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
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)
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
There exist fast DCT algorithms in analogy to the Fast Fourier Transform.
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discrete Fourier transform
<mathematics> (DFT) A Fourier transform, specialized to the case where
the abscissas are integers.
The DFT is central to many kinds of signal processing, including the analysis
and compression of video and sound information.
A common implementation of the DFT is the Fast Fourier Transform (FFT).
See also discrete cosine transform.
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<mathematics> A preorder is said to be discrete if any two of its
elements are incomparable.
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