## 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:

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 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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Time Domain Filter Estimation

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Inverse Filters