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showing explicitly only the terms of interest. The fourth moment required in expres-
sion (2.367) is then supplied by
E
g
1
g
∗
2
g
3
g
∗
4
4
Φ
∂ω
∗
1
∂ω
2
∂ω
∗
3
∂ω
4
,
∂
=
16
(2.372)
evaluated at ω
1
erentiation of the first expression (2.369)
for the characteristic function, and settingω
1
=
ω
2
=
ω
3
=
ω
4
=
0. Di
ff
=
ω
2
=
ω
3
=
ω
4
=
0, gives the required
fourth moment as
E
g
1
g
∗
2
g
3
g
∗
4
=
c
14
c
32
=
φ
gg
(τ
1
)φ
gg
(τ
2
)
c
12
c
34
+
+
φ
gg
(τ
3
)φ
gg
(τ
4
),
(2.373)
with τ
1
t
. Substitution of this expression for
the fourth moment in (2.367), and changing variables of integration, produces the
formula
E
S
gg,
n
(
f
)
S
gg,
n
+
m
(
f
)
=
s
−
t
, τ
2
=
u
−
v, τ
3
=
s
−
v, τ
4
=
u
−
I
2
∞
i
2π
f
τ
1
)
φ
gg
(τ
1
)
∞
−∞
1
w
n
(
s
)w
∗
n
(
s
exp
(
=
−
−
τ
1
)
dsd
τ
1
−∞
∞
i
2π
f
τ
2
)
φ
gg
(τ
2
)
∞
−∞
w
n
+
m
(
u
)w
∗
n
+
m
(
u
·
exp
(
−
−
τ
2
)
dud
τ
2
−∞
I
2
∞
−
i
2π
f
τ
3
)
φ
gg
(
τ
3
)
∞
−∞
1
w
n
(
s
)w
∗
n
+
m
(
s
−
τ
3
)
dsd
τ
3
+
exp
(
−∞
∞
−
i
2π
f
τ
4
) φ
gg
(τ
4
)
∞
∞
w
n
+
m
(
u
)w
∗
n
(
u
−
τ
4
)
dud
τ
4
,
·
exp (
(2.374)
−∞
for the expected value of the product of spectral estimates, based on identical
windows w
n
(
t
)andw
n
+
m
(
t
) shifted to centre on
b
n
and
b
n
+
m
on the time axis,
respectively. Each of the inner integrals on the right-hand side of expression (2.374)
represents a convolution of a shifted window with the time reverse (complex con-
jugate of the function obtained by switching the sign of the argument) of another
shifted window. If the Fourier transform of the window w(
t
)is
W
(
f
), the Fourier
transform of a window shifted to
b
on the time axis is
∞
−∞
w(
t
∞
−∞
w(
t
)
e
−
i
2π
f
(
t
+
b
)
dt
=
b
)
e
−
i
2π
ft
dt
e
−
i
2π
fb
W
(
f
),
−
=
(2.375)
a general property of Fourier transforms known as the
shifting theorem
. Further,
the complex conjugate of
W
(
f
)is
∞
∞
t
)
e
−
i
2π
ft
dt
,
W
∗
(
f
)
w
∗
(
t
)
e
i
2π
ft
dt
w
∗
(
=
=
−
(2.376)
−∞
−∞
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