 ## discrete cosine transform

(DCT) A technique for expressing a waveform as a weighted sum of cosines.

The DCT is central to many kinds of signal processing, especially video compression.

Given data A(i), where i is an integer in the range 0 to N-1, the forward DCT (which would be used e.g. by an encoder) is:

``` B(k) =    sum    A(i) cos((pi k/N) (2 i + 1)/2)
i=0 to N-1

```
B(k) is defined for all values of the frequency-space variable k, but we only care about integer k in the range 0 to N-1. The inverse DCT (which would be used e.g. by a decoder) is:

``` AA(i)=    sum    B(k) (2-delta(k-0)) cos((pi k/N)(2 i + 1)/2)
k=0 to N-1

```
where delta(k) is the Kronecker delta.

The main difference between this and a discrete Fourier transform (DFT) is that the DFT traditionally assumes that the data A(i) is periodically continued with a period of N, whereas the DCT assumes that the data is continued with its mirror image, then periodically continued with a period of 2N.

Mathematically, this transform pair is exact, i.e. AA(i) == A(i), resulting in lossless coding; only when some of the coefficients are approximated does compression occur.

There exist fast DCT algorithms in analogy to the Fast Fourier Transform.

Last updated: 1997-03-10

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