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# Online Computer Terms Dictionary - D

## 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)

Nearby terms: disclaimer « disconnect « Discordianism « discrete cosine transform » discrete Fourier transform » discrete preorder » discriminated union

## 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).

(1997-03-10)

Nearby terms: disconnect « Discordianism « discrete cosine transform « discrete Fourier transform » discrete preorder » discriminated union » discussion group

## discrete preorder

<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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