Estimates for the stability of low-pass filters

A measure of stability is defined for digital filters. It is shown that the stability of low-pass filters goes to zero as the filter characteristics approach that of an ideal low-pass filter. More precisely, estimates for the rate of convergence are given. It is shown that, except for constants, the stability of a low-pass filter is no greater than the square root of the reciprocal of the logarithm of the transition bandwidth. For linear phase filters the square root may be removed. These estimates are shown to be optimal for the case of linear phase filters.