Modified Discrete Cosine Transform - Relationship To DCT-IV and Origin of TDAC

Relationship To DCT-IV and Origin of TDAC

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=–1/2), odd at its right boundary (around n=N–1/2), and so on (instead of periodic boundaries as for a DFT). This follows from the identities and . Thus, if its inputs are an array x of length N, we can imagine extending this array to (x, –x_R, –x, x_R, ...) and so on, where x_R denotes x in reverse order.

Consider an MDCT with 2N 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 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.

Thus, the MDCT of 2N inputs (a, b, c, d) is exactly equivalent to a DCT-IV of the N inputs: (–c_R–d, a–b_R), where R denotes reversal as above.

(In this way, any algorithm to compute the DCT-IV can be trivially applied to the MDCT.)

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

IMDCT(MDCT(a, b, c, d)) = (a–b_R, b–a_R, c+d_R, d+c_R) / 2.

Half of the IMDCT outputs are thus redundant, as b–a_R = –(a–b_R)_R, and likewise for the last two terms.

One can now understand how TDAC works. Suppose that one computes the MDCT of the subsequent, 50% overlapped, 2N block (c, d, e, f). The IMDCT will then yield, analogous to the above: (c–d_R, d–c_R, e+f_R, e_R+f) / 2. When this is added with the previous IMDCT result in the overlapping half, the reversed terms cancel and one obtains simply (c, d), recovering the original data.