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#### State SpaceFilter Realization Example

The digital filter having difference equation can be realized in state-space form as follows:6.11      (6.13)

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 §E.7.8 (after we have covered transfer functions).

A general procedure for converting any difference equation to state-space form is described in §E.7. The particular state-space model shown in Eq. (5.13) happens to be called controllable canonical form, for reasons discussed in Appendix E. 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. (5.13). Similarity transformations are discussed in §E.8, and in books on linear algebra .

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. (5.12) reduces to (6.14)

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

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