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Kaiser Windows and Transforms

Figure 3.24 plots the Kaiser window and its transform for $ \alpha = \beta/\pi = 1,2,3$ . Note how increasing $ \alpha $ causes the side-lobes to fall away from the main lobe. The curvature at the main lobe peak also decreases somewhat.

Figure 3.24: Kaiser window and transform for $ \alpha =1,2,3$ .
\includegraphics[width=\twidth]{eps/kaiser123}

Figure 3.25 shows a plot of the Kaiser window for various values of $ \beta = [0,2,4,6,8,10]$ . Note that for $ \beta=0$ , the Kaiser window reduces to the rectangular window.

Figure 3.25: The Kaiser window for various values of the time-bandwidth parameter $ \beta $ .
\includegraphics[width=\twidth]{eps/KaiserTBetas}

Figure 3.26 shows a plot of the Kaiser window transforms for $ \beta = [0,2,4,6]$ . For $ \beta=0$ (top plot), we see the dB magnitude of the aliased sinc function. As $ \beta $ increases the main-lobe widens and the side lobes go lower, reaching almost 50 dB down for $ \beta=6$ .

Figure 3.26: Kaiser window transform magnitude for various $ \beta $ .
\includegraphics[width=\twidth]{eps/KaiserFBetas}

Figure 3.27 shows the effect of increasing window length for the Kaiser window. The window lengths are $ M = [20,30,40,50]$ from the top to the bottom plot. As with all windows, increasing the length decreases the main-lobe width, while the side-lobe level remains essentially unchanged.

Figure 3.27: Kaiser window transform magnitudes for various window lengths.
\includegraphics[width=\twidth]{eps/KaiserFLengths}

Figure 3.28 shows a plot of the Kaiser window side-lobe level for various values of $ \alpha = [0,0.5,1,1.5,\ldots,4]$ . For $ \beta=0$ , the Kaiser window reduces to the rectangular window, and we expect the side-lobe level to be about 13 dB below the main lobe (upper-lefthand corner of Fig.3.28). As $ \alpha =\beta /\pi $ increases, the dB side-lobe level reduces approximately linearly with main-lobe width increase (approximately a 25 dB drop in side-lobe level for each main-lobe width increase by one sinc-main-lobe).

Figure 3.28: Kaiser window side-lobe level for various values of $ \alpha =\beta /\pi $ .
\includegraphics[width=\twidth]{eps/kaiserBeta}


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2015-02-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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