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Periodicity in Frequency and Time Domains

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:

X[k]=n=0N1x[n]ej2πkNn,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+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} ...