Digital Signal Processing Reference
In-Depth Information
1
1
(
a
)
(
b
)
P
= 25
L
+1 = 99
0.9
0.9
0.8
0.8
P
= 10
0.7
0.7
0.6
0.6
L
+1 = 55
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Normalized Frequency,
F
Normalised Frequency,
F
Normalised Frequency,
F
Normalized Frequency,
F
Figure 6.2
(a)
Frequency response of a Type III full-band differentiator of lengths 55 and 99 designed
with the window technique. (b) Type IV full-band differentiator of order
P
= 10 and 25 based on the
method of
[8].
Second order differentiating filters will also be described and presented.
Additionally, aspects of their application and gross performance features will be
provided.
6.2 FILTER CLASSIFICATION
Digital differentiators are usually placed in one of four categories, namely, Types
I, II, III, and IV (e.g.,
[5])
. For reasons of consistency with the rest of the topic, we
opted to design and present only Type III differentiators. A differentiator is
classified as Type III if it has an odd number of antisymmetric coefficients, and its
magnitude response | | is zero at both
F
= 0 and 1. However, note that the
Type IV differentiator comes closest to the ideal differentiator because its
magnitude response is zero at
F
= 0 and
D
(
F
)
at
F
= 1. Figure 6.2 shows typical
Types III and IV differentiator response characteristics.
Several classic methods can be used to realize a particular set of Type III
differentiator coefficients. The technique used by Kumar and Dutta Roy
[6]
and
Carlsson
[7]
is rather useful since the coefficients are recursively generated. This
is similar to the maximally flat recursive approach of Khan and others [8,9]. The
latter method of Khan and colleagues is principally a Type IV, but could be useful
in noiseless data applications. The least squares method of coefficient generation
(e.g.,
[12])
is also useful when pass-band requirements are less restrictive but edge
transition sharpness is important. The optimal design method of McClellan and
Parks
[10],
also a least squares technique, produces differentiator coefficients
