Adaptive Filters

CHAPTER 5 Linear Models

We apply the linear estimation theory of the previous two chapters to the important special case of linear models, which arises often in applications. Specifically, we now assume that the zero-mean random vectors {x,y} are related via a linear model of the form

(5.1)

for some q × p matrix H. Here v denotes a zero-mean random noise vector with known covariance matrix, R_v = E vv*. The covariance matrix of x is also assumed to be known, say, Exx* = R_x. Both {x, v} are uncorrelated, i.e., Exv* = 0, and we further assume that R_x > 0 and R_v > 0.

5.1 ESTIMATION USING LINEAR RELATIONS

According to Thm. 3.1, when R_y > 0, the linear least-mean-squares estimator of x given y is

(5.2)

Because of (5.1), the covariances {R_xy, R_y} can be determined in terms of the given matrices {H, R_X, R_v}. Indeed, the uncorrelatedness of {x, v} gives

Moreover, since R_v > 0 we get R_y > 0. The expression (5.2) for then becomes

(5.3)

This expression can be rewritten in an equivalent form ...