02. Relation to Stochastic Processes

GUIDE: Elementary Digital Filter Theory. Relation to Stochastic Processes

It appears that you are using AdBlocking software. The cost of running this website is covered by advertisements. If you like it please feel free to a small amount of money to secure the future of this website.

<< Previous page  TOC  INDEX  Next page >>

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.


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


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



<< Previous page  TOC  INDEX  Next page >>


© 1998-2017 – Nicola Asuni - Tecnick.com - All rights reserved.
about - disclaimer - privacy