Digital Signal Processing Reference
In-Depth Information
0
1.4
(
b
)
(
a
)
-20
1.2
-40
1
-60
0.8
-80
0.6
-100
0.4
-120
0.2
-140
14-bit
Full Precision
0
-160
0.01
0.1
1
0
0.2
0.4
0.6
0.8
1
Normalised Frequency,
F
Normalized Frequency,
F
Normalized Cut-Off Frequency,
F
c
Normalised Frequency,
F
c
d
peak
(
)
Figure 6.6
(a)
Dependence of first and largest peak of the impulse response function plotted
against normalized cut-off frequency for a first-order differentiator. (b) Frequency response of a 14-bit
and full-precision first order differentiator.
frequency response of a 14-bit differentiator using (6.11) and (6.12).
6.3.8
Filter Gain
G
Since we have dilated the filter to fit the range of integer values, in order to
achieve unity gain we must reduce the output by the same factor. The filter gain
G
from (6.12) is
(
2
B
1
1
(6.13)
G
=
(
+
Q
/
100
)
512
d
(
peak
Thus dividing the filtered output by
G
produces a properly scaled derivative of the
signal.
6.4 SECOND-ORDER DIFFERENTIATING FILTERS
F
)
2
. There are
several techniques for the design of higher-order differentiators, such as [14,15].
In this work, we will concentrate on the window technique with some minor
modifications to the coefficients. The approach is simple but problematic. As we
The ideal second-order differentiator
D
2
(
F
) is proportional to (
