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In applied mathematics, a DFT matrix is an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal through matrix multiplication. Definition [ edit ]
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration ...
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.
From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function. In simpler terms, when you take the DTFT of regularly-spaced samples of a continuous signal, you get repeating (and possibly overlapping) copies of the signal's frequency ...
The development of fast algorithms for DFT was prefigured in Carl Friedrich Gauss's unpublished 1805 work on the orbits of asteroids Pallas and Juno.Gauss wanted to interpolate the orbits from sample observations; [6] [7] his method was very similar to the one that would be published in 1965 by James Cooley and John Tukey, who are generally credited for the invention of the modern generic FFT ...
There is a direct relationship between the Fourier transform on finite groups and the representation theory of finite groups.The set of complex-valued functions on a finite group, , together with the operations of pointwise addition and convolution, form a ring that is naturally identified with the group ring of over the complex numbers, [].
A decimation-in-time radix-2 FFT breaks a length-N DFT into two length-N/2 DFTs followed by a combining stage consisting of many butterfly operations. More specifically, a radix-2 decimation-in-time FFT algorithm on n = 2 p inputs with respect to a primitive n -th root of unity ω n k = e − 2 π i k n {\displaystyle \omega _{n}^{k}=e^{-{\frac ...
When the DFT is used for spectral analysis, the {x n} sequence usually represents a finite set of uniformly spaced time-samples of some signal x(t) where t represents time. The conversion from continuous time to samples (discrete-time) changes the underlying Fourier transform of x ( t ) into a discrete-time Fourier transform (DTFT), which ...