Digital Signal Processing Reference
In-Depth Information
however, that complete cancellation of self-interference is a strongly desirable property.
Otherwise, the symbol-by-symbol detector loses its optimality [59].
To ensure that the self-interference component is completely cancelled, the following
additional zero-forcing constraints can be added to (8.67):
T
aw
1
=
0
forall
mlml
≠ , , =,, , .
1 22
…J
(8.70)
,
l
m
Problem (8.67) with the additional constraints (8.70) can be expressed as
ˆ
T
T
min{
tr
WW
R
}
subjectto
A
= ,
WI
(8 .71)
W
where tr{·} denotes the trace of a matrix.
The solution to (8.71) is given by [59]
ˆ
ˆ
−
1
T
−
1
−
1
W
MV
=
RAARA
(
)
.
(8.72)
Clearly, this MV receiver can be interpreted as a combination of the prewhitener
R
ˆ
-1/2
and decorrelator receiver
R
ˆ
-1/2
A
(
A
T
R
ˆ
-1
A
)
-1
. Furthermore, using the property (8.59), it
can be seen that in the specific case of
R
ˆ
∝
I
, the MV receiver (8.72) simplifies to the MF
receiver (8.62). The latter property is in agreement with the well-known fact that the MF
receiver ignores the effect of MAI treating it as a white noise.
Several robust modifications of the MV receivers (8.69) and (8.72) have been con-
sidered in the literature to improve their robustness in the case of imperfect receive
channel state information (CSI). In [59], diagonally loaded modifications of (8.69) and
(8.72) have been developed. In [63], the approach of [38] has been used to design worst-
case optimization-based RMV receivers that explicitly account for norm-bounded CSI
errors. For example, the following worst-case modification of (8.67) has been proposed
in [63]:
ˆ
(
ˆ
T
T
min
ww
R
subjectto
|| ||≤
min
wa H
1
≥
+
∆
)
1
forall
l
= , ,
1
…J
,
2
(8.73)
l
l
k
l
w
∆
ε
l
where, for the sake of clarity, the user-of-interest spatio-temporal signature
a
l
,1
is explic-
itly denoted as
a
l
(
H
1
) to stress that it is a function of the user-of-interest channel
H
1
.
In (8.73), another version of this signature,
a
l
(
ˆ
1
+
Δ
), is used, where
Δ
stands for the
receive CSI error of the user-of-interest:
−
ˆ
∆
HH
1
,
(8.74)
1
and
H
1
and
ˆ
1
denote the actual and presumed (estimated) channel matrices of the user
of interest, respectively. Similar to [38], it is assumed in (8.73) that the CSI error is norm-
bounded by some known constant ε, that is,
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