Chemistry - Kaiser window

The Kaiser window is a nearly optimal window function w_k used for digital signal processing, and is defined by the formula:

$w_k = \left\{ \begin{matrix} \frac{I_0(\pi\alpha \sqrt{1 - (2k/n-1)^2})} {I_0(\pi\alpha)} & \mbox{if } 0 \leq k \leq n \\ \\ 0 & \mbox{otherwise} \\ \end{matrix} \right.$

where I₀ is the zeroth order modified Bessel function of the first kind, α is an arbitrary real number that determines the shape of the window, and the integer n is the length of the window.

By construction, this function peaks at unity for k = n/2, i.e. at the center of the window, and decays exponentially towards the window edges.

The larger the value of |α|, the narrower the window becomes; α = 0 corresponds to a rectangular window. Conversely, for larger |α| the width of the main lobe increases in the Fourier transform of w_k, while the side lobes decrease in amplitude. Thus, this parameter controls the tradeoff between main-lobe width and side-lobe area. For large α, the shape of the Kaiser window tends to a Gaussian curve.

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 d_k is defined in terms of the Kaiser window w_k by the formula:

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

This defines a window of length 2n, where by construction d_k satisfies the Princen-Bradley condition for the MDCT (using the fact that w_n−k = w_k): d_k² + d_k + n² = 1 (interpreting k and k + n modulo 2n). The KBD window is also symmetric in the proper manner for the MDCT: d_k = d_2n−1−k.

References

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice-Hall, 1999).
J. F. Kaiser, "Digital Filters," System Analysis by Digital Computer chap. 7 (Wiley: New York, 1966); F. F. Kuo and J. F. Kaiser, eds.
Marina Bosi, Kaiser-Bessel Derived Window, Music 422 / EE 367C: Perceptual Audio Coding (Stanford University course page, 2005).