1348 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 5, MAY 2011
bounded as follows:
P
s−err
≤
𝑠∕=𝑠
′
Pr(𝑠 → 𝑠
′
)
=
𝑠∕=𝑠
′
𝒬
∣𝜌∣
2
∣𝑠 − 𝑠
′
∣
2
𝐿
𝑙=1
∣𝑤
∗
𝑙
𝑔
𝑙
∣
2
𝜎
2
𝑙
+ 𝜎
2
0
, (4)
where 𝑠, 𝑠
′
∈𝒮, Pr(𝑠 → 𝑠
′
) is the pairwise error probability
(PEP), and 𝒬(𝑥)=
∞
𝑥
1
√
2𝜋
𝑒
−
𝑧
2
2
𝑑𝑧.
It is noteworthy that the destination node only needs to
know the overall channel coefficient, 𝜌, for decoding. Thus,
the resulting complexity of the destination node can be low
compared with distributed space-time coding approaches. Fur-
thermore, if the relay weights are decided for coherent combin-
ing so that 𝜌 has a positive large value, the destination node
can have a good performance without having any diversity
schemes and high computational complexity. In addition to
power efficiency, this could be another major advantage of
beamforming in relay networks over d istributed space-time
coding (i.e., the computational burden and diversity operations
are transferred from the destination node to relay nodes).
Therefore, for downlink channels
5
of a cellular system, relay
networks could be an effective means to provide reliable
transmissions when user terminals have single antenna and
limited co mputing power.
B. MSNR Distributed Beamforming
The channel coefficients, ℎ
𝑙
and 𝑔
𝑙
, are referred to as the
incoming and outgoing channel coefficients (from r elay nodes’
point of view) in this paper. The weight at relay nodes, 𝑤
𝑙
,is
to compensate channel gains or maximize the array gain.
In this section, we consider the following assumptions:
A1) 𝔼[s]=0 and 𝔼[ss
H
]=𝑃
𝑠
I,where𝑃
𝑠
is the power of
symbol.
A2) Each relay node knows its instantaneous incoming and
outgoing channel coefficients, which is referred to as
local CSI (the channel estimation will be discussed in
Section IV). Furthermore, the signal power, 𝑃
𝑠
, and noise
variances, 𝜎
2
𝑙
and 𝜎
2
0
, are known. However, the CSI of
the other relay nodes are assumed to be not known.
If the relay nodes can exchange the CSI of their incoming
and outgoing channels, they can have global CSI (as op-
posed to local CSI) and find optimal beamforming vectors
with global CSI under various criteria. However, if the CSI
exchange between relay nodes is not possible or limited,
the distributed beamforming approaches in [9] [10] are more
appealing and practical. Throughout the paper, we assume that
only local CSI is available at relay nodes as in A2).
For normalization purpose, let
𝑤
𝑙
=
𝑣
𝑙
∣ℎ
𝑙
∣
2
𝑃
𝑠
+ 𝜎
2
𝑙
,
where 𝑣
𝑙
is the normalized beamforming weight at relay node
𝑙. From this, we can easily show that the transmission power
per symbol is given by
1
𝑀
𝔼[∣∣𝑤
𝑙
r
𝑙
∣∣
2
]=∣𝑤
𝑙
∣
2
(∣ℎ
𝑙
∣
2
𝑃
𝑠
+ 𝜎
2
𝑙
)=∣𝑣
𝑙
∣
2
.
5
In this case, each user has his/her own dedicated channel and relay nodes
can support multiusers of different channels.
Let f =[𝑓
1
𝑓
2
... 𝑓
𝐿
]
T
,where𝑓
𝑙
=
ℎ
𝑙
𝑔
𝑙
√
∣ℎ
𝑙
∣
2
𝑃
𝑠
+𝜎
2
𝑙
,and
q =[𝑞
1
𝑞
2
... 𝑞
𝐿
]
T
,where𝑞
𝑙
=
𝑔
𝑙
𝜎
𝑙
√
∣ℎ
𝑙
∣
2
𝑃
𝑠
+𝜎
2
𝑙
.Furthermore,
let Q = diag(q). Then, the SNR at the destination node is
given by
𝛾(v)=
v
H
ff
H
v𝑃
𝑠
v
H
QQ
H
v + 𝜎
2
0
. (5)
As shown in (4), if the beamforming weights maximize the
SNR, we can see that the error probab ility is minimized. Thus,
the maximum SNR (MSNR) beamforming can also minimize
the (upper-bound on) error probability.
1) Short-Term Total Power Constrained MSNR Beamform-
ing: With the following constraint on the short-term total
transmission power per symbol from 𝐿 relay nodes:
𝐸
rel
≥
1
𝑀
𝐿
𝑙=1
𝔼[∣∣𝑤
𝑙
r
𝑙
∣∣
2
]=∣∣v∣∣
2
, (6)
where 𝐸
rel
is the maximum total transmission power per
symbol from 𝐿 relay nodes, a b eamforming problem can be
formulated as
max
∣∣v∣∣
2
≤𝐸
rel
𝛾(v)= max
∣∣v∣∣
2
≤𝐸
rel
v
H
ff
H
v𝑃
𝑠
v
H
QQ
H
v + 𝜎
2
0
. (7)
Noting that 𝜎
2
0
≥ v
H
v
𝜎
2
0
𝐸
rel
,wehave
max
∣∣v∣∣
2
≤𝐸
rel
v
H
ff
H
v𝑃
𝑠
v
H
QQ
H
v + 𝜎
2
0
≤ max
∣∣v∣∣
2
≤𝐸
rel
v
H
ff
H
v𝑃
𝑠
v
H
(QQ
H
+
𝜎
2
0
𝐸
rel
I)v
.
Since the equality is achieved when ∣∣v∣∣
2
= 𝐸
rel
, we can
modify the beamforming problem to maximize the SNR as
follows:
v
st
=arg max
∣∣v∣∣
2
=𝐸
rel
v
H
ff
H
v𝑃
𝑠
v
H
(QQ
H
+
𝜎
2
0
𝐸
rel
I)v
. (8)
This beamforming problem is identical to a well-known
beamforming, namely the maximum signal-to-interference-
noise ratio (MSINR) beamforming [16, p.128]. The solution
of the MSNR beamforming with a total transmission power
constraint in (6) is found in [10] as follows:
v
st
= 𝜅
QQ
H
+
𝜎
2
0
𝐸
rel
I
−1
f, (9)
where 𝜅 is the normalization constant. Alternatively, we have
𝑤
st,𝑙
= 𝜅
𝐸
rel
ℎ
𝑙
𝑔
𝑙
∣𝑔
𝑙
∣
2
𝐸
rel
𝜎
2
𝑙
+ ∣ℎ
𝑙
∣
2
𝑃
𝑠
𝜎
2
0
+ 𝜎
2
𝑙
𝜎
2
0
,𝑙=1, 2,...,𝐿.
(10)
The resulting beamforming is referred to as the short-term
total power constrained MSNR (STPC-MSNR) beamforming.
In (10), while the phase compensatio n is carried out with local
CSI, the normalization coefficient, 𝜅, is required for the power
allocation. Since 𝜅 depends on the channel coefficients of
all the relay nodes, it is a global CSI variable and central
processing with the knowledge of all the channel coefficients
or exchange of the channel coefficients between relay nodes
is necessary to decide 𝜅. This is certainly not desirable in
distributed implementation for beamforming.