Digital Signal Processing Reference
In-Depth Information
the filter when the input is linear of slope 1. The output is the step response
function
U
k
and is determined using
k
=
∑
=
(
U
d
(
k
j
+
1
k
=
1
2
,
L
+
1
(6.7)
k
j
j
1
Thus, any unit step change in
slope
will yield an output similar to that shown in
Figure 6.5 (b), and described by (6.7). The rise time, settling time, and overshoot
have been tabulated.
6.3.5
DC Accuracy
The dc accuracy
of a first-order differentiator in the time domain is the
difference between the ideal settling value of unity, and the actual settling value of
the slope response function,
U
L+1
. Thus, we have
ε
=
1
U
(6.8)
L
+
1
is a few parts in 10
4
(typically 0.03%) for standard
floating-point precision coefficients. Furthermore, the dc accuracy was further
investigated when 10
4
samples of the unit slope function are offered to the
differentiator. For floating-point precision, first order differentiators show no
appreciable error. However, in passing, the dc accuracy is usually sensitive for
second-order differentiator operations and could display significant inaccuracies
over long steady slopes. This will be discussed later. Note also that the ac
accuracy of the differentiator is characterized by the pass-band ripple
R
(
F
).
For most differentiators,
6.3.6
Band-Pass Differentiating Filter Coefficients
The coefficients corresponding to band-pass differentiating filters are similar in
form to unity-gain band-pass filters and are given by
(
(
(
d
=
d
(
F
)
d
(
F
)
(6.9)
c,
off
c,
on
k
k
k
where
F
c,off
and
F
c,on
are the normalized cut-off and cut-on frequency,
respectively. Other classes of filters such as high-pass and band-stop
differentiators can be designed in a similar manner to unity-gain filters.
6.3.7
Quantization of Low-Pass Filter Differentiator Coefficients
The quantization of differentiator coefficients is slightly more involved that its
unity-gain counterpart. Previously, we used the peak value attained by the impulse
