Adaptive Filters

CHAPTER 9 Transient Behavior

In order to gain further insight into the workings of steepest-descent methods, we shall continue to examine recursion (8.20), namely,

(9.1)

which pertains to the quadratic cost function (8.8). In particular, we shall now study more closely the manner by which the weight-error vector of (8.21) tends to zero. We repeat the weight-error vector recursion here for ease of reference,

(9.2)

Along with its transformed version (8.24):

(9.3)

9.1 MODES OF CONVERGENCE

To begin with, it is clear from (9.3) that the form of the exponential decay of the k–th entry of x_i, namely, x_k(i), to zero depends on the value of the mode 1 – μλ_k. For instance, the sign of 1 – μλ_k determines whether the convergence of x_k(i) to zero occurs with or without oscillation. When 0 ≤ 1 – μλ_k < 1 the decay of x_k(i) to zero is monotonic. On the other hand, when −1 < 1 − μλ_k < 0 the decay of x_k(i) to zero is oscillatory.

Example 9.1 (Exponential decay)

Consider a two-dimensional data vector u, i.e., M = 2 and R_u is 2 × 2. Assume the eigenvalues of R_u are λ_min = 1 and λ_max = 4. Then