DISCRETE COSINE TRANSFORM
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Definitions of DISCRETE COSINE TRANSFORM
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(DCT) A technique for expressing a waveform as aweighted 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-1B(k) is defined for all values of the frequency-space variablek, 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-1where delta(k) is the Kronecker delta.The main difference between this and a discrete Fouriertransform (DFT) is that the DFT traditionally assumes thatthe data A(i) is periodically continued with a period of N,whereas the DCT assumes that the data is continued with itsmirror 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 thecoefficients are approximated does compression occur.There exist fast DCT algorithms in analogy to the FastFourier Transform.
By Denis Howe