Editing Modified discrete cosine transform (section)

=== Origin of TDAC ===

The origin of the term "time-domain aliasing cancellation" is now clear.  The use of input data that extend beyond the boundaries of the logical DCT-IV causes the data to be ''aliased'' in the same way that frequencies beyond the [[Nyquist frequency]] are [[aliasing|aliased]] to lower frequencies, except that this aliasing occurs in the time domain instead of the frequency domain: we cannot distinguish the contributions of
''a'' and of ''b''<sub>''R''</sub> to the MDCT of (''a'', ''b'', ''c'', ''d''), or equivalently, to
the result of
: IMDCT(MDCT(''a'', ''b'', ''c'', ''d''))= (''a'' − ''b''<sub>''R''</sub>, ''b'' − ''a''<sub>''R''</sub>, ''c'' + ''d''<sub>''R''</sub>, ''d'' + ''c''<sub>''R''</sub>)/2.
The combinations ''c'' − ''d''<sub>''R''</sub> and so on have precisely the right signs for the combinations to cancel when they are added.

For ''odd'' ''N'' (which are rarely used in practice), ''N''/2 is not an integer, so the MDCT is not simply a shift permutation of a DCT-IV.  In this case, the additional shift by half a sample means that the MDCT/IMDCT becomes equivalent to the DCT-III/II, and the analysis is analogous to the above.