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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.9For 1D signals, one of several DCT definitions (the one called DCT-II)A.10is given by

$\displaystyle \hbox{\sc DCT}_k(x)$ $\displaystyle =$ $\displaystyle 2\sum_{n=0}^{N-1} x(n) \cos\left[\frac{\pi k}{2N}(2n+1)\right],
\quad k=0,1,2,\ldots,N-1$  
  $\displaystyle =$ $\displaystyle 2\sum_{n=0}^{N-1} x(n) \cos\left[\omega_k\left(n+\frac{1}{2}\right)\right]
\protect$ (A.2)

where

$\displaystyle \omega_k \isdef \frac{2\pi}{2N}k.
$

Note that $ \omega_k$ is the DFT frequency for a length $ 2N$ DFT (as opposed to $ N$ ).

For real signals, the real part of the DFT is a kind of DCT:

\begin{eqnarray*}
\mbox{re}\left\{\hbox{\sc DFT}_k(x)\right\}
&\isdef & \mbox{re}\left\{\sum_{n=0}^{N-1} x(n) e^{-j\omega_k n}\right\}\\
&=& \sum_{n=0}^{N-1} x(n) \mbox{re}\left\{e^{-j\omega_k n}\right\}\\
&=& \sum_{n=0}^{N-1} x(n) \cos\left(\omega_k n\right)
\end{eqnarray*}

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 $ s_k(n) = e^{j\omega_k n}$ (and a scaling by 2 which is unimportant).

In practice, the DCT is normally implemented using the same basic efficiency techniques as in FFT algorithms. In Matlab and Octave (Octave-Forge), 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:

$\displaystyle \hbox{\sc DCT}_{N,k} =
e^{j\omega_k\frac{1}{2}} \overline{X_{2N}(\omega_k)}
+e^{-j\omega_k\frac{1}{2}} X_{2N}(\omega_k)
$

where $ X_{2N}(\omega_k)=\hbox{\sc DFT}_{2N,k}(x)$ denotes the length $ 2N$ DFT of $ x$ .


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA