This paper investigates optimal relay selection and power allocation under an aggregate power constraint for cooperative wireless sensor networks assisted by amplifyandforward relay nodes. By considering both transmission power and circuit power consumptions, the received signaltonoise ratio (SNR) at the destination node is calculated, based on which, a relay selection and power allocation scheme is developed. The core idea is to adaptively adjust the selected relays and their transmission power to maximize the received SNR according to the channel state information. The proposed scheme is derived by recasting the optimization problem into a threelayered problem―determining the number of relays to be activated, selecting the active relays, and performing power allocation among the selected relays. Monte Carlo simulation results demonstrate that the proposed scheme provides a higher received SNR and a lower bit error rate as compared to the average power allocation scheme.
1. Introduction
W
ireless sensor networks (WSNs) can provide an efficient sensing and computing solution for various applications ranging from environment monitoring to outer space exploration. Wireless sensor nodes are equipped with a radio transceiver and a set of transducers for data acquisition, transmission and processing. Since sensor nodes are usually powered by batteries which are difficult to change or recharge
[1]
, the pursuit of better performance under given energy consumption is a critical issue in the design of WSNs.
Cooperative communication, as a popular technique for providing reliable transmission and high throughput in networks with single antenna nodes
[2]

[5]
, has been widely used in WSNs. Various relay selection (RS) and power allocation (PA) schemes have been studied in the cooperative communication scenario
[6]

[12]
. Some of these schemes have considered individual power constraint, for example, the power allocation strategy for networks with perfect channel state information in
[6]
and that for networks with statistical channel information in
[7]
. Some of them assume aggregate power constraint, for example, the spacetime coded cooperation with two relays in
[10]
.
However, most of the earlier works on individual or aggregate power constrained communication have considered the transmission power only. This is reasonable in traditional wireless networks where a large amount of transmission power is required due to the large transmission distance and thus the power consumption on circuits is negligible. In a WSN, however, the dense distribution of sensor nodes makes the transmission power much smaller than that of a traditional cooperative network and hence the circuit power consumption cannot be ignored. Moreover, the circuit power consumed by the relays depends on their transmission power. Therefore, relay selection and power allocation with consideration of the circuit power consumption are more challenging than that without considering circuit power consumption, especially for cooperative networks under aggregate power constraint.
In fact, some optimization strategies, considering both transmission power and circuit power consumption, have been proposed for WSNs
[13]

[16]
. In
[13]
and
[14]
, the best modulation and transmission strategy were analyzed to minimize the total energy consumption required to send a given number of bits and it was also confirmed that in some shortrange communications, SingleInput SingleOutput (SISO) systems may beat MultipleInput MultipleOutput (MIMO) systems. In
[15]
, a cooperative communication architecture with feedback was proposed to minimize the energy consumption per bit. Moreover, Shuguang Cui and Andrea J. Goldsmith have proved that having more relay nodes may lead to poorer performance due to the extra circuit power consumed by extra relays
[14]
,
[16]
. However, to the best of our knowledge, no existing work on RS and PA which takes circuit power consumption into account has been found for cooperative WSNs.
In this paper, we propose a RS and PA scheme for a cooperative WSN with both transmission power and circuit power consumptions being considered. The scheme aims to maximize the received signaltonoise ratio (SNR), which means more information bits will be correctly transmitted under certain aggregate power constraint, thus making the network more energy efficient. We derive the RS and PA scheme by recasting the optimization problem into a threelayered problem, namely, determination of the number of active relays, active relay selection, and power allocation of active relays. In our scheme, not only the transmission power of the relays but also the relays themselves are adaptively adjusted according to the channel state information. Simulation results showing the higher received SNR and lower bit error rate (BER) of the proposed scheme are provided.
The remainder of this paper is organized as follows. Section II describes the system model of a WSN using beamforming technique
[6]
at the relays. The received SNR is calculated in Section III, based on which we derive a RS and PA scheme. Simulation results are given in Section IV to demonstrate the advantage of the proposed scheme. Finally, Section V provides the conclusion.
Throughout the paper, the following notations are adopted: · denotes the module of a complex number.
is the ceiling function. 〈⋅,⋅〉 denotes the inner product and ║.║ is the 2norm.
2. System Model
Consider a wireless network consisting of two endusers, called the source node S and the destination node D, and
M
relay nodes R
_{i}
,
i
= 1,…,
M
, as shown in
Fig.1
. Each node is equipped with a single antenna working in halfduplex mode. Assume that there is no direct link from the source to the destination due to the poor quality of the channel between them.
A cooperative wireless sensor network
Denote the channel between S and R
_{i}
as
f_{i}
and the channel between R
_{i}
and D as
g_{i}
. Suppose each link has a flat Rayleigh fading and the channels are independent of each other. We also assume that R
_{i}
knows its own channel while D knows all channels. A relay node is called an active relay if it is selected to cooperate.
A twostep amplifyandforward (AF) protocol
[2]
is used in this network. During the first step, the source S broadcasts signal
to all relays where the information symbol
s
is selected randomly from a codebook and is normalized as E(
s

^{2}
) = 1 , and
P
_{s}
is the transmission power used at S. Relay R
_{i}
receives
where
v_{i}
is the white Gaussian noise at R
_{i}
with zero mean and unit variance.
In the second step, by using an AF protocol and beamforming technique, R
_{i}
sends the following signal
to the destination, where
P_{ti}
is the transmission power of R
_{i}
and is set to zero if R
_{i}
is not activated. The signals from relays are added up at the destination D:
where
w
is the noise at D and also assumed to be CN (0,1) .
v_{i}
and
w
are independent.
It has been proved that the optimal choice of the angles is
θ_{i}
= (arg
f_{i}
+ arg
g_{i}
)
[6]
. That means matched filters should be used at the relays to compensate for the phases of their channels. Thus,
This is a general model for cooperative network using beamforming technique with no power constraint. However, in this paper, we aim to derive a RS and PA scheme under an aggregate power constraint with consideration of circuit power consumption. Thus, the model in (4) should be modified.
In order to consider the circuit power consumption, all signal processing blocks at the transmitter and receiver of each relay should be included. However, to avoid overcomplicated modeling, some blocks are intentionally omitted, e.g., source coding, pulse shaping, digital modulation and AD/DA conversion. Thus, the constituent modules of the transmitter and that of the receiver are shown in
Fig. 2
and
Fig. 3
, respectively.
Transmitter block diagram
Receiver block diagram
The circuit power consumption at R
_{i}
, denoted as
P_{ci}
, consists of the mixer power consumption
P
_{mix}
, the frequency synthesizer power consumption
P
_{syn}
, the low noise amplifier (LNA) power consumption
P
_{LNA}
, the active filters power consumption
P
_{fil}
, the intermediate frequency amplifier (IFA) power consumption
P
_{IFA}
and the power amplifier power consumption P
_{amp,i}
=
μP
_{ti}
, where
μ
=
with
η
being the drain efficiency of the RF power amplifier and
ξ
being the peak to average ratio (PAR), which is dependent on the modulation scheme and the associated constellation size. Therefore, the total power consumed by R
_{i}
can be written as
P
_{ti}
+
P
_{ci}
= (1 +
μ
)
P
_{ti}
+
P
_{0i}
where
P
_{0i}
denotes the power consumption of all other circuit blocks except the power amplifier. Set
P
_{0}
= 2
P
_{mix}
+ 2
P
_{syn}
+
P
_{LNA}
+
P
_{fil}
+
P
_{IFA}
, then we have
P
_{0i}
=
P
_{0}
R
_{i}
is activated and
P
_{0i}
= 0 if it is not.
The aggregate power constraint can be expressed as
Accordingly, the aggregate transmission power of all the relays can be written as
For the sake of convenience, we introduce a power control variable
α_{i}
into the model and let
be the power control coefficient of R
_{i}
, i.e.,
By defining
we have
P
_{0i}
=
ε
(
α_{i}
)
P
_{0}
,
and
where
Apparently,
n
is the number of active relays and
n
≤
M
.
In order to guarantee that there is power left for relays to transmit,
is needed, which means
Let
then
N
≥
n
is the largest number of relays that can be activated under aggregate power
P
.
Combining equations (4) and (7), the received signal at D is given by
By Shannon’s information capacity theorem, the instantaneous capacity of the network is governed by
W
log
_{2}
(1 + SNR ). Since the base2 logarithm is an increasing function, the maximization of the received SNR is equivalent to the maximization of the capacity. It is also equivalent to error rate minimization.
3. Relay Selection and Power Allocation
In Section II, the network model of a cooperative WSN with consideration of circuit power consumption has been established. For selectiverelay scenario with an aggregate power constraint, relay selection and power allocation are closely related. Here, we propose an optimal RS and PA scheme, i.e., determine the best value of
α_{i}
, which is the main contribution in this paper.
 3.1 Optimal RS and PA Scheme
From (8), the received SNR can be calculated as
By defining
and
equation (9) can be simplified to
which can be more conveniently rewritten as
where
and
The SNR optimization problem can now be expressed as
To maximize an inner product of two vectors, we normally set the angle between the two vectors to 0˚. However, this does not work for the above optimization problem. Denoting the angle between γ and
ρ
by
ϕ
, we have SNR = ║
ρ
║
^{2}
cos
^{2}
ϕ
. Note that ║
ρ
║
^{2}
is a function of
n
, so both ║
ρ
║
^{2}
and cos
^{2}
ϕ
are functions of
If cos
^{2}
ϕ
= 1, γ and
ρ
should be in the same or opposite direction, which means there is no zerovalue element in γ since the elements in
ρ
are all nonzero, i.e.,
n
=
M
. However,
n
=
M
may not yield the maximal ║
ρ
║
^{2}
, not even to mention that the aggregate power may not allow all relays to be activated. Thus, we divide the problem in (11) into several subproblems, and solve it by an alternative method.
Suppose the number of relays to be selected is
n
, namely, there are
n
nonzero elementsin γ . Then the above optimization problem can be seen as, for a given value of
n
, selecting the best
n
relays, allocating power properly among them to obtain the maximal received SNR, and determining the best value of
n
through comparing the maximal received SNRs. Therefore, instead of optimizing SNR over all
α_{i}
, we first determine the value of nonzero
α_{i}
, then their indices and finally the number of them. This being said, the optimization problem in (11) can be recast as
where
is a vector consisting of all the nonzero elements of γ and
.
Recalling that
n
≤
M
, we have
n
≤ min (
M
,
N
) . So the optimization problem in (11) has been decomposed into min (
M
,
N
) subproblems. We now work on the
n
th subproblem of (12):
Theorem 1
: The solution of the inner optimization problem
denoted as
α_{ki}
^{*}
, is given by
where Κ is an arbitrary nonzero constant.
Proof:
Denote the angle between γ′ and
ρ
′ as
ϕ
′ Then we have
Obviously, the maximal value is attained iff γ′ and
ρ
′ are in the same or opposite direction, i.e.,
ϕ
′ = 0˚ or 180˚. This indicates that
should be a nonzero constant (denoted by Κ). Thus, we have the optimal PA solution
where Κ can take any nonzero value since it does not affect the power control coefficient
α_{k}
^{2}
.
Using the obtained results, one can derive that the outer optimization problem of the
n
th subproblem is
This is a relay selection problem which can obviously be solved by the exhaustive method. But there are
combinations of relays, which requires a large amount of computation. By looking into the structure of ║
ρ
′║
^{2}
, we may find a way to reduce the computation.
Note that
The maximal received SNR, by using R
_{k1}
, R
_{k2}
, ⋯, R
_{kn}
as active relays and the above PA scheme, is simply the sum of each relay…s contribution. Hence, we can order the relays according to their contributions.
By defining
, as the SNR contribution of R
_{i}
using the above PA scheme if it is one of the
n
active relays. Arrange
ρ_{i(n)}
in nonincreasing order as
then
is a permutation of (1,2,…,
M
). Apparently, the
n
best relays are
Define
and
is the maximal received SNR that can be attained by using
n
relays. Now that we have solved the
n
th subproblem, what is left is to optimize ║
ρ
_{An}
║
^{2}
, i.e.,
s.t.
n
≤
N
. Since ║
ρ
_{An}
║
^{2}
is a function of both
n
and Α
_{n}
(which cannot be expressed by
n
in a closed form), it is impossible to derive
n
^{*}
analytically. Thus, we can determine
n
^{*}
by comparing ║
ρ
_{A1}
║
^{2}
, ║
ρ
_{A2}
║
^{2}
. …, ║
ρ
_{Amin(M,N)}
║
^{2}
.
The proposed RS and PA scheme is described in the following table.
The above algorithm gives the optimal solution as
The power control coefficient of R
_{i}
is given by
The destination can calculate
and broadcast them to every relay. After receiving that, relay R
_{i}
can find out whether it should cooperate and calculate the transmission power by using its own channel state information.
 3.2 Discussion
It is to be noted that in obtaining the received SNR in (14) we have actually performed a maximalratiocombining (MRC)
[17]
through power allocation.
It is also to be noted that there are
combinations of relays to be considered if an exhaustive method is used in relay selection. However, the scheme proposed above, due to the decomposition of the received SNR and the ordering of each relay’s maximal contribution, drops the number of combinations to min (
M
,
N
) and hence reduces the computational complexity greatly.
Another thing to be noted is that, to apply the solution in (15) requires the destination to estimate the channels and each relay to know its own channel state information. Hence, this scheme should be implemented to cooperative networks where training sequences and feedback are allowed.
4. Simulation Results
In this section, we verify the proposed RS and PA scheme by computer simulation. The simulation experiments are conducted using MATLAB R2010a. In our simulations, the source signal is a binary sequence following the discrete uniform distribution and it is then modulated by 4QAM. All the channels are generated as i.i.d. normalized Rayleigh fading channels obeying CN (0,1) . The noises are also generated as i.i.d. complex Gaussian distribution following CN (0,1) . The source uses 20 percent of the aggregate power. The power consumption of each circuit block is given in the following table
[13]
:
Circuit power parameters
As the optimal number of active relays cannot be obtained in closed form, we show the relationship between the optimal number and the aggregate power through Monte Carlo simulation.
Fig.4
shows the simulation results on the average number of best active relays of two networks with 8 and 4 relays, respectively. As described above, the channels were generated following CN (0,1), the number of best relays was derived for every realization of
f_{i}
and
g_{i}
by using the algorithm in subsection 3.1, and all results were averaged over 10,000 independent realizations of the channels. From
Fig.4
, we can see that the number grows with the aggregate power and tends to the number of relays available. This is reasonable because the proportion of extra circuit power consumption in the aggregate power gradually reduces when the aggregate power increases and the benefit due to using more relays can make up for the disadvantage brought by extra consumption. When the aggregate power is large enough, all relays in the network should be activated. For a network with 4 relays, this happens when the aggregate power is higher than 47dBm and this value changes to 50dBm for a network with 8 relays, exactly twice of the former value.
Average number of best active relays
Fig.5
provides the comparison of the proposed scheme with average power allocation among all relays in terms of the received SNR. The advantage of the proposed scheme is obvious and is especially high when the aggregate power is small. For example, the proposed scheme has 10dB improvement, i.e., 10 times the received SNR of average power allocation, when
M
= 8 and
P
= 33dBm. This attributes to the overlarge circuit power consumption which leaves no much power for relays to transmit when all relays are activated. We can also see from
Fig.4
that the number of best active relays is about 3 in this case. Thus, in our scheme, the extra circuit power consumed by the other 5 relays is used on the 3 activated ones to achieve higher received SNR. Besides, by comparing the two curves of average power allocation scheme, we can see that using 4 relays achieves higher received SNR than using 8 relays when
P
＜ 35dBm which would not be possible if the circuit power consumption is negelected.
Average received SNR versus aggregate power
Moreover, we can estimate the power saving and the network lifetime prolonging from
Fig.4
. Suppose that the aggregate power needed in our scheme and in the average power allocation scheme are
P
_{o}
and
P
_{a}
(in mW), respectively, to guarantee a certain received SNR (say 10dB) when
M
= 8 . From the two curves of
M
= 8 , we can see that about 10log
_{10}
P
a 10log
_{10}
P
_{o}
≈ 43.8  42.6 = 1.2 dBm can be saved, which means
P
_{a}
/
P
_{o}
≈ 1.32 . So the average power allocation scheme needs to use about 32% more power than the proposed scheme to achieve the same received SNR and therefore a network using the proposed scheme has about 32% more lifetime.
Fig.6
shows the BER curves of our scheme and of the average power allocation scheme in WSNs with 8 relays and 4 relays, respectively. It can be seen that the proposed scheme outperforms the average power allocation scheme in both cases. Thus, by using the proposed scheme, more information bits can be correctly transmitted which also indicates the effectivemess of our scheme.
Bit error rate versus aggregate power
5. Conclusion
In this paper, we have proposed a RS and PA scheme for a cooperative WSN with consideration of circuit power consumption. Our new scheme selects active relays and allocate their transmission power according to the channel state information to maximize the received SNR. The power allocation among active relays has been given in a closed form. The relay selection has been simplified to relay ordering by using the result of power allocation. This ordering reduces the computational complexity greatly due to the reduction in the number of relay combinations to be chosen. The simulation results have demonstrated that the new scheme provides a higher received SNR and a lower BER as compared to the simple average power allocation scheme.
As our scheme is only for twohop cooperative networks with singleantena nodes while a practical WSN might be a multihop one or/and the nodes may be equipped with multiple antennas, a new RS and PA scheme for a more general network model shall be further investigated.
BIO
Mujun Qian received the B.S. and M.S. degrees in telecommunication and information systems from Nanjing University of Posts and Telecommunications (NUPT), Nanjing, China, in 2008 and 2011, repesctively. She is now a Ph.D. candidate at NUPT. Her current research interests include cooperative communications, wireless sensor networks and cognitive radio systems.
Chen Liu received the B.S. degree in electrical and information engineering from Nanjing Institute of Technology (Southeast University), China, in 1985 and the M.S. degree in circuits and systems from Anhui University, China, in 1988. He received the Ph.D. degree in signal and information processing from Southeast University, China, in 2005. He joined NUPT in 1988 where he has been a professor since 2002. His current research interests include multiple antenna systems, spacetime coding, and cooperative communication. He is a member of IEEE.
Youhua Fu received the B.S. degree in communication engineering and the M.S. degree in communication and information systems from the School of Information Science and Technology, Southwest Jiaotong University, Chengdu, China in 2000 and 2003, respectively. She received the Ph.D. degree in signal and information processing from the School of Information Science and Engineering, Southeast University, Nanjing, China, in 2010. She has been with NUPT as a faculty since 2010. Her current research interests include MIMO communication systems and cooperative communications. She is a member of IEEE.
WeiPing Zhu received the B.E. and M.E. degrees from NUPT, and the Ph.D. degree from Southeast University, Nanjing, China in 1982, 1985 and 1991, respectively, all in electrical engineering. He was a postdoctoral fellow from 1991 to 1992 and a research associate from 1996 to 1998 in Concordia University, Montreal, Canada. During 19931996, he was an associate professor at NUPT. Since July 2001, he has been with Concordia University as a fulltime faculty, where he is now a full professor. His research interests include digital signal processing fundamentals, speech and audio processing, and signal processing for wireless communication.
He was an Associate Editor of IEEE Transactions on Circuits and Systems Part I: Fundamental Theory and Applications from 2001 to 2003, and an Associate Editor of Circuits, Systems and Signal Processing from 2006 to 2009. He currently serves as a Guest Editor for IEEE Journal on Selected Areas in Communications: Broadband Wireless Communications for High Speed Vehicles.He currently serves as a Guest Editor for IEEE Journal on Selected Areas in Communications: Broadband Wireless Communications for High Speed Vehicles.
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