This letter presents an extension of the wellknown Whittle's formula for the asymptotic Fisher information matrix (FIM) on the power spectrum parameters of zeromean stationary Gaussian processes to compound Gaussian processes (CGP). The new formula includes a corrective factor that depends on the considered CG distribution, in addition to the usual Gaussian term.

Linear beamformers are optimal, in a mean square (MS) sense, when the signal of interest (SOI) and observations are jointly Gaussian and circular. Otherwise, optimal beamformers become non-linear with a structure depending on the unknown joint probability distribution of the SOI and observations. In this context, third-order Volterra minimum variance distortionless response (MVDR) beamformers have been proposed recently to improve the performance of linear beamformers in the presence of non-Gaussian and potentially non-circular interference, omnipresent in practical situations. High performance gains have been obtained for binary phase shift keying (BPSK) and quadrature phase shift keying (QPSK) interference having a square pulse shaping filter in particular. However in practice, for spectral efficiency reasons, most of signals use non-square pulse shaping filters, such as square root raised cosine filter. It is then important to analyze the sensitivity of third-order MVDR beamformers to interference pulse shaping filter, which is the purpose of this paper.

This letter is mainly dedicated to an extension of the Slepian-Bangs formula to non-circular complex elliptical symmetric (NC-CES) distributions, which is derived from a new stochastic representation theorem. This formula includes the non-circular complex Gaussian and the circular CES (C-CES) distributions. Some general relations between the Cramér Rao bound (CRB) under CES and Gaussian distributions are deduced. It is proved in particular that the Gaussian distribution does not always lead to the largest stochastic CRB (SCRB) as many authors tend to believe it. Finally a particular attention is paid to the noisy mixture where closed-form expressions for the SCRBs of the parameters of interest are derived.

Widely linear (WL) filters have the capability to perform single antenna interference cancellation (SAIC) of one rectilinear or quasi-rectilinear co-channel interference (CCI). The SAIC technology for quasi-rectilinear signals is operational in GSM handsets but requires enhancements for both VAMOS standard, an evolution of GSM/EDGE standard, and FBMC-OQAM networks, which are candidate for 5G mobile networks, in particular. For this reason, we propose in this paper a SAIC enhancement based on the use of third-order complex Volterra (CV) filtering, exploiting both the non-Gaussianity and the non-circularity of the signals up to the 6th-order. Limiting the analysis to rectilinear signals for space limitations, the performance of the proposed receiver are proved to outperform those of WL receivers for SAIC of one CCI.

Several direction of arrival (DOA) estimation algorithms have been proposed to exploit the structure of rectilinear or strictly second-order noncircular signals. But until now, only the compact closed-form expressions of the corresponding deterministic Cramér-Rao bound (DCRB) have been derived because it is much easier to derive than the stochastic CRB (SCRB). As this latter bound is asymptotically achievable by the maximum likelihood estimator, while the DCRB is unattainable, it is important to have a compact closed-form expression for this SCRB to assess the performance of DOA estimation algorithms for rectilinear signals. The aim of this letter is to derive this expression directly from the Slepian-Bangs formula including in particular the case of prior knowledge of uncorrelated or coherent sources. Some properties of these SCRBs are proved and numerical illustrations are given.

Widely linear (WL) filters have the capability to perform single antenna interference cancellation (SAIC) of one rectilinear (R) or quasi-rectilinear (QR) co-channel interference (CCI). The SAIC technology for QR signals is operational in GSM handsets but requires enhancements for both VAMOS standard, an evolution of GSM/EDGE standard, and FBMC-OQAM networks, which are candidate for 5G mobile networks. For this reason, we propose and analyze in this paper, for QR signals, a SAIC/MAIC enhancement based on the concept of three inputs WL FRESH filtering, exploiting almost exhaustively both the non-circularity and the cyclostationarity of QR signals, contrary to classical approaches which only exploit very partially these properties.

An antenna array of directional sensors offers more degrees-of-freedom to improve the source localization accuracy. The originally sophisticated expression of the CRB turns into a factorized one at high SNR, perfectly suitable to derive an objective function that depends on the array geometry and the source direction PDF. Analytically untractable, optimization is conducted by systematic search for a two-sensor array. The so-optimized array has a finite accuracy in all directions, contrarily to fixed-geometry linear arrays, and a performance comparable to larger-sized fixed-geometry circular arrays, and so at all SNR levels.

Arrays of sensors freely located along an axis are considered in this paper that are used to localize a near-field emitting source. Using Taylor expansion and a suitable coordinate system, simple, yet rich to interpret, Cramer-Rao bounds relative to the direction and range parameters are derived. Our analysis allows in particular to unveil a family of non-uniform linear arrays with better near field estimation capabilities, compared to the well-established uniform linear arrays.

This paper is devoted to the Cramer Rao bound (CRB) on the azimuth, elevation and range of a narrow-band near-field source localized by means of a uniform circular array (UCA), using the exact expression of the time delay parameter. After proving that the conditional and unconditional CRB are generally proportional for constant modulus steering vectors, we specify conditions of isotropy w.r.t. the distance and the number of sensors. Then we derive very simple, yet very accurate non-matrix closed-form expressions of different approximations of the CRBs.

This paper is devoted to time delay estimation for wide sense stationary complex circular or noncircular Gaussian signals. Using a theorem by Whittle that we have extended to complex data, closed-form expressions of the Cramer Rao bound (CRB) are given for the time delay alone in presence of nuisance parameters. In particular, we prove that the CRB for the time delay is weakly reduced for noncircular signals w.r.t. circular signals, except for very low signal to noise ratios (SNR), for which the CRB for rectilinear signals is half of the CRB for circular signals. Then, the maximum likelihood (ML) estimate that extends the generalized cross correlation (GCC) estimate is derived.

It is now well known that time invariant (TI) linear beamformers, such as the Capon's beamformer, are only optimal for stationary Gaussian observations whose complex envelope is necessarily second order (SO) circular. However in many applications such as in radiocommunications, most of the signals are nonGaussian and their complex envelope presents very often some SO and/or higher order (HO) non circularity properties. For this reason we propose in this paper a third order widely non linear Volterra minimum variance distortionnless response (MVDR) beamformer taking into account both the potential non Gaussian character and the potential HO non circularity (up to sixth order) of interferences. Some properties, performance and adaptive implementation of this new beamformer are presented in the paper. Illustrations shows the great interest, with respect to the existing beamformers, of this new beamformer for non Gaussian and HO non circular interferences, omnipresent in practice.

This paper presents an asymptotic analysis of the eigen value decomposition (EVD) of the sample covariance matrix associated with independent identically distributed (IID) non necessarily circular and Gaussian data that extends the well known analysis presented in the literature for circular and Gaussian data. Closed-form expressions of the asymptotic bias and variance of the sample eigenvalues and eigenvectors are given. As an application of these extended expressions, the statistical performance analysis of the minimum description length (MDL) criterion applied to the detection of the number of noncircular or/and nonGaussian components is considered.

In this paper, we address the problem of the sensor placement for estimating the direction of a narrow-band source, randomly located in the far-field of a planar antenna array. Estimation performance is evaluated by means of the expectation of the conditional Cramer Rao bound (ECRB), which depends on the prior probabilistic distribution of the DOA angles. We study the particular, but practical, case where the azimuth angle is uniformly distributed. Surprisingly, it turns out that the optimal arrays are not isotropic, i.e. they do not have the same accuracy in all possible look directions. In fact, optimal arrays computed here increase performance by about 10% compared to optimal isotropic arrays computed in a previous work.

In this paper, we consider the problem of testing impropriety (i.e., second-order noncircularity) of a complex-valued random variable (RV) based on the generalized likelihood ratio test (GLRT) derived under Gaussian distributions. Asymptotic (w.r.t. the data length) distributions of the GLR are given under the hypothesis that data are proper or improper, and under the true, not necessarily Gaussian distribution of the data. This enables us to derive in particular the receiver operating characteristics (ROC) of this test, an issue previously overlooked. Finally illustrative examples are presented in order to strengthen the obtained theoretical results.

This paper presents a performance analysis of likelihood ratio test (LRT)-based and generalized likelihood ratio test (GLRT)-based array receivers for the detection of a known signal corrupted by a potentially noncircular interference. Studying the distribution of the statistics associated with the LRT and GLRT, expressions of the probability of detection (PD) and false alarm (PFA) are given. In particular, an exact closed-form expression of PDand PFAare given for two LRT-based receivers and asymptotic (with respect to the data length) closed-form expression are given for PDand PFAfor four GLRT-based receivers. Finally illustrative examples are presented in order to strengthen the obtained results.

This paper introduces the optimal widely linear (WL) minimum variance distorsionless response (MVDR) beamformer for the reception of an unknown signal of interest (SOI) corrupted by potentially second order (SO) noncircular background noise and interference. The SOI, whose waveform is unknown, is assumed to be SO noncircular with arbitrary noncircular properties. In the steady state and for SO noncircular SOI and/or interference, this new WL beamformer, that is derived from an original orthogonal decomposition, is shown to always improve the performance of both the well-known Capon's beamformer and a WL MVDR beamformer introduced recently in the literature. This optimal WL MVDR beamformer is first introduced and some of its performance are analyzed. Then, several adaptive implementations of this optimal WL beamformer are presented.

This paper studies the asymptotic distribution of the eigenvectors estimated by some PCA algorithms based on the weighted subspace criterion. This enables us to analyse how the choice of the weighting matrix affects the algorithm's performance, an issue previously overlooked.

This paper addresses the degree of second-order non-circularity or impropriety of complex random variables and its purpose is to complement previously available theoretical results. New properties of the non-circularity rate (also called circularity spectrum) are given for scalar and multidimensional complex random variables with a particular attention paid to rectilinear random variables, i.e., with maximum circularity spectrum. Finally, the maximum likelihood estimate of the circularity spectrum in the Gaussian case and asymptotic distribution of this estimate for arbitrary distributions are given.

In many detection and estimation problems associated with processing of second order stationary 2-D discrete random processes, the observation data are the sum of two zero-mean second order stationary processes: the process of interest and the noise process. In particular, the main performance criterion is the signal to noise ratio (SNR). After linear filtering, the optimal SNR corresponds to the maximal value of a Rayleigh quotient which can be interpreted as the largest generalized eigenvalue of the covariance matrices associated with the signal and noise processes, which are Toeplitz block Toeplitz structured. In this paper, an extension of Szego's theorem to the generalized eigenvalues of Hermitian Toeplitz block Toeplitz matrices is given, under the hypothesis of absolutely summable elements, providing information about the asymptotic distribution of those generalized eigenvalues and in particular of the optimal SNR after linear filtering.

This letter addresses the stochastic Cramer-Rao bound (CRB) pertaining to the joint estimation of the carrier frequency offset, the carrier phase and the noise and signal powers of binary phase-shift keying (BPSK), minimum shift keying (MSK), and quaternary phase-shift keying (QPSK) modulated signals corrupted by additive white circular Gaussian noise. Because the associated models are governed by simple Gaussian mixture distributions, an explicit expression of the Fisher information matrix is given and an explicit expression for the stochastic CRB of these four parameters are deduced. Specialized expressions for low and high SNR are presented as well. Finally, these expressions are related to the modified CRB and our proposed analytical expressions are numerically compared with the approximate expressions previously given in the literature.

This paper addresses the resolution of the standard and non-circular MUSIC algorithms for arbitrary distribution and non-circularity of two closely spaced transmitters. Using an analysis based on perturbations of the noise projector instead of those of the eigenvectors, interpretable closed-form expressions of the threshold array signal to noise ratios (ASNR) at which these two algorithms are able to resolve the transmitters along the Cox and the Sharman and Durrani criteria are given. We prove in particular that the threshold ASNRs given by the noncircular MUSIC algorithm are sensitive to the non-circularity phase separation of the sources and are comfortably smaller that those given by the standard MUSIC algorithm. Numerical examples illustrate these results.

This paper examines the stochastic Cramer-Rao bound (CRB) of direction of arrival (DOA) estimates for binary phase-shift keying (BPSK), minimum shift keying (MSK) and quaternary phase-shift keying (QPSK) modulated signals in the presence of unknown nonuniform Gaussian noise. After deriving closed-form expressions of the CRB, the statistical resolution limit, defined as the source separation that equals its own CRB is given. It is shown that this highest achievable resolution is proportional to the reciprocal of the fourth root of the product of the number of snapshots by an extended signal to noise ratio (SNR), in contrast to the square root dependence for circular Gaussian sources

This paper considers interference cancellation in radar systems when the signal environment is non-stationary and focuses on the extended sample matrix inversion algorithm (ESMI) first proposed by Hay ward (1996). An explicit expression of the signal to noise plus interference ratio (SINR) obtained by this ESMI algorithm is given and analyzed. Compared to the SMI algorithm, it is shown that the performance improves for mainlobe jammers but degrades for sidelobe jammers. To overcome this drawback, an alternative constraint strategy is proposed which attains the good performances of the standard ESMI algorithm whatever the position of the jammers. Finally, the explicit expressions of these SINR are compared to Monte Carlo simulations w.r.t. implementation conditions

This paper gives explicit closed-form expressions of the stochastic Cramer-Rao bound (CRB) on direction of arrival (DOA) estimation accuracy for non-circular Gaussian sources in the case of an arbitrary unknown noise field parameterized by a vector of unknowns from the Slepian-Bangs formula. As a special case, the CRB under the nonuniform white noise assumption is derived. Our expressions can be viewed as extensions of the well-known results by Stoica and Nehorai (1990), Weiss and Friedlander (1993), Ottersten et al. (1992), and Gershman et al. (2002). Some properties of these CRB are proved, then these bounds are numerically compared with the conventional CRB under the circular complex Gaussian distribution for different unknown noise field models.

This paper focuses on the stochastic Cramer-Rao bound (CRB) on direction of arrival (DOA) estimation accuracy for non-circular Gaussian sources. We derive an explicit expression of the CRB for DOA parameters alone in the case of non-circular complex Gaussian sources by two different methods. One of them consists of computing the asymptotic covariance matrix of the maximum likelihood (ML) estimator, and the other is obtained directly from an extended Slepian-Bangs formula. Finally some properties of this CRB are proved.

The paper focuses on the stochastic Cramer-Rao bound (CRB) of direction of arrival (DOA) estimates for binary phase-shift keying (BPSK) and quaternary phase-shift keying (QPSK) modulated signals corrupted by additive circular complex Gaussian noise. Explicit expressions of the CRB for the DOA parameter alone in the case of a single signal waveform are given. Finally, these results are extended to the case of two independent BPSK distributed sources where an explicit expression of the DOA parameters alone is given for large SNR.

This paper addresses an asymptotically minimum variance (AMV) algorithm within the class of algorithms based on second-order statistics for estimating direction of arrival (DOA) parameters of possibly spatially correlated (even coherent) narrowband non-circular sources impinging on arbitrary array structures. To reduce the computational complexity due to the nonlinear minimization required by the matching approach, the covariance matching estimation techniques (COMET) are included in the algorithm. A numerical example illustrates the performance of the AMV algorithm.

This paper focuses on the extension of the asymptotic covariance of the sample covariance (denoted Bartlett's formula) of linear processes to third-order sample cumulant and to noisy linear processes. Thanks to a matrix polyspectral representation, closed-form expressions of the asymptotic covariance and cross covariance of the sample second and third moments are derived in a straightforward manner. As an application of these extended formulae, we enhance the sensitivity of the asymptotic performance of estimated ARMA parameters by an arbitrary third order-based algorithm to the spectrum of colored additive noise.

We study the asymptotic behavior of the smallest singular value of the single input multiple output (SIMO) channel filtering matrix. We prove that this can be expressed in terms of the sub-channel transfer functions. We apply this result to study the identifiability of bandlimited channels from their (estimated) second order statistics (SOS). We prove, and verify through examples, that SOS based algorithms are unable to identify frequency selective channels regardless of the assumed channel order.

This correspondence addresses the asymptotic normal distribution of the sample mean and the sample covariance matrix of mixed spectra time series containing a sum of sinusoids and a moving average (MA) process. Two central limit (CL) theorems are proved. As an application of this result, the asymptotic normal distribution of any sinusoidal frequencies estimator of such time series based on second-order statistics is deduced.

Szego's (1984) theorem states that the asymptotic behavior of the eigenvalues of a Hermitian Toeplitz matrix is linked to the Fourier transform of its entries. This result was later extended to block Toeplitz matrices, i.e., covariance matrices of multivariate stationary processes. The present work gives a new proof of Szego's theorem applied to block Toeplitz matrices. We focus on a particular class of Toeplitz matrices, those corresponding to covariance matrices of single-input multiple-output (SIMO) channels. They satisfy some factorization properties that lead to a simpler form of Szego's theorem and allow one to deduce results on the asymptotic behavior of the lowest nonzero eigenvalue for which an upper bound is developed and expressed in terms of the subchannels frequency responses. This bound is interpreted in the context of blind channel identification using second-order algorithms, and more particularly in the case of band-limited channels.

This paper improves and extends bounds on the numbers of sensors, redundancies, and holes for sparse linear arrays to sparse planar and volume arrays. As an application, the efficiency of regular planar and volume arrays with redundancies but no holes is deduced. Also, examples of new redundancy and hole square arrays, found by exhaustive computer search, are given.

Active research in blind identification of single input multiple output (SIMO) channels has led to a variety of second order statistics based algorithms, mainly the subspace and the linear prediction approaches. The subspace algorithm shows good performance, although it requires exact knowledge of the channel order, which is not guaranteed by current order detection algorithms. The linear prediction algorithm is sensitive to observation noise while its robustness to channel order over estimation is only theoretical. We propose a new second order statistics based blind channel identification algorithm using a shifted version of the channel output covariance matrix. It proves to be truly robust to channel order over estimation i.e., able to estimate the channel impulse response when the assumed channel order is greater than the exact order and when channel output is corrupted by additive noise and observed over finite time duration. Moreover, the proposed algorithm shows clearly better performance than the linear prediction algorithm.

In this paper, we perform a complete asymptotic performance analysis of the stochastic approximation algorithm (denoted subspace network learning algorithm) derived from Oja's learning equation, in the case where the learning rate is constant and a large number of patterns is available. This algorithm drives the connection weight matrix W to an orthonormal basis of a dominant invariant subspace of a covariance matrix. Our approach consists in associating to this algorithm a second stochastic approximation algorithm that governs the evolution of WWT to the projection matrix onto this dominant invariant subspace. Then, using a general result of Gaussian approximation theory, we derive the asymptotic distribution of the estimated projection matrix. Closed form expressions of the asymptotic covariance of the projection matrix estimated by the SNL algorithm, and by the smoothed SNL algorithm that we introduce, are given in case of independent or correlated learning patterns and are further analyzed. It is found that the structures of these asymptotic covariance matrices are similar to those describing batch estimation techniques. The accuracy or our asymptotic analysis is checked by numerical simulations and it is found to be valid not only for a "small" learning rate but in a very large domain. Finally, improvements brought by our smoothed SNL algorithm are shown, such as the learning speed/misadjustment tradeoff and the deviation from orthonormality.

We address adaptive estimation methods of eigenspaces of covariance matrices. We are interested in methods based on several coupled maximizations or minimizations of Rayleigh ratios where the constraints are replaced by appropriate parameterizations (Givens and mixed Givens/Householder). We prove the convergence of these algorithms with the help of the associated ordinary differential equation, and we propose an evaluation of the performance by computing the variances of the estimated eigenvectors for fixed gain factors. We show that these variances are very sensitive to the difference between two consecutive eigenvalues. Moreover, they also depend on whether the successive analyzed vector signals are correlated or not, and thus greatly depend on the origin of the covariance matrices of interest (spatial, temporal, spatio-temporal). Finally we show that the performance can be improved when the centro-symmetric property of some of those covariance matrices is taken into account.