Translations:Delay in general/33/en

The inverse of a minimum-delay filter A(Z) is $A^{-{\rm {l}}}\left(Z\right)={\rm {l}}lA\left(Z\right)$, which also is a minimum-delay filter. An important type of stable causal IIR (infinite-impulse-response) filter is the recursive filter, which has its transfer function given by the rational function $H\left(Z\right)=B\left(Z\right)/A\left(Z\right)$ with the provision that the denominator polynomial A(Z) is a minimum-delay polynomial (i.e., all the roots of the polynomial lie outside of the unit circle). This provision ensures that the recursive filter is stable (Treitel and Robinson, 1964). The locations of the zeros of the numerator polynomial B(Z) do not need to be specified for the recursive filter to be stable. However, if it is further specified that the numerator polynomial B(Z) is a minimum-delay polynomial, the recursive filter is a minimum-delay filter.