**<< Previous page TOC INDEX Next page >>**

## Relation to Stochastic Processes

Theorem.If a stationary random process has a rational power spectral density corresponding to an autocorrelation function , then

is positive real.

Proof.By the representation theorem [Astrom 1970, pp. 98-103] there exists an asymptotically stable filter which will produce a realization of when driven by white noise, and we have . We define the analytic continuation of by . Decomposing into a sum ofcausal and anti-causal components gives

(27) (28)

where is found by equating coefficients of like powers of inSince the poles of and are the same, it only remains to be shown that .

Since spectral power is nonnegative, for all , and so

(29) (30) (31) (32)