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In applied mathematics, a DFT matrix is an expression of a discrete Fourier transform ... For other properties of the DFT matrix, including its eigenvalues, ...
A useful property of the DFT is that the inverse DFT can be easily expressed in terms of the (forward) DFT, via several well-known "tricks". (For example, in computations, it is often convenient to only implement a fast Fourier transform corresponding to one transform direction and then to get the other transform direction from the first.)
The lower right corner depicts samples of the DTFT that are computed by a discrete Fourier transform (DFT). The utility of the DTFT is rooted in the Poisson summation formula, which tells us that the periodic function represented by the Fourier series is a periodic summation of the continuous Fourier transform: [b]
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa.
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.
The discrete Fourier transform is defined by a specific Vandermonde matrix, the DFT matrix, where the are chosen to be n th roots of unity. The Fast Fourier transform computes the product of this matrix with a vector in O ( n log 2 n ) {\displaystyle O(n\log ^{2}n)} time.
An example application of the Fourier transform is determining the constituent pitches in a musical waveform.This image is the result of applying a constant-Q transform (a Fourier-related transform) to the waveform of a C major piano chord.
An circulant matrix takes the form = [] or the transpose of this form (by choice of notation). If each is a square matrix, then the matrix is called a block-circulant matrix.. A circulant matrix is fully specified by one vector, , which appears as the first column (or row) of .