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where
(
f
) is the unit-step function. The C-PSD is
P
xx
(
f
)
H
(
f
)
P
uu
(
f
)
H
∗
(
f
)
P
uu
(
f
)
H
∗
(
f
)
=
P
uu
(
f
)
−
+
2j Re
{
}=
0
.
(8.30)
So the analytic signal constructed from a WSS real signal has zero PSD for negative
frequencies and four times the PSD of the real signal for positive frequencies, and it is
proper
. Propriety also follows from the bound (
8.5
) since
|
P
xx
(
f
)
2
|
≤
P
xx
(
f
)
P
xx
(
−
f
)
=
0
.
(8.31)
There is also a close interplay between wide-sense stationarity and propriety for equiv-
alent complex baseband signals. As in Section
1.4
,let
p
(
t
)
x
(
t
)e
j2
π
f
0
t
denote a
real passband signal obtained by complex modulation of the complex baseband signal
x
(
t
) with bandwidth
=
Re
{
}
f
0
. It is easy to see that a complex-modulated signal
x
(
t
)e
j2
π
f
0
t
is proper if and only if
x
(
t
) is proper. If
x
(
t
)e
j2
π
f
0
t
<
is the analytic signal obtained from
p
(
t
), we find the following.
Result 8.2.
A real passband signal p
(
t
)
is WSS if and only if the equivalent complex
baseband signal x
(
t
)
is WSS and proper.
An alternative way to prove this is to express the covariance function of
p
(
t
)as
Re
r
xx
(
t
)e
j2
π
f
0
τ
+
Re
r
xx
(
t
)e
j2
π
f
0
(2
t
+
τ
)
.
r
pp
(
t
,τ
)
=
,τ
,τ
(8.32)
If
r
pp
(
t
,τ
) is to be independent of
t
, we need the covariance function
r
xx
(
t
,τ
)tobe
independent of
t
and the complementary covariance function
r
xx
(
t
0.
We note a subtle difference between analytic signals and complex baseband signals:
while there are no improper WSS analytic signals, improper WSS complex baseband
signals do exist. However, the real passband signal produced from an improper WSS
complex baseband signal is
cyclostationary
rather than WSS. Result
8.2
is important
for communications because passband thermal noise is modeled as WSS. Hence, its
complex baseband representation is WSS and proper.
,τ
)
≡
8.2.2
Noncausal Wiener filter
The celebrated Wiener filter produces a linear estimate
x
(
t
)ofa
message signal x
(
t
)
from an
observation
(or measurement)
y
(
t
) based on the PSD of
y
(
t
), denoted
P
yy
(
f
),
and the cross-PSD between
x
(
t
) and
y
(
t
), denoted
P
xy
(
f
). This assumes that
x
(
t
) and
y
(
t
) are jointly WSS. The estimate
x
(
t
) is optimal in the sense that it minimizes the
mean-squared error
E
2
, which is independent of
t
because
x
(
t
) and
y
(
t
)are
jointly WSS. The frequency response of the noncausal Wiener filter is
|
x
(
t
)
−
x
(
t
)
|
P
xy
(
f
)
P
yy
(
f
)
.
H
(
f
)
=
(8.33)
If
y
(
t
)
=
x
(
t
)
+
n
(
t
), with uncorrelated noise
n
(
t
) of PSD
P
nn
(
f
), then the frequency
response is
P
xx
(
f
)
P
xx
(
f
)
H
(
f
)
=
P
nn
(
f
)
.
(8.34)
+
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