Periodicity in Frequency and Time Domains

Discover how a DFT gives rise to a periodic time-domain signal.

Frequency domain

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


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

ej2πk+NNn=ej2πkNnej2πn=ej2πkNne^{-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 ej2πn=cos2πnjsin2πn=1e^{-j2\pi n}=\cos 2\pi n-j\sin 2\pi n=1.

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

Time domain

The inverse DFT is defined as:


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

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

This can be understood as follows. While taking the DFT of an input signal x[n]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]X[k] would be the same if the input signal x[n]x[n] was periodic with period NN. This nature of periodicity is a matter of debate in the DSP community.

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