Low-transient high-pass filters

A new and simple method is given for transforming a low-pass filter (LPF) to a high-pass filter (HPF) whose impulse and step-function responses have the same quality (in terms of overshoot, delay, settling time, transition time, etc.) as that of the LPF. The transfer functions, impulse, and step responses of these two filters are related by
H_h (s) = 1 - H_l (s), h_h (t) = \delta(t) - h_l (t)
, and
s_h (A) = 1 - s_l (t)
, respectively. This immediately enables us to define new classes of low-transient HPFs having little or no overshoot. One new class is obtained by transforming all-pole Bessel filters, but other low-pass families can be transformed to obtain similar results. It is also shown that if the quality of an all-pole LPF transient response is to be retained by the HPF, then the maximum attenuation rate of the HPF magnitude response near
\omega
= 0 is 6 dB/octave. These results are additionally beneficial for designing low-transient narrow-band notch filters.