## 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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Inverse Filters