Syllabus

Review of Correlation matrix and its properties, its physical significance. Eigen analysis of matrix, structure of matrix and relation with its Eigen values and Eigen vectors. Spectral decomposition of correlation matrix, positive definite matrices and their properties and physical significance. Complex Gaussian processes. LMMSE Filters: Goal of adaptive signal processing, some application scenarios, problem formulation, MMSE predictors, LMMSE predictor, orthogonality theorem (concept of innovation processes), Wiener filter, Yule-walker equation, unconstrained Wiener filter, recursive Wiener filter (using innovation process). Kalman filter, recursions in Kalman filter, Extended Kalman filter, comparison of Kalman and Wiener filters. Adaptive Filters-Filters with recursions based on the steepest descent and Newton's method, criteria for the convergence, rate of convergence. LMS filter, mean and variance of LMS, the MSE of LMS and mis adjustment, Convergence of LMS. RLS recursions, assumptions for RLS, convergence of RLS coefficients and MSE. Lattice Filters-Filter based on innovations generation of forward and backward innovations, forward and reverse error recursions. Implementation of Weiner, LMS and RLS filters using lattice filters, Levinson Durbin algorithm, reverse Levinson Durbin algorithm. Tracking performance of the time varying filters –Tracking performance of LMS and RLS filters. Degree of stationarity and misadjustment, MSE derivations. Applications: System identification, channel equalization, noise and echo cancellation. Applications in array processing, beam forming.