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Transfer Function of a State Space Filter

The transfer function can be defined as the $ z$ transform of the impulse response:

$\displaystyle H(z) \isdef \sum_{n=0}^{\infty} h(n) z^{-n}
= D + \sum_{n=1}^{\infty} \left(C A^{n-1} B \right) z^{-n}
= D + z^{-1}C \left[\sum_{n=0}^{\infty} \left(z^{-1}A\right)^n\right] B
$

Using the closed-form sum of a matrix geometric series,G.4we obtain

$\displaystyle \fbox{$\displaystyle H(z) = D + C \left(zI - A\right)^{-1}B.$} \protect$ (G.5)

Note that if there are $ p$ inputs and $ q$ outputs, $ H(z)$ is a $ p\times q$ transfer-function matrix (or ``matrix transfer function'').



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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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