why real part of fft (exp(at)) is negative in some parts in contrary to analytical calculation of fft(exp(at))?

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Neda 2012-6-14

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Hi, I want to calculate the real part of fourier transform of A = exp(at) where "a" is a constant and t = [0 T]. when I calculate it from the analytical definition of the fourier transform, the real( A(f)) >0 ; however, when I use Matlab for some parts real (A(f))<0. In other words, the graphs from two methods are not identical for some ranges. Can you please help me to understand why FFT from Matlab and analytical calculation doesn't match? I did the same procedure for B =exp(-at) and in this case both results had a good agreement. Thank you

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Answer 1

Dr. Seis 2012-6-15

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https://ww2.mathworks.cn/matlabcentral/answers/41177-why-real-part-of-fft-exp-at-is-negative-in-some-parts-in-contrary-to-analytical-calculation-of-f#answer_50819

Try plotting the abs result from FFT. Basically, you will not get a purely real, all positive frequency spectrum unless your time domain signal is symmetric and also all positive (or at least nearly all positive).

Kind of a similar situation as here:

http://www.mathworks.com/matlabcentral/answers/40257-gaussian-fft

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Answer 2

Neda 2012-6-17

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Thanks for the reply. I agree with you that symmetry is important. But my problem is about the real values calculated from FFT. Let me express my question in another way. Assume that we have exp(at) which is in the range of [-T/2 T/2], then I compared its fourier, by 1)doing analytical calculation, 2) writing a code to calculate the fourier series coefficiencts and 3) using Matlab FFT. when I compared the abs values from these three ways my results are in good agreement (Of course, the amplitudes are different). But when I compared the real values, the result from Matlab FFT (third method) is very different from the other two. In other words, the occurance of negative and positive real points form (1) and (2) are at the same but from (3) my results are different. (It is good to add that here by making my function symmetrical, I observed negative real values from analytical calculation in contrary to previous asymetrical case where the real part was positive ) Can you please help me to understand the reason for this difference? my expectation was to have similar results from 3 methods for the real part as well.