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 models (or ``realizations''). They are especially convenient for representing filters with multiple inputs and multiple outputs (MIMO filters). An order $N$ digital filter with $p$ inputs and $q$ outputs can be written in state-space form as follows:

$${\underline{x}}(n+1) = A {\underline{x}}(n) + B \underline{u}(n)$$

$$\underline{y}(n) = C {\underline{x}}(n) + D\underline{u}(n) \protect$$ (6.12)

where ${\underline{x}}(n)$ is the length $N$ state vector at discrete time $n$, $\underline{u}(n)$ is a $p\times 1$ vector of inputs, and $\underline{y}(n)$ the $q\times 1$ output vector. $A$ is the $N\times N$ state transition matrix, and it determines the dynamics of the system (its poles, or resonant modes).

State-space models are described further in Appendix E. Here, we will only give an illustrative example and make a few observations: