Research On The Steady-state Performance For Adaptive Filters

The performance of an adaptive filter is generally measured in terms of its transient behavior and its steady-state behavior. The former provides information about the stability and the convergence rate of an adaptive filter, whereas the latter provides information about the mean square error (MSE) of the filter once it reaches steady state. In general, the steady state performance can be obtained as the limiting case of a transient analysis, but it encounters some difficulties. Firstly, transient analyses tend to be laborious, especially for adaptive filters with nonlinear estimation error signals. Secondly, transient analyses tend to require some simplifying assumptions, which at times can be restrictive. Finally, it is common in the literature to perform transient analyses of different adaptive filters separately by studying each nonlinear update form individually. These points motivate the development of a unified approach in the thesis to the steady-state performance for a large class of adaptive filters that bypass several of the difficulties encountered in obtaining steady-state results as the limiting case of a transient analysis.In the thesis, based on the real form and complex form Taylor series expansions and the energy conservation relation during two successive iteration update, the steady-state MSE and tracking performance analyses for the adaptive filters with nonlinear estimation error signals are studied. Firstly, based on separation principle, some unified closed analytical expressions for the steady-state MSE, the tracking MSE, the optimal step-size and the minimum MSE, respectively, are derived for adaptive filters with nonlinear estimation error signals, and the restrictive conditions for these expressions are also given. The steady-state performance analyses for some common algorithms, such as least mean square (LMS) algorithm, least mean forth (LMF) algorithm and least mean mixed norm (LMMN) algorithm are special cases for these general expressions. In addition, the extensive computational simulations for adaptive filters validate these theoretical resutls. Secondly, rather using separation principle, the steady-state performance analyses can be obtained while the regressors are white white Gaussian noise, and the restrictive conditions for these expressions are also given. Comparing with the results obtained by separation principle, it can be found that only small distinction exists. In addition, in view of small step-size enough, their expressions are same. Thirdly, the steady-state performance analyses for least mean p-order (LMP) algorithm with different parameter p and LMMN algorithm are presented in both Gaussian noise environments and uniformly distributed noise environments. The experimental and theoretical results are matched reasonable well. Finally, comparison of the tracking performance between the LMP algorithm and LMS algorithm are implemented both in Gaussian noise environments and in uniformly distributed noise environments. Simulations show the superiority of the LMS algorithm over LMP with parameter p>2 for tracking nonstationary systems in Gaussian noise environments, and the superiority of the LMP algorithm over LMS in uniformly distributed noise environments.Most of modulation signals, such as QPSK, 8PSK, 4QAM, 8QAM, 16QAM, 64QAM, 256QAM, and so on, all have symmetrical constellations. In the thesis, based on the constellation symmetric characteristics of the modulated signals, some expressions for the steady-state EMSE of the Bussgang algorithms (BA) are derived without appealing to the circularity assumption, and are proved that they are same as the results derived by A. Goupil and J. Palicot in [127].The concurrent constant modulus algorithm and decision-directed scheme (CCMA+SDD) blind equalization achieves a considerable improvement in equalization performance over the constant modulus algorithm (CMA) for high-order quadrature amplitude modulation (QAM) channels, such as faster convergence and lower steady-state MSE. However, the actual steady-state performance has largely been left undone because of its concurrent adaptive filter structure and the complexity to analyze the time evolution of the weight update estimation error that arises from the nonlinear estimation error. In the thesis, an equivalent blind equalizer described by a hybrid cost function of CMA and SDD scheme is proposed. Then, based on the previous steady-state EMSE expressions for BA, some closed analytical expressions for EMSE for CCMA+SDD algorithm are given, for real-valued cases and for complex-valued cases, respectively. Extensive computational simulations for CCMA+SDD algorithm validate these theoretical results.