Adaptive Filters

CHAPTER 31 Kalman Filtering and RLS

There is a close relation between regularized least-squares problems, as studied in the previous chapter, and linear least-mean-squares estimation problems, as studied in Part II (Linear Estimation). Although the former class of problems deals with deterministic variables and the latter class of problems deals with random variables, both classes turn out to be equivalent in the sense that solving a problem from one class also solves a problem from the other class and vice-versa.

31.1 EQUIVALENCE IN LINEAR ESTIMATION

Stochastic Problem

Let x and y be zero-mean random variables that are related via a linear model of the form

(31.1)

for some known matrix H and where v denotes a zero-mean random noise vector with known covariance matrix, say, R_v = Evv*. The covariance matrix of x is also known and denoted by Exx* = R_x. Both {x,v} are uncorrelated, i.e., Exv* = 0, and we further assume that R_x > 0 and R_v > 0. We established in Thm. 5.1 that the linear least-mean-squares estimator of x given y is

(31.2)

and that the resulting minimum mean-square error matrix is

(31.3)

Deterministic Problem

Now consider instead deterministic variables {x, y} and ...