Digital Signal Processing Reference
In-Depth Information
0.1
0.1
(
a
)
(
b
)
0.09
0.09
0.08
0.08
ideal
0.07
0.07
0.06
0.06
0.05
0.05
0.04
0.04
mismatch
0.03
0.03
0.02
0.02
0.01
0.01
dc offset
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Normalised Frequency,
F
Normalised Frequency,
F
Normalized Frequency,
F
Normalized Frequency,
F
Figure 6.7
The dc distortion in a second-order differentiator for low filter order
L
= 55.
(a) Uncorrected (b) dc corrected.
will see, the frequency response of the second-order differentiator exhibits a dc
offset as well as a mismatch between the attenuation and zero positions.
The coefficients of the second-order differentiator are found by taking the
second derivative of the impulse response function of the unity-gain low-pass
filter with respect to
k
. Alternatively, we can take the first derivative of the first
order impulse response function. The second-order coefficients are therefore given
by
[
(
)
]
NW
(
2
)
k
2
2
2
d
(
F
)
=
2
F
n
cos
F
n
+
F
n
2
sin
F
n
k
c
c
c
c
c
3
2
n
N
(6.14)
(
2
)
2
3
d
=
F
c
L
+
1
2
2
(
2
)
(
2
)
L
L
d
=
d
n
=
k
+
1
k
=
1
2
,
L
k
+
2
k
2
2
Unfortunately, this result leads to distortions in the response of the differentiating
filters. The dominant distortion appears as a dc offset in the frequency domain.
This can be removed by tweaking the central value
(
2
2
)
1
d
, as discussed in Section
L
+
2.11. The second distortion appears to be a mismatch between the response at dc,
and that across the attenuation band. However, this second distortion cannot be
removed by simple techniques. In principle, if a differentiator is to exhibit both
frequency-based dc accuracy and good attenuation, then the magnitude in the
attenuation band should not be less than the dc offset as happens in these
