Digital Signal Processing Reference
In-Depth Information
∆
≤ε
(8.75)
The essence of the RMV receiver design problem in (8.73) is to minimize the receiver
output power while maintaining the distortionless response for the worst-case CSI errors.
It has been proved in [63] that for any OSTBC,
∆
=
δ
l
forall
l
= ,…,
12,
J
(8.76)
where
() (
ˆ
δ
l
aH
−
aH
)
(8.77)
l
1
l
1
is the error between the actual spatiotemporal signature
a
l
(
H
1
) and its mismatched (pre-
sumed) value
a
l
(
ˆ
1
). Therefore, problem (8.73) can be rewritten as
ˆ
(
ˆ
min
ww
T
R
subjectto
|| ||
δ
ε
min(
waH
T
1
++≥
)
δ
l
)1
forall
l
=, , .
12
…J
(8.78)
l
l
l
l
w
l
l
This problem is mathematically equivalent to 2
J
decoupled RMV beamforming prob-
lems (8.32). Hence, using the results of [38], problem (8.78) can be reformulated as [63]
ˆ
(
ˆ
T
T
min
ww
R
subjectto
wa Hw
)
≥
ε
+
1
for al
l
l
=, 12,
…J
(8.79)
l
l
l
l
1
l
w
l
where the constraint can be shown to be satisfied with equality. Similar to (8.33), the
problem in (8.79) belongs to the class of standard SOCP problems and can be efficiently
solved using either modern convex optimization tools [50] or Newton-type algorithms
[46, 51, 52].
In addition to (8.79), one more RMV receiver has been developed in [63] as a robust
extension of the MV receiver (8.72). The later RMV receiver enables better cancellation
of self-interference and, as a result, offers an improved performance compared to (8.79).
Using the concept of probabilistically constrained RMV beamforming, several prom-
ising extensions of multiuser RMV MIMO receivers of [63] have been developed in [55,
64-67] using soft (outage probability-based) rather that deterministic worst-case distor-
tionless response constraints. An interesting open problem is how to extend these receiver
techniques to other than orthogonal space-time codes with higher transmission rates.
8.4
Transmit Beamforming
In wireless communications, there are two basic ways to use transmit antenna arrays: for
space-time data encoding [8, 9, 16] and spatial data multiplexing (beamforming) [11, 14].
Although space-time coding techniques do not require any CSI at the transmitter,
beamforming methods typically require an accurate knowledge of the transmit CSI.
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