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.