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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 ...
One can ask whether the DFT matrix is unitary over a finite field. If the matrix entries are over F q {\displaystyle F_{q}} , then one must ensure q {\displaystyle q} is a perfect square or extend to F q 2 {\displaystyle F_{q^{2}}} in order to define the order two automorphism x ↦ x q {\displaystyle x\mapsto x^{q}} .
The conversion from continuous time to samples (discrete-time) changes the underlying Fourier transform of () into a discrete-time Fourier transform (DTFT), which generally entails a type of distortion called aliasing. Choice of an appropriate sample-rate (see Nyquist rate) is the key to minimizing that distortion.
Both transforms are invertible. The inverse DTFT reconstructs the original sampled data sequence, while the inverse DFT produces a periodic summation of the original sequence. The Fast Fourier Transform (FFT) is an algorithm for computing one cycle of the DFT, and its inverse produces one cycle of the inverse DFT.
They may also include density functional theory (DFT), molecular mechanics or semi-empirical quantum chemistry methods. The programs include both open source and commercial software. Most of them are large, often containing several separate programs, and have been developed over many years.
which is the inverse DFT of one cycle of the periodic summation, . [1]: p.542 (eq 8.4) [3]: p.77 (eq 4.24 ...
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.
The DFT is (or can be, through appropriate selection of scaling) a unitary transform, i.e., one that preserves energy. The appropriate choice of scaling to achieve unitarity is 1 / N {\displaystyle 1/{\sqrt {N}}} , so that the energy in the physical domain will be the same as the energy in the Fourier domain, i.e., to satisfy Parseval's theorem .
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