musicdsp.org source code archive - WSInf

Butterworth (click this to go back to the index)
Type : LPF 24dB/Oct
References : Posted by Christian[at]savioursofsoul[dot]de
Code :
First calculate the prewarped digital frequency:
K = tan(Pi * Frequency / Samplerate);
Now calc some intermediate variables: (see 'Factors of Polynoms' at
http://en.wikipedia.org/wiki/Butterworth_filter, especially if you want a higher order like 48dB/Oct)
a = 0.76536686473 * Q * K;
b = 1.84775906502 * Q * K;
K = K*K; (to optimize it a little bit)
Calculate the first biquad:
A0 = (K+a+1);
A1 = 2*(1-K);
A2 =(a-K-1);
B0 = K;
B1 = 2*B0;
B2 = B0;
Calculate the second biquad:
A3 = (K+b+1);
A4 = 2*(1-K);
A5 = (b-K-1);
B3 = K;
B4 = 2*B3;
B5 = B3;
Then calculate the output as follows:
Stage1 = B0*Input + State0;
State0 = B1*Input + A1/A0*Stage1 + State1;
State1 = B2*Input + A2/A0*Stage1;
Output = B3*Stage1 + State2;
State2 = B4*Stage1 + A4/A3*Output + State2;
State3 = B5*Stage1 + A5/A3*Output;
Comments
from : Christian@savioursofsoul.de
comment : Once you figured it out, it's even possible to do higher order butterworth shelving filters. Here's an example of an 8th order lowshelf.
First we start as usual prewarping the cutoff frequency:
K = tan(fW0*0.5);
Then we settle up the Coefficient V:
V = Power(GainFactor,-1/4)-1;
Finally here's the loop to calculate the filter coefficients:
for i = 0 to 3
{
cm = cos(PI*(i*2+1) / (2*8) );
B[3*i+0] = 1/ ( 1 + 2*K*cm + K*K + 2*V*K*K + 2*V*K*cm + V*V*K*K);
B[3*i+1] = 2 * ( 1 - K*K - 2*V*K*K - V*V*K*K);
B[3*i+2] = (-1 + 2*K*cm - K*K - 2*V*K*K + 2*V*K*cm - V*V*K*K);
A[3*i+0] = ( 1-2*K*cm+K*K);
A[3*i+1] = 2*(-1 +K*K);
A[3*i+2] = ( 1+2*K*cm+K*K);
}
from : scoofy@inf.elte.hu
comment : Hmm... interesting. I guess the phase response/group delay gets quite funky, which is generally unwanted for an equalizer.
I think the 1/ is not necessary for the first B coefficient! (of course you divide all the other coeffs with the inverse of that coeff at the end...)
I guess the next will be Chebyshev shelving filters ;)
BTW did you check whether my 4 pole highpass Butterworth code is correct?
Peter
from : Christian@savioursofsoul.de
comment : The 1/ is of course an error here. It's left of my own implementation, where I divide directly. Also I think A and B is exchanged.