#### Relation to Stochastic Processes

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 [19, 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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