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While Analyzing the DFT Plot for a signal with N samples, say we find a peak in the magnitude of the DFT plot at index $k$, this implies that our signal has a high amount of similarity with an exponential phasor rotating at $\omega = (2\pi/N)k$ or equivalently a period of $N/k$ samples.

But since we are in discrete time, the fundamental period must be an integer i.e. let the fundamental period be $T$ samples $ \implies T/m = N/k$ (where T and m are relatively prime)

Hence shouldn't the period be $Nm/k$ and hence the frequency would be $(2\pi/N)(k/m)$ ? (where $m$ is the smallest integer that makes $T$ a positive integer.)

Thank you

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Your assumption that the fundamental period has to be an integer since we are in discrete time is wrong. It can be any real number, nobody says, there has to be an integer number of periods in a signal of finite length. Also, if there is no $k$ representing this exact period, the peak in the spectrum will smear across neighboring bins.

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  • $\begingroup$ But a fractional period of say T=4.5 would suggest that the exponential phasor completes one full rotation in 4.5 samples. How can the sample index ever be fractional? $\endgroup$
    – Nemin
    Oct 22 '20 at 10:26
  • $\begingroup$ dsp.stackexchange.com/a/43784/53811 states that the period of a discrete time periodic signal must be an integer. $\endgroup$
    – Nemin
    Oct 22 '20 at 10:43
  • $\begingroup$ @Nemin, a discrete sinusoid that makes 2 revolutions in 9 samples would have a period of 4.5. This signal when transformed into it's Frequency domain representation, would have more than one FFT bin that is non-zero. I think what might be missing from your understanding are the concepts of spectral leakage and scalloping. So if you do a 16 point FFT, thinking about the signal, you have 2 cycles/9 samples * 16 samples, = 3.55 revolutions. So bins 3 and 4 would both have half the energy. I think there is some abuse of notation in the answer you reference. $\endgroup$
    – Jotorious
    Oct 22 '20 at 13:36
  • $\begingroup$ Thank you @Jotorious for your explanation and help, I am indeed missing some concepts. Thank you for drawing my attention. $\endgroup$
    – Nemin
    Oct 22 '20 at 14:00

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