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Fourier Representations

Module by: Mark A. Davenport. E-mail the author

Fourier Representations

Throughout the course we have been alluding to various Fourier representations. We first recall the appropriate transforms:

  • Fourier Series (CTFS): x(t)x(t): continuous-time, finite/periodic on [-π,π][-π,π]
  • Discrete-Time Fourier Transform (DTFT): x[n]x[n]: infinite, discrete-time
  • Discrete Fourier Transform (DFT): x[n]x[n]: finite, discrete-time
  • Continuous-Time Fourier Transform (CTFT): x(t)x(t): infinite, continuous-time

We will think of Fourier representations in two complimentary senses:

  1. “Eigenbasis” representations: Each Fourier transform pair is very naturally related to an appropriate class of LTI systems. In some cases we can think of a Fourier transform as a change of basis.
  2. Unitary operators: While we often use Fourier transforms to analyze certain operators, we can also think of a Fourier transform as itself being an operator.
Figure 1
Various venn diagrams showing the relationship between finite and infinite discrete time, infinite continuous time, DFT, IDFT, CTFS, ICTFS, and others.

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