fourier_demo
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% cs515 Demo: Smoothness of f ~ decay of f^
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Useful data for first two examples:
n = [-128:127];              % range of fourier coefficients
t = n / 128 * pi;            % define functions on [-pi,pi)
ttt = [t-2*pi,t,t+2*pi];     % time axis for periodic extension
N = length(n);    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Ex 1: A piecewise constant function related to H(t)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
f = (t<0) - (t>=0);                      % evaluate f(t)
% F = (2*pi)*fftshift(fft(f))/N;         % FFT evaluation: F(n) = f^(n)
F = (2i*pi) * sin(n*pi/2) .* sinc(n/2);  % Use analytic form of f^(n)

echo off;
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Ex 2: A piecewise linear function related to B2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
f = (t/pi+1).*(t<0) + (1-t/pi).*(t>=0);  % evaluate f(t) 
% F = (2*pi)*fftshift(fft(f))/N;         % FFT evaluation: F(n) = f^(n)
F = pi*sinc(n/2).^2;                     % Use analytic form of f^(n)

echo off;
Hit Enter to continue
Hit Enter to continue
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Hit Enter for next example

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Example 3: Reconstruction of a 'smooth' function %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
n = [-128:127];   % range of Fourier coefficients
t = n/128*pi;
N = length(n);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% A piecewise cubic related to B4 %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
f = (2*t/pi+2).^3/6 .* (t>=-pi & t<-pi/2) ... % ... continued
    -(3*(2*t/pi+2).^3 - 12*(2*t/pi+2).^2+12*(2*t/pi+2)-4)/6 .* (t>=-pi/2 & t<0) ...
    +(3*(2*t/pi+2).^3 -24*(2*t/pi+2).^2 + 60*(2*t/pi+2)-44)/6 .*(t>=0 & t<pi/2) ...
    -(2*t/pi-2).^3/6 .* (t>=pi/2 & t<pi);
s = max(f); f = f/s;

F = (2*pi)*fftshift(fft(f))/N;         % FFT evaluation: F(n) = f^(n)
echo off
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Hit Enter to continue
Hit Enter to continue
Hit Enter to continue
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% A C_inf function with compact support
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
t = linspace(0,N,N*(a/pi));
g = zeros(size(t));
index = find(t>0&t<1);
g(index) = exp(1./(t(index)-1)-1./t(index));
s = max(g); g = g / s;
G = fftshift(fft(g))/N; 

echo off;
Hit Enter to continue
Hit Enter to continue
Hit Enter to continue
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??? Error using ==> get
Invalid handle object.

Error in ==> <a href="error:/s/matlab-7.5/toolbox/matlab/uitools/btndown.m,21,0">btndown at 21</a>
axesHandle = findobj(get(figHandle, 'children'), 'flat', ...

Error in ==> <a href="error:/u/a/m/amos/public/html/515/demos/fourier/fourier_demo.m,352,0">fourier_demo at 352</a>
btndown(5,'TestGroup', 'Next>>');


fourier_demo
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% cs515 Demo: Smoothness of f ~ decay of f^
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Useful data for first two examples:
n = [-128:127];              % range of fourier coefficients
t = n / 128 * pi;            % define functions on [-pi,pi)
ttt = [t-2*pi,t,t+2*pi];     % time axis for periodic extension
N = length(n);    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Ex 1: A piecewise constant function related to H(t)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
f = (t<0) - (t>=0);                      % evaluate f(t)
% F = (2*pi)*fftshift(fft(f))/N;         % FFT evaluation: F(n) = f^(n)
F = (2i*pi) * sin(n*pi/2) .* sinc(n/2);  % Use analytic form of f^(n)

echo off;
Hit Enter to continue
Hit Enter to continue
Hit Enter to continue
Hit Enter to continue
Hit Enter to continue
Hit Enter for next example
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Ex 2: A piecewise linear function related to B2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
f = (t/pi+1).*(t<0) + (1-t/pi).*(t>=0);  % evaluate f(t) 
% F = (2*pi)*fftshift(fft(f))/N;         % FFT evaluation: F(n) = f^(n)
F = pi*sinc(n/2).^2;                     % Use analytic form of f^(n)

echo off;
Hit Enter to continue
Hit Enter to continue
Hit Enter to continue
Hit Enter to continue
Hit Enter to continue
Hit Enter for next example

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Example 3: Reconstruction of a 'smooth' function %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
n = [-128:127];   % range of Fourier coefficients
t = n/128*pi;
N = length(n);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% A piecewise cubic related to B4 %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
f = (2*t/pi+2).^3/6 .* (t>=-pi & t<-pi/2) ... % ... continued
    -(3*(2*t/pi+2).^3 - 12*(2*t/pi+2).^2+12*(2*t/pi+2)-4)/6 .* (t>=-pi/2 & t<0) ...
    +(3*(2*t/pi+2).^3 -24*(2*t/pi+2).^2 + 60*(2*t/pi+2)-44)/6 .*(t>=0 & t<pi/2) ...
    -(2*t/pi-2).^3/6 .* (t>=pi/2 & t<pi);
s = max(f); f = f/s;

F = (2*pi)*fftshift(fft(f))/N;         % FFT evaluation: F(n) = f^(n)
echo off
Hit Enter to continue
Hit Enter to continue
Hit Enter to continue
Hit Enter to continue
Hit Enter to continue
Hit Enter to continue
Hit Enter to continue
Hit Enter for next example
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% A C_inf function with compact support
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
t = linspace(0,N,N*(a/pi));
g = zeros(size(t));
index = find(t>0&t<1);
g(index) = exp(1./(t(index)-1)-1./t(index));
s = max(g); g = g / s;
G = fftshift(fft(g))/N; 

echo off;
Hit Enter to continue
Hit Enter to continue
Hit Enter to continue
Hit Enter to continue