Jan Eriksson

Complex random vectors and ICA models: identifiability, uniqueness, and separability

In this paper, the conditions for identifiability, separability and uniqueness of linear complex valued independent component analysis (ICA) models are established. These results extend the well-known conditions for solving real-valued ICA problems to complex-valued models. Relevant properties of complex random vectors are described in order to extend the Darmois-Skitovich theorem for complex-valued models. This theorem is used to construct a proof of a theorem for each of the above ICA model concepts. Both circular and noncircular complex random vectors are covered. Examples clarifying the above concepts are presented

Identifiability, separability, and uniqueness of linear ICA models

In this letter, we give the conditions for identifiability, separability and uniqueness of linear real valued independent component analysis (ICA) models. A theorem is formulated and a proof is provided for each of the above concepts. These results extend the conditions for solving ICA problems, originally established by Comon , to wider class of mixing models and source distributions. Examples clarifying the above concepts are presented as well.

Steepest Descent Algorithms for Optimization Under Unitary Matrix Constraint

In many engineering applications we deal with constrained optimization problems with respect to complex-valued matrices. This paper proposes a Riemannian geometry approach for optimization of a real-valued cost function T of complex-valued matrix argument W, under the constraint that W is an n times n unitary matrix. We derive steepest descent (SD) algorithms on the Lie group of unitary matrices U(n). The proposed algorithms move towards the optimum along the geodesics, but other alternatives are also considered. We also address the computational complexity and the numerical stability issues considering both the geodesic and the nongeodesic SD algorithms. Armijo step size [1] adaptation rule is used similarly to [2], but with reduced complexity. The theoretical results are validated by computer simulations. The proposed algorithms are applied to blind source separation in MIMO systems by using the joint diagonalization approach [3]. We show that the proposed algorithms outperform other widely used algorithms.

Characteristic-function-based independent component analysis

A novel characteristic-function-based method for blind separation of statistically independent source signals is proposed in the independent component analysis (ICA) framework. The definition of independence may be given in terms of factorization of joint characteristic function. Three criteria for ICA are derived based on this property. These criteria always exist and two of them have desirable large sample properties. An objective function for estimating the independence criteria directly from data is proposed. An efficient algorithm using Fourier coefficients is developed for minimizing the objective function. Simulation results demonstrate that the method performs reliably even in such situations where many widely used ICA methods may fail.

Adaptive Score Functions for Maximum Likelihood ICA

We propose Blind Source Separation (BSS) techniques that are applicable to a wide class of source distributions that may be skewed or symmetric and may even have zero kurtosis. Skewed distributions are encountered in many important application areas such as communications and biomedical signal processing. The methods stem from maximum likelihood approach. Powerful parametric models based on the Extended Generalized Lambda Distribution (EGLD) and the Pearson system are employed in finding the score function. Model parameters are adaptively estimated using conventional moments or linear combinations of order statistics (L-moments). The developed methods are compared with the existing methods quantitatively. Simulation examples demonstrate the good performance of the proposed methods in the cases where the standard Independent Component Analysis (ICA) methods perform poorly.

On the Cramér-Rao bound for the constrained and unconstrained complex parameters

We derive a complex form of the unconstrained and constrained Cramer-Rao lower bound (CRB) of composite real parameters formed by stacking the real and imaginary part of the complex parameters. The derived complex constrained and unconstrained CRB is easy to calculate and possesses similar structure as in the real parameter case but with the real covariance, Jacobian and the Fisher information matrix replaced by complex matrices with analogous interpretations. The advantage of the complex CRB is that it is oftentimes easier to calculate than its real form. It is highlighted that a statistic that attains a bound on the complex covariance matrix alone do not necessarily attain the CRB since complex covariance matrix does not provide a full second-order description of a complex statistic since also the pseudo-covariance matrix is needed. Our derivations also lead to some new insights and theory that are similar to real CRB theory.

Maximum likelihood estimation of ICA model for wide class of source distributions

We propose two blind source separation techniques that are applicable to a wide class of source distributions that may also be skewed and may even have zero kurtosis. Skewed distributions are encountered in many important application areas such as communications and biomedical signal processing. The methods are based on maximum likelihood approach where source distributions are modeled adaptively by the Pearson system and the extended generalized lambda distribution (EGLD). To compare the developed methods with the existing methods, quantitative measures for the quality of separation are used. Simulation experiments demonstrate the good performance of proposed methods in the cases where the standard BSS methods perform poorly.

Statistics for complex random variables revisited

Complex random signals play an increasingly important role in array, communications, and biomedical signal processing and related fields. However, the mathematical foundations of complex-valued signals and tools developed for handling them are scattered in literature. There appears to be a need for a concise, unified, and rigorous treatment of such topics. In this paper such a treatment is provided. Moreover, we establish connections between seemingly unrelated objects such as real differentiability and circularity. In addition, a novel complex-valued extension of Taylor series is presented and a measure for circularity is proposed.

Equivariant algorithms for estimating the strong-uncorrelating transform in complex independent component analysis

For complex-valued multidimensional signals, conventional decorrelation methods do not completely specify the covariance structure of the whitened measurements. In recent work [1,2], the concept of strong-uncorrelation and its importance for complex-valued independent component analysis has been identified. Few algorithms for estimating the strong-uncorrelating transform currently exist. This paper presents two novel algorithms for estimating and computing the strong uncorrelating transform. The first algorithm uses estimated covariance and pseudo-covariance matrices, and the second algorithm estimates the strong uncorrelating transform directly from measurements. An analysis shows that the only stable stationary point of both algorithms produces the strong uncorrelating transform when the circularity coefficients of the sources are distinct and positive. Simulations show the efficacy of the approach in a source clustering task for wireless communications.

Adaptive Estimation of the Strong Uncorrelating Transform with Applications to Subspace Tracking

In some signal processing tasks involving complex-valued multichannel measurements, classical whitening approaches do not completely remove the second-order statistical dependencies of the data. This paper describes adaptive procedures for estimating the strong uncorrelating transform for jointly diagonalizing the covariance and pseudo-covariance matrices of multidimensional signals. Novel algorithms are derived that extend and combine the power method and orthogonal iterations with ordinary fixed and iterative whitening procedures. Finally, we show how to combine our procedures with orthogonal PAST algorithms to perform subspace tracking and source signal clustering based on non-circularity