High-pass filter step-response energy: A new performance measure

A new performance criterion for comparing high-pass filters (HPF\´s) is proposed; namely, , the unit-step-response energy. Increased interest in narrow-bandwidth notch filters with minimum energy of the envelope response to the center frequency amplitudemodulated by a step function prompted this choice, for this envelope energy is proportional to of the equivalent HPF. Initially, it is shown that is identical to the prototype low-pass filter (LPF) impulse-response energy, thus relating the transient characteristics of the two filters. It is further shown that the so-called "unknown energy" referred to in the literature is actually . A new expression is derived for the step-response energy of a lossless passive singly terminated network, whose input impedance is minimum reactance. This formula simplifies for HPF\´s so that is easily computed and then plotted for well-known responses. The noise bandwidth of an LPF is shown to be proportional to ; thus the curves for also yield the noise bandwidths. This consolidates much of the scattered information on this subject. It is then established that of all second-order systems, the Chebyshev filters possess the lowest value of ; in fact, the energy decreases as the passband ripple increases.