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A DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. The DCTs are generally related to Fourier series coefficients of a periodically and symmetrically extended sequence whereas DFTs are related to Fourier series coefficients of only periodically extended sequences.
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 ...
For example, for the Fourier transform the counterpart is the discrete Fourier transform. In addition to spectral analysis of signals, discrete transforms play important role in data compression , signal detection , digital filtering and correlation analysis. [ 2 ]
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 ]
The DCT and DFT are often used in signal processing [6] and image processing, and they are also used to efficiently solve partial differential equations by spectral methods. The DFT can also be used to perform other operations such as convolutions or multiplying large integers.
In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using a purely real matrix.It is equivalent to the imaginary parts of a DFT of roughly twice the length, operating on real data with odd symmetry (since the Fourier transform of a real and odd function is imaginary and odd), where in some variants the input and ...
Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period.
In mathematics the finite Fourier transform may refer to either . another name for discrete-time Fourier transform (DTFT) of a finite-length series. E.g., F.J.Harris (pp. 52–53) describes the finite Fourier transform as a "continuous periodic function" and the discrete Fourier transform (DFT) as "a set of samples of the finite Fourier transform".