The discrete Fourier transform (DFT) is the primary transform used for numerical computation in digital signal processing.
It is very widely used for spectrum analysis,
fast convolution, and many other applications.
The DFT transforms
Inside Collection (Course): Digital Signal Processing: A User's Guide
Summary: The discrete Fourier transform (DFT) and its inverse (IDFT) are the primary numerical transforms relating time and frequency in digital signal processing. The DFT has a number of important properties relating time and frequency, including shift, circular convolution, multiplication, timereversal and conjugation properties, as well as Parseval's theorem equating time and frequency energy.
The discrete Fourier transform (DFT) is the primary transform used for numerical computation in digital signal processing.
It is very widely used for spectrum analysis,
fast convolution, and many other applications.
The DFT transforms
The inverse DFT (IDFT) transforms
Due to the
A shift in time corresponds to a phase shift that is linear in frequency.
Because of the periodicity induced by the DFT and IDFT, the shift is circular, or modulo

Circular convolution is defined as
Circular convolution of two discretetime signals corresponds
to multiplication of their DFTs:
A similar property relates multiplication in time to circular convolution in frequency.
Parseval's theorem relates the energy of a length
The continuoustime Fourier transform, the DTFT, and DFT are all defined as transforms of complexvalued data to complexvalued spectra. However, in practice signals are often realvalued. The DFT of a realvalued discretetime signal has a special symmetry, in which the real part of the transform values are DFT even symmetric and the imaginary part is DFT odd symmetric, as illustrated in the equation and figure below.
