The distribution of singular values of the propagation operator in a random medium is investigated, in a backscattering configuration. Experiments are carried out with pulsed ultrasonic waves around 3 MHz, using an array of 64 programmable transducers placed in front of a random scattering medium. The impulse responses between each pair of transducers are measured and form the response matrix. The evolution of its singular values with time and frequency is computed by means of a short-time Fourier analysis. The mean distribution of singular values exhibits very different behaviours in the single and multiple scattering regimes. The results are compared with random matrix theory. Once the experimental matrix coefficients are renormalized, experimental results and theoretical predictions are found to be in a very good agreement. Two kinds of random media have been investigated: a highly scattering medium in which multiple scattering predominates and a weakly scattering medium. In both cases, residual correlations that may exist between matrix elements are shown to be a key parameter. Finally, the possibility of detecting a target embedded in a random scattering medium based on the statistical properties of the strongest singular value is discussed.
Deep Dive into Singular value distribution of the propagation matrix in random scattering media.
The distribution of singular values of the propagation operator in a random medium is investigated, in a backscattering configuration. Experiments are carried out with pulsed ultrasonic waves around 3 MHz, using an array of 64 programmable transducers placed in front of a random scattering medium. The impulse responses between each pair of transducers are measured and form the response matrix. The evolution of its singular values with time and frequency is computed by means of a short-time Fourier analysis. The mean distribution of singular values exhibits very different behaviours in the single and multiple scattering regimes. The results are compared with random matrix theory. Once the experimental matrix coefficients are renormalized, experimental results and theoretical predictions are found to be in a very good agreement. Two kinds of random media have been investigated: a highly scattering medium in which multiple scattering predominates and a weakly scattering medium. In both ca
Wave propagation in a multiple scattering environment has been an interdisciplinary subject of interest in a huge variety of domains ranging, e.g., from solid state physics to optics, electromagnetism or seismology since multiple scattering can occur with all kinds of waves, whether quantum or classical. Among all areas of mesoscopic wave physics, some (like acoustics, seismology, microwaves) have the experimental advantage to offer controllable multi-element arrays of quasi-pointlike emitters/receivers. In such a case, the propagation between two arrays is best described by a matrix, termed the propagation operator K. At each frequency, its coefficients k ij correspond to the complex response between array elements i and j. Despite their diversity all practical applications of wave physics (communication, detection, imaging, characterization...), have one thing in common: all the available information is contained in the array response matrix K. Once K is known, the rest is only post-processing. Therefore, in a random scattering environment it is essential to study the statistical properties of K, and their relation to field correlations, weak localisation, single versus multiple scattering.
Previous works have been performed in a transmission context, whether it be for communication purposes [1][2][3][4] or scattering problems [5,6]. In this paper, we will consider backscattering configurations : the same array of N independent elements is used to transmit and receive waves. In that case, K is a square matrix of dimension N × N, and it is symmetric if the medium is reciprocal. We are particularly interested in the singular value decomposition (SVD) of the propagation operator, which amounts to write K as the product of three matrices: K = UΛV † . Λ is a diagonal matrix whose nonzero elements λ i are called the singular values of K. They are always real and positive, and arranged in a decreasing order (λ 1 > λ 2 > … > λ N ). U and V are unitary matrices whose columns correspond to the singular vectors.
It is now well known that in the case of point-like scatterers in a homogeneous medium, each scatterer is mainly associated to one non-zero singular value of K [7,8], as long as the number of scatterers is smaller than N and multiple scattering is neglected [9,10]. So a singular value decomposition of K (or equivalently a diagonalisation of the so-called time-reversal operator KK * ) allows the selective detection of several targets, each being associated to a singular value of K. This is the core of a detection method named DORT (French acronym for Decomposition of the Time Reversal Operator) [7,8]. DORT has shown its efficiency in detecting and separating the responses of several scatterers in homogeneous or weakly heterogeneous media [7,11] as well as in waveguides [12][13][14]. It has found applications in non-destructive evaluation [15], underwater acoustics [16,17], electromagnetism [18][19][20][21] and in radar applied to forest environments [22][23][24].
In this paper we will deal with random scattering media, consisting of a large number (» N) of randomly distributed scatterers, showing possibly multiple scattering between them. We will also address the issue of detecting a stronger reflector hidden in a statistically homogenous scattering medium, based on the strongest singular value λ 1 . Since the propagation medium is considered as one realisation of a random process, some general results of random matrix theory (RMT) may be fruitfully applied.
RMT has been widely used in physics, statistics and engineering. The domain of applications are numerous, ranging from nuclear physics [25] or chaotic systems [26] to neural networks [27], telecommunications [1] or financial analysis [28]. RMT predicts general behaviours of stochastic systems as, for instance, determining the Shannon capacity for MIMO communications in random media [3,4] or statistical properties of highly excited energy levels for heavy nuclei [25]. Another direct application of RMT is to separate the deterministic and random contributions in multivariate data analysis [29][30][31].
Here, the experimental configuration we consider uses a piezo-electric array, with a finite (N =64) numbers of elements sending wide-band ultrasonic waves around 3 MHz in an a priori unknown scattering medium. The main issues we address in this work are the applicability of RMT [1] to this experimental context, and its interest to establish a detection criterion based on the statistical properties of λ 1 . Since K is random, the relevant observable is the probability distribution function ρ(λ) of its singular values. Recent experiments [32] indicate that in a multiple scattering regime, the distribution of the singular values is in good agreement with a simple law derived from RMT, the so-called “quarter circle law” [1,33]. In theory, the quarter circle law applies to square matrices of infinite dimensions, containing independently and identically dis
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
