S> (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}. (1997-03-10)
