Esa Ollila

Complex Elliptically Symmetric Distributions: Survey, New Results and Applications

Complex elliptically symmetric (CES) distributions have been widely used in various engineering applications for which non-Gaussian models are needed. In this overview, circular CES distributions are surveyed, some new results are derived and their applications e.g., in radar and array signal processing are discussed and illustrated with theoretical examples, simulations and analysis of real radar data. The maximum likelihood (ML) estimator of the scatter matrix parameter is derived and general conditions for its existence and uniqueness, and for convergence of the iterative fixed point algorithm are established. Specific ML-estimators for several CES distributions that are widely used in the signal processing literature are discussed in depth, including the complex t -distribution, K-distribution, the generalized Gaussian distribution and the closely related angular central Gaussian distribution. A generalization of ML-estimators, the M-estimators of the scatter matrix, are also discussed and asymptotic analysis is provided. Applications of CES distributions and the adaptive signal processors based on ML- and M-estimators of the scatter matrix are illustrated in radar detection problems and in array signal processing applications for Direction-of-Arrival (DOA) estimation and beamforming. Furthermore, experimental validation of the usefulness of CES distributions for modelling real radar data is given.

On the Circularity of a Complex Random Variable

An important characteristic of a complex random variable z is the so-called circularity property or lack of it. We study the properties of the degree of circularity based on second-order moments, called circularity quotient, that is shown to possess an intuitive geometrical interpretation: the modulus and phase of its principal square-root are equal to the eccentricity and angle of orientation of the ellipse defined by the covariance matrix of the real and imaginary part of z. Hence, when the eccentricity approaches the minimum zero (ellipse is a circle), the circularity quotient vanishes; when the eccentricity approaches the maximum one, the circularity quotient lies on the unit complex circle. Connection with the correlation coefficient rho is established and bounds on rho given the circularity quotient (and vice versa) are derived. A generalized likelihood ratio test (GLRT) of circularity assuming complex normal sample is shown to be a function of the modulus of the circularity quotient with asymptotic chi2 2 distribution.

Robust antenna array processing using M-estimators of pseudo-covariance

This paper addresses the problem of antenna array processing in nonGaussian noise and interference conditions. Such conditions arise due to man-made interference in indoor and outdoor mobile communication channels as well as in military communications. In this paper M-estimators of the array (pseudo-)covariance matrix based upon complex data set are introduced. Estimates of the noise and signal subspaces based on M-estimators are then used to robustify the subspace direction of arrival (DOA) estimation methods. In addition, eigenvalues based on M-estimators are used in MDL criterion, thus yielding a robust signal detection method. The reliable performance of the proposed methods are shown by simulations.

The Deflation-Based FastICA Estimator: Statistical Analysis Revisited

This paper provides a rigorous statistical analysis of the deflation-based FastICA estimator, where the independent components (ICs) are extracted sequentially. The focus is on two aspects of the estimator: robustness against outliers as measured by the influence function (IF) and on its asymptotic relative efficiency (ARE) as measured by the ratio of the asymptotic variance of the FastICA w.r.t. the optimal maximum likelihood estimator (MLE). The derived compact closed-form expression of the IF reveals the vulnerability of the FastICA estimator to outliers regardless of the used nonlinearity. A cautionary finding is that even a moderate observation towards certain directions can render the estimator deficient in the sense that its separation performance degrades worse than a plain guess. The IF allows the derivation of a compact closed-form expression for the asymptotic covariance matrix of the FastICA estimator and subsequently its asymptotic relative efficiencies (AREs). The ARE figures calculated for some selected source distributions illustrate the fact that the order which the ICs are found is crucial as the accuracy of the previously extracted components can dominantly affect the accuracy of the successive deflation stages.

Essential Statistics and Tools for Complex Random Variables

Complex random signals play an increasingly important role in array, communications, and biomedical signal processing and related fields. However, the fundamental properties of complex-valued signals and mathematical tools needed to process them are scattered in literature. We provide a concise, unified, and rigorous treatment of essential properties and tools of complex random variables, and apply these fundamentals to derive complex extensions of Leibniz rule, Faá di Brunos formula, and Taylors series. The extensions allow establishing relationships among complex moments and cumulants, and characterizing the circularity property. We propose measures for testing and quantifying circularity, and observe that non-circularity may be more common in practical applications than previously thought. All results are rigorously proved and supplemented with clarifying examples.

Compound-Gaussian Clutter Modeling With an Inverse Gaussian Texture Distribution

The compound-Gaussian (CG) distributions have been successfully used for modelling the non-Gaussian clutter measured by high-resolution radars. Within the CG class, the complex K -distribution and the complex t-distribution have been used for modelling sea clutter which is often heavy-tailed or spiky in nature. In this paper, a heavy-tailed CG model with an inverse Gaussian texture distribution is proposed and its distributional properties such as closed-form expressions for its probability density function (p.d.f.) as well as its amplitude p.d.f., amplitude cumulative distribution function and its kurtosis parameter are derived. Experimental validation of its usefulness for modelling measured real-world radar lake-clutter is provided where it is shown to yield better fits than its widely used competitors.

Generalized complex elliptical distributions

We introduce a new class of distributions called generalized complex elliptically symmetric distributions. Several distributions commonly used in the literature, for example, the multivariate complex normal and Cauchy and the generalized complex normal distribution, are prominent members of this class. The treatment covers both proper and improper random vectors and goes beyond second-order concepts in defining the distribution model. Some properties of these distributions are studied and illustrative examples of their applications in multichannel signal processing are presented such as tests for circularity.

Complex Elliptically Symmetric Random Variables—Generation, Characterization, and Circularity Tests

Complex elliptically symmetric (CES) distributions constitute a flexible and broad class of distributions for many engineering applications and include the widely used complex Gaussian, complex -, complex generalized Gaussian and symmetric -stable distributions for example. Their careful statistical characterization is needed. Moreover, the mechanisms for generating random variables from these distributions are not well defined in literature. In this paper we provide such treatment in order to provide a better insight on their statistical properties and simplifying proofs and derivations. For example, insightful expressions for complex kurtosis coefficients of CES distributions are derived providing an interpretation on what complex kurtosis really measures. Also derived are asymptotic distributions of circularity measures, the sample circularity quotient and coefficient, assuming i.i.d. samples from an unspecified CES distribution. Also new Walds type detectors of circularity are proposed that are valid within CES distributions with finite fourth-order moments. These results are accompanied with examples that are of interest in developing signal processing algorithms for complex-valued signals.

Complex-valued ICA based on a pair of generalized covariance matrices

It is shown that any pair of scatter and spatial scatter matrices yields an estimator of the separating matrix for complex-valued independent component analysis (ICA). Scatter (resp. spatial scatter) matrix is a generalized covariance matrix in the sense that it is a positive definite hermitian matrix functional that satisfies the same affine (resp. unitary) equivariance property as does the covariance matrix and possesses an additional IC-property, namely, it reduces to a diagonal matrix at distributions with independent marginals. Scatter matrix is used to decorrelate the data and the eigenvalue decomposition of the spatial scatter matrix is used to find the unitary mixing matrix of the uncorrelated data. The method is a generalization of the FOBI algorithm, where a conventional covariance matrix and a certain fourth-order moment matrix take the place of the scatter and spatial scatter matrices, respectively. Emphasis is put on estimators employing robust scatter and spatial scatter matrices. The proposed approach is one among the computationally most attractive ones, and a new efficient algorithm that avoids decorrelation of the data is also proposed. Moreover, the method does not rely upon the commonly made assumption of complex circularity of the sources. Simulations and examples are used to confirm the reliable performance of our method.

Regularized M-estimators of scatter matrix

In this paper, a general class of regularized M-estimators of scatter matrix are proposed that are suitable also for low or insufficient sample support (small n and large p) problems. The considered class constitutes a natural generalization of M-estimators of scatter matrix (Maronna, 1976) and are defined as a solution to a penalized M-estimation cost function. Using the concept of geodesic convexity, we prove the existence and uniqueness of the regularized M-estimators of scatter and the existence and uniqueness of the solution to the corresponding M-estimating equations under general conditions. Unlike the non-regularized M-estimators of scatter, the regularized estimators are shown to exist for any data configuration. An iterative algorithm with proven convergence to the solution of the regularized M-estimating equation is also given. Since the conditions for uniqueness do not include the regularized versions of Tylers M-estimator, necessary and sufficient conditions for their uniqueness are established separately. For the regularized Tylers M-estimators, we also derive a simple, closed form, and data-dependent solution for choosing the regularization parameter based on shape matrix matching in the mean-squared sense. Finally, some simulations studies illustrate the improved accuracy of the proposed regularized M-estimators of scatter compared to their non-regularized counterparts in low sample support problems. An example of radar detection using normalized matched filter (NMF) illustrate that an adaptive NMF detector based on regularized M-estimators are able to maintain accurately the preset CFAR level.