Modified discrete cosine transform

MDCTMDCT-basedMDCTs
The modified discrete cosine transform (MDCT) is a lapped 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.wikipedia
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Discrete cosine transform

DCTiDCTinverse discrete cosine transform
The modified discrete cosine transform (MDCT) is a lapped 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,
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.

Lapped transform

discrete cosine transform with overlapping blocks
The modified discrete cosine transform (MDCT) is a lapped 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,
The best known example is the modified discrete cosine transform used in the MP3, Vorbis, AAC, and Opus audio codecs.

Vorbis

Ogg VorbisOGGaoTuV
As a result of these advantages, the MDCT is employed in most modern lossy audio formats, including MP3, AC-3, Vorbis, Windows Media Audio, ATRAC, Cook, AAC, Opus, and LDAC.
Vorbis had been shown to perform significantly better than many other lossy audio formats in the past in that it produced smaller files at equivalent or higher quality while retaining computational complexity comparable to other MDCT formats such as AAC or Windows Media Audio.

Windows Media Audio

WMAXMA.wma
As a result of these advantages, the MDCT is employed in most modern lossy audio formats, including MP3, AC-3, Vorbis, Windows Media Audio, ATRAC, Cook, AAC, Opus, and LDAC.
Fundamentally, WMA is a transform coder based on modified discrete cosine transform (MDCT), somewhat similar to AAC, Cook and Vorbis.

Adaptive Transform Acoustic Coding

ATRACATRAC3Adaptive Transform Acoustic Coding (ATRAC)
As a result of these advantages, the MDCT is employed in most modern lossy audio formats, including MP3, AC-3, Vorbis, Windows Media Audio, ATRAC, Cook, AAC, Opus, and LDAC. Similar to MP3, ATRAC uses stacked quadrature mirror filters (QMF) followed by an MDCT.
Like ATRAC1 and MP3, ATRAC3 is also a hybrid subband-MDCT encoder, but with several differences.

Compression artifact

artifactscompression artifactsmosquito noise
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.
Lossy audio formats typically involve the use of a time/frequency domain transform, such as a modified discrete cosine transform.

Advanced Audio Coding

AACAAC/AAC+/eAAC+.aac
As a result of these advantages, the MDCT is employed in most modern lossy audio formats, including MP3, AC-3, Vorbis, Windows Media Audio, ATRAC, Cook, AAC, Opus, and LDAC.
higher efficiency and simpler filter bank (rather than MP3's hybrid coding, AAC uses a pure MDCT);

Polyphase quadrature filter

polyphased
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.
Similar to the MDCT time domain alias cancellation the aliasing of polyphase quadrature filters is canceled by neighbouring sub-bands, i.e. signals are typically stored in two sub-bands.

Cook Codec

CookCook'' codecRealAudio Gecko
As a result of these advantages, the MDCT is employed in most modern lossy audio formats, including MP3, AC-3, Vorbis, Windows Media Audio, ATRAC, Cook, AAC, Opus, and LDAC.
It is a pure transform codec based on the modified discrete cosine transform with a single block size.

Dolby Digital

Dolby Digital 5.1AC3AC-3
As a result of these advantages, the MDCT is employed in most modern lossy audio formats, including MP3, AC-3, Vorbis, Windows Media Audio, ATRAC, Cook, AAC, Opus, and LDAC.
Channel blocks can be either long, in which case the entire block is processed as single modified discrete cosine transform or short, in which case two half length transforms are performed on the block.

MPEG-4 Part 3

AACAAC+ISO/IEC 14496-3
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.
Then these 4 bands are further split using MDCTs with a size k of 32 or 256 samples.

Short-time Fourier transform

STFTresolution issues of the short-time Fourier transformshort time Fourier transform
Short-time Fourier transform
See also the modified discrete cosine transform (MDCT), which is also a Fourier-related transform that uses overlapping windows.

Discrete Fourier transform

DFTcircular convolution theoremFourier transform
In order to define the precise relationship to the DCT-IV, one must realize that the DCT-IV corresponds to alternating even/odd boundary conditions: even at its left boundary (around n=−1/2), odd at its right boundary (around n=N−1/2), and so on (instead of periodic boundaries as for a DFT).
(Compression applications often use a specialized form of the DFT, the discrete cosine transform or sometimes the modified discrete cosine transform.)

Window function

window functionswindowingwindowed
In typical signal-compression applications, the transform properties are further improved by using a window function w n (n = 0, ..., 2N−1) that is multiplied with x n and y n in the MDCT and IMDCT formulas, above, in order to avoid discontinuities at the n = 0 and 2N boundaries by making the function go smoothly to zero at those points.
See Welch method of power spectral analysis and the modified discrete cosine transform.

Modulated complex lapped transform

Modulated complex lapped transform
The modulated complex lapped transform (MCLT) is a lapped transform, similar to the modified discrete cosine transform, that explicitly represents the phase (complex values) of the signal.

Data set

datasetdatasetsdata
The modified discrete cosine transform (MDCT) is a lapped 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,

MP3

.mp3MP3 downloadmp3s
As a result of these advantages, the MDCT is employed in most modern lossy audio formats, including MP3, AC-3, Vorbis, Windows Media Audio, ATRAC, Cook, AAC, Opus, and LDAC.

Discrete sine transform

DSTsine
(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.)

Sony

Sony CorporationSony BMG MusicSony Music
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.

Quadrature mirror filter

QMF
Similar to MP3, ATRAC uses stacked quadrature mirror filters (QMF) followed by an MDCT.

Linear function

linearlinear functionslinear factor
In particular, it is a linear function (where R denotes the set of real numbers).

Real number

realrealsreal-valued
In particular, it is a linear function (where R denotes the set of real numbers).

Fast Fourier transform

FFTFast Fourier Transform (FFT)Fourier
Although the direct application of the MDCT formula would require O(N 2 ) operations, it is possible to compute the same thing with only O(N log N) complexity by recursively factorizing the computation, as in the fast Fourier transform (FFT).

Nyquist frequency

NyquistN/2 different frequenciesNyquist component
The use of input data that extend beyond the boundaries of the logical DCT-IV causes the data to be aliased in the same way that frequencies beyond the Nyquist frequency are aliased to lower frequencies, except that this aliasing occurs in the time domain instead of the frequency domain: we cannot distinguish the contributions of