%% FFT for Spectral Analysis
% This example shows the use of the FFT function for spectral  analysis.  A
% common use of FFT's is to find the frequency components of a signal buried in
% a noisy time domain signal.
%
% Copyright 1984-2002 The MathWorks, Inc. 
% $Revision: 5.8 $  $Date: 2002/04/15 03:36:57 $

%%
% First create some data.  Consider data sampled at 1000 Hz.  Start by forming a
% time axis for our data, running from t=0 until t=.25 in steps of 1
% millisecond.  Then form a signal, x, containing sine waves at 50 Hz and 120
% Hz.

t = 0:.001:.25;
x = sin(2*pi*50*t) + sin(2*pi*120*t);

%%
% Add some random noise with a standard deviation of 2 to  produce a noisy
% signal y.  Take a look at this noisy  signal y by plotting it.

y = x + 2*randn(size(t));
plot(y(1:50))
title('Noisy time domain signal')

%%
% Clearly, it is difficult to identify the frequency components from looking at
% this signal; that's why spectral analysis is so popular.
%
% Finding the discrete Fourier transform of the noisy signal y is easy; just
% take the fast-Fourier transform (FFT).

Y = fft(y,256);

%%
% Compute the power spectral density, a measurement of the energy at various
% frequencies, using the complex conjugate (CONJ).  Form a  frequency axis for
% the first 127 points and use it to plot the result.  (The remainder of the 256
% points are symmetric.)

Pyy = Y.*conj(Y)/256;
f = 1000/256*(0:127);
plot(f,Pyy(1:128))
title('Power spectral density')
xlabel('Frequency (Hz)')

%%
% Zoom in and plot only up to 200 Hz.  Notice the peaks at 50 Hz and  120 Hz.
% These are the frequencies of the original signal.

plot(f(1:50),Pyy(1:50))
title('Power spectral density')
xlabel('Frequency (Hz)')