Stochastic Algorithms

Abstract

For non-stationary white noise ξ(n ) the covariance matrix D_ξ is diagonal with elements d _m,m =σξ²(m). The minimising function (see Eq. (4.45), Lecture 4) takes the form

$$J^\circ (w) = \sum\limits_{m = 1}^n {(y*(m) - w){\,^T}\,u(m){)^2}{\sigma _\xi }^{ - 2}} (m)$$

where σξ⁻²(m) plays the role of weights in the sum (5.1). The estimate w(n) that provides the minimum for function (5.1) satisfies the equation

$$\sum\limits_{m = 1}^n {(y*(m) - w{\,^T}\,(n)u(m))u(m){\sigma _\xi }^{ - 2}} (m) = 0$$

and the estimate w(n-1), obviously, satisfies the equation

$$\sum\limits_{m = 1}^{n - 1} {(y*(m) - w{\,^T}\,(n - 1)u(m))u(m){\sigma _\xi }^{ - 2}} (m) = 0$$

.