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alternative is the Rayleigh-Ritz technique, which numerically solves operator equations.
The Rayleigh-Ritz technique is discussed by
Chen
et al
. (1997
);
Navarro-Moreno
et al
.
processes.
Example 9.1.
To gain more insight into the improper KL expansion, consider the follow-
ing communications example. Suppose we want to detect a
real
waveform
u
(
t
) that is
transmitted over a channel that rotates it by some random phase
φ
and adds complex
white Gaussian noise
n
(
t
). The observations are then given by
u
(
t
)e
j
φ
+
y
(
t
)
=
n
(
t
)
,
(9.24)
where we assume pairwise mutual independence of
u
(
t
),
n
(
t
), and
φ
. Furthermore,
u
(
t
)e
j
φ
. Its covariance is given by
r
xx
(
t
1
,
we denote the rotated signal by
x
(
t
)
=
t
2
)
=
E
e
j2
φ
.
E
[
u
(
t
1
)
u
(
t
2
)] and its complementary covariance is
r
xx
(
t
1
,
t
2
)
=
E
[
u
(
t
1
)
u
(
t
2
)]
·
There are two important special cases. If the phase
φ
is uniformly distributed,
then
r
xx
(
t
1
,
0 and detection is
noncoherent
. The eigenvalues of the augmented
covariance of
x
(
t
) satisfy
t
2
)
≡
λ
2
n
−
1
=
λ
2
n
=
µ
n
. On the other hand, if
φ
is known, then
e
j2
φ
r
xx
(
t
1
,
r
xx
(
t
1
,
t
2
)
≡
t
2
) and detection is
coherent
. If we order the eigenvalues appro-
priately, we have
0. Therefore, the coherent case is the most
improper case under the power constraint
λ
2
n
−
1
=
2
µ
n
and
λ
2
n
=
λ
2
n
−
1
+
λ
2
n
=
2
µ
n
. These comments are
clarified by noting that
j
2
λ
2
n
=
1
2
λ
2
n
−
1
=
E
{
x
(
t
)Re
x
n
}
,
E
{
x
(
t
)Im
x
n
}
.
(9.25)
Thus,
λ
2
n
−
1
measures the covariance between the real part of the observable coordinate
x
n
and the continuous-time signal, and
λ
2
n
does so for the imaginary part.
In the noncoherent version of (
9.24
), these two covariances are equal, suggesting that
the information is carried equally in the real and imaginary parts of
x
n
. In the coherent
version,
0 shows that the information is carried exclusively in the
real part of
x
n
, making Re
x
n
a
sufficient statistic
for the decision on
x
(
t
). Therefore,
in the coherent problem, WL processing amounts to considering only the real part of
the internal description. The more interesting applications of WL filtering, however, lie
between the coherent and the noncoherent case, being characterized by a nonuniform
phase distribution, or in adaptive realizations of coherent algorithms.
λ
2
n
−
1
=
2
µ
n
,
λ
2
n
=
9.1.1
Estimation
The KL expansion can be used to solve the following problem: widely linearly estimate a
nonstationary improper complex zero-mean random signal
x
(
t
) with augmented covari-
ance
R
xx
(
t
1
,
t
2
) in complex white (i.e., proper) noise
n
(
t
) with power-spectral density
N
0
. The observations are
y
(
t
)
=
x
(
t
)
+
n
(
t
)
,
0
≤
t
≤
T
,
(9.26)
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