MIMO instantaneous blind identification and separation based on arbitrary order temporal structure in the data

This thesis is concerned with three closely related problems. The first one is called Multiple-Input Multiple-Output (MIMO) Instantaneous Blind Identification, which we denote by MIBI. In this problem a number of mutually statistically independent source signals are mixed by a MIMO instantaneous mixing system and only the mixed signals are observed, i.e. both the mixing system and the original sources are unknown or ‘blind’. The goal of MIBI is to identify the MIMO system from the observed mixtures of the source signals only. The second problem is called Instantaneous Blind Signal Separation (IBSS) and deals with recovering mutually statistically independent source signals from their observed instantaneous mixtures only. The observation model and assumptions on the signals and mixing system are the same as those of MIBI. However, the main purpose of IBSS is the estimation of the source signals, whereas the main purpose of MIBI is the estimation of the mixing system. If the number of source signals is not larger than the number of sensors, the source signals can be recovered by applying the inverse of the estimated mixing system to the observed mixtures. Hence, from this point of view IBSS is merely a direct application of MIBI. The third problem is called Instantaneous Semi-Blind Source Localization (ISBSL) and concerns the localization of a set of narrowband sources from their observed instantaneous mixtures only. In this case, the instantaneous mixing system is parameterized by the source position parameters. Hence, narrowband ISBSL can be considered as a parameterized version of MIBI in which more a priori knowledge is available than in the general ‘fully blind’ MIBI case. For this reason we call this problem ‘semi-blind’. Because MIBI is a kind of abstraction or generalization of both IBSS and ISBSL, the main focus in this work is on MIBI, while IBSS and ISBSL are considered as applications or examples. Many methods and algorithms for performing MIBI, IBSS and ISBSL have been developed during the last decade. Until now, mainly three different approaches have been used. These are based on the following properties of the source signals, several of which are related to the guiding principle of statistical independence: non-Gaussianity, second order spatial uncorrelatedness in combination with temporal non-whiteness/correlatedness, and second order spatial uncorrelatedness in combination with second order non-stationarity. What is lacking from those approaches is the exploitation of higher order temporal structure in the data, such as higher order non-whiteness/correlatedness and non-stationarity. Some methods for exploiting Higher Order Temporal Structure (HOTS) exist, but usually these are quite specific. In addition, most blind methods described in the literature cannot deal with a MIBI scenario of great interest, viz. one with more sources than sensors. In this work we present a unifying framework for exploiting arbitrary order temporal structure in the signals by means of cumulant functions, which possess convenient mathematical properties such as multilinearity. The MIBI problem is formulated in such a way that any kind of temporal structure in the data, such as arbitrary order non-stationarity and non-whiteness, is exploited in a unified manner. Based on physically plausible assumptions on the temporal structure of the source and noise signals, and applying subspace techniques to a subspace matrix containing cumulant values arranged in a specific manner, it is shown that the MIBI problem can be ‘projected onto’ two dual mathematical problems in the sense that solving these problems solves the MIBI problem. In the first problem, MIBI is projected onto the problem of solving a system of multivariate homogeneous polynomial or so-called polyconjugal (polynomial-like) equations. The number of variables and the degree (of homogeneity) of the functions in the system equal the number of sensors and the order of the considered statistics, respectively. In the second problem, MIBI is projected onto the problem of solving a Multi-Matrix Generalized Eigenvalue Decomposition (MMGEVD) problem that is dual to the first problem. Possible solution approaches for those problems are described. In particular, a so-called homotopy method is used for solving the system of equations. Because of the connection between the system of homogeneous equations on the one hand, and the MMGEVD problem on the other hand, this in fact solves both mathematical problems. The theory developed in this thesis is unifying in several senses. Firstly, it is general with respect to the order of the exploited temporal structure in the sense that the mathematical problem formulation has the same structure for any considered order. Secondly, all types of statistical variability in the data, such as arbitrary order non-stationarity and non-whiteness, are exploited in a unified manner. Finally, for complex-valued signals, the conjugation pattern of the arguments of the involved cumulant functions can be chosen arbitrarily. In practice, this should be done in accordance with the characteristics of the involved signals. Our approach allows the identification of a mixing system with more sources than sensors, even with second order statistics. Depending on the number of sensors, the order and type of the exploited temporal structure, the chosen conjugation pattern, and the arrangement of the statistics in the subspace matrix, a certain maximum number of mixing matrix columns can be determined that usually exceeds the number of sensors for orders larger than one. We provide insight into the computation of the maximum number of identifiable columns as a function of the number of sensors, the employed statistical order, and the conjugation pattern. The rationale behind the work presented in this thesis is based on providing insight and highlighting the geometric and algebraic structure of the different problem formulations that are developed. Therefore, the practical problems associated with the use of estimated sensor cumulants are not discussed in great detail. Nevertheless, the theory is directly applicable to many practical scenarios. This is demonstrated by various examples, including the identification of an instantaneous mixing system for different types of signals, the separation of instantaneously mixed artificial random signals, speech signals and images, Direction Of Arrival (DOA) estimation, etcetera. Finally, the theory allows us to make trade-offs between various related quantities such as the arrangement of the statistics in the subspace matrix, maximum number of identifiable mixing matrix columns, number of sensors, exploited type(s) of temporal structure, exploited order(s) of temporal structure, employed conjugation pattern(s), number of samples required for reliable estimation of the involved statistics, and so on.