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Relation to Stochastic Processes

Property. If a stationary random process $ \{x_n\}$ has a rational power spectral density $ Re^{j\omega}$ corresponding to an autocorrelation function $ r(k)={\cal E}\left\{x_nx_{n+k}\right\}$ , then

$\displaystyle R_+(z)\isdef \frac{r(0)}{ 2} + \sum_{n=1}^\infty r(n)z^{-n}
$

is positive real.

Proof.

By the representation theorem [19, pp. 98-103] there exists an asymptotically stable filter $ H(z)=b(z)/a(z)$ which will produce a realization of $ \{x_n\}$ when driven by white noise, and we have $ Re^{j\omega}
= H(e^{j\omega})H(e^{-j\omega})$ . We define the analytic continuation of $ Re^{j\omega}$ by $ R(z) = H(z)H(z^{-1})$ . Decomposing $ R(z)$ into a sum of causal and anti-causal components gives

\begin{eqnarray*}
R(z) = \frac{b(z)b(z^{-1})}{ a(z)a(z^{-1})}
&=&R_+(z) + R_-(z) \\
&=& \frac{q(z)}{ a(z)}+\frac{q(z^{-1})}{ a(z^{-1})}
\end{eqnarray*}

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

$\displaystyle b(z)b(z^{-1})=q(z)a(z^{-1}) + a(z)q(z^{-1}).
$

Since the poles of $ H(z)$ and $ R_+(z)$ are the same, it only remains to be shown that re$ \left\{R_+(e^{j\omega})\right\}\geq 0,\;0\leq \omega\leq \pi$ .

Since spectral power is nonnegative, $ Re^{j\omega}\geq 0$ for all $ \omega $ , and so

\begin{eqnarray*}
Re^{j\omega}&\isdef & \sum_{n=-\infty }^\infty r(n)\,e^{j\omega n}\\
&=&r(0) + 2\sum_{n=1}^\infty r(n)\,\cos(\omega n)\\
&=&2\mbox{re}\left\{R_+(e^{j\omega})\right\}\\
&\geq& 0.
\end{eqnarray*}

$ \Box$


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2015-05-22 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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