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 digital filter with inputs and outputs can be written in state-space form as follows:

where is the length state vector at discrete time , is a vector of inputs, and the output vector. is the 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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Subsections

• State Space Filter Realization Example

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written by 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.