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Its inverse, the type-III DCT, is correspondingly often called simply the inverse DCT or the IDCT. Two related transforms are the discrete sine transform (DST), which is equivalent to a DFT of real and odd functions, and the modified discrete cosine transform (MDCT), which is based on a DCT of overlapping data. Multidimensional DCTs (MD DCTs ...
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.
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.
Two-dimensional DCT frequencies from the JPEG DCT. 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 ...
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 ...
The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size = in terms of N 1 smaller DFTs of sizes N 2, recursively, to reduce the computation time to O(N log N) for highly composite N (smooth numbers).
The development of fast algorithms for DFT can be traced to Carl Friedrich Gauss's unpublished 1805 work on the orbits of asteroids Pallas and Juno.Gauss wanted to interpolate the orbits from sample observations; [6] [7] his method was very similar to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 ...
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 .