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 Hz= b 0 + b 1 z-1+…+ b m z-m1+ a 1 z-1+ a 2 z-2+…+ a n z-n H z b 0 b 1 z … b m z m 1 a 1 z a 2 z -2 … a n z n Assuming no pole-zero cancellations, Hz H z is FIR if ∀i,i>0: a i =0 i i 0 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, different filter strucures may produce very different errors with quantized data and finite-precision or fixed-point arithmetic. The computational expense and memory usage may also differ greatly. Knowledge of different filter structures allows DSP engineers to trade off these factors to create the best implementation.