Chalmers University of Technology G Oteborg Sweden on Signal Separation by Second Order Statistics on Signal Separation by Second Order Statistics

The problem of separating two or more uncorrelated signals from equally many observed mixtures is considered in this thesis. The observed signals are modeled as a sum of original signals ltered through linear lters. Various kinds of mixing lters 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. Two types of algorithms, both based on second order statistics, are presented in order to estimate the coe cients of the lters in the separation structure. The rst type of algorithms are based on minimizing a criterion which is the sum over di erent lags of squared cross-correlations of the separation structure output. The second type of algorithm is based on a system identi cation approach, using the Recursive Prediction Error Method (RPEM). The Cram er 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, eg. images. In this case non-causal and two-dimensional lters are used. In a system identi cation approach, applied to a TITO scenario, both the channel lters and the color of the sources are estimated. It is shown that the TITO system, under various weak conditions, is identi able using second order statistics only.