Periodicity in Frequency and Time Domains

Frequency domain

From the DFT definition seen below, the DFT periodicity arises naturally:

$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi\frac{k}{N}n},$$

To see why, consider the DFT analysis of sinusoids at a frequency of $k+N$.

$$e^{-j2\pi\frac{k+N}{N}n}=e^{-j2\pi\frac{k}{N}n}\cdot e^{-j2\pi n}=e^{-j2\pi\frac{k}{N}n}$$

because $e^{-j2\pi n}=\cos 2\pi n-j\sin 2\pi n=1$.

Therefore, the DFT $X[k]$ is periodic with period $N$. This can be traced back to the fact that the discrete frequency axis is periodic, i.e., the frequency index $k$ and $k+N$ are essentially the same. This is a result of sampling the aliases outside the primary zone between $-\pi$ and $+\pi$.

Time domain

The inverse DFT is defined as:

$$x[n]=\frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{+j2\pi\frac{k}{N}n}$$

A similar derivation proves that the input signal $x[n]$ is also periodic.

$$x[n + N] = x[n]$$

This can be understood as follows. While taking the DFT of an input signal $x[n]$, a finite number of samples are required, and there is no inherent periodicity visible.

However, because of the way these complex sinusoids are defined, the DFT output $X[k]$ would be the same if the input signal $x[n]$ was periodic with period $N$. This nature of periodicity is a matter of debate in the DSP community.