Why DFT exact for periodic signals?

1. Why is DFT periodic?
2. Does DFT assume periodicity?
3. Is DFT of a finite duration sequence is periodic?
4. Is discrete Fourier transform exact?

Why is DFT periodic?

the DFT spectrum is periodic with period N (which is expected, since the DTFT spectrum is periodic as well, but with period 2π). Example: DFT of a rectangular pulse: x(n) = 1, 0 ≤ n ≤ (N − 1), 0, otherwise.

Does DFT assume periodicity?

In many signal processing books, it is claimed that the DFT assumes the transformed signal to be periodic (and that this is the reason why spectral leakage for example may occur).

Is DFT of a finite duration sequence is periodic?

Hence, the DFT represents one period of a periodic discrete-time signal by one period of its discrete-frequency periodic spectrum. Both the “actually” transformed signal and its spectrum are therefore periodic.

Is discrete Fourier transform exact?

No approximation is needed. In the continuous-time case, the Fourier series representation is accurate only for certain signals. For the discrete-time case, it is always accurate.