Undergrad Predicting the decay time of a resonant bandpass filter

[mikejm]

In a typical resonant bandpass filter, resonance is set as none at 0 and full at 1, such that the filter rings infinitely at 1 and not at all at 0.

If there is a two-pole resonant bandpass filter, with an impulse excitation to a maximum amplitude of "1" and a resonance setting between 0 and 1:

What is the mathematical equation that would allow you to predict the decay time of the resonant filter based on its resonance setting? For example, to an arbitrary level of "1/e"? ie. To 36.7879% its original amplitude?

Thanks.

 

[mikejm]

This issue is solved. Thanks. Can the thread be closed? I can no longer edit or delete my OP.