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.

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Formally, the discrete cosine transform is a linear, invertible function *F* : **R**^{n} `->` **R**^{n} (where **R** denotes the set of real numbers), or equivalently an *n* × *n* square matrix. There are several variants of the DCT with slightly modified definitions. The *n* real numbers *x*_{0}, ..., *x*_{n-1} are transformed into the *n* real numbers *f*_{0}, ..., *f*_{n-1} according to one of the formulas:

A DCT-I of *n*=5 real numbers *abcde* is exactly equivalent to a DFT of eight real numbers *abcdedcb* (even symmetry), divided by two. (In contrast, DCT types II-IV involve a half-sample shift in the equivalent DFT.) Note, however, that the DCT-I is not defined for *n* less than 2. (All other DCT types are defined for any positive *n*.)

Thus, the DCT-I corresponds to the boundary conditions: *x*_{k} is even around *k*=0 and even around *k*=*n*-1; similarly for *f*_{j}.

Some authors further multiply the *f*_{0} term by 1/√2 (see below for the corresponding change in DCT-III). This makes the DCT-II matrix orthogonal (up to a scale factor), but breaks the direct correspondence with a real-even DFT of half-shifted input.

The DCT-II implies the boundary conditions: *x*_{k} is even around *k*=-1/2 and even around *k*=*n*-1/2; *f*_{j} is even around *j*=0 and odd around *j*=*n*.

Some authors further multiply the *x*_{0} term by √2 (see above for the corresponding change in DCT-II). This makes the DCT-III matrix orthogonal (up to a scale factor), but breaks the direct correspondence with a real-even DFT of half-shifted output.

The DCT-III implies the boundary conditions: *x*_{k} is even around *k*=0 and odd around *k*=*n*; *f*_{j} is even around *j*=-1/2 and odd around *j*=*n*-1/2.

A variant of the DCT-IV, where data from different transforms are *overlapped*, is called the modified discrete cosine transform (MDCT).

The DCT-IV implies the boundary conditions: *x*_{k} is even around *k*=-1/2 and odd around *k*=*n*-1/2; similarly for *f*_{j}.

(The trivial real-even array, a length-one DFT (odd length) of a single number *a*, corresponds to a DCT-V of length *n*=1.)

Like for the DFT, the normalization factor in front of these transform definitions is merely a convention and differs between treatments. For example, some authors multiply the transforms by so that the inverse does not require any additional multiplicative factor.

Although the direct application of these formulas would require O(*n*^{2}) operations, as in the fast Fourier transform (FFT) it is possible to compute the same thing with only O(*n* log *n*) complexity by factorizing the computation. (One can also compute DCTs via FFTs combined with O(*n*) pre- and post-processing steps.)

The DCT, and in particular the DCT-II, is often used in signal and image processing, especially for lossy data compression, because it has a strong "energy compaction" property: most of the signal information tends to be concentrated in a few low-frequency components of the DCT, approaching the optimal Karhunen-Loève transform for signals based on certain limits of Markov processes.

For example, the DCT is used in JPEG image compression, MJPEG video compression, and MPEG video compression. There, the two-dimensional DCT-II of 8x8 blocks is computed and the results are filtered to discard small (difficult-to-see) components. That is, *n* is 8 and the DCT-II formula is applied to each row and column of the block. The result is an array in which the top left corner is the DC (zero-frequency) component and lower and rightmore entries represent larger vertical and horizontal spatial frequencies. For the chrominance components, *n* is 16 but the frequency components beyond the first 8 are discarded.

A related transform, the modified discrete cosine transform (MDCT), is used in AAC, Vorbis, and MP3 audio compression.

DCTs are also widely employed in solving partial differential equations by spectral methods, where the different variants of the DCT correspond to slightly different even/odd boundary conditions at the two ends of the array.

- K. R. Rao and P. Yip,
*Discrete Cosine Transform: Algorithms, Advantages, Applications*(Academic Press, Boston, 1990). - A. V. Oppenheim, R. W. Schafer, and J. R. Buck,
*Discrete-Time Signal Processing*, second edition (Prentice-Hall, New Jersey, 1999). - S. A. Martucci, "Symmetric convolution and the discrete sine and cosine transforms,"
*IEEE Trans. Sig. Processing***SP-42**, 1038-1051 (1994). - Matteo Frigo and Steven G. Johnson:
*FFTW*, http://www.fftw.org/. A free (GPL) C library that can compute fast DCTs (types I-IV) in one or more dimensions, of arbitrary size.