Kaiser window

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The Kaiser window is a window function w_k used for digital signal processing, and is defined by the formula: Image:Kaiser-window.png

<math>

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.

</math>

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, as is illustrated in the plot of the frequency spectra below. For large α, the shape of the Kaiser window (in both time and frequency domain) tends to a Gaussian curve. The Kaiser window is nearly optimal in the sense of its peak's concentration around ω=0 (Oppenheim et al., 1999).

Image:Kaiser-window-spectra.png

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_n is defined in terms of the Kaiser window w_k by the formula: Image:Kbd-window.png

<math>

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.

</math>

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

• Template:Cite book
• Kaiser, J. F. (1966). Digital Filters. In Kuo, F. F. and Kaiser, J. F. (Eds.), System Analysis by Digital Computer, chap. 7. New York, Wiley.
• Marina Bosi, Kaiser-Bessel Derived Window, Music 422 / EE 367C: Perceptual Audio Coding (Stanford University course page, 2005).es:Ventana de Kaiser

ja:カイザー窓