discrete cosine transform definition
<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)
Nearby terms:
disclaimer « disconnect « Discordianism « discrete cosine transform » discrete Fourier transform » discrete preorder » discriminated union
discrete Fourier transform definition
<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.
(1997-03-10)
Nearby terms:
disconnect « Discordianism « discrete cosine transform « discrete Fourier transform » discrete preorder » discriminated union » discussion group
discrete preorder definition
<mathematics> A preorder is said to be discrete if any two
of its elements are incomparable.
(1995-09-21)
Nearby terms:
Discordianism « discrete cosine transform « discrete Fourier transform « discrete preorder » discriminated union » discussion group » Disiple
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