Modified discrete cosine transform: Difference between revisions

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This overlapping, in addition to the energy-compaction qualities of the DCT, makes the MDCT especially attractive for signal compression applications, since it helps to avoid [[compression artifact|artifacts]] stemming from the block boundaries. As a result of these advantages, the MDCT is employed in most modern lossy audio formats, including [[MP3]], [[Dolby AC-3|AC-3]], [[Vorbis]], [[Windows Media Audio]], [[ATRAC]], [[cook codec|Cook]], [[advanced audio coding|AAC]], [[Opus_(audio_format)|Opus]], and [[LDAC_(codec)|LDAC]].

Discrete cosine transform (DCT) was introduced by [[N. Ahmed|Nasir Ahmed]], T. Natarajan and [[K. R. Rao]] in 1974.<ref name="pubDCT">{{Citation |first=Nasir |last=Ahmed |author1-link=N. Ahmed |first2=T. |last2=Natarajan |first3=K. R. |last3=Rao |title=Discrete Cosine Transform |journal=IEEE Transactions on Computers |date=January 1974 |volume=C-23 |issue=1 |pages=90–93 |doi=10.1109/T-C.1974.223784}}</ref> The MDCT was proposed by J. P. Princen, A. W. Johnson, and A. B. Bradley in 1987,<ref>J. P. Princen, A. W. Johnson und A. B. Bradley: ''Subband/transform coding using filter bank designs based on time ___domain aliasing cancellation'', IEEE Proc. Intl. Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2161–2164, 1987. Initial description of what is now called the MDCT.</ref> in 1987, following earlier (1986) work by Princen and Bradley (1986)<ref>John P. Princen, Alan B. Bradley: ''Analysis/synthesis filter bank design based on time ___domain aliasing cancellation'', IEEE Trans. Acoust. Speech Signal Processing, ''ASSP-34'' (5), 1153–1161, 1986. Described a precursor to the MDCT using a combination of discrete cosine and sine transforms.</ref> to develop the MDCT's underlying principle of '''time-___domain aliasing cancellation''' (TDAC), described below. (There also exists an analogous transform, the MDST, based on the [[discrete sine transform]], as well as other, rarely used, forms of the MDCT based on different types of DCT or DCT/DST combinations.)

In MP3, the MDCT is not applied to the audio signal directly, but rather to the output of a 32-band [[polyphase quadrature filter]] (PQF) bank. The output of this MDCT is postprocessed by an alias reduction formula to reduce the typical aliasing of the PQF filter bank. Such a combination of a filter bank with an MDCT is called a ''hybrid'' filter bank or a ''subband'' MDCT. AAC, on the other hand, normally uses a pure MDCT; only the (rarely used) [[MPEG-4 AAC-SSR]] variant (by [[Sony]]) uses a four-band PQF bank followed by an MDCT. Similar to MP3, [[ATRAC]] uses stacked [[quadrature mirror filter]]s (QMF) followed by an MDCT.