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Kaiser Window Beta Parameter

The $ \beta $ parameter of the Kaiser window provides a convenient continuous control over the fundamental window trade-off between side-lobe level and main-lobe width. Larger $ \beta $ values give lower side-lobe levels, but at the price of a wider main lobe. As discussed in §5.4.1, widening the main lobe reduces frequency resolution when the window is used for spectrum analysis. As explored in Chapter 9, reducing the side lobes reduces ``channel cross talk'' in an FFT-based filter-bank implementation.

The Kaiser beta parameter can be interpreted as 1/4 of the ``time-bandwidth product'' $ \Delta t\cdot \Delta \omega$ of the window in radians (seconds times radians-per-second).4.13 Sometimes the Kaiser window is parametrized by $ \alpha\isdef \beta/\pi$ instead of $ \beta $ . The $ \alpha $ parameter is therefore half the window's time-bandwidth product $ \Delta
t\cdot \Delta f$ in cycles (seconds times cycles-per-second).

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