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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 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 ]
In applied mathematics, the sliding discrete Fourier transform is a recursive algorithm to compute successive STFTs of input data frames that are a single sample apart (hopsize − 1). [1] The calculation for the sliding DFT is closely related to Goertzel algorithm .
In addition to spectral analysis of signals, discrete transforms play important role in data compression, signal detection, digital filtering and correlation analysis. [2] The discrete cosine transform (DCT) is the most widely used transform coding compression algorithm in digital media , followed by the discrete wavelet transform (DWT).
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 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.
The development of fast algorithms for DFT can be traced to Carl Friedrich Gauss's unpublished 1805 work on the orbits of asteroids Pallas and Juno.Gauss wanted to interpolate the orbits from sample observations; [6] [7] his method was very similar to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 .