Transfer function between two audio signals

Hi, i'm trying to calculate a transfer function between 2 audio signals (x and x2). I applied an octave filter to each signal so I have the power for each band of the signals (x_oct_pow and x2_oct_pow), I want to know the transfer function. An example below:

[x_oct_pow, x2_oct_pow] = octave(x, x2, Fs);

What is the correct method to get the transfer function between x_oct_pow and x2_oct_pow? I have to divide the vectors:

tr_f = x2_oct_pow ./ x_oct_pow;

or, since I'm working with a power, I have to subtract them?

tr_f = x2_oct_pow - x_oct_pow;

Thank you for your help

Answer 1

[Ankit Bhatnagar]

I believe that you are using the Octave Filter function as there is no 'Octave' function in MATLAB. The usage of the Octave Filter is of the form: octFilt = octaveFilter(1000,'1/3 octave','SampleRate',96000) which creates a System object, octFilt, with a center frequency of 1000 Hz, a 1/3 octave filter bandwidth, and a sample rate of 96,000 Hz. For details on the output and other properties please refer to:

https://www.mathworks.com/help/audio/ref/octavefilter-class.html

As far as transfer function for the signals, it can be obtained by dividing the signals not subtracting.

[Luca Mastrangelo]

Hi Ankit, thank you for your answer. I wrote my own function to apply the octave filter. Taking two audio signals x and x2 as inputs it returns two vectors, x_oct_pow and x2_oct_pow, containing the power for each band. Below the code of the function:

function [x_pow_oct, x2_pow_oct, Fc] = octave(x, x2, Fs)
    lx = length(x);

      Fc = [20 25 31.5 40 50 63 80 100 125 160 200 250 315 400 ...
          500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 ... 
          6300 8000 10000 12500 16000 20000];% ANSI central frequencies
      Fc = Fc(Fc<Fs/2);
      fl = Fc*2^(-1/6); % lower frequencies
      fu = Fc*2^(1/6); % upper frequencies
      fu(fu>Fs/2) = Fs/2 -1; 
      numBands = length(Fc);
      b = cell(numBands,1);
      a = cell(numBands,1);
      for n = 1:length(Fc)
          [b{n},a{n}] = butter(2, [fl(n) fu(n)]/(Fs/2), 'bandpass');
      end

      % apply filter
      x_oct = zeros(lx, numBands);
      x2_oct = zeros(lx, numBands);
      for i = 1:numBands
          x_oct(:, i) = filtfilt(b{i}, a{i}, x);
          x2_oct(:, i) = filtfilt(b{i}, a{i}, x2);
      end
      x_pow_oct = zeros(1, numBands);
      x2_pow_oct = zeros(1, numBands);
      for k = 1:numBands
          s1 = sum(abs(fft(x_oct(:,k))).^2)/lx;
          s2 = sum(abs(fft(x2_oct(:,k))).^2)/lx;
          x_pow_oct(k) = 10*log10(s1/(lx/Fs));
          x2_pow_oct(k) = 10*log10(s2/(lx/Fs));
      end
  end

Unfortunately this code doesn't work perfectly. With a pink noise it returns the same value for each band. But if I compare the results with a professional octave filter, applying both filters to a real signal, I obtain very different values. Can you check if I made any mistake?

Luca