stephenHartzell

%% FFT Demo - Information

% Stephen Hartzell
% 5-13-2012
% Last Revised: 1-15-2012
%
% DESCRIPTION
% This script demonstrates the use of the discrete fourier transform (DFT)
% via the Fast Fourier Transform (FFT). Note that the fft only operates on
% signals with a total number of samples equal to an integer power of 2. If

stephenHartzell

stephenHartzell

% Define parameters
Fs = 10;                        % Sampling Frequency
Ts = 1/Fs;                      % Sampling Period
N = 32;                         % Total Number of Samples
n = 0:N-1;                      % Samples
t = Ts*(n);                     % Sampled times

y = cos(2*pi*t);                % Discrete signal
% Plot the signal and its discrete counter-part
figure

stephenHartzell

Y = fft(y);                     % The fft of the signal
df = 1/(N*Ts);                  % The frequency resolution
k = n;                          % Frequency spectrum samples
f = k*df;                       % Frequency values at the k samples    
% Plot the fft of the signal
figure
stem(k,abs(Y),'filled')
title('Raw Fourier Transform of y')
ylabel('|fft(y)|')
xlabel('k')

stephenHartzell

% Plot the useful fft of the signal
figure
stem(f,abs(Y)./N,'filled')
title('Useful Fourier Transform of y')
ylabel('Scaled |fft(y)|')
xlabel('frequency (Hz)')

stephenHartzell

Fs = 10;                        % Sampling frequency
Ts = 1/Fs;                      % Sampling period
N = 32;                         % Total number of samples
n = 0:N-1;                      % Samples
t = Ts*(n);                     % Sampled times
y = cos(2*pi*t);                % Discrete 1 Hz signal
yp = cos(18*pi*t);              % Discrete 9 Hz signal
ypp = cos(20*pi*t);             % Discrete 10 Hz signal

% Plot the 1 Hz signal and its discrete counter-part

stephenHartzell

% Define parameters
Fs = 10;                        % Sampling Frequency
Ts = 1/Fs;                      % Sampling Period
N = 32;                         % Total Number of Samples
n = 0:N-1;                      % Samples
t = Ts*(n);                     % Sampled times

y = cos(2*pi*t);                % Discrete signal
% Plot the signal and its discrete counter-part
figure

stephenHartzell

% Define parameters
Fs = 16;                        % Sampling Frequency
Ts = 1/Fs;                      % Sampling Period
N = 64;                         % Total Number of Samples
n = 0:N-1;                      % Samples
t = Ts*(n);                     % Sampled times

y = cos(2*pi*t);                % Discrete signal
Y = fft(y);                     % The fft of the signal
df = 1/(N*Ts);                  % The frequency resolution

stephenHartzell

% The f and g signals
f = [1,-1,4,8];
g = [2,1,5];

% Convolution of f and g
f_conv_g = conv(f,g);
figure, stem(f_conv_g,'filled')
xlim([-1,length(f_conv_g)+1])
ylim([-20,50])
title('Convolution of f and g')

stephenHartzell

% The convolution is calculated as shown in this figure
f_pad = f;
g_pad = [g, zeros(1,length(f)-length(g))];
g_pad = circshift(fliplr(g_pad)',1)';

MF = zeros(1,length(f_pad));
figure('Position',[100 100 850 600])
v = VideoWriter('test.avi','Uncompressed AVI');
v.FrameRate = 1;
open(v)