Blind Signal Separation by Second Order Statistics

Doktorsavhandling, 1998

The problem of separating uncorrelated signals from equally many observed mixtures is considered in this thesis. Each observed signal is modeled as a sum of original signals filtered through unknown linear filters. Various kinds of mixing filters are considered: Finite Impulse Response (FIR) and Auto Regressive Moving Average (ARMA), causal and non-causal, one and two-dimensional. A separation structure is used in order to extract the original signals from the observed signals. Separation structures are presented both for a Two Input Two Output (TITO) scenario and for a Multi Input Multi Output (MIMO) scenario. The proposed separation structure is parameterized in such a way that the minimum number of parameters are used. Two types of algorithms, both based on second order statistics, are presented in order to estimate the coefficients of the filters in the separation structure. The first type of algorithms are based on minimizing a criterion which is the sum over different lags of squared cross-correlations of the separation structure output. Expressions for the gradient and Hessian of the criterion are derived for a MIMO FIR channel system. The second type of algorithm is based on a system identification approach, using the Recursive Prediction Error Method (RPEM). The Cramer Rao Lower Bound is derived for the signal separation problem. This bound is the lowest possible variance achievable of the estimated parameters, given Gaussian signals. A compact and computationally appealing formula for this bound is presented. The bound is computed for some scenarios and compared with the results from signal separation algorithms. The signals to be separated can also be multidimensional, e.g. images. In this case non-causal and two-dimensional filters are used. In a system identification approach, applied to a TITO scenario, both the channel filters and the color of the sources are estimated. It is shown that the TITO system, under various weak conditions, is identifiable using second order statistics only. Results from a project of collecting and separating real world signals are presented. Both the abovementioned signal separation methods are used. When using the algorithm based on squared crosscorrelations, a regularization term is added which improve the Signal to Noise Ratio (SNR) of the separated signals.