Discrete Fourier Transform

Updated 02 Oct 2026

•
What it is
- •
  represents a finite collection of samples using complex sinusoidal components
•
Two-dimensional formula
- •
  for an $M\times N$ image, $F[u,v]=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}f[m,n]e^{-2\pi i(um/M+vn/N)}$ .
- •
  the corresponding inverse includes a factor $1/(MN)$ under this convention.
•
Interpretation
- •
  low spatial frequencies describe slowly varying structure
- •
  high spatial frequencies describe rapid changes, including edges, fine texture, and some nosie
- •
  the dc coefficient satisfies $F[0,0]=\sum_{m,n}f[m,n]$ .
•
Implementation
- •
  FFT is an efficient algorithm for computing the DFT, not a different transform.
- •
  fftshift rearranges frequency bins to place zero frequency centrally; it does not alter their values.
- •
  undo that rearrangement with ifftshift before the inverse transform.
•
Related
- •
  Fourier Series
- •
  Unavailable reference
- •
  Frequency-Domain Filtering
•
Reference
- •
  https://numpy.org/doc/stable/reference/routines.fft.html

Linked references 2