Editing Discrete Fourier transform (section)

=== Other fields ===
{{Main|Discrete Fourier transform (general)|Number-theoretic transform}}
Many of the properties of the DFT only depend on the fact that <math>e^{-\frac{i 2 \pi}{N}}</math> is a [[primitive root of unity]], sometimes denoted <math>\omega_N</math> or <math>W_N</math> (so that <math>\omega_N^N = 1</math>).  Such properties include the completeness, orthogonality, Plancherel/Parseval, periodicity, shift, convolution, and unitarity properties above, as well as many FFT algorithms. For this reason, the discrete Fourier transform can be defined by using roots of unity in [[field (mathematics)|fields]] other than the complex numbers, and such generalizations are commonly called ''number-theoretic transforms'' (NTTs) in the case of [[finite field]]s. For more information, see [[number-theoretic transform]] and [[discrete Fourier transform (general)]].