Editing Modified discrete cosine transform (section)

=== TDAC for the windowed MDCT ===

Above, the TDAC property was proved for the ordinary MDCT, showing that adding IMDCTs of subsequent blocks in their overlapping half recovers the original data.  The derivation of this inverse property for the windowed MDCT is only slightly more complicated.

Consider two overlapping consecutive sets of 2''N'' inputs (''A'',''B'') and (''B'',''C''), for blocks ''A'',''B'',''C'' of size ''N''.
Recall from above that when <math>(A,B)</math> and <math>(B,C)</math> are MDCTed, IMDCTed, and added in their overlapping half, we obtain <math>(B+B_R) / 2 + (B-B_R) / 2 = B</math>, the original data.

Now we suppose that we multiply ''both'' the MDCT inputs ''and'' the IMDCT outputs by a window function of length 2''N''.  As above, we assume a symmetric window function, which is therefore of the form <math>(W,W_R)</math> where ''W'' is a length-''N'' vector and ''R'' denotes reversal as before.  Then the Princen-Bradley condition can be written as <math>W^2 + W_R^2 = (1,1,\ldots)</math>, with the squares and additions performed elementwise.

Therefore, instead of MDCTing <math>(A,B)</math>, we now MDCT <math>(WA,W_R B)</math> (with all multiplications performed elementwise).  When this is IMDCTed and multiplied again (elementwise) by the window function, the last-''N'' half becomes:
:<math>W_R \cdot (W_R B+(W_R B)_R) =W_R \cdot (W_R B+W B_R) = W_R^2 B+WW_R B_R</math>.
(Note that we no longer have the multiplication by 1/2, because the IMDCT normalization differs by a factor of 2 in the windowed case.)

Similarly, the windowed MDCT and IMDCT of <math>(B,C)</math> yields, in its first-''N'' half:
:<math>W \cdot (WB - W_R B_R) = W^2 B - W W_R B_R</math>.

When we add these two halves together, we obtain:

:<math>(W_R^2 B+WW_R B_R) + (W^2 B - W W_R B_R)= \left(W_R^2 + W^2\right)B = B,</math>

recovering the original data.