The Fourier transform of the Kaiser window (where is treated as continuous) is given by
- Reduces to rectangular window for
- Asymptotic roll-off is 6 dB/octave
- First null in window transform is at
- Time-bandwidth product radians if bandwidths are measured from 0 to positive band-limit
- Full time-bandwidth product radians when frequency bandwidth is defined as main-lobe width out to first null
- Sometimes the Kaiser window is parameterized by
Kaiser Window Beta Parameter
The 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 values give lower side-lobe levels, but at the price of a wider main lobe. As discussed in §4.5.2, widening the main lobe reduces frequency resolution when the window is used for spectrum analysis. As explored in Chapter 8, 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'' of the window in radians (seconds times radians-per-second).4.8 Sometimes the Kaiser window is parameterized by instead of . The parameter is therefore half the window's time-bandwidth product in cycles (seconds times cycles-per-second).
Figure 3.19 shows a plot of the Kaiser window for various values of . Note that for , the Kaiser window reduces to the rectangular window.
Figure 3.20 shows a plot of the Kaiser window transforms for . For (top plot), we see the dB magnitude of the aliased sinc function. As increases the main-lobe widens and the side lobes go lower, reaching almost 50 dB down for .
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 3.22 shows a plot of the Kaiser window side-lobe level for various values of . For , 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 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).
The requirements on window length for resolving closely tuned sinusoids was discussed in §4.6.2. This section considers this issue for the Kaiser window. Table 3.1 lists the parameter required for a Kaiser window to resolve equal-amplitude sinusoids with a frequency spacing of rad/sample [1, Table 8-9]. Recall from §3.9 that can be interpreted as half of the time-bandwidth of the window (in cycles).
Figure 3.23 shows an overlay of DPSS and Kaiser windows for some different values. In all cases, the window length was . Note how the two windows become more similar as increases. The Matlab for computing the windows is as follows:
w1 = dpss(M,alpha,1); % discrete prolate spheroidal sequence w2 = kaiser(M,alpha*pi); % corresponding kaiser window
The following Matlab comparison of the DPSS and Kaiser windows illustrates the interpretation of as the bin number of the edge of the critically sampled window main lobe, i.e., when the DFT length equals the window length:
format long; M=17; alpha=5; abs(fft([ dpss(M,alpha,1), kaiser(M,pi*alpha)/2])) ans = 2.82707022360190 2.50908747431366 2.00652719015325 1.92930705688346 0.68469697658600 0.85272343521683 0.09415916813555 0.19546670371747 0.00311639169878 0.01773139505899 0.00000050775691 0.00022611995322 0.00000003737279 0.00000123787805 0.00000000262633 0.00000066206722 0.00000007448708 0.00000034793207 0.00000007448708 0.00000034793207 0.00000000262633 0.00000066206722 0.00000003737279 0.00000123787805 0.00000050775691 0.00022611995322 0.00311639169878 0.01773139505899 0.09415916813555 0.19546670371747 0.68469697658600 0.85272343521683 2.00652719015325 1.92930705688346
Finally, Fig.3.24 shows a comparison of DPSS and Kaiser window transforms, where the DPSS window was computed using the simple method listed in §G.1.2. We see that the DPSS window has a slightly narrower main lobe and lower overall side-lobe levels, although its side lobes are higher far from the main lobe. Thus, the DPSS window has slightly better overall specifications, while Kaiser-window side lobes have a steeper roll off.
Slepian or DPSS Window