Fourier transform of a function differentiated with respect to time.

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Erkan
Erkan 2024-2-9
评论： Daniel 2024-2-9
Hi everybody, I have a question about the Fourier transform of a function differentiated with respect to time. Let the time derivative of a function be df(t)/dt. If we want to examine the behavior of the f(t) function in the frequency domain; From the basic properties of the Fourier transform, the Fourier transform of this derivative expression is becomes -jwF(w). How can I calculate the value or values ​​of w(=2*pi*frequency) in this -jwF(w) expression? So, how can I derive these frequency values? Thanks in advanced.

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Daniel
Daniel 2024-2-9
If you're invoking fft you're using the Discrete Fourier Transform, and your w values coming out of that function are evenly spaced, starting from 0, at intervals of where N is the length of your FFT. I'm not sure if you would need to normalize those values to your sampling frequency or not. I suppose it would depend on the context...
If you're doing a Fourier transform of a continuous signal (which you can't do with fft), or computing the Fourier transform of an infinite-duration signal (which would have to be analytical), you'll follow whatever the definition for w is in that context.
Does that help at all?
  2 个评论
Erkan
Erkan 2024-2-9
Hi Daniel, thanks for your answer. Signal is discrete. If fs is the sampling frequency, Would the equation w=((0:(2*pi)/N:N)*fs be correct?
Daniel
Daniel 2024-2-9
Close! There are a couple of clerical issues with it, so I'll provide what I would use directly:
(0:N-1)/N*fs
Issue 1: 2*pi gives you the sort of abstract mathematical frequencies, so to re-normalize to fs you would just have to divide that back out anyway.
Issue 2: Also, you would need to reach for an upper frequency of 1-1/N, not N, if you wrote 0:1/N:<bound>.
Are you dealing with a real or a complex input? If it's complex it's often useful to view the upper half of the fft as negative frequencies rather than as high positive frequencies. In that case you would need to do a little extra manipulation to both the fft output and the w vector, I think. That being said, I'm not really sure if there would be mathematical repercussions to that---I am pretty familiar with the mechanics of the fft but not so sure what the real-world applications are for translating the derivative into the frequency domain.

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