### State Space Filter Realization Example

The digital filter having difference equation

^{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

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:

*zero-input response*of the filter,

*i.e.*, .) Similarly, setting to in Eq.(F.6) yields

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Effect of Measurement Noise

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Lossless Analog Filters