Optimal Estimation of Dynamic Systems, 2nd Edition

696 Optimal Estimation of Dynamic Systems

Consider the following discrete-time process:

x

k+1

= x

k

+ w

k

(C.71)

where w

k

is a zero-mean Gaussian white-noise process. Equation (C.71) indicates

that the change x

k+1

−x

k

is a random process. Thus, the best prediction of x for the

next period is the current value. It can be shown that the mean of a random walk

process is constant but its variance is not. Therefore, a random walk process is non-

stationary and its variance increases with k.

The Wiener process is the limiting form of the random walk. This process, denoted

by

β

(t) for a single variable, has the following pdf:

p(

β

(t)) = N(0, qt) (C.72)

where q is a constant. ...