Digital Signal Processing Reference
In-Depth Information
58 dB from the peak
of the mainlobe. While the Blackman window produces the largest reduction in the
sidelobe compared with the previous window functions, it has the widest mainlobe.
As with the previous windows, the width of the mainlobe can be decreased by
increasing the width of the window.
which has the highest sidelobe level down to approximately
-
4.6.4 Kaiser Window
The design of FIR filters with the Kaiser window has become very popular in recent
years. It has a variable parameter to control the size of the sidelobe with respect to
the mainlobe. The Kaiser window function is
()
()
IbIa
n Q
£
Ó
()
=
0
0
wn
(4.74)
K
0
otherwise
where
a
is an empirically determined variable, and
b
=
a
[1
-
(
n
/
Q
)
2
]
1/2
.
I
0
(
x
) is the
modified Bessel function of the first kind defined by
2
2
+
(
)
()
•
Â
n
È
(
)
˘
025
1
.
x
2
025
2
.
x
2
x
2
()
=+
Ix
1
+◊◊◊=
1
+
(4.75)
Í
˙
0
2
2
n
!
()
!
!
Î
˚
n
=
1
which converges rapidly. A trade-off between the size of the sidelobe and the width
of the mainlobe can be achieved by changing the length of the window and the
parameter
a
.
4.6.5 Computer-Aided Approximation
An efficient technique is the computer-aided iterative design based on the Remez
exchange algorithm, which produces equiripple approximation of FIR filters [5,6].
The order of the filter and the edges of both passbands and stopbands are fixed, and
the coefficients are varied to provide this equiripple approximation. This minimizes
the ripple in both the passbands and the stopbands. The transition regions are left
unconstrained and are considered “don't care” regions, where the solution may fail.
Several commercial filter design packages include the Parks-McClellan algorithm
for the design of an FIR filter.
4.7 PROGRAMMING EXAMPLES USING C AND ASM CODE
Within minutes, an FIR filter can be designed and implemented in real time. Several
filter design packages are available for the design of FIR filters. They are described
in Appendix D using MATLAB [50] and in Appendix E using DigiFilter and a
homemade package (on the accompanying CD).
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