Discrete Cosine Transform

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}.

   (1997-03-10)