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Eli Bendersky7 min readadvanced

Notes on discrete-time Fourier series and transform

Read on thegreenplace.netHN 15·0 comments

Summary

This post details the Discrete-Time Fourier Series (DTFS) for periodic discrete signals and the Discrete-Time Fourier Transform (DTFT) for non-periodic ones. It derives their formulas, shows their properties, and explains their close relationship as theoretical foundations for digital signal processing.

  • DTFS represents N-periodic discrete signals as a finite sum of N distinct complex exponentials.
  • Unlike continuous Fourier series, DTFS has no convergence issues due to finite summation.
  • The DTFT extends Fourier analysis to non-periodic discrete signals, resulting in a continuous, 2pi-periodic frequency domain function.
  • DTFS coefficients are scaled, sampled versions of the DTFT of the signal's periodic extension.

Engineers working with digital signal processing, audio, or image processing will find this a rigorous theoretical foundation for understanding algorithms like the DFT.

7/10
mathematicssignal processingfourier transformdspdiscrete math

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