Digital Signal Processing Reference
In-Depth Information
whose characteristics are the best against the performance requirements supplied
to the algorithm.
The Window technique is the simplest, but very little control is achieved over
the filter design; there is usually a trade-off between transition width and filter
length. Pass-band errors for the window technique are often small in comparison
to other techniques. Moreover, the technique is particularly useful for
instrumentation prototyping when quick demonstration of an operational principle
is necessary. The filter design process often gravitates towards one with stringent
filtering requirements as product performance limitations are understood and
matched against performance goals.
The window technique has been used to design the first-order differentiators
presented in this chapter. In the case of second-order differentiators, these have
been designed using both the former technique and a new algorithm based on the
conversion of unity-gain filters (UGF). Although there is no strict requirement on
the type of UGF coefficients to use apart from the fact that they must be stable, we
have restricted the design of second-order differentiators to those derived from
Gaussian windowed filter coefficients.
6.3 FIRST-ORDER DIFFERENTIATING FILTERS
In this section, a brief description of the design of first-order differentiators will be
given. The discussion will be limited to the filters designed using the window
technique. Some emphasis will be given to the practical aspects of the
implementation of these filters.
6.3.1
Low-Pass First-Order Differentiating Filters
(
k
First-order differentiator coefficients are found by taking the first derivative,
with respect to
k
, of the impulse response function of the low-pass filter
h
d
k
given
in (2.1). The differentiator coefficients are therefore given by
[11]
W
K
{
}
(
k
d
(
F
)
=
sin
F
n
F
n
cos
F
n
k
c
c
c
c
2
n
(6.2)
(
d
=
0
L
+
1
2
(
(
L
L
d
=
d
n
=
k
+
1
k
=
1
2
,
L
k
+
2
k
2
2
where
F
c
indicates the low-pass cut-off of the differentiator, and
W
k
is the
Gaussian weighting factor. The coefficients in (6.2) may be normalized by
dividing by
K
= 2
9
. Equation (6.2) above has been condensed into a few steps
where, in comparison to unity-gain filters,
N
/2 has been replaced by
L
/2 to reflect
the truncation process. In principle, the design process is identical to the design of
