Int. J. Communications, Network and System Sciences, 2011, 4, 447-451
doi:10.4236/ijcns.2011.47053 Published Online July 2011 (http://www.SciRP.org/journal/ijcns)
Joint Power Allocation and Beamforming
for Cooperative Networks
Sondes Maadi1,2, Noureddine Hamdi2
1The National School of Engineering of Tunis, El-Manar University, Tunis, Tunisia
2The Communication System Laboratory, The National School of Engineering of Tunis, El-Manar University, Tunis, Tunisia
Received May 6, 2011; revised June 13, 2011; accepted June 25, 2011
Abstract
In this paper, we investigate power allocation problem with the use of transmit beamforming in a dual hop
MISO (multiple input single output) relay channel. We consider either amplify and forward (AF) or decode
and forward (DF) cooperative protocols at the relay and optimize the power allocated to the relay and the
source, under total transmit power constraint, to minimize the bit error rate (BER) of relaying system. Coop-
erative communication is viewed as a method for increasing diversity gain and reducing end to end path loss.
The use of relay can create a virtual antenna array so that it allows users to exploit the advantages of multiple
input multiple output (MIMO) techniques. In this work, we solve cooperative ratio, which is defined as the
ratio power used for cooperative transmission over the total power. This approach is then compared to an
equal power assignment method and its performance enhancement is verified by simulation results.
Keywords: Amplify and Forward, Decode and Forward, Transmit Beamforming, Power Allocation, MISO,
Cooperative Communication
1. Introduction
In recent years, cooperative diversity is considered as a
potential transmit strategy for wireless networks. The
basic idea of cooperative transmission is to allow several
transmit nodes in the network to help in order to create a
virtual antenna array and exploit spatial diversity at the
destination [1-3]. It has been shown in the literature [4],
[5] that cooperative communication can avoid the diffi-
culties of implementing actual antenna arrays and con-
vert the single input single output (SISO) system into a
virtual multiple input multiple output (MIMO) system. In
this way, cooperation between users allows them to ex-
ploit the diversity gain and others advantages of MIMO
system in a SISO wireless network. Several protocols have
been proposed to achieve the gains promised by the use
of node cooperation. Two main relay protocols are widely
known: amplify and forward (AF) and decode and for-
ward (DF). AF means that the received signal is multi-
plied by an amplification gain and then retransmitted by
the relay without performing any decoding. In contrast to
this, the signal is decoded at the relay and re-encoded for
retransmission in (DF) strategy.
An important question in cooperative communication
is power allocation. Given the same total power, how
much should be allocated to source information trans-
mission and how much to relaying information transmis-
sion. In [6], efficient power allocation strategy is inves-
tigated in an orthogonal AF network to satisfy the target
SNR requirement. In [7], the achievable information ca-
pacity is analyzed to obtain optimal power allocation and
partner selection with the total power constraints for
SIMO relay channels. In [8], optimal power allocation is
studied to minimize the total energy consumption satis-
fying the BER target of the cooperative system. Most cur-
rent research considers transmit beamforming and distrib-
uted beamforming in relaying system. In [9], the per-
formance of the DF protocol in multiple relay channels
with maximal ratio combining (MRC) and transmit
beamforming has been analyzed. In [10], the performance
of a two hop relay network with transmit beamforming at
the source and MRC at the destination has been analyzed.
Also, distributed beamforming over multiple relays is
widely studied. For example, several distributed beam-
forming techniques using AF strategy have been recently
proposed in [11,12] that minimize the total relay transmit
power subject to the receive signal-to- noise ratio (SNR)
constraint.

M. MAADI ET AL. 448
In this paper, we focus on the optimal cooperative ra-
tio in the sense to minimize the average BER. For both
AF and DF relays, we solve cooperative ratio, which is
defined as the ratio power used for cooperative transmis-
sion over the total power.
In particular, we focus on AF/DF cooperation scheme
in an environment with one source equipped with multi-
ple antennas and using beamforming transmitting to the
destination through a relay equipped both with single
antenna.
The remainder of this paper is organized as follows:
Section 2 deals with system model in consideration. In
Section 3, we formulate an optimal power allocation so-
lution for AF protocol and then in Section 4 for DF pro-
tocol. Section 5 presents simulations results. The paper
finally draws the conclusions in Section 6.
2. System Model
Consider a cooperative MISO networks, where the source s
equipped with N antennas and using transmit beamforming
and communicates with a destination d with the help of
relay r equipped both with a single antenna. We assume
that the source does not have a direct link to the destina-
tion due to the large distance or the fading obstacle. The
whole transmission is accomplished in two phases: in the
first phase, the source transmits to the relay, and the re-
lay receives. In the second phase, the relay amplifies (AF)
or decode (DF) the received signal and forwards to the
destination. We assume that the relay is located in the
straight line between the source and the destination. The
distance between s to r, s to d and r to d are denoted as
dsr, dsd and drd respectively. We define a ratio r as follows:
dsr = r dsd and dsr = (1 – r) dsd. In the first phase, the
source uses beamforming to transmit to the relay. The
source-relay r signal model ysr is given by:
s
rssr srsr
y
PKdh wxn
 
 (1)
where 12 ,,,
s
rrr Nr
hhh h

 is (1 × N) channel vector
from the source to the relay with complex Gaussian en-
tries with zero mean and unit variance. α is the path loss
exponent and K is a constant due to path loss at the ref-
erence distance. w is (N × 1) beamforming vector and
satisfies (w* w = 1), where (.)* denotes the conjugate
transpose. x is the modulated transmitted symbol, and nsr
is the channel noise drawn from an ensemble of inde-
pendent and identically distributed (i.i.d) complex Gaus-
sian random variables with zero mean and variance 2
n

.
Ps is the transmitting energy used at the source. During
the second phase, the relay transmits to the destination.
The relay-destination yrd signal model is given as:
rdrsrrd rrd
yPKdhxn
 
 (2)
where hrd is the channel gain of relay-destination path
which is complex gaussien with zero mean and unit
variance and nrd is the channel noise drawn from an en-
semble of i.i.d complex Gaussian random variables with
zero mean and variance 2
n

. Pr is the transmitting en-
ergy used at the relay and xr is the relay signal. Either AF
or DF can be used in this phase. We discuss them sepa-
rately in the next two sections.
3. Amplify and Forward and Optimum
Power Allocation
If the relay use AF protocol,
rsr
x
Gy

(4)
The relay-destination yrd signal model is rewritten as:
rdrrd rdsrrd
y
PKdh Gyn
 
 (5)
where
22
1
s
rd srn
G
PKdh w





(6)
The end to end SNR (Signal to Noise Ratio) at the
destination is given by:
22
22
12
222
1
ssrsrrrd rd
nn
AF
rrdrd
nn
PKd hwPKd h
PKd h
G









(7)
The end to end SNR can be simplified as:
12
12
1
AF

  (8)
where 2
12
ssr
sr
n
PKd hw



 and 2
22
rrd
rd
n
PKd h



.
The form of the end to end SNR in (8) can be upper
bounded by:
12
12
1
AF


 (9)
where


11
E

  and


22
E

 . When the signal is
transmitted with coherent beamforming, the beamform-
ing weight is
s
r
s
r
h
wh

 and the beamforming gain is
2
sr
Ehw N



, we can rewrite 1

and 2

as:
12
ssr
n
PKd N



 and 22
rrd
n
PKd




The goal of optimal power allocation is to maximize
the total system SNR under the total power constraint P

M. MAADI ET AL. 449
as:

max ,
..
AFs r
sr
PP
s
tP PP


The constraint can be reformulated by introducing new
variable

, 01


and s PP

 and

1
r
PP   . The average end to end SNR can then be
calculated as follows:



0
2
1
1
sr rd
AF
s
rrdn
PKd NPKd
PKdNPKd









 (10)
where 02
1
n


.
Optimum power allocation can be performed by taking
the first derivation of Equation (10) and equate it to zero.
0
AF



 (11)
This will result in polynomial degree 2 in the form of
22 0abc
 
where

00
0
0
21
1
s
rrd
rd
rd
aNPKd PKd
bPKd
cPKd










 
 

(12)
Solving the above second degree polynomial, opti-
mum power allocation ratio is obtained as the positive
real root which lies between zero and one.
For a BPSK modulated signal, the BER is calculated
as follows:

2AF BER Q   (13)
4. Decode and Forward and Optimum
Power Allocation
As in the case of AF the relay receives the broadcast
from the source in Phase 1, but instead simply amplify-
ing and forwarding the signal, the relay tries to decode
the signal and then forward it. If the relay decodes the
signal correctly (xr = x).
Considering that the message will be received cor-
rectly at destination when transmission of both phases is
correct, the end to end error probability can be found
using:
 
111
e esrrd
eee
srrdsr rd
eeee
PPP
PPPP


(14)
where
s
r
e
P and rd
e
P are respectively the instantaneous
probability of decoding error at source-relay link and
relay-destination link and can be calculated as follows:
2
0
2
sr
essr sr
PQ PKdh


 


(15)
2
0
2
rd
errdrd
PQPKdh






(16)
where

2
1exp d
2
2π
x
t
Qx t
 


 (17)
Since 2
sr
h is the sum of squared N i.i.d. zero mean
circularly symmetric complex Gaussian random vari-
ables, 2
s
r
h   has a chi-square distribution with 2N
degrees of freedom and 2
rd
X
h has an exponential
distribution.
After applying Chernoff Bound to (15) and (16) and
integrating them with respect to distributions of their
terms, we obtain the average error probability:



0
0
0
1expexp d
2
1
21
rd
errd
rrd
PPKdXXX
PKd









(18)
We recall that the Chernoff Bound for a Gaussian
random variable is:

2
1e
2
x
Qx

 (19)




0
0
1
0
1exp d
2
1
20.5
sr
essr
N
N
ssr
PPKdp
PKd


 






(20)
The pdf (probability density function) of the random
variable

is given by:
 
1
1exp 2
2
N
N
pN






(21)
Substituting (18) and (20) into (14), we obtain the av-
erage end to end error probability as:






(1)
0
0
2
00
1
1
22
1
21 1
1
211
ee
eN
N
sr
rd
N
N
rd rd
P
PKd
PKd
PKd PKd

















(22)
Again, optimal power allocation can be similarly

M. MAADI ET AL. 450
found by replacing Ps and Pr respectively by P

and

1P  . Optimum power allocation can be found from
Equation (22). Taking the first derivation and equate it to
zero, the optimization problem translates into solving
polynomial degree N + 1. In this paper, we consider N =
4. So, power allocation problem consists of solving a
polynomial degree equal to 5
(543 20abcd ef
 
)











5
00
4
00
3
00
2
00
00
00
2
00
2
00
00
64
160
160
80
16
42
32
16
24
rd sr
rd sr
rd sr
rd sr
rd sr
rd sr
rd sr
rd sr
rd
aPKdPKd
bPKdPKd
cPKdPKd
dPKdPKd
PKd PKd
ePKdPKd
PKdPKd
fPKdPKd
PKdPKd






































5
00
8
sr
sr rd
PKd PKd






(23)
5. Simulation Results
In this section, we evaluate the performance of our scheme
in term of end-to-end BER. We consider equal and opti-
mal power allocations for both AF and DF and evaluate
their performance assuming that the relay is allowed to
be in any position along a straight line between the
source and the destination. The simulation is conducted
for N = 4 transmit antennas at the source and we recall
that the relay and the destination are equipped both of a
single receive antenna.
Throughout the simulation, the parameters dsd and α
are set to 10 km and 3.6 respectively. We define the SNR
(Signal to Noise Ratio) as the ratio between system total
power P and noise variance 2
n

and we replace dsr by
rdsd and drd by (1 – r) dsd in the different expressions of
BER.
Figure 1 shows the average bit error rate (BER) ver-
sus r for AF protocol at SNR = 25 dB. It is observed that
amplify and forward with optimal power allocation can
bring significant performance improvement compared to
equal power allocation with and without beamforming
(BF) especially when the relay is away from the source.
The lowest BER is achieved when r = 0.7 whereas with
equal power allocation and without beamforming (single
antenna at the source) r = 0.5 and r = 0.6 with beam-
forming at the source.
Figure 2 displays the average BER versus r for DF
protocol at SNR = 25 dB. Similar observations about
BER performance compared to equal power allocation
can also be made. It is interesting to see that the amelio-
ration in performance is still observed as r increases. The
lowest BER is achieved by optimal power allocation
when r = 0.8
Figure 3 shows the difference in average BER be-
tween AF and DF evaluated at SNR = 25 dB. We note
that DF protocol outperform AF protocol for r from 0.01
to 0.99.
6. Conclusions
In this work, we studied the performance of optimal
power allocation with transmit beamforming in two hop
AF/DF relay. We first derive an analytical study of op-
timal power allocation of the proposed system. Next, our
Figure 1. BER performance of AF with equal and optimal
power allocation.
Figure 2. BER performance of DF with equal and optimal
power allocation.

M. MAADI ET AL. 451
Figure 3. BER performance of AF and DF with equal and
optimal power allocation.
analytical results are confirmed by simulations. It is shown
that by using our optimized method, we provide better
performance than equal power allocation with and with-
out beamforming. Especially, when the relay is away
from middle point between the source and the destination,
optimal power allocation technique should be used to en-
sure better performance. Moreover, in our case the BER
can be made low by moving the relay closer to the desti-
nation.
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