Kaiser and DPSS Windows Compared

Figure 3.29 shows an overlay of DPSS and Kaiser windows for some different $ \alpha $ values. In all cases, the window length was $ M=51$ . Note how the two windows become more similar as $ \alpha $ increases. The Matlab for computing the windows is as follows:

  w1 = dpss(M,alpha,1);    % discrete prolate spheroidal seq.
  w2 = kaiser(M,alpha*pi); % corresponding kaiser window

Figure: Comparison of length 51 DPSS and Kaiser windows for $ \alpha =1,3,5$ .
\includegraphics[width=\twidth]{eps/dpsstest}

The following Matlab comparison of the DPSS and Kaiser windows illustrates the interpretation of $ \alpha $ 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.3.30 shows a comparison of DPSS and Kaiser window transforms, where the DPSS window was computed using the simple method listed in §F.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.

Figure: DPSS and Kaiser window transforms, for length $ M=61$ , DPSS cut-off $ \omega _c = 2\pi 3.5/M$ , and Kaiser $ \beta =(M-1)\omega _c/2$ .
\includegraphics[width=\twidth]{eps/dpsskaiser-fd}


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Minimum Frequency Separation vs. Window Length