Discrete Cosine Transform
The DFT transforms a complex signal into its complex spectrum. However, if the signal is real (as in most of applications), half of the data is redundant, and half of the computation is wasted. (In time domain the imaginary part of the signal is all zero, and in frequency domain, both real and imaginary parts of the spectrum are symmetry.) As a real transform, Discrete cosine transform (DCT) generates real spectrum of a real signal and thereby avoids redundant data and computation. Also as DCT is derived from DFT, all the desirable properties of DFT are preserved, and the fast algorithm exists. The DCT of a real sequence![$\{x[m],\;\;\;(m=0,\cdots,N-1)\;\}$](http://fourier.eng.hmc.edu/e101/lectures/Image_Processing/img349.png){kind=small}
is defined as:![\begin{displaymath}
X[n] = c(n) \sum_{m=0}^{N-1}x[m]\;cos(\frac{ (2m+1)n\pi}{2N})
\;\;\;\;\;\;\;(n=0,\cdots,N-1) \nonumber
\end{displaymath}](http://fourier.eng.hmc.edu/e101/lectures/Image_Processing/img350.png)
and the inverse DCT is
![\begin{displaymath}
x[m] = \sum_{n=0}^{N-1} c(n) X[n]\;cos(\frac{ (2m+1)n\pi}{2N})
\;\;\;\;\;\;\;(m=0,\cdots,N-1) \nonumber
\end{displaymath}](http://fourier.eng.hmc.edu/e101/lectures/Image_Processing/img351.png)
where
is a normalization coefficient defined as
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