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Markov Parameters

The impulse response of a state-space model is easily found by direct calculation using Eq. (E.1):

\mathbf{h}(0) &=& C {\underline{x}}(0) + D\,\underline{\delta}...
... B\\ [5pt]
\mathbf{h}(n) &=& C A^{n-1} B, \quad n>0

Note that we have assumed $ {\underline{x}}(0)=0$ (zero initial state or zero initial conditions). The notation $ \underline{\delta}(n)$ denotes a $ q\times q$ matrix having $ \delta (n)$ along the diagonal and zeros elsewhere.E.2

The impulse response of the state-space model can be summarized as

$\displaystyle \fbox{$\displaystyle \mathbf{h}(n) = \left\{\begin{array}{ll} D, & n=0 \\ [5pt] CA^{n-1}B, & n>0 \\ \end{array} \right.$}$ (E.2)

The impulse response terms $ C A^n B$ for $ n\geq 0$ are known as the Markov parameters of the state-space model.

Note that each sample of the impulse response $ \mathbf{h}(n)$ is a $ p\times q$ matrix.E.3 Therefore, it is not a possible output signal, except when $ q=1$. A better name might be ``impulse-matrix response''. In §E.4 below, we'll see that $ \mathbf{h}(n)$ is the inverse z transform of the matrix transfer-function of the system.

Given an arbitrary input signal $ \underline{u}(n)$ (and zero intial conditions $ {\underline{x}}(0)=0$), the output signal is given by the convolution of the input signal with the impulse response:

$\displaystyle \underline{y}_u(n) = (\mathbf{h}\ast \underline{u})(n) = \left\{\...
... \sum_{m=0}^nCA^{m-1}B\underline{u}(n-m), & n>0 \\ \end{array} \right. \protect$ (E.3)

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (August 2006 Edition).
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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