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I have some queries about the Fourier transform

  1. In most of the cases, the Fourier transform of a signal is symmetric with respect to positive and negative frequency. I think the computational complexity increases because only half of the symmetric spectrum (i.e. spectrum for only positive frequencies) is useful. Also, while working in the frequency domain we could get wrong value of energy or power due to the spectrum on negative axis.

  2. In the Fourier transform formula the limits of integration are from $-\infty$ to $\infty$. However, for a signal which is continuously or exponentially increasing with time, one can't compute it's Fourier transform.

  3. After computation of Fourier transform of a signal, we get phase and frequency spectrum of the whole signal which is localized in frequency domain only. But from both of these spectrum, we don't get any spatial component features like which frequency component is present at which time (and same with the phase value).

  4. After computation of Fourier transform of signal, with dc and positive frequencies we also get unnecessary negative frequency components. I think concept of negative frequency doesn't exists practically.

I feel that these are the "shortcomings" of Fourier transform but don't know whether they really are or not.

So can anybody give explanation on any of above doubts?

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  • $\begingroup$ You have some in accuracies in your claims and misinterpretations: $\endgroup$ – Moti May 7 '15 at 18:46
  • $\begingroup$ You are not specific about your problem: 1. Symmetry is not an issue and does not impact computation - you might get a DC value. 2. ? 3. Wrong 4. concept exist, practical interpretation is the challange $\endgroup$ – Moti May 7 '15 at 19:09
  • $\begingroup$ @Moti sir i have edited my 2nd question .Also for doubt 1 ,using FT we unneccessarily compute negative component.So while calculating energy in frequency domain we get wrong value due to that negative component and for doubt 4, how would u convince on concept of negative frequency $\endgroup$ – pandu May 7 '15 at 19:20
  • $\begingroup$ what is that you try to solve? FT is a tool that may be used in various ways. As transform, it has also an inverse one. Would you use it? All depends on the application. $\endgroup$ – Moti May 8 '15 at 22:44
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1) For real signals you can skip the computations of the negative part if you want, and the energy on it will always be the same as in the positive part

2) Fourier analysis is not aimed at this kind of signals. See Laplace Transforms for that

3) You can get information from both time and frequency domains by windowing the signal in many frames and computing the FT for each of them. Take a look into spectrograms and how people do it

4) In most cases you will only work with real signals so we do ignore the negative part because it is redundant. Telecommunications and electronics engineers sometimes make use of complex signals as a shortcut even if these don't exist in reality

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  • $\begingroup$ sir i have edited my 3rd question $\endgroup$ – pandu May 7 '15 at 20:05
  • $\begingroup$ Yes, I see. Look at this and this $\endgroup$ – gsmafra May 7 '15 at 20:07
  • $\begingroup$ If you plan to do some processing in the frequency domain and then inverse FT to time domain, you can not neglect the negative direction. Actually the "negative direction" for time limited/spectrum limited are frequencies over the f/2 range. $\endgroup$ – Moti May 8 '15 at 22:41
  • $\begingroup$ You can sort of synthesize the negative part from the positive part when going back to time and save some computation. And if the negative part were frequencies over the f/2 range then the Nyquist Theorem would be wrong. They just appear as if they were because of the periodicity of the frequency domain when working with discrete time signals $\endgroup$ – gsmafra May 9 '15 at 8:45

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