Modified Discrete Cosine Transform - Window Functions

Window Functions

In typical signal-compression applications, the transform properties are further improved by using a window function w_n (n = 0, ..., 2N-1) that is multiplied with x_n and y_n in the MDCT and IMDCT formulas, above, in order to avoid discontinuities at the n = 0 and 2N boundaries by making the function go smoothly to zero at those points. (That is, we window the data before the MDCT and after the IMDCT.) In principle, x and y could have different window functions, and the window function could also change from one block to the next (especially for the case where data blocks of different sizes are combined), but for simplicity we consider the common case of identical window functions for equal-sized blocks.

The transform remains invertible (that is, TDAC works), for a symmetric window w_n = w_2N-1-n, as long as w satisfies the Princen-Bradley condition:

.

various window functions are common, e.g.

for MP3 and MPEG-2 AAC, and

for Vorbis. AC-3 uses a Kaiser-Bessel derived (KBD) window, and MPEG-4 AAC can also use a KBD window.

Note that windows applied to the MDCT are different from windows used for other types of signal analysis, since they must fulfill the Princen-Bradley condition. One of the reasons for this difference is that MDCT windows are applied twice, for both the MDCT (analysis) and the IMDCT (synthesis).