Editing Discrete cosine transform (section)

{{Short description|Technique used in signal processing and data compression}}

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 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.

There are eight standard DCT variants, of which four are common.
The most common variant of discrete cosine transform is the type-II DCT, which is often called simply ''the DCT''. This was the original DCT as first proposed by Ahmed. 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 function]]s, and the [[modified discrete cosine transform]]   (MDCT), which is based on a DCT of overlapping data. Multidimensional DCTs (MD DCTs) are developed to extend the concept of DCT to multidimensional signals. A variety of fast algorithms have been developed to reduce the computational complexity of implementing DCT. One of these is the integer DCT (IntDCT),<ref name="Stankovic"/> an [[integer]] approximation of the standard DCT,<ref name="Britanak2010" />{{rp|pages= [https://books.google.com/books?id=iRlQHcK-r_kC&pg=PA141 ix, xiii, 1, 141–304]}} used in several [[ISO/IEC]] and [[ITU-T]] international standards.<ref name="Stankovic"/><ref name="Britanak2010"/>

DCT compression, also known as block compression, compresses data in sets of discrete DCT blocks.<ref name="Alikhani"/> DCT blocks sizes including 8x8 [[pixels]] for the standard DCT, and varied integer DCT sizes between 4x4 and 32x32 pixels.<ref name="Stankovic"/><ref name="apple"/> The DCT has a strong ''energy compaction'' property,<ref name="pubDCT"/><ref name="pubRaoYip"/> capable of achieving high quality at high [[data compression ratio]]s.<ref name="Barbero"/><ref name="Lea">{{cite journal|last1=Lea|first1=William|date=1994|title=Video on demand: Research Paper 94/68|url=https://researchbriefings.parliament.uk/ResearchBriefing/Summary/RP94-68|journal=[[House of Commons Library]]|access-date=20 September 2019}}</ref> However, blocky [[compression artifacts]] can appear when heavy DCT compression is applied.