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Physical Audio Signal Processing
    Digital Waveguide Theory
       Properties of Passive Impedances
          Positive Real Functions
             Relation to Stochastic Processes

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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\omeg... ...)\\ &=&2\mbox{re}\left\{R_+(e^{j\omega})\right\}\\ &\geq& 0. \end{eqnarray*}

$ \Box$

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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