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

[Magnus Danielson]

Hi Richie,

On 13/07/11 17:29, Richie Burnett wrote:
> 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.)

Yes, that the trivial stuff. I completely ducked the Elliptic/Cauer 
filters. You can drop in some sharp notches at each end of the bandpass 
section, but you still want at least one or two poles at 0 and infinity 
for the overall roll-off properties, so you will not gain too much by 
doing it, you are still looking on at least a 6th degree system.

> 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.

You can be fooled to beleive that... yes.

But my point is that the response time do depend on the filter type. 
This is however not very often documented, but I do have suitable 
documentation... at home where I can't get to it right now.

The overall property of scaling relative bandwidth is there, but there 
is a scaling factor due to filter-type. For instance, Butterworth and 
Bessel-Thompson filters does not have the same response-time for the 
same degree, center frequency and bandwidth.

> 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.

Yes, yes... that's the obvious stuff.

> 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.

I've not had time to cook up a simulation displaying this, even if I 
made some stabs at it.

I already have the basic knowledge that the difference is there in the 
back of my head, so I wanted to bring this up to see if it was a know 
issue or not.

I know how to simulate this so I will see how it turns out. I will see 
if I can make a stab at it on the analysis approach too. Always nice to 
make well-founded proofs.

Cheers,
Magnus