On the Fisher information matrix for multivariate elliptically contoured distributions

The Slepian-Bangs formula provides a very convenient way to compute the Fisher information matrix (FIM) for Gaussian distributed data. The aim of this letter is to extend it to a larger family of distributions, namely elliptically contoured (EC) distributions. More precisely, we derive a closed-form expression of the FIM in this case. This new expression involves the usual term of the Gaussian FIM plus some corrective factors that depend only on the expectations of some functions of the so-called modular variate. Hence, for most distributions in the EC family, derivation of the FIM from its Gaussian counterpart involves slight additional derivations. We show that the new formula reduces to the Slepian-Bangs formula in the Gaussian case and we provide an illustrative example with Student distributions on how it can be used

Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 2: The Under-Sampled Case

In the first part of this series of two papers, we extended the expected likelihood approach originally developed in the Gaussian case, to the broader class of complex elliptically symmetric (CES) distributions and complex angular central Gaussian (ACG) distributions. More precisely, we demonstrated that the probability density function (p.d.f.) of the likelihood ratio (LR) for the (unknown) actual scatter matrix \mSigma_{0} does not depend on the latter: it only depends on the density generator for the CES distribution and is distribution-free in the case of ACG distributed data, i.e., it only depends on the matrix dimension $M$ and the number of independent training samples $T$, assuming that $T \geq M$. Additionally, regularized scatter matrix estimates based on the EL methodology were derived. In this second part, we consider the under-sampled scenario ($T \leq M$) which deserves a specific treatment since conventional maximum likelihood estimates do not exist. Indeed, inference about the scatter matrix can only be made in the $T$-dimensional subspace spanned by the columns of the data matrix. We extend the results derived under the Gaussian assumption to the CES and ACG class of distributions. Invariance properties of the under-sampled likelihood ratio evaluated at \mSigma_{0} are presented. Remarkably enough, in the ACG case, the p.d.f. of this LR can be written in a rather simple form as a product of beta distributed random variables. The regularized schemes derived in the first part, based on the EL principle, are extended to the under-sampled scenario and assessed through numerical simulations

Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 1: The Over-Sampled Case

In \cite{Abramovich04}, it was demonstrated that the likelihood ratio (LR) for multivariate complex Gaussian distribution has the invariance property that can be exploited in many applications. Specifically, the probability density function (p.d.f.) of this LR for the (unknown) actual covariance matrix $\R_{0}$ does not depend on this matrix and is fully specified by the matrix dimension $M$ and the number of independent training samples $T$. Since this p.d.f. could therefore be pre-calculated for any a priori known $(M,T)$, one gets a possibility to compare the LR of any derived covariance matrix estimate against this p.d.f., and eventually get an estimate that is statistically ``as likely'' as the a priori unknown actual covariance matrix. This ``expected likelihood'' (EL) quality assessment allows for significant improvement of MUSIC DOA estimation performance in the so-called ``threshold area'' \cite{Abramovich04,Abramovich07d}, and for diagonal loading and TVAR model order selection in adaptive detectors \cite{Abramovich07,Abramovich07b}. Recently, a broad class of the so-called complex elliptically symmetric (CES) distributions has been introduced for description of highly in-homogeneous clutter returns. The aim of this series of two papers is to extend the EL approach to this class of CES distributions as well as to a particularly important derivative of CES, namely the complex angular central distribution (ACG). For both cases, we demonstrate a similar invariance property for the LR associated with the true scatter matrix \mSigma_{0}. Furthermore, we derive fixed point regularized covariance matrix estimates using the generalized expected likelihood methodology. This first part is devoted to the conventional scenario ($T \geq M$) while Part 2 deals with the under-sampled scenario ($T \leq M$)

When each continuous operator is regular, II

The following theorem is essentially due to L.~Kantorovich and B. Vulikh and it describes one of the most important classes of Banach lattices between which each continuous operator is regular. {\bf Theorem 1.1.} {\sl Let $E$ be an arbitrary L-space and $F$ be an arbitrary Banach lattice with Levi norm. Then ${\cal L}(E,F)={\cal L}^r(E,F),\ (\star)$ that is, every continuous operator from $E$ to $F$ is regular.} In spite of the importance of this theorem it has not yet been determined to what extent the Levi condition is essential for the validity of equality $(\star)$. Our main aim in this work is to prove a converse to this theorem by showing that for a Dedekind complete $F$ the Levi condition is necessary for the validity of $(\star)$. As a sample of other results we mention the following. {\bf Theorem~3.6.} {\sl For a Banach lattice $F$ the following are equivalent: {\rm (a)} $F$ is Dedekind complete; {\rm (b)} For all Banach lattices $E$, the space ${\cal L}^r(E,F)$ is a Dedekind complete vector lattice; {\rm (c)} For all L-spaces $E$, the space ${\cal L}^r(E,F)$ is a vector lattice.

Spectrally Limited Periodic Waveforms for HF OTHR Applications

The problem of a constant modulus (CM) continuous wave (CW) waveform design with the thumb-tack ambiguity function that meets the NTIA RSEC requirements is addressed. The ad-hoc and alternating projection techniques are proposed to modify the spectrum of a prototype waveform to meet the NTIA RSEC requirements, retaining the "thumb-tack" property of the original waveform ambiguity function. Periodic binary frequency shift keying (BFSK), Costas FSK, and noise-like waveforms are modified to meet CM and NTIA RSEC requirements. Introduced examples demonstrate the spectrum modification consequences and the proposed technique's efficiency for generating the CM spectrum-controlled waveforms with the thumb-tack ambiguity function

Invariance properties of the likelihood ratio for covariance matrix estimation in some complex elliptically contoured distributions

The likelihood ratio (LR) for testing if the covariance matrix of the observation matrix X is R has some invariance properties that can be exploited for covariance matrix estimation purposes. More precisely, it was shown in Abramovich et al.(2004, 2007, 2007) that, in the Gaussian case, LR(\R0|X), where R0 stands for the true covariance matrix of the observations \X, has a distribution which does not depend on R0 but only on known parameters. This paved the way to the expected likelihood (EL) approach, which aims at assessing, and possibly enhancing the quality of any covariance matrix estimate (CME) by comparing its LR to that of R0. Such invariance properties of LR (R0|X) were recently proven for a class of elliptically contoured distributions (ECD) in Abramovich and Besson (2013) and Bession and Abramovich (2013) where regularized CME were also presented. The aim of this paper is to derive the distribution of LR(R0|X) for other classes of ECD not covered yet, so as to make the EL approach feasible for a larger class of distributions

Sensitivity analysis of likelihood ratio test in K distributed and/or Gaussian noise

In a recent letter we addressed the problem of detecting a fluctuating target in $K$ distributed noise using multiple coherent processing intervals. It was shown through simulations that the performance of the likelihood ratio test is dominated by the snapshot which corresponds to the minimal value of the texture. However, for this particular snapshot the clutter to thermal noise ratio is not large and hence thermal noise cannot be neglected. In the present paper, we continue our investigation with a view to consider detection in a mixture of $K$ distributed and Gaussian noise. Towards this end we study the sensitivity of our previously derived detectors. First, we provide stochastic representations that allow to evaluate their performance in $K$ distributed noise only or Gaussian noise only. Then, their robustness to a mixture is assessed

Fluctuating target detection in fluctuating K-distributed clutter

This letter deals with the problem of fluctuating target detection in heavy-tailed $K$-distributed clutter over a number $T$ of independent coherent intervals, e.g., in the case of a long observation interval (``stare mode''), or that of independent (range) resolution cells as per the track before detect techniques. The generalized likelihood ratio test for the problem at hand is derived, as well as an approximation of it, whose distribution under the null hypothesis is derived. We also show some significant differences as compared to the usual Gaussian case, in particular the influence of $T$ and of the shape parameter of the $K$ distribution

Adaptive detection in elliptically distributed noise and under-sampled scenario

The problem of adaptive detection of a signal of interest embedded in elliptically distributed noise with unknown scatter matrix $\R$ is addressed, in the specific case where the number of training samples $T$ is less than the dimension $M$ of the observations. In this under-sampled scenario, whenever $\R$ is treated as an arbitrary positive definite Hermitian matrix, one cannot resort directly to the generalized likelihood ratio test (GLRT) since the maximum likelihood estimate (MLE) of $\R$ is not well-defined, the likelihood function being unbounded. Indeed, inference of $\R$ can only be made in the subspace spanned by the observations. In this letter, we present a modification of the GLRT which takes into account the specific features of under-sampled scenarios. We come up with a test statistic that, surprisingly enough, coincides with a subspace detector of Scharf and Friedlander: the detector proceeds in the subspace orthogonal to the training samples and then compares the energy along the signal of interest to the total energy. Moreover, this detector does not depend on the density generator of the noise elliptical distribution. Numerical simulations illustrate the performance of the test and compare it with schemes based on regularized estimates of $\R$

Toeplitz Inverse Eigenvalue Problem: Application to the Uniform Linear Antenna Array Calibration

The inverse Toeplitz eigenvalue problem (ToIEP) concerns finding a vector that specifies the real-valued symmetric Toeplitz matrix with the prescribed set of eigenvalues. Since phase "calibration" errors in uniform linear antenna arrays (ULAs) do not change the covariance matrix eigenvalues and the moduli of the covariance matrix elements, we formulate a number of the new ToIEP problems of the Hermitian Toeplitz matrix reconstruction, given the moduli of the matrix elements and the matrix eigenvalues. We demonstrate that for the real-valued case, only two solutions to this problem exist, with the "non-physical" one that in most practical cases could be easily disregarded. The computational algorithm for the real-valued case is quite simple. For the complex-valued case, we demonstrate that the family of solutions is broader and includes solutions inappropriate for calibration. For this reason, we modified this ToIEP problem to match the covariance matrix of the uncalibrated ULA. We investigate the statistical convergence of the ad-hoc algorithm with the sample matrices instead of the true ones. The proposed ad-hoc algorithms require the so-called "strong" or "argumental" convergence, which means a large enough required sample volume that reduces the errors in the estimated covariance matrix elements. Along with the ULA arrays, we also considered the fully augmentable minimum redundancy arrays that generate the same (full) set of covariance lags as the uniform linear arrays, and we specified the conditions when the ULA Toeplitz covariance matrix may be reconstructed given the M-variate MRA covariance matrix.Comment: 27 pages, 40 figure