Geology Reference
In-Depth Information
5
f
N
4
f
N
3
f
N
2
f
N
f
N
0
-
f
N
-2
f
N
-3
f
N
-4
f
N
-5
f
N
Figure 2.15 Folding of the content on the frequency axis causing
aliasing
in the
case of time sequences not limited in their frequency content to the principal
Nyquist band,
−
f
N
≤
≤
f
f
N
.
2.5 Power spectral density estimation
The power spectral density, which displays how power in a sequence is distributed
over the frequency axis, is a very useful tool in the search for new signals in back-
ground noise. Usually, a new signal, to be observed for the first time, will be obs-
cured by background noise, otherwise it would have been found previously. Thus,
the estimation of power spectral density is a very important topic for consideration.
2.5.1 Autocorrelation and spectral density
In our earlier discussion of discrete time sequences (Section 2.1.3), we defined the
autocorrelation at lag
j
, time index
k
, of an equispaced time sequence
f
,as
=
E
f
k
f
k
−
j
.
φ
ff
(
j
,
k
)
(2.303)
The equivalent definition for a continuous sequence at lag τ, time
t
,is
E
f
(
t
)
f
∗
(
t
−
τ)
.
φ
ff
(τ,
t
)
=
(2.304)
For stationary sequences, the autocorrelation becomes a function of lag alone,
E
f
(
t
)
f
∗
(
t
−
τ)
.
φ
ff
(τ)
=
(2.305)
Again, for a stationary process, the expectation over an ensemble of realisations
may be replaced, by the ergodic hypothesis, with the average over time of an infin-
ite record. Hence,
T
/2
1
T
f
(
t
)
f
∗
(
t
φ
ff
(τ)
=
lim
T
→∞
−
τ)
dt
.
(2.306)
−
T
/2
Search WWH ::
Custom Search