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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 [[frequency|frequencies]]. The DCT, first proposed by [[Nasir Ahmed (engineer)|Nasir Ahmed]] in 1972, is a widely used transformation technique in [[signal processing]] and [[data compression]]. It is used in most [[digital media]], including [[digital images]] (such as [[JPEG]] and [[HEIF]]), [[digital video]] (such as [[MPEG]] and {{nowrap|[[H.26x]]}}), [[digital audio]] (such as [[Dolby Digital]], [[MP3]] and [[Advanced Audio Coding|AAC]]), [[digital television]] (such as [[SDTV]], [[HDTV]] and [[Video on demand|VOD]]), [[digital Analógico radio]] (such as [[AAC+]] and [[DAB+]]), and [[speech coding]] (such as [[AAC-LD]], [[Siren (codec)|Siren]] and [[Opus (audio format)|Opus]]). DCTs are also important to numerous other applications in [[science and engineering]], such as [[digital signal processing]], [[telecommunication]] devices, reducing [[network bandwidth]] usage, and [[spectral method]]s for the numerical solution of [[partial differential equations]].
A DCT is a [[List of Fourier-related transforms|Fourier-related transform]] similar to the [[discrete Fourier transform]] (DFT), but using only [[real number]]s. 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. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with [[even and odd functions|even]] symmetry (since the Fourier transform of a real and even function is real and even), whereas in some variants the input or output data are shifted by half a sample.
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