FFT result looks nothing like analytic result

Question

James Kirk 2015-11-29

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/258071-fft-result-looks-nothing-like-analytic-result

编辑： Rick Rosson 2016-3-14

Forgive me if I am missing understanding something simple here but I am confused by exactly what the FFT function returns in Matlab. I have code to compare the result of FFT to the analytic result of the fourier transform of a gaussian:

steps = 2^10; lim = 4;
x = linspace(-lim, lim, steps);
w = lim/4; A = 1;
func = A*exp((-1).*(x./w).^2);            % Gaussian
Analytic = sqrt(pi)*w*exp(-(w*pi.*x).^2); % Analytic Transform
Func = fftshift(fft(func));               % FFT transform

When I plot this, the fft function looks nothing like the analytic result:

If anyone could shed some light on why this is and how I can adapt my use of the FFT function to give me back the analytic result I would greatly appreciate it!

I am asking this question as I have been developing some code to model the propagation of various beams through a turbulent medium using phase screens. For the code to be valid it is important that the Fourier Transforms produced agree with Physics.

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Answer 1

Rick Rosson 2016-1-6

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https://ww2.mathworks.cn/matlabcentral/answers/258071-fft-result-looks-nothing-like-analytic-result#answer_205198

编辑：Rick Rosson 2016-3-14

在 MATLAB Online 中打开

steps = 2^10; lim = 4;
dx = 2*lim/steps;
x = -lim:dx:lim-dx;
% x = linspace(-lim, lim, steps);
Fs = 1/dx;
dF = Fs/steps;
f = -Fs/2:dF:Fs/2-dF;
w = lim/4; A = 1;
func = A*exp((-1).*(x./w).^2);            % Gaussian
% Analytic = sqrt(pi)*w*exp(-(w*pi.*x).^2); % Analytic Transform
Analytic = sqrt(pi)*w*exp(-(w*pi.*f).^2);
% Func = fftshift(fft(func));               % FFT transform
Func = dx*fftshift(fft(ifftshift(func)));      % FFT transform
diff = abs(Func) - Analytic;
figure;
subplot(2,1,1);
plot(f,abs(Func),f,Analytic);
xlim([-4 4]);
subplot(2,1,2);
plot(f,diff);
xlim([-4 4]);

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Luna 2016-2-29

编辑：Luna 2016-2-29

Dear Rick,

I have troubles with exactly the same problem. I just tested your code and it produces, as in James example, an output fft function that do not agree with the analytical one. Can you explain why? Is there a reason? Thank you very much

Luna

Rick Rosson 2016-3-14