Number Theoretic Transform

Other FFT Algorithms

Other FFT Algorithms
** Index**
**Search**

##

The Discrete
Cosine Transform (DCT)

In image coding (such as MPEG and JPEG), and many audio coding
algorithms (MPEG), the *discrete cosine transform* (DCT) is used
because of its nearly optimal asymptotic theoretical
*coding gain*.^{A.3}For 1D signals, one of several essentially equivalent DCT definitions is given by

where
Note that is the DFT frequency for a length DFT (as opposed to ).
For real signals, the real part of the DFT is a kind of DCT:

Thus, the real part of a double-length FFT is the same as the DCT
except for the half-sample phase shift in the sinusoidal basis
functions
(and a scaling by 2 which is
unimportant).

In practice, the DCT is implemented using the same basic efficiency
techniques as the FFT. In Matlab, the functions `dct` and
`dct2` are available for the 1D and 2D cases, respectively.

**Exercise:** Using Euler's identity, expand the cosine
in the DCT defined by Eq. (A.2) above into a sum of complex
sinusoids, and show that the DCT can be rewritten as the sum of two
phase-modulated DFTs:

DCT

where
DFT denotes the length DFT of .

Number Theoretic Transform

Other FFT Algorithms

Other FFT Algorithms
** Index**
**Search**

``Mathematics of the Discrete Fourier Transform (DFT), with
Music and Audio Applications'',
by Julius O. Smith III,
W3K Publishing, 2003, ISBN 0-9745607-0-7.

(Browser settings for best viewing results)

(How to cite this work)

(Order a printed hardcopy)

Copyright © *2004-09-24* by Julius O. Smith III

W3K Publishing,