I need to get the difference equation of a specific elliptic filter. I calculated the transfer function coefficients in MATLAB with:

  %% Low pass design
  n = 10;
  passband_ripple = 1;
  stopband_attenuation = 60;
  f = 4750;
  fs = 44100;
  [B,A] = ellip(n,passband_ripple,stopband_attenuation,f/(0.5*fs),'low');

However, when I use the difference equation, I calculated from the transfer function, the samples go to +-inf. Here is the code where I create the noise input sound and filter it with difference eq.

  xin = rand(1,fs/20);     
  xin = xin - mean(xin);         
  xin = [xin zeros(1,fs*2)];  

  x = xin;
  y = zeros(1, length(x));

  %% Low pass
  b1 = 0.0027;
  b2 = -0.0134;
  b3 = 0.0362;
  b4 = -0.0651;
  b5 = 0.089;
  b6 = -0.0978;

  a1 = -7.953;
  a2 = 29.7485;
  a3 = -68.5714;
  a4 = 107.5494;
  a5 = -119.7162;
  a6 = 95.6848;
  a7 = -54.2043;
  a8 = 20.8340;
  a9 = -4.9109;
  a10 = 0.5401;

  for n = 11:length(y)
      y(n) = b1*x(n) - b2*x(n-1) + b3*x(n-2) - b4*x(n-3) + b5*x(n-4) - ...
      b6*x(n-5) + b5*x(n-6) - b4*x(n-7) + b3*x(n-8)- b2*x(n-9) + ...
      b1*x(n-10) - ...
      a1*y(n-1) - a2*y(n-2) - a3*y(n-3) - a4*y(n-4) - a5*y(n-5) - ...
      a6*y(n-6) - a7*y(n-7) - a8*y(n-8) - a9*y(n-9) - a10*y(n-10);

The filter function does what I want, but I need to be able to filter the sound sample-by-sample.

y = filter(B,A,x);

How can I obtain the right difference equation of my filter?


1 Answer 1


There are two problems with your code. First, you should not invert every other $b$ coefficient, they should all have a positive sign, just like the $a$ coefficients that all have a negative sign. Second, - and that's the main problem -, why do you quantize the coefficients? Your actual coefficient $b[1]$ is 0.00266475348894505 (and even longer), and you rounded it to 0.0027. That's bad for the $b$ (numerator) coefficients, but detrimental for the $a$ (denominator) coefficients, because the filter's characteristic is significantly changed, and the filter can become unstable. You should just use the coefficients as stored in the vectors A and B.

Implementing the difference equation is a nice exercise to understand what's going on, but I think you'd be better off using filter.m.

  • $\begingroup$ Using the actual coefficient instead of rounded solved the problem! Thank you! I didn't realize that the coefficients are rounded as I was transcribing them from Variables window in Matlab. I am sorry about the coefficients inverting mistake in my original code. I was trying different things out of craziness after the code that was supposed to be working didn't work and I forgot to fix the experimental code. $\endgroup$ Jan 8, 2018 at 17:46

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