Kaiser Window - Kaiser-Bessel Derived (KBD) Window

Kaiser-Bessel Derived (KBD) Window

A related window function is the Kaiser-Bessel derived (KBD) window, which is designed to be suitable for use with the modified discrete cosine transform (MDCT). The KBD window function is defined in terms of the Kaiser window of length M+1, by the formula:

$d_n = \left\{ \begin{matrix} \sqrt{\frac{\sum_{j=0}^{n} w_j} {\sum_{j=0}^{M} w_j}} & \mbox{if } 0 \leq n < M \\ \\ \sqrt{\frac{\sum_{j=0}^{2M-1-n} w_j} {\sum_{j=0}^{M} w_j}} & \mbox{if } M \leq n < 2M \\ \\ 0 & \mbox{otherwise} \\ \end{matrix} \right.$

This defines a window of length 2M, where by construction d_n satisfies the Princen-Bradley condition for the MDCT (using the fact that w_M−n = w_n): d_n2 + d_{n + M}2 = 1 (interpreting n and n + M modulo 2M). The KBD window is also symmetric in the proper manner for the MDCT: d_n = d_2M−1−n.

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