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the simplified expression
N
2
|
ρ
xx
(
t
¯
,
f
)
|
=
(9.130)
−|
S
xx
(2
f
r
xx
(
t
,
0)(
S
xx
(0
,
f
)
,
f
)
|
2
)
,
where
+|
V
xx
(
t
2
2
)
=
,
|
,
|
,
|
N
S
xx
(0
f
)(
V
xx
(
t
f
)
f
)
f
)
V
xx
(
t
f
)
S
xx
(2
f
2Re[
V
xx
(
t
f
)e
j4
π
ft
]
−
,
,
,
,
=|
S
xx
(2
f
S
xx
(0
2
. Both in the proper case, which is characterized by
if
,
f
)
,
f
)
|
V
xx
(
t
0 and
S
xx
(2
f
,
f
)
=
,
f
)
=
0, and in the maximally improper case, characterized by
=|
S
xx
(2
f
2
, the magnitude-squared time-frequency coherence simplifies
to the magnitude-squared rotational time-frequency coherence (
9.93
), i.e.,
S
xx
(0
,
f
)
,
f
)
|
2
|
V
xx
(
t
,
f
)
|
2
2
|
ρ
xx
(
t
¯
,
f
)
|
=|
ρ
xx
(
t
,
f
)
|
=
f
)
.
(9.131)
r
xx
(
t
,
0)
S
xx
(0
,
In these cases
x
(
t
) can be estimated from counterclockwise rotating phasors only. In other
words, the optimum WLMMSE estimator is the LMMSE estimator, i.e.,
W
2
(
t
,
−
f
)
=
0.
9.5
Higher-order statistics
We conclude this chapter with a very brief introduction to the higher-order statistics of
a continuous-time signal
x
(
t
).
15
We denote the
n
th-order moment function by
E
x
n
(
t
)
+
τ
i
)
n
−
1
x
i
(
t
m
x
,
♦
(
t
,
)
=
,
(9.132)
i
=
1
τ
1
,...,τ
n
−
1
]
T
, and
n
,
1
,
2
,...,
n
−
1
]
T
contains
where, as in Section
8.5
,
=
[
♦=
[
i
that are either 1 or the conjugating star
∗
. This leads to 2
n
different
n
th-order
moment functions, depending on which terms are conjugated. As explained in Section
8.5
, not all of these functions are required for a complete statistical description.
We assume that
x
(
t
) can be represented as in the CL spectral representation (
9.47
),
elements
∞
(
f
)e
j2
π
ft
x
(
t
)
=
d
ξ
,
(9.133)
−∞
but
(
f
)isnowan
N
th-order spectral process with moments defined and bounded up
to
N
th order. That is, the moment functions can be expressed in terms of the increment
process d
ξ
ξ
(
f
)as
E
d
i
f
i
)
e
j2
π
[
f
T
n
−
1
ξ
n
(
ξ
i
(
+
(
f
1
+···+
f
n
−
1
−
f
n
)
t
]
m
x
,
♦
(
t
,
)
=
−
n
f
n
)
d
,
(9.134)
IR
n
i
=
1
f
n
−
1
]
T
, and
i
is the conjugation star
∗
.
where
n
=
1
,...,
N
,
f
=
[
f
1
,...,
i
f
i
=−
1if
The “
”signfor
f
n
is to ensure that (
9.134
) complies with (
9.50
) and (
9.51
)inthe
second-order case
n
−
=
2. Since
t
is a global time variable and
contains local time
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