Why does the DFT assume the transformed signal is periodic?

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).

Now, if you look at the definition of the DFT, there is simply no that kind of assumption. However, in the Wikipedia article about the discrete-time Fourier transform (DTFT), it is stated that

When the input data sequence $x[n]$ is $N$-periodic, Eq.2 can be computationally reduced to a discrete Fourier transform (DFT)

• So, does this assumption stems from the DTFT?


• Actually, when calculating the DFT, am I in fact calculating the DTFT with the assumption that the signal is periodic?