 1 3 + 1 π µ 4 µ 2 2 1 M M , (4)
with M M = mB M the average number of modes overlapping within the average -3 dB bandwidth B M = f /Q centered around the working frequency, for a modal density m(f ) and an average composite quality factor Q. The terms µ n refer to the moments of the modal weights {γ i }

µ n = E [|γ i | n ] , (5)
so that µ 4 /µ 2 2 = 2 for the case of normally distributed modal weights, which is the usual assumption applied for these quantities. Other distribution laws do not have much of an effect on this ratio, so that the final result is weakly dependent on any assumption on the nature of the modal weights.

Comparing (4) to the standardized variance ς 2 χ 2 6 = 1/3 obtained for the ideal diffused field scenario shows what is needed for an RC to be as close as possible to the ideal case, i.e., presenting a large number of overlapped modes M M . Whence, it is clear that there is no reason for assuming that any threshold frequency ensures an RC to support a diffused field, as M M is actually a complex function of frequency, with a non monotonous trend. As a matter of fact, both the composite quality factor Q and the modal density m are non-monotonous; attention must be paid to the fact that the often used Weyl's approximation is indeed a smooth fitting curve to the real m(f ), which is generally not known. Local variations around this smooth function can be far from negligible, leading to modal depletion or excess: it can therefore be intuitively understood that M M follows a similar trend, whose range of variations is made further wider and unpredictable by the dominant role of the composite quality factor Q.

As a result, the standardized variance ς 2 W can get quite larger than the 1/3 value expected in the ideal case. This value is attained only asymptotically as M M → ∞, a condition that can be thought as well approximated as the working frequency increases, since the modal density will increase (on average) quadratically with the frequency, while the composite quality factor has typically a slower rate of growth. In any case, (4) links the rate of convergence of the statistics of W to the average number of overlapped modes M M : hence, by settling on an acceptable error between the real and ideal ς 2 , we can come up with a threshold on the minimum number of modes that should overlap on each other bandwidth, as a way to ensure this maximum error.

Since M M (f ) can be reasonably expected to broadly increase with the frequency, (4) predicts that the standardized variance of the energy density should converge to the asymptotic value 1/3 at high frequency. This fact is demonstrated in Fig. 1a, where experimentally determined estimates of ς 2 W are shown over a wide frequency range, as obtained with a 100-step mechanical stirrer in an RC with a volume of 13.8 m 3 . The RC was excited by means of a log-periodic dipole antenna pointed at one corner, while the field samples were measured with an electro-optical probe positioned at the center of the cavity. Fig. 1 confirms that indeed the relative error ǫ ς 2 = (ς 2 W -1/3)/(1/3) broadly decreases with the frequency, while it is far from negligible in the lower frequency range. The thicker line in Fig. 1 has been obtained by applying (4), using a loose estimate for M M . This latter quantity was derived by using the average modal density predicted by Weyl's formula and a smooth majorant of an experimentally-derived estimate of the composite quality factor. This operation was aimed at assessing whether (4) is capable of providing a loose upper-bound for the error observed in practice on the standardized variance ς 2 W . Indeed, a good agreement between the prediction of the maximum error yielded by (4) and the experimental results is observed, with an envelop well identified by the former. The case treated in Fig. 1b deals with the introduction of additional losses: in this case too our model predicts in a fairly good way the reduction of the variability of W . Further details about the validity of (4) are provided in [START_REF] Cozza | The role of losses in the definition of the overmoded condition for reverberation chambers and their statistics[END_REF], particularly about the role of losses in the statistical compliance of the energy density in an RC.

Figure 1 :

 1 Figure 1: Estimates of the standardized variance of the energy density W , as assessed from experimental results obtained for : (a) an empty cavity (apart for the excitation antenna) and (b) one loaded with a set of 4 pyramidal absorbers about 30 cm high.

Conclusions

In this paper we have briefly recalled the reasons of our introducing a physically-motivated definition of the overmoded condition for an RC. The use of a modal approach, though inevitably based on simplifying assumptions, has led to pointing out that there is no threshold frequency for a real RC to behave as an ideal one. Local disagreements for the statistics of field-related quantities can appear at any frequency, depending on eventual configurations featuring a weak overlapping of modes. The probability for these events is intuitively expected to decrease for an increasing frequency, and the proposed model indeed supports this idea. Therefore, it is uncorrect to consider that departures from the asymptotic statistical behavior of the field statistics be due to an imperfect design of the RC setup, be it the design of the cavity excitation or the stirring technique.


Nicolas P Rougier
email: nicolas.rougier@inria.fr

Georgios Is
email: georgios.detorakis@inria.fr

Detorakis

Self-Organizing Dynamic Neural Fields

In this paper, we propose a model of cortical self-organization based on the dynamic field theory. Learning is made through the modification of feed-forward connections using a time invariant learning rule that allows for dynamic (or life-long) learning. This preliminary model suggests that cortical plasticity may be conveyed through feed-forward connections only while cortico-cortical connections role would be to ensure dynamic competition among cortical columns.

Introduction

We introduced in [START_REF] Rougier | Dynamic selforganising map[END_REF] the dynamic self-organized map architecture that is a variation of the selforganizing map algorithm [START_REF] Kohonen | Self-organized formation of topologically correct feature maps[END_REF] where the original time-dependent (learning rate and neighborhood) learning function has been replaced by a timeinvariant learning rule. This modification allows the network to support life-long learning and may explain to some extent cortical plasticity. However, current implementation is not biologically plausible since it requires a central supervisor (to designate the winning unit) and the neighborhood influence is computed using a function. This article introduces preliminary results concerning a biologically plausible implementation using numerical, distributed and local computations, based on the original dynamic neural field definition [START_REF] Amari | Dynamic of pattern formation in lateral-inhibition type neural fields[END_REF].

The concept of self-organization using lateral connections is well-known and well-investigated, especially by [START_REF] Sirosh | Selforganizing feature maps with lateral connections: Modeling ocular dominance[END_REF][START_REF] Bednar | Self-organization of color opponent receptive fields and laterally connected orientation maps[END_REF]. In those works authors provide an algorithm for self-organization learning lateral weights using a Hebbian-like learning rule. On the other hand, we put forward a new approach of self-organization using a combination of a dynamic neural field and a Hebbian-like learning rule. Thus, self-organization can be achieved, learning only the feed-forward weights. Because of that property our model is quite simple, straightforward to implementation and it does not require any outlandish handling.

Model

The neural field theory has been introduced by [START_REF] Wilson | Excitatory and inhibitory interactions in localized populations of model[END_REF][START_REF] Wilson | A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue[END_REF] and latter formalized by [START_REF] Amari | Dynamic of pattern formation in lateral-inhibition type neural fields[END_REF][START_REF] Taylor | Neural bubble dynamics in two dimensions: foundations[END_REF]. This theory introduces a model at the level of a population of neurons in the form of an integro-differential equation describing the spatio-temporal evolution of coarse-grained variables such as synaptic or firing rate value [START_REF] Coombes | Neural fields[END_REF]. We will use notations introduced by Amari and consider the membrane potential to be governed by the following equation:

τ ∂V (x, t) ∂t = -V (x, t) + h + I(x, t) + M W (|x -y|)f (V (y, t))dy (1)
where V (x, t) designates the membrane potential at position x and time t, W (|x -y|) is the lateral connection weight function between position x and y (we assume here that the system is spatially homogeneous and isotropic), f is the mean firing rate function, I(x, t) is the input at position x and h is the resting potential.

Fitting input

Depending on the firing rate function f , the lateral connectivity function W and the input I, such fields are known to exhibit a range of dynamic behaviors going from spatially and/or temporally periodic patterns to localized regions of activity. This latter case has been extensively used in a number of work to model visual attention [START_REF] Rougier | Emergence of attention within a neural population[END_REF], motor control [START_REF] Erlhagen | The dynamic neural field approach to cognitive robotics[END_REF], working memory [START_REF] Spencer | An embodied approach to cognitive systems: A dynamic neural field theory of spatial working memory[END_REF], etc. In the following, we will consider the lateral connectivity function to be a Difference of Gaussians (DoG) of the form:

W (x) = Ae -x 2 /2σ A 2 -Be -x 2 /2σ B 2
and the firing rate function to be a simple positive linear function:

f (x) = x if x > 0 0 if x ≤ 0
The numerical simulation of such fields requires the spatial discretization of the domain M into n spatial elements while the temporal integration can be made using classical integration schemes. Considering M = [0..1] and h = 0, we can thus rewrite equation 1 as:

τ ∆V (x i , t) ∆t = -V (x i , t) + I(x i , t) + n j=0 W (|x i -x j |)f (V (x j , t))
with x i = i/(n -1). In the following, we will use the forward Euler integration scheme. Under these assumptions, we have been studying a set of parameters for the lateral weight connection that exhibit the following property: for any uniform and positive input I, the neural field converges towards a single localized packet of activity whose maximum is approximately equal to I. We do not have yet the formal proof of such behavior but we found the property to be very consistent over a wide set of numerical simulations using different parameters (n, I, ∆t). We report on figure (1) such a simulation where the field is able to fit a constant input of level 0.45 after convergence. Instead of a stereotyped packet of activity with a constant maximum, the field activation represents a measure of the input. We will now explain how to use such property to ensure self-organization.

Self-Organization

Let us now consider the slightly modified equation: where I is now considered to be uniform over M while a set of feed-forward weights W F has been added such that the actual input for any position

τ ∂V (x, t) ∂t = -V (x, t) + 1 - M |I(t) -W F (x)|dx + M W L (|x -y|)f (V (y, t))dy (2)
x is 1 -|I(t) -W F (x)|.
Considering the set of parameters given in the previous subsection and considering a given input I, if we have W F (x) = I, the actual input of the field would be 1. In such a case, we explained that the field maximum activity would match this value [START_REF] Rougier | Dynamic selforganising map[END_REF]. However, it is not possible to have such equality for any value of I because it would means to change all feed-forward weights at once. Nonetheless, we can restrict this equality to the support of the localized packet of activity at the equilibrium point. Said differently, it is sufficient to have W F (x) = I for x such that V (x, t) is not null at the equilibrium point. The goal of the learning rule is thus to reach such a state.

We considered the learning rule introduced in [START_REF] Rougier | Dynamic selforganising map[END_REF] that reflect two main ideas:

• If a neuron is close enough to the data, there is no need for others to learn anything: the winner can represent the data.

• If there is no neuron close enough to the data, any neuron learns the data according to its own distance to the data.

To achieve such behavior, we propose to consider the following learning rule:

∂W F (x, t) ∂t = ηL e (x, t) I(t) -W F (x, t) (3)
where L e (x, t) = M W Le (|x -y|)f (V (y, t))dy denotes the excitatory part of the lateral interaction such that W L = W Le -W Li and η is a constant learning rate. The modification of any feedforward weights is thus directly correlated with the closeness of weights to the input and this is modulated by the amount of lateral excitation. Since we know the final state of the model is a localized Gaussian-shaped packet of activity, learning occurs maximally in this vicinity.

Results

Experimental Setup

A

x, t + dt) -V (x, t)| ≤ ε.
Another sample is then drawn and the procedure is repeated for a given number of epochs.

Learning discrete values

We trained a network of 100 neurons for 2500 epochs and using as parameters, τ = 10, A = 1.5, σ A = 0.1, B = 0.75, σ B = 1.0, self-organization was obtained. In figure [START_REF] Kohonen | Self-organized formation of topologically correct feature maps[END_REF], is illustrated the results of the simulation. The feed-forward weights were randomly initialized (the red line) and after 2500 epochs the feed-forward weights were organized (the blue line), as the network learned the three input values (0, 1 /2,1). Hence, that step-like shape of feed-forward weights were caused because the first 20 neurons learned the value 0, the 20 middle neurons learned the value 1 /2, and the last 20 neurons learned the value 1. Moreover, the rest of the neurons learned different values from the input and may be that provide a smooth drift from one batch of neurons to another one.

Learning continuous values

We used the same network architecture as in introduced in previous paragraph but it has been trained on the S ∞ set. After 2500 epochs the network has learned the feed-forward weights. In ad- dition, the so-called step-like shape of feed-forward weights was substituted by an almost linear shape. That's because of the continuous nature of the input. It is to be noted that almost all values are represented but the bounds due to the non-toric nature of the network. This side effect is well-known in the case of Kohonen's map. Results are depicted in figure 3 while figure 4 shows the evolution of the receptive fields of unit #50 that slowly drifts from a weak and random response to a sharp localized one.

Discussion

We introduced a one dimensional dynamic neural field that can continuously and dynamically self-organize itself around a set of one-dimensional discrete or uniform values by modifying its feedforward connections and using the lateral weighted sum of excitation as a modulation signal for learn- ing. We are currently investigating the twodimensional case, due to obtain a more biological plausible model since the cortex can be modeled as a two-dimensional sheet of neurons. In this context, we would like to investigate meta-plasticity and/or homeostatic plasticity as it has been reported in [START_REF] Buonomano | Cortical plasticity: from synapses to maps[END_REF].

Finally, we would like to examine the properties of cortical reorganization under the presence of a lesion. Such lesions have been extensively investigated by [START_REF] Kaas | Plasticity of sensory and motor maps in adult mammals[END_REF] and others from a neurophysiological point of view and there is consequently a strong and detailed experimental background providing significant data which could feed our computational model.

Figure 1 :

 1 Figure 1: One dimensional neural field using n = 100, τ = 10, A = 1.5, σ A = 0.1, B = 0.75, σ B = 1.0. For any uniform and positive input level I, the neural field maximum activity is approximately equal to I after convergence.

Figure 2 :

 2 Figure2: A neural field has been trained for 2500 epochs on S 3 (0, 1 /2 and 1). Blue and red curves respectively show initial and final set of weights.

Figure 3 :

 3 Figure 3: A neural field has been trained for 2500 epochs on S ∞ . Blue and red curves respectively show initial and final set of weights.

  Figure 4: Evolution through time (t) of the receptive field of unit #50 from a network learning S ∞ .At each record time, learning is frozen and the network is presented successively with values ranging from 0 to 1 with step of 0.01. Activity level of unit #50 is recorded (after convergence) for each of these input values and is plotted for each record time.

	Unit activity	0.3 0.4 0.5 0.6 0.7	t = 25 t = 50 t = 100 t = 250 t = 500 t = 1000
		0.2
		0.1
		0.0 0.0	0.2	0.4	I	0.6	0.8	1.0