Adaptive Filters

CHAPTER 15 Energy Conservation

In Part III (Stochastic–Gradient Algorithms) and its problems, we developed stochastic gradient approximations for several steepest-descent methods. The approximations were obtained by replacing exact covanance and cross-correlation quantities by instantaneous estimates. The resulting algorithms operate on actual data realizations and they lead to adaptive filter implementations. However, stochastic approximations introduce gradient noise and, consequently, the performance of adaptive filters will degrade in comparison with the performance of the original steepest-descent methods.

The purpose of this chapter, and of the subsequent chapters in this part (Mean-Square Performance) and in Part V (Transient Performance), is to describe a unifying framework for the evaluation of the performance of adaptive filters. This objective is rather challenging, especially since adaptive filters are, by design, time-variant, stochastic, and nonlinear systems. Their update recursions not only depend on the reference and regression data in a nonlinear and time-variant manner, but the data they employ are also stochastic in nature. For this reason, the study of the performance of adaptive algorithms is a formidable task, so much so that exact performance analyses are rare and limited to special cases. It is customary to introduce simplifying assumptions in order to make the performance analyses more tractable. Fortunately, most assumptions tend to lead to reasonable ...