[music-dsp] biquad filters formulae

[Nicolas Lhommet]

Hello everybody,

I looking for some good explanations about differents biquad filters
formulae and implementations.
Pure Data and Jmax are two realtime audio synthesis software, similar to
Max/MSP :

biquad~ object formula in Jmax is :
y[n] = c1*x[n] + c2*x[n-1] + c3*x[n-2] - c4*y[n-1] - c5*y[n-2]

This is the implementation proposed by Robert Bristow-Johnson on this page :
http://www.harmony-central.com/Computer/Programming/Audio-EQ-Cookbook.txt

biquad~  object formula in Pure Data is explained in documentation like this
:
y(n) = ff1 * w(n) + ff2 * w(n-1) + ff3 * w(n-2)
w(n) = x(n) + fb1 * w(n-1) + fb2 * w(n-2)

and the code looks like this :
for (i = 0; i < n; i++)
{
  float output =  *in++ + fb1 * last + fb2 * prev;
  *out++ = ff1 * output + ff2 * last + ff3 * prev;

  prev = last;
  last = output;
}

What is the mathematical relation between these two formulae (and so between
Pure Data and Jmax biquad object coefficients) ?

Thanks for help !

nicolas



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