Effects of Node Geometry on Cooperative Distributed AF Wireless Relay Network

Transcription

1 Effects of Node Geometry on Cooperative Distributed AF Wireless Relay Network Wenhao Xiong, Hyuck M Kwon, Yazan Ibdah, Kanghee Lee, and Yu Bi Department of Electrical Engineering and Computer Science, Wichita State University, Wichita, Kansas s: {wxxiong, hyuckkwon, yxibdah, kxlee1, yxbi}@wichitaedu Abstract This paper presents the effects of relay-node geometry on bit error rate (BER) performance of an amplifying and forward (AF) relay cooperative distributed wireless network This paper considers one-source-one-destination pair and N relay nodes In addition, this paper assumes that all relay nodes share their received signals from the source node and the channel coefficients for cooperation Then, an optimal relay-amplifying matrix is derived using the minimum mean square error (MMSE) criterion under a total power constraint at the relay nodes Simulation results are presented using the analytically derived amplifying matrix Index Terms Amplify-and-forward, minimum mean square error, relay-amplifying matrix, node geometry, wireless relay networks I INTRODUCTION To overcome the limitation with respect to communication distance of the small-scaled nodes in wireless relay networks, there have been proposing lots of various schemes associated with the various relay protocols Relay protocols in wireless networks can be usually classified as amplify-and-forward (AF), decode-and-forward (DF), and compress-and-forward (CF) [2] [6] The AF relay protocol, which is employed in this current paper, only forwards an amplified version of its received signals from a source node according to its power constraints In [7], the AF relay protocol was employed but did not employ cooperation among the distributed relay nodes In addition, power constraint was not included in the derivation of relay-amplifying matrix as shown in (6) of [7] While, the relay-amplifying matrix in (84) of [7] was obtained by approximations, which were not justified In contrast to [7], in [8], [9], all relay nodes can exchange their received signals from a source node for the relay cooperation However, the authors in [8], [9] did not consider relay-node geometry under power constraints at the relay nodes In other words, the effects of the relay node locations have not been studied in [8], [9] In addition, noncooperative distributed AF relay schemes for This work was partly sponsored by the Army Research Office under DEPSCoR ARO Grant W911NF , and by NASA under EPSCoR CAN Grant NNX08AV84A single-input single-output and single-input multiple-output wireless systems without/with power constraints at the destination node have recently been studied by the authors of this current paper in [10] [12] In other words, effects of node geometry on a cooperative distributed AF wireless relay network under power constraint at relay nodes was not studied yet Therefore, this current paper is to consider a cooperative distributed wireless relay network and present an optimum relay-amplifying matrix for an AF distributed relay network under realistic power constraints at the relays, and node geometry effects without approximations In practice, node geometry in wireless relay networks is an essential element to be considered The cooperative distributed relay nodes may be operated as transmit antennas if they are close to the source node On the other hand, they may function as receiver antennas if they are close to the destination node Taking into consideration of the influence of node geometry on wireless relay networks, this paper uses an aggregate channel model composed of both large-scaled path loss and small-scaled fading [13] The large-scaled path loss is due to the decay of the signal power strength On the other hand, the small-scaled fading is due to the randomly changing multipath in wireless relay networks In [14], the authors proposed the effect of node geometry on the performance of two-phase protocols with regard to the outage probability of mutual information in wireless networks with the direct link between the source and the destination node They claimed that performance of outage probability can achieve better if the relay nodes are close to the source and destination nodes This claim supports the better understanding of the node geometry effects This current paper derives an efficient relay-amplifying matrix based on minimum mean square error (MMSE) criterion for cooperative distributed wireless relay networks including the effects of node geometry and power constraints with no approximation To achieve this goal, it is assumed that all relays share their received signals from the source node and that the channel coefficients are known This cooperation requires sufficient power which may be available in a relay

2 network For example, cell phone can be used as a relay It is also assumed that there is no direct link between the source and the destination node for a worst case Also an AF protocol is employed at each relay The remaining paper is organized into four sections Section II describes the system model and symbol transmission scheme applied Section III derives an optimal relayamplifying matrix for the cooperative distributed MMSE relay scheme by considering node geometry and power constraints at the relays Section IV shows the simulation results Finally, Section V concludes the paper Notation: Matrices and vectors are denoted, respectively, by uppercase and lowercase boldface characters (eg, A and a) The transpose, complex conjugate, inverse, and Hermitian of A are denoted, respectively, by A T, A, A 1, and A H An n n identity matrix is denoted by I N The expectation operator is E[ ] Notations a, a, and A F denote the absolute value of a for any scalar, 2-norm of a, and Frobenius-norm of A, respectively nodes α and β In order to consider the influence of node geometry on the wireless relay networks, it is assumed that all channel node links in Fig 1 consist of large-scaled path loss The path loss between two nodes α and β is expressed by L αβ = K/d ε αβ (1) where K is a constant that depends on the environment, and ε is the path-loss exponent [14] For free space path loss, this paper considers ε = 2 and K = G t G r λ 2 /(4π 2 ), where G t and G r are, respectively, antenna gains at a transmitter and receiver node, and λ is the wavelength Furthermore, it is also assumed that, for all channel node links, ε and K are the same, even though they may be different from each link during symbol transmission The relationship among the channel links based on the law of cosines can be expressed as d 2 SD = d 2 SR i + d 2 R id 2d SRi d RiD cos θ i (2) Let η i denote the ratio of d SRi /d RiD Then, the db value of 10 log 10 η i indicates the relative location of the i-th relay node with regard to the source and destination node The more positive db values represent relay nodes closer to the destination node than to the source node Using (1), (2), and the ratio η i, L Ri D = L SD (1 2η i cos θ i + η 2 i ) ε/2 (3) Similarly, Fig 1 Wireless relay network for one-source-one-destination pair and N relay nodes with a given relay-node geometry II SYSTEM MODEL Figure 1 shows a wireless relay network with N cooperative distributed relay nodes between a source node and a destination node The relay-node geometries are specified with the distance and angles As illustrated in Fig 1, there are two stages for symbol transmission where a source node in the wireless relay network transmits a signal s in Stage I, and the relay nodes retransmit their signals to a destination node in Stage II This paper focuses on the case where the direct link between the source and destination node is considerably weak and can be negligible However, the doted line in Fig 1 showing the link between the source and destination node is for the path loss computation Let θ i denote the angle between the two links from a source node to the i-th relay node and from the i-th relay node to the destination node (0 < θ i π), i = 1,, N Let d αβ be the Euclidean distance between L SRi = L SD (1 2η 1 i cos θ i + η 2 i ) ε/2 (4) Consequently, when ε = 2, the geometrical large-scaled path loss matrix W s C N N from the source node to the relay nodes is modeled by W s = diag ( L SR1,, L SRN ) (5) Let h s C N 1 denote the channel small-scaled fading coefficient column vector from the source node to the relay nodes as h s = [h s,1, h s,2,, h s,n ] T (6) where h s,i, i = 1,, N, is the i-th element of h s, which represents the channel small-scaled fading coefficient from the source node to the i-th relay node The received signal column vector r C N 1 at the relay nodes is written as r = W s h s s + v s (7) where v s C N 1 is a zero-mean complex additive white Gaussian noise vector with covariance matrix σ 2 v s I N Each channel coefficient h s,i is assumed to be independent identically distributed with zero-mean circularly symmetric complex Gaussian of unit variance and quasi-static Rayleigh fading

3 so that they are constant during symbol transmission For the relay cooperation, it is also assumed that all relays can communicate their received signals from the source node with each other with negligible errors The amplified signal column vector x C N 1 at the relay nodes is given by x = Fr = FW s h s s + Fv s (8) where F C N N is a relay-amplifying matrix applied at the relay nodes to enhance performance at the destination node Similarly, when ε = 2, the geometrical large-scaled path loss matrix W d C N N from the relay nodes to the destination node is generated by W d = diag ( L R1 D,, L RN D) (9) Let h y C 1 N denote the channel small-scaled fading coefficient row vector from the relay nodes to the destination node h y = [h y,1, h y,2,, h y,n ] (10) where h y,i, i = 1,, N, is the i-th element of h y, which stands for the channel coefficient from the i-th relay node to the destination node Each channel h y,i is also assumed to be independent identically distributed with zero-mean circularly symmetric complex Gaussian of unit variance and quasi-static Rayleigh fading so that they are constant during the symbol transmission The received signal y C 1 1 at the destination node can be represented as y = h y W d x + v y = h y W d FW s h s s + h y W d Fv s + v y (11) where v y C 1 1 is a zero-mean complex additive white Gaussian noise with variance σ 2 v y It is assumed that all relay nodes have an accurate knowledge of their local channels, both from the source node to the relay nodes and from the relay nodes to the destination node In the next section, the optimal relay-amplifying matrix F will be found using the MMSE criterion for a given node geometry in wireless relay networks Note that the authors in [7] proposed a non-cooperative MMSE relay strategy for wireless sensor networks In other words, the linear transformation matrix F in [7] is a diagonal matrix, while, the relay-amplifying matrix F in this current paper can be a nondiagonal matrix due to cooperation among the relay nodes III COOPERATIVE DISTRIBUTED MMSE RELAY SCHEME WITH NODE GEOMETRY In this section, the optimum relaying matrix for the AF relay scheme based on the MMSE criterion including node geometry in the cooperative distributed wireless relay networks is found In addition, because the total power usage within relay nodes in a network is fixed, a constraint on the power usage at the relay nodes is also included in the analysis Note that this power constraint was not considered in (6) of [7] Minimizing the mean square error between the signal component h y H d x of the received signal at the destination node and the transmitted signal s at the source node under the relay node power constraint can be written as F = arg min F J(F) (12) st E [ x 2] = P where P is the total power usage at the relay nodes The objective function J(F) can be written as J(F) = E{ h y W d x s 2 } = σ 2 sh y W d FW s h s h H s W H s F H W H d h H y σ 2 sh H s W H s F H W H d h H y + σ 2 v s h y W d FF H W H d h H y σ 2 sh y W d FW s h s + σ 2 s, (13) and the total power constraint at the relay nodes can be written as P = E [ x 2] = σ 2 s FW s h s 2 + σ 2 v s F 2 F (14) In order to determine a constrained optimization problem, a Lagrangian multiplier λ is used in [15] Therefore, the constrained optimization objective function can be written as L(F, λ) = J(F) + λ ( E [ x 2] P ) (15) For simple matrix derivative computation, the complex conjugate of F, ie, F, is used By taking derivative of L(F, λ) with respect to F, and λ, respectively, and using the properties of the complex derivative matrix in [16], [17], the optimum relayamplifying matrix F using the matrix inversion lemma [18] can be rewritten as F σsw 2 H d h H y h H s W H s = ( λ + h y W d 2)( σ 2 vs + σs 2 W s h s 2) (16) Substituting (16) into (14), the power can be written as P= σ6 s W H d h H y h H s W H s W s h s 2 +σ 4 s σv 2 s W H d h H y h H s W H 2 s F [ (λ + h y W d 2)( σ 2 + σ2 vs s W s h s 2) ] 2 (17) Using (17), the Lagrangian multiplier λ can be found as λ=± σ6 s W H d h H y h H s W H s W s h s 2 +σ 4 s σv 2 s W H d h H y h H s W H 2 s F P ( σv 2 s + σs 2 W s h s 2) 2 h y W d 2 (18) Substituting (18) into (16), the optimum relay-amplifying matrix for the cooperative relay networks can be written as F W H d h H y h H s W H s P = σs 2 W H d h H y h H s W H s W s h s 2 +σ 2 vs W H d h H y h H s W H 2 s F (19)

4 Either a positive or negative sign in (18) yields the same BER Hence, only the positive sign of λ is used in the simulation For comparison, (84) of [7] is employed for the noncooperative wireless relay networks with node geometry as LSRi LRiDh y,i h s,i P f i N L SRk L Rk D h y,k 2 h s,k ( 2 k=1 L SRk h s,k 2 σs 2+σ2 vs ) ( L SRi h s,i 2 σ 2 s + σ 2 v s ) (20) However, the relay-amplifying matrix presented in (84) of [7] is suboptimum for the noncooperative relay networks because multiple approximations and assumptions have been used Therefore, this paper also presents an optimum relayamplifying matrix for the noncooperative relay networks as LSRi LRiDh y,i f i = h s,i P N k=1 L SR k L Rk D h y,k 2 h s,k 2( ) L SRk h s,k 2 σs 2 +σv 2 s (21) which is derived using similar steps in (12)-(19) with a diagonal F in mind IV SIMULATION RESULTS This section presents the Monte-Carlo BER simulation results for the cooperative distributed AF relay networks The MMSE-based optimum relay-amplifying matrix derived in (19) was used for a given relay-node geometry specified by W s and W d in (5) and (9) All simulations are performed for a one-source-one-destination pair with a different number of cooperative distributed relay nodes, N = 2 or 3, for path loss exponent ε = 2 First, the case of the relay nodes located equidistance from the source node to the destination node, ie, η i = 0 db is considered, with different values of the angle θ i between two links from a source node to the i-th relay node and from the i-th relay node to the destination node Then, it is also considered that they are located closer to or farther away from the source node, ie, η i = 10 and 10 db, respectively All nodes employ only one antenna It is assumed that the transmitted signal at the source node is modulated by quadrature phase shift keying (QPSK) with unity power The channel column vector h s and the channel row vector h y are generated from independent complex Gaussian random variables with zero-mean and unity variance Also, it is assumed that all nodes have the same thermal noise power, ie, σv 2 s =σv 2 y Figure 2 shows the BER performance comparisons for N = 2 cooperative distributed relay networks Three relaynode geometries of θ 1 = θ 2 = π/3, π/2, and 2π/3 are considered with η 1 = η 2 = 0 db As the angle θ i (0 < θ i π) increases when the relay nodes are located at the middle place between the source and destination node, it is observed that BER performance gets better This is because the distances d SRi and d Ri D become smaller, and hence, the path losses become smaller when d SD is fixed and the relay nodes are placed equidistance Fig 2 BER performance comparisons for N = 2 cooperative distributed relay networks Three relay-node geometries of θ 1 =θ 2 =π/3, π/2, and 2π/3 are considered with η i =0 db, i=1, 2 Figure 3 shows BER performance for N = 2 cooperative distributed relay networks Six different relay-node geometries are considered: θ 1 = θ 2 = π/3 or 2π/3 and η 1 = η 2 = 10 db, 0 db or 10 db It is observed that as the relay nodes get closer to either the source node or the destination node, ie, η i = 10 db or 10 db, BER gets smaller than that for η i = 0 db This is because the network can be viewed as a multiple-input single-output and single-input multiple-output, respectively, and the diversity gain can be achieved, when η i =- 10 db and 10 db Fig 3 BER performance comparisons for N = 2 cooperative distributed relay networks Six different relay-node geometries, θ 1 =θ 2 =π/3 or 2π/3 and η 1 =η 2 = 10 db, 0 db or 10 db, are considered Figure 4 shows BER performance for N = 2 and 3 nonco-

5 operative/cooperative distributed relay networks For N = 2, a geometry of θ 1 = θ 2 = π/2 and η 1 = η 2 = 0 db is considered For N = 3, a geometry of θ 1 = θ 3 = π/2, θ 2 = π and η 1 = η 2 = η 3 = 0 db is considered It is found that the cooperative distributed relay strategy accomplishes a better BER performance than the noncooperative one In addition, as expected, the optimum relay-amplifying matrix achieves a better BER performance than the suboptimum one for the noncooperative relay networks For example, at BER = 10 3, using the proposed optimum relay-amplifying coefficient in (21) can show 4 db better performance than that using the suboptimum in (84) of [7] (or (20)) for the noncooperative relay networks when N = 3, θ 1 = θ 3 = π/2, θ 2 = π, and η 1 = η 2 = η 3 =0 db Also, it is observed that using the optimum relay-amplifying matrix in (19) proposed for the cooperative relay networks can show another 3 db improvement over the noncooperative relay network with the proposed optimum relay-amplifying coefficient in (21) Finally, it is observed that the N = 3 case can be 6 db better than the N = 2 case at BER = 10 3 This is because not only the number of relay nodes is increased from 2 to 3 and hence the diversity gain is enhanced, but also the path loss can be significantly reduced for relay node 2 with θ 2 = π Fig 4 BER performance comparisons for N = 2 or 3 noncooperative/cooperative distributed relay networks For N = 2, a geometry of θ 1 = θ 2 = π/2 and η 1 = η 2 = 0 db is considered For N = 3, a geometry of θ 1 =θ 3 =π/2, θ 2 =π and η 1 =η 2 =η 3 =0 db is considered V CONCLUSION This paper derived an optimum relay-amplifying matrix using the MMSE criteria for both cooperative and noncooperative distributed wireless relay networks, including the effects of relay-node geometry and total power constraints at the relay nodes This paper also presented BER simulation results using the derived optimum relay-amplifying matrix It was found that the BER improves as the angle becomes larger when the relay nodes are located at equidistance from the source node and the destination node because the distance becomes smaller Also, it was observed that the BER gets smaller as the number of relay nodes increases Furthermore, the proposed optimum relay-amplifying matrix for cooperative relay networks can show significant improvement, eg, 7 db, over the suboptimum one proposed for noncooperative networks in [7] REFERENCES [1] Y W Hong, W J Huang, F H Chiu, and C C J Kuo, Cooperative communications in resource-constrained wireless networks, IEEE Signal Process Mag, vol 24, no 3, pp 47-57, May 2007 [2] A Saadani and O Traoré, Orthogonal or non-orthogonal amplify and forward protocol: How to cooperate?, IEEE WCNC 2008, Las Vegas, NV, Mar 2008, pp [3] J Laneman, D N C Tse, and G W Wornell, Cooperative diversity in wireless networks: Efficient protocols and outage behaviour, IEEE Transactions on Information Theory, vol 50, no 12, pp , Dec 2004 [4] S Lee and S Chung, When is compress-and-forward optimal?, IEEE ITA 2010, San Diego, CA, Jan 2010 [5] G Kramer, M Gastpar, and P Gupta, Cooperative strategies and capacity theorems for relay networks, IEEE Transactions on Information Theory, vol 51, no 9, pp , Sep 2005 [6] S Berger and A Wittneben, Cooperative distributed multiuser MMSE relaying in wireless ad-hoc networks, IEEE 39th Asilomar Conference, Oct 2005, pp [7] N Khajehnouri and A H Sayed, Distributed MMSE relay strategies for wireless sensor networks, IEEE Transactions on Signal Processing, vol 55, no 7, Jul 2007 [8] S Berger and A Wittneben, Cooperative distributed multiuser MMSE relaying in wireless ad-hoc networks, IEEE 39th Asilomar Conf, Oct 2005, pp [9] R Krishna, Z Xiong, and S Lambotharan, A cooperative MMSE relay strategy for wireless sensor networks, IEEE Signal Processing Letters, vol 15, pp , 2008 [10] K Lee, H M Kwon, Y Ding, Y Ibdah, and Z Wang, Noncooperative distributed MMSE relay schemes under jamming environment and node geometry in wireless relay network, IEEE WTS 2011, New York, NY, April 13-15, 2011 [11] K Lee, H M Kwon, Y Ding, Y Ibdah, Z Wang, and Y Bi, Node geometry and broadband jamming in noncooperative relay networks under received power constraint, IEEE SARNOFF 2011, Princeton, NJ, May 3-4, 2011 [12] K Lee, H M Kwon, Y Ding, Y Ibdah, and Z Wang, Effects of node geometry on noncooperative distributed SIMO wireless relay networks, IEEE AMS 2011, Manila, Philippines, May 23-27, 2011 [13] T Rappaport, W ireless Communications : P rinciples and P r- actice, 2nd ed Englewood Cliffs, NJ: Prentice-Hall, 2001 [14] H Ochiai, P Mitran, and V Tarokh, Variable-rate two-phase collaborative communication protocols for wireless networks, IEEE Transactions on Information Theory, vol 52, no 9, pp , Sept 2006 [15] S Boyd and L Vandenberghe, Convex Optimization Cambridge, UK: Cambridge University Press, 1985 [16] R A Horn and C R Johnson, Matrix Analysis 1st ed Cambridge, MA: Cambridge University Press, 1985 [17] The Matrix Cookbook, [On-line] Available: bookcom [18] A H Sayed, Fundamentals of Adaptive Filtering Hoboken, NJ: John Wiley & Sons, 2003