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Kaiser Window

Jim Kaiser discovered a simple approximation to the DPSS window based upon Bessel functions [115], generally known as the Kaiser window (or Kaiser-Bessel window).


Definition:

$\displaystyle w_K(n) \isdefs \left\{ \begin{array}{ll} \frac{ I_0 \left( \beta \sqrt{ 1 - \left( \frac{n}{M/2}\right)^2} \right)}{ I_0(\beta) }, & -\frac{M-1}{2} \leq n \leq \frac{M-1}{2} \\ 0, & \mbox{elsewhere} \\ \end{array} \right.$ (4.39)


Window transform:

The Fourier transform of the Kaiser window $ w_K(t)$ (where $ t$ is treated as continuous) is given by4.11

$\displaystyle W(\omega) = \frac{M}{I_0(\beta)} \frac{\sinh\left(\sqrt{\beta^2 - \left(\frac{M \omega}{2}\right)^2}\right)} {\sqrt{ \beta^2 - \left(\frac{M\omega}{2}\right)^2}} = \frac{M}{I_0(\beta)} \frac{\sin\left(\sqrt{\left(\frac{M \omega}{2}\right)^2-\beta^2}\right)} {\sqrt{\left(\frac{M\omega}{2}\right)^2 - \beta^2}}$ (4.40)

where $ I_0$ is the zero-order modified Bessel function of the first kind:4.12

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

Notes:



Subsections
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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2014-06-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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