In this paper, we address the problem of detection, in the frequency domain, of a M-dimensional time series modeled as the output of a M × K MIMO filter driven by a K-dimensional Gaussian white noise, and disturbed by an additive M-dimensional Gaussian colored noise. We consider the study of test statistics based of the Spectral Coherence Matrix (SCM) obtained as renormalization of the smoothed periodogram matrix of the observed time series over N samples, and with smoothing span B. To that purpose, we consider the asymptotic regime in which M, B, N all converge to infinity at certain specific rates, while K remains fixed. We prove that the SCM may be approximated in operator norm by a correlated Wishart matrix, for which Random Matrix Theory (RMT) provides a precise description of the asymptotic behaviour of the eigenvalues. These results are then exploited to study the consistency of a test based on the largest eigenvalue of the SCM, and provide some numerical illustrations to evaluate the statistical performance of such a test.

This paper introduces an improved Low Rank Adaptive Normalized Matched Filter (LR-ANMF) detector in a high dimensional (HD) context where the observation dimension is large and of the same order of magnitude than the sample size. To that end, the statistical analysis of the LR-ANMF, in a context where the target signal is disturbed by a spatially correlated Gaussian clutter and a spatially white Gaussian noise, is addressed. More specifically, the asymptotic distribution under the null hypothesis is derived, in the regime where both the dimension M of the observations and the number N of samples converge to infinity at the same rate and when the clutter covariance matrix has fixed rank K. In particular, it is shown that the LR-ANMF test statistic does not exhibit the CFAR property in the previous asymptotic regime. A correction to the LR-ANMF test is then proposed to ensure the asymptotic CFAR property, providing the improved LR-ANMF, termed as HD-LR-ANMF. Its asymptotic distribution is derived under both the null and the alternative hypotheses. Numerical simulations illustrate the fact that, despite the asymptotic nature of the analysis, the results obtained are accurate for reasonable values of M, N.

Testing the independence of the entries of multidimensional Gaussian observations is a very important problem in statistics, with a number of applications in signal processing, radar, cognitive radio, seismography, and multiple other fields. Typically, the problem is formulated as a binary hypothesis test, whereby the presence of correlation is declared when the value of a certain statistic is higher than a certain predetermined threshold. Most of the statistics for correlation tests are constructed from the sample correlation matrix (also known as sample coherence matrix in signal processing), which is defined as a power-normalized version of the sample covariance matrix. In this paper, correlation tests constructed from linear spectral statistics (LSS) of the sample correlation matrix are analyzed under the asymptotic framework where both sample size and observation dimension become large but comparable in magnitude. A central limit theorem (CLT) is established on this class of statistics, which is valid for generally correlated Gaussian observations. Results show that LSS asymptotically fluctuate as Gaussian random variables under both the hypotheses, with an asymptotic mean and variance that can be established for each particular test. In particular, this general CLT can be used to establish the asymptotic behavior of two of the most important correlation test statistics, namely the generalized likelihood ratio test and the Frobenius norm test, under both null and alternative hypotheses. As a by-product, it is established that LSS of sample covariance and sample correlation matrices have exactly the same first order behavior, but quite different asymptotic fluctuations in the second-order regime. In both the cases, the LSS asymptotically behave as Gaussian random variables, although with quite different asymptotic means and variances.

Correlation tests of multiple Gaussian signals are typically formulated as linear spectral statistics on the eigenvalues of the sample coherence matrix. This is the case of the Generalized Likelihood Ratio Test (GLRT), which is formulated as the determinant of the sample coherence matrix, or the locally most powerful invariant test (LMPIT), which is formulated as the Frobenius norm of this matrix. In this paper, the asymptotic behavior of general linear spectral statistics is analyzed assuming that both the sample size and the observation dimension increase without bound at the same rate. More specifically, almost sure convergence of a general class of linear spectral statistics is established, and an associated central limit theorem is formulated. These asymptotic results are shown to provide an accurate statistical description of the behavior of the GLRT and the LMPIT in situations where the sample size and the observation dimension are both large but comparable in magnitude.

This paper addresses the statistical behaviour of spatial smoothing subspace DoA estimation schemes using a sensor array in the case where the number of observations N is significantly smaller than the number of sensors M, and that the number of virtual arrays L is such that M and NL are of the same order of magnitude. This context is modelled by an asymptotic regime in which NL and M both converge towards 1 at the same rate. As in recent works devoted to the study of (unsmoothed) subspace methods in the case where M and N are of the same order of magnitude, it is shown that it is still possible to derive improved DoA estimators termed as Generalized-MUSIC (G-MUSIC). The key ingredient of this work is a technical result showing that the largest singular values and corresponding singular vectors of low rank deterministic perturbation of certain Gaussian block-Hankel large random matrices behave as if the entries of the latter random matrices were independent identically distributed.

This paper deals with a phase-coded waveform which combines two binary phase codes, each impacting on specific properties of the radar receiving channel. After giving a detailed analysis of the expression of the received signal after processing when the Gaussian clutter is modeled by a pth-order autoregressive process, we focus our attention on the choice of the two phase codes: one aims at increasing the unambiguous range whereas the other is chosen by taking into account several criteria such as the detection performance. For this purpose, we suggest determining the Pareto fronts of 1st, 2ndand 3rdorders by means of an exhaustive search. Given the three Pareto fronts for different types of clutters simulated by making the AR parameters vary, we provide an automatic way to determine, before embedding it, the most robust phase codes using a fuzzy logic operator.

This paper addresses the statistical performance of subspace DoA estimation using a sensor array, in the asymptotic regime where the number of samples and sensors both converge to infinity at the same rate. Improved subspace DoA estimators were derived (termed as G-MUSIC) in previous works, and were shown to be consistent and asymptotically Gaussian distributed in the case where the number of sources and their DoA remain fixed. In this case, which models widely spaced DoA scenarios, it is established that the traditional MUSIC method also provides consistent DoA estimates having the same asymptotic MSE as the G-MUSIC estimates. In the case of closely spaced DoA (i.e. with a spacing of the order of a beamwidth), it is shown that G-MUSIC is still able to consistently separate the sources, while it is no longer the case for MUSIC.

The MUSIC method is widely used in the field of DoA estimation using an array of M sensors, and is known to perform well as long as the number of available samples N is much larger than M. Nevertheless, in the scenario where N is of the same order of magnitude than M, its performance degrades, essentially because the sample covariance matrix (SCM) is no more a good estimator. A classical improvement, known as "rectification", consists in forcing the SCM to have a Toeplitz structure. In this paper, we analyze this method, by considering the asymptotic regime where M, N both converge to infinity at the same rate, and by studying consistency and asymptotic normality of the related DoA estimates.

We consider the problem of subspace estimation in situations where the number of available snapshots and the observation dimension are comparable in magnitude. In this context, traditional subspace methods tend to fail because the eigenvectors of the sample correlation matrix are heavily biased with respect to the true ones. It has recently been suggested that this situation (where the sample size is small compared to the observation dimension) can be very accurately modeled by considering the asymptotic regime where the observation dimensionMand the number of snapshotsNconverge to +∞ at the same rate. Using large random matrix theory results, it can be shown that traditional subspace estimates are not consistent in this asymptotic regime. Furthermore, new consistent subspace estimate can be proposed, which outperform the standard subspace methods for realistic values ofMandN. The work carried out so far in this area has always been based on the assumption that the observations are random, independent and identically distributed in the time domain. The goal of this paper is to propose new consistent subspace estimators for the case where the source signals are modelled as unknown deterministic signals. In practice, this allows to use the proposed approach regardless of the statistical properties of the source signals. In order to construct the proposed estimators, new technical results concerning the almost sure location of the eigenvalues of sample covariance matrices of Information plus Noise complex Gaussian models are established. These results are believed to be of independent interest.

This paper is devoted to the subspace DoA (direction-of-arrival) estimation using a large antennas array when the number of available snapshots is of the same order of magnitude than the number of sensors. In this context, the traditional subspace methods fail because the empirical covariance matrix of the observations is a poor estimate of the true covariance matrix. Mestre et al. proposed recently to study the behaviour of the traditional estimators when the number of antennas M and the number of snapshots N converge to +infin at the same rate. Using large random matrix theory results, they showed that the traditional subspace estimate is not consistent in the above asymptotic regime and they proposed a new consistent subspace estimate which outperforms the standard subspace method for realistic values of M and N. However, the work of Mestre et al. assumes that the source signals are independent and identically distributed in the time domain. The goal of the present paper is to propose new consistent estimators of the DoAs in the case where the source signals are modelled as unknown deterministic signals. This, in practice, allows to use the proposed approach whatever the statistical properties of the source signals are.

In this paper, we deal with the estimation of the ergodic capacity of large MIMO systems, using training sequences whose lengths are of the same order of magnitude than the number of antennas. In this context, the traditional estimator becomes inconsistent. Following the ideas developed by Girko in the context of the so-called theory of G-estimation, we propose a new estimator. We analyze its asymptotic behaviour and show, using numerical simulations, that it tends to improve significantly the performance of the standard estimate for a realistic number of antennas.