## 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:

 (F.4)

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:

### State SpaceFilter Realization Example

The digital filter having difference equation

can be realized in state-space form as follows:F.5
 (F.5)

Thus, is the vector of state variables at time , is the state-input gain vector, is the vector of state-gains for the output, and the direct-path gain is .

This example is repeated using matlab in §G.7.8 (after we have covered transfer functions).

A general procedure for converting any difference equation to state-space form is described in §G.7. The particular state-space model shown in Eq.(F.5) happens to be called controller canonical form, for reasons discussed in Appendix G. The set of all state-space realizations of this filter is given by exploring the set of all similarity transformations applied to any particular realization, such as the control-canonical form in Eq.(F.5). Similarity transformations are discussed in §G.8, and in books on linear algebra [58].

Note that the state-space model replaces an th-order difference equation by a vector first-order difference equation. This provides elegant simplifications in the theory and analysis of digital filters. For example, consider the case , and , so that Eq.(F.4) reduces to

 (F.6)

where is the transition matrix, and both and are signal vectors. (This filter has inputs and outputs.) This vector first-order difference equation is analogous to the following scalar first-order difference equation:

The response of this filter to its initial state is given by

(This is the zero-input response of the filter, i.e., .) Similarly, setting to in Eq.(F.6) yields

Thus, an th-order digital filter looks like'' a first-order digital filter when cast in state-space form.

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Inverse Filters