[sdiy] SVF LP trapezoidal

[Andrew Simper]

Hi Tom,

The standard Chamberlin digital SVF is a slightly modified forward
euler integration, so it isn't stable. This is a trapezoidal
integration, so is stable from dc right up to just below nyquist, and
the response is also much neater - it is what people call "cramped"
since the trapezoidal integration warps infinite frequency down to
nyquist, but preserves the phase and amplitude at cutoff.

It is up to you how long you want to keep cutoff and damping fixed
for, it doesn't matter to the algorithm, just how smooth modulation
sounds. You may be able to come up with a nice approximation that for
the division term, I haven't bothered looking into it, but some
initial observations:

g = tan (pi * cutoff / samplerate);
gv1 = g / (1 + g*(g + k))
gv1 = 1 / (k + 2 / sin (2 * pi * cutoff / samplerate))

so for k = 0  you have:
gv1 = 1/2 * sin (2 * pi * cutoff / samplerate)


Andrew Simper
--
cytomic - sound music software




On 21 August 2011 17:20, Tom Wiltshire <tom at electricdruid.net> wrote:
> Hi Andrew,
>
> A few questions, if I may.
>
> How does this "trapezoidal" filter differ from other digital SVF implementations? (like the Chamberlin one, for example).
>
> Does it avoid the 1/6th sample rate limitation of that filter? (the chamberlin one explodes).
>
> Finally, when you say "If the dampling and cutoff remain fixed" how long do they have to be fixed for? Eliminating the division from the calculation of v1 would be high on my list of priorities.
>
> Thanks,
> Tom
>
>
> On 21 Aug 2011, at 04:50, Andrew Simper wrote:
>
>> Hi Guys,
>>
>> Since there is so much "dsp" stuff going on here I thought I would
>> post something as well. I'm doing some analog modelling plugins, and
>> have been investigating all possible variations on digitising
>> different non-linear circuit topologies with some promising results.
>>
>> For starters I would like to share with you the linear trapezoidal
>> integrated SVF, which has excellent stability properties even when
>> being modulated very fast, and it has a very wide range of frequency,
>> and you can also have arbitrary values of damping, so you can use it
>> for eq as well. I haven't really had a chance to study the numerical
>> properties, but I am guessing it should fare better than direct form
>> one ugliness. I will eventually write some papers on all this going
>> through the forward euler, backward euler, and trapezoidal integration
>> of the SVF, Sallen-Key, and 4 one pole cascade structures and possibly
>> more if anyone can suggest other musical filters.
>>
>> Here is the basic analysis from the SEM 1A SVF
>> http://dl.dropbox.com/u/14219031/Dsp/sem-1a-linear-svf.jpg --
>>
>> vota1 = v0 - k*v1 - v2;
>> vota2 = v1;
>> vc1 = v1 - 0;
>> vc2 = v2 - 0;
>> iota1 = gr * vota1;
>> iota2 = gr * vota2;
>> gc = s*c;
>> gr = g = 1/r;
>> ic1 = gc * vc1;
>> ic2 = gc * vc2;
>>
>> then solve the two equations which is just kirchoff's current law at each node:
>> 0 = -iota1 + ic1
>> 0 = -iota2 + ic2
>>
>> and then the output responses are just the output divided by the input
>> (setting c = 1 since it doesn't matter in this case as we just want
>> the shape of the plot):
>> low = v2/v0 = (g^2)/(g^2 + g*k*s + s^2)
>> band = v1/v0 = (g*s)/(g^2 + g*k*s + s^2)
>> high = (v0 - k*v1 - v2)/v0 = (s^2)/(g^2 + g*k*s + s^2)
>> notch = (v0 - k*v1)/v0 = 1 - (g*k*s)/(g^2 + g*k*s + s^2)
>> peak = (v0 - k*v1 - 2*v2) = (-g^2 + s^2)/(g^2 + g*k*s + s^2)
>>
>>
>> And now for the digital implementation --
>>
>> init:
>> v0 = v1 = v2 = 0;
>> v0z = v1z = v2z = 0;
>>
>> process:
>> g = tan (pi * cutoff / samplerate);
>> k = damping factor (typically in the range 2 to 0, but damping can be > 2);
>> v0 = input;
>> v1 = v1z + g * (v0 + v0z - 2*(g + k)*v1z - 2*v2z) / (1 + g*(g + k));
>> v2 = v2z + g * (v1 + v1z);
>> v0z = v0;
>> v1z = v1;
>> v2z = v2;
>>
>> outputs (the same as the analog circuit):
>> band = v1;
>> low = v2;
>> high = v0 - k*v1 - v2;
>> notch = high + low;
>> peak = high - low;
>>
>>
>> If the dampling and cutoff remain fixed then you can pre-compute the following:
>> gpk = 2*(g + k);
>> gv1 = g/(1 + g*(g+k));
>>
>> and then process becomes:
>> v1z = v1;
>> v2z = v2;
>> v0 = input;
>> v1 = v1z + gv1 * (v0 + v0z - gpk*v1z - v2z - v2z);
>> v2 = v2z + g * (v1 + v1z);
>> v0z = v0;
>>
>> which is is 7 adds and 3 mult
>>
>>
>> Andrew Simper
>> --
>> cytomic - sound music software
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