MATLAB Language Fourier Transforms and Inverse Fourier Transforms Implement a simple Fourier Transform in Matlab


Fourier Transform is probably the first lesson in Digital Signal Processing, it's application is everywhere and it is a powerful tool when it comes to analyze data (in all sectors) or signals. Matlab has a set of powerful toolboxes for Fourier Transform. In this example, we will use Fourier Transform to analyze a basic sine-wave signal and generate what is sometimes known as a Periodogram using FFT:

%Signal Generation
A1=10;                % Amplitude 1
A2=10;                % Amplitude 2
w1=2*pi*0.2;          % Angular frequency 1
w2=2*pi*0.225;        % Angular frequency 2
Ts=1;                 % Sampling time
N=64;                 % Number of process samples to be generated
K=5;                  % Number of independent process realizations
sgm=1;                % Standard deviation of the noise
n=repmat([0:N-1].',1,K);             % Generate resolution
phi1=repmat(rand(1,K)*2*pi,N,1);     % Random phase matrix 1
phi2=repmat(rand(1,K)*2*pi,N,1);     % Random phase matrix 2
x=A1*sin(w1*n*Ts+phi1)+A2*sin(w2*n*Ts+phi2)+sgm*randn(N,K);   % Resulting Signal

NFFT=256;            % FFT length
F=fft(x,NFFT);       % Fast Fourier Transform Result
Z=1/N*abs(F).^2;     % Convert FFT result into a Periodogram

Periodogram of the 5 realizations

Note that the Discrete Fourier Transform is implemented by Fast Fourier Transform (fft) in Matlab, both will yield the same result, but FFT is a fast implementation of DFT.

xlabel('w [\pi rad/s]')
ylabel('Z(f) [dB]')
title('Frequency Range: [ 0 ,  \omega_s ]')