Digital Signal Processing Reference
In-Depth Information
As in the Kaiser window case, an empirical formulation of the required length
of an FIR filter designed using the PM algorithm has been developed, as shown in
(7.39). Although somewhat extensive in its presentation, it does provide an
accurate estimate of the required length.
2
K
−
K
⋅
∆
f
1
2
N
=
+
1
∆
f
(7.39)
where
2
K
=
[
0
.
005309
⋅
(log
δ
)
+
0
.
07114
⋅
log
δ
−
0
.
4761
]
⋅
log
δ
1
p
p
s
(7.40)
2
−
[
0
.
00266
⋅
(log
δ
)
+
0
.
5941
⋅
log
δ
+
0
.
42781
]
p
p
K
2
=
0
51244
⋅
(log
δ
−
log
δ
)
+
11
.
012
p
s
(7.41)
∆
f
=
(
f
stop
−
f
)
/
f
pass
s
(7.42)
Example 7.4 Determining Parks-McClellan Coefficients for FIR Filter
Problem:
Determine the impulse response coefficients for a bandpass filter
using the same specifications as in Example 7.3 (as indicated below), except use
the Parks-McClellan algorithm for coefficient determination.
f
pass1
= 4 kHz,
f
pass2
= 5 kHz,
f
stop1
= 2 kHz,
f
stop2
= 8 kHz,
a
pass
= −0.5 dB,
a
stop1
=
a
stop2
= −50 dB, and
f
samp
= 20 kHz
Solution:
We will use the WFilter program for this example. The same input
parameters as in the previous example are specified, except the approximation
type has been changed to Parks-McClellan. Using the specified parameters, the
first design attempt resulted in an estimated length of 19, which produced a filter
with passband edge gains of −0.54 dB and stopband edge gains of −49.38 dB.
These values are certainly very close to the design specifications and might be
acceptable in many designs. However, for comparison purposes, the filter was
redesigned using a filter length of 21 and produced passband gains of −0.23 dB
and stopband gains of −56.70 dB. The resulting coefficients and frequency
response curve are shown in Figures 7.15 and 7.16.
We should notice the equal ripple in the stopbands for the PM filter and the
fact that it is implemented in one-third fewer coefficients than the Kaiser filter. As
a comparison to IIR filters, a sixth-order elliptic or an eighth-order Butterworth
