FFT vs Table Fourier Pairs

Question

Eugene 2012-2-5

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/28120-fft-vs-table-fourier-pairs

回答： Sadiq Nasser 2021-1-8

Dear friends, I done FFTs of good known functions from Fourier Transform pairs table. The result of FFT doesn't match analytic transform in most cases by magnitude value, and in example 4 also by envelopment form. Code examples below. Where I wrong, please ?

fs=1000;            % sampling rate
dt=1/fs;            % time step
Sec=5;              % Signal length
N=Sec*fs;           % number of samples
n=(0:N-1)-N/2;
tn=n*dt;            % Time axis
df = fs/N;          % Frequency resolution
fk=n*df;            % Frecuency axis [Hz]
wk=fk*2*pi;         % Angular frequency axis [Rad/sec]
%--------------
% Example 1
% Time domain
alfa=7;
ex1=exp(-alfa.*abs(tn));
% FFT result
Ex1a=fftshift(fft(ifftshift(ex1)));
% Analytics Fourier transform
Ex1b=(2*alfa)./((alfa^2)+(wk.^2));
figure(1);
subplot(3,1,1);
plot(tn,ex1); 
grid;
title('e^{-a|t|}');
subplot(3,1,2);
plot(wk,real(Ex1a));
grid;
title('Ex1a');
subplot(3,1,3);
plot(wk,real(Ex1b));
grid;
title('2a/(a^2+w^2)');
%--------------
% Example 2
% Time domain
sigma=0.5;
ex2=exp(-(tn.^2)/(2*(sigma^2)));
% FFT result
Ex2a=fftshift(fft(ifftshift(ex2)));
% Analytics Fourier transform
Ex2b=sigma*sqrt(2*pi)*exp(-(sigma^2).*(wk.^2)/2);
figure(2);
subplot(3,1,1);
plot(tn,ex2);
grid;
title('e^{-t^2/2s^2}');
subplot(3,1,2);
plot(wk,real(Ex2a));
grid;
title('Ex2a');
subplot(3,1,3);
plot(wk,real(Ex2b));
grid;
title('s*sqrt(2pi)*e^{-s^2w^2/2}');
%--------------
% Example 3
% Time domain
ex3=1i./(pi.*tn);
% FFT result
Ex3a=fftshift(fft(ifftshift(ex3)));
% Analytics Fourier transform
Ex3b=-1*(wk<0)+1*(wk>0);
figure(3);
subplot(3,1,1);
plot(tn,abs(ex3));
grid;
title('-i/(t*pi)');
subplot(3,1,2);
plot(wk,real(Ex3a));
grid;
title('Ex3a');
subplot(3,1,3);
plot(wk,real(Ex3b));
grid;
title('(w<0)=>-1;(w>0)=>1');
%--------------
% Example 4
% Time domain
ex4=cos(alfa.*(tn.^2));
% FFT result
Ex4a=fftshift(fft(ifftshift(ex4)));
% Analytics Fourier transform
Ex4b=sqrt(pi/alfa)*cos((wk.^2)/4/alfa-pi/4);
figure(4);
subplot(3,1,1);
plot(tn,ex4);
grid;
title('cos(a*t^2)');
subplot(3,1,2);
plot(wk,abs(Ex4a));
grid;
title('Ex4a');
subplot(3,1,3);
plot(wk,abs(Ex4b));
grid;
title('sqrt(pi/a)*cos(w^2/4a-pi/4)');
%--------------
% Example 5
% Time domain
ex5=2*cos(2*pi*4*tn)+3*cos(2*pi*7*tn);
% FFT result
Ex5a=fftshift(fft(ifftshift(ex5)));
figure(5);
subplot(2,1,1);
plot(tn,ex5);
grid;
title('2cos(4wt)+3cos(7wt)');
subplot(2,1,2);
plot(wk,abs(Ex5a));
grid;
title('Delta(-14pi)+Delta(-8pi)+Delta(8pi)+Delta(14pi)');