Adaptive Filters

CHAPTER 7 Kalman Filter

The theory developed in Chapters 3–5 on linear estimation can be used to introduce one of the most celebrated tools in linear least-mean-squares estimation theory, namely the Kalman filter. The filter has an intimate relation with adaptive filter theory, so much so that a solid understanding of its functionality can suggest extensions of classical adaptive schemes. A demonstration to this effect will be given later in Chapter 31, after we have progressed sufficiently enough in our treatment of adaptive filters. At that stage, we shall tie up the Kalman filter with adaptive least-squares theory and show how it can motivate useful extensions. For the time being, it suffices to treat the material in this chapter as simply an application of linear least-mean-squares estimation theory.

7.1 INNOVATIONS PROCESS

Consider two zero-mean random variables {x, y}. We already know from Thm. 3.1 that the linear least-mean-squares estimator of x given y is , where K_o is any solution to the normal equations

(7.1)

In the sequel we assume that R_y is positive-definite so that K_o is uniquely defined as .

Usually, the variable y is vector-valued, say, y = col{y₀, y₁, …, y_N}, ...