I am doing a lot of audio synthesis DSP where I need to use low pass filters to shape the decay of an impulse. I understand that a one pole filter fed a discontinuity, say from a steady state of 1's to a steady state of 0's, will decay with a perfectly exponential curve. I believe the decay will be such that the signal will reach an amplitude of $1/e$ at $time = 1/(2*pi*f)$.
This makes conceptualizing one pole low pass impulse modeling very easy. But I am trying to understand how other filter orders will do the same.
I understand that a second order filter can be approximated running two one pole filters in sequence (ie. each will attenuate by 6 dB/oct, leading to a 12 dB/oct curve). If that is the case, then I would expect the step response of a second order filter without resonance to behave the same as a one pole filter and remain exponential, only that much faster.
So perhaps the time to 1/e would be:
$time = 1/(2*pi*f)^2$
Would that be correct?
I asked here about a lower slope filter and no one has replied so I am guessing it is not common knowledge: A one-pole LPF (6 dB/oct) has a step response to $1/e$ amplitude of $time = 1/(2∗pi∗f)$. What would the response time be of a 3 dB/oct filter?
But I presume the same principle would then apply. A 3 dB/oct low pass filter would have a time constant of:
$time = 1/(2*pi*f)^{0.5}$
And if this is all correct, then we could say the time constants of any non-resonant low pass filter would be roughly:
$time = 1/(2*pi*f)^{filter-order}$
What do you think? Does all this sound correct? If so, there is no difference between low pass filtering with any order of non-resonant filter, as they can all be set up to create the same outcome with different parameters.
Lastly, then the remaining question would be: What does the step response of a resonant second order low pass filter look like, again going from steady 1's to steady 0's? I presume the resonance ruins the precisely exponential decay, but in what manner? Does it create a more compressed curve? Will it resonate creating a wobbling of the output? Will it dip below zero as it tries to settle out?
I tried testing it with a resonant second order LPF but I was just getting almost random maximum amplitudes coming out. Very unpredictable. I'm not sure if I did something wrong or that's to be expected from the resonance.
Thanks for any help understanding all this.