### 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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