TABLE OF CONTENTS In this section we expand upon the topics briefly introduced in Sections 12.1.3 through 12.1.5, that is, LMS adaptation algorithms, their convergence properties and excess MSE. The effect of finite precision in digital implementations is also discussed. The results of this section are applicable to other forms of adaptive filters, for example, an echo canceler (see also Chapter 4), with appropriate reinterpretation of terms.
The deterministic gradient algorithm, which is of little practical interest, is presented first to set the stage for a discussion of the LMS or stochastic gradient algorithm.
When the equalizer input covariance matrix A and the crosscorrelation vector ? (see Section 12.2.3) are known, one can write the MSE as a function of A, ?, and the equalizer coefficient vector c k according to
Taking the gradient of the MSE with respect to c k, a deterministic (or exact) gradient algorithm for adjusting c k to minimize ? k. can be written as
where ? is the step-size parameter. (This update procedure is also known as the steepest descent algorithm). Using the fact that ?= A c opt, from (12.11) or (12.25), and subtracting c opt both sides of (12.44), we obtain
In order to analyze the stability and convergence of the deterministic gradient algorithm, we use coordinate transformation to diagonalize the set of equations (12.45) so that
where the transformed coefficient deviation vector is...
