Digital Music Programming II: Biquad filter



A biquad filter is a linear filter, defined by the following flowgraph or difference equation:

The name biquad is an abbreviation of the word bi-quadratic because the transfer-function (which can be derived from the difference equation or flowgraph) contains two quadratic functions:

Here is an example set of values for the filter coefficients: 

       a0 =  1
   a1 =  0
   a2 = -1
   b1 =  0.1
   b2 =  0.9

These values will generate the following z-plane magnitude plot:

Examing the unit circle in the above z-plane figure, you can determine the spectrum of the filter:

Further Reading:

Biquad filtering program

Here is a command-line program called biquad.cpp which implements a biquad filter. Save it to your computer and compile it. You can download it directly onto mambo by typing the following command on mambo in the terminal: 

   wget http://peabody.sapp.org/class/dmp2/lab/biquad/biquad.cpp
If you need to review how to compile programs, refer to Lab 1. 

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#include "soundfile.h"

int main(int argc, char** argv) {
   Options options;
   options.define("a0=d:1", "first  feedforward term");
   options.define("a1=d:0", "second feedforward term");
   options.define("a2=d:-1", "third  feedforward term");
   options.define("b1=d:0.1", "first  feedback term");
   options.define("b2=d:0.9", "second feedback term");
   options.process(argc, argv);

   double a0 = options.getDouble("a0");
   double a1 = options.getDouble("a1");
   double a2 = options.getDouble("a2");
   double b1 = options.getDouble("b1");
   double b2 = options.getDouble("b2");

   SoundFileRead  insound(options.getArg(1));
   int sampleCount = insound.getSamples();
   SoundHeader    header = insound;
   header.setChannels(1);  // write only mono files
   SoundFileWrite outsound(options.getArg(2), header);

   double currin   = 0.0;   // x[n]
   double lastin   = 0.0;   // x[n-1]
   double lastin2  = 0.0;   // x[n-2]
   double currout  = 0.0;   // y[n]
   double lastout  = 0.0;   // y[n-1]
   double lastout2 = 0.0;   // y[n-2]

   // the for-loop implements the difference equation for a biquad filter: 
   //    y[n] = a0*x[n] + a1*x[n-1] + a2*x[n-2] - b1*y[n-1] - b2*y[n-2]
   for (int i=0; i<sampleCount; i++) {
      insound.incrementSample();
      lastin2  = lastin;
      lastin   = currin;
      currin   = insound.getCurrentSampleDouble(0);
      lastout2 = lastout;
      lastout  = currout;
      currout  = a0*currin + a1*lastin + a2*lastin2 - b1*lastout - b2*lastout2;
      outsound.writeSampleDouble(currout);
   }

   return 0;
}

Exercises

  • Try different values for the filter coeffients. Use whitenoise as an input to the filter program. You can make whitenoise soundfiles on mambo by typing: 
       whitenoise -d 5.0 noise.wav
    This example would create a five second whitenoise soundfile.

    You can set the filter coefficients like this: 

       biquad input.wav output.wav --a0 1 --a1 0 --a2 -1 --b1 0.1 --b2 0.9
or
   biquad --a0 1 --a1 0 --a2 -1 --b1 0.1 --b2 0.9 input.wav output.wav