Geology Reference
In-Depth Information
Writing
l
=
j
−
k
, this becomes
−∞
∞
∞
h
j
=
f
j
−
l
g
l
=
g
l
f
j
−
l
=
g
k
f
j
−
k
.
(2.11)
l
=∞
l
=−∞
k
=−∞
Thus, the convolution of
f
j
with g
j
is the same as the convolution of g
j
with
f
j
.
The convolution of
two energy signals
produces an
energy signal
, and the con-
volution of an
energy signal
with a
power signal
results in a
power signal
, while
the convolution of
two power signals does not exist
.
Convolution, and many other operations with time sequences, are facilitated by
the use of the
z-transform
. For the time sequence
...,
f
−
1
,
f
0
↑
,...,
f
j
,...,
(2.12)
the z-transform is defined as
f
−
1
z
+
f
j
z
j
F
(
z
)
=···+
f
0
+
f
1
z
+···+
+···
(2.13)
If
h
j
is the sequence resulting from the convolution of the sequence
f
j
with the
sequence g
j
, it has the z-transform
H
(
z
)
=
F
(
z
)
·
G
(
z
).
(2.14)
Hence, the z-transform of the convolution of
f
j
with g
j
is the product of the
z-transforms of
f
j
and g
j
. The product of the z-transforms is
f
−
1
z
+
f
0
+
f
1
z
+···
···+
g
−
1
F
(
z
)
·
G
(
z
)
=
···+
z
+
g
0
+
g
1
z
+···
···+
···
f
−
1
g
−
1
···
z
2
+
···
f
−
1
g
0
+
f
0
g
−
1
···
z
=
+ ···
f
−
1
g
1
+
f
0
g
0
+
f
1
g
−
1
···
)
z
2
+
(
···
f
0
g
1
+
f
1
g
0
···
)
z
+
(
···
f
1
g
1
···
+···
,
(2.15)
where we have included only terms involving time indices
−
1, 0 and 1 in the
cient of the term in
z
j
sequences
f
j
and g
j
. If all terms are included, the coe
takes the form
···+
f
−
1
g
j
+
1
+
f
0
g
j
+
f
1
g
j
−
1
+···+
f
k
g
j
−
k
+···
,
(2.16)
in conformity with the general term in expression (2.10) for the convolution.
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