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.
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
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
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