Convergence analysis for efficient adaptive digital filtering algorithms and structures

This dissertation presents analyses of various convergence properties of efficient adaptive filtering algorithms and structures. Under the assumption that the signals involved are zero-mean and Gaussian, and further employing the independence assumption, the mean and mean-squared convergence properties of the popular least-mean-square (LMS) algorithm are first reviewed. The sign algorithm is a stochastic gradient method for adaptive filtering that requires only about half the computational complexity of the basic LMS adaptive filter. The convergence behavior of the sign algorithm is analyzed by deriving a set of nonlinear difference equations that characterizes the mean and mean-squared behavior of the coefficient misalignment vector. In particular, these results are obtained by successfully relaxing the white signal assumption that was used in earlier works.; A fairly comprehensive study of the tracking properties of the sign algorithm when it is operating in nonstationary environments is presented. By modeling the nonstationarity as that due to a random disturbance, it is shown that the long-term time-average of the mean-absolute estimation error is bounded for all finite and positive values of the convergence parameter and that there exists an optimal value of the parameter that minimizes this bound. Under a set of more stringent conditions, it is then shown that the probability distribution functions of successive coefficient misalignment vectors converge to a stationary limiting distribution when the adaptive filter is used in the application of system identification. A set of nonlinear evolution equations is also derived to describe the mean and mean-squared behavior of the filter coefficients during the adaptation and tracking when the signals involved are zero-mean and Gaussian random processes.; Finally, a convergence analysis of a multiplication-free adaptive digital filter structure is presented. Expressions for the mean and mean-squared behavior of the filter coefficients are derived. Savings in computational complexity of the multiplication-free adaptive digital filter structure are discussed by comparing them with the complexities of the LMS and the sign algorithms.; Computer simulation examples are included to demonstrate the validity of all the theoretical results presented in this dissertation.