[sdiy] Bandwidth vs. resonance in the world of vocoders

[Richie Burnett]

Hi Magnus and all,

The design decision between the flat butterworth or the multiple resonant
chebyshev response is a tradeoff.  You are trading off pass-band smoothness for
increased initial transition-band steepness.  In other words, if you make the
decision that +/- 1dB of passband ripple is inaudible (or at least "acceptable")
then you can achieve a steeper initial rolloff at the edges of each band.  As
you correctly said, the ultimate rolloff slope well away from the centre
frequency is still determined solely by the filter order though.  (For what it's
worth, there are still sharper filters called Eliptic or Cauer filters that
place zeros just outside the passband so as to create deep notches and a very
abrupt transition from passband to stopband.)

Regarding response time or settling time of the two filter types...
I think the answer is that it simply depends on the bandwidth regardless of how
you actually achieve that bandwidth.  So if you have a 1/3 octave filter with
-3dB points at 1000Hz and 1260Hz, then the bandwidth is 260Hz, and the response
time is roughly equal to the reciprocal of the bandwidth.  In this case in the
order of 4ms.

For a vocoder the response time for the LF bands will be much slower than the HF
bands.  But this is purely a symptom of the narrower absolute bandwidth that the
LF bands must have for a logarithmically spaced filter bank.  In general systems
that have a narrow feature in the frequency domain have a spread out behaviour
in the time domain and vice versa.  So a high Q filter has a narrow bandwidth in
the freq domain but prolonged ringing in the time domain.  Conversely a wideband
gradual filter has a short transient response with little ringing in the time
domain.

If you have access to a tool like MATLAB you could easily generate models for
various filter types, (Butterworth, Cheby, Cauer) with the same design
bandwidths and then simulate their behaviours when excited by an impulse.  You
can then look at their responses and grade them subjectively in terms of
settling time.

-Richie,