Digital Signal Processing Reference
In-Depth Information
6.6 CONVERSION OF UNITY GAIN FILTERS INTO
DIFFERENTIATORS
6.6.1
First-Order Differentiating Filters
The conversion of unity gain filters into differentiators is a quick and simple way
of realizing low-pass differentiators. Given a low-pass unity-gain filter
h
for
example, of length
L
+1, the corresponding first-order differentiator
d
(1)
is given by
(
d
=
h
h
k
=
2
,
L
+
1
(6.23)
k
1
k
k
1
If the filter length
L
+1 of
h
is odd, then the differentiator
d
will be of even length
and one shorter than its progenitor; that is, a new length of
L
. On the other hand, if
the length of the unity gain filter is even, then the corresponding differentiator will
be odd and one shorter than the filter. This approach is useful for fast generation
of first-order low-pass differentiating filters up to
F
c
0.15, showing no Gibbs
ringing in the pass-band error
R
(
F
), good dc accuracy, and linear in phase if
h
is a
linear-phase filter. However, for increasing
F
c
, it exhibits increasingly poorer ac
accuracy than standard techniques across the differentiator passband. This has a
straightforward explanation.
In terms of the of the Z-transform, (6.23) could be written as
(
1
d
=
(
z
)
h
(6.24)
k
k
1
which is easily converted into the frequency domain by substituting
j
F
z
=
e
. The
normalized magnitude in the frequency domain is therefore given by
2
π
F
D
(
F
)
=
sin
H
(
F
)
(6.25)
π
2
where
H
(
F
) and
D
(
F
)
are the frequency responses of the unity gain low-pass filter
and corresponding first-order differentiator, respectively. The result shows that
(6.24) is the convolution of the sine approximation of a first-order differentiator
and a low-pass filter. This explains why the pass-band of the new differentiator is
modified by the envelope function 2sin(
F
/2) and the ac error increases with
frequency. Figure 6.12 shows the ideal full-band differentiator, the term in
(2/
F
/2), and two low-pass differentiating filters with cut-off at 0.2 and 0.5
determined from (6.23). Note that the magnitude of both differentiating filters
follows the sine term, which results in large inaccuracies for relatively large cut-
off frequencies. As such, it is recommended that the said differentiators be
)sin(
