Digital Signal Processing Reference
In-Depth Information
responses. To illustrate these effects, Figure 6.7 (a) shows the frequency response
of a 55-tap differentiator for
F
c
= 0.3, using (6.14). In Figure 6.7 (b), the dc offset
in the frequency domain has been corrected, using the procedure described in
Section 2.11, so that the filter is dc accurate (i.e.,
D
2
(
F
) = 0 at
F
= 0). Once the dc
accuracy has been restored, there is usually very good correlation between the
design and the ideal second-order differentiator. However, because of the
mismatch problem, the filter response in the rejection band is reflected in the x-
axis, showing poor attenuation in the passband (typically 50 dB).
The consequence of the dc correction is a slight shift in the ripple function
R
(
F
) across the pass-band and appears asymmetric about zero error. This shift
ensures that the error is zero at dc. Moreover, the peak-to-peak error is still of the
order of 10
-5
. In general, the second-order differentiator designed in this way is
both ac and dc accurate. However, it displays rather poor attenuation. Figure 6.8
(a) shows the shift in pass-band error while (b) shows the corrected impulse
response. Because of the attenuation problem, another second-order differentiator
design has been provided. This will be discussed in Section 6.6.
6.4.1
Second-Order Slope Response and Stability
The second order slope response of a differentiator is its response to the function
x
2
/2, where ideally, its second derivative is unity. In fact, this concept could be
extended to the
m
th order response via the input slope function
x
m
/
m
!. In principle,
the
m
th order slope response is similar in nature to the step response
U
k
in unity
gain filters. The approach is useful because it allows us to compare various
0.25
40
(
a
)
(
b
)
0.2
30
0.15
20
0.1
10
0.05
0
0
Not
symmetric
about zero
-10
-0.05
-20
-0.1
-30
-0.15
-40
-0.2
-0.25
-50
0
0.12
0.24
0.36
0.48
0.6
0
50
100
Normalised Frequency,
F
Index,
k
Normalized Frequency,
F
Figure 6.8
(a) Pass-band error for a 55-point second-order differentiator. (b) Impulse response
function.
