The Fourier Transform

  • The 1D Continuous Fourier Transform
  • The 2D Continuous Fourier Transform
  • The 1D Discrete Fourier Transform (1D DFT)
  • The 2D Discrete Fourier Transform (2D DFT)
  • Review Euler's Formula
  • Review some important relevant DFT symmetry theorems.
  • Definition of magnitude response and phase response.
  • Definition of power spectrum.
  • Block diagram of time domain filtering and corresponding diagram of frequency domain filtering.
  • Frequency domain filtering, i.e., convolution in the time domain is multiplication in the frequency domain.
  • Zero padding signals, what it does and why one needs to do it for frequency domain filtering.
  • 2D DFT Properties, i.e., the location of a specific frequency by DFT sample index and the ordering of frequencies as a function of the sample index.
  • Modulation of the 2D dft to place the DC component at DFT sample (M/2,N/2) for an (M,N) image.
  • What this means for a geometric interpretation of the 2D DFT.
  • What the 2D DFT arguments mean in terms of different quantities, i.e., radians/sample, cycles/sample, and cycles/image row or cycles/image column.
  • Review of Nyquist's sampling theorem and its relevance to the DFT.