A Systematic Approach to Adaptive Algorithms for Multichannel System Identification, Inverse Modeling, and Blind Identification

In many situations related to acoustics and data communications we are confronted with multiple signals received from a multipath mixture, e.g., the famous cocktail-party problem. A multipath mixture can be described by a mixing matrix, whose elements are the individual transfer functions between a source and a sensor. The mixing matrix is usually unknown, and so are sometimes also the source signals. Depending on the application, different parameters are of interest: the mixing matrix for system identification, the inverse mixing matrix for inverse modeling, or the source signals for system equalization. This thesis gives a systematic approach to the aforementioned problems in a multipath mixing environment. To this end, we investigate the multichannel-mixing problem and the single-channel multipath problem separately. Based on a mean-squared-error (MSE) cost function, several stochasticgradient update equations, which are related to the least-mean-square (LMS) and the recursive least-squares (RLS) algorithm, are derived for the instantaneous mixing case. Thereby the matrix-inversion lemma has shown to be a very powerful tool to transform an algorithm which estimates the mixing matrix (system identification) into an algorithm which estimates the inverse mixing matrix (inverse modeling). With the help of circulant matrices, the adaptive algorithms for the multichannel instantaneous mixing case are transformed to cope with the singlechannel multipath case. Block processing techniques are used, allowing efficient implementation of the filtering and adaptation in the frequency domain. The Fast Fourier Transform (FFT) plays a crucial role, owing to its close relationship to circulant matrices.