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State Space Realization

Above, we used a matrix multiply to represent convolution of the filter input signal with the filter's impulse response. This only works for FIR filters since an IIR filter would require an infinite impulse-response matrix. IIR filters have an extensively used matrix representation called state space form (or ``state space realizations''). They are especially convenient for representing filters with multiple inputs and multiple outputs (MIMO filters). An order $ N$ digital filter with $ p$ inputs and $ q$ outputs can be written in state-space form as follows:

$\displaystyle {\underline{x}}(n+1)$ $\displaystyle =$ $\displaystyle A {\underline{x}}(n) + B \underline{u}(n)$  
$\displaystyle \underline{y}(n)$ $\displaystyle =$ $\displaystyle C {\underline{x}}(n) + D\underline{u}(n)
\protect$ (F.4)

where $ {\underline{x}}(n)$ is the length $ N$ state vector at discrete time $ n$ , $ \underline{u}(n)$ is a $ p\times 1$ vector of inputs, and $ \underline{y}(n)$ the $ q\times 1$ output vector. $ A$ is the $ N\times N$ state transition matrix, and it determines the dynamics of the system (its poles, or resonant modes).

State-space models are described further in Appendix G. Here, we will only give an illustrative example and make a few observations:

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University