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An N-point DFT is expressed as the multiplication =, where is the original input signal, is the N-by-N square DFT matrix, and is the DFT of the signal. The transformation matrix W {\displaystyle W} can be defined as W = ( ω j k N ) j , k = 0 , … , N − 1 {\displaystyle W=\left({\frac {\omega ^{jk}}{\sqrt {N}}}\right)_{j,k=0,\ldots ,N-1 ...
Over the complex numbers, it is often customary to normalize the formulas for the DFT and inverse DFT by using the scalar factor in both formulas, rather than in the formula for the DFT and in the formula for the inverse DFT. With this normalization, the DFT matrix is then unitary.
An example FFT algorithm structure, using a decomposition into half-size FFTs A discrete Fourier analysis of a sum of cosine waves at 10, 20, 30, 40, and 50 Hz. A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT).
The split-radix FFT is a fast Fourier transform (FFT) algorithm for computing the discrete Fourier transform (DFT), and was first described in an initially little-appreciated paper by R. Yavne (1968) and subsequently rediscovered simultaneously by various authors in 1984.
For this reason, the discrete Fourier transform can be defined by using roots of unity in fields other than the complex numbers, and such generalizations are commonly called number-theoretic transforms (NTTs) in the case of finite fields. For more information, see number-theoretic transform and discrete Fourier transform (general).
Density functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases.
In applied mathematics, the non-uniform discrete Fourier transform (NUDFT or NDFT) of a signal is a type of Fourier transform, related to a discrete Fourier transform or discrete-time Fourier transform, but in which the input signal is not sampled at equally spaced points or frequencies (or both).
The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calculated via the discrete Fourier transform (DFT) (see § Sampling the DTFT), which is by far the most common method of modern Fourier analysis. Both transforms are invertible.