Filter structures

A realizable filter must require only a finite number of computations per output sample. For linear, causal, time-Invariant filters,this restricts one to rational transfer functions of the form $$H(z)=\frac{b_0+b_{1}z^{(-1)}++b_{m}z^{-m}}{1+a_{1}z^{(-1)}+a_{2}z^{-2}++a_{n}z^{-n}}$$ Assuming no pole-zero cancellations, $H(z)$ is FIR if $\forall i, i> 0\colon a_i=0$ , and IIR otherwise. Filter structures usually implement rational transfer functions as difference equations.

Whether FIR or IIR, a given transfer function can be implemented with many different filter structures. With infinite-precision data,coefficients, and arithmetic, all filter structures implementing the same transfer function produce the same output. However, differentfilter strucures may produce very different errors with quantized data and finite-precision or fixed-point arithmetic. The computationalexpense and memory usage may also differ greatly. Knowledge of different filter structures allows DSP engineers to trade off these factors to createthe best implementation.