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Zero-State Impulse Response
(Markov Parameters)

Linear State-Space Model:

\begin{eqnarray*}
\underline{y}(n) & = & C \underline{x}(n) + D\underline{u}(n)\nonumber \\
\underline{x}(n+1) & = & A \underline{x}(n) + B \underline{u}(n)
\end{eqnarray*}

The zero-state impulse response of a state-space model is easily found by direct calculation: Let $ \underline{x}(0)\isdef \underline{0}$ . Then

\begin{eqnarray*}
{\mathbf{h}}(0) &=& C \underline{x}(0) B + D = D\\ [5pt]
\underline{x}(1) &=& A\, \underline{x}(0) + B\,\underline{\delta}(0) = B\\
{\mathbf{h}}(1) &=& C B\\ [5pt]
\underline{x}(2) &=& A\,\underline{x}(1) + B\,\underline{\delta}(1) = AB\\
{\mathbf{h}}(2) &=& C A B\\ [5pt]
\underline{x}(3) &=& A\,\underline{x}(1) + B\,\underline{\delta}(1) = A^2B\\
{\mathbf{h}}(3) &=& C A^2 B\\ [5pt]
&\vdots&\\
{\mathbf{h}}(n) &=& C A^{n-1} B, \quad n>0
\end{eqnarray*}

Thus, 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.$}
$


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``Introduction to State Space Models'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2015-03-29 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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