Digital Signal Processing Reference
In-Depth Information
unity-gain filters where subscript
k
was previously allowed to take on values, say
between 1 and
N
, (which was 1024) then windowed and truncated to length
L
+1.
Note that the center coefficient (i.e., the (
L
/2+1)th value), is zero when
L
is even.
Moreover, since this is a Type III system, the coefficients are antisymmetric; that
is,
for and of odd length. This antisymmetry means
that there is only a slight change in sign in the implementation involving the
standard time domain filtering process, and as such only the first half of the
coefficients are recorded in the tables that follow.
(
1
)
(
)
d
=
d
k
k
=
1
,
L
L
k
+
2
2
6.3.2
Implementation
Given an input data sequence
x
i
, sampled at interval
T
, an
m
th order differentiating
filter
d
(m)
will produce the
m
th derivative
y
from the input sequence by
L
1
2
(6.3)
∑
=
y
=
d
(
m
)
(
x
+
(
1
m
x
)
j
j
+
k
1
L
j
+
k
+
1
L
m
+
k
(
T
)
K
2
j
1
where
K
= 2
9
is a normalizing factor for our approach and
m
= 1 for the first
derivative. The start position of the differentiated result is similar to that of unity
gain filters for reasons discussed earlier in Chapter 2.
Figure 6.3 describes an
algorithm that implements this differentiation operation. The user is reminded of
the antisymmetry if using coefficient values directly from the tables, or when
using alternative implementations for the said filters. Filtering can also be carried
out in the frequency domain as discussed in Chapter 2.
for
k
= 1 to
N
(
L+
1)
y
=
0
L
+
k
2
L
for
j
=
1 to
2
(
m
)
m
y
=
y
+
d
[
x
+
(
1
x
]
L
L
j
j
+
k
1
L
j
+
k
+
1
+
k
+
k
2
2
end
j
y
=
1
y
L
m
L
+
k
(
T
)
K
+
k
2
2
end
k
Figure 6.3
Algorithm to implement
m
th-order differentiation in the time domain. Note that
m
= 0 for
unity gain filters.
