The digital filter having difference equation
can be realized in state-space form as follows:^{F.5}
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
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.