October – Fast Fourier Transform in GNU Octave

Fast fourier transform (FFT) is an algorithm that computes the discrete Fourier transform.

FFT is especially handy in real-time digital signal processing. Digital computers are working on discrete data, thus a input signal is always sampled with some sampling rate $F_s$ (Hz). By the Nyquist-Shannon sampling theorem, sampling captures frequencies under $F_s/2$.

Octave calculates the FFT of a discrete signal. In the following code, we calculate the normalized FFT and plot the frequency spectrum w.r.t. the frequency. Signal- and sample time vectors are given as input.

##Calculates the FFT and plots the frequency spectrum of a signal. Sampletimes have to start at time 0.

function fftvector = plotfft(signal, sampletimes)
  N = length(sampletimes); #Signal length.
  Fs = (N - 1)/(sampletimes(N)) #Sampling rate. We have N - 1 "hops" in sampletimes(N) seconds.
  FFT =  fft(signal);
  if(mod(N,2) == 0) #Check if signal length is odd or even.
    FFT = 2*FFT(1 : N/2)/N;
    f = Fs*(0 : (N - 1)/2)/(N - 1); 
    FFT = 2*FFT(1 : (N - 1)/2)/N;
    f = Fs*(0 : (N - 2)/2)/(N - 1);
  fftvector = [f; FFT];
  plot(f, abs(FFT));
  title('Fast fourier transform')
  xlabel('Frequency (Hz)')
  print  plot.jpg

The following figure shows my track Tappimarssi in the frequency domain. The track was imported to Octave by

[signal, fs] = audioread('Tappimarssi.wav');
sampletimes = 0 : 1/fs : 1/fs*(length(signal) - 1);

Image tappimarssi