Why is dft periodic

1. Why is discrete time Fourier transform periodic?
2. Is DFT is a periodic function?
3. Why is DFT not continuous?
4. Is DFT of a finite duration sequence is periodic?

Why is discrete time Fourier transform periodic?

This periodicity is due to the discrete-time nature of the signal. Thus, when working with DTFT's, we only need to look at the range 0 ≤ Ω ≤ 2π (or −π ≤ Ω ≤ π). X(Ω)ejnΩdΩ. Like its continuous-time counterpart, the DTFT has several very useful properties.

Is DFT is a periodic function?

synthesis. the DFT spectrum is periodic with period N (which is expected, since the DTFT spectrum is periodic as well, but with period 2π).

Why is DFT not continuous?

Because DFT Frequencies are discrete because k in ω=2πkN can take only integers from 0,1,2,...,N−1 and DTFT frequencies can take any value on real line, so they are continuous frequencies.

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.