# discrete cosine transform

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## discrete cosine transform

(mathematics)(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.

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.

This article is provided by FOLDOC - Free Online Dictionary of Computing (

**foldoc.org**)## DCT

(**D**iscrete

**C**osine

**T**ransform) An algorithm that is widely used for data compression. Similar to Fast Fourier Transform, DCT converts data (pixels, waveforms, etc.) into sets of frequencies. The first frequencies in the set are the most meaningful; the latter, the least. To compress data, the least meaningful frequencies are stripped away based on allowable resolution loss. DCT is used to compress JPEG, MPEG, DV and H.263 frames.

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