<figure><img src="image_35.png" /><figcaption>FIG. 2. Influence of the frequency range $k_M B_M$ on the relative variance of the mean-squared pressure (y-axis) obtained by MC simulation as a function of the average number $M_M$ of overlapping modes. Several values of $k_M$ have been considered $k_M = 1$ (bold solid black line), $k_M = 2$ (bold dashed black line), $k_M = 5$ (bold dotted line), $k_M = 10$ (bold dashed-dotted line), $k_M = 20$ (solid line), $k_M = 30$ (dashed black line), $k_M = 40$ (dotted black line), $k_M = 50$ (extra bold dashed line).</figcaption></figure>

the relative variance for different frequency ranges, i.e., different values of $k_M$. We can see that 50 modal bandwidths are enough to obtain non significant changes in the statistics. Note that $k_M$ is all the more important to assess that an insufficient frequency range would lead to overestimated values of the variability of the mean-squared pressure.

For reverberation chambers whose reverberation time is around 10 s, this leads to a relative bandwidth of about 1%, which is sufficiently narrow to assume a local constant modal density. The average modal overlap parameter $M_s$ was controlled by emulating different volumes at fixed working frequency of 1 kHz with an average damping constant of 0.7 rad/sec.

In order to check that the MC simulations are trustworthy we plot the mean values $\langle U^2(\omega) \rangle$ and $\langle W^2(\omega) \rangle$ that equal $n\pi/2\gamma$; these mean values were expressed by relations (A45) and (A46) in Davy's work, respectively. Fig. 3 shows the values obtained by MC simulation of $\langle U^2(\omega) \rangle$ (circles) and $\langle W^2(\omega) \rangle$ (squares); the analytical expressions given by (A45) and (A46) are also reported (solid line), and this, as a function of the average number of modes $M_M$ overlapping in the -3-dB modal bandwidth. The good agreement found allows us to consider that (A45) and (A46) are consistent.

The same simulations are used now to check the mean value of the cross product $U^2(\omega)W^2(\omega)$ given by (A50/A53). Fig. 4 shows the values of $\langle U^2(\omega)W^2(\omega) \rangle$ obtained by MC simulation (circles) and by using the analytical expression (solid line) given by (A53). The relative error (dashed line) is also reported in Fig. 4 (right y-axis). In order to raise possible doubts about the ana-

<figure><img src="image_36.png" /><figcaption>FIG. 3. MC simulation results of $\langle U^2 \rangle$ (circles) and $\langle W^2 \rangle$ (squares) as a function of the average number $M_M$ of overlapping modes. The analytical expressions (which are identical for both quantities) have also been reported (solid line) and are given by eq. (A45) and (A46) of Davy's work$^2$, respectively.</figcaption></figure>

<figure><img src="image_37.png" /><figcaption>FIG. 4. Evolution of $\langle U^2 W^2 \rangle$ (left y-axis) as a function of the average number $M_M$ of overlapping modes obtained by MC simulation (circles) and by using the analytical expression (solid line) given by (A53) in Davy's work$^2$. The relative error (right y-axis) is also reported (dashed line).</figcaption></figure>

Analytical expression of (A53) reported in Fig. 4, the reader can check that, in the case of single-tone excitation, equation (A53) can be simplified as follows,

$$ \langle U^2(\omega) W^2(\omega) \rangle = n \langle p_m^4(\mathbf{x}) \rangle \langle p_m^4(\mathbf{y}) \rangle (h^2 \# g^2(0)) + n^2 \langle p_m^2(\mathbf{x}) \rangle^2 \langle p_m^2(\mathbf{y}) \rangle^2 (h \# h(0) g \# g(0)), \quad (34) $$

since only one receiver and one emitter are considered and $h \# g(0) = 0$ from (A39); note that # refers to the convolution operator in Davy's notations.

We clearly see that the analytical expression of Davy overestimates $\langle U^2(\omega)W^2(\omega) \rangle$. The fact that we deal with a single receiver and a single source has the advantage of simplifying (A50) (from which ensues (A53)).
<figure><img src="image_38.png" /><figcaption>FIG. 5. Relative variances obtained by Monte Carlo simulation (black dots), Davy's model (dashed line) and the model derived in the present work (solid line) as a function of the average number $M_M$ of overlapping modes. The shift shows that Davy's$^2$ and Lyon's$^8$ results are not consistent with the model assumed in their derivation, and overestimates the dispersion of the mean-squared pressure.</figcaption></figure>

If we thoroughly analyze (A50), it appears that Davy double counted some elements in the last term. This double counting is at the origin of the overestimation of $\langle U^2 W^2 \rangle$. In other terms the double sum $\sum_{m,l=1}^K$ should be replaced by $\sum \sum_{m,l=1,m \neq l}$, where $K$ stands for a given number of modes spreading over an angular frequency range $\Omega$. This “local” overestimation is the cause of the relative-variance discrepancy.

Indeed, Fig. 5 shows resulting relative variances obtained by MC simulations (black dots) and Davy's expression (dashed line) given by (33). We can clearly see a shift that confirms the inconsistency due to the elements pointed out previously and an overestimation of the expected relative variance. Conversely, we can see that the model derived herein (solid line) is consistent with the MC simulations performed on the basis of the assumptions made upon the different modal parameters.

In order to be more accurate in the analysis of the inconsistency, the number of terms counted in the sum was $K^2$ (that explains the $n^2$ term in (A53)) instead of $K(K-1)$. It is clear that at high modal overlap both can be confounded, but not at low modal overlap, which is the region of interest. This explains that the error is all the more important as the modal overlap lowers (see Fig. 4).

Now, we may wonder about the possibility to correct Davy's expression by taking into account $K(K-1)$ terms instead of $K^2$ terms. If do so, given that Davy related the ratio $K/\Omega$ to the modal density such that $K/\Omega = n$, it follows that relation (34) reads,

$$ \langle U^2(\omega) W^2(\omega) \rangle = n \langle p_m^4(\mathbf{x}) \rangle \langle p_m^4(\mathbf{y}) \rangle (h^2 \# g^2(0)) + \left(n^2 - \frac{n}{\Omega}\right) \langle p_m^2(\mathbf{x}) \rangle^2 \langle p_m^2(\mathbf{y}) \rangle^2 (h \# h(0) g \# g(0)), \quad (35) $$

with the inevitable question of assessing $\Omega$. In order to include all the modes in the modal expansion of the sound

field, $\Omega$ was logically taken to infinity by Davy, but with the inability to isolate only the cross terms in the double sum of (A50/A53). So, where is the problem ?

The problem lies in the definition of the ensemble average of the frequency response that was defined, in the case of $h^2(\omega - \omega_m)$ for instance, as follows

$$ \langle h^2(\omega - \omega_m) \rangle = \frac{1}{\Omega} \int_0^\Omega h^2(\omega - \omega_m) d\omega_m. \quad (36) $$

This expression means that an eigenfrequency is uniformly distributed over $\Omega$ defined as the frequency range over which the modal expansion of the sound field is considered. Recalling that the ensemble average applies to an ensemble of rooms that share approximately the same characteristics (volume, reverberation time,...), relation (36) means that the $10^{th}$ and the $10000^{th}$ eigenvalues, taken out from this ensemble, would be found to be uniformly distributed over the same frequency range $\Omega$ that turns out to be infinite ! It is clear that it not so. Instead, each of these will be found to be uniformly distributed (on an ensemble of rooms) over frequency ranges that will likely not overlap. This is why the ensemble average of the frequency responses must be defined as in (25).

Moreover, setting $K/\Omega = n$ is clearly incorrect for 3D structures for which the modal density is a quadratic function of frequency, i.e., the modal density is non linear. But one may wonder why the analytical expression of $\langle U^2 \rangle$ and $\langle W^2 \rangle$ were found to agree with MC simulation as shown in Fig. 3 ? This is due to the fact that $n$ is indeed the right probability density function to set, but not for the reasons exposed in Ref.2. So, to deal with the cross terms, one need to follow the approach presented in appendix.

The next step consists in checking if the assumptions made on the different modal quantities allows us to find a good agreement with the relative variance of the mean-squared pressure estimated from experimental data.

## V. MODEL VALIDATION

Although the consistency of the model was checked, we need to verify if (32) still overestimates relative variances with respect to those obtained in practice.

To this end we performed measurements in a reverberation chamber at the Laboratoire National de métrologie et d'Essais, whose volume is 218 m$^3$.

The ensemble average estimation was based on 64 realizations that consisted in displacing pairs of receiver/source positions. Although it is of common practice to characterize the chamber with noise signals such as the interrupted noise method$^{19}$, we chose to use a calibrated pulse with a “witness” microphone close to the source in order to check the reproducibility of the emitted pulse. This pulse was such that no zero appears in the [0:10 kHz] frequency range.

The exits of the microphones were amplified and then plugged into a digital scope configured in such a way as to access 4 million temporal samples with a sampling frequency of 200 kHz. A 50 kHz anti-aliasing filter was used
<figure><img src="image_39.png" /><figcaption>FIG. 6. Estimated reverberation times obtained when the re-
verberation chamber is empty (circles) and loaded (squares),
respectively</figcaption></figure>

<figure><img src="image_40.png" /><figcaption>FIG. 7. Estimation of the average number of overlapping modes in the half-width bandwidth of a mode when the re-verberation chamber is empty (circles) and loaded (squares), respectively.</figcaption></figure>

and the spectra were computed by using Fast Fourier
Transform. The resulting resolution was 0.05 Hz in the
frequency domain.

The frequency range of interest has been reduced to
[250:10kHz] in order to visualize the diffuse and non-
diffuse regimes in the chamber, and this, with a minimum
of one mode overlapping in the modal bandwidth.

The chamber was used in two configurations. In the
first configuration the chamber was empty and in the sec-
ond case, some absorbers were introduced (loaded case).
Reverberation times, classically referred to as T<sub>60</sub>, were
computed over one-third octave bands and are shown in
Fig. 6 for the empty (circles) and loaded (squares) cases,
respectively.

Using the modal density given by (24), we estimated
the average number *M* of overlapping modes; these
are shown in Fig. 7 for the empty (circles) and loaded
(squares) cases, respectively. The effect of the losses is
especially observed at low frequencies.

As relative variances are estimated with no spatial averag-
ing, the unavoidable dispersion is expected to be
large. In order to lower this dispersion and to be able to
extract the mean trend, we made use of frequency averag-

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /><figcaption>FIG. 8. Relative variances of the mean-squared pressure ob-
tained experimentally (thin solid line), by the analytical ex-
pression given by (32) (bold solid line) and by MC simulation
(black dots) with the corresponding uncertainty bars (verti-
cal bars), for the empty (upper plot) and loaded cases (lower
plot), respectively. Weaver’s model (dashed line) given by
(37) and Davy’s/Lyon’s model (dotted line) given by (33) are
also reported.</figcaption></figure>

ing. To that end, we assessed the larger modal bandwidth
on the [250:1kHz] frequency range which turned out to be
around 0.5 Hz. Consequently, we decimated the data by
2 Hz steps, i.e., four times the modal bandwidth in the
worst case (and more in other cases), which allowed to
consider uncorrelated data in the frequency domain. To
reduce the experimental uncertainty we processed mov-
ing averages over ten contiguous points. We obtained the
relative variances given by Fig. 8.

By minimizing the distance between experimental data
and (32), i.e., by using a least-square method, we found
optimized values for the K-factor of 2.75 and 2.63 for
the empty and loaded cases, respectively. It turns out
that these values are close to the value of 2.7 found by
Langley and Brown¹⁸. Accordingly, we adopted the same
value in (32). By using the results given in Fig. 7 we
managed to assess *M*<sub>s</sub>(*f*) to use in (32). The resulting
curve is reported in Fig. 8 (solid line) for the empty case
(upper plot) and the loaded case (lower plot). Note that
the slight irregularity of solid lines is directly due to the
slight irregularity of *M*<sub>s</sub>(*f*) observed in Fig. 7.

We can note that the analytical result matches quite
well with the average trend obtained experimentally. In
order to check the consistency of our model, we per-
formed MC simulations based on 50000 realizations. As
$K^2 \simeq 7.3$, modal weights at the source and at the receiver, were assumed to follow triangular and normal distributions, respectively. It is worth emphasizing that this choice only aimed at ensuring $K^2 \simeq 7.3$ and must not be regarded as a choice supported by any physical argument. Mean values of the resulting relative variances are reported in Fig. 8 (black dots) where quite a good consistency with the analytical model can be observed. The slight shift is likely due to the estimation uncertainty of the reverberation time which impacts on the estimation of $M_s$.

As experimental variances are deduced from sets made up of 64 samples, an unavoidable uncertainty has to be taken into account and is highlighted by the experimental dispersion observed in Fig. 8. To estimate this dispersion, MC simulations were used again; this merely consisted in computing 50000 times estimated relative variances obtained over sets composed of 64 realizations. In order to take into account the moving average over 10 (uncorrelated) contiguous points the data were grouped by 10 and averaged, so that 5000 resulting realizations were available. From these sets of data 95% confidence intervals were computed and reported by vertical bars for both configurations in Fig. 8. We can see that the experimental dispersion is quite coherent with the one obtained by MC simulation. In other terms, the assumptions of independence between the different modal parameters, including the independence between eigenfrequencies, on the one hand, and the $K$ factor of 2.7, on the other hand, allows us to observe a good agreement, not only with mean values but also with the order of magnitude of the unavoidable experimental and statistical dispersion. As $K=3$ is the commonly accepted value, we plotted (32) in Fig. 8 for such case (dashed-dotted line). As it could be expected, the difference with respect to the previous $K$ value, i.e., $K=2.7$, is all the more visible that the number of overlapping modes is weak, i.e., for the empty case (upper plot).

We superpose in Fig. 8 the expression of the relative variance derived by Weaver, whose expression is$^4$,

$$ \mathrm{RelVar}(|p(x, \omega_w)|^2) = 1 + \frac{K^2 - 3}{M_s}, \quad (37) $$

for $K=3$. We observe the good agreement which made this model accepted in the community. However, as pointed out in Sec. II, the model cannot be regarded as able to properly take into account the repulsion effect of GOE systems. In terms of agreement, we can see that the model derived herein, based on an independence assumption between the eigenfrequencies, is able to predict the relative variance decrease observed at low modal overlap. Conversely, if we report Lyon and Davy's formula, given by (33), in Fig. 8 (dotted line), we can see that it overestimates indeed the relative variance obtained in practice and that the overestimation is statistically significant for reasons explained in section IV.

# VI. CONCLUSION

By revisiting Lyon and Davy's derivation of the relative variance of the mean-squared pressure where the mean-spacing distribution is assumed to follow a Poisson distribution, an inconsistency in the model was highlighted and was proved to be at the origin of the overestimation observed when compared to experimental data.

The resulting corrected model was analytically derived and comparisons with experimental and simulation data were found to be in good agreement. A special care to the dispersion was also taken into account and a good agreement was also found. This agreement shows that the repulsion effect has likely little influence at low modal overlap. Lyon had already proven that this effect was weak for the case of input impedances. The present work confirms in a way this finding for the case of the relative variance of transmission functions.

Finally, it was found that $K \simeq 2.7$ provides the best fit to the experimental data and converges towards the figure proposed by Langley and Brown. The value of the $K$ factor is however not yet understood and is still under investigation.

# APPENDIX: A

In order to derive $E_{|\psi|^2|\psi|^2}$ defined by (18), it is worth noting that,

$$ \langle \sum_n |\psi_n|^2 \rangle^2 = \sum_m \langle |\psi_m|^2 \rangle^2 + \sum_{i \neq j} \langle |\psi_i|^2 \rangle \langle |\psi_j|^2 \rangle. \quad (A.1) $$

Recalling the assumption of independence and considering (A.1) allows us to restate $E_{|\psi|^2|\psi|^2}$ as follows,

$$ \begin{aligned} E_{|\psi|^2|\psi|^2} &= \sum_{i \neq j} \langle |\psi_i|^2 \rangle \langle |\psi_j|^2 \rangle \\ &= \langle \sum_n |\psi_n|^2 \rangle^2 - \sum_m \langle |\psi_m|^2 \rangle^2 \quad (A.2) \\ &= E_{|\psi|^2}^2 - \sum_m \beta_m^2, \end{aligned} $$

where,

$$ \begin{aligned} \beta_m &= \langle |\psi_m|^2 \rangle = \int_{\omega_k - \frac{\Delta\omega}{2}}^{\omega_k + \frac{\Delta\omega}{2}} \frac{1}{\Delta\omega} |\psi_k(\omega)|^2 d\omega_k \quad (A.3) \\ &\approx \int_{\omega_k - \frac{\Delta\omega}{2}}^{\omega_k + \frac{\Delta\omega}{2}} \frac{1}{\Delta\omega} |\psi(\omega_k - \omega_w)|^2 d\omega \end{aligned} $$

A Taylor's expansion taken at first-order allows us to simplify $\beta_m$ such that,

$$ \beta_m = |\psi(\omega_k - \omega_w)|^2. \quad (A.4) $$

From a graphical point of view, it is worth noting that $\beta_m$ corresponds to an area on a $\Delta\omega$ interval centered at $\omega_k$. Consequently, we obtain,

$$ \sum_m \beta_m^2 = E_{|\psi|^4}. \quad (A.5) $$
Finally, $E_{|\psi|^2|\psi|^2}$ reads,

$$E_{|\psi|^2|\psi|^2} = E_{|\psi|^4} \left( \frac{E_{|\psi|^2}}{E_{|\psi|^4}} - 1 \right) \quad (A.6)$$

## ACKNOWLEDGMENTS

Thanks to Christophe Conessa for his help during the measurements.

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# Average Number of Significant Modes Excited in a Mode-Stirred Reverberation Chamber

Florian Monsef, Member, IEEE, and Andrea Cozza, Senior Member, IEEE

**Abstract**—Although the number of significant modes is intuitive, this concept has never been clearly defined, mainly because of the unbound number of modes involved in modal overlap. In the present paper, we show that, for a perfect stirring process, the effect of modal overlap can be modeled as an equivalent filtering formulation. By introducing the statistical-bandwidth concept, we show that the electromagnetic field statistics due to an infinite number of modes can be summarized by a finite number of significant modes. The case of the electric-energy density in an mode-stirred reverberation chamber has been considered and a new expression of its variability has been established. The good agreement found between the new expression and the experimental and simulation results support the several concepts introduced in this paper.

**Index Terms**—Cavity resonators, electromagnetic compatibility (EMC), modal analysis, parametric statistics, Reverberation chamber (RC).

## I. INTRODUCTION

MODE-STIRRED reverberation chambers (MSRC) are microwave complex environments in which it is of common use to regard the electromagnetic field as a random process. This approach is all the more justified when the chamber is overmoded, i.e., when a large number of modes of the chamber are excited. Indeed, in that case, tests indicate that the ratios of peak to average values of the field remain approximately the same [1]–[3] throughout the central part of the chamber. However, for metrology purposes, the degree of uniformity of such ratios need to be assessed, and this, even for the undermoded case, i.e., near the lowest usable frequency (LUF), for which field statistics differ from the overmoded case.

An important part of the measured-field uncertainty is directly due to the medium complexity. This is why the study of the MSRC is based on statistical tools [4]–[6] and aims at establishing statistical models able to describe the electromagnetic field from the undermoded to the overmoded regime. The most common approach in the community consists in finding probability density functions (pdf) able to describe the statistics of field-related quantities [7]–[9] from the LUF to the overmoded case. An alternative approach consists in modeling the field by

using a modal expansion [10], [11]; the pdfs to choose are this time those applied to modal parameters. In [10], the derivation was based on a finite number of modes to which a limit to infinity was applied as required by the modal theory, whereas in [11], the modal expansion was only including a finite number of contributing modes.

The notion of contributing mode is intuitively linked to the modal density but the concept has never been clearly defined. In [11] for instance, this number was arbitrarily set to the average number of modes overlapping in the average modal bandwidth. In [10], the bandwidth over which this number had to be considered was defined but not quantitatively specified as the aim was to include all the modes. The need to determine this significant number of modes is all the more important that it would provide a metric linked to the degree of overmodedness (or undermodedness) of an MSRC.

To clarify the significant mode concept, we need to regard modal overlap from an alternative point of view; to this end, we will present in Section II how the overlap in the frequency domain of an infinite number of modes can be summed up to the frequency response of a single mode. In Section III, we will show that the concept of statistical bandwidth provides a statistical criterion that highlights the frequency range over which the number of significant modes must be considered. This finding allows revising the expression of the variability of the electric energy density, referred to as $\zeta_W^2$, derived in [10], by highlighting some key terms intervening in its expression. However, the number of significant modes cannot be directly assessed experimentally. This notwithstanding, as shown in Section IV, the validation of $\zeta_W^2$'s law allows us to validate the assessment of the number of significant modes.

## II. REVISITING MODAL OVERLAP

The aim of this section is to revisit the way in which we commonly represent and regard the modal overlap effect. To this end, it is convenient to consider the statistics of a field-related quantity, such as the square modulus of the field in an MSRC or, equivalently, its corresponding electric-energy density referred to as $W$. This choice has the advantage to provide a link with a previous work [10] dealing with the problem of considering modal overlap in the derivations of the statistical moments of $W$; to improve the readability with that previous work, we adopt the same notations as much as possible and we invite the reader to refer to the aforementioned paper.

To conduct this study, we need to consider a modal approach in which the electric field $E$ at a position $r$ and at an operating

Manuscript received February 17, 2013; revised July 26, 2013; accepted September 19, 2013. Date of publication January 9, 2014; date of current version March 24, 2014.

The authors are with the Département de Recherche en Electromagnétisme, Laboratoire des Signaux et Systèmes, UMR8506, Univ Paris-Sud, SUPELEC, CNRS, 91190 Gif-sur-Yvette, France (e-mail: florian.monsef@lss.supelec.fr; andrea.cozza@supelec.fr).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEMC.2013.2284924
frequency $f_w$ can be expanded as follows:

$$ \mathbf{E}(\mathbf{r}, f_w) = \sum_{n=1}^{\infty} \gamma_n \psi_n(f_w) \mathbf{e}_n(\mathbf{r}), \quad (1) $$

where $\mathbf{e}_n(\mathbf{r})$ is the eigenvector of the nth mode, $\gamma_n$ is the modal weight, i.e., the coupling constant of the excitation source to the nth eigenmode, and $\psi_n$ is the frequency response of the nth mode.

If we express, on the one hand, $\mathbf{e}_n(\mathbf{r}) = e_n(\mathbf{r})\hat{\xi}_n(\mathbf{r})$, where $\hat{\xi}_n(\mathbf{r})$ is the unitary polarization vector of the nth mode, and use, on the other hand, $\{\tilde{\gamma}_n\}$ referred to as equivalent modal weights and defined as

$$ \tilde{\gamma}_n(\mathbf{r}) = \gamma_n e_n(\mathbf{r}), \quad (2) $$

expression (1) can be expressed in the following convenient form

$$ \mathbf{E}(\mathbf{r}, f_w) = \sum_{n=1}^{\infty} \tilde{\gamma}_n(\mathbf{r}) \psi_n(f_w) \hat{\xi}_n(\mathbf{r}), \quad (3) $$

where the frequency response $\psi_n(f)$ reads

$$ \psi_n(f) = \frac{-jf}{f^2 - f_n^2 - jf f_n / Q_n}. \quad (4) $$

Under the composite quality factor approximation [12], $Q_n \approx Q(f_n)$, where $Q(f)$ is the composite quality factor, a function of frequency. High Q resonances obtained in the MSRC are such that the most contributing part of $\psi_n(f)$ come from working frequencies “close” to the eigenfrequency $f_n$, so that (4) can be approximated by

$$ \psi_n(f) \approx \frac{2(f - f_n) + jB_M}{4(f - f_n)^2 + B_M^2} \quad (5) $$

where $B_M$ is the modal bandwidth related to the composite quality factor. As it will be stressed hereafter, $B_M$ will be approximated by the modal bandwidth assessed at working frequency, i.e., $B_M \approx f_w/Q(f_w)$.

The concept of significant mode is studied here in a statistical sense. To position the problem of modal overlap within this framework, let us consider the average of $W$ (see [10, eq. (19)]) expressed herein as follows:

$$ \begin{aligned} E[W] &= E_{f_n, |\tilde{\gamma}_n|^2}[W] \\ &= E_{f_n, |\tilde{\gamma}_n|^2} \left[ \sum_{n=1}^{\infty} |\tilde{\gamma}_n|^2 |\psi_n(f_w)| \right] \end{aligned} \quad (6) $$

where $E_x[\cdot]$ stands for the ensemble-average operator linked to the stirring process and applied to the random variable $x$; note that the spatial dependence of the $|\tilde{\gamma}_n|^2$ has been omitted for the sake of brevity.

The difficulty of defining the number of significant modes lies in the infinite number of tails of frequency responses that overlap at the working frequency. The upper plot in Fig. 1 depicts the case of a random realization based on a set of frequency responses of excited modes, weighted by a set of equivalent squared modal weights $|\tilde{\gamma}_n|^2$ (vertical arrows); the excitation

<figure><img src="image_43.png" /><figcaption>Fig. 1. Due to modal overlap, all resonant frequencies $f_n$ are involved in the ensemble statistics of a given field-related quantity at given working frequency $f_w$ (black dots in the upper plot). Each frequency response is weighted by a corresponding $|\gamma_n|^2$ (vertical arrows). As illustrated, the black dots of the upper plot are contained in the template $|\psi_0(f - f_w)|^2$ (lower plot). This illustrates the ergodic principle stating that expected values on the eigenfrequencies ensemble converges to the one found using a single-mode frequency response in the whole frequency domain.</figcaption></figure>

level of the frequency responses at working frequency are highlighted by black dots.

Now, the discrete sum in (6) consists in summing the values indicated by the black dots weighted by the corresponding $|\tilde{\gamma}_n|^2$. As already done in [10], each frequency response in Fig. 1 is expressed by means of a frequency template $\psi_0(f)$ defined as

$$ \psi_n(f) \approx \psi_0(f - f_n). \quad (7) $$

The use of such template could appear as a way of simplifying the derivation with a degree of approximation that can be questioned, because of the well known variation of the composite Q-factor with frequency. In fact, as shown in (4) and (5), the important parameter to consider in $\psi_n(f)$ is the modal bandwidth (i.e., the ratio $f_n/Q_n$) which, given the local quasi-linear variation of $Q$ with frequency, does not change significantly. The experimental data presented in Section V show that the deviation of the average modal bandwidth is of 0.7% over a 100-MHz bandwidth centered about 800 MHz. So, the results obtained by using (7) are expected to ensure a reasonable level of accuracy.
Using (7) in (6) allows us to approximate the discrete sum (of (6)) as follows:

$$ \sum_{n=1}^{\infty} |\tilde{\gamma}_n|^2 |\psi_n(f_w)|^2 \approx |\tilde{\gamma}(f)|^2 |\psi_0(f - f_w)|^2, \quad (8) $$

where $|\tilde{\gamma}(f)|^2$ is a discrete random signal for a given realization.

In order to further recast (6), it is worth recalling that the ensemble-average operator stands for an average over an ideally infinite number of stirring states. Moreover, since a stirring process consists in displacing resonance frequencies over a small frequency range, referred to as $\overline{\Delta f}$ (see lower plot of Fig. 1), the vertical dashed line in the upper plot of Fig. 1, as well as the initially discrete frequency template, will both become continuous under perfect stirring conditions.

Note that the random signal $|\tilde{\gamma}(f)|^2$ is analog to a power spectrum density (PSD). This one will become continuous in the frequency domain with an average amplitude $\mu_2 = E[|\tilde{\gamma}_n|^2]$.

Accordingly, the ensemble average over $W$ can be approximated by a frequency average expressed as follows:

$$ E[W] \approx \lim_{B_c \to \infty} \frac{1}{\Delta f} \int_{(B_c)} |\tilde{\gamma}(f)|^2 |\psi_0(f - f_w)|^2 df, \quad (9) $$

where $B_c$ is the frequency range centered about $f_w$; an infinite $B_c$ allows including all the modes.

If (6) is compared to (9), an ergodic-like property is exhibited. When dealing with the first-order moment, this property consists usually (in its strict sense) in considering that a statistical average coincide with a temporal average; in the present case, the statistical average can be approximately assessed by a frequency average.

Equation (9) shows that the modal overlap effect can be restated as a simple filtering formulation of a random signal characterized by a PSD $|\tilde{\gamma}(f)|^2$ applied to a filter with a frequency response $|\psi_0(f - f_w)|^2$.

The approach presented in this section can be extended to other statistical moments. For instance, it is easy to show that $E[W^2]$ can be restated in a similar way to (9) where the filter response would be $|\psi_0(f - f_w)|^4$ and the PSD would be $|\tilde{\gamma}(f)|^4$ with an average value $\mu_4 = E[|\tilde{\gamma}_n|^4]$.

Note that if one is interested in the derivation of relative or normalized variance of $W$, one will have to deal with the following typical ratio

$$ \mathcal{R} = \frac{\langle |\tilde{\gamma}(f)|^4 |\psi_0(f - f_w)|^4 \rangle_{\mathcal{F}}}{\langle |\tilde{\gamma}(f)|^2 |\psi_0(f - f_w)|^2 \rangle_{\mathcal{F}}} \quad (10) $$

where $\langle \cdot \rangle_{\mathcal{F}}$ stands for the frequency average as defined in (9).

### III. STATISTICAL BANDWIDTH AND SIGNIFICANT MODES

According to the previous section, $|\psi_0(f - f_w)|^2$ and $|\psi_0(f - f_w)|^4$ can be regarded as weighting functions acting on $|\tilde{\gamma}(f)|^2$ and $|\tilde{\gamma}(f)|^4$, respectively.

The weighting functions are common tools in signal processing and instrumentation. A classic application deals with the estimation of the PSD of a signal [13] based on analog signal processing. The estimation of the PSD at a given frequency $f_w$ consists in measuring the power of the signal at the output of a

perfect filter, i.e., a filter with an ideally flat frequency response of width $B_1$. Due to the finite bandwidth of the filter, the estimation error is unavoidable and is characterized by its normalized relative error. In practice however, the filter used for such estimation is of Lorentzian shape and the resulting estimated error may vary. To relate the impact of using such filter instead of the ideal one, the concept of statistical bandwidth, referred to as $B_s$, is introduced. The latter corresponds to the bandwidth $B_1$ that would provide the same relative variance (RV) than the one obtained in practice.

Although the application is different in the present context, the scenario is quite similar if the normalized relative error is regarded as the RV of an unbiased estimator. Moreover, as stated previously, $|\tilde{\gamma}(f)|^2$ being the PSD of a random signal, we can easily show that the RV to compute corresponds to $\mathcal{R}$ given by (10). By using the statistical bandwidth concept it follows that

$$ \mathcal{R} = \frac{\overline{\Delta f} \mu_4}{B_s \mu_2^2} \quad (11) $$

where $B_s$ is the statistical bandwidth defined as [13], [14]

$$ B_s = \frac{\left( \int_0^\infty |\psi_0(f - f_w)|^2 df \right)^2}{\int_0^\infty |\psi_0(f - f_w)|^4 df} \quad (12) $$

Considering the frequency response given by (5) we obtain

$$ B_s = \pi B_M \quad (13) $$

The interest of the statistical bandwidth lies in the fact that the contribution, in a statistical sense, of an infinite number of $|\tilde{\gamma}_n|^2$ weighted by a Lorentzian spreading on an infinite frequency range, can be summed up by a finite number of unweighted $|\tilde{\gamma}_n|^2$ in a finite bandwidth $B_s$.

Within the statistical bandwidth model, it follows that the number of significant modes, referred to as $M$, reads finally

$$ M = \pi M_M \quad (14) $$

where $M_M$ refers to the number of modes overlapping in the modal bandwidth $B_M$ that can be expressed as

$$ M_M = m(f) B_M \quad (15) $$

where $m(f)$ is the modal density in Hz$^{-1}$.

The condition $V(f/c)^3 \gg 1$ being met, the simplest form of the modal density [15] can be used, such that

$$ m(f) \approx \frac{8\pi V f^2}{c^3}, \quad (16) $$

where $V$ and $c$ are the volume of the MSRC and the speed of light, respectively.

It is worth stressing that the modal density is, in a strict way, a fluctuating quantity whose variance will not be taken into account in the present study. We will only consider its approximated median value given by (16).

### IV. ASSESSMENT OF $M$

The concept and the number of significant modes is difficult to validate since it cannot be “counted” in practice. However, as shown in [10], the variability of $W$ is governed by the number
of modes $M_M$. With the concepts introduced in the previous sections, studying the variability of $W$ consists in assessing $\zeta_W^2$ for a finite number $M$ of modes spreading over $B_s$. This is exactly the result given by (24) in [10], where $B_e$ was at that stage regarded as a finite bandwidth over which $M$ modes spread. Adapting the result is then straightforward, yielding

$$ \begin{aligned} \zeta_W^2 &= \frac{B_s}{M} \frac{\mu_4}{\mu_2^2 \pi B_M} + \frac{M-4}{3M} \\ &= \frac{1}{3} + \frac{2}{3\pi M_M} \end{aligned} \qquad (17) $$

for $\mu_4/\mu_2^2 = 2$, keeping the assumption that $\tilde{\gamma}_i$ are normally-distributed complex random variables.

A quick glance at the final expression of $\zeta_W^2$ in [10] shows a difference of a factor 3 in the second term of the right-hand term. In fact, in order to include all the modes, the initially finite bandwidth $B_e$ was logically extended to infinity, but $M/B_e$ was improperly substituted by the modal density $m(f)$—indeed this substitution can only be justified if $B_e$ is sufficiently small. This is the advantage provided by the statistical-bandwidth concept which, given the order of magnitude of $B_s$, allows substituting $B_s/M$ by $1/m(f)$ in (17).

Since $B_s$ and $M$ only exist in the framework of an equivalent statistical model, the validation of these as pertinent (and useful) quantities cannot be performed, but indirectly, by validating the analytical expression of $\zeta_W^2$.

## V. CONCEPTS AND MODEL VALIDATION

In order to validate the analytical expression of $\zeta_W^2$ found in (17), we will compare the RV to the one obtained, on the one hand, by Monte Carlo (MC) simulations and on the other hand, experimentally.

### A. MC Simulation Setup

The importance of the results obtained by the MC simulation should not be underestimated over those obtained experimentally. Indeed, beyond the flexibility that the MC method allows, it is a very convenient way to check the self-consistency of our analytical expressions with respect to the assumptions made on the different parameters intervening in (3).

The statistical distributions are such that the real and imaginary parts of the equivalent modal weights were assumed to follow a normal law; a uniform distribution was assumed for both the polarization of the modes, over $4\pi$ sr.

As stated by (17), the only parameter that can vary is the number of modes overlapping in the modal bandwidth $B_M$. As shown by (16), this number can be modified by considering a variable volume and/or a variable frequency. As shown in [16], the two approaches are equivalent. In order to simplify the MC computations, we pick out the method consisting in fixing a modal bandwidth and adapting a (virtual) volume to ensure the desired number of overlapping modes. In the present study, a 1-MHz modal bandwidth is arbitrarily set.

Each MC simulation consisted in generating twenty sets of 50 000 independent random realizations of the electric field de-

scribed by (3), and this for the following values of $M_M$: 1, 2, 3, 5, 10, 15, 20, 25, 30, 35. The number of simulated modes was taken over a bandwidth of $51B_M$ [16]. The resulting estimated variance was averaged over the twenty values.

### B. Experimental Setup and Chamber Characterization

In order to make the comparison more sensible with [10], the same measurements are considered herein. As a brief reminder, and to provide some more details, the setup takes place in the 13.3 m³ (3.08 m × 1.84 m × 2.44 m) RC equipped with a 100-step mechanical stirrer blade of 50 cm width; its LUF is around 550 MHz. The RV is studied over the frequency range of 0.7–3 GHz.

The reverberation chamber (RC) was used in two configurations. In the first one, the RC was empty; in the second one, it was loaded by inserting an hybrid absorber made up of four pyramids of about 30 cm high, standing in the center of the RC.

The loaded and empty cases provide respective advantages. When the RC is not loaded, the losses are minimized, allowing us to visualize, as clearly as possible, the transition of $\zeta_W^2$ toward its well known asymptotic value of $1/3$. The interest of the loaded case is to provide another configuration of the chamber and to have enough overlapping modes in order to approach the asymptotic value, i.e., to attain very well-overmoded conditions.

For the empty and loaded scenarios, we can show that the maximum average modal bandwidths are 150 and 375 kHz, respectively. On the frequency range of interest, 1000 linearly-spaced frequency bins are used, inducing a 2.3-MHz frequency space between each point. This frequency space is much larger than the maximum value of the modal bandwidths previously mentioned, allowing each measured point to be considered as uncorrelated between one another.

The variability $\zeta_W^2$ will be estimated with an inevitable uncertainty. The latter, i.e., the uncertainty, can be minimized by applying a moving average over five contiguous points, followed by a decimation whose factor equals 5 accordingly. The average values are therefore obtained over 10-MHz bandwidths.

It is worth recalling that the modal bandwidths and the modal density are two parameters that can fluctuate considerably. The moving average followed by the decimation aim at extracting the average trend, in order to be consistent with the model used herein that neither takes into account the variance of the number of modes [17] nor the variance of the quality factors [15], [18].

In order to make the validation reliable, we need to estimate some useful preliminary quantities such as the number of uncorrelated stirrer positions, referred to as $N_{sp}$.

The number of uncorrelated stirrer positions can be assessed when the correlation coefficient is below a certain threshold that depends on the number of stirrer positions [1]. For 100 positions, the threshold is found around 20%. For this correlation level, Fig. 2 shows the resulting number of “independent” number of stirrer positions as a function of frequency for the empty (solid line) and loaded (dashed line) scenarios, respectively.

In the validation process, the number of overlapping modes $M_M$ is also a key quantity resulting from the knowledge of the composite quality factor. Accordingly, a special care must
<figure><img src="image_44.png" /><figcaption>Fig. 2. Estimated number of independent stirrer positions as a function of frequency for the empty (solid line) and the loaded (dashed line) cases, respectively.</figcaption></figure>

<figure><img src="image_45.png" /><figcaption>Fig. 3. Composite quality factor obtained by averaging over 10-MHz bandwidths in the empty case (black upper curve) and the loaded case (grey lower curve), respectively. The straight lines stand for first-order approximated median values in a least-square sense.</figcaption></figure>

be taken with the estimation of $Q$. As the field probe used was phase sensitive, we were able to compute the composite quality factor for the chamber over the entire frequency range of test, by postprocessing the frequency-spectrum data in time domain. The frequency spectrum used was made up of 60 000 bins over the entire frequency range of interest. The composite quality factors were estimated on 1-MHz frequency bands. In order to be consistent with the process applied to $\zeta_W^2$, $Q$ values were averaged on 10-MHz frequency bands; the uncertainty was further minimized by using the data obtained on the three field components.

Fig. 3 shows the resulting composite quality factors obtained for the empty (black upper curve) and loaded cases (grey lower curve), respectively. In order to extract the mean trend for both cases, a first-order fit was performed in a least-square sense (straight lines). The measured quality factors, as a function of frequency, allow to compute modal bandwidths.

The knowledge of (average) modal bandwidths allowed us to derive the number $M_M$ of overlapping modes by using (15). Accordingly, Fig. 4 shows the number $M_M$ for the empty (black lower curve) and loaded cases (grey upper curve), respectively. The solid lines provide the mean trends resulting from those obtained for $Q$. Note that the fluctuations of $M_M$ observed in Fig. 4 results from the fluctuations of the estimated composite

<figure><img src="image_46.png" /><figcaption>Fig. 4. Number of overlapping modes in the -3-dB bandwidth obtained when the RC is empty (black lower curve) and loaded (grey upper curve). The solid lines are the first-order approximated median values in a least-square sense.</figcaption></figure>

<figure><img src="image_47.png" /><figcaption>Fig. 5. Fluctuations of the estimated $M_M$ values, obtained in Fig. 4, about the approximated median values, for the empty case (solid grey line) and the loaded case (black dotted line), respectively.</figcaption></figure>

quality factor and are not due to local modal-density fluctuations; as explained in section III, these have not been considered in the present study.

In order to have an estimation of the fluctuations of $M_M$ about mean values, Fig. 5 shows the normalized dispersion obtained for the empty case (solid grey line) and the loaded case (black dotted line), respectively. A dispersion of about ±7% is observed for both cases.

This characterization of the chamber in both scenarios allows us to proceed to the validation of the RV of the electric-energy density.

### C. Results

We present in Fig. 6 the estimated values of $\zeta_W^2$, referred to as $\zeta_W^2$, obtained experimentally (left y-axis) as a function of frequency (grey markers) with the corresponding relative deviation from its asymptotic value (right y-axis). For clarity, and in order to well visualize the transition of the RVs toward their asymptotic values, only the empty case is reported in Fig. 6. We superimposed the analytical expression obtained in (17) (solid line) and the analytical result obtained in [10] (dashed line). We can observe that the expression given by (17) is in very good agreement with measurements, whereas the RV $\zeta_W^2$
<figure><img src="image_48.png" /><figcaption>Fig. 6. Estimated RV $\hat{\zeta}_W^2$ of the electric-energy density (left y-axis) and the relative deviation from the asymptotic value of 1/3 (right y-axis), as a function of frequency. The experimental results (grey markers) and analytical results (solid line) are reported; dashed line stands for the analytical expression derived in [10].</figcaption></figure>

obtained in [10] (dashed line) tends to overestimate the degree of nonuniformity of $W$.

In order to compare these results to those obtained with the MC approach, the number of modes assessed experimentally for the empty and loaded scenarios (see Fig. 4) must be used. This allows to transpose and superimpose the experimental results to those obtained by the MC simulation. The variabilities are shown (left y-axis) accordingly in Fig. 7, where the experimental results (grey markers), MC results (black dots), and analytical results (solid line) have been reported. The deviation of $\hat{\zeta}_W^2$ from its asymptotic value is also shown (right y-axis).

However, when experimental variances are estimated from sets made up of finite number of samples, directly related herein to the number $N_{sp}$ of independent stirrer positions, an unavoidable uncertainty has to be taken into account. To estimate the latter, the MC simulations were used again; this merely consisted in computing 5000 times, for a given $M_M$ value, estimated the RVs of $W$ obtained over sets composed of $N_{sp}$ realizations.

Recalling that $\hat{\zeta}_W^2$ plotted in Fig. 6 was deduced from averages performed over five contiguous points, the 5000 MC values were rearranged in $5 \times 1000$ values in order to perform averages over five contiguous points. From these averages, 95% confidence intervals of $\hat{\zeta}_W^2$ have been computed, and correspond to the uncertainty bars reported in Fig. 7, for the empty case (upper plot) and the loaded case (lower plot), respectively.

Finally, we need to recall that $M_M$ are mean estimated values for which 7% fluctuations were observed (see Fig. 5); accordingly, horizontal uncertainty bars on $M_M$ have been added in Fig. 7.

We can observe a very good agreement between the results obtained analytically, numerically, and those obtained experimentally. For the latter, we note remarkable agreement even with uncertainty bars, especially for the empty case.

We stress the interest of using an MC approach, since the good agreement found ensures in some way that the analytical results are consistent with the assumptions made on the different parameters and their related statistical laws.

<figure><img src="image_49.png" /><figcaption>Fig. 7. Estimated RV $\hat{\zeta}_W^2$ (left y-axis) of the electric-energy density, as a function of the number of modes $M_M$ for the empty (upper plot) and loaded (lower plot) scenarios, respectively. The corresponding relative deviation from the asymptotic value of 1/3 (right y-axis). The experimental results (grey symbols), MC results (black dots) and analytical results (solid line) are reported. The dashed line stands for the analytical expression derived in [10]. The vertical bars are related to the uncertainty of the experimental measurements considered on a 95% confidence interval. The horizontal bars stand for the dispersion of $M_M$ around its mean value.</figcaption></figure>

## VI. CONCLUSION

The concept of significant modes has been revisited and highlighted. The bandwidth over which this number is defined, depends inevitably on the statistical quantity of interest. In the present study, a quantitative assessment of this number has been carried out on the basis of the variability of the electric-energy density. This notwithstanding, the approach presented in the present study can be extended to other statistical quantities.

Linking the concept of significant modes to previous work allowed us to derive a new expression of the variability of the electric-energy density. The good agreement found between the experimental results and simulation supports the different concepts introduced in the present paper. From a more practical point of view, it provides an answer to the pending question dealing with the assessment of the average number of significant modes to consider at a given frequency in an MSRC.

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[6] R. Serra, F. Leferink, and F. Canavero, ""Good-but-imperfect" electromagnetic reverberation in a VIRC," in *Proc. Int. Symp. Electromagn. Compat.*, 2011, pp. 954–959.

[7] C. F. Bunting, "Statistical characterization and the simulation of a reverberation chamber using finite-element techniques," *IEEE Trans. Electromagn. Compat.*, vol. 44, no. 1, pp. 214–221, Feb. 2002.

[8] L. Arnaut, "Compound exponential distributions for undermoded reverberation chambers," *IEEE Trans. Electromagn. Compat.*, vol. 44, no. 3, pp. 442–457, Aug. 2002.

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[10] A. Cozza, "The role of losses in the definition of the overmoded condition for reverberation chambers and their statistics," *IEEE Trans. Electromagn. Compat.*, vol. 53, no. 2, pp. 296–307, May 2011.

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[12] B. H. Liu, D. C. Chang, and M. T. Ma, "Eigenmodes and the composite quality factor of a reverberating chamber," Tech. Rep. Tech. Note 1066, U.S. National Bureau of Standards, 1983.

[13] J. Bendat and A. Piersol, *Random Data: Analysis and Measurement Procedures*. New York, NY, USA: Wiley-Interscience, 1971.

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[15] L. Arnaut and G. Gradoni, "Probability distribution of the quality factor of a mode-stirred reverberation chamber," *IEEE Trans. Electromagn. Compat.*, vol. 55, pp. 35–44, Feb. 2013.

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[17] A. Cozza, "Probability distributions of local modal-density fluctuations in an electromagnetic cavity," *IEEE Trans. Electromagn. Compat.*, vol. 54, no. 5, pp. 954–967, Oct. 2012.

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<figure><img src="image_50.png" /></figure>

**Florian Monsef** (M'12) received the Masters degree from the Université Paris-Sud, Orsay, France. He entered the Ecole Normale Supérieure de Cachan, Cachan, France, in electrical engineering and computer science. He did a thesis on electron transport in IV–IV heterostructures and received the Ph.D. degree in electronics from the Université Paris-Sud, Orsay, France, in 2002.

From 2002 to 2009, he was a Teacher in electrical engineering at the Technical Institute of the Université Paris X, Nanterre, France. Since 2009, he is currently an Assistant Professor with the Laboratoire des Signaux et Systèmes (Université Paris-Sud). His topics in research since 2009 include EMC, reverberation chambers, and time-reversal applications.

<figure><img src="image_51.png" /></figure>

**Andrea Cozza** (S'02–M'05–SM'12) received the Laurea degree (summa cum laude) in electronic engineering from Politecnico di Torino, Turin, Italy, in 2001, and the Ph.D. degree in electronic engineering jointly from the Politecnico di Torino and the University of Lille, Lille, France, in 2005.

In 2007, he joined the Département de Recherche en Électromagnétisme, SUPELEC, Gif sur Yvette, France, where since 2013, he is a Full Professor. He is a reviewer for several scientific journals, including those of the IET and IEEE. His current research interests include reverberation chambers, statistical electromagnetics, wave propagation through complex media, and applications of time reversal to electromagnetics.

Dr. Cozza was awarded the 2012 Prix Coron-Thévenet by the Académie des Sciences, in France.
# A Possible Minimum Relevance Requirement for a Statistical Approach in a Reverberation Chamber

Florian Monsef, Member, IEEE, and Andrea Cozza, Senior Member, IEEE

**Abstract**—Modal overlap in a reverberation chamber (RC) involves, strictly speaking, an infinite number of modes in the field statistics. The degree of overmodedness of an RC is often assessed by introducing an arbitrary reference bandwidth in which a number of overlapping modes is computed. We introduce a statistical model of the field in which the number of modes is rigorously finite. The model is chosen such that the number of degrees of freedom (DOF) of the field can be assessed. The number of DOF is assessed by considering the degree of homogeneity of the average power received by a linearly polarized antenna. It is considered that a statistical approach requires at least a single degree of freedom. Based on this criterion, the minimum frequency at which a statistical approach would make sense is evaluated and compared to the lowest useable frequency (LUF) commonly considered in EMC.

**Index Terms**—Cavity resonators, electromagnetic compatibility (EMC), modal analysis, parametric statistics, reverberation chamber (RC).

## I. INTRODUCTION

In order to deal with the complexity of the electromagnetic (EM) field inside reverberation chambers (RCs), a statistical approach is commonly used to estimate the degree of homogeneity of the mean and maximum power level that can be expected [1]–[3]. The power, referred to as *P*, is meant as being measured on a given rectangular field component by a linearly polarized antenna.

Due to wall losses and other loss mechanisms, the EM field results from the superposition of modes regarded as random normal modes. As soon as a certain number of modes is efficiently excited, the field can be assumed to follow a Gaussian law, a direct result from the central-limit theorem [4]. If the number of excited modes happened to raise, fluctuations of *P* about its mean value, i.e., its relative variance (or variability), would remain steady. It corresponds to an overmoded regime for which an infinite number of modes is ideally and theoretically assumed. The relative variance of the measured mean power tends to unity, as *P* follows a $\chi_2^2$ law. A recent study [5] has highlighted the rate at which power variability tends toward its asymptotic value.

Each contributing mode could be thought of acting as a degree of freedom (DOF). The larger the number of DOF, the more Gaussian the field will be. Although intuitive, the correspondence between the number of excited mode and the number of DOF is not straightforward for the following reason. The frequency response of the modes are of Lorentzian shape. This means that the number of overlapping modes is rigorously infinite. One could bound the number of overlapping modes by setting a bandwidth of reference. In [6] for instance, the –3 dB modal bandwidth was chosen. This choice can be a metric that provides the degree of modal overlap, but does not provide the number of modes that contribute to a given level of field homogeneity [7]. In any case, whatever the chosen bandwidth, the modal overlap involves theoretically an infinite number of modes.

This study tackles the problem in another way. It proposes an equivalent statistical model that, on the one hand, makes not possible the modal overlap to be infinite, and on the other hand, provides the same degree of homogeneity in a chamber, i.e., leads to the expected power variability. It is essentially a way to circumvent the problem of the tails of the frequency response of the modes. As we will see, the bandwidth over which the modal overlap is regarded is however related to the choice of some model parameters. Interestingly, this proposed approach allows us to define properly the concept of DOF and highlights the results derived in [8]. Finally, the model allows us to propose a criterion that could be a possible answer to the pending question dealing with the frequency from which a statistical approach makes sense in an RC.

After defining what is meant by DOF in Section II, the equivalent statistical model of the electric field will be exposed. The assessment of the number of DOF will also be presented. Section III will focus on the validation of the proposed model by means of Monte Carlo (MC) simulation. Finally, the frequency from which a statistical approach makes sense in an RC will be presented in Section IV.

## II. MODAL OVERLAP AND DEGREES OF FREEDOM

### A. Degrees of Freedom: Definition

It is important to define what is meant by DOF in order to understand the interest of the approach presented herein. Let us consider a random variable *Y* expressed as follows:

$$Y = \sum_{i=1}^{N} X_{i} \qquad (1)$$

Manuscript received June 29, 2015; accepted July 15, 2015.

The authors are at the GeePS Lab, UMR8507, Université Paris-Sud, CentraleSupelec, CNRS, UPMC, 91192 Gif-sur-Yvette, France (e-mail: florian.monsef@u-psud.fr).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEMC.2015.2464318
where $X_i$ are independent and identically distributed (i.i.d.) random variables. The number $N$ is defined as the number of DOF of $Y$.

A typical example of such definition is met with a $\chi_N^2$ variable where the $X_i$ are $\chi_1^2$ distributed.

The basic idea of the paper is to introduce a way to express the electric field on the model given by (1), the interest of it will become clearer thereafter.

## B. Matching a Physical Model to a Statistical Field Model

The first step of the proposed approach consists in recalling the physics that describes the electric field in an RC. To that aim, a modal approach is used, and the electric field $E$ at a position $\mathbf{r}$ and at a frequency $f$ can be expressed as follows [8], [9]:

$$ \mathbf{E}(\mathbf{r}, f) = \sum_{i=1}^{\infty} \gamma_i \psi_i(f) \mathbf{e}_i(\mathbf{r}) \quad (2) $$

where $\mathbf{e}_i$ is the eigenvector of the $i$th mode, which is assumed not to have a specific form, $\psi_i$ is the frequency response of the $i$th mode, and $\gamma_i$ is the modal weight, i.e., the coupling constant of the source to the $i$th eigenmode.

Limiting the analysis to a single position $\mathbf{r}$, the following factorization for the modal topographies can be used:

$$ \mathbf{e}_i(\mathbf{r}) = e_i(\mathbf{r}) \hat{\xi}_i(\mathbf{r}) \quad (3) $$

where $\hat{\xi}_i(\mathbf{r})$ is a unitary polarization vector assumed to be uniformly distributed over $4\pi$ sr.

As explained in [9], it is convenient to regard the set $\{\gamma_i e_i(\mathbf{r})\}$ as a set of a single random variable $\tilde{\gamma}_i$ defined as

$$ \tilde{\gamma}_i = \gamma_i e_i(\mathbf{r}) \quad (4) $$

referred to as equivalent modal weights [9]. The real and imaginary parts of $\tilde{\gamma}_i$ are assumed to follow a centered normal law.

As done in [5], [8], and [9], we consider three sets of modal parameters, i.e., $\tilde{\gamma}_i$, $\psi_i$, and $\hat{\xi}_i$ as random variables. The main assumptions required are at two levels. First, the modal parameters of different sets are independent, second, the parameters within the same set are i.i.d.

Now, the electric field described by (2) corresponds to a sum of weighted random variables, the weight function being $\psi_i(f)$. If we want to highlight the number of DOF for the case of the electric field, a rectangular shaped weight function has to be considered; the weight function, referred to as $\psi_{i,e}(f)$, would be of width $B_w$ and of height $A_{eq} \forall i$.

By doing so, we are introducing fictitious modes that have rectangular shaped frequency response. So, these modes are, or totally excited, or not at all. When these are excited, their contribution is equal to their neighbors.

In other terms, we are introducing an arbitrary electric field model. Accordingly, an equivalent electric field, referred to as $\mathbf{E}_e(\mathbf{r}, f)$ must be considered; it is still expressed as a modal expansion as follows:

$$ \mathbf{E}_e(\mathbf{r}, f) = \sum_{i=1}^{\infty} \tilde{\gamma}_{i,e} \psi_{i,e}(f) \hat{\xi}_{i,e}(\mathbf{r}) \quad (5) $$

<figure><img src="image_52.png" /><figcaption>Fig. 1. Illustration of the overlap using the fictitious-modes model. Given the shape of their frequency response, only a finite number of those is excited—two in the illustrated case.</figcaption></figure>

where $\tilde{\gamma}_{i,e}$, $\hat{\xi}_{i,e}(\mathbf{r})$ and $\psi_{i,e}(f)$ refer to equivalent modal quantities. However, these do not follow necessarily the same statistics as the modal quantities in (2) as they are related to fictitious modes.

The infinite modal-overlap issue at given frequency $f_w$ is then circumvented as shown in Fig. 1. The sketch highlights indeed that only a finite number of fictitious modes intervene in (5). It follows that $\mathbf{E}_e(\mathbf{r}, f)$ can be simplified and reads

$$ \mathbf{E}_e(\mathbf{r}, f) = \sum_{i=1}^{N} \tilde{\gamma}_{i,e} \hat{\xi}_{i,e}(\mathbf{r}) \quad (6) $$

where $N$ is the number of DOF, i.e., of fictitious modes, whose number will be specified in the next section. It is worth noting that the expression in (6) follows indeed the form given by (1).

Provided that the degree of homogeneity of the average power is the quantity of interest, its variability, referred to as $\zeta_P^2$, must remain the same no matter what model is used. In other terms, (2) and (6) must lead to the same expression of the variability. This one was shown to be expressed in [5] as

$$ \zeta_P^2 = 1 + \frac{8}{5\pi M_M} \quad (7) $$

where $M_M$ stands for the average number of real modes overlapping in the -3dB modal bandwidth.

We chose $\tilde{\gamma}_{i,e}$ and $\hat{\xi}_{i,e}(\mathbf{r})$ to share $\tilde{\gamma}_i$ and $\hat{\xi}_i(\mathbf{r})$ properties, respectively. This choice is motivated by the fact that the main issue is only linked to the tails of the frequency responses. Accordingly, we do not have any reason to consider that $\tilde{\gamma}_{i,e}$ and $\hat{\xi}_{i,e}(\mathbf{r})$ follow different probability laws than $\tilde{\gamma}_i$ and $\hat{\xi}_i(\mathbf{r})$.

Now, if (6) is considered, by using the approach used in [5], it is easy to show that $\zeta_P^2$ reads

$$ \zeta_P^2 = 1 + \frac{\Gamma - 2}{N} \quad (8) $$

where $\Gamma$ is defined as

$$ \Gamma = \frac{\mu_4 \nu_4 \kappa_4}{\mu_2^2 \nu_2^2 \kappa_2^2} \quad (9) $$

From [5] and [9], $\mu_n = E_{\tilde{\gamma}_{i,e}} [|\tilde{\gamma}_{i,e}|^n]$, $\kappa_n = E_{\psi_i}[|\psi_{i,e}(f)|^n]$, and $\nu_n = E_{\psi_i}[|\psi_{i,e}(f)|^n]$, where $E_x[\cdot]$ is the ensemble average operator applied to $x$.
Given that $\tilde{\gamma}_{i,e}$ and $u_{i,e}$ follow a Gaussian and a uniform law,<fnref n="1" /> respectively, we can show that $\Gamma$ reads

$$ \Gamma = \frac{18 \nu_4}{5 \nu_2^2}. \qquad (10) $$

Moreover, the frequency responses of the model being of rectangular shape, we can easily show that

$$ \nu_2 = m(f_w) A_{eq}^2 B_w \qquad (11a) $$

$$ \nu_4 = m(f_w) A_{eq}^4 B_w \qquad (11b) $$

where $m(f_w)$ is the modal density in mode/Hz at working frequency $f_w$.

If we use the real-modes model, we have $\nu_2$ and $\nu_4$ such that [5]

$$ \nu_2 = \frac{\pi m(f_w)}{2B_M} \qquad (12a) $$

$$ \nu_4 = \frac{\pi m(f_w)}{4B_M^3}. \qquad (12b) $$

In order to match (11) to (12), the frequency responses of the fictitious modes must be chosen such that

$$ A_{eq} = \frac{1}{B_M \sqrt{2}} \qquad (13) $$

and

$$ B_w = \pi B_M. \qquad (14) $$

We can make a total analogy with the equivalent noise bandwidth, referred to as $B_N$, of a filter; $B_N$ is the bandwidth of an ideal filter (rectangular response) for which the power noise is the same as the one that we have with a real filter. This concept has been transposed herein such that the equivalent bandwidth provides the same statistical moments. This bandwidth may be referred to as the statistical bandwidth as proposed in [8].

If we had taken the electric energy density $W$ as quantity of interest, as done in [9] for instance, the related variability, referred to as $\zeta_W^2$ would be the metric to consider. In order to match the two first moments of $W$, we can show that the conditions needed on $A_{eq}$ and $B_w$ remain unchanged.

It is worth stressing that other shapes (triangles, etc.) of frequency responses could have led to finite modal overlap, but those would still have behaved as weighting functions. In a strict sense, the bandwidth, over which modal overlap is estimated, depends on the shape of the frequency response. However, the assessment of the number of DOF would not be possible for nonrectangular shapes. So the coupling of the DOF constraint enforces a single bandwidth value, the one given by (14).

### C. Number of Degrees of Freedom

Now that the bandwidth $B_w$ is identified as being $\pi$ times the modal bandwidth, the sketch of Fig. 1 allows us to assess easily the number of fictitious modes, or in other words, the number of DOF.

<figure><img src="image_53.png" /><figcaption>Fig. 2. Relative variances of the power measured along a given cartesian component (upper curve) and of the electric-energy density (lower curve) obtained by MC simulations using (2) (circles) and (6) (triangles). These are plotted as a function of the number of degrees of freedom of the field.</figcaption></figure>

The average distance between continuous modes being $1/m(f_w)$, the number $N$ of DOF is such that

$$ N = m(f_w) B_w = \pi M_M. \qquad (15) $$

This number corresponds to what was called the “effective number” of excited modes in [8]. The problem is tackled here from another angle, highlighting that these “modes” correspond in fact, from a statistical point of view, to a number of DOF.

We emphasize that the number $N$ must not be regarded as a number of real modes that contribute to the field statistics otherwise leading to an improper variability level as shown in [7]. This is especially striking at low modal overlap. For this case, the modes that contribute to the statistics are poorly excited due to their Lorentzian-shaped frequency response. This impose to consider real modes spreading over a wide frequency band (50$B_M$ according to [7] for $M_M = 1$); with the DOF model, the fictitious modes are entirely excited and the statistics does not need such a wide frequency band to converge.

### III. MODEL VALIDATION

The form obtained in (6) can make one skeptical about the validity of such a simple equivalent model of the electric field. In order to raise doubts, we have computed, on the one hand, $\zeta_P^2$ and $\zeta_W^2$ using MC simulations for the following number of DOF: 1, 2, 3, 5, 10, 15, 20, 25, 30, 50, and 90. On the other hand, we have computed MC simulations by using the real-modes model given by (2). Fig. 2 shows the resulting relative variances, i.e., those obtained using the physical model (circles) and the equivalent model (triangles), respectively. In this procedure, 5000 realizations and 500 independent stirrer states were considered.

When using physical modes, the computation of the moments of $P$ assumes that a sufficient number of (real) modes is considered in the simulation as explained in [7]. In a strict manner, the number of real modes included is always truncated. This does not happen with the equivalent model, since this one is defined as being composed of a finite number of DOF as expressed in

<footnote n="1">¹See [5, Sec. III].</footnote>
(6). Besides, it is clear that the number of DOF taken into account in the simulations are integer values since the sum in (6) is discrete. It allows us not to run into the question of setting a sufficiently large bandwidth.

Relative variances obtained with the physical model on the one hand, and with the statistical model on the other hand, are found to be in good agreement. This allows us to validate the fact that the relative variance obtained using (6) matches the “real” one obtained by using classical modal theory.

## IV. DOF AND RELEVANCE OF A STATISTICAL APPROACH

The advantage of the statistical model lies in the simple overlap model that it provides. If a single DOF is excited would not it be a reasonable criterion to consider it as the minimum requirement to consider the relevance of a statistical approach? We propose to regard the frequency for which a single DOF is considered as the minimum frequency from which a statistical modeling of the field makes sense.

We emphasize to distinguish this frequency, referred to as $f_m$, from the lowest useable frequency (LUF). This one is indeed defined over a homogeneity criterion of the maximum stress [10]. The link between the variabilities considered herein and the maximum stress being not straightforward, $f_m$ and the LUF are clearly different quantities. Note however that $f_m$ is likely to be lower than the LUF.

In the light of the above, $f_m$ is such that

$$ \mathbf{E}_e(\mathbf{r}, f_m) \triangleq \mathbf{E}_e(\mathbf{r}, f), N = 1. \quad (16) $$

In order to determine $f_m$, we must solve the following equation:

$$ N(f_m) = m(f_m) \pi B_M = 1. \quad (17) $$

Recalling that $B_M = f_m/Q$, where $Q \equiv Q(f_m)$ is the composite quality factor [11], relation (17) can be recast and $f_m$ must satisfy

$$ m(f_m) \pi f_m = Q(f_m). \quad (18) $$

The frequency range under consideration corresponds to the case where the RC is undermoded. Accordingly, we prefer to adopt the following modal-density expression:

$$ m(f) \approx \frac{8\pi V f^2}{c_0^3} - \frac{a+b+c}{c_0} \quad (19) $$

valid for the cuboid case, note that the second term is a correcting term for low frequencies [11] where a, b, and c correspond to the cuboid edges.

We observed in [8] that the quality factor of the chamber was quite linear with the frequency. The study was performed for a minimum frequency of 700 MHz, i.e., for a frequency above the 550 MHz LUF of the chamber. In order to have a more accurate insight on the value of $f_m$, we processed measurements between 200 MHz and 1 GHz. Note that for these measurements, the RC was empty. Fig. 3 shows the evolution of the quality factor, the left-hand term of (18) is superimposed. The intersection point provides the solution for $f_m$ which turns out to be about 250 MHz, i.e., well below the LUF as expected.

<figure><img src="image_54.png" /><figcaption>Fig. 3. Estimated $Q$ factor (symbols) with the average trend (bold line). $m(f) * \pi f$ (thin solid line) has been superposed. The intersection point highlights the frequency for which a single DOF is found.</figcaption></figure>

## V. CONCLUSION

A statistical model of the electric field has been presented. It is based on a square frequency response of the modes which allows us to highlight the number of DOF of the field. The interest of this choice lies in the fact that only a strictly finite number of modes overlap. The number of DOF is defined over a criterion based on the variability of the power measured along a linearly polarized antenna and/or of the electric energy density.

If less than a single DOF is considered, the relevance of a statistical approach is cut down. Accordingly the frequency at which a single DOF is excited was proposed to be regarded as the lower bound from which a statistical approach would make sense in a reverberation chamber. The frequency is expected to be smaller than the LUF commonly considered in EMC.

## REFERENCES

[1] Reverberation Chamber Test Methods, IEC Standard 61000-4-21, 2003.

[2] T. H. Lehman, “A statistical theory of electromagnetic fields in complex cavities,” Interaction Notes, USAF Phillips Laboratory, NM, USA, Note 494, May 1993.

[3] J. Kostas and B. Boverie, “Statistical model for a mode-stirred chamber,” IEEE Trans. Electromagn. Compat., vol. 33, no. 4, pp. 366–370, Nov. 1991.

[4] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed. New York, NY, USA: McGraw-Hill, 1984, p. 168.

[5] F. Monsef and A. Cozza, “Variability and confidence intervals of the power measured in a reverberation chamber,” IEEE Trans. Electromagn. Compat., vol. 56, no. 5, pp. 1238–1241, Oct. 2014.

[6] L. Arnaut and G. Gradoni, “Probability distribution of the quality factor of a mode-stirred reverberation chamber,” IEEE Trans. Electromagn. Compat., vol. 55, no. 1, pp. 35–44, Feb. 2013.

[7] F. Monsef, “Why a reverberation chamber works at low modal overlap,” IEEE Trans. Electromagn. Compat., vol. 54, no. 6, pp. 1314–1317, Dec. 2012.

[8] F. Monsef and A. Cozza, “Average number of significant modes excited in a mode-stirred reverberation chamber,” IEEE Trans. Electromagn. Compat., vol. 56, no. 2, pp. 259–265, Apr. 2014.

[9] A. Cozza, “The role of losses in the definition of the overmoded condition for reverberation chambers and their statistics,” IEEE Trans. Electromagn. Compat., vol. 53, no. 2, pp. 296–307, May 2011.

[10] Reverberation Chamber Test Methods, IEC Standard 61000-4-21, 2011.

[11] B. Liu, D. Chang, and M. Ma, “Eigenmodes and the composite quality factor of a reverberating chamber,” U.S. Nat. Bur. Stand., Tech. Rep. 1066, 1983.
Les travaux sur le retournement temporel (RT) se sont intégrés dans les travaux amorcés au laboratoire par Andrea Cozza. Ces travaux portent sur le RT classique dans deux types d'environnements principaux : les milieux indoor et les chambres réverbérantes.

Un deuxième volet concerne le développement du RT dit généralisé qui peut être vu comme un prolongement et/ou une généralisation du RT classique. On entend par RT classique la technique développée dans les années 90 par l'équipe de Matthias Fink de l'ESPCI et permettant de focaliser une onde en un point (FINK 1992; WU, THOMAS et al. 1992; CASSEREAU et FINK 1992).

### I Retournement Temporel classique

Pour situer l'intérêt des travaux menés dans le cadre du RT, il convient de rappeler les performances offertes par le RT classique et les questions ouvertes dans ce domaine.

Le RT classique est une technique dont le protocole classique s'effectue en deux phases. La première phase consiste à émettre un signal par le biais d'une source (acoustique ou électromagnétique). Dans sa version d'origine, on considère cette source entourée de transducteurs qui reçoivent et enregistrent les signaux propagés<fnref n="9" />; Dans la partie gauche de la Fig. 3.1, on représente cette première phase. Ces transducteurs sont communément appelés *Miroirs à Retournement Temporel (MRT)*. Si $x(t)$ est le signal envoyé, $y(t)$ correspond au signal reçu par un transducteur et $h(t)$ correspond à la réponse impulsionnelle, on a,

$$y(t) = h(t) * x(t), \qquad (3.1)$$

où * est l'opération de convolution.

La deuxième phase consiste à renverser les signaux acquis dans le temps et à les réinjecter dans les *MRT* entourant la source initiale, comme illustré dans la partie droite de la Fig. 3.1. On réinjecte alors $y(-t)$; si le milieu est réciproque la réponse impulsionnelle (R.I) est inchangée et on obtient alors un signal, noté $s(t)$, au niveau du point source initial tel que,

$$s(t) = h(t) * h(-t) * x(-t), \qquad (3.2)$$

Une onde se propage alors vers la source initiale autour de laquelle elle se focalise sur le point source (puis diffracte). On peut y voir une forme de rétro propagation sous réserve que la R.I

<footnote n="9">9. définie par M-Fink comme Cavité dite à *Retournement Temporel cf. Ref.* (CASSEREAU et FINK 1992)</footnote>
### 3 Retournement Temporel

est une distribution de Dirac. En toute rigueur, ce n'en est pas une dans la mesure où le signal qui se focalise est affecté par le milieu et n'est donc pas une réplique exacte de ce qui a été émis.

<figure><img src="image_55.png" /><figcaption>FIGURE 3.1 – Phase d'émission et d'enregistrement des signaux par les MRT (gauche) et réémission par réinjection des signaux dans les MRT (droit) dans le protocole du RT classique (d'après Tourin (ESPCI)).</figcaption></figure>

Le milieu de propagation ne permet pas en effet de transmettre les composantes réactives du champ, i.e., les composantes au nombre d'onde élevé. On pourrait donc modéliser le signal focalisant comme le signal initial entâché d'un bruit de reconstruction, noté $N(f)$ dans l'espace des fréquences et $n(t)$ dans l'espace des temps.

Il a été montré que le cas de la cavité à RT peut être drastiquement simplifié dans des milieux complexes (LEROSEY, DE ROSNY et al. 2004) ... tels qu'une chambre réverbérante. La simplification réside dans le fait de ne plus avoir besoin d'autant de transducteurs. L'idée est que l'on puisse collecter le maximum d'informations sur l'onde émise. Un raisonnement avec le principe des images permet de mettre en évidence cet atout. Ainsi, dans un milieu complexe, la deuxième phase, peut, à l'extrême, s'appuyer sur un seul récepteur ! L'apparition d'un champ focalisant se fera durant un laps de temps, ou fenêtre temporelle, pendant lequel aucune réflection sur les parois n'aura lieu.

Dans le cas de milieux complexes tels que les chambres réverbérantes le bruit de reconstruction est encore plus important du fait des résonances du milieu : certaines fréquences sont filtrées par le milieu. Le signal focalisant pourra d'autant moins être une réplique du signal émis initialement. Le fait que certaines fréquences soient filtrées amène à ne pas considérer le RT en régime monochromatique, le risque étant de ne rien obtenir dans le cas où la fréquence coïncide avec un zéro de transmission ; le fonctionnement du RT dans un tel régime serait très aléatoire et fortement lié à l'emplacement de la source et du récepteur.

Plus la bande passante du signal augmente, plus la fonction de transfert est complexe. Il est alors adapté de la décrire de manière statistique. L'augmentation de la bande passante accroît la complexité de la réponse en fréquence. Toutefois les moments de celle-ci deviennent indépendants du point source et du TRM. Ceci est illustré Fig. 3.2 pour laquelle 4 fonctions de transfert $|S_{21}|^2$ sont reportées. Celles-ci ont été prises en quatre points distant d'un mètre les uns des autres ; on considère une bande de 300 MHz centrée autour de 1.5 GHz. On voit que les moments d'ordre 1 et 2 fluctuent de 10 à 15% - la variance relative (variabilité) y figure également car on évalue un peu plus loin son influence sur le bruit de reconstruction. On peut percevoir que les variabilités abordées au chapitre précédent se retrouvent dans le cadre de cette application.

L'étude des signaux dans l'espace des fréquences met en évidence que le RT classique est un
<figure><img src="image_56.png" /><figcaption>FIGURE 3.2 – Relevés expérimentaux de $|S_{21}|^2$ en quatre points de la CR pour une position de brasseur fixe. On constate que les moments statistiques ont des valeurs proches pour ces quatre positions.</figcaption></figure>

filtre adapté pour lequel la conjugaison de phase permet de rattraper le déphasage subi lors de la première phase. Il est donc commode de travailler avec la transformée de Fourier de la relation (3.2), et obtenir la réponse suivante dans l'espace des fréquences,

$$S(f) = |H(f)|^2 X^*(f), \qquad (3.3)$$

où $S(f)$, $H(f)$ et $X(f)$ sont respectivement les transformées de Fourier de $s(t)$, $h(t)$ et $x(t)$. Si l'on pose,

$$|H(f)|^2 = \bar{H}^2 + \delta H^2, \qquad (3.4)$$

où $\bar{H}^2$ est une constante, le signal refocalisé peut être vu comme la superposition d'une image du template initial $(\bar{H}^2 X^*(f))$ et d'un bruit de reconstruction ($N(f) = \delta H^2 X^*(f)$) à valeur moyenne nulle. Ce bruit est multiplicatif, i.e., augmenter la puissance de $x(t)$ amènerait à amplifier également le bruit. La partie utile dans le domaine des fréquences, notée $S_c(f) = |\bar{H}^2|X(f)$, est qualifiée de partie cohérente au sens de la cohérence de phase inhérente au filtre adapté et la partie bruit est notée $N(f)$ comme introduit précédemment; $s_c(t)$ et $n(t)$ correspondent aux notations temporelles.
Il est intéressant de souligner que l’intégralité de mes travaux sur le RT s’est faite à partir
de mesures dans l’espace des fréquences. L’intérêt est de s’affranchir des bruits autres que le
bruit intrinsèque à la technique elle-même. Tous les signaux temporels sont donc obtenus par
post-traitement en prenant une transformée de Fourier inverse. Afin de juger de l’allure des
signaux émis et focalisé, on donne un exemple Fig. 3.3 pour laquelle on reporte la fonction de
transfert pondérée par une fonction gaussienne (haut), le template temporel, le signal focalisé.
On constate que le signal focalisé porte un bruit que l’on isole dans le graphe donnant $n(t)$.

<figure><img src="image_57.png" /><figcaption>FIGURE 3.3 – La fonction de transfert (haut) est pondérée par une gaussienne. On obtient le signal focalisé pour un template du type impulsion gaussienne. Le bruit du signal focalisé est extrait et représenté sur le graphe du bas.</figcaption></figure>

II Métrique utile : le contraste en énergie et le contraste pic

Suivant les applications, deux visions sont à considérer quant aux métriques utiles. Pour les applications pour lesquelles le critère important est de pouvoir délivrer une énergie intense à l'instant de focalisation, on cherche à quantifier de combien l'énergie à l'instant de focalisation sort du niveau du bruit ; on parle alors de *contraste pic*, noté $\Lambda_p$ dans la suite du manuscrit. Ce contraste est défini tel que,

$$
\Lambda_p = \frac{s_c^2(0)}{n_{eff}^2} \tag{3.5}
$$

où $t = 0$ est pris comme l'instant de focalisation, $n_{eff}^2$ représente la puissance de bruit et $s_c^2(0)$ la puissance instantanée de la partie cohérente à l'instant de focalisation.

Pour les applications pour lesquelles la forme de l'impulsion à l'instant de focalisation a
une importance, le contraste en énergie, noté $\Lambda$ est alors plus utile pour quantifier ce que l'on
qualifierait de SNR dans les télécommunications. Si l'on considère $\mathcal{E}_c$ l'énergie de la partie
cohérente et $\mathcal{E}_f$ l'énergie de la partie fluctuante (bruit), on a $\Lambda$ tel que,

$$
\Lambda = \frac{\mathcal{E}_c}{\mathcal{E}_f}. \tag{3.6}
$$
Les deux contrastes ont été reliés dans (COZZA 2009); il y a été montré comment la forme
de l'impulsion envoyée impacte les contrastes. Nous verrons une manière différente de les relier,
notamment en présence de plusieurs MRT.

III Contributions au RT classique

Dans le cadre des travaux menés, une partie a porté sur le RT classique dans plusieurs contextes. Le cas des milieux hétérogènes ou multi-trajets à été considéré avec une attention particulière portant sur l'influence du nombre de MRT. Le cas de milieux indoor a également été étudié pour analyser les performances de transmissions point à point basées sur le RT pour des applications Telecom. Enfin, le cas de perturbation dans un milieu a été dérivé afin d'en mesurer l'impact sur les performances de RT.

### III.1 Influence du nombre de sources

L'étude de l'influence du nombre de sources nous a amené à reconsidérer la manière d'exprimer le contraste en énergie et le contraste pic. Pour une source, le contraste en énergie est par définition le rapport de l'énergie du template sur l'énergie de bruit, soit une image de $\bar{H}^2/\delta H^2$. Il en découle aisément que le contraste en énergie est directement relié à la variabilité du milieu tel que : $\Lambda = 1/\zeta_W^2(f_c)$, où $f_c$ est la fréquence centrale de la bande occupée de l'impulsion émise. Le calcul mené dans (COZZA et MONSEF 2014) met en évidence que ce résultat suppose que le spectre de l'impulsion soit lentement variable devant la bande de cohérence $B_c$.

En outre pour relier les contrastes entre eux, on a montré que $\Lambda_p = \Lambda \times B_T/B_c$ (COZZA et MONSEF 2014). On peut voir dans le rapport $B_T/B_c$ le nombre de degrés de liberté (modes significatifs) du champ EM généré. A l'instant de focalisation, le bruit étant nul, la partie cohérente bénéficie des contributions en phase de chaque mode équivalent. Ainsi, on observe que le contraste pic est lié à la bande occupée de l'impulsion, contrairement à $\Lambda$. Pour ce dernier on s'est intéressé à sa dépendance avec le nombre de MRT.

Intuitivement, plus le nombre de MRT augmente plus on pourrait s'attendre à une amélioration monotone des contrastes. Pour s'en convaincre, un déploiement des images de ces MRT permettrait de mettre en évidence que le système se ramènerait au cas de la cavité à retournement temporel de la Fig. 3.1. La différence est toutefois que les transducteurs seraient agencés de manière désordonnée et donc pas forcément optimale; chaque miroir aurait néanmoins une contribution. Cette intuition s'avèrerait vraie dans le cas où les MRT seraient totalement indépendantes. Or dans un milieu réverbérant, il existe une corrélation spatiale (HILL et LADBURY 2002). En d'autres termes, les signaux reçus par certains MRT portent une information partiellement contenue dans le signal reçu par le MRT voisin. Si l'on considère un niveau de corrélation moyen $\bar{\mu}_r$ l'intérêt d'augmenter le nombre $N_A$ de MRT va diminuer car une partie de l'information portée par chaque MRT deviendra redondante. Vu d'un point de vue fluctuations, celles-ci deviendront corrélées de telle sorte que la somme globale $\delta H^2$ ne sera pas à moyenne nulle, menant à une performance « plafond »; au contraire si elles étaient vraiment indépendants, la somme tendrait vers zéro.

Pour rendre compte de ce phénomène, nous nous sommes intéressés à l'amélioration du contraste en énergie obtenue après l'introduction d'un MRT supplémentaire; on s'y réfère à travers la notion d'amélioration *incrémentale* sachant $N_A$ sources déjà présentes. Ce rapport
est tel que (COZZA et MONSEF 2014)

$$ \frac{\Delta\Lambda}{\Lambda^0} = \frac{1 - \bar{\mu}_r}{1 - \bar{\mu}_r + N_A \bar{\mu}_r (2 - \bar{\mu}_r)}, \qquad (3.7) $$

où $\Lambda^0$ représente le contraste mono-MRT, i.e., pour $N_A = 1$. Pour la validation du modèle

<figure><img src="image_58.png" /><figcaption>FIGURE 3.4 – Setup expérimental : (a) une des 12 antennes papillons, montées sur les parois de la chambre réverbérante, avec un espaceur de 3 cm en polystyrène ; (b) La sonde électro-optique Enprobe's EFS105, équipée d'une tête de mesure de 6.6 mm de large.</figcaption></figure>

trouvé, on a procédé à un protocole expérimental incluant douze antennes papillons (cf. Fig. 3.4). Ces antennes ont été conçues pour travailler sur une plage de 500 MHz à 2 GHz. Pour une position de brasseur, douze fonctions de transfert ont ainsi été mesurées sur cette plage de fréquence. Afin de générer un ensemble de réalisations, 50 positions du brasseur ont été utilisées. Il est important de souligner qu'un post-traitement des fonctions de transfert a été effectué afin de soustraire la contribution de ligne directe (Line of sight) commune aux fonctions de transfert, et ce, afin de respecter l'hypothèse de milieu diffusif. Pour plus de détails, on renvoie à la publication jointe en annexe.

Un des paramètres important qui ressort de (3.7) est la corrélation moyenne existant entre les différentes antennes. On montre Fig. 3.5 l'évolution du gain incrémental en fonction du nombre d'antennes présentes dans la chambre, les courbes étant paramétrées par le niveau moyen de corrélation. On peut voir à quelle vitesse l'introduction d'un nouveau MRT perd de son efficacité sur le contraste en fonction du niveau de corrélation moyen.

Pour estimer le gain incrémental attendu dans le cas du setup proposé, nous avons estimé le niveau de corrélation moyen entre les fonctions de transfert sur des bandes de fréquence de 5%. On montre l'estimation de niveau de corrélation sur la Fig. 3.6. Comme attendu, le niveau de corrélation baisse avec la fréquence, et ce, à cause du degré de surmodage qui augmente jusqu'à 1,5 GHz puis plafonne. Dans la configuration considérée, on passe d'un niveau moyen d'environ 25% vers 500 MHz pour plafonner autour de 5% à plus haute fréquence.

De telles performances sont possibles dans un milieu réverbérant. Un certain nombre d'autres, voient dans les milieux multi-trajets une forme de réverbération. Nous avons rejoint l'intérêt porté à l'étude de la viabilité du retournement temporel dans des milieux indoors.

## Transmissions par RT en milieux indoor

L'étude sur la viabilité du RT en milieu indoor a été essentiellement expérimentale et a donné lieu à deux conférences internationales (MONSEF, COZZA et ABBOUD 2010; MONSEF et COZZA
<figure><img src="image_59.png" /><figcaption>FIGURE 3.5 – Apport incrémental à l’ajout d’un MRT sachant $N_A$ MRT déjà présents, en considérant un niveau moyen de cohérence $\bar{\mu}_r$ (en pourcentage sur chaque ligne).</figcaption></figure>

<figure><img src="image_60.png" /><figcaption>FIGURE 3.6 – Correlation moyenne estimée entre différentes fonction de transfert (trait plein) ; régression au sens des moindres carrés est superposée (pointillés).</figcaption></figure>

2011).

Les études trouvées dans la littérature portent sur l'utilisation des caractéristiques de focalisation du RT pour permettre des transmissions point à point, sécurisées le cas échéant. L'étude a consisté à considérer des milieux standards tels que des couloirs, des halls et des bureaux pour juger de la qualité de focalisation et d'étudier de plus près les aspects « transmissions sécurisées ».

Plusieurs milieux indoor ont été étudiés. La richesse du canal est directement liée à la complexité et à la durée $T_{RMS}$ de la réponse impulsionnelle (prise à -60dB) du milieu considéré; on obtient les réponses impulsionnelles Fig. 3.8.

Nous introduisons deux métriques afin d'estimer les performances du RT. La première est un SNR défini comme le taux de réjection du deuxième écho par rapport à l'impulsion utile (cf Fig. 3.9). Ce critère est utile pour limiter les interférences inter-symboles. La deuxième métrique est un gain défini comme le rapport du SNR obtenu par RT et du SNR obtenu sans RT.

Si l'on considère que le temps de réponse est un paramètre associé à la complexité du milieu,
<figure><img src="image_61.png" /></figure>

<figure><img src="image_62.png" /><figcaption>FIGURE 3.7 – Exemple de milieu indoor dans lequel les performances du RT ont été étudiées (gauche) avec les dimensions et positionement(s) des émetteurs/récepteur (droite). On constate des trajets directs (LOS) et des parcours multi-trajets (NLOS).</figcaption></figure>

<figure><img src="image_63.png" /><figcaption>FIGURE 3.8 – Réponses impulsionnelles des différents milieux étudiés : en-dehors de la chambre (en haut à gauche), dans un couloir (en haut) droite), dans la chambre réverbérante (en bas à gauche) et dans le hall illustré Fig. 3.7.</figcaption></figure>

Il est intéressant d'évaluer le SNR obtenu pour le RT. On résume les résultats Fig. 3.10. Les mesures ont été effectuées pour trois fréquences centrales avec des bandes relatives de ? ?

Le degré de complexité étant illustré par le temps de réverbération $T_{RMS}$, on observe Fig. 3.10 une certaine corrélation entre la degré de complexité du milieu et la valeur du SNR (gauche) ainsi que le gain (droite).

Il s'avère que la richesse des échos en milieu indoor est insuffisante pour que l'effet automoyennant du RT permette de focaliser le signal de manière à pouvoir transmettre sur des récepteurs proches les uns des autres sans « baver » sur le récepteur adjacent.
<figure><img src="image_64.png" /><figcaption>FIGURE 3.9 – Illustration du SNR obtenu par RT avec une fréquence de 5 GHz : rapport des deux pics considérés (écho et impulsion utile).</figcaption></figure>

<figure><img src="image_65.png" /><figcaption>FIGURE 3.10 – SNR obtenus (gauche) dans les différents milieux indoor étudiés et le gain qu'apporte le RT (droite) par rapport une transmission classique.</figcaption></figure>

### III.2 RT pour la synthèse de front d'ondes : la TREC

Le RT a été étendu à l'idée de pouvoir générer un front d'onde contrôlé en termes de polarisation, direction d'incidence et de forme, ce dernier point sera discuté plus en détail. Ayant vu précédemment que la complexité du milieu était synonyme de bonnes performance du RT, une chambre réverbérante a été considérée pour l'étude de cette extension du RT. Ce dispositif est appelé TREC pour (Time-Reversal Electromagnetic Chamber) (COZZA et MOUSSA 2015) et le principe a été formalisé par Andrea Cozza dans (COZZA 2012).

Contrairement au RT classique il n'y a pas d'étape initiale pendant laquelle une vraie source émet. En fait, une manière simple d'appréhender l'idée derrière la TREC est de partir de ce que l'on veut : créer un front d'onde incident sur un dispositif sous test. L'étape initiale du RT suppose que ce front d'onde est émis par une source ... virtuelle !

Dans un tel scénario, considérons une surface (S) que le front d'onde traverse. En discrétisant cette surface, on peut y mesurer le champ électrique $E_s$ et magnétique $H_s$, l'indice $s$ renvoyant à la surface considérée. En utilisant le principe d'équivalence, celà revient à considérer que des sources électriques et magnétiques, notées respectivement $J_s$ et $M_s$, placées en ces points géné-
reraient le front d'onde à l'extérieur de cette surface ... dans l'étape initiale du RT. Finalement, les « vraies » sources à considérer dans le cas de la TREC sont en fait ces sources $J_s$ et $M_s$. Dans le cas de front d'onde directif, la surface (S) peut être vue comme un ensemble de sources électriques et magnétiques croisées, ou coupleurs directifs, qui rayonnent le champ désiré. Cette phase doit rester néanmoins virtuelle sous peine de ne pouvoir créer le front d'onde incident sur l'objet sous test, car la surface (S) doit évidemment restée électromagnétiquement « transparente », i.e., ne doit pas comporter d'éléments diffractants. « L'astuce » consiste alors à utiliser le principe de réciprocité : les éléments rayonnants seront les MRT et la mesure du champ se fera sur la surface (S) ; cette mesure doit être la moins perturbante possible.

<figure><img src="image_66.png" /><figcaption>FIGURE 3.11 – L'arche constituant le coeur de la TREC consiste en un échantillonneur spatial du champ suivant les coordonnées polaires $\theta$ et $\varphi$ (a) - Un zoom met en évidence le système de courroie utilisé ainsi que le chariot accueillant la sonde électro-optique (b).</figcaption></figure>

Comme montré Fig. 3.11 la TREC est constitué d'un échantillonneur du champ sur une surface de Huyghens hémisphérique. Pour remplir la contrainte d'être le moins perturbant possible, l'arche est constitué de matériaux dont la permittivité est proche de l'air et la mesure du champ se fait via la sonde électro-optique présentée Fig. 3.4. Le but est de pouvoir retrouver des conditions d'espace libre pendant un laps de temps donné en recréant un front d'onde émis par une source. En pratique, ceci nécessite d'échantillonner le champ sur une surface de Huyghens et ainsi, mesurer un ensemble de fonctions de transfert entre les points d'échantillonnage de cette surface et les MRT. Il est important de rappeler que les bras du système robotisé sont en diélectrique. De ce fait, le déplacement de ceux-ci engendrent en toute rigueur une perturbation. Le degré de perturbation est présenté en détail dans (COZZA, MASCIOVECCHIO et al. 2017) et permet de conclure à la viabilité du dispositif.

La mesure d'un ensemble de fonctions de transfert est le prix à payer pour pouvoir générer des fronts d'onde arbitraires. Dit autrement, c'est la contrainte à considérer pour pouvoir émuler un milieu anéchoïque dans un environnement tel qu'une chambre réverbérante. La contrepartie positive est de pouvoir générer des séries de tests où les directions d'incidence, les polarisations, les fronts d'onde sont configurables électroniquement.

On montre Fig. 3.12a un exemple de front d'onde gaussien 3D obtenu dans la TREC. Deux angles d'ouverture ont été considérés : 10° et 40°. Pour un détail des conditions expérimentales on renvoie à l'article (COZZA et MONSEF 2017b) joint en annexe. On illustre également la qualité de synthèse de front d'onde avec la TREC en présentant les coupes de composantes cartésiennes du champ Fig. 3.12b ; les topographies attendues sont sur la partie gauche tandis
que les topographies obtenues sont sur la partie droite respectivement. Un coefficient linéaire
de corrélation de 97% témoigne de la bonne qualité de synthèse des fronts d'onde.

<figure><img src="image_67.png" /><figcaption>(a) Distribution des composantes horizontales et verticales du champ ($E_x$ et $E_y$ resp.) atten-dues (gauche) et mesurées (droite) - le co-efficient de corrélation linéaire est de 97%. Les résulats sont normalisés au maximum de champ.</figcaption></figure>

<caption>(b) Distribution 2D d'un faisceau Gaussien obte-nue avec la TREC pour un angle d'ouverture de 40° (gauche) et 10° (droite)</caption>

FIGURE 3.12 – Topographies 3D (gauche) et coupe 2D (droite) obtenues avec la TREC

Une des applications intéressantes de la TREC porte sur la mesure d'antenne hors chambre anéchoïque. Pour ce type d'application, un front d'onde plan (type onde plane) est souvent dé-siré. La thèse menée par Philippe Méton a permis d'explorer la faisabilité en la matière : générer un front d'onde plan avec une extension spatiale maîtrisée ... dans une chambre réverbérante ...

Le concept d'onde plane n'intègre pas la notion d'extension spatiale. En pratique, on consi-
dère souvent des ondes localement planes, qui pré-suppose que l'onde émise est (en général)
divergente. Cela pose le problème d'une intensité qui diminue avec la distance. Si l'on veut
contraindre l'extension spatiale du front d'onde, cela posera inévitablement, de par le principe
d'incertitude, un étalement spectral théoriquement infini.

La problématique posée a donc porté sur la possibilité de décomposer les signaux émis dans les MRT sur une base de fonctions dont les extensions spatiales et spectrales puissent être limitées. Pour simplifier la formalisation, on a considéré l'étude de la concentration de l'onde incidente dans un plan focal 1D comme montré Fig. 3.13 où le champ est polarisé suivant z et un demi-angle d'ouverture Θ. Le champ électrique, noté **E** (r), étant confiné suivant x uniquement, peut être alors décomposé en un spectre d'ondes planes, tel que,

$$
\mathbf{E}(\mathbf{r}) = \int \tilde{\mathbf{E}}(k_x) \exp(-jk_x x) \exp\left(-j\sqrt{k_0^2 - k_x^2}y\right) dk_x, \quad (3.8)
$$

où $k_0$ est le nombre d'onde, $\tilde{\mathbf{E}}(k_x)$ est le spectre onde plane à déterminer, relié au champ dans
le plan focal $\mathbf{E}(x, y = 0)$, tel que : $\tilde{\mathbf{E}}(k_x) = \mathcal{F}(\mathbf{E}(x, y = 0))$, où $\mathcal{F}(\cdot)$ est la transformée de
Fourier.

Pour que le champ soit d'une part, confiné spatialement, et d'autre part confiné spectralement
pour éviter toute dispersion du milieu, on introduit un opérateur respectant ces deux critères.
Si l'on note C l'opérateur de concentration, la fonction recherchée devra obéir à l'équation aux
valeurs propres suivantes (SLEPIAN et POLLAK 1961) :

$$CE_{x,n}(x) = \lambda_n E_{x,n}(x) \qquad (3.9)$$

où la valeur propre $\lambda_n$ correspond à la fraction d'énergie de la fonction confinée sur un intervalle fermé (spectral ou spatial). Les fonctions propres $E_{x,n}(x)$ sélectionnées seront donc celles pour lesquelles $\lambda_n \simeq 1$. La notion d'énergie est entendue ici au sens d'une norme L2.

<figure><img src="image_68.png" /><figcaption>FIGURE 3.13 – Topographie de champ visé où la concentration de l'onde est confinée spatialement et localement plane.</figcaption></figure>

Le nombre $N_{sl}$ de Slepian « parfaitement » concentrées dans un intervalle [-W;+W] peut être estimé en sommant les valeurs propres, tel que (SIMONS 2010) :

$$N_{sl} \approx \sum_{n=1}^{\infty} \lambda_n = 2W \sin \Theta \qquad (3.10)$$

Pour $W = 5$ et un angle $\Theta = 20^\circ$, on montre Fig. 3.14 les fonctions satisfaisant l'équation aux valeurs propres; on peut constater que 5 fonctions peuvent être vues comme concentrées. Pour chaque fonction, on a reporté la valeur propre associée. On peut remarquer que pour une valeur propre inférieure à 0.5 les fonctions débordent en-dehors de la zone de confinement attendue. En fonction de la répartition optimale désirée $E_{opt}$, on en déduit les coefficients $\alpha_n$ de projection de $E_{opt}$ sur les fonctions de base. On obtiendra alors un champ $E \simeq \sum_n \alpha_n E_n$.

Supposons que l'on souhaite que $E_{opt}$ corresponde à une fonction porte de largeur 2W. Pour les 4 Slepian considérées Fig. 3.15, on détermine que le jeu de coefficients doit avoir les valeurs suivantes : $\alpha_n = 0.2241, 0.1522, 0.1238, 0.08$; on obtient alors la topographie du champ total comme montrée Fig. 3.16.

Cette procédure 1D est à transposer en 3D pour le cas d'une TREC réelle. La synthèse des signaux à générer dans les MRT prendra comme base le champ échantillonné sur la surface de Huyghens évoqué lors de la présentation de la TREC. Le cas 3D est laissé en perspective.
<figure><img src="image_69.png" /><figcaption>FIGURE 3.14 – Pour une largeur W=5 et un angle Θ = 30°, on illustre les fonctions propres résultantes avec les valeurs propres λₙ correspondantes.</figcaption></figure>

<figure><img src="image_70.png" /><figcaption>FIGURE 3.15 – Topographies des fonctions Slepian pour W = 5 et un angle Θ = 20°.</figcaption></figure>

L'étude s'est limitée au cas de milieux complexes tels que les chambres réverbérantes à
brassage de modes. Lorsque un objet est déplacé dans un tel milieu, le déplacement ne modifie
pas la statistique du milieu ; en revanche, les fonctions de transfert considérées pour une phase
<figure><img src="image_71.png" /><figcaption>FIGURE 3.16 – Topographie de champ obtenue en incluant 4 Slepians</figcaption></figure>

de RT sont modifiées. Suivant le degré de perturbation, la question est de savoir comment les énergies des parties cohérentes et fluctuantes sont modifiées par rapport au cas non perturbé.

Formellement celà revient à considérer la fonction de transfert modifiée, notée $H(\nu)$, en fonction de la fonction de transfert non perturbée, notée quant à elle $H_0(\nu)$, telle que

$$H(\nu) = \rho H_0(\nu) + \delta H(\nu) \qquad (3.11)$$

où $\rho$ est le facteur de corrélation linéaire entre la fonction de transfert perturbée et non perturbée ; $\delta H(\nu)$ et $H_0(\nu)$ sont donc orthogonaux. Dans le cadre de ce modèle, il est aisé de montrer que :

$$\mathcal{E}_c = \rho^2 \mathcal{E}_{c_0}, \qquad (3.12)$$

où $\mathcal{E}_{c_0}$ et $\mathcal{E}_c$ correspondent aux énergies de la partie cohérente avant et après perturbation. Dans le cadre de l'étude menée la perturbation était associée à la rotation du brasseur.

De la même manière on peut montrer (COZZA et MONSEF 2017a) que l'évolution de l'énergie des fluctuations est telle que :

$$\mathcal{E}_f / \mathcal{E}_{f_0} = \rho^2 (1 - \Lambda_0) + \Lambda_0, \qquad (3.13)$$

où $\Lambda_0$ est le contraste du milieu initial, i.e., du milieu non perturbé.

Pour cette étude, trois configurations de la chambre ont été adoptées menant à des contrastes initiaux de 0.32, 0.67 et 1.32. Pour ce dernier cas, la chambre est surmodée ou diffusive. Pour le détail de la façon dont ces configurations ont été mises en place on renvoie à la Ref. (COZZA et MONSEF 2017a). On représente Fig. 3.17a l'évolution du rapport exprimé dans (III.2) (tirets), en attirant l'attention que l'axe des abscisses est gradué en fonction de $(1 - \rho^2)$, i.e., gradué en fonction du niveau de perturbation introduit dans le milieu ; les résultats expérimentaux sont représentés par des cercles pleins de couleurs rouge, noir et bleu associés respectivement aux contrastes initiaux de 0.32, 0.67 et 1.32.
<figure><img src="image_72.png" /></figure>

(a) Evolution du rapport de l'énergie de la partie cohérente en fonction du degré de perturbation pour trois conditionnements de la chambre. Les couleurs rouge, noire et bleue sont associées respectivement à des contrastes initiaux $\Lambda_0$ de 0.32, 0.67 et 1.32.

(b) Evolution de l'énergie des fluctuations en fonction du degré de perturbation pour trois conditionnements de la chambre donnant lieu à des contrastes initiaux $\Lambda_0$ différents.

FIGURE 3.17 – Evolution du rapport des parties cohérentes et fluctuantes en fonction du degré de perturbation illustré par le coefficient $(1 - \rho^2)$. Les résultats expérimentaux sont représentés par les symboles pour trois statistiques différentes de chambre ; les modèles théoriques sont représentés par les courbes en tirets.

<figure><img src="image_73.png" /></figure>

FIGURE 3.18 – Evolution relative du constraste en fonction du niveau de perturbation de la chambre, et ce, pour trois statistiques différentes.

On constate, que quelque soit la statistique initiale de la chambre (surmodée, partiellement surmodée,...), l'énergie de la partie cohérente décroît avec le niveau de perturbation comme on pouvait intuitivement s'y attendre. La Fig. 3.17b, quant à elle, met en évidence l'évolution de l'énergie de la partie fluctuations (en fonction du degré de perturbation également). On constate, comme montré par (III.2) (tirets), que l'évolution de celle-ci obtenue expérimentalement est fonction de la statistique initiale de la chambre (cercles de couleurs). Plus le milieu est diffusif
au départ, plus l'énergie des fluctuations augmente (par rapport à ce qu'elle était initialement) avec le degré de perturbation ; a contrario, l'énergie des fluctuations tend à diminuer si le milieu n'est pas complètement diffusif au départ, i.e., si le contraste $\Lambda_0$ est inférieur à 1.

Sachant que dans certains cas, l'énergie des fluctuations peut augmenter ou au contraire diminuer par rapport à la situation initiale, on peut se demander comment évolue le contraste. La Fig. 3.18 met en évidence l'évolution de $\Lambda/\Lambda_0$ en fonction du degré de perturbation, et ce, avec comme paramètre le contraste initial de la TREC. Il en ressort que le contraste en cas de perturbation se dégrade mais la vitesse à laquelle se produit cette dégradation est fonction de la statistique initiale du milieu.

# IV Conclusion et perspectives

Les travaux menés sur les milieux indoors standard ont permis de montrer que le degré de complexité du milieu est vraiment fondamental dans l'obtention d'une focalisation spatiale satisfaisante.

Les travaux sur le retournement temporel ont permis de mettre en évidence les limites mais aussi les possibilités offertes par cette technique dans le cadre de la TREC notamment. Le travail sur l'intérêt du multi-source a permis quant à lui de mettre en évidence comment la statistique des grandeurs liées au champ, vues au chapitre 2, est directement impliquée dans le contraste de la partie cohérente et du bruit de reconstruction. De plus, bien que restreinte à un cas 2D, on a pu voir de quelle manière confiner une extension spatiale d'un front d'onde. Le cas 3D est laissé en perspective.

# Références

* CASSEREAU, D. et M. FINK (1992). « Time-Reversal of ultrasonic fields. III. Theory of the closed Time-Reversal cavity ». *IEEE Trans Ultrason Ferroelectr Freq Controll* 39.5, p. 579-592 (cf. p. 65).

* COZZA, A. (2009). « Statistics of the performance of Time Reversal in a lossy reverberating medium ». *Phys Rev E* 80.5, p. 56604 (cf. p. 69).

* COZZA, A. et F. MONSEF (2017a). « Influence of Medium Statistics on Robustness of Time-Reversal Transmissions ». *IEEE Antennas Wireless Propagat. Lett.* 16, p. 38-41. ISSN : 1536-1225. DOI : 10.1109/LAWP.2016.2553148 (cf. p. 78).

* COZZA, A. (2012). « Emulating an anechoic environment in a wave-diffusive medium through an extended Time-Reversal approach ». *IEEE Trans. Antennas Propagat.* 60.8, p. 3838-3852 (cf. p. 73).

* COZZA, A., F. MASCIOVECCHIO, C. DORGAN, M. SERHIR, F. MONSEF et D. LECOINTE (2017). « A Dielectric Low-Perturbation Field Scanner for Multi-Path Environments ». *IEEE Trans. Antennas Propagat.* 65.4, p. 1978-1987 (cf. p. 74).

* COZZA, A. et F. MONSEF (2014). « Multiple-source Time-Reversal transmissions in random media ». *IEEE Trans. Antennas Propagat.* 62.8, p. 4269-4281 (cf. p. 69, 70).

* COZZA, A. et F. MONSEF (2017b). « Steering Focusing Waves in a Reverberation Chamber with Generalized Time Reversal ». *IEEE Trans. Antennas Propagat.* 65.3, p. 1349-1356 (cf. p. 74).

* COZZA, A. et M. H. MOUSSA (2015). *Method of checking the directivity and polarization of coherent field distributions in a reverberant medium*. US Patent 9,103,862 (cf. p. 73).
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# Multiple-Source Time-Reversal Transmissions in Random Media

Andrea Cozza, *Senior Member, IEEE* and Florian Monsef, *Member, IEEE*

**Abstract**—The ability of time-reversed signals in reproducing a given time-dependence of the electromagnetic field within random media is investigated. A general setup consisting of multiple sources cooperating in providing the best transmission is considered, where the constructive interference of their individual contributions is meant to improve the quality of the field generation with respect to a single-source setup. The medium response is described by means of tools from random-process theory, for the case of stationary media complex enough to ensure a large number of multi-path contributions. It is shown that even a very weak spatial coherence in the medium is sufficient to significantly hinder the improvement expected from the use of multiple-source scenarios. Experimental results obtained in a reverberation chamber support the validity of the proposed theory. Direct applications of these results can be found in recent proposals about the potential benefits of time-reversed signals used in wireless communications, imaging techniques, as well as in pulsed-field generation devices based on energy compression through dispersive media.

***Index Terms***—Time-reversal transmissions, spatial correlation, complex media, multi-path propagation, statistical electromagnetics.

## I. INTRODUCTION

Time-reversal transmissions were first introduced in acoustics [1], [2] and initial applications dealt mainly with open media with local inhomogeneities leading to relatively weak propagation aberrations with respect to free-space configurations. In the case of free-space configurations time-reversal can be interpreted as a direct application of Huygens' principle [3]: as well known from the equivalence theorem [4], sampling over a closed surface the field distribution generated by a radiating source allows defining equivalent currents that can recreate on their own a wavefront identical to the one originally generated by the source, be it outward or inward propagating, according to the sign of the time variable. In this respect, time reversal is predated by earlier work in nonlinear optics, particularly on phase-conjugation techniques [5], reminiscent of the ideas behind retrodirective (also known as Van Atta) arrays [6].

For the large family of quasi free-space (or weakly echoic) configurations, time-reversed wavefronts will therefore be a close (time-reversed) replica of the propagative part of the original wavefront, as long as the spatial sampling criterium [7] is fulfilled, by using a sufficient number of field transducers/sources [1], [8]. Conversely, in the case of more complex media, e.g., multiple-scattering media [9]–[14], dominated

by discrete scattering phenomena (e.g., collections of small scatterers [15], [16], multi-path environments [17]) or spatially distributed ones (e.g., turbulent media [16], inhomogeneous soils [18]), where field propagation can be modeled by means of random-process tools [16], [19], time-reversed wavefronts are affected by incoherent fluctuations [9], [11], [20], [21].

Although the appearance of these fluctuations could put off users of time-reversed signals in complex media, this nuisance is counterbalanced by a major advantage, since it is no longer necessary to satisfy the spatial sampling criterium. As a matter of fact, single-source configurations have been demonstrated to be capable of recreating high-fidelity versions of a wavefront, when operated by time-reversed signals in a complex medium [20], [22]–[26]. In this case the use of more than one emitter is no longer justified on the basis of the equivalence theorem, but rather on the need to mitigate the level of these fluctuations, by averaging them out thanks to spatial/angle/polarization diversity. Previous results about this topic are available in the literature [27]–[30], but are more concerned with the ability to obtain space-time focusing within random media, rather than assessing the quality of signals thus received.

It is the aim of this paper to understand how multiple-source implementations of time-reversed transmissions in complex media allow to control the relative intensity of the fluctuations, depending on the number of emitters and the statistical properties of the propagation medium, in particular spatial and frequency coherence (or correlation). To this end, a general theoretical approach is developed in section III, based on the concept of energy contrast introduced in section II.

The proposed theory confirms that while multiple sources can improve the quality of the transmission, residual spatial coherence in the medium can significantly reduce the overall effectiveness in mitigating the level of fluctuations. A quantitative analysis of the effects of these predictions is discussed in section IV; experimental results validating them are presented in section V. The results shown are intended to serve as quantitative predictive tools in the design of applications based on the transmission of time-reversed signals, such as in wireless communications, non-destructive testing, remote-sensing and imaging techniques.

In this respect, two families of setups for the application of time-reversal transmissions (TRTs) can be defined, both involving the need to generate a faithful reproduction of a template signal $p(t)$ at the receiver-end: (a) a scalar component of the electromagnetic field is the quantity of interest or (b) the output signal observed at the electric port of a receiver. Clearly, these two setups can be regarded as belonging to the same class of problem, by considering that the first case implies the use of an ideal receiver generating an output

<footnote>A. Cozza and F. Monsef are with the Département de Recherche en Électromagnétisme, Laboratoire des Signaux et Systèmes (L2S), UMR 8506 SUPELEC - Univ Paris-Sud - CNRS, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, France.
Contact e-mail: andrea.cozza@supelec.fr</footnote>
signal $b(t) = h \cdot E(r, t)$, where $h$ is the vector effective length of the receiving transducer found at the position $r$. This kind of response translates the use of an ideal non-dispersive elementary dipole, oriented along the electric-field component of interest $h/\|h\|$.

## II. TRT: COHERENT AND INCOHERENT CONTRIBUTIONS

We consider a set of $N_A$ sources, not necessarily identical, found at the positions $\{r_m\}$. Depending on the application, an observer or a receiver is present at the position $r_o$; no assumption is made on the reciprocal orientations of the sources and the observer. Input signals $\{a_m(t)\}$ applied to the transmitting antennas and the overall output signal $b(t)$ are chosen to be represented as power waves, in accordance with standard scattering-parameter formalism [31]. Assuming the medium to be linear, the individual contributions $\{b_m(t)\}$ provided by the excitation of the $m$th source is given by $b_m(t) = a_m(t) \star h_m(t)$, with $\star$ the time-convolution integral operator and $h_m(t)$ the impulse response between the $m$th source and the receiver output.

The goal of transmission systems is typically to ensure the ability to generate at the receiver output a signal as close as possible to a reference signal $p(t)$, hereafter referred to as the template signal, which is typically taken to be a short pulse in time-reversal literature, but it is not necessarily so; no assumption will be made about this point in this paper since in practical settings, particularly for wireless communications, waveforms are typically more complex and not pulse-like.

When dealing with complex media, if the template signal $p(t)$ were directly applied to an emitter, as soon as its time support $T_p$ is shorter than the relaxation time of the medium, any received signal would risk being significantly distorted. Time-reversal transmissions act as a generalization of matched filter, by requiring the use of excitation signals

$$a_m(t) = p(t) \star h_m(-t). \quad (1)$$

This is the standard definition of time-reversed excitation signals, as originally formulated in acoustics [1]. Unlike for matched-filter theory, the reason for this result is not the maximization of the output signal-to-noise ratio for an additive white gaussian noise channel, but rather to produce the strongest focusing of energy around the receiver [27], [29].

It is well-known that applications of time-reversed signals are limited by the assumption of a stationary medium, in order to ensure the best performance with respect to the quality of reproduction of the template signal; please refer to the papers cited in the introduction for more details. Hence, this property will be assumed throughout this paper.

In order to assess the quality of the output signals generated by this kind of procedure, the mathematical analysis presented in this paper is developed within the framework introduced in [23], where each individual contribution $b_m(t)$ to the output signal is expressed as

$$b_m(t) = \alpha_m p(t) + f_m(t), \quad (2)$$

i.e., as composed of a coherent part $\alpha_m p(t)$, with $\alpha_m \ge 0$, proportional to the template signal and a residual part $f_m(t)$, orthogonal to $p(t)$.

The overall signal $b(t)$ resulting from the superposition of the $N_A$ contributions can therefore be written as

$$b(t) = p(t)\alpha + f(t) = p(t) \sum_{m=1}^{N_A} \alpha_m + \sum_{m=1}^{N_A} f_m(t). \quad (3)$$

From (1), $b_m(t) = h_m(t) \star h_m(-t) \star p(t)$, so that for $H_m(\nu)$ the Fourier transform of the impulse response of the medium, with $\nu$ the frequency variable, it is convenient to introduce the equivalent transfer functions $W_m(\nu) = |H_m(\nu)|^2$, relating the Fourier spectrum $P(\nu)$ of the template signal to the actual output signals received when using (1); $P(\nu)$ will be assumed to have a compact support, centered at the frequency $\nu_c$ and covering a bandwidth $B_T$. The coherent coefficients $\{\alpha_m\}$ can therefore be computed by projecting $b(t)$ over $p(t)$, i.e.,

$$\alpha_m = \mathcal{E}_p^{-1} \int_{B_T} W_m(\nu) |P(\nu)|^2 d\nu, \quad (4)$$

with $\mathcal{E}_p$ the energy of the template signal $p(t)$.

## III. THEORY OF PEAK AND ENERGY CONTRASTS

Arguably, the most important figures of merit in TRTs are those measuring its ability to generate a received signal dominated by the coherent part $\alpha p(t)$, at the expenses of the residual part $f(t)$. This feature is commonly assessed by measuring the ratio between the peak value of the received signal and the average intensity of the fluctuations [20], [32]. While this kind of definition makes sense as long as $p(t)$ is expected to be a narrow pulse, it is not general enough in electrical engineering, since one could be interested in transmitting more complex signals, e.g., when dealing with wireless communications; it is therefore necessary to apply a different definition, as the one introduced in section III-A, which can be extended to any kind of signals. More importantly, it is simply related to another figure of merit, the energy contrast. This last quantity is shown in section III-B to only depend on the statistical behavior of the medium's Green's function, independently from the specific choice of the template signal $p(t)$ and more importantly from its bandwidth.

### A. Peak contrast

For the sake of simplicity, it will be assumed that $p(t)$ reaches its peak intensity at $t = 0$. We can then introduce the following definition of the peak contrast,

$$\Lambda_p = \frac{\alpha^2 p^2(0)}{\max_t \langle f^2(t) \rangle}, \quad (5)$$

as a direct measure of the relative contribution of the coherent and incoherent parts; the latter is measured by the average power of the fluctuations, defined by means of an ensemble-average operator, represented by the brackets. As proven by our derivation, the shape of the template has a non-negligible impact on the performance of time reversal and its role should
therefore be explicitly accounted for. For this reason we
introduce the shape factors $\chi_p$ and $\kappa_2$, defined as

$$
\chi_p = \frac{|p(0)|^2}{\mathcal{E}_p} \qquad (6a)
$$

$$
\kappa_2^2 = \frac{1}{B_T} \frac{\left( \int_{B_T} |P(\nu)|^2 d\nu \right)^2}{\int_{B_T} |P(\nu)|^4 d\nu}, \quad (6b)
$$

where Cauchy-Schwarz inequality indicates that $\kappa_2 \le 1$;
$\chi_p$ measures whether the energy of the template signal is
distributed in a narrow span of time as a pulse, or rather
as a longer signal. It has dimensions of a bandwidth and its
definition implies that $\chi_p \le B_T$, where the extreme value
is reached only for a sinc pulse, corresponding to a constant
power spectral density over $B_T$.

In a similar manner, we introduce $\chi_f$, defined as

$$
\chi_f = \frac{\max_t \langle f^2(t) \rangle}{\langle \mathcal{E}_f \rangle} \qquad (7)
$$

in order to relate the overall energy of the residual fluctuations
to their peak average intensity. The use of ensemble averages
in (7) is due to the random nature of the residual fluctuations
$f(t)$, inherited from the description of the medium Green's
functions as random processes. The major difference between
(6) and (7) is that the latter is mostly dependent on the medium
and not on the template signal.

From (5), (6) and (7)

$$
\langle \Lambda_p \rangle = \langle \Lambda \rangle \frac{\chi_p}{\chi_f}, \qquad (8)
$$

where

$$
\Lambda = \frac{\mathcal{E}_c}{\mathcal{E}_f} = \frac{\alpha^2 \mathcal{E}_p}{\int_{B_T} |F(\nu)|^2 d\nu}, \quad (9)
$$

is the energy contrast, defined as the ratio of the energy
E_c of the coherent portion of the received signal and the
energy E_f of the fluctuating residual part; F(ν) in (9) is
the Fourier transform of the overall fluctuating part f(t).
The energy contrast, as it will be proven in section III-B,
is practically independent from the choice of the template
signal, but strongly related to the statistical properties of the
medium and the number of sources used. As a result, (8)
allows separating the respective contributions of the medium
and the sources from the choice of the template signal, with
respect to the quality of the received signals. In particular, the
impact of the number of sources used in the transmission only
appears in Λ, hence it is sufficient to study this quantity, as
done in the next section.

B. Energy contrast

In order to compute the ensemble average of the energy
contrast Λ, the following approximation of (9) will be applied:

$$
\langle \Lambda \rangle \approx \frac{\langle \mathcal{E}_c \rangle}{\langle \mathcal{E}_f \rangle} = \frac{\langle \alpha^2 \rangle \mathcal{E}_p}{\int_{B_T} \langle |F(\nu)|^2 \rangle d\nu}, \quad (10)
$$

justified by applying the delta method [33] to the definition of
Λ, with respect to the energies of the coherent and fluctuating
parts. It is expected to hold with good precision as long as the
number of degrees of freedom underlying the field distribution
is much larger than one, as shown in [23]. This condition is not
conservative, since it is required when time reversal is applied
to complex media, as discussed in [20], [32]. The experimental
results presented in V-B prove that this approximation is very
robust, as they are derived frequency by frequency, without
requiring the definition of a bandwidth, as initially stated in
the introduction.

From (3)

$$
\langle \alpha^2 \rangle = \left\langle \left( \sum_{i=1}^{N_A} \alpha_i \right)^2 \right\rangle = \sum_{i=1}^{N_A} \langle \alpha_i^2 \rangle + \sum_{i \neq j} \langle \alpha_i \alpha_j \rangle, \quad (11)
$$

while

$$
\int_{B_T} \langle |F(\nu)|^2 \rangle d\nu = \sum_{i=1}^{N_A} \int_{B_T} \langle |F_i(\nu)|^2 \rangle d\nu \\
+ \sum_{i \neq j} \int_{B_T} \langle F_i(\nu) F_j^*(\nu) \rangle d\nu. \tag{12}
$$

As it will be shown later, the residual correlation existing
between mixed terms in (12) should not be neglected, lest
we overestimate the actual performance of the transmission
system.

In order to compute these four sets of terms, it is necessary
to consider the mutual moments $\langle W_i(\nu_1)W_j(\nu_2) \rangle$. As argued
in the appendix, the functions involved in these moments,
though defined for discrete indexes (i.e., the source identi-
fiers) can be interpreted as continuous parametric fields, i.e.,
$W_i(\nu) = W(\mathbf{r}_i, \nu; \mathbf{h}_i)$, parameterized by the vector effective
height of each respective source. While this would mean
dealing with a set of $N_A$ different fields, it is possible to
simplify our derivation by noticing that a common feature of
complex media is that they generate depolarized fields, i.e., on
average they do not present dominant polarization components
[16], [34], independently from the way the medium was first
excited; non-line-of-sight propagation channels comply with
this kind of description, which is one of the reasons for the
use of reverberation chambers as channel emulators [35]. This
property leads to a strong simplification in the statistical rep-
resentation of Green's function. Under this assumption, if the
sources are arbitrarily oriented versions of the same antenna,
all the parametric fields share the same statistical moments.
This property, discussed in the appendix, is instrumental in
our derivation, since it allows dropping the effective heights
as parameters, i.e.,

$$
\langle W_i(\nu_1) W_j(\nu_2) \rangle = \langle W(\mathbf{r}_i, \nu_1) W(\mathbf{r}_j, \nu_2) \rangle. \quad (13)
$$
As a result the moments $\langle W_i(\nu_1)W_j(\nu_2) \rangle$ can now be related to the space-frequency coherence function of what will be referred to as the TRT field $W(r, v)$. There should be no confusion at this point that even though (13) is no more dependent on the effective heights $\{h_m\}$, there is no such a thing as a single TRT field, but a set of fields, depending on the characteristics of each source. The TRT field $W(r, v)$ considered hereafter should therefore be regarded as an auxiliary function that makes sense only when considering its statistical moments, as those in (13).

We need to recall that

$$ \langle W(\mathbf{r}_i, \nu_1)W(\mathbf{r}_j, \nu_2) \rangle = \langle W(\mathbf{r}_i, \nu_1) \rangle \langle W(\mathbf{r}_j, \nu_2) \rangle + \mu(\mathbf{r}_i, \mathbf{r}_j, \nu_1, \nu_2) \sqrt{\sigma_W^2(\mathbf{r}_i, \nu_1)\sigma_W^2(\mathbf{r}_j, \nu_2)}, \quad (14) $$

where $\sigma_W^2(r, v) = \langle |W(r, v)|^2 \rangle - |\langle W(r, v) \rangle|^2$ is the variance of the TRT field and $\mu(r_i, r_j, \nu_1, \nu_2)$ is its spectral degree of coherence [36].

In practical scenarios involving complex media where the field can be represented as a large collection of plane waves, e.g., those involving multiple-scattering propagation and interactions with rough surfaces, some further properties can be invoked. First, statistical moments of the Green's dyadic functions can be approximated as being independent from the point of observation, in the same way as wide-sense stationarity is assumed for random time-series, but with respect to space rather than time. As a result $\sigma_W^2(r, v) \simeq \sigma_W^2(r_o, v)$, $\forall r$ so that in the following the argument $r$ will be dropped.

Furthermore, by introducing $\nu_d = \nu_1 - \nu_2$, $\nu_c = (\nu_1 + \nu_2)/2$, $\mathbf{r}_c = (\mathbf{r}_i + \mathbf{r}_j)/2$ and $\Delta r_{ij} = \|\mathbf{r}_i - \mathbf{r}_j\|$, the degree of coherence can be factorized as

$$ \mu(\mathbf{r}_i, \mathbf{r}_j, \nu_1, \nu_2) \approx \mu_\nu(\mathbf{r}_c, \nu_d)\mu_\mathbf{r}(\nu_c, \Delta r_{ij}), \quad (15) $$

as argued in [36, section 5.3] for quasi-homogeneous random fields. The two resulting functions in (15), $\mu_\nu(\mathbf{r}_c, \nu_d)$ and $\mu_\mathbf{r}(\nu_c, \Delta r_{ij})$, separately account for the degree of frequency and spatial coherence, respectively. While this property can be expected to hold over relatively wide regions of the medium when observed at a single frequency at the time, frequency invariance is a reasonable approximation only when required over a limited bandwidth $B_W$. We will assume in the rest of this paper that $B_W \gg B_T$.

The rationale for approximation (15) is provided by the observation that for complex media the degree of coherence falls to negligible levels if one of two conditions is met: a) as soon as $|\nu_2 - \nu_1| > B_c$, with $B_c$ the coherence bandwidth of the medium, as defined later on, or; b) when $\Delta r_{ij} > D_c$, the coherence distance (or length) of the medium. When $|\nu_2 - \nu_1| \simeq 0$, the degree of coherence is dominated by the spatial coherence, while the opposite holds when $\Delta r_{ij} \simeq 0$.

Finally, by assuming that the template signal $p(t)$ operates over a frequency range $B_T \gg B_c$, the frequency degree of coherence will dominate (15), independently of the observer position, by virtue of the stationarity property; this condition is known to be necessary to ensure effective TRTs in complex media [32]. Hence, $\mu_\nu(\mathbf{r}_c, \nu_d) = \mu_\nu(\nu_d)$ in the following. The existence of three separate frequency scales is therefore

assumed throughout this work, subject to the condition

$$ B_W \gg B_T \gg B_c. \quad (16) $$

The term $\langle \alpha_i \alpha_j \rangle$ in (11) can now be computed by using (4), (14) and (15)

$$ \begin{aligned} \mathcal{E}_p^2 \langle \alpha_i \alpha_j \rangle = & \iint_{B_T} |P(\nu_1)|^2 |P(\nu_2)|^2 \times \\ & \left\{ \sigma_W^2(\nu_c) \mu_\nu(\nu_d) \mu_\mathbf{r}(\nu_c, \Delta r_{ij}) + \langle W(\nu_1) \rangle \langle W(\nu_2) \rangle \right\} d\nu_1 d\nu_2. \end{aligned} \quad (17) $$

Switching to $\nu_c$ and $\nu_d$, the first part of the integral can be written as

$$ \begin{aligned} & \iint_{B_T} |P(\nu_1)|^2 |P(\nu_2)|^2 \sigma_W^2(\nu_c) \mu_\nu(\nu_d) \mu_\mathbf{r}(\nu_c, \Delta r_{ij}) d\nu_1 d\nu_2 = \\ & \qquad \int \sigma_W^2(\nu_c) \mu_\mathbf{r}(\nu_c, \Delta r_{ij}) \times \\ & \qquad \int |P(\nu_c - \nu_d/2)|^2 |P(\nu_c + \nu_d/2)|^2 \mu_\nu(\nu_d) d\nu_d d\nu_c \end{aligned} \quad (18) $$

where the second integral is significantly different from zero only for $\nu_d < B_c \ll B_T$, resulting into

$$ \begin{aligned} & \iint_{B_T} |P(\nu_1)|^2 |P(\nu_2)|^2 \sigma_W^2(\nu_c) \mu_\nu(\nu_d) \mu_\mathbf{r}(\nu_c, \Delta r_{ij}) d\nu_1 d\nu_2 \approx \\ & \qquad \sigma_W^2(\nu_c) B_c \int \mu_\mathbf{r}(\nu_c, \Delta r_{ij}) |P(\nu_c)|^4 d\nu_c, \end{aligned} \quad (19) $$

having extracted $\sigma_W^2(\nu_c)$ from the integral because of (16), while

$$ B_c = \int \mu_\nu(\nu_d) d\nu_d \quad (20) $$

is the coherence bandwidth of the medium. By introducing the variability $\zeta_W^2(\nu)$ of the TRT field

$$ \zeta_W^2(\nu) = \sigma_W^2(\nu) / \langle W(\nu) \rangle^2 \quad (21) $$

we obtain

$$ \langle \alpha_i \alpha_j \rangle \approx \langle W(\nu_c) \rangle^2 \left[ 1 + \mathcal{E}_p^{-2} B_c \zeta_W^2(\nu_c) \int_{B_T} \mu_\mathbf{r}(\nu_c, \Delta r_{ij}) |P(\nu_c)|^4 d\nu_c \right]. \quad (22) $$

This expression can be further simplified by noticing that

$$ \mathcal{E}_p^{-2} B_c \int_{B_T} \mu_\mathbf{r}(\nu_c, \Delta r_{ij}) |P(\nu_c)|^4 d\nu_c \le \frac{B_c}{B_T \kappa_2^2}. \quad (23) $$

We can therefore conclude that

$$ \langle \alpha_i \alpha_j \rangle \approx \langle W(\nu_c) \rangle^2, \quad (24) $$

as long as $\kappa_2^2 \gg B_c/B_T$. The computation of $\langle \alpha_i^2 \rangle$ goes along the same lines, with the difference that now $\Delta r_{ij} = 0$. It results into $\langle \alpha_i^2 \rangle = \langle W(\nu_c) \rangle^2$, so that the average coherent energy is

$$ \langle \mathcal{E}_c \rangle = N_A^2 \langle W(\nu_c) \rangle^2, \quad (25) $$
a result that corresponds to the case of individual coherent contributions {$\alpha_i$} always constructively interfering, implying a unitary positive correlation. In other words, the spatial invariance of the moments of the TRT field translates into individual coherent contributions that asymptotically behave as deterministic rather than random quantities; this result can be ascribed to the property of self-averaging well known in time-reversal applications [32].

From (2) and (4), the term involving mutual moments in (12) is made up of four contributions

$$ 
\begin{aligned}
\int_{B_T} \langle F(\mathbf{r}_i, \nu) F^*(\mathbf{r}_j, \nu) \rangle \, d\nu = & \\
& \int_{B_T} \langle W(\mathbf{r}_i, \nu) W(\mathbf{r}_j, \nu) \rangle |P(\nu)|^2 d\nu + \int_{B_T} \langle \alpha_i \alpha_j \rangle |P(\nu)|^2 d\nu \\
& - \int_{B_T} \langle W(\mathbf{r}_i, \nu) \alpha_j \rangle |P(\nu)|^2 d\nu - \int_{B_T} \langle W(\mathbf{r}_j, \nu) \alpha_i \rangle |P(\nu)|^2 d\nu.
\end{aligned}
\quad (26) $$

Recalling (14) and (15)

$$ 
\begin{aligned}
\int_{B_T} \langle W(\mathbf{r}_i, \nu) W(\mathbf{r}_j, \nu) \rangle |P(\nu)|^2 d\nu = & \\
& \sigma_W^2(\nu_c) \int_{B_T} \mu_r(\Delta r_{ij}, \nu) |P(\nu)|^2 d\nu + \langle W(\nu_c) \rangle^2 \mathcal{E}_p,
\end{aligned}
\quad (27) $$

a result also appearing in the integrand functions in the last two terms in (26)

$$ 
\begin{aligned}
\langle \alpha_i W(\mathbf{r}_j, \nu) \rangle &= \mathcal{E}_p^{-1} \int_{B_T} \langle W(\mathbf{r}_j, \nu) W(\mathbf{r}_i, \nu') \rangle |P(\nu')|^2 d\nu' \\
&\approx \mathcal{E}_p^{-1} \sigma_W^2(\nu) \mu_r(\Delta r_{ij}, \nu) B_c |P(\nu)|^2 + \langle W(\nu) \rangle^2, 
\end{aligned}
\quad (28) $$

having applied the same line of reasoning as in the derivation of (24). Hence,

$$ 
\begin{aligned}
\int_{B_T} \langle \alpha_i W(\mathbf{r}_j, \nu) \rangle |P(\nu)|^2 d\nu = & \\
& = \mathcal{E}_p^{-1} \sigma_W^2(\nu_c) B_c \int_{B_T} \mu_r(\Delta r_{ij}, \nu_c) |P(\nu_c)|^4 d\nu_c + \langle W(\nu_c) \rangle^2 \mathcal{E}_p \\
& \approx \mathcal{E}_p \langle W(\nu_c) \rangle^2, 
\end{aligned}
\quad (29) $$

as long as $\kappa_2^2 \gg B_c/B_T$; this result also holds when the indexes i and j are switched. Stitching together these results yields

$$ \int_{B_T} \langle F(\mathbf{r}_i, \nu) F^*(\mathbf{r}_j, \nu) \rangle d\nu \approx \sigma_W^2(\nu_c) \int_{B_T} \mu_r(\Delta r_{ij}, \nu) |P(\nu)|^2 d\nu. \quad (30) $$

Applying essentially the same procedure, the remaining term needed is

$$ \int_{B_T} \langle |F(\mathbf{r}_i, \nu)|^2 \rangle d\nu \approx \sigma_W^2(\nu_c) \mathcal{E}_p. \quad (31) $$

Finally, (10) becomes

$$ \langle \Lambda \rangle \approx \frac{\zeta_W^{-2}(\nu_c) N_A}{1 + \mathcal{E}_p^{-1} N_A^{-1} \sum_{i \neq j} \int_{B_T} \mu_r(\Delta r_{ij}, \nu) |P(\nu)|^2 d\nu}. \quad (32) $$

It can be noticed that

$$ \mu_{ij} = \mathcal{E}_p^{-1} \int_{B_T} \mu_r(\Delta r_{ij}, \nu) |P(\nu)|^2 d\nu \quad (33) $$

in (32) represents the degree of spatial coherence between the single-source contributions averaged over $B_T$, weighted by $|P(\nu)|^2$, while the average energy contrast expected for a single-source configuration is

$$ \langle \Lambda^o \rangle = \langle \Lambda \rangle \Big|_{N_A=1} = \zeta_W^{-2}(\nu_c), \quad (34) $$

allowing to restate (32) as

$$ \langle \Lambda \rangle \approx \langle \Lambda^o \rangle D(N_A, \bar{\mu}_r), \quad (35) $$

with

$$ D(N_A, \bar{\mu}_r) = \frac{N_A}{1 + (N_A - 1)\bar{\mu}_r} \quad (36) $$

the improvement provided by exploiting spatial/polarization diversity from $N_A$ sources, in an arrangement presenting an average spatial coherence $\bar{\mu}_r$

$$ \bar{\mu}_r = \frac{\sum_{i \neq j} \mu_{ij}}{N_A(N_A - 1)}. \quad (37) $$

## IV. PERFORMANCE ANALYSIS

The derivation of (35) highlights how the average energy contrast obtained by means of TRT depends on two independent contributions: the average single-source contribution, $\langle \Lambda^o \rangle$, and a multiplicative factor $D(N_A, \bar{\mu}_r)$, hereafter referred to as the diversity factor. In this section the relative roles of these two terms are discussed.

### A. Single-source configurations

The individual contributions $\langle \Lambda^o \rangle$ appear only to depend on the statistical variability $\zeta_W^2(\nu)$ of the TRT field and therefore on the statistical properties of the Green's function of the medium, and not on the characteristics of the template signal (bandwidth, shape factor $\chi_p$). The fact that the bandwidth of the template has no impact on $\langle \Lambda^o \rangle$ implies that the energy efficiency of TRT is firmly constrained by the type of medium we have to deal with.

These limitations in the energy performance (energy contrast) do not apply to the peak contrast, which on the contrary yields more easily to design needs. As a matter of fact, the shape factor $\chi_p$ introduced in (6) allows to increase the average peak contrast, since Wiener-Khinchin theorem implies that (8) goes like

$$ \langle \Lambda_p \rangle \sim \langle \Lambda^o \rangle \frac{T_f}{T_p} \sim \langle \Lambda^o \rangle \frac{B_T}{B_c}, \quad (38) $$

where $T_p$ and $T_f$ are the effective time-domain supports (e.g., the half-power durations) of the template (coherent part)
<figure><img src="image_74.png" /><figcaption>Fig. 1: The diversity factor $D(N_A, \bar{\mu}_r)$ as a function of the number of sources $N_A$ and the average spatial degree of coherence $\bar{\mu}_r$ (in percent units over each line).</figcaption></figure>

and residual fluctuation signals (incoherent part related to the relaxation time of the medium), respectively. In other words, the closer the template to a short pulse, the better the performance of TRTs. The interest of our approach is that it makes clear that while a higher peak contrast can be generated by acting on $B_T$ (hence reducing $T_p$), the overall energy efficiency of TRTs is not under control in complex media; these two quantities act as complementary metrics of the performance of TRTs.

## B. Multiple-source configurations

The function $D(N_A, \bar{\mu}_r)$ can have a multiplicative effect on the energy contrast of a single-source configuration, since (36) implies that $D(N_A, \bar{\mu}_r) \ge 1$. In the ideal case of $\bar{\mu}_r = 0$, we expect $D(N_A, \bar{\mu}_r) = N_A$, resulting in a potentially dramatic improvement of the energy contrast. Since (25) proves that the coherent parts of each individual contribution are hardly distinguishable, the improvement of the energy contrast should not be interpreted as solely due to an increase in the overall coherent transmission. The reasons for the improvement in the overall energy contrast is rather to be found in the perfect decorrelation between the residual fluctuations contributed by each source. Under these circumstances, the average energy of the incoherent fluctuations will sum up as a linear function of $N_A$, to be compared with the $N_A^2$ increase expected for the coherent part.

In the more realistic case of $\bar{\mu}_r > 0$, the trend of $D(N_A, \bar{\mu}_r)$ as a function of $N_A$ and $\bar{\mu}_r$ is shown in Fig. 1. It can be observed that as the number of sources increases, the incremental improvement in the overall energy contrast decreases with $N_A$. Moreover, as $N_A \to \infty$, there exists a maximum attainable improvement given by

$$ \lim_{N_A \to \infty} D(N_A, \bar{\mu}_r) = \frac{1}{\bar{\mu}_r}. \qquad (39) $$

Even a relatively weak average spatial coherence leads to an upper limit to the improvement: e.g., $\bar{\mu}_r = 0.1$, usually regarded as a negligible level of coherence, directly leads to a maximum improvement of a factor 10 which, though not negligible, implies that it is not possible to do better than the performance that would be obtained with 10 independent

<figure><img src="image_75.png" /><figcaption>Fig. 2: Incremental contribution to the average overall energy contrast $\langle \Lambda \rangle$ from a source added to $N_A$ pre-existing sources, as a function of the average spatial degree of coherence $\bar{\mu}_r$ (in percent units over each line).</figcaption></figure>

sources in an idealized setup. Hence, $D(N_A, \bar{\mu}_r)$ can be interpreted as an equivalent number of uncorrelated sources taking part to the transmission.

By far worse is the fact that (39) requires an infinite number of real sources excited at the same time, thus implying a huge redundancy level. The limited contribution of each individual source is better observed by computing the incremental improvement due to a single source added to $N_A$ already existing, normalized to that of a single source, i.e.,

$$ \frac{\Delta\langle\Lambda\rangle}{\langle\Lambda^o\rangle} = D(N_A + 1, \bar{\mu}_r) - D(N_A, \bar{\mu}_r), \qquad (40) $$

hence

$$ \frac{\Delta\langle\Lambda\rangle}{\langle\Lambda^o\rangle} = \frac{1 - \bar{\mu}_r}{1 - \bar{\mu}_r + N_A\bar{\mu}_r(2 - \bar{\mu}_r) + N_A^2\bar{\mu}_r^2}. \qquad (41) $$

Since (41) goes like $O(N_A^{-2})$, it appears that the realized improvement in the performance of TRT after adding a new source decreases relatively fast as $N_A$ increases, even for rather low values of $\bar{\mu}_r$, as shown in Fig. 2.

It is important to stress the fact that spatial coherence has no link with the more common concept of antenna coupling. Spatial coherence rather implies that the random TRT between one source and the receiver/observer and those of other sources are not entirely uncorrelated random processes. Moreover, spatial coherence is a property that is independent of the structure of the sources and rather depends on the complexity of the medium.

In the design of TRT schemes, it is therefore important to acknowledge this loss of effectiveness of the sources. As a result, there exists a maximum number of sources $N_A^*$ such that each relative contribution to the overall contrast is higher than a given value $s < 1$. Fig. 3 show how $N_A^*$ evolves with $\bar{\mu}_r$ and $s$. Notice how for a requirement of a loss of less than 10 % in the incremental improvement, i.e., $s \ge 90 \%$, it is worthwhile to switch from 1 to 2 sources only if $\bar{\mu}_r \le 5 \%$, a very low level of residual correlation/coherence.

In the case of $\bar{\mu}_r \ll 1$ the results in Fig. 3 are well approximated by

$$ N_A^* \approx 1 + \frac{s^{-1/2} - 1}{\bar{\mu}_r}. \qquad (42) $$
<figure><img src="image_76.png" /><figcaption>Fig. 3: The maximum number of sources allowed in order to ensure that all of them provide a contribution to the overall contrast at least equal to $s \langle \Lambda^o \rangle$. The 9 curves correspond to values of $s$ going from 10% to 90%, with a 10% step.</figcaption></figure>

## V. EXPERIMENTAL VALIDATION

In order to assess the validity of (35) we chose a reverberation chamber as test case. The rationale for such a choice is threefold: 1) reverberation chambers are routinely used to emulate the effects of wave propagation in complex media of practical interest, such as indoor and outdoor environments, e.g., for hand-held mobile set testing [35], [37], [38]. In this respect, they provide a natural setting for testing our theory; 2) reverberation chambers often come with mechanical field stirrers, intended to modify the boundary conditions for waves propagating through them. A stirrer comes in handy here, as a simple way of generating a large number of random configurations, i.e., an ensemble of media sharing the same statistical properties; 3) it was found in [39] that the variability of the field within a reverberation chamber is far from being a constant value, and can be expected to take rapidly changing values in the lower-frequency range. This property is here interesting as it allows testing our predictions against different values of $\zeta_W^2$, i.e., varying random-field statistics, depending on the central frequency of operation; by the same token, spatial coherence is also expected to be a function of frequency [40], so that our predictions can be tested against a whole range of values of $\bar{\mu}_r$.

The effectiveness of TRTs is better appreciated in the time domain. In order to simplify the experimental validations, we rather carried out the measurements in the frequency domain; time-domain signals were then post-processed by means of discrete Fourier transforms. Transfer functions were measured with a vector network analyzer, as discussed in section V-A.

Mechanical stirring can only affect that portion of the field (and therefore of the transfer functions) that interacts with it, i.e., submitted to multiple scattering interactions. As a result, line-of-sight propagation between the sources and the receiver position appear as a deterministic contribution shared by each realization of {$H_m(\nu)$} [35]. In order to ensure realizations as independent as possible, line-of-sight contributions were subtracted from the original transfer functions, defining a new

<figure><img src="image_77.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_78.png" /><figcaption>(b)</figcaption></figure>

<caption>Fig. 4: The experimental setup used for the validation of the proposed theory. A total of 12 transmitting antennas (see Fig. 5(a)) were mounted over the 4 walls of a reverberation chamber, equipped with a field stirrer (paddle): (a) top view; (b) front view from the AA cut.</caption>

set of results as {$H_m(\nu) - \langle H_m(\nu) \rangle$}. It is important to stress that this procedure is in no way related to the assumptions invoked throughout our derivation, but is merely a matter of ensuring a set of independent realizations. At the same time this procedure also removes eventual couplings appearing through the excitation circuit.

### A. Experimental setup

A schematic description of the chamber is given in Fig. 4. A total of 12 sources were included into the chamber; the sources were identical and consisted of printed bow-tie antennas perpendicularly mounted over the walls of the chamber, as shown in Fig. 5(a). Their input ports were connected to a multiplexer through coaxial cables. These antennas where chosen in order to cover the frequency range from 0.5 to 2.0 GHz.

The receiver was an electro-optical probe (Enprobe's EFS-105) positioned at the center of the chamber, resting on a 1 m high styrofoam column. The probe, shown in Fig. 5(b), has
<figure><img src="image_79.png" /><figcaption>Fig. 5: Experimental setup: (a) one of the 12 bow-tie transmitting antennas, mounted over the walls of the reverberation chamber, kept 3 cm away from them by a piece of styrofoam; (b) Enprobe's EFS105 optical probe, measuring head 6.6 mm wide, connected to an optical-fiber cable.</figcaption></figure>

<figure><img src="image_80.png" /><figcaption>Fig. 6: Estimate of $\zeta_W^2$, averaged over the results obtained with 12 sources, smoothed over a 5 % bandwidth.</figcaption></figure>

<figure><img src="image_81.png" /><figcaption>Fig. 7: Plots of the 50 realizations of $W_1(\nu) = |H_1(\nu)|^2$ (gray) and of their ensemble average $\langle W_1(\nu) \rangle$ (black curve). Fast fluctuations on a short range in the $\{W_1(\nu)\}$ are a consequence of the narrow coherence bandwidth of the medium, as opposed to the smoother evolution of $\langle W_1(\nu) \rangle$, a direct indication of the existence of two separate scales of frequency-related phenomena.</figcaption></figure>

the particularity of presenting a negligible scattering cross-section, while behaving as an ideal Hertzian dipole. Its ability to measure phase-shift angles makes it ideally suited for time-domain post-processing. This configuration corresponds to the case where the sources are employed in the generation of a field at the probe position. The probe was operated as a receiver by connecting it to a vector network analyzer (Rohde & Schwarz ZVB8) through an optic-to-electric converter, while the sources were independently excited one-by-one by means of a multiplexer, as shown in Fig. 4. In this way 12 transfer functions were collected for each realization, corresponding to a single stirrer position. The stirrer consisted of four tilted paddles mounted around a vertical axis; 50 angular positions were considered, in order to generate a corresponding number of sets of transfer functions between the sources and the observation point.

Preliminary results from this setup are given in Fig. 6, where an estimate of the variability of the TRT is shown. This quantity spans values going from 1.1 to 1.7; the lower value is close to what is expected for an ideal diffuse-field configuration, related to Rayleigh fading in wireless communications [17], [41], while the higher value is a measure that field propagation is less complex than expected, with a reduced number of degrees of freedom [39]. Moreover, $\zeta_W^2 \neq 1$ implies that the underlying field cannot be described as a Gaussian process, and therefore neither as a Rayleigh propagation. This fact is important in proving that the agreement between results discussed in the next section and the proposed theory is not limited to simplified configurations (e.g., Rayleigh propagation), but also in intermediary cases closer to practical settings.

Complementary information is provided by the power density results shown in Fig. 7 as a black line, estimated from the 50 realizations of the first of the 12 sources. The received fields appear as essentially behaving as random processes, weakly dependent on the frequency of observation (frequency stationarity). These results are of interest since they imply that condition (16) is satisfied: as a matter of fact, the average power density (black curve) evolves rather smoothly along the frequency axis, while the fast random fluctuations of each realization provide a direct assessment of the frequency coherence of the medium, which is of the order of 100 kHz. Furthermore, Fig. 7 does not give away any clue about the fact that the statistics of the medium is fundamentally different at the two ends of the frequency axis, as opposed to Fig. 6. The importance of the information derived from the variability, already exposed in section III-B, will be confirmed in the next section.

## B. Output field statistics

Throughout this section we will consider a gaussian template signal $p(t)$, with a -20 dB bandwidth $B_T$ modulating a harmonic carrier at the frequency $\nu_c$; the values of $B_T$ were chosen in order to be much larger than the coherence bandwidth $B_c$, as required in the derivation of our theory. The received signals were computed by means of inverse discrete Fourier transforms of the spectra $B_m(\nu) = |H_m(\nu)|^2 P(\nu)$;
<figure><img src="image_82.png" /></figure>

<figure><img src="image_83.png" /><figcaption>Fig. 8: A single realization of signals received for an increasing number of active sources, for $\nu_c = 0.7$ GHz and: (a) $B_T/\nu_c = 5\%$; (b) $B_T/\nu_c = 20\%$. The peak values attained by the signals are practically independent of the number of sources involved.</figcaption></figure>

overall signals were then computed by summing over a chosen subset of individual contributions. For the sake of simplicity, the following convention will be used: for a configuration with $N_A$ active sources, the resulting signals correspond to the sum of the first $N_A$ sources, without considering all the available combinations. In order to allow a direct comparison of signal amplitude in different configurations, the output signals are always normalized to $N_A$, while $p(t)$ has unit energy.

Examples of these signals are given in Fig. 8, for $\nu_c = 0.7$ GHz and two relative bandwidths $B_T/\nu_c$, namely 5% and 20%, when exciting an increasing number of sources. While increasing $B_T$ leads to a clear improvement in the peak contrast, as already well documented in the available literature about TRT, a further important information is that the resulting fields are practically independent of the number of sources involved. This observation confirms the results obtained in (25), stating that the peak-value of the coherent part is proportional to $N_A$, i.e., the individual contributions are practically identical. Still, Fig. 8 also shows that the ratio between the peak coherent value and the background fluctuations does not improve as fast when $N_A$ increases; in other words, while the peak received signal increases, its quality is not improving as fast as would be expected for ideally incoherent fluctuations. This behavior is better understood by switching from the intuitive peak contrast to the energy contrast, as argued in section III.

In this respect, we needed to compute the empirical probability density functions (epdf) obtained for the single-source

<figure><img src="image_84.png" /></figure>

<figure><img src="image_85.png" /></figure>

<figure><img src="image_86.png" /><figcaption>Fig. 9: Empirical probability density functions for $\Lambda^o$ for three values of the central frequency $\nu_c$: (a) 0.7 GHz; (b) 1.3 GHz and (c) 2.0 GHz, for a 5% relative bandwidth. Twelve results are shown on each graph, corresponding to the results obtained from the use of each transmitting antenna.</figcaption></figure>

$\Lambda^o$, shown in Fig. 9, derived for the 12 sources and the data collected for the 50 realizations. These results confirm that the epdfs are practically independent of the source, supporting the arguments presented in the appendix.

The main interest in the use of the energy contrast is that its average value $\langle\Lambda^o(\nu)\rangle$ does not depend on the specific choice of $p(t)$, but only on the medium; evidence for this claim is provided in Fig. 10, where $\langle\Lambda^o(\nu)\rangle$ is computed either applying its definition (9), based on output signals, or from (34), based on estimates of medium statistics; a good agreement is found within a few percent points over the entire frequency range and for three different relative bandwidths, while minor disagreements are likely due to the approximation applied in (10). In particular, going from $B_T/\nu_c = 5\%$ to 20%, e.g., a fourfold increase, does not lead to any improvement in $\langle\Lambda^o(\nu)\rangle$, as predicted from the proposed theory. Of notable interest is the fact that the agreement is extended to the lower
<figure><img src="image_87.png" /><figcaption>Fig. 10: Comparisons between several estimates of $\langle \Lambda^o \rangle$, computed from : the arithmetic average over the 50-realization dataset (solid line); the reciprocal of the variability, i.e., $\zeta_W^{-2}(\nu)$ (gray dots), as predicted by (34); its smoothed version computed by averaging with $|P(\nu)|$ as a weighting function (dashed line). The three graphs relate to three different relative bandwidths: (a) 5 %; (b) 10 %; (c) 20 %.</figcaption></figure>

frequency range, where the field statistics are not compatible with those of a gaussian distribution law, since $\zeta_W^2 \neq 1$.

Equation (36) states that in order to predict the improvement brought in by the use of multiple-source excitations, the only needed quantity to be computed is $\bar{\mu}_r$, according to (37), starting from the residual correlations between the $\{W_m(\nu)\}$ equivalent transfer functions; results are shown in Fig. 11. As expected for a complex medium, the residual spatial correlation is rather low and could be dismissed as negligible.

The predictions given by (36) about the improvement $D(N_A, \bar{\mu}_r)$ are compared in Fig. 12 against the measured ratio $\langle \Lambda \rangle / \langle \Lambda^o \rangle$, for three values of $\nu_c$ and three relative bandwidths. As predicted, the realized improvement is smaller what would be expected from the deterministic behavior of the individual contributions to the coherent part, i.e.,

<figure><img src="image_88.png" /><figcaption>Fig. 11: Estimate of the mean-value of $\bar{\mu}_r(\nu)$, computed over the 50 realizations, obtained from the $\{W_m(\nu)\}$ for a 5 % relative bandwidth (solid line), compared to the least-square estimate $\bar{\mu}_{LS}(\nu)$ derived by fitting the diversity factor to the measured data.</figcaption></figure>

$D(N_A, \bar{\mu}_r) = N_A$. The reduction in $\langle \Lambda \rangle / \langle \Lambda^o \rangle$ is accurately predicted by (36), based on the $\bar{\mu}_r$ estimates in Fig. 11. The divergence between the theoretical predictions and the experimental results can be explained in terms of residual uncertainties in the estimation of $\bar{\mu}_r$: assuming a 10 % uncertainty provides mutually compatible results.

A more general comparison is possible by computing the least-squares fit of $\langle \Lambda \rangle / \langle \Lambda^o \rangle$, yielding an optimal $\bar{\mu}_{LS}$, compared in Fig. 11 and Fig. 12 with the estimate of $\bar{\mu}_r$. A good agreement is observed over the entire frequency range, supporting the proposed theory as capable of predicting the performance of TRTs in complex media.

## VI. CONCLUSIONS

In this paper we have introduced a theory describing the ability of multiple-source TRT in generating coherent signals in complex media. This theory shows how inevitable spatial correlation puts an upper bound on the improvement brought by the introduction of additional transmitting sources, even for values of coherence that are often regarded as negligible.

The introduction of the energy contrast, as opposed to the more intuitive peak contrast, allows appreciating limitations of the energy efficiency that have been seldom discussed; this topic deserves further research for its practical importance in the optimal design of transmission systems.

Overall, experimental results strongly support the theoretical model presented in section III, and in particular the possibility of factorizing the peak contrast into three separate contributions: a) the ratio between the template bandwidth $B_T$ and the coherence bandwidth of the medium; b) the variability of the TRT field supported by the medium; c) the diversity factor $D(N_A, \bar{\mu}_r)$, depending on the residual spatial coherence and the number of sources.

These two last points make clear that albeit allowing remarkable performances, time reversal is profoundly affected by physical limitations that cannot be easily compensated by means of signal-processing techniques.
# ACKNOWLEDGEMENTS

Part of the work here reported was funded by the French National Research Agency through the grant ANR-12-ASTR-0005-01, within the framework of the 2012 ASTRID program.

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[41] M. K. Simon and M.-S. Alouini, *Digital communication over fading channels*. Wiley-Interscience, 2005, vol. 95.

# APPENDIX

The derivation presented in section III-B rests on the idea that while each transmitting antenna generates a different time-reversal field distribution, their statistical moments are independent of the orientation of the sources, as long as they are rotated replicas of the same type of source. In this respect, it is necessary to prove that the first- and second-order moments of the TRT fields only depend on {$||h_m||$} and are rather dominated by the medium statistics. The main assumption is that the medium generates depolarized fields.

The transfer functions {$H_m(\nu)$} can be related to the vector effective heights of the transmitters and receiver, {$h_m$} and $h_o$, respectively. For the sake of simplicity it will be assumed that the transmitters and the receiver are electrically small, whence

$$H_m(\nu) = C(\nu)h_o(\nu) \cdot \underline{G}_{ee}(r_o, r_m, \nu) \cdot h_m(\nu), \quad (43)$$
where $G_{ee}$ is the electric-electric Green's dyadic function of the medium, while $C(\nu)$ is a function depending on the reference impedance used in the definition of the scattering parameters; in the following we will set $C(\nu) = 1$, with no loss of generality. From (43), we can interpret each transfer function as a continuous parametric field

$$H_m(\nu) = H(\mathbf{r}_m, \nu; \mathbf{h}_m), \quad (44)$$

not only defined for the actual position of the mth source, but for a general $\mathbf{r}_m$.

With no loss of generality, in our derivation it will be assumed that the reference system relating to the observer in $\mathbf{r}_o$ is defined such that $\mathbf{h}_o = \mathbf{h}_o \hat{\mathbf{u}}_k$, i.e., the observer is aligned to the kth axis $\hat{\mathbf{u}}_k$. The frequency variable is here dropped for the sake of space. From (43)

$$W_m = \left| \sum_{i,j=1}^{3} h_o^i h_m^j G_{ee}^{ij}(\mathbf{r}_o, \mathbf{r}_m) \right|^2, \quad (45)$$

where {$G_{ee}^{ij}$} are the scalar components of the dyadic Green's function. For $\mathbf{h}_o = \mathbf{h}_o \hat{\mathbf{u}}_k$ (45) reduces to

$$W_m = |h_o|^2 \left| \sum_{i=1}^{3} h_m^i G_{ee}^{ki}(\mathbf{r}_o, \mathbf{r}_m) \right|^2, \quad (46)$$

which, using the convention $X_i = G_{ee}^{ki}(\mathbf{r}_o, \mathbf{r}_m)$, leads to

$$W_m = |h_o|^2 \sum_{i,j} h_m^i h_m^{j*} X_i X_j^*, \quad (47)$$

where * stands for the complex conjugate. The ensemble average $\langle W_m(\nu) \rangle$ is therefore

$$\langle W_m \rangle = |h_o|^2 \sum_{i,j} h_m^i h_m^{j*} \langle X_i X_j^* \rangle. \quad (48)$$

Following the depolarized-field assumption, $\langle G_{ee}^{ij}(\mathbf{r}_o, \mathbf{r}_m) G_{ee}^{pq*}(\mathbf{r}_o, \mathbf{r}_m) \rangle = \delta_{ij} \delta_{pq} \sigma_G^2$, with $\sigma_G^2$ the variance of each scalar component of the dyadic Green's function; in practical terms, a rotational symmetry or isotropy is introduced in the statistics of the components of the dyadic Green's function. Hence

$$\langle W_m \rangle = |h_o|^2 \|\mathbf{h}_m\|^2 \sigma_G^2, \quad (49)$$

which does not depend on the orientation of the mth source. Field statistics for complex media can be assumed, at least at the first order, as following a complex-valued circular normal distribution, i.e., with zero-mean real and imaginary parts normal distributed and iid, with variances $\sigma_0^2$, thus $\sigma_G^2 = 2\sigma_0^2$, quantities directly related to the spectral density $\sigma_W^2(\mathbf{r})$. As argued in section III-B, spatial dependence is here assumed to be negligible at least in the region of space where the sources are found, so that the argument $\mathbf{r}$ will be dropped.

In the same way the second-order moment $\langle |W_m|^2 \rangle$ can be computed as

$$\langle |W_m|^2 \rangle = |h_o|^4 \sum_{i,j,p,q} h_m^i h_m^{j*} h_p^p h_m^{q*} \langle X_i X_j^* X_p X_q^* \rangle. \quad (50)$$

The assumption of a depolarized field implies that of the four sums above only the terms involving $\{i = j = p = q\}$, $\{i = j \wedge p = q\}$ and $\{i = q \wedge j = p\}$ contribute non-zero moments

$$\langle |W_m|^2 \rangle |h_o|^{-4} = \sum_i |h_m^i|^4 \langle |X_i|^4 \rangle + 2 \sum_{i \neq j} |h_m^i|^2 |h_m^j|^2 \langle |X_i|^2 |X_j|^2 \rangle, \quad (51)$$

which can be shown to yield

$$\langle |W_m|^2 \rangle = 8 \sigma_0^4 |h_o|^4 \|\mathbf{h}_m\|^4 \quad (52)$$

and hence

$$S_W = 4\sigma_0^4 |h_o|^4 \|\mathbf{h}_m\|^4 \quad (53)$$

The last moment to consider is $\langle W_m(\nu)W_n(\nu) \rangle$, which will bring into play the spatial degree of coherence of the medium Green's function. First, it can be shown that

$$\langle W_m W_n \rangle |h_o|^{-2} = \sum_i |h_m^i|^2 |h_n^i|^2 \langle |X_i(\mathbf{r}_m)|^2 |X_i(\mathbf{r}_n)|^2 \rangle + \\ \sum_{i \neq j} |h_m^i|^2 |h_n^i|^2 \langle |X_i(\mathbf{r}_m)|^2 \rangle \langle |X_j(\mathbf{r}_n)|^2 \rangle + \\ \sum_{i \neq j} h_m^i h_n^{i*} h_m^{j*} h_n^{j*} \langle X_i(\mathbf{r}_m) X_i^*(\mathbf{r}_n) \rangle \langle X_j^*(\mathbf{r}_m) X_j(\mathbf{r}_n) \rangle. \quad (54)$$

This expression can be simplified by recalling that

$$\langle |X_i(\mathbf{r})|^2 \rangle = 2\sigma_0^2 \quad \forall \mathbf{r} \quad (55a)$$

$$\langle X_i(\mathbf{r}_m) X_i^*(\mathbf{r}_n) \rangle = 2\sigma_0^2 \rho_i'(\mathbf{r}_m, \mathbf{r}_n) \quad (55b)$$

$$\langle |X_i(\mathbf{r}_m)|^2 |X_i(\mathbf{r}_n)|^2 \rangle = 4\sigma_0^4 [1 + \rho_i''(\mathbf{r}_m, \mathbf{r}_n)] \quad (55c)$$

where $\rho_i'(\mathbf{r}_m, \mathbf{r}_n)$ and $\rho_i''(\mathbf{r}_m, \mathbf{r}_n)$ are the spatial degree of coherence of the i-th scalar component of the electric field and its squared modulus, respectively. In order to further simplify our derivation, it is convenient to assume a diffuse-field propagation [16]; in this case, by choosing a reference system pertaining to the source part of Green's dyadic function such that the i-th axis is either parallel or perpendicular to the direction $\mathbf{r}_m - \mathbf{r}_n$, then $\rho_i'(\mathbf{r}_m, \mathbf{r}_n) = [\rho_i'(\mathbf{r}_m, \mathbf{r}_n)]^2$, as demonstrated in [40].

This results, after some algebraic manipulations, into

$$\frac{\langle W_m W_n \rangle}{4 |h_o|^4 \sigma_0^4} = \|\mathbf{h}_m\|^2 \|\mathbf{h}_n\|^2 + \left[ \sum_{i=1}^{3} h_m^i h_n^{i*} \rho_i'(\mathbf{r}_m, \mathbf{r}_n) \right]^2. \quad (56)$$

This last expression is actually dependent on the angles between $\mathbf{h}_m$ and $\mathbf{h}_n$; depending on the direction cosines of $\mathbf{h}_m$ and $\mathbf{h}_n$, the functions $\rho_i'(\mathbf{r}_m, \mathbf{r}_n)$ will be linearly combined according to different weights. Two special cases can be invoked in order to understand when the final result is not affected by the orientation of the sources.

First, consider the case of identical sources, aligned along a straight line. In this case the direction cosines will be the same independently of the couple $(m, n)$, hence $\langle W_m(\nu)W_n(\nu) \rangle$ will only depend on the medium coherence functions. This configuration is often employed in practice, particularly when dealing with linear arrays of sources.
The second case is that of a more complex geometry, where sources are not necessarily parallel to each other and distributed either along a two- or three-dimensional grid, or randomly. In either case, depending on $(m, n)$, the direction cosines will span any possible value, since even for identical sources their projections over the local reference system defined around the direction $\mathbf{r}_m - \mathbf{r}_n$ will change. By modeling the angles as uniformly distributed over all possible directions, the average value of $h_m^i h_n^{i*}$ observed for any $(m, n)$ is expected to be $\langle h_m^i h_n^{i*} \rangle = \|\mathbf{r}_m\| \|\mathbf{r}_n\|/3$, hence (56) will be, according to the law of large numbers, asymptotically equal to

$$ \frac{\langle W_m W_n \rangle}{4 |h_o|^4 \sigma_0^4} = \|\mathbf{h}_m\|^2 \|\mathbf{h}_n\|^2 \left[ 1 + \left( \frac{1}{3} \sum_{i=1}^3 \rho_i'(\mathbf{r}_m, \mathbf{r}_n) \right)^2 \right], \quad (57) $$

which only depends on the degree of coherence of the medium.

Recalling the definition of the spatial coherence functions in (15) and making use of (49), (53) and (56) yields

$$ \mu_r(\Delta r_{mn}) = \left[ \sum_{i=1}^{3} h_m^i h_n^{i*} \rho_i'(\mathbf{r}_m, \mathbf{r}_n) \right]^2 = \left[ \frac{\sin(k_o \Delta r_{mn})}{k_o \Delta r_{mn}} \right]^2 \quad (58) $$

where the last result holds for the case of a diffusive medium [16], as demonstrated in [40], with $k_o = \nu/c_o$ the wavenumber and $c_o$ the average speed of light through the medium. Equation (58) also proves that $\mu_{mn} \ge 0, \forall m, n$, so that the $\bar{\mu}_r$ cannot be smaller than the smallest individual coherences $\{\mu_{ij}\}$, i.e., they cannot compensate themselves reducing the overall average coherence of the transmission system.
<figure><img src="image_89.png" /><figcaption>Fig. 12: Equivalent number of independent sources, or diversity factor, as estimated from measurements, for three values of the central frequency $\nu_c$ (top to bottom: 0.7, 1.3 and 2.0 GHz) and three relative bandwidths (left to right: 5, 10 and 20 %). Each graph presents experimental results as circles, to be compared with the theoretical prediction from (36), shown as a thick solid line, based on the average coherence $\bar{\mu}_r$ estimated in Fig. 11. Dotted lines correspond to the use of an optimal average coherence, derived by least-square fitting, while dashed lines are the results that would be expected if the propagation medium presented no spatial coherence. The effects of experimental uncertainties are shown at two levels: 1) a 10 % uncertainty over the estimate of $\bar{\mu}_r$, affecting the model prediction, as shown by the shaded area; 2) a 10 % uncertainty over the estimate of $\langle\Lambda\rangle$, as vertical bars.</figcaption></figure>
# Steering Focusing Waves in a Reverberation Chamber with Generalized Time Reversal

Andrea Cozza, Senior Member, IEEE, Florian Monsef, Member, IEEE

*Abstract*—Generalized time reversal was introduced in a previous paper from a theoretical point of view. In this paper experiments are conducted to demonstrate its application to a reverberation chamber, as a method for generating coherent wavefronts in a medium displaying random propagation. Wavefronts thus generated were sampled over a planar surface, confirming that they propagate as if in a free-space environment. The accuracy with which they match their free-space counterpart is not affected by changing their features, e.g., direction of arrival and focus. These results prove that a single excitation antenna can generate complex wavefronts when coupled to diffusive wave propagation.

*Index Terms*—Time reversal, diffusive media, reverberation chamber, wavefront synthesis.

## I. BACKGROUND AND MOTIVATIONS

Time reversal (TR) was first introduced in acoustics as a method for generating wavefronts focusing back the field distribution initially radiated by a source [1]. Its intrinsical ability to compensate distortions introduced by heterogeneous and complex media with no need for active equalization is perhaps its most striking feature [2]–[4]. Indeed, beam-forming methods require a priori information about the propagation of waves in a given medium; this need is avoided in TR applications by the fact that the wave radiated by the source (which would afterwards be focused back at it) acts as a sort of medium sounding, and thus brings in itself information about the medium. Moreover, contrarily to methods based on inverse problems, TR is not affected by any ill-posedness, as opposed to dereverberation methods [5]. Other beam-forming techniques, such as wave field synthesis [6], require a very large number of synchronous sources, which would typically be prohibitively expensive at microwave frequencies.

Still, TR suffers from an obvious drawback: focusing waves cannot be synthesized arbitrarily, as they can only reproduce previously radiated wavefronts. This is particularly problematic when thinking about applications of TR for testing the response to impinging wavefronts of a device under test.

Generalized TR (GTR) was first tested in an embryonic version in [7], [8], where direct wavefront synthesis was not yet formalized, as later done from a theoretical point of view in [9]. It is based on the use of the equivalence theorem as a way of avoiding the introduction of a source, in order to directly synthesize signals capable of generating focusing

wavefronts with arbitrary features. GTR is entirely dependent on the availability of a medium capable of supporting diffuse-field propagation, a statistically isotropic case of multipath propagation, also known as Rayleigh diffusion. Reverberation chambers comply with these requirements and bring at the same time a further advantage, energy efficiency, being closed systems with weak losses; when used with TR signals their efficiency can be further increased [10].

This paper shows validation results obtained with a fully automated system. Experimental results of wavefronts generated within a reverberation chamber are compared to theoretically computed distributions. Sequences of wavefronts varying in the position of their focal region and direction of arrival are generated, confirming that the diffusive nature of the medium ensures an excellent reproducibility of wavefronts at any position.

The conclusions of this study imply that GTR can accurately generate arbitrary pulsed focusing wavefronts with a single antenna. Moreover, generating similar focusing wavefronts in open media could require a potentially large number of sources, depending on the angular spectrum of the wavefront, i.e., its focusing power. These sources would need to be excited coherently over a relatively large bandwidth, an operation which is complex and expensive, as it requires synchronizing a number of modulators or signal generators, which can hardly handle high peak-power levels, were they required.

Applications of GTR-based synthesis are currently in progress and range from hardening shielded systems to high-power microwaves, imaging and generating stable high-intensity fields without the statistical uncertainty found in standard reverberation chambers [10].

The paper is organized as follows. GTR is briefly explained in Sec. II, as an intuitive application of first principles well-known in electromagnetics. Sec. III provides some details about the way wavefront-related data is generated. The experimental setup is described in Sec. IV, while Sec. V explains how generated fields are imaged. Reproducibility and robustness to varying conditions are tested and commented in Sec. VI.

## II. A SHORT SUMMARY OF GTR

This section presents a very short description of how GTR works. The simplest way to understand it is by going through the different steps at its origin. The interested reader can refer to [9] for a much more detailed discussion.

Suppose a source radiates a free-space field distribution $E_s(r, t)$. When excited within a medium supporting diffusive propagation, the resulting field distribution would be

Part of the work here reported was funded by the French National Research Agency through the grant ANR-12-ASTR-0005, MIMOCHIC project, within the framework of the 2012 ASTRID program. The design and manufacturing of the robot were funded by Digiteo, under the AUTOTREC project.

A. Cozza and F. Monsef are with PIEM | GeePs, UMR 8507, CentraleSupélec, Univ Paris Sud, UPMC, CNRS, 1 rue Joliot-Curie, 91192 Gif-sur-Yvette, France. Contact e-mail: andrea.cozza@ieee.org.
<figure><img src="image_90.png" /><figcaption>Fig. 1: Relevant quantities in the derivation of direct synthesis of excitation signals with GTR. The auxiliary surfaces are shown to act as an interface between the virtual source radiation (1) and its propagation through the medium (2).</figcaption></figure>

underpinned by a rich plane-wave spectrum, due to a large number of scattering events over its boundaries [11]–[13]. Standard time-reversal would require measuring the signal $y(t)$ received by a further antenna present in the medium [1], [14]. Injecting the time-reversed signal $x(t) = y(-t)$ into what was the receiving antenna, eventually corrected for its dispersive response, can generate a close replica of $E_s(r, -t)$, i.e., the converging counterpart to the originally diverging wavefront $E_s(r, t)$, stripped of its reactive, i.e., non-propagating, components. A single source can support an entire wavefront thanks to self-averaging ensured by the multipath propagation of the medium, if excitation bandwidths cover several coherence bandwidths of the medium [15]. Stated in other terms, complex media can present frequency responses that become weakly interdependent when observed at frequencies spanning at least one coherence bandwidth. The result is the existence of degrees of freedom that can be roughly measured by taking the ratio of the bandwidth of the excitation signal divided by the coherence bandwidth of the medium. TR signals work by controlling these degrees of freedom such that at the time of focusing they cooperate to create coherent wavefronts, out of individual random contributions from each frequency. This property, known as self-averaging, is fundamentally equivalent to the mechanisms behind Van Cittert-Zernike theorem [16], where degrees of freedom are in space rather than in the frequency domain, and result in overall coherent field distribution generated from random spatial distributions.

The problem with the TR approach is that a focusing wavefront can be generated only if a source radiates a corresponding divergent wavefront in the first place. Clearly, if several focusing wavefronts were to be generated, not only the source would have to be modified (e.g., orientation, position) and made to radiate, but also an array of sources with different characteristics may be required (e.g., directivity, polarization).

GTR bypasses this problem by getting rid of the need for physical sources. The excitation signal $x(t)$ is directly synthesized, according to the wished characteristics of the focusing wavefront. In order to do that, $y(t)$ is obtained in

two steps. With reference to Fig. 1, first the field radiated by the source is sampled over (at least) an auxiliary surface $\Sigma$. From these samples, equivalent currents are defined, capable of reproducing the outgoing field that would have reached the receiving antenna, were the source real. In the second step, $y(t)$ is computed from an assumed knowledge of the Green's functions on the medium, acting as transfer functions between the equivalent sources over $\Sigma$ and the receiving antenna, hereafter referred to as $N_e(r, \omega)$ and $N_m(r, \omega)$, respectively for the electric and magnetic field Green's functions. The result is a direct synthesis formula for the Fourier spectrum of the received signal $y(t)$ [9]

$$Y(\omega) = \int_{\Sigma} d\mathbf{r} \left[ \frac{N_e(\mathbf{r}, \omega)}{\zeta} + \hat{\mathbf{r}} \times N_m(\mathbf{r}, \omega) \right] \cdot \mathbf{E}_s(\mathbf{r}, \omega). \quad (1)$$

The derivation of (1) means that the auxiliary surfaces act as a relay in the synthesis of the signal $y(t)$ generated by a given virtual source. Moreover, since the field radiated by the virtual source is not sampled over $\Sigma$ for its own sake, but rather in order to propagate it to the receiver, there is no need to include reactive contributions [17], [18], due to an eventually near-field configuration. As a result, far-field representations of $E_s(r, t)$ can be used, as done in Sec. V.

If the Green's functions were known, then the first step could be accomplished numerically, with no need of a source in the first place, just by defining firsthand the field distribution expected over $\Sigma$. In fact, since the medium is reciprocal, the Green's functions can be measured by exciting the receiving antenna and sampling the electromagnetic field distribution it excites over $\Sigma$. This operation is carried out in our case by a low-perturbation robot developed in our laboratory, as discussed in Sec. IV.

In practice, weakly perturbative field probes are needed. Since most often electric-field probes have a better performance in this respect, equivalent currents can be defined from electric field samples only, if a further auxiliary surface is introduced. As demonstrated in [19], this is possible if specific weights $\{A_i\}, i \in [1, L]$ are applied to the field distributions sampled over each auxiliary surface $\Sigma_l$, yielding

$$Y(\omega) = \zeta^{-1} \sum_{l=1}^{L} A_l \int_{\Sigma_l} d\mathbf{r} \, N_e(\mathbf{r}, \omega) \cdot E_s(\mathbf{r}, \omega) \quad (2)$$

For the case $L = 2$, $A_1 = -\exp(-2jk\Delta R)/[1 - \exp(-2jk\Delta R)]$, $A_2 = -A_1 \exp(2jk\Delta R)$, with $\Delta R$ the difference between the radii of the two spherical auxiliary surfaces, required to satisfy $\Delta R < \lambda/2$, with $\lambda$ the working wavelength.

### III. REFERENCE FIELD DISTRIBUTIONS

With reference to Fig. 2, the definition of the virtual source is based on a Gaussian far-field radiation pattern, with phase center $r_s$

$$E_s(\mathbf{r}, \omega) = \hat{\varphi} G(\mathbf{r} - \mathbf{r}_s, \omega) e^{-\psi^2/2\psi_s^2} P(\omega) \quad (3)$$

and $\psi = \cos^{-1}(\hat{\mathbf{r}}' \cdot \hat{\mathbf{y}})$, for an angular divergence $\psi_s$; $G(r, \omega)$ is the scalar Green's function for free space; the polarization is set equal to $\hat{\varphi}$; $P(\omega)$ is the Fourier spectrum of $p(t)$, the time evolution of the radiated field, usually chosen to be a pulse.