Geology Reference
In-Depth Information
In general, this is not a satisfactory definition for energy signals since it would
give zero for all lags. For energy signals, the autocorrelation is given the relaxed
definition
∞
f
l
f
l
−
k
.
φ
ff
(
k
)
=
(2.22)
l
=−∞
While this definition might appear to be inconsistent with the ensemble definition
of autocorrelation, we shall see later that if we form a stochastic process in which
the deterministic energy signal is taken to begin at random times with random
amplitudes, it becomes a consistent definition for such a process.
The
crosscorrelation
of the sequence
f
j
with the sequence g
j
at lag
k
, time
index
l
,isdefinedtobe
E
f
l
g
∗
l
−
k
.
φ
f
g
(
k
,
l
)
=
(2.23)
Once again, for stationary sequences, the dependence on time index disappears and
E
f
l
g
∗
l
−
k
E
g
∗
m
f
m
+
k
=
φ
∗
g
f
(
φ
f
g
(
k
)
=
=
−
k
).
(2.24)
Accepting the ergodic hypothesis for two stationary power signals yields the equi-
valent definition
N
1
f
l
g
∗
l
−
k
.
φ
f
g
(
k
)
=
lim
N
→∞
(2.25)
2
N
+
1
l
=−
N
For two energy signals, or one energy signal and one power signal, the definition
is relaxed to
∞
f
l
g
∗
l
−
k
.
φ
f
g
(
k
)
=
(2.26)
l
=−∞
Again, we shall see later how this definition can be made consistent with the
ensemble average definition.
We can make a connection between autocorrelation, crosscorrelation and con-
volution by introducing the
time reverse
of a sequence. The time reverse of the
sequence
f
j
is simply defined as the sequence
f
−
j
.If
f
j
and g
j
are two energy
signals, the crosscorrelation of
f
j
with g
j
is
∞
∞
f
l
g
∗
l
−
k
=
φ
f
g
(
k
)
=
f
l
h
k
−
l
,
(2.27)
l
=−∞
l
=−∞
=
g
∗
−
j
, the time reverse of g
j
. Hence, the crosscorrelation of
f
j
with g
j
is the same as the convolution of
f
j
with the time reverse of g
j
. Similarly,
the autocorrelation of an energy signal is the same as its convolution with its own
time reverse.
=
g
∗
l
−
k
or
h
j
with
h
k
−
l
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