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 Fouriertransform (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 FastFourier Transform.

(1997-03-10)