Modified discrete cosine transform

The modified discrete cosine transform (MDCT) is a frequency transform similar to the discrete Cosine transform (DCT), but it is slighly different.

Definition

$f_{j}={\frac {1}{2}}(x_{0}+(-1)^{j}x_{n-1})+\sum _{k=1}^{n-2}x_{k}\cos \left[{\frac {\pi }{n-1}}jk\right]$

A DCT-I of n=5 real numbers abcde is exactly equivalent to a DFT of eight real numbers abcdedcb (even symmetry). (In contrast, DCT types II-IV involve a half-sample shift in the equivalent DFT.) Note, however, that the DCT-I is not defined for n less than 2. (All other DCT types are defined for any positive n.)

Inverse Transforms

The inverse of DCT-I is DCT-I multiplied by 2/(n-1). The inverse of DCT-IV is DCT-IV multiplied by 2/n. The inverse of DCT-II is DCT-III multiplied by 2/n (and vice versa).

Like for the DFT, the normalization factor in front of these transform definitions is merely a convention and differs between treatments. For example, some authors multiply the transforms by ${\sqrt {2/n}}$ so that the inverse does not require any additional multiplicative factor.

Applications

The MDCT, is often used in audio signal and image processing, especially for lossy data compression, because it has a strong "energy compaction" property: most of the signal information tends to be concentrated in a few low-frequency components of the MDCT, approaching the optimal Karhunen-Loeve transform for signals based on certain limits of Markov processes.

For example, the DCT is used in AAC audio compression.

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