Kaiser Window
Jim Kaiser discovered a simple approximation to the DPSS window based
upon
Bessel functions [
115], generally known as the Kaiser
window (or
Kaiser-Bessel window).

Definition:
 |
(4.39) |
Window transform:
The
Fourier transform of the Kaiser window

(where

is
treated as continuous) is given by
4.11
 |
(4.40) |
where

is the zero-order modified Bessel function of the first
kind:
4.12
Notes:
- 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 parametrized by
, where
 |
(4.42) |
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 §
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''

of the window
in radians (seconds times radians-per-second).
4.13 Sometimes the Kaiser window is
parametrized by

instead of

. The

parameter is therefore half the window's time-bandwidth
product

in cycles (seconds times
cycles-per-second).
Figure
3.24 plots the
Kaiser window and its transform for

. Note how increasing

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
.
![\includegraphics[width=\twidth]{eps/kaiser123}](http://www.dsprelated.com/josimages_new/sasp2/img499.png) |
Figure
3.25 shows a plot of the Kaiser window
for various values of
![$ \beta = [0,2,4,6,8,10]$](http://www.dsprelated.com/josimages_new/sasp2/img500.png)
. Note that for

, the
Kaiser window reduces to the rectangular window.
Figure 3.25:
The Kaiser window for various
values of the time-bandwidth parameter
.
![\includegraphics[width=\twidth]{eps/KaiserTBetas}](http://www.dsprelated.com/josimages_new/sasp2/img501.png) |
Figure
3.26 shows a plot of the Kaiser window
transforms for
![$ \beta = [0,2,4,6]$](http://www.dsprelated.com/josimages_new/sasp2/img502.png)
. 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 3.26:
Kaiser window transform
magnitude for various
.
![\includegraphics[width=\twidth]{eps/KaiserFBetas}](http://www.dsprelated.com/josimages_new/sasp2/img504.png) |
Figure
3.27 shows the effect of increasing window length
for the Kaiser window. The window lengths are
![$ M = [20,30,40,50]$](http://www.dsprelated.com/josimages_new/sasp2/img505.png)
from the top to the bottom plot. As with all windows, increasing the
length decreases the main-lobe width, while the side-lobe level
remains essentially unchanged.
Figure 3.27:
Kaiser window transform
magnitudes for various window lengths.
![\includegraphics[width=\twidth]{eps/KaiserFLengths}](http://www.dsprelated.com/josimages_new/sasp2/img506.png) |
Figure
3.28 shows a plot of the Kaiser window side-lobe level
for various values of
![$ \alpha = [0,0.5,1,1.5,\ldots,4]$](http://www.dsprelated.com/josimages_new/sasp2/img507.png)
. 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.
3.28). 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).
Figure 3.28:
Kaiser window side-lobe level for various values of
.
![\includegraphics[width=\twidth]{eps/kaiserBeta}](http://www.dsprelated.com/josimages_new/sasp2/img508.png) |
Minimum Frequency Separation vs. Window Length
The requirements on window length for resolving closely tuned
sinusoids was discussed in §
5.5.2. 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).
Table:
Kaiser
parameter for various frequency resolutions,
assuming an FFT zero-padding factor of at least 3.5.
|
Figure
3.29 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 seq.
w2 = kaiser(M,alpha*pi); % corresponding kaiser window
Figure:
Comparison of length 51
DPSS and Kaiser windows for
.
![\includegraphics[width=\twidth]{eps/dpsstest}](http://www.dsprelated.com/josimages_new/sasp2/img515.png) |
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.
3.30 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
, DPSS cut-off
, and Kaiser
.
![\includegraphics[width=\twidth]{eps/dpsskaiser-fd}](http://www.dsprelated.com/josimages_new/sasp2/img516.png) |
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