Joint Maximum Sum-Rate Receiver Design and Power Adjustment for Multihop Wireless Sensor Networks
aa r X i v : . [ c s . I T ] M a r JOINT MAXIMUM SUM-RATE RECEIVER DESIGN AND POWER ADJUSTMENT FORMULTIHOP WIRELESS SENSOR NETWORKS
Tong Wang, Rodrigo C. de Lamare
Department of ElectronicsUniversity of York, UKEmail: [email protected]@ohm.york.ac.uk
Anke Schmeink
UMIC Research CentreRWTH Aachen UniversityD-52056 Aachen, GermanyEmail: [email protected]
ABSTRACT
In this paper, we consider a multihop wireless sensor network(WSN) with multiple relay nodes for each hop where the amplify-and-forward (AF) scheme is employed. We present a strategy tojointly design the linear receiver and the power allocation parame-ters via an alternating optimization approach that maximizes the sumrate of the WSN. We derive constrained maximum sum-rate (MSR)expressions along with an algorithm to compute the linear receiverand the power allocation parameters with the optimal complex am-plification coefficients for each relay node. Computer simulationsshow good performance of our proposed methods in terms of sumrate compared to the method with equal power allocation.
Index Terms — Maximum sum-rate (MSR), power allocation,multihop, wireless sensor networks (WSNs)
1. INTRODUCTION
Recently, there has been a growing research interest in wireless sen-sor networks (WSNs) as their unique features allow a wide range ofapplications in the areas of defence, environment, health and home[1]. They are usually composed of a large number of densely de-ployed sensing devices which can transmit their data to the desireduser through multihop relays [2]. Low complexity and high energyefficiency are the most important design characteristics of commu-nication protocols [3] and physical layer techniques employed forWSNs. The performance and capacity of WSNs can be significantlyenhanced through exploitation of spatial diversity with cooperationbetween the nodes [2]. In a cooperative WSN, nodes relay signals toeach other in order to propagate redundant copies of the same signalsto the destination nodes. Among the existing relaying schemes, theamplify-and-forward (AF) and the decode-and-forward (DF) are themost popular approaches [4].Due to limitations in sensor node power, computational capac-ity and memory [1], some power allocation methods have been pro-posed for WSNs to obtain the best possible SNR or best possiblequality of service (QoS) [5] at the destinations. The majority of theprevious literature considers a source and destination pair, with oneor more randomly placed relay nodes. These relay nodes are usuallyplaced with uniform distribution [6], equal distance [7], or in line [8]with the source and destination. The reason of these simple consider-ations is that they can simplify complex problems and obtain closed-form solutions. A single relay AF system using mean channel gainchannel state information (CSI) is analyzed in [9], where the outageprobability is the criterion used for optimization. For DF systems, anear-optimal power allocation strategy called the Fixed-Sum-Power with Equal-Ratio (FSP-ER) scheme based on partial CSI has beendeveloped in [6]. This near-optimal scheme allocates one half of thetotal power to the source node and splits the remaining half equallyamong selected relay nodes. A node is selected for relay if its meanchannel gain to the destination is above a threshold. Simulationsshow that this scheme significantly outperforms existing power al-location schemes. One is the ’Constant-Power scheme’ where allnodes serve as relay nodes and all nodes including the source nodeand relay nodes transmit with the same power. The other is the ’Best-Select scheme’ where only the node with the largest mean channelgain to the destination is chosen as the relay node.In this paper, we consider a general multihop wireless sensornetworks where the AF relaying scheme is employed. Our strategyis to jointly design the linear maximum sum-rate (MSR) receiver( w ) and the power allocation parameter ( a ) that contains the optimalcomplex amplification coefficients for each relay node via an alter-nating optimization approach. It can be considered as a constrainedoptimization problem where the objective function is the sum-rate(SR) and the constraint is a bound on the power levels among therelay nodes. Then the constrained MSR solutions for the linear re-ceiver and the power allocation parameter can be derived. The pro-posed strategy and algorithm are not only applicable to simple 2-hopWSNs but also to general multihop WSNs with multi relay nodesand destination nodes. Another novelty is that we make use of theGeneralized Rayleigh Quotient [13] to solve the optimization prob-lem in an alternating fashion.This paper is organized as follows. Section 2 describes the mul-tihop WSN system model. Section 3 develops the joint MSR re-ceiver design and power allocation strategy. Section 4 presents theproposed alternating optimization algorithm to maximize the sumrate. Section 5 presents and discusses the simulation results, whileSection 6 provides some concluding remarks.
2. SYSTEM MODEL
Consider a general m-hop wireless sensor network (WSN) with mul-tiple parallel relay nodes for each hop, as shown in Fig. 1. The WSNconsists of N source nodes, N m destination nodes and N r relaynodes which are separated into m − groups: N , N , ... , N m − .We will focus on a time division scheme with perfect synchroniza-tion, for which all signals are transmitted and received in separatetime slots. The sources first broadcast the N × signal vector s tothe first group of relay nodes. We consider an amplify-and-forward(AF) cooperation protocol. Each group of relay nodes receives thesignal, amplifies and rebroadcasts them to the next group of relaynodes (or the destination nodes). In practice, we need to considerhe constraints on the transmission policy. For example, each trans-mitting node would transmit during only one phase. In our WSNsystem, we assume that each group of relay nodes transmits the sig-nal to the nearest group of relay nodes (or destination nodes) directly. Sourcenodes DestinationnodesCooperativerelay nodes N N N m N N m- FeedbackChannel
Fig. 1 . m -hop WSN with N sources, N m destinations and N r relays. Let H s denote the N × N channel matrix between the sourcenodes and the first group of relay nodes, H d denote the N m × N m − channel matrix between the ( m − th group of relay nodes anddestination nodes, and H i − ,i denote the N i × N i − channel matrixbetween two groups of relay nodes as described by H s = h s, h s, ... h s,N , H d = h m − , h m − , ... h m − ,N m , H i − ,i = h i − , h i − , ... h i − ,N i , (1)where h s,j = [ h s,j, , h s,j, , ..., h s,j,N ] for j = 1 , , ..., N is a rowvector between source nodes and the j th relay of the first group ofrelay nodes, h m − ,j = [ h m − ,j, , h m − ,j, , ..., h m − ,j,N m − ] for j = 1 , , ..., N m is a row vector between the ( m − thgroup of relay nodes and the j th destination node and h i − ,j =[ h i − ,j, , h i − ,j, , ..., h i − ,j,N i − ] for j = 1 , , ..., N i is a rowvector between the ( i − th group of relay nodes and the j th relayof the i th group of relay nodes. The received signal at the i th groupof relay nodes ( x i ) for each phase can be expressed as:Phase 1: x = H s s + v , (2) y = F x , (3)Phase 2: x = H , A y + v , (4) y = F x , (5)...Phase i : ( i = 2 , , ..., m − ) x i = H i − ,i A i − y i − + v i , (6) y i = F i x i , (7)At the destination nodes, the received signal can be expressed as d = H d A m − y m − + v d , (8) where v is a zero-mean circularly symmetric complex additive whiteGaussian noise (AWGN) vector with covariance matrix σ I . A i =diag { a i, , a i, , ..., a i,N i } is a diagonal matrix whose elements rep-resent the amplification coefficient of each relay of the i th group. F i denotes the normalization matrix which can normalize the power ofthe received signal for each relay of the i th group of relays.(see theappendix to find the expression of F i .) Please note that the propertyof the matrix vector multiplication Ay = Ya will be used in the nextsection, where Y is the diagonal matrix form of the vector y and a is the vector form of the diagonal matrix A . At the receiver, a linearMMSE detector is considered where the optimal filter and optimalamplification coefficients are calculated. The optimal amplificationcoefficients are transmitted to the relays through the feedback chan-nel. And the block marked with a Q[ · ] represents a decision device.
3. PROPOSED JOINT MAXIMUM SUM-RATE DESIGN OFTHE RECEIVER AND THE POWER ALLOCATION
By substituting (2)-(7) into (8), we can get d = C ,m − s + C ,m − v + C ,m − v + ... + C m − ,m − v m − + v d = C ,m − s + m − X i =1 C i,m − v i + v d (9)where C i,j = (cid:26) Q jk = i B k , if i j , I , if i > j. (10)and B = H s (11) B i = H i,i +1 A i F i for i = 1 , , ..., m − (12) B m − = H d A m − y m − (13)We focus on the system which consists of one source node. There-fore, the expression of the Sum Rate (SR) for our m-hop WSNs isexpressed as SR = 1 m log " σ s σ n w H C ,m − C H ,m − ww H ( P mi =1 C i,m − C Hi,m − ) w (14)where w is the linear receiver, and ( · ) H denotes the complex-conjugate (Hermitian) transpose. Let φ = C ,m − C H ,m − (15)and Z = m X i =1 C i,m − C Hi,m − (16)The expression for the sum-rate can be written as SR = 1 m log (cid:18) σ s σ n w H φ ww H Zw (cid:19) = 1 m log (1 + ax ) (17)where a = σ s σ n (18)and x = w H φ ww H Zw (19)Since m log (1 + ax ) is a monotonically increasing function of x ( a > ), the problem of maximizing the sum rate is equivalent toaximizing x . In this section, we consider the case where the totalpower of the relay nodes in each group is limited to some value P T,i (local constraint). The proposed method can be considered as thefollowing optimization problem: [ w opt , a ,opt , ..., a m − ,opt ] = arg max w , a ,..., a m − w H φ ww H Zw , subject to P i = P T,i , i = 1 , , ..., m − . (20)where P i as defined above is the transmitted power of the i th groupof relays, and P i = N i +1 a Hi a i . We can notice that the expression w H φ ww H Zw in (20) is the Generalized Rayleigh Quotient, therefore theoptimal solution of our maximization problem can be solved: w opt isany eigenvector corresponding to the dominant eigenvalue of Z − φ .In order to obtain the optimal power allocation vector a opt , werewrite w H φ ww H Zw and the expression is given by w H φ ww H Zw = a Hi M i a i a Hi diag { w Hi P i w i } a i + w Hi T i w i , for i = 1 , , ..., m − , (21)where M i = diag { w Hi C i +1 ,m − H i,i +1 F i } C ,i − C H ,i − × diag { F Hi H Hi,i +1 C Hi +1 ,m − w i } , P i = C i +1 ,m − H i,i +1 F i } ( P ik =1 C k,i − C Hk,i − ) × diag { F Hi H Hi,i +1 C Hi +1 ,m − } and T i = ( P mk = i +1 C k,m − C Hk,m − ) .Since the multiplication of any constant value and a eigenvectoris still the eigenvector of the matrix, we can express the receive filteras w i = w opt q w Hopt ( P mk = i +1 C k,m − C Hk,m − ) w opt (22)Therefore, we can obtain w Hi ( m X k = i +1 C k,m − C Hk,m − ) w i = 1 = N i +1 a Hi a i P T,i (23)By substituting (23) into (21) and using M i and N i given above tosimplify the expression of (21), we obtain w H φ ww H Zw = a Hi M i a i a Hi N i a i for i = 1 , , ..., m − (24)We can notice that the expression a H M i aa Hi N i a i in (26) is the GeneralizedRayleigh Quotient, therefore the optimal solution of our maximiza-tion problem can be solved: a i,opt is any eigenvector correspondingto the dominant eigenvalue of N − i M i , and satisfying a Hi,opt a i,opt = P T,i N i +1 . The solutions of w opt and a i,opt depend on each other. There-fore it is necessary to iterate them with an initial value of a i ( i =1 , , ..., m − ) to obtain the optimum solutions.
4. PROPOSED ALTERNATING MAXIMIZATIONALGORITHM
In this section, we devise our proposed alternating maximization al-gorithm which computes the linear receive filter and the power allo-cation parameters that maximize the sum rate of the WSN. In partic-ular, we employ two methods to calculate the dominant eigenvectors.
Table 1 . Summary of the Proposed AlgorithmInitialize the algorithm by setting A i = q P T,i N i N i +1 I for i = 1 , , ..., m − For each iteration:1. Compute φ and Z in (15) and (16).2. Using QR algorithm or power method to compute thedominate eigenvector of Z − φ , denoted as w opt .3. For i = 1 , , ..., m − a) Compute M i and N i in (24) and (25).b) Using the QR algorithm or power method to compute thedominate eigenvector of N − i M i , denoted as a i .c) To ensure the local power constraint a Hi,opt a i,opt = P T,i N i +1 ,compute a i,opt = r P T,i N i +1 a Hi a i a i .The first one is the QR algorithm [18] which calculates all the eigen-values and eigenvectors of a matrix. We can choose the dominanteigenvector among them. The second one is the power method [18]which only calculates the dominant eigenvector of a matrix. There-fore, the computational complexity can be reduced. Table 1 showsa summary of our proposed algorithm which will be used for thesimulations.Tong, we should include a short paragraph here discussing therequired complexity and the convergence issues.
5. SIMULATIONS
In this section, we numerically study the sum-rate performance ofour proposed joint MSR design of the receiver and power allocationmethods and compare them with the equal power allocation method[6] which allocates the same power level equally for all links fromthe relay nodes. We consider a 3-hop ( m =3) wireless sensor net-work. The number of source nodes ( N ), two groups of relay nodes( N , N ) and destination nodes ( N ) are 1, 4, 4, 2 respectively. Weconsider an AF cooperation protocol. The quasi-static fading chan-nel (block fading channel) is considered in our simulations whose el-ements are Rayleigh random variables (with zero mean and unit vari-ance) and assumed to be invariant during the transmission of eachpacket. In our simulations, the channel is assumed to be known at thedestination nodes. For channel estimation algorithms for WSNs andother low-complexity parameter estimation algorithms, one can referto [15] and [16]. During each phase, the sources transmit the QPSKmodulated packets with 1500 symbols. The noise at the destinationnodes is modeled as circularly symmetric complex Gaussian randomvariables with zero mean. When perfect (error free) feedback chan-nel between destination nodes and relay nodes is assumed to trans-mit the amplification coefficients, it can be seen from Fig. 3 that ourproposed method can achieve better sum-rate performance than theequal power allocation method. When using the power method tocalculate the dominant eigenvector, it can get a very similar result tothe QR algorithm. In practice, the feedback channel can not be errorfree. In order to study the impact of feedback channel errors on theperformance, we employ the binary symmetric channel (BSC) as themodel for the feedback channel and quantize each complex ampli-fication coefficient to an 8-bit binary value (4 bits for the real part,4 bits for the imaginary part). Vector quantization methods [17] canalso be employed for increased spectral efficiency. The error prob-ability (Pe) of BSC is fixed at − . The dashed curves in Fig. 3show the performance degradation compared with the performancehen using a perfect feedback channel. To show the performancetendency of the BSC for other values of Pe, we fix the SNR at 10 dBand choose Pe ranging form 0 to − . The performance curves areshown in Fig. 4, which illustrates the sum-rate performance versusPe of our two proposed methods. It can be seen that along with theincrease in Pe, their performance becomes worse. S u m − R a t e Equal Power Allocation, QR AlgorithmEqual Power Allocation, Power MethodLocal Constraints, QA Algorithm (Perfect Feedback Channel)Local Constraints, Power Method (Perfect Feedback Channel)Local Constraints, QA Algorithm (BSC 8bits Pe=10e−3)Local Constraints, Power Method (BSC 8bits Pe=10e−3)
Fig. 2 . Sum-rate performance versus SNR of our proposed joint maximumsum-rate design of the receiver and power allocation strategy for a 3-hopWSN, compared with equal power allocation method. S u m − R a t e Local Constraint, QR Algorithm (BSC 8Bits)Local Constraint, Power Method (BSC 8Bits)
Fig. 3 . Sum-rate performance versus Pe of our proposed joint strategy whenemploying BSC as the model for the feedback channel.
6. CONCLUSIONS
A joint MSR receiver design and power allocation strategy has beenproposed for general multihop WSNs. It has been shown that ourproposed strategy achieves a significantly better performance thanthe equal power allocation method. Possible extensions to this workmay include the study of the complexity and the requirement for thefeedback channel.
7. APPENDIX
Here, we derive the expression of F i which is denoted in Section 2. F i = diag { E ( | x i, | ) , E ( | x i, | ) , ..., E ( | x i,N i | ) } − (25) where E ( | x i,j | = σ s | h s,j | + σ n , for i = 1 , h i − ,j A i − E ( y i − y Hi − ) A Hi − h Hi − ,j + σ n , for i = 2 , , ..., m. (26) E ( y i y Hi ) = F i ( σ s H s H Hs + σ n I ) F Hi , for i = 1 , F i [ H i − ,i A i − E ( y i − y Hi − ) A Hi − H Hi − ,i + σ n I ] F Hi for i = 2 , , ..., m. (27)
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