Non-Gaussian, Non-stationary and Nonlinear Signal Processing Methods - with Applications to Speech Processing and Channel Estimation

The Gaussian statistic model, despite its mathematical ele gance, is found to be too factitious for many real world signals, as manifested by its uns atisfactory performance when applied to non-Gaussian signals. Traditional non-Gau ssi n signal processing techniques, on the other hand, are usually associated with h igh complexities and low data efficiencies. This thesis addresses the problem of opti mum estimation of nonGaussian signals in computation-efficient and data-efficie nt ways. The approaches that we have taken exploit the high temporal-resolution non-sta tion rity or the underlying dynamics of the signals. The sub-topics being treated inclu de: joint MMSE estimation of the signal DTFT magnitude and phase, high temporal-resol ution Kalman filtering, blind de-convolution and blind system identification, and o ptimum non-linear estimation. Applications of the proposed algorithms to speech enh ancement, non-Gaussian spectral analysis, noise-robust spectrum estimation, and bli channel equalization are demonstrated. The thesis consists of two parts, the Introduction and the Pa pers. The Introduction gives background information of the problems at hand, s tate the motivation of approaches taken, summarizes the state-of-the-art in lite ratur , and describes our contributions briefly. The Papers presents our contributions i n the form of published papers. The first part of the Papers (paper A and B) deals with the impor tance of phase in non-Gaussian signal estimation. Joint MMSE estimators of b oth magnitude spectra and phase spectra are developed. Application to the enhancemen t of noisy speech signals results in clearer sounds and higher SNR than frequency doma in MMSE estimators. Here the non-Gaussianity of the speech signal is modeled by t he linearity in the phase spectrum, and is enhanced by the joint estimator. This is in c ontrast to the spectral domain MMSE estimator (e.g., the Wiener filter), which is zer o-phase. The second part of the Papers (paper C and D) attacks the non-G aussian estimation problem with a purely temporal domain approach. It is recogn ized that a temporaldomain high-resolution non-stationary LMMSE estimator is able to extract structures in both magnitude and phase spectra at a lower complexity. Fo r speech signals, the non-Gaussianity is represented by an excitation sequence w ith a rapidly varying vari-