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are approximately 1.82/
M
apart on the frequency axis. If the true spectral density
varies only slightly over this range of frequencies,
E
S
gg,
n
(
f
)
≈
S
gg
(
f
),
(2.361)
and the estimator, on average, gives an
unbiased
estimate of the true spectral
density.
In the overlapping segment analysis, the final spectral density estimate,
S
gg
(
f
),
is taken as the average of the estimates (2.360) over κ overlapping segments of
length
M
of a complete record of length
T
. Thus,
κ
κ
S
gg,
j
(
f
)
S
gg
(
f
)
1
κ
1
κ
S
gg
(
f
)
S
gg,
j
(
f
)
=
=
−
+
S
gg
(
f
).
(2.362)
j
=
1
j
=
1
The variance of the final spectral density estimate is then
E
S
gg
(
f
)
S
gg
(
f
)
2
var
S
gg
(
f
)
=
−
E
S
gg,
j
(
f
)
S
gg
(
f
)
κ
κ
S
gg
(
f
)
S
gg,
k
(
f
)
1
κ
=
−
−
.
(2.363)
j
=
1
k
=
1
The process generating g(
t
) is assumed to be stationary, so that the terms in the
double sum depend only on the unsigned di
ff
erence of the indices
j
and
k
. Taking
m
as the non-negative value of this di
ff
erence, the double sum can be rearranged
to give
var
S
gg
(
f
)
var
S
gg,
n
(
f
)
1
κ
=
κ
−
1
m
) cov
S
gg,
n
(
f
),
S
gg,
n
+
m
(
f
)
,
2
κ
+
(κ
−
(2.364)
2
m
=
1
where the covariance of
S
gg,
n
(
f
)and
S
gg,
n
+
m
(
f
)is
E
S
gg,
n
(
f
)
S
gg
(
f
)
cov
S
gg,
n
(
f
),
S
gg,
n
+
m
(
f
)
S
gg
(
f
)
S
gg,
n
+
m
(
f
)
=
−
−
E
S
gg,
n
(
f
)
S
gg,
n
+
m
(
f
)
E
S
gg,
n
(
f
)
2
=
−
, (2.365)
recognising that both
E
S
gg,
n
(
f
)
and
E
S
gg,
n
+
m
(
f
)
are
≈
S
gg
(
f
).
To evaluate the covariance, we require the expected value of the product of a
spectral estimate based on a window w
n
(
t
) of the data centred at
b
n
on the time
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