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Zero-State Impulse Response
(Markov Parameters)

Linear State-Space Model:

\begin{eqnarray*}
\underline{y}(n) & = & \mathbf{C}\underline{x}(n) + \mathbf{D}\underline{u}(n)\nonumber \\
\underline{x}(n+1) & = & \mathbf{A}\underline{x}(n) + \mathbf{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}$ and $ \underline{u}=\mathbf{I}_p\delta(n) =$   diag$ (\delta(n),\ldots,\delta(n))$ . Then

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



Subsections
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Download StateSpace.pdf
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Download StateSpace_4up.pdf

``Introduction to State Space Models'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2015-04-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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