ratio to about 57 dB. The Blackman window provides 74 dB of stopband attenuation,
but with a transition width six times that of the rectangular window.
The Kaiser window function is defined as
wn
I
0
b
1 À n À M
2
=M
2
q
!
I
0
b
, n 0, 1, . . . , L À 1,
5:3:26a
where b is an adjustable (shape) parameter and
I
0
b
I
k0
b=2
k
k!
4
5
2
5:3:26b
is the zero-order modified Bessel function of the first kind. In practice, it is sufficient to
keep only the first 25 terms in the summation of (5.3.26b). Because I
0
0 1, the Kaiser
window has the value 1=I
0
b at the end points n 0 and n L À 1, and is symmetric
about its middle n M. This is a useful and very flexible family of window functions.
MATLAB provides Kaiser window function as
kaiser(L, beta);
The window function and its magnitude response are shown in Figure 5.19 for L 41
and b 8 using the MATLAB script ksw.m given in the software package. The Kaiser
window is nearly optimum in the sense of having the most energy in the mainlobe for a
given peak sidelobe level. Providing a large mainlobe width for the given stopband
attenuation implies the sharpness transition width. This window can provide different
transition widths for the same L by choosing the parameter b to determine the trade-off
between the mainlobe width and the peak sidelobe level.
As shown in (5.3.26), the Kaiser window is more complicated to generate, but the
window function coefficients are generated only once during filter design. Since a window
is applied to each filter coefficient when designing a filter, windowing will not affect the
run-time complexity of the designed FIR filter.
Although d
p
and d
s
can be specified independently as given in (5.1.8) and (5.1.9), FIR
filters designed by all windows will have equal passband and stopband ripples. There-
fore we must design the filter based on the smaller of the two ripples expressed as
d mind
p
, d
s
:
5:3:27
The designed filter will have passband and stopband ripples equal to d. The value of d
can be expressed in dB scale as
A À20 log
10
d dB:
5:3:28
In practice, the design is usually based on the stopband ripple, i.e., d d
s
. This is because
any reasonably good choices for the passband and stopband attenuation (such as
A
p
0:1 dB and A
s
60 dB) will result in d
s
< d
p
.
212
DESIGN AND IMPLEMENTATION OF FIR FILTERS