I would like to know, why does the periodic signal in time always give a discrete frequency spectrum in FT?

I know the equations, but I simply dont understand why is it so.



1 Answer 1


Here's an intuitive explanation if the convolution theorem is taken for granted:

Since the time-domain signal is periodic, one can say that it can be built by "copying and pasting" the same block of signal every period: your periodic signal can be expressed as a little block of signal (spanning one period) convolved with a dirac comb.

Thus, its Fourier transform will be the Fourier transform of the little block multiplied by the Fourier transform of a Dirac comb (which is another Dirac comb). Multiplying a continuous signal by a Dirac comb yields a discrete signal.

The same reasoning is also true the other way round (discrete in time implies periodic in frequency).

  • $\begingroup$ THanks man for fast reply. I understod your explanation time to frequency domain, but I have problems understandig frequency to time. I dont understand how do you (or is it even possible to) break discrete signal in small blocks and use "copy paste" method. $\endgroup$ Jun 26, 2014 at 8:08
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    $\begingroup$ All properties of the Fourier transform also apply to its inverse (up to normalizing constants), so proving that a "the FT of signal discrete in time is a signal periodic in frequency" is equivalent to proving that "the inverse FT of a signal periodic in frequency is a signal discrete in time". $\endgroup$ Jun 26, 2014 at 8:27

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