Kaiser Windows and Transforms

International Standard Book Number — Click for https://mathworld.wolfram.com/ISBN.html

The 'Cardinal sine function' is defined as sin(pi x)/(pi x) — Click for https://ccrma.stanford.edu/~jos/mdft/Sinc_Function.html

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The aliased sinc function is obtained from the sinc function by applying the aliasing operator. It arises in digital audio as the spectrum of a sampled rectangular pulse. — Click for https://ccrma.stanford.edu/~jos/sasp/Rectangular_Window.html

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The size of the passband, typically expressed in Hz. — Click for http://en.wikipedia.org/wiki/Bandpass

The main lobe in the Fourier transform of a window function, such as the Chebyshev window, is the central local maximum in the transform magnitude (response about dc). Outside of the main lobe are usually side lobes (ripples in the magnitude spectrum). — Click for https://ccrma.stanford.edu/~jos/sasp/Rectangular_Window_Side_Lobes.html

A side lobe in the Fourier transform of a window function, such as the Chebyshev window, is any local maximum in the transform magnitude outside of the main lobe (response about dc). — Click for https://ccrma.stanford.edu/~jos/sasp/Rectangular_Window_Side_Lobes.html

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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

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 $\alpha =\beta /\pi$ .
$\includegraphics[width=\twidth]{eps/kaiserBeta}$

Kaiser Windows and Transforms

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1. Copyright © 2022-02-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University