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Spectral Audio Signal Processing
    Spectrum Analysis Windows
       Kaiser Window

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Kaiser Window

Jim Kaiser discovered a simple approximation to the DPSS window based upon Bessel functions [110], generally known as the Kaiser window (or Kaiser-Bessel window).


Definition:

$\displaystyle w_K(n) \isdef \left\{ \begin{array}{ll} \frac{ I_0 \left( \beta \... ...2} \leq n \leq \frac{M-1}{2} \\ 0, & \mbox{elsewhere} \\ \end{array} \right. $


Window transform:

The Fourier transform of the Kaiser window $ w_K(t)$ (where $ t$ is treated as continuous) is given by

$\displaystyle W(\omega) = \frac{M}{I_0(\beta)} \frac{\sinh\left(\sqrt{\beta^2... ...\right)^2-\beta^2}\right)} {\sqrt{\left(\frac{M\omega}{2}\right)^2 - \beta^2}} $

where $ I_0$ is the zero-order modified Bessel function of the first kind:4.7

$\displaystyle I_0(x) \isdef \sum_{k=0}^{\infty} \left[ \frac{\left(\frac{x}{2}\right)^k}{k!} \right]^2 $

Notes:
  • Reduces to rectangular window for $ \beta=0$
  • Asymptotic roll-off is 6 dB/octave
  • First null in window transform is at $ \omega_0=2\beta/M$
  • Time-bandwidth product $ \omega_0 (M/2) = \beta$ radians if bandwidths are measured from 0 to positive band-limit
  • Full time-bandwidth product $ (2\omega_0) M = 4\beta$ radians when frequency bandwidth is defined as main-lobe width out to first null
  • Sometimes the Kaiser window is parameterized by $ \alpha $, where

    $\displaystyle \beta\isdef \pi\alpha $


Subsections
Previous: Matlab for the DPSS Window
Next: Kaiser Window Beta Parameter

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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