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Fundamentals of Digital Signal Processing

Periodicity in Frequency and Time Domains

Explore the concept of periodicity in digital signal processing by analyzing how the Discrete Fourier Transform causes signals to repeat in both time and frequency domains. Learn how this periodicity affects signal analysis, particularly shifts and aliasing, and how choosing appropriate DFT sizes ensures accurate signal representation.

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Frequency domain

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

X[k]=∑n=0N−1x[n]e−j2πkNn,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.

e−j2πk+NNn=e−j2πkNn⋅e−j2πn=e−j2πkNne^{-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} ...