02. Relation to Stochastic Processes

GUIDE: Elementary Digital Filter Theory. Relation to Stochastic Processes

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Relation to Stochastic Processes

Theorem. If a stationary random process $\ has a rational power spectral density $R(e^{j\ corresponding to an autocorrelation function $r(k)={\, then

\

is positive real.

Proof.

By the representation theorem [Astrom 1970, pp. 98-103] there exists an asymptotically stable filter $H(z)=b(z)/a(z)$ which will produce a realization of $\ when driven by white noise, and we have $R(e^{j\. We define the analytic continuation of $R(e^{j\ by $R(z) = H(z)H(z^{-1})$. Decomposing $R(z)$ into a sum ofcausal and anti-causal components gives

$\ $\ $\ (27)
  $\ $\ (28)

where $q(z)$ is found by equating coefficients of like powers of $z$ in
\

Since the poles of $H(z)$ and $R_+(z)$ are the same, it only remains to be shown that $\.

Since spectral power is nonnegative, $R(e^{j\ for all $\, and so

$\ $\ $\ (29)
  $\ $\ (30)
  $\ $\ (31)
  $\ $\ (32)

$\

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