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.

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.

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.

This paper addresses the joint path delay and time-varying complex gain estimation for continuous phase modulation (CPM) over a time-selective slowly varying flat Rayleigh fading channel. We propose an expectation-maximization (EM) algorithm for path delay estimation in a Kalman smoother framework. The time-varying complex gain is modeled by a first order autoregressive (AR) process. Such a modeling yields to the representation of the problem by a dynamic Bayesian system in a state-space form that allows the application of EM algorithm in the context of unobserved data for obtaining an estimate of the path delay. This is used with Kalman smoother for state estimation. We derive analytically a closed-form expression of the modified hybrid Cramer-Rao bound (MHCRB) for path delay and complex gain parameters. Finally, some numerical examples are presented to illustrate the performance of the proposed algorithm compared to the conventional generalized correlation method and to the MHCRB.

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.

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 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.