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Also , is the inverse Fourier transform of a periodic signal also periodic ?

hakunamatata
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  • Do you want to know about the fourier transform of a periodic signal or the inverse fourier transform of a periodic signal? Your title and actual question conflict with each other – Kvothe Sep 16 '15 at 22:18
  • I am sorry what I meant to ask is when is the Fourier transform of any signal periodic. I am talking about the periodicity of the Fourier transform not the signal. – hakunamatata Sep 16 '15 at 22:21
  • I would need to check but I believe the Fourier transform of a periodic signal would be periodic since the Fourier transform expresses the original signal as a sum of sinusoidal functions - which is periodic – Kvothe Sep 16 '15 at 22:31
  • I thought Fourier transform is a countinuous time representation of Fourier series coefficients. Transform of a sine is an impulse - not periodic . – hakunamatata Sep 16 '15 at 22:39
  • That's a good point (I was wrong, ignore what I said). And you have just answered your question - the fourier transform of sine in not periodic and therefore the fourier transform of a periodic function is not (always) periodic – Kvothe Sep 16 '15 at 22:46

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I'm going to assume that we can start with a signal for which a Fourier transform exists (such as an absolutely integrable function). If we construct another signal by sampling this original signal at regular time intervals, then the Fourier transform of that newly constructed signal would corresponds to the Discrete-Time Fourier Transform (DTFT) which would be periodic. Note that if the original signal's bandwidth is less than the Nyquist frequency then the time-sampled signal can also be recovered from this frequency-domain representation.

Conversely, if we evenly sample a continuous frequency-domain function, the corresponding inverse transform would be a periodic signal in the time domain. Correspondingly an evenly time-sampled and periodic signal would have a periodic evenly frequency sampled signal (ie. the inverse Fourier transform of that periodic signal would also be periodic).

SleuthEye
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