# Spatio-temporal analysis of electromagnetic field coherence in complex media

Thomas Fromenteze  University of Limoges, CNRS, XLIM, UMR 7252, F-87000
Limoges, France  Matthieu Davy  Univ Rennes, CNRS, IETR, UMR6164, F-35000
Rennes, France  Okan Yurduseven  Institute of Electronics, Communications and
Information Technology (ECIT), Queen’s University Belfast, Belfast BT3 9DT,
United Kingdom  Yann Marie-Joseph  Cyril Decroze  University of Limoges, CNRS,
XLIM, UMR 7252, F-87000 Limoges, France  corresponding author:
<EMAIL_ADDRESS>
(September 2021)

###### Abstract

We study the coherence in time and space of electromagnetic fields propagated
through complex media. Whether for localization, imaging or telecommunication,
the development of dedicated numerical techniques is generally based on the
exploitation of simplified models considering either coherent or diffuse
fields. The optimization of such applications in conditions of partial
coherence can therefore be particularly challenging, requiring the development
of hybrid algorithms adaptable to prior knowledge on the processed fields. The
objective of this work is to provide numerical techniques for decomposing an
electromagnetic field into subspaces that can then be filtered according to
their level of spatial and temporal coherence. In contrast to the studies
carried out on space-space transfer matrices notably used for the calculation
of Wigner-Smith operators, these decompositions are carried out on space-time
matrices in order to facilitate the study of temporal dispersion. The theory
is developed for illustrative purposes using experimental results from a leaky
resonant system but seem to be applicable to any scattering and reverberating
media capable of transforming localized and coherent excitations into complex
and diffuse distributions. To conclude this work, the proposed technique is
exploited to improve image reconstruction in a millimeter-wave computational
imaging demonstration. In the studied context and from a more general
perspective, we propose a technique to select the most suitable subspaces for
each application operating under conditions of partial coherence, whether
these correspond in the most extreme cases to ballistic paths or to diffuse
fields.

##  I Introduction

The study of electromagnetic wave propagation in complex media is motivated by
both physical considerations and technological challenges involved in a
growing number of practical applications [1, 2, 3]. Whether one considers
strongly scattering and/or reverberating environments, the excitation of
spatially and temporally coherent waves within their boundaries shortly gives
way to multimode regimes whose diversity tends to increase with dwell time
[4]. Apart from integrable boundary conditions which can lead to periodic
solutions, the more general complex and weakly symmetrical geometries of
disordered propagation media are generally associated with chaotic or
diffusive dynamics [5, 6]. The sensitivity to initial conditions makes the
long-term prediction of field distributions particularly delicate. The
characterization of such media thus requires the definition of asymptotic and
statistical properties such as the reverberation time [7] and the mean free
path [8]. In contexts where the analytical description of electromagnetic
fields is notoriously challenging, it remains nevertheless necessary to
promote operating conditions adapted to the constraints of each practical
application. While free space propagation models can be considered in many
cases for simplicity, the recent exploitation of new degrees of freedom
offered by the diversity of complex propagation channels receives a growing
interest from the scientific community. These efforts notably allow the
optimization of telecommunication systems [9, 10] and enable the focusing of
energy [11, 12] and the localization of objects beyond the theoretical
diffraction limits defined in free space [13]. In associated contexts such as
imaging through biological tissues, multiple scatterings are the source of
limitations that impose the development of adapted processing techniques
notably based on embedded detectors [14, 15], non-linear materials [16, 17] or
fluorescence [18].

In this paper, we show that in open systems it is possible to discriminate the
ballistic and diffuse contributions of a field transmitted through a complex
scattering system using a singular value decomposition of the frequency-space
transfer matrix. We determine the level of temporal and spatial coherence of
each subspace composing the studied field, facilitating the identification and
extraction of independent structures, arranged from the first ballistic paths
to the most complex and diffuse distributions.

This work differs from most techniques developed for the study of complex
media properties, which generally focus on the analysis of transmission
matrices linking only spatial variables or the associated modes at the input
and output of the considered systems [19, 20, 21]. The temporal dimension is
thus generally not considered, apart from couples of frequency samples
allowing the calculation of partial derivatives for the evaluation of Wigner-
Smith time-delay operators [22, 23, 24], facilitating the identification of
eigenstates of scattering matrices [25, 26]. Focusing this work on the
decomposition of transmission matrices linking input excitation points to the
spatial and frequency dimensions of propagated fields, we propose here to
study the evolution of coherence within complex media. Frequency-resolved
measurements over large spectral bands thus provide an additional dimension to
the transmission matrices, opening the way to new perspectives in the analysis
of the propagation of electromagnetic fields in complex media. Significant
advances have recently been published to reveal the information associated
with this additional temporal dimension, in a context notably oriented towards
the study of the modal diversity of optical fibers. Using swept-wavelength
interferometric techniques, it has been demonstrated that it is possible to
extract and amplify characteristic sub-spaces associated with propagation in
studied complex media [27, 28]. In particular, the reference [28] has direct
connections to the technique studied in this work, focusing on the eigenmodes
of the covariance matrix computed from spatial transmission matrices
determined at given propagation times. This work builds on previous
contributions also based on singular value analysis and eigendecomposition,
studying the characteristic spatial structures of matrices relating variables
at the input and output of the studied media for the spatial focusing of waves
[29] and the transmission of information through complex media [30]. In our
work, we show that by adapting such formalisms to space-time transfer
matrices, such spatial structures can be associated with temporal singular
vectors, determining both the spatial and temporal coherence of the subspaces
describing the propagation of waves through complex media. A more detailed
mathematical description of these references will be made after the definition
of the formalisms associated with the proposed technique, allowing to better
position our work in relation to the literature.
Without loss of generality, we focus on the particular case of an electrically
large metallic cavity coupled to the external world with radiating irises,
motivated by computational microwave and millimeter-wave imaging applications
presented in the last part of this study [31, 32, 33]. This work thus
introduces methods for processing fields in complex environments using
techniques that are as simple, efficient and intuitive as possible while
remaining original, to the authors’ knowledge. These efforts, however, fall
within a context where singular value decompositions are applied in various
research and engineering works such as blind source separation [34, 35], the
identification and extraction of acoustic and radar signatures [36, 37], the
separation of diffracted waves in geophysics [38, 39], deep imaging in complex
media [40, 41, 42] and speckle engineering in optics [43]. This work could
potentially find an application in the identification and study of regimes
specific to disordered media such as localization and trapping phenomena
characterized by a strong attenuation of the diffusion mechanisms [44, 45,
46]. The extension of this work to the analysis and identification of
particular regimes could then be explored by varying the local order level of
the considered environments [47, 2].
This paper is structured as follows: we first describe in the next section the
considered cavity to facilitate the introduction of the coherence analysis
techniques proposed in this work. The associated formalisms are developed in
section 3, introducing coherence metrics and providing intuitive
interpretations to the singular value decomposition. Section 4 presents a
practical application of this work to the improvement of source localization
experiments using frequency-diverse diffuse field radiation. The last section
concludes this paper and discusses potential application perspectives.

##  II Experimental conditions

In this work, we investigate the temporal and spatial evolution of an
electromagnetic field bouncing into a metallic cavity. All the measurements
presented in this article are thus derived from data captured in an
experimental context. Excitation ports allow the injection of waves at two
locations, which after a certain number of reflections are coupled towards the
external world by the intermediary of irises (Fig. 1). The latter is realized
with polished aluminum panels, forming a parallelepiped whose internal
dimensions are $251\text{mm}\times 66\text{mm}\times 251\text{mm}$. The
excitation is achieved with two WR-10 guide ports electrically polarized along
the $z$ axis, with dimensions $2.54\text{mm}\times 1.27\text{mm}$. These are
arranged on the rear face according to the arbitrarily chosen positions
$(x,z)=(-37.52\text{mm},-39.54\text{mm})$ and
$(x,z)=(76.63\text{mm},0\text{mm})$. An eighth of an aluminum sphere of 40mm
radius is placed in one of the corners to limit the symmetries, ensuring a
more homogeneous spatial distribution of the modes by combining regular and
convex boundary conditions. The front face is perforated by a uniform array of
$28\times 28$ circular irises of 3mm diameter arranged along the $x$ and $z$
axes every 8mm. A 45∘ milling is performed for all the irises on the 5mm
thickness of the front face. The radiation of this assembly is characterized
by means of a near field scanner, moving a mono-polarized probe in a plane
located at 14mm from the radiating surface. The characterization is carried
out in the millimeter band from 70GHz to 100GHz, sampled by $n_{\nu}=$4001
measurement points. The scan is performed on a surface of $300\text{mm}\times
300\text{mm}$, spatially sampled by 201 points per side corresponding to a
total number of $n_{r}$=40401 scanned positions. The space $\mathbf{r}$ swept
by the field probe is discretized according to a sampling constraint of 1.5mm
per transverse axis imposed by the highest frequency of the $\nu$ vector,
corresponding to half a wavelength at 100GHz. The electric near field
$E_{i}^{(p)}(\mathbf{r},\nu)$ radiated by the front panel is thus determined
in a plane parallel to the latter according to one of the two transverse
polarizations $i=(x,z)$ and input port $p=(1,2)$. Four days were necessary for
the complete characterization of the radiation of this cavity, performing
intermediate verifications of the measured data for each excitation port and
each transverse polarization. Due to the sequential Cartesian positioning of
the field probe, these characterization times are necessarily much longer than
those involving the use of spatial light modulators at optical wavelengths. As
a comparison, the measurement of a transfer matrix determining the interaction
between 254 input spatial modes, 650 output spatial modes for 1273 wavelengths
is achieved in a little more than one hour in [28].

Figure 1: Near field scan of a leaky reverberant medium. This aluminum cavity
is excited at two locations by z-polarized WR-10 waveguides. Circular slits on
the front panel radiate fields measured by a single-polarized scanning probe,
oriented according to one of the two transverse polarizations $x$ and $z$.

The evolution of a temporally and spatially localized excitation within such a
medium tends to generate an increasingly complex field distribution, coupled
to the external environment through the circular irises. The near field thus
includes components reflecting a direct, strong and coherent coupling with the
scanning probe as well as a succession of diffuse contributions, linked to an
accumulation of wavefronts reflected in this reverberant environment. These
characteristics are directly visible in the measured field at each frequency
(Fig. 2).

Figure 2: Electric fields $E^{(1)}_{z}(\mathbf{r},\nu)$ measured at 14 mm from
the radiating aperture following the excitation of port 1. The phase (in
radians) is represented in color and the opacity corresponds to the amplitude.
The field region surrounded by a dashed line has a spatial structure common to
several frequency scans and appears to be related to a ballistic path. The
field distributions represented in this study are based on scientific
colormaps optimized for color perception impairments [48].

In the given examples, the direct and co-polarized coupling between the cavity
excitation port and the measurement probe forms a zone where the phase varies
more weakly, around the location of the first port
$(x,z)=(76.63\text{mm},0\text{mm})$. The speckle pattern around this area
changes rapidly with the analysis frequency and corresponds to random
magnitude and phase distributions associated with the superposition of a large
number of incident wavefronts. The electromagnetic field in perfectly chaotic
cavities is indeed statistically isotropic, uniform and depolarized with
universal statistics [49].

Having defined the experimental conditions in which the electric field is
measured in space at each frequency of an operating band, a technique allowing
the extraction of ordered subspaces according to their level of spatial and
temporal coherence is presented in the next section.

##  III Subspace decomposition by level of spatial and temporal coherence

In the given context, the singular value decomposition helps highlighting the
predominant independent structures characterizing a measured field. We define
a first complex matrix $\mathbf{E}_{z}^{(1)}\in\mathbb{C}^{n_{\nu}\times
n_{r}}$, with $n_{\nu}$ and $n_{r}$ the respective numbers of elements of
$\nu$ and $\mathbf{r}$. Throughout this paper, we use bold notation for
matrices and vectors which are distinguished with upper and lower case letters
respectively. In the first part of this study, we omit the systematic writing
of the index of the excitation port $(1)$ and of the polarization $z$ to
lighten the notations of the next developments applied in an identical manner
to all the near fields studied thereafter. The singular value decomposition is
defined as follows:

$\displaystyle\mathbf{E}=\sum_{n}\sigma_{n}\mathbf{u}_{n}\mathbf{v}_{n}^{\dagger}$
(1)

where $\mathbf{u}_{n}\in\mathbb{C}^{n_{\nu}\times 1}$ and
$\mathbf{v}_{n}\in\mathbb{C}^{n_{r}\times 1}$ are the singular vectors whose
outer product forms an orthonormal basis, weighted by each associated singular
value $\sigma_{n}\in\mathbb{R}$. The symbol $.^{\dagger}$ represents the
conjugate-transpose operator. The near field matrix is thus decomposed into a
set of orthogonal structures, sorted by amplitude contribution of the singular
values and their associated vectors. This factorization thus represents an
algebraic tool revealing the effective rank of a field matrix, exploited
notably to define diversity metrics [50, 51, 52]. We note at this stage that
the $\mathbf{u}_{n}$ vectors allow the definition of the frequency dimension
of the field matrix to be characterized, thus revealing the connection between
the main spatial field structures of $\mathbf{v}_{n}$ and the associated
frequencies. In order to analyze the evolution of these field distributions in
the time domain and to identify the associated level of temporal coherence, an
inverse Fourier transform, denoted by $\mathfrak{F}^{-1}$, is applied to these
singular vectors as $\mathfrak{u}_{n}(t)=\mathfrak{F}^{-1}(u_{n}(\nu))$. The
attention of the readers must be drawn to the notation of the temporal vectors
$\mathbf{\mathfrak{u}}_{n}$, stylized in order to limit a possible confusion
with their frequency analogues $\mathbf{u}_{n}$. It seems worthwhile to point
out that the singular values $\sigma_{n}$ and singular vectors
$\mathbf{v}_{n}$ are identical whether a captured field is expressed in the
frequency domain or the time domain. This property is justified insofar as the
$\mathbf{v}_{n}$ vectors represent only spatial field structures and as
unitary matrices preserve norms, thus not impacting the magnitude of the
singular values. This invariance can also be illustrated by applying a
discrete inverse Fourier transform matrix $\mathbf{W^{-1}}$ to the frequency
domain field $\mathbf{E}$:

$\displaystyle\mathbf{E}_{t}$ $\displaystyle=\mathbf{W}^{\dagger}\mathbf{E}$
(2)
$\displaystyle=\mathbf{W}^{\dagger}\mathbf{U}\mathbf{S}\mathbf{V}^{\dagger}$
(3) $\displaystyle=\mathbf{U}_{t}\mathbf{S}\mathbf{V}^{\dagger}$ (4)

where $\mathbf{E}_{t}$ corresponds to the temporal field matrix at the output
of the considered complex medium. The unitary matrices $\mathbf{U}$ and
$\mathbf{V}$ are respectively composed of the singular vectors
$\mathbf{u}_{n}$ and $\mathbf{v}_{n}$. The diagonal matrix
$\mathbf{S}=\text{diag}(\bm{\sigma})$ is constituted by the singular values
arranged in descending order. Finally,
$\mathbf{U}_{t}=\mathbf{W}^{-1}\mathbf{U}$ corresponds to the matrix of
temporal singular vectors $\bm{\mathfrak{u}}_{n}$. These transformations
finally allow the analysis of the evolution of the temporal singular vectors
according to the level of the associated singular values (Fig. 3).

Figure 3: Top: Spectrum of singular values, highlighting the particularly
significant contributions of a set of orthogonal subspaces associated with the
first singular values. Bottom: Singular vectors $\bm{\mathfrak{u}}_{n}$
corresponding to the values highlighted by square markers.

The analyzed electric field is dominated by first components with limited
temporal spread, corresponding to strong wavefronts that have weakly
interacted with the considered medium. One can notice that the temporal
coherence of the singular vectors then decreases rapidly, transiting from high
amplitude pulses to diffuse distributions whose spread and average propagation
time increase with the singular vector index, studied more carefully in the
next part of this paper.

The previous developments have highlighted the existing links between the
temporal and spatial singular vectors, justifying that the matrix
$\mathbf{U}_{t}$ allows the identification of the quantity of independent
spatial structures necessary to characterize the field at each instant $t$
(Fig. 4).

Figure 4: Matrix $\mathbf{U}_{t}$ of temporal singular vectors
$\bm{\mathfrak{u}}_{n}$ highlighting the dwell time of the different
associated spatial vectors $\mathbf{v}_{n}$ as well as the increasing modal
diversity within this complex environment. The matrix is normalized and its
magnitude is displayed in decibels.

Such a simple visualization thus reveals crucial information on the temporal
evolution of the spatial diversity of the field propagating within this
environment, corresponding to the columns of this matrix where each sample is
associated to a spatial singular vector. The distribution of values within
this matrix also reveals the evolution of the dwell time of spatial structures
associated with the same indices. Counter-intuitively, we observe that the
first instants are associated with space vectors $\mathbf{v}_{n}$ whose
indices are particularly high. Corresponding to propagation times lower than
required to reach the front panel of the cavity, these non-zero values seem to
be caused by a temporal aliasing phenomenon induced by the inverse Fourier
transform associated with an incomplete damping of the impulse responses of
this medium. An in-depth analysis is proposed by studying the distribution of
spatial singular vectors $\mathbf{v}_{n}$, de-vectorized according to the two
dimensions $x$ and $z$ to form associated matrices noted $\mathbf{V}_{n}$ and
defined as $\mathbf{v}_{n}=\text{vec}(\mathbf{V}_{n})$ (Fig. 5).

Figure 5: Four singular vectors $\mathbf{V}_{n}$ reshaped according to the two
spatial dimensions $x$ and $z$. The phase (in radians) is represented in color
and the opacity is defined by the magnitude.

These analyses depict the progressive increase in complexity of the measured
field distributions in space. The growing number of independent structures
required for their characterization represents as many degrees of freedom
exploited by various applications of communication, localization and imaging.
It seems meaningful at this point to emphasize more directly the link between
our work and previous papers mentioned in the introductory section [29, 30,
28]. These references exploit decompositions of transmission matrices noted
here $\mathbf{T}\in\mathbb{C}^{N_{\text{out}}\times N_{\text{in}}}$, linking
$N_{\text{in}}$ spatial variables at the input and $N_{\text{out}}$ at the
output of a complex propagation medium, determined for a given wavelength or
time. These studies notably focus on the eigenvalue decomposition of
$\mathbf{T}^{\dagger}\mathbf{T}\in\mathbb{C}^{N_{\text{in}}\times
N_{\text{in}}}$, where $\dagger$ corresponds to the transpose-conjugate
operator, revealing the associated principal spatial field structures. By
realizing a singular value decomposition such that $\mathbf{T}$ =
$\mathbf{U}_{T}\,\mathbf{S}_{T}\mathbf{V}_{T}^{\dagger}$ and exploiting the
properties of the unitary matrices $\mathbf{U}_{T}$ and $\mathbf{V}_{T}$, we
have:

$\displaystyle\mathbf{T}^{\dagger}\mathbf{T}$
$\displaystyle=\mathbf{V}_{T}\,\mathbf{S}_{T}^{\dagger}\mathbf{U}_{T}^{\dagger}\mathbf{U}_{T}\,\mathbf{S}_{T}^{\dagger}\mathbf{V}_{T}^{\dagger}$
(5)
$\displaystyle=\mathbf{V}_{T}\,\mathbf{S}_{T}^{\dagger}\,\mathbf{S}_{T}\,\mathbf{V}_{T}^{\dagger}$
(6)

The eigenvectors $\mathbf{V}_{T}$ of $\mathbf{T}^{\dagger}\mathbf{T}$ then
also correspond to the spatial singular vectors at the system input et les
valeurs propres sont équivalentes au module carré des valeurs singulières. The
singular vectors gathered in $\mathbf{U}_{T}$ correspond to the spatial
structures characterizing the field at the output of the medium, orthogonal to
each other according to the properties of unitary matrices. The injection of a
singular vector $\mathbf{v}_{T,k}$ at the input of the system, corresponding
to the singular value $\sigma_{k,T}$ extracted from the diagonal of the matrix
$\mathbf{S}_{T}$, allows one to obtain at the output a field distribution
$\hat{\mathbf{u}}_{T,k}$ such as:

$\displaystyle\hat{\mathbf{u}}_{T,k}$
$\displaystyle=\mathbf{T}\,\mathbf{v}_{T,k}$ (7)
$\displaystyle=\mathbf{U}_{T}\,\mathbf{S}_{T}\mathbf{V}_{T}^{\dagger}\,\mathbf{v}_{T,k}$
(8) $\displaystyle=\sigma_{T,k}\,\mathbf{u}_{T,k}$ (9)

A measurement of the field distribution at the system output allows the
retrieval of the singular vector $\mathbf{u}_{T,k}$ weighted by the associated
singular value. In this context, these studies generally focused on the
identification of the most significant singular values or eigenvalues,
allowing to excite the most significant field structures. There does not seem
to be a direct correspondence between our spatial singular vectors and those
studied in these earlier works because the decompositions are performed here
on space-time matrices. However, we find common characteristics associated to
the largest singular values being exploited for the excitation of quasi-
ballistic subspaces.

We now proceed to analyze the results associated with our model. Having
highlighted the link between the spatial and temporal characteristics of the
field under consideration, it now seems useful to define metrics allowing the
determination of the level of coherence of the associated subspaces. The
temporal singular vectors $\mathbf{\mathfrak{u}}_{n}$ form wave packets
localized around average propagation times $\bar{t}_{n}$ which increases with
the index $n$ of the associated singular values. The latter can be determined
by calculating the first temporal moment of each vector, also referred in the
literature as the time centroid [53, 54, 55]:

$\displaystyle\bar{t}_{n}=\frac{\int_{0}^{+\infty}t\,\lvert\mathfrak{u}_{n}(t)\lvert^{2}dt}{\int_{0}^{+\infty}\lvert\mathfrak{u}_{n}(t)\lvert^{2}dt}$
(10)

The results shown in Fig. 6 show quasi-linear dependency on the time centroid
of each singular vector according to the associated indices.

Figure 6: Time centroids $\bar{t}_{n}$ (top) and spread times $\tau_{n}$
(bottom) of the singular vectors $\mathbf{\mathfrak{u}}_{n}$.

Considering a physical optics approximation to the propagation within this
reverberant environment, we deduce that the set of rays received at a given
time $\bar{t}_{n}$ travels on average a distance corresponding to
$c\,\bar{t}_{n}$, where $c$ corresponds to the speed of light. An average
propagation time of 50 ns corresponds to a distance of about 15m, representing
several tens of reflections within this medium and 5000 times the smallest
wavelength considered for this study. The only information of spread time of
each singular vectors $\mathbf{\mathfrak{u}}_{n}$ seems sufficient to
determine the level of temporal coherence of the associated subspace. The
extraction of these time constants can be achieved by considering exponential
decay models related to the energy leakage from the studied domain at each
time step [56, 57], involving envelope variations of the form
$\exp(-t/\tau_{n})$ in intensity, or $\exp(-t/(2\tau_{n}))$ in magnitude of
the associated fields. This operation can be quite tedious and imprecise,
achieving linear regressions on weighted random distributions expressed on a
logarithmic scale. Adapted from the acoustic field, the backward integration
method initially developed by Schroeder helps with the determination of
reverberation times [58]. He astutely considered the following identity for
this evaluation:

$\displaystyle\int_{0}^{t}\lvert\mathfrak{u}_{n}(t^{\prime})\lvert^{2}dt^{\prime}=\int_{0}^{+\infty}\lvert\mathfrak{u}_{n}(t^{\prime})\lvert^{2}dt^{\prime}-\int_{t}^{+\infty}\lvert\mathfrak{u}_{n}(t^{\prime})\lvert^{2}dt^{\prime}$
(11)

noting that one can substitute the averaging of a set of damped responses by a
temporal averaging performed on the duration of a single acquisition. The
envelope of each vector, expressed on a logarithmic scale, is then determined
with this technique to extract the associated spread times:

$\displaystyle
A_{n}(t)=10\log_{10}\frac{\int_{t}^{+\infty}\lvert\mathfrak{u}_{n}(t^{\prime})\lvert^{2}dt^{\prime}}{\int_{0}^{+\infty}\lvert\mathfrak{u}_{n}(t)\lvert^{2}dt}$
(12)

It is then possible for each vector $\mathfrak{u}_{n}(t)$ to determine an
associated spread time $\tau_{n}$ by linear regression, considering around the
centroid time $\bar{t}_{n}$ (Fig. 6). The extraction of these parameters
allows the quantification of the spread of each subspace, reflecting the
progressive decrease of the temporal coherence within this environment. The
analysis window being determined by the frequency sampling considered for
these measurements, these approaches remain nevertheless limited by the time
truncation of the vectors of highest indices. The spread of singular vectors,
as observed in Fig. 3, is indeed limited beyond half of the indices by the
frequency sampling $d\nu\approx 7.5\text{MHz}$, defining the maximum
acquisition time $T_{\text{max}}=1/d\nu\approx 133\text{ns}$. It will thus be
necessary to keep a critical eye on the processing carried out on the time
vectors impacted by this truncation, represented by a red gradient on Figs. 6
and 7. The definition of a frequency resolution appropriate to the decay time
of the cavity under study ensures that the singular vectors beyond this
threshold contribute only to one hundredth of that of the most significant
subspaces. The use of linear regressions also implies a certain number of
limitations linked to the determination of short reverberation times as well
as uncertainties directly visible through the fluctuations of the values of
$\tau_{n}$. As an alternative, it is proposed to consider the $\ell_{1}$ norm
for the determination of a temporal coherence metric of each vector, stating
that the latter will necessarily increase with the spread of the considered
responses (Fig. 7).

Figure 7: Evolution of the $\ell_{1}$ norm of the $\mathbf{\mathfrak{u}}_{n}$
vectors, defined as a temporal coherence metric.

The $\ell_{1}$ norm of the $\mathbf{\mathfrak{u}}_{n}$ singular vectors more
clearly reveals a rapid spreading of the temporal responses from the first
singular values, corresponding to the conversion from quasi-ballistic states
to highly reverberating subspaces. The effect of time gating can also be
observed, causing a decrease in this metric beyond half of the singular vector
indices. Having presented the temporal characteristics of the subspaces
describing such fields at the output of complex media, it now seems
interesting to study the spatial coherence of these same structures. The
processing of singular vectors $\mathbf{v}_{n}$ must be adapted to the fact
that the spatial field distributions are vectorized. As previously introduced
in Fig. 5, it is possible by reshaping the $\mathbf{V}_{n}$ spatial matrices
from the associated $\mathbf{v}_{n}$ vectors to take into account the
interaction of adjacent pixels to determine the level of coherence of these
complex spatial distributions. It is proposed in this context to use again the
singular value decomposition to determine the level of spatial coherence of
each $\mathbf{V}_{n}$ matrix. The spectrum of singular values determines by
its shape the number of independent structures of a matrix, and tends towards
a set of identical values in the case of a full rank square matrix where all
rows and columns are perfectly uncorrelated. Localized and/or spatially
correlated field distributions in phase will thus tend to decrease the number
of subspaces needed to describe these field matrices. The SVD entropy $h_{n}$
seems to be a particularly well suited metric for this work [59], defined as
follows for each matrix $\mathbf{V}_{n}$:

$\displaystyle h_{n}=-\sum_{i}\sigma_{n,i}\log_{2}(\sigma_{n,i})$ (13)

where $\sigma_{n,i}$ corresponds to the singular values of index $i$ for each
matrix $\mathbf{V}_{n}$. This metric allows the determination of the number of
independent structures required to describe the considered spatial matrices,
each $\mathbf{V}_{n}$-matrix being associated to a corresponding value
$h_{n}$. The evolution of the SVD entropy for all these matrices is given in
Fig. 8.

Figure 8: SVD entropy $h_{n}$ computed for the set of matrices
$\mathbf{V}_{n}$.

An inflection point appearing at the first singular values clearly helps
identifying the change of propagation regime within this environment. This
section will have introduced coherence metrics based on the computation of the
$\ell_{1}$-norm of the temporal vectors and the entropy SVD of the spatial
vectors, facilitating the study of the characteristics of the considered
fields.

This paper is completed with the excitation of a new cavity port (Fig. 1) and
by optionally rotating the scanning probe. In such conditions, it is therefore
four matrices noted respectively $\mathbf{E}_{x}^{(1)}$,
$\mathbf{E}_{z}^{(1)}$, $\mathbf{E}_{x}^{(2)}$ and $\mathbf{E}_{z}^{(2)}$
which can be determined with the excitation of input ports of index 1 and 2
and field measurement along the two transverse field polarization $x$ and $z$
(Fig. 9)

Figure 9: Singular value spectra of the fields
$\mathbf{E}_{x}^{(1)}$,$\mathbf{E}_{z}^{(1)}$, $\mathbf{E}_{x}^{(2)}$ and
$\mathbf{E}_{z}^{(2)}$. The spectra are normalized by the largest singular
value obtained with $\mathbf{E}_{z}^{(1)}$. The cavity ports, whose electric
field is oriented along the vertical axis $z$, couple more easily to the
measurement probe when the latter is co-polarized.

The $\mathbf{E}_{z}^{(1)}$ and $\mathbf{E}_{z}^{(2)}$ fields are measured when
the scanning probe is co-polarized with the cavity’s excitation ports 1 and 2.
The amplitude of the first singular values, corresponding the contribution of
a direct path, is then a factor of 2 times larger than those of the cross-
polarized fields $\mathbf{E}_{x}^{(1)}$ and $\mathbf{E}_{x}^{(2)}$, reflecting
the impact of early polarization conversion that occurs in the cavity.
Following the previous analysis, it is possible to anticipate an impact on the
spatial structuring of the first singular vectors $\mathbf{v}_{n}$ associated
with each measured near field. We thus compare the SVD entropy for each case,
considering again the $\mathbf{V}_{n}$ matrices obtained by reshaping the
$\mathbf{v}_{n}$ vectors (Fig. 10).

Figure 10: SVD entropy of the $\mathbf{V}_{n}$ matrices derived from the
$\mathbf{E}_{x}^{(1)}$,$\mathbf{E}_{z}^{(1)}$, $\mathbf{E}_{x}^{(2)}$, and
$\mathbf{E}_{z}^{(2)}$ fields. The number of spatial structures needed to
describe the electric fields depends directly on the polarization conversion
of the quasi-ballistic paths. The cross-polarized fields
$\mathbf{E}_{x}^{(1)}$ and $\mathbf{E}_{x}^{(2)}$ thus require more orthogonal
subspaces to describe their distribution.

The effect of polarization conversion is again clearly visible in this
analysis. The first singular vectors require twice as many independent spatial
structures when the direct coupling is measured through polarization
conversion, reflecting a reduced spatial coherence. After the inflection point
corresponding to the change of propagation regime within the cavity, it is
interesting to note that the entropy of the singular vectors tends towards an
identical distribution, whatever the excitation port and the polarization of
the measured electric field. The relative difference between the smallest and
largest value of $h_{n}$ determined with our four electric field distributions
is thus of the order of 1% beyond half of the indices of singular values. Over
long timescales, the characteristics of the spatial field distributions do not
depend on the initial excitation conditions and seem dominated by the geometry
of the considered environment. The impact of diffraction on the first singular
vectors can be illustrated more directly by representing the $\mathbf{V}_{1}$
matrices associated to each field distribution (Fig. 11).

Figure 11: Matrices obtained by de-vectorizing the singular vectors
$\mathbf{v}_{1}$ derived from the four studied electric fields studied. The
action of the polarization conversion by diffraction on the cavity irises is
clearly visible in the first singular vectors associated with $x$-polarized
fields.

The extraction of these dominant spatial structures in the description of the
electric fields allows to highlight the nature of the ballistic paths in each
considered case. The spatial distributions of the cross-polarized cases
($\mathbf{E}_{x}^{(1)}$ and $\mathbf{E}_{x}^{(2)}$) can be used to guess the
mechanism of polarization conversion by diffraction. The projection of the
electric field along the longitudinal dimension $y$ associated with the
excitation of oblique incidence wave vectors then undergoes a conversion along
the two transverse dimensions, justifying the absence of field in the axis of
the excitation ports. These tools help highlighting the level of spatial
coherence of the main structures characterizing the fields radiated at the
output of this reverberant medium. This study is finally completed by an
analysis of the temporal characteristics associated with the four electric
fields analyzed (Fig. 12).

Figure 12: Evolution of the temporal coherence of the singular vectors
$\mathbf{u}_{n}$, determined through their norm $\ell_{1}$ for the four
electric fields considered.

The computation of $|\mathbf{u}_{n}\lvert_{1}$ for all the considered cases
highlights the weak impact of the polarization conversions and of the position
of the excitation ports on the temporal spread of the singular vectors,
confirming that spatial coherence characteristics can be affected without
major impact on the associated temporal coherence metrics.

This section has introduced a number of useful methods for the
characterization of a field propagated through a complex medium, decomposing
the latter into subspaces ordered by level of contribution and coherence. In
the next section of this paper, an application of this work is proposed in the
context of source localization experiments using frequency-diverse and
spatially diffuse fields.

##  IV Application to source localization by frequency-diverse diffuse field
radiation

In this paper, numerical techniques are studied to characterize singular
vectors describing the temporal and spatial evolution of fields in complex
media. Following the identification of the coherence level of each subspace,
it is now possible to filter those which are the most favorable to the
targeted applications. We anticipate, for example, a possible application of
this work to the characterization of antenna radiation in environments where
the control of propagation channels cannot be simply ensured with absorbing
materials. It would then be possible to limit the impact of weakly coherent
subspaces, preserving the structures associated with the direct path. In an
opposite context, it is sometimes desirable to suppress the temporally
coherent structures of radiated fields. An application example is thus
proposed in a context of computational imaging performed with the same cavity
[32, 60, 61]. The field probe initially used for the radiated near field
measurement is now placed at a longer distance from the radiating plane.
Therefore, our objective is to locate the source from the sole measurement of
the transfer function between the latter and one of the ports of the cavity.
The success of such an operation relies initially on the computation of the
field radiated in space, discretized into a certain number of voxels according
to the resolution limits imposed by the dimensions of the radiating aperture
and by the operating bandwidth [62]. If the electric field expressed in the
frequency domain at the source location is sufficiently different from all
other positions in the region of interest, a simple correlation calculation
then allows its localization. In such conditions, the optimization of the
decoherence of the radiated fields minimizes the possible similarities of the
frequency distributions computed at the different locations, thus limiting the
ambiguity associated with the determination of the source position.

The radiated electric field is computed by propagation of the near field
measured by feeding the first excitation port of the cavity. The propagation
of the near-field scans performed for the localization experiments presented
in this work is carried out using the discrete dipole approximation [63, 64,
65]. The measured tangential electric field vector $\mathbf{E}_{\text{tan}}$
is thus transformed into a set of point sources, considering that the spatial
mesh is chosen fine enough so that the field is constant at the scale of each
cell. We thus determine for each elementary surface of area $da$ the electric
$\mathbf{p}$ and magnetic $\mathbf{m}$ dipoles according to the following
relations:

$\displaystyle\mathbf{m}=\frac{2}{i2\pi\nu\mu_{0}}\int\hat{\mathbf{n}}\times\mathbf{E}_{\text{tan}}\,da$
(14)
$\displaystyle\mathbf{p}=\epsilon_{0}\hat{\mathbf{n}}\int\hat{\mathbf{n}}\cdot\mathbf{E}_{\text{tan}}\,da$
(15)

The vector $\hat{\mathbf{n}}$ refers to the normal to the surface on which the
tangential field $\mathbf{E}_{\text{tan}}$ is determined, i.e.
$\hat{\mathbf{y}}$ in the reference frame chosen for this work. The constants
$\mu_{0}$ and $\epsilon_{0}$ correspond respectively to the magnetic
permeability and the dielectric permittivity of vacuum. The dipoles can then
facilitate the determination of the electric $\mathbf{E}$ field radiated in
space:

$\displaystyle\mathbf{E}=Z_{0}\mathbf{G}^{\text{me}}\mathbf{m}+\mathbf{G}^{\text{ee}}\mathbf{p}$
(16)

where the $\mathbf{G}$ matrices are the dyadic Green’s functions propagating
the different electric and magnetic quantities. The contribution of the
electric dipoles $\mathbf{p}$ can be neglected in the context of this work,
considering that the scalar product
$\hat{\mathbf{n}}\cdot\mathbf{E}_{\text{tan}}$ tends to zero if the normal to
the characterization plane is orthogonal to the polarization axes of the
transverse electric field. In practice, the various surface irregularities
(screws, iris milling) can induce the generation of effective electric
dipoles, but their contribution along the axis of propagation remains
negligible compared to that of the magnetic sources on the entire
characterized surface. The expression of the radiated electric field then
takes the following form [64, 66]:

$\displaystyle\mathbf{E}(\mathbf{r}_{j})=-Z_{0}\,k^{2}\sum_{i}(\hat{\mathbf{r}}_{ij}\times\mathbf{m}_{i})\,\frac{e^{-ik\mathbf{r}_{ij}}}{4\pi\mathbf{r}_{ij}}\,\left(1-\frac{i}{k\,\mathbf{r}_{ij}}\right)$
(17)

The sum of the contributions of magnetic dipoles of index $i$ then allows the
determination of the field at each position $\mathbf{r}_{j}$, considering
Euclidean distances
$\mathbf{r}_{ij}=\lVert\mathbf{r}_{i}-\mathbf{r}_{j}\lVert$ and unit vectors
$\hat{\mathbf{r}}_{ij}=\frac{\mathbf{r}_{i}-\mathbf{r}_{j}}{\lVert\mathbf{r}_{i}-\mathbf{r}_{j}\lVert}$.
The wavenumber $k=2\pi\nu/c$ is defined for all the frequency samples of the
operating bandwidth, set between 70GHz and 100 GHz. We recall that the
localization experiments performed in this work require the computation of the
radiated field when the first port of the cavity is fed. This field is
determined at a distance $y=R=25\text{cm}$ from the front panel of the cavity,
on a surface $\mathbf{r}^{\prime}$ of $15\text{cm}\times 15\text{cm}$. Each
axis is sampled by 51 points, i.e. $n_{r^{\prime}}=2601$. The diffraction
limit is defined at the highest frequency (100GHz), determining the smallest
radiated wavelength $\lambda_{\text{min}}$. Considering a distance
$R=25\text{cm}$ and a square radiating aperture of $L_{x,z}=220\text{mm}$
side, the Rayleigh diffraction limit $\delta_{x,z}$ is calculated as follows:

$\displaystyle\delta_{x,z}=\frac{\lambda_{\text{min}}R}{L_{x,z}}\approx
3.4\text{mm}$ (18)

The imaged space $\mathbf{r}^{\prime}$ is defined by pixel size of 2.5mm,
chosen slightly oversampled compared to the resolution limits in order to
mitigate the effects of grid shift between the target position and the nearest
pixels.

In the absence of filtering of the subspaces according to their level of
coherence, we first determine the field matrix
$\mathbf{E}_{0}\in\mathbb{C}^{n_{\nu}\times n_{r^{\prime}}}$. The singular
value decomposition filtering is then performed directly on the tangential
electric field $\mathbf{E}_{\text{tan}}$ measured by the scanner. Using the
notations introduced in the paper, the data are arranged to form two scalar
near-field matrices $\mathbf{E}_{x}^{(1)}$ and $\mathbf{E}_{z}^{(1)}$, both
complex and of dimensions $n_{\nu}\times n_{r}$. Each matrix
$\mathbf{E}_{x,z}^{[m]}$ is then diagonalized using a singular value
decomposition, in order to remove the $m$ first most significant and coherent
components :

$\displaystyle\mathbf{E}_{x,z}^{[m]}=\sum_{n=m+1}^{n_{\nu}}\sigma_{n}\mathbf{u}_{n}\mathbf{v}_{n}^{\dagger}$
(19)

where the singular vectors $\mathbf{u}_{n}$ and $\mathbf{v}_{n}$ correspond to
the polarizations $x$ and $z$ considered. Following the suppression of the
most coherent subspaces, the filtered tangential fields allow the
determination of point sources enabling the propagation of electric fields
following Eqs. (14) to (19). In the context of this study, we are specifically
focused on the field matrix $\mathbf{E}_{m}\in\mathbb{C}^{n_{\nu}\times
n_{r^{\prime}}}$ determined in the imaged space $\mathbf{r}^{\prime}$
according to the polarization $z$ to match to that of the source. The
measurements of transfer functions $s^{(q)}(\nu)$ are conducted for two source
positions of index $q$, successively arranged at
$(x,z)=(2\text{cm},0\text{cm})$ and $(x,z)=(0\text{cm},2\text{cm})$ in the
plane $y=25\text{cm}$. In the context of a simplified scalar model, the
emissivity of the source $\rho^{(q)}(r^{\prime})$, considered independent of
the frequency on the analysis band, is then related to the measured signals
according to the relation $\mathbf{s}^{(q)}=\mathbf{E}_{0}\,\bm{\rho}^{(q)}$.
The emissivity can then be estimated by compensating the phase of the field
distributions in space and frequency, thus providing a spatial correlation of
the measured signal with the pre-determined field:

$\displaystyle\hat{\bm{\rho}}_{m}^{(q)}$
$\displaystyle=\mathbf{E}_{m}^{\dagger}\,\mathbf{s}^{(q)}=\mathbf{E}_{m}^{\dagger}\mathbf{E}_{0}\,\bm{\rho}^{(q)}$
(20)

The results obtained for different filtering indices $m$ are presented in Fig.
13. This demonstration highlights the effect of filtering the first indices to
remove the influence of the most direct and coherent field paths in time and
space. It is even possible for the $\hat{\rho}_{0}^{(1)}$ reconstruction in
the first source position to identify quite distinctly the diffraction pattern
caused by the radiation from the excitation port through the iris grid. While
a single source is arranged facing the cavity in the case of the second
localization experiment, an artifact is clearly visible in
$\hat{\rho}_{0}^{(2)}$ and $\hat{\rho}_{100}^{(2)}$, at a position symmetric
with respect to the $z=0$ axis. The latter appeared for several source
positions in our experiments and seems to be related to the geometry of the
cavity, still presenting symmetries despite the presence of a portion of metal
sphere at one of its corners. The progressive increase of the filtering index
$m$ allows nevertheless to remove these artifacts. Considering that higher
indices are associated with longer mean propagation times (Fig. 6), it seems
intuitive that the associated spatial field distributions are increasingly
exposed to the impact of the convex boundary condition imposed in this
environment, limiting the ambiguity related to the response of this artifact
with respect to the real position of the source. Such properties are studied
in more detail in the context of time-reversal experiments, where the ergodic
properties of the media considered ensure that all locations in an environment
will be visited by incident waves over large propagation times [67].

Figure 13: Localization experiments of radiating sources set at
$(x,z)=(2\text{cm},0\text{cm})$ and $(x,z)=(0\text{cm},2\text{cm})$,
illustrating the deleterious impact of the most coherent subspaces on the
ambiguity of the reconstructed positions.

The progressive increase of the filtering index $m$ does not allow to
continuously improve the quality of the reconstructed images. It is indeed
necessary to consider the effect of the level differences of the singular
values associated with these subspaces, of which the example $m=3000$
illustrated in Fig. 13 corresponds to amplitudes more than 100 times lower
than the first singular values (Fig. 9). In the presence of additive noise
inherent to any measurement, the latter are indeed the most likely to be
corrupted, justifying the development of pseudo-inversion by regularization
based on the suppression of the highest indices [68, 69].

This analysis is finally completed by a remark. The monotonic decay of the
spatial and temporal coherence of the considered subspaces implies a possible
substitution of the detailed processing by a simple time gating applied
directly to the near field matrix. In such conditions, the cancellation of the
first temporal samples limits the impact of the strongly correlated subspaces
according to the spatial and frequency dimensions. We thus define a time
gating matrix $\mathbf{T}_{m}=\text{diag}(\textbf{q}_{m})$, where
$\text{diag}(\textbf{q}_{m})$ is a vector composed of zeros except for the
samples of index $m+1$ to $n_{\nu}$ which are set to 1. A temporal filtering
of the first $m$ indices is applied on the near field matrix $\mathbf{E}$ as
follows :

$\displaystyle\mathbf{E}_{tg-m}$
$\displaystyle=\mathbf{W}\,\mathbf{T}_{m}\mathbf{W}^{\dagger}\mathbf{E}$ (21)
$\displaystyle=\mathbf{W}\,\mathbf{T}_{m}\mathbf{W}^{\dagger}\mathbf{U}\mathbf{S}\mathbf{V}^{\dagger}$
(22)
$\displaystyle=\mathbf{W}\,\mathbf{T}_{m}\mathbf{U_{t}}\mathbf{S}\mathbf{V}^{\dagger}$
(23)

This decomposition allows us to highlight that the application of filtering by
$\mathbf{T}_{m}$ is performed indirectly on the singular vectors of the
$\mathbf{U_{t}}$ matrix. The diagonal form of the latter (Fig. 4) implies that
a suppression of the $m$ first temporal samples will have an impact comparable
to the suppression of the $m$ first singular values. The near field
$\mathbf{E}_{tg-m}$ is thus propagated using equations (14) and (17) and used
for the estimation of the position of the sources $\bm{\rho}_{tg-m}$ following
Eq. (20). The results for $m=100$ ($t=30$ ns) and $m=1000$ ($t=300$ ns) are
given in Fig. 14.

Figure 14: Source localization achieved with time gated near fields.

Benefiting from the previous analyses, image reconstruction can be greatly
accelerated by substituting the singular value decomposition of the near field
by a succession of direct and inverse Fourier transforms, ideally performed
with fast Fourier transforms, and simple temporal filtering. It is
nevertheless necessary to maintain a critical eye on the use of such a method,
based on a monotonic decrease of the temporal and spatial coherence of the
field during its propagation. As an example, the Saleh-Venenzuela model in the
field of indoor and broadband telecommunications describes stochastically the
impulse responses between transmitter and receiver formed by a certain number
of temporal clusters constituting a succession of damped responses separable
in time [70, 71]. The study of scattering along quasi one-dimensional media
similarly allows the identification of configurations where the contribution
of ballistic paths is much smaller than that of more complex paths [72],
implying a potentially more complex mapping of coherence states as a function
of the singular value index. A deeper analysis of the spatial and temporal
coherence thus allows in all situations and with the help of the previously
introduced metrics to identify and select the most suitable subspaces for the
considered applications.

##  V Conclusion

In conclusion, this study has allowed the development of techniques for the
decomposition of electromagnetic fields propagated through a complex medium,
discriminating orthogonal structures composing the latter according to their
level of spatial and temporal coherence. Many references in the scientific
literature focus on techniques involving space-space transfer matrices for the
calculation and analysis of Wigner-Smith operators. In a similar context, this
work has investigated the decomposition of space-frequency or space-time
transfer matrices, paving the way for an in-depth analysis of the temporal
spreading phenomena occurring during interaction with a complex medium. Beyond
the only perspectives of characterization of these challenging propagation
environments, these methods can also be exploited in applications whose
operation in conditions of partial coherence may be compromised. Such
thresholding on the first singular values and vectors of a linear application
can be counter-intuitive for scientists initiated to conditioning and
regularization problems [68]. The most significant subspaces are by definition
the most immune to noise, limiting the amplification of the most diffuse
structures under poor signal-to-noise conditions. A thresholding of the
strongest and weakest singular values could thus be considered in more
disturbed operating conditions. Previous analyses have also shown that the
position and polarization of the excitation port within these environments
could have a direct effect on the coherence of the most significant orthogonal
structures. Coupled with numerical tools, the optimization of applications
requiring the use of diffuse fields must therefore necessarily be carried out
by limiting as much as possible the direct interaction between the excitation
ports of the environments considered and the analysis plane. Finally, we
anticipate possible applications of this work to the shaping of
electromagnetic fields, exploiting the many degrees of freedom offered by
complex media and allowing the local focusing or cancellation of energy with
linear combinations of pre-selected subspaces.

## ACKNOWLEDGMENTS

T. Fromenteze acknowledges the support of the French Agence Nationale de la
Recherche (ANR) under reference ANR-21-JCJC-0027-01. The work of O. Yurduseven
was supported by a research grant from the Leverhulme Trust under the Research
Leadership Award RL-2019-019.

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