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State Space Filters

An important representation for discrete-time linear systems is the state-space formulation

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

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

The state-space representation is especially powerful for multi-input, multi-output (MIMO) linear systems, and also for time-varying linear systems (in which case any or all of the matrices in Eq.(G.1) may have time subscripts $ n$ ) [37]. State-space models are also used extensively in the field of control systems [28].

An example of a Single-Input, Single-Ouput (SISO) state-space model appears in §F.6.

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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