Geology Reference
In-Depth Information
In conventional spectral density estimation, resolution in the frequency domain
is limited to the order of the reciprocal of the record length. For an equally spaced
time sequence, the discrete Fourier transform (DFT) representation in the fre-
quency domain is periodic in the record length. In the time domain, the sequence
is assumed to vanish outside the finite record. For actual time sequences neither of
these properties hold. Information theory suggests that data outside the finite record
should not add entropy, as a measure of information, to the result of analysis. In
other words, a spectral estimate should maximise the entropy of the available data.
This leads to the maximum entropy method (MEM) of spectral analysis, due to
Burg, which we describe in detail.
2.1 Time domain analysis
We begin with consideration of the properties and analysis of time sequences in
the time domain. Generally, any operation in the time domain has a counterpart in
the frequency domain, but we will delay discussion of such relationships until we
describe their analysis in the frequency domain.
2.1.1 Classification of time sequences
Perhaps the most common time sequences, and the easiest to treat, are those that
arise from equally spaced sampling of a continuous function of time. In general, a
time sequence is denoted by the indefinite sequence of complex numbers
...,
f
−
1
,
f
0
↑
,
f
1
,...,
f
j
,...
(2.1)
The time index is subscripted, an arrow underneath indicating the origin of the time
axis. The whole time sequence, for brevity, may be indicated by its general term
f
j
. The samples,
f
j
, will be assumed to be equally spaced, unless otherwise stated.
The average
power
in the finite time sequence
f
−
N
,
f
−
N
+
1
,...,
f
0
↑
,...,
f
N
−
1
,
f
N
(2.2)
is
N
N
f
j
1
2
N
+
1
2
N
+
2
f
j
f
∗
j
,
=
(2.3)
1
1
j
=−
N
j
=−
N
where the superscript asterisk denotes complex conjugation. Time sequences that
obey the restriction
N
f
j
1
2
lim
N
→∞
<
∞
,
(2.4)
2
N
+
1
j
=−
N
Search WWH ::
Custom Search