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State Space Filters

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

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

where $ {\underline{x}}(n)$ is the length $ N$ state vector at discrete time $ n$, $ \underline{u}(n)$ is a $ 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,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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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