Geology Reference
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or equivalently
N
1
f
j
f
∗
j
<
∞
,
lim
N
→∞
(2.5)
2
N
+
1
j
=−
N
are called
power signals
. Power signals are the most intense time sequences we
can expect to meet in practice. The Earth tide signal registering on gravimeters is
an example of a power signal.
Other signals build up from zero or a very low level and then fade away. Aver-
aged over all time, such signals would have zero power. Instead, they have finite
energy and are called
energy signals
. These are such that
∞
f
j
=
∞
f
j
f
∗
j
<
∞
,
(2.6)
j
=−∞
j
=−∞
and, hence, have finite energy. An example of an energy signal would be the baro-
metric pressure signal associated with the passage of a storm front.
A third class of time sequences are zero until a certain time, often when an
earthquake or other energy releasing event takes place. They are one-sided and
take the form
...,0,0,0,
f
0
↑
,
f
1
,
f
2
,...,
(2.7)
and are called
wavelets
.
We have left unspecified the duration of particular time sequences. Of course, all
time sequences resulting from actual records are of
finite duration
.Intheoretical
discussions they may be taken to be of
unlimited duration
.
2.1.2 Convolution and the z-transform
Perhaps the most common operation that can be performed with two time sequences
is their
convolution
. For two time sequences with general terms
f
j
and g
j
or
...,
f
−
1
,
f
0
↑
,
f
1
,...,
f
j
,...
(2.8)
and
...,g
−
1
,g
0
↑
,g
1
,...,g
j
,...,
(2.9)
the
convolution
of
f
j
with g
j
is
∞
h
j
=
f
k
g
j
−
k
.
(2.10)
k
=−∞
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