Digital Signal Processing Reference
In-Depth Information
The sum capacity of such a vector Gaussian channel
z
=
Hy
+
n
(
t
) with the total trans-
mit power constrained to P is given by [79]
{
}
H
C
=
maxlog
r
det
I
+
HH
,
r
(8 .91)
y
y
where
r
y
E{
yy
H
} and the maximization in (8.91) is performed over positive semi-
definite matrices that satisfy the power constraint
tr{}
r
y
≤P
(8.92)
The SINR of the
l
th
user can be written as
2
H
hw
l
l
SINR
l
=
.
(8.93)
∑
2
2
H
σ
+
hw
l
l
m
ml
≠
Using (8.90)-(8.93), the transmit beamforming problem can be formulated as [79]
L
∑
+
2
max
log INRsubject to
W
(
1
)
≤ .
P
(8.94)
l
W
l
=
1
The essence of (8.94) is to maximize the total system throughput under the transmit
power constraint. In simple words, the maximum in (8.94) is achieved by means of
exploiting
multiuser diversity
that suggests always to transmit to the strongest-channel
users. Several computationally efficient algorithms to solve (8.94) have been proposed in
the literature; see [
77
-
79
] and
references
therein.
8.4.1.2 Robust Extensions
The idea of incorporating robustness against transmitter CSI errors in the downlink
beamforming problem (8.82) has been discussed in [14] and [80]. This approach uses
an idea related to that used in [44] for receive RMV beamforming. More specifically,
the following upper and lower bounds on downlink channel correlation matrices are
considered in [80]:
lower
upper
,
r
r
r
(8.95)
icm
,
()
i cm
,
()
icm
,
()
where
lower
upper
r
=
r
− ,
ξ
I
r
=
r
)
+ξ
I
,
(8.96)
icmi cm
,
()
,
()
icm
,
()
icm
,
(
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