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Chapters

Spectral Audio Signal Processing
    Spectrum Analysis Windows
       Kaiser Window
             Kaiser Windows and Transforms

Chapter Contents:

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Kaiser Windows and Transforms

Figure 3.18 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 3.18:** Kaiser window and transform for $\alpha =1,2,3$ .
$\includegraphics[width=\twidth]{eps/kaiser123}$

Figure 3.19 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 3.19:** The Kaiser window for various values of the time-bandwidth parameter $\beta$ .
$\includegraphics[width=\twidth]{eps/KaiserTBetas}$

Figure 3.20 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 3.20:** Kaiser window transform magnitude for various $\beta$ .
$\includegraphics[width=\twidth]{eps/KaiserFBetas}$

Figure 3.21 shows the effect of increasing window length for the Kaiser window. The window lengths are 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 3.21:** Kaiser window transform magnitudes for various window lengths.
$\includegraphics[width=\twidth]{eps/KaiserFLengths}$

Figure 3.22 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.3.22). 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 3.22:** Kaiser window side-lobe level for various values of $\alpha =\beta /\pi$ .
$\includegraphics[width=\twidth]{eps/kaiserBeta}$

Previous: Kaiser Window Beta Parameter
Next: Minimum Frequency Separation vs. Window Length

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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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