ABSTRACT
CONTENTS 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Data Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Energy-Conservation Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Weighted Variance Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Mean-Square Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Mean-Square Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.7 Steady-State Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 1.8 Small-Step-Size Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 1.9 Applications to Selected Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 1.10 Fourth-Order Moment Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22 1.11 Long Filter Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 1.12 Adaptive Filters with Error Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . .23 1.13 An Interpretation of the Energy Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 1.14 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
The study of the steady-state and transient performance of adaptive filters is a challenging task because of the nonlinear and stochastic nature of their update equations (e.g., References 1 to 4). The purpose of this chapter is to provide an overview of an energy-conservation approach to study the performance of adaptive filters in a unified manner.4 The approach is based on showing that certain a priori and a posteriori errors maintain an energy balance for all time instants.5-7 When examined under expectation, this energy balance leads to a variance relation that characterizes the dynamics of an adaptive filter.10-14 An advantage of the energy framework is that it allows us to push
