Modified Discrete Cosine Transform

2016-10-05T13:07:00Z

168.1.3.36:

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The '''Modified Discrete Cosine Transform''' ('''MDCT''') is a [[DCT]]-IV transform. The MDCT tries to minimize blocking artifacts. It is common in lossy audio codecs including [[MP3]], [[Vorbis]], and [[AAC]].

As a lapped transform, the MDCT is a bit unusual compared to other Fourier-related transforms in that it has half as many outputs as inputs (instead of the same number). In particular, it is a linear function ''F'' : '''R'''2''n'' <tt>-></tt> '''R'''''n'' (where '''R''' denotes the set of real numbers). The 2''n'' real numbers ''x''0, ..., ''x''2''n''-1 are transformed into the ''n'' real numbers ''f''0, ..., ''f''''n''-1 according to the formula:

<center>[[Image:MDCT.png|MDCT formula]]</center>

The inverse MDCT is known as the '''IMDCT'''. Because there are different numbers of inputs and outputs, at first glance it might seem that the MDCT should not be invertible. However, perfect invertibility is achieved by ''adding'' the overlapped IMDCTs of subsequent overlapping blocks, causing the errors to ''cancel'' and the original data to be retrieved; this technique is known as ''time-domain aliasing cancellation'' (TDAC).

The IMDCT transforms ''n'' real numbers ''f''0, ..., ''f''''n''-1 into 2''n'' real numbers ''y''0, ..., ''y''2''n''-1 according to the formula:

<center>[[Image:IMDCT.png|IMDCT formula]]</center>

The MDCT was developed by Henrique Malvar, an engineer now working for Microsoft who had considerable participation in creating the [[WMA]] format.

== References ==
* [http://www.dspguide.com/ch8.htm Discrete Fourier Transform] it's basic properties and applications.
* [http://ccrma-www.stanford.edu/~bosse/proj/node27.html Planet CCRMA] description of the DCT-IV.
* [http://www.comp.nus.edu.sg/~wangye/publications/Conference%20papers/00027_aes22.pdf Modified Discrete Cosine Transform] a complete research paper on MDCT presented at the 22nd AES conference.

[[Category:Signal Processing]]

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Modified Discrete Cosine Transform