Modified discrete cosine transform: Difference between revisions

Revision as of 11:49, 28 August 2021

Revision as of 00:51, 17 September 2021

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As can be seen by inspection of the definitions, for '''even''' ''N'' the MDCT is essentially equivalent to a DCT-IV, where the input is shifted by ''N''/2 and two ''N''-blocks of data are transformed at once. By examining this equivalence more carefully, important properties like TDAC can be easily derived.

In order to define the precise relationship to the DCT-IV, one must realize that the DCT-IV corresponds to alternating even/odd boundary conditions: even at its left boundary (around ''n''&#8239nbsp;=&#8239nbsp;&minus;1/2), odd at its right boundary (around ''n''&#8239nbsp;=&#8239nbsp;''N''&#8239nbsp;&minus;&#8239nbsp;1/2), and so on (instead of periodic boundaries as for a [[discrete Fourier transform|DFT]]). This follows from the identities <math>\cos\left[\frac{\pi}{N} \left(-n-1+\frac{1}{2}\right) \left(k+\frac{1}{2}\right)\right] = \cos\left[\frac{\pi}{N} \left(n+\frac{1}{2}\right) \left(k+\frac{1}{2}\right)\right]</math> and <math>\cos\left[\frac{\pi}{N} \left(2N-n-1+\frac{1}{2}\right) \left(k+\frac{1}{2}\right)\right] = -\cos\left[\frac{\pi}{N} \left(n+\frac{1}{2}\right) \left(k+\frac{1}{2}\right)\right]</math>. Thus, if its inputs are an array ''x'' of length ''N'', we can imagine extending this array to (''x'', &minus;''x''_''R'', &minus;''x'', ''x''_''R'', ...) and so on, where ''x''_''R'' denotes ''x'' in reverse order.

Consider an MDCT with 2''N'' inputs and ''N'' outputs, where we divide the inputs into four blocks (''a'', ''b'', ''c'', ''d'') each of size ''N''/2. If we shift these to the right by ''N''/2 (from the +''N''/2 term in the MDCT definition), then (''b'', ''c'', ''d'') extend past the end of the ''N'' DCT-IV inputs, so we must "fold" them back according to the boundary conditions described above.

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Similarly, the IMDCT formula above is precisely 1/2 of the DCT-IV (which is its own inverse), where the output is extended (via the boundary conditions) to a length 2''N'' and shifted back to the left by ''N''/2. The inverse DCT-IV would simply give back the inputs (&minus;''c''_''R''&minus;''d'', ''a''&minus;''b''_''R'') from above. When this is extended via the boundary conditions and shifted, one obtains:

:IMDCT&#8239nbsp;(MDCT&#8239nbsp;(''a'', ''b'', ''c'', ''d''))&#8239nbsp;=&#8239nbsp;(''a''&minus;''b''_''R'', ''b''&minus;''a''_''R'', ''c''+''d''_''R'', ''d''+''c''_''R'')&#8239nbsp;/&#8239nbsp;2.

Half of the IMDCT outputs are thus redundant, as ''b''&minus;''a''_''R'' = &minus;(''a''&minus;''b''_''R'')_''R'', and likewise for the last two terms. If we group the input into bigger blocks ''A'',''B'' of size ''N'', where ''A''&#8239nbsp;=&#8239nbsp;(''a'', ''b'') and ''B''&#8239nbsp;=&#8239nbsp;(''c'', ''d''), we can write this result in a simpler way:

:IMDCT&#8239nbsp;(MDCT&#8239nbsp;(''A'', ''B''))&#8239nbsp;=&#8239nbsp;(''A''&minus;''A''_''R'', ''B''+''B''_''R'')&#8239nbsp;/&#8239nbsp;2

One can now understand how TDAC works. Suppose that one computes the MDCT of the subsequent, 50% overlapped, 2''N'' block (''B'', ''C''). The IMDCT will then yield, analogous to the above: (''B''&minus;''B''_''R'', ''C''+''C''_''R'')&#8239nbsp;/&#8239nbsp;2. When this is added with the previous IMDCT result in the overlapping half, the reversed terms cancel and one obtains simply ''B'', recovering the original data.

=== Origin of TDAC ===

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''a'' and of ''b''_''R'' to the MDCT of (''a'', ''b'', ''c'', ''d''), or equivalently, to

the result of

:IMDCT&#8239nbsp;(MDCT&#8239nbsp;(''a'', ''b'', ''c'', ''d''))&#8239nbsp;=&#8239nbsp;(''a''&minus;''b''_''R'', ''b''&minus;''a''_''R'', ''c''+''d''_''R'', ''d''+''c''_''R'') / 2.

The combinations ''c''&minus;''d''_''R'' and so on, have precisely the right signs for the combinations to cancel when they are added.