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. 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
artifacts stemming from the block boundaries. As a result of these advantages, the MDCT is the most widely used
lossy compression
In information technology, lossy compression or irreversible compression is the class of data compression methods that uses inexact approximations and partial data discarding to represent the content. These techniques are used to reduce data size ...
technique in
audio data compression. It is employed in most modern
audio coding standards, including
MP3
MP3 (formally MPEG-1 Audio Layer III or MPEG-2 Audio Layer III) is a coding format for digital audio developed largely by the Fraunhofer Society in Germany, with support from other digital scientists in the United States and elsewhere. Orig ...
,
Dolby Digital
Dolby Digital, originally synonymous with Dolby AC-3, is the name for what has now become a family of audio compression (data), audio compression technologies developed by Dolby Laboratories. Formerly named Dolby Stereo Digital until 1995 in film, ...
(AC-3),
Vorbis
Vorbis is a free and open-source software project headed by the Xiph.Org Foundation. The project produces an audio coding format and software reference encoder/decoder (codec) for lossy audio compression. Vorbis is most commonly used in con ...
(Ogg),
Windows Media Audio
Windows Media Audio (WMA) is a series of audio codecs and their corresponding audio coding formats developed by Microsoft. It is a proprietary technology that forms part of the Windows Media framework. WMA consists of four distinct codecs. The ...
(WMA),
ATRAC,
Cook,
Advanced Audio Coding
Advanced Audio Coding (AAC) is an audio coding standard for lossy digital audio compression. Designed to be the successor of the MP3 format, AAC generally achieves higher sound quality than MP3 encoders at the same bit rate.
AAC has been sta ...
(AAC),
High-Definition Coding HDC (Hybrid Digital Coding or High-Definition Coding) with SBR ( spectral band replication) is a proprietary lossy audio compression codec developed by iBiquity for use with HD Radio. It replaced the earlier PAC codec in 2003. In June 2017, the fo ...
(HDC),
LDAC,
Dolby AC-4, and
MPEG-H 3D Audio, as well as
speech coding
Speech coding is an application of data compression of digital audio signals containing speech. Speech coding uses speech-specific parameter estimation using audio signal processing techniques to model the speech signal, combined with generic ...
standards such as
AAC-LD (LD-MDCT),
G.722.1,
G.729.1,
CELT
The Celts (, see pronunciation for different usages) or Celtic peoples () are. "CELTS location: Greater Europe time period: Second millennium B.C.E. to present ancestry: Celtic a collection of Indo-European peoples. "The Celts, an ancien ...
,
[Presentation of the CELT codec](_blank)
by Timothy B. Terriberry (65 minutes of video, see als
presentation slides
in PDF) and
Opus.
The
discrete cosine transform (DCT) was first proposed by
Nasir Ahmed in 1972,
and demonstrated by Ahmed with T. Natarajan and
K. R. Rao
Kamisetty Ramamohan Rao was an Indian-American electrical engineer. He was a professor of Electrical Engineering at the University of Texas at Arlington (UT Arlington). Academically known as K. R. Rao, he is credited with the co-invention of di ...
in 1974.
The MDCT was later proposed by John P. Princen, A.W. Johnson and Alan B. Bradley at the
University of Surrey
The University of Surrey is a public research university in Guildford, Surrey, England. The university received its royal charter in 1966, along with a number of other institutions following recommendations in the Robbins Report. The institu ...
in 1987, following earlier work by Princen and Bradley (1986) 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 In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using a purely real matrix. It is equivalent to the imaginary parts of a DFT of roughly twice the length, operating ...
, 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
MPEG-4 Part 3 or MPEG-4 Audio (formally ISO/IEC 14496-3) is the third part of the ISO/IEC MPEG-4 international standard developed by Moving Picture Experts Group. It specifies audio coding methods. The first version of ISO/IEC 14496-3 was publish ...
variant (by
Sony
, commonly stylized as SONY, is a Japanese multinational conglomerate corporation headquartered in Minato, Tokyo, Japan. As a major technology company, it operates as one of the world's largest manufacturers of consumer and professional ...
) uses a four-band PQF bank followed by an MDCT. Similar to MP3,
ATRAC uses stacked
quadrature mirror filters (QMF) followed by an MDCT.
Definition
As a lapped transform, the MDCT is a bit unusual compared to other Fourier-related transforms in that it has half as many outputs as inputs (instead of the same number). In particular, it is a
linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...
(where R denotes the set of
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s). The 2''N'' real numbers ''x''
0, ..., ''x''
2''N''-1 are transformed into the ''N'' real numbers ''X''
0, ..., ''X''
''N''-1 according to the formula:
: