**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Introduction to Digital
Filters with Audio Applications", by Julius O. Smith III,
Copyright © *2017-11-26* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## 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 in

Since 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)