**Property. **
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 [#!AstromB!#, 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 of causal and anti-causal components gives

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 re .

Since spectral power is nonnegative, for all , and so

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University