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narrowband interferer
f
−
1
T
0
1
T
2
T
Figure 10.3
The solid dark line shows the support of d
ξ
(
f
), and the dashed dark line the support
of frequency-shifted d
ξ
(
f
−
1
/
T
). The interferer disturbs d
ξ
(
f
)at
f
=
1
/
(2
T
)and
d
ξ
(
f
−
1
/
T
)at
f
=
3
/
(2
T
). Since d
ξ
(
f
)andd
ξ
(
f
−
1
/
T
) correlate perfectly on [0
,
1
/
T
],
d
ξ
(
f
−
1
/
T
) can be used to compensate for the effects of the narrowband interferer.
2
d
2
d
where
σ
and
σ
are the variance and complementary variance of the data sequence
d
k
. Hence, if
d
k
is maximally improper, there is complete reflectional coherence
2
1 as long as
B
(
f
)
B
∗
(
f
|
ρ
xx
(
n
/
T
,
f
)
|
=
−
n
/
T
)
=
0. Thus, d
ξ
(
f
) can be perfectly
ξ
∗
(
n
estimated from frequency-shifted conjugated versions d
f
). This spectral
redundancy comes on top of the spectral redundancy already discussed above. For
instance, the spectrum of baseband BPSK, using a Nyquist roll-off pulse with
/
T
−
1,
contains the information about the transmitted real-valued data sequence
d
k
exactly
four times. On the other hand, if
d
k
is proper (as in QPSK), then
α
=
0 and
there is no additional complementary spectral redundancy that can be exploited using
conjugate-linear operations.
ρ
xx
(
n
/
T
,
f
)
=
10.3
Cyclic Wiener filter
The problem discussed in the previous subsection can be generalized to the setting where
we observe a random process
y
(
t
), with corresponding spectral process
(
f
), that is
a noisy measurement of a message
x
(
t
). Measurement and message are assumed to
be individually and jointly (almost) CS. We are interested in constructing a filter that
produces an estimate
x
(
t
) of the message
x
(
t
) from the measurement
y
(
t
) by exploiting
CS properties. In the frequency domain, such a filter adds suitably weighted spectral
components of
y
(
t
), which have been frequency-shifted by the cyclic frequencies, to
produce an estimate of the spectral process:
υ
N
N
n
=
1
G
n
(
f
)d
d
ˆ
υ
∗
(˜
ξ
(
f
)
=
G
n
(
f
)d
υ
(
f
−
ν
n
)
+
ν
n
−
f
)
.
(10.41)
n
=
1
Such a filter is called a (widely linear)
frequency-shift (FRESH) filter
. The corresponding
noncausal time-domain estimate is
N
N
(
y
(
t
)e
j2
πν
n
t
)
(
y
∗
(
t
)e
j2
π ν
n
t
)
x
(
t
)
=
∗
+
g
n
(
t
)
∗
.
g
n
(
t
)
(10.42)
n
=
1
n
=
1
N
N
n
=
1
Since this FRESH filter allows arbitrary frequency shifts
n
=
1
, with
N
and
N
possibly infinite, it can be applied to CS or almost CS signals. Of course, only
those frequency shifts that correspond to nonzero cyclic PSD/C-PSD are actually useful.
{
ν
n
}
and
{
ν
n
}
˜
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