Digital Signal Processing Reference
In-Depth Information
E
max
satisfies the upper bound imposed by the specifications; when this is
not the case, the degree of the polynomial (and therefore the length of the
filter) must be increased and the procedure must be restarted. Once the
conditions on the error are satisfied, the filter coefficients can be obtained
by inverting the Chebyshev expansion.
As a final note, an initial guess for the number of taps can be obtained
using the empirical formula by Kaiser; for an
M
-tap FIR
h
l
g
r
,
y
i
d
.
,
©
,
L
s
[
n
]
,
n
=
0,...,
M
−
1:
−
10 log
10
(
δ
δ
)
−
13
p
s
+
M
1
2.324
Ω
where
δ
p
is the passband tolerance,
δ
s
is the stopband tolerance and
Ω=
ω
−
ω
p
is the width of the transition band.
s
The Final Design.
We now summarize the design steps for the specifica-
tions in Figure 7.1. We use a Type I FIR. We start by using Kaiser's formula to
obtain an estimate of the number of taps: since
δ
δ
=
10
−
3
and
Ω=
π
0.2
,
p
s
we obtain
M
12.6 which we round up to 13 taps. At this point we can
use any numerical package for filter design to run the Parks-McClellan algo-
rithm. In Matlab this would be
=
[h, err] = remez(12, [0 0.4 0.6 1], [1 1 0 0], [1 10]);
The resulting frequency response is plotted in Figure 7.14; please note
that we are plotting the frequency responses of the zero-centered filter
h
d
[
n
]
,
ω
which is a real function of
. We can verify that the filter has indeed
(
6 alternation by looking at enlarged picture of the passband and
the stopband, as in Figure 7.15. The maximum error as returned by Mat-
lab is however 0.102 which is larger than what our specifications called for,
M
−
1
)
/
2
=
1
0
0
π/
4
2
π/
4
3
π/
4
π
Figure 7.14
An optimal 13-tap Type I filter which does not meet the error specifi-
cations.
