Digital Signal Processing Reference
In-Depth Information
1
0.9
ideal
0.8
0.7
0.6
0.5
(2/
)sin(
F
/2)
0.4
0.3
0.2
F
c
= 0.5
0.1
F
c
= 0.2
0
0
0.2
0.4
0.6
0.8
1
Normalised Frequency,
F
Normalized Frequency,
F
Figure 6.12
First-order differentiators generated from low-pass filters. The sine term determines the ac
accuracy.
designed with bandwidths no larger than 0.15. In principle, they are maximally
smooth and accurate only at dc. In this context, the realization of band-pass
differentiators outside this region should also be avoided.
The technique is similar in some respect to that developed by Kumar and
approach could realize arbitrary low-pass filter characteristics. Because of the
simplicity of the approach, we have not archived low-pass differentiators based on
this method. With regard to pass-band errors, corrections could be made using
methods similar to that described in
[13].
6.6.2
Second-Order Differentiating Filters
Second-order differentiators are more involved than their first-order forerunners,
but they are still manageable. For second-order differentiators, a similar difference
approach yields
k
2
∑
(
2
)
k
+
j
d
=
2
h
6
h
+
(
1
8
h
L
k
+
2
k
k
1
j
(6.26)
j
=
1
(
2
)
(
1
2
)
d
=
d
=
2
h
k
=
2
,
L
+
1
1
L
+
2
where
h
is a low-pass unity-gain filter. As before, the symmetry condition,
, holds for
m
th-order FIR differentiating filters.
Differentiators formed in this way exhibit a wider acceptable passband than
(
m
)
(
m
)
m
d
d
=
(
1
)
L
k
+
1
k
