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A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT, first proposed by Nasir Ahmed in 1972, is a widely used transformation technique in signal processing and data compression .
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 .
The DCT is used in JPEG image compression, MJPEG, MPEG, DV, Daala, and Theora video compression. There, the two-dimensional DCT-II of NxN blocks are computed and the results are quantized and entropy coded. In this case, N is typically 8 and the DCT-II formula is applied to each row and column of the block. The result is an 8x8 transform ...
The plot shows the differences between a DFT and a DCT of a generic signal. The first plot is a sampled ramp in the time domain. The second one represents the modulus of its DFT. The third one the plot of its DCT. I obtained it in a two step process. First I ran the following Matlab code: thus creating a file called example_dft_dct.dat.
An example of both these distributions is depicted in the two traces of Fig 1. [ e ] [ f ] [ g ] Aliasing occurs when adjacent copies of X ( f ) overlap. The purpose of the anti-aliasing filter is to ensure that the reduced periodicity does not create overlap.
In applied mathematics, a discrete Chebyshev transform (DCT) is an analog of the discrete Fourier transform for a function of a real interval, converting in either direction between function values at a set of Chebyshev nodes and coefficients of a function in Chebyshev polynomial basis.
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 modified discrete cosine transform (MDCT) is a transform based on the type-IV discrete cosine transform (DCT-IV), with the additional property of being lapped: it is designed to be performed on consecutive blocks of a larger dataset, where subsequent blocks are overlapped so that the last half of one block coincides with the first half of the next block.