# [PD] convert reson filter bandwidth to decay time

Libero Mureddu libero.mureddu at gmail.com
Wed Jan 23 23:25:20 CET 2008

```Hi Charles,
thanks, this is exactly the solution I was looking for.
Thanks a lot for the detailed math explanation too.
ciao

Libero

On Jan 22, 2008 9:12 PM, Charles Henry <czhenry at gmail.com> wrote:
> It depends upon the order of the filter... by resonant filter, I
> assume you mean a two-pole bandpass filter.  Here's the math for
> converting between bandwidth and exponential decay
>
> Take a function, g(t)=1000^-(t/r)
> where r is the -60 dB ring time.  I used a base of 1000, because a
> factor of 1000 = 60 dB
>
> g(t)=e^-(t*ln(1000)/r)
>
> Now we take a one-sided fourier transform of this function:
>
> G(f)=integral(0, inf;  e^(-2*pi*i*f*t)*e^-|t*ln(1000)/r|)
> G(f)=integral(0, inf;  e^((-2*pi*i*f-ln(1000)/r)*t)
> G(f)=1/(-2*pi*i*f-ln(1000)/r) * e^((-2*pi*i*f-ln(1000)/r)*t), eval at t=0, t=inf
>
> G(f)= - 1/(-2*pi*i*f-ln(1000)/r)
> G(f)=1/(2*pi*i*f+ln(1000)/r)
>
> G(f)=r/ln(1000)  / [2*pi*i*f*r/ln(1000) + 1]
>
> now, it's just a simple matter of finding the -3 dB points to find
> bandwidth of this function and interpreting the result of this
> function...
>
> |G(f)|^2=(r/ln(1000))^2 / [1 + (2*pi*f*r/ln(1000))^2]
>
> 0.5 = 1 / [1 + (2*pi*f*r/ln(1000))^2]
> (2*pi*f*r/ln(1000))^2 = 1
>
> f = +/-  ln(1000)/(2*pi*r)
> bandwidth
> bw= 2 * ln(1000)/(2*pi*r)
>
> conversions:
> bw = ln(1000) / (pi*r)
> r = ln(1000) / (pi*bw)
> r is in seconds and bw is in Hz
>
> Try out these relations and see if they work.
>
> Chuck
>
>
> On Jan 22, 2008 12:11 PM, Libero Mureddu <libero.mureddu at gmail.com> wrote:
> > Hi,
> > I have  a simple patch with a [click~] object connected to a [reson~]
> > filter. I'd like to know how it is possible to convert the bandwith
> > parameter of the filter to decay lenght (in milliseconds).
> > In SuperCollider there is a particular version of the "resonz" filter
> > called "ringz", here is the description:
> > Ringz.ar(in, freq, decaytime, mul, add)
> >
> > This is the same as Resonz, except that instead of a resonance
> > parameter, the bandwidth is
> > specified in a 60dB ring decay time.
> >
> > My knowledge of filters is too little to be able to convert the
> > between the two parameters by myself, any help appreciated!
> >
> > thanks,
> >
> > libero
> >
> > _______________________________________________
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> >
>

--
Libero Mureddu
Vanha Viertotie, 21 as 417
00350 Helsinki
Finland
http://webusers.siba.fi/~limuredd/
http://www.myspace.com/liberomureddu

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