Minimum Power Allocation for Cooperative Routing in ... - Unik

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The total transmitted power for the multipoint-to-point caseis ∑ mi=1 P i. Therefore, the power allocation problem, whichhas a required outage probability constraint on the receivingnode, can be formulated asm∑min P i ,s.t.i=1(m∏1 − ei=1)−γ th N 0mP i σh 2 i≤ ρ 0 ,P i ≥ 0, for i = 1, . . . , m. (24)The solution of the power allocation problem given in (24)is found in the following theorem:Theorem 2: For the set of m transmitters, which send acommon signal toward the destination, the optimum transmitpower coefficients of the problem stated in (24) satisfy thefollowing equations⎤P k = −γ th N 0σ 2 h k⎡ln −1 ⎢⎣ 1 − ρ( 01 − e∏ mi=1i≠k)−γ th N 0mP i σh 2 i⎥⎦ , (25)for k = 1, . . . , m.Proof: For the problem stated in (24), we redefine theconstraint function of f(P 1 , . . . , P m ) as()−γ m∏th N 0mPf(P 1 , . . . , P m ) = 1 − e i σh 2 i − ρ 0 , (26)i=1and its corresponding derivative can be calculated ( as∂f(P 1 , . . . , P m ) = −γ −γth Nth N 00∏m mP∂P k mPk 2 e k σh 2 k 1 − eσ2 h kCombining (16) and (27), we have(λ −1 = γ th N 0m Pk 2 eσ2 h k−γ th N 0mP k σ 2 h km ∏i=1i≠k1 − ei=1i≠k)−γ th N 0mP i σh 2 i)−γ th N 0mP i σh 2 i.(27). (28)By assuming that the equality in the first constraint in (24)is satisfied, we have ()−γ m∏th N 0mPρ 0 = 1 − e i σh 2 i . (29)i=1Dividing both sides of equalities (28) and (29), we can findthe Lagrange multiplier asλ = mP ()k 2σ2 γ th N 0h k mPe k σh 2 k − 1 . (30)γ th N 0 ρ 0Substituting λ from (30) into (23), we get (25). Moreover,using (29), it can be verified that the arguments of thelogarithm in (25) are between zero and one, and thus, P k in(25) are positive. Hence, the second set of constraints in (24)are satisfied.The optimal power allocation scheme proposed in Theorem2 can be easily solved with initializing some positivevalues for P i , i = 1, . . . , m, and using (25) in a fixed-pointiteration.IV. ENERGY SAVINGS VIA COOPERATIVE ROUTINGThe problem of finding the optimal cooperative route fromthe source node to the destination node can be mapped toa Dynamic Programming (DP) problem [4]. As the networknodes are allowed only to either fully cooperate or broadcast,finding the best cooperative path from the source node to thedestination has a special layered structure. In [4], it is shownthat in a network with N + 1 nodes, which has 2 N nodes inthe cooperation graph, standard shortest path algorithms havea complexity of O(2 N ). Hence, finding the optimal cooperativeroute in an arbitrary network becomes computationallyintractable for larger networks. For this reason, we restrictthe cooperation to nodes along the optimal noncooperativeroute. That is, at each transmission slot, all nodes that havereceived the information cooperate to send the informationto the next node along the minimum energy noncooperativeroute [4]. Therefore, with the help of the link cost expressed inSubsection III-A, the minimum-energy non-cooperative routeis first selected, which has N intermediate relays. Then,nodes along the optimal non-cooperative route cooperate totransmit the source information toward the destination. Thatis, at each transmission slot, all nodes that have received theinformation cooperate to send the information to the next nodealong the minimum energy non-cooperative route. In the nthtransmission slot of full-cooperation scheme, the reliable setis Tx n = {s, r 1 , . . . , r n−1 }, which includes the source nodeand the previous relays r i , i = 1, . . . , n − 1. The link costassociated with the nodes in Tx n , which cooperate to sendthe∑information to the next node n, is given by C(Tx n , n) =ni=1 P i(n), where P 1 (n) is the transmit power from thesource node in the nth phase, and P i (n), i = 2, . . . , n are thetransmit power from r i−1 th node, and they can be estimatedwith the optimization problems stated in Subsection III-B andC. Note that the nth node denotes the nth relay when n ≤ N,and the destination node when n = N +1. Therefore, the totaltransmission power for the cooperative multihop system isN+1∑N+1∑ n∑P T (coop) = C(Tx n , n) = P i (n). (31)n=1n=1 i=1For the case of m-cooperation scheme, in which justprevious closest nodes cooperate to transmit along the noncooperativeroute, P T (cooperative) in (31) can be modified toP T (m-coop) ==N+1∑n=1m∑n=1 i=1C m (Tx n , n)n∑P i (n) +N+1∑n∑n=m+1 i=n−m+1P i (n). (32)The energy savings for a cooperative routing strategy relativeto the optimal non-cooperative strategy is defined asEnergy Savings = P T (non-coop) − P T (coop), (33)P T (non-coop)where P T (coop) is computed in (31) and (32) for the caseof full-cooperation and m-cooperation routings, respectively.
Eergy Savings10.90.80.70.60.5ρ 0=10 −7ρ 0=10 −4Opt. power alloc. with full−coop.Subopt. power alloc. with full−coop.Opt. power alloc. with m=3−coop.Subopt. power alloc. with m=3−coop.0.42 4 6 8 10 12 14Number of nodes (N+1)Fig. 2. The average energy savings curves versus the number of transmittingnodes, (N + 1), employing full-cooperation and m-cooperation with twodifferent BER max constraints.P T (non-coop) denotes the total transmission power for thenon-cooperative multihop strategy which is obtained in (6).Now, assume a connection from the source node to thedestination via the m-cooperation routing around the best noncooperativepath with N intermediate nodes. For decoding themessage reliably, the outage probability at the destination mustbe less than the desired end-to-end outage probability ρ max .We denote the required outage probability at the nth phase asρ 0 . The outage probability ρ o n at nth node is affected by allprevious n−1 hops and can be iteratively calculated accordingto the recursionρ o n = 1 − (1 − ρ o n−1,n)n−1∏i=max{1,n−m}(1 − ρ o i ), (34)with ρ0b = 0, where ρn−1,n b is the outage probability of thenth transmission phase and can be calculated as (10) or (23).The end-to-end outage probability at the destination is givenby using n = N + 1 in (34). If the power allocation strategyderived in Section III is used, (34) can be rewritten asρ o n = 1 − (1 − ρ 0 )n−1∏i=max{1,n−m}(1 − ρ o i ). (35)To get an insight into the relationship between the end-toendoutage probability ρ max = ρ o N+1 and ρ 0, we have ρ o max =1−(1 − ρ 0 ) 2N , when full cooperation is used. Thus, the targetoutage probability at each hop, i.e., ρ 0 , can be represented interm of end-to-end desired outage probability ρ max .It is important to note that based on (34), outage at thedestination occurs even if an intermediate node experience anoutage. This guarantees that by using the power allocationstrategies investigated in Section III, the outage probabilityQoS at the destination is satisfied.V. SIMULATION RESULTSIn this section, we present some simulation results toquantify the energy savings due to the proposed cooperativerouting scheme. We consider a regular line topology wherenodes are located at unit distance from each other on a straightline. The optimal non-cooperative routing in this network isto always send the information to the next nearest node inthe direction of the destination. Assume γ th = 1 and noisepower N 0 is normalized to 1. From (6), and by assumingthat σi,j 2 is proportional to the inverse of the squared distance,the total power required for non-cooperative transmission canbe calculated as P T (non-coop) = −(N+1)ln(1−ρ 0 ). Since we restrictthe cooperation to nodes along the optimal non-cooperativeroute, the total transmitted power for full- cooperation and m-cooperation in line networks can be obtained from (31) and(32), respectively.In Fig. 2, we compare the achieved energy savings ofthe proposed cooperative routings with respect to the noncooperativemultihop scenario, in which satisfying the requiredoutage probability ρ 0 at each step. For ρ 0 = 10 −4 , it can beobserved that using the full cooperation scheme around 90%saving in energy is achieved when 10 relays are employed.Fig. 2 confirms that the optimal power allocation based onoptimization problem stated in (13) has the same performanceas the suboptimal solution expressed in Theorem 2. Furthermore,we have plotted the case of m = 3 cooperation. Onecan observe that even with m = 3 cooperation, a substantialgain can be achieved comparing to the non-cooperativetransmission.Using (35), we can represent ρ 0 in terms of the endto-endoutage probability requirement ρ max . Thus, curvescorresponding to the outage requirement ρ max have the samebehavior as the curves demonstrated in Fig. 2.In Fig. 2, we also studied the energy saving curves for thecase of ρ 0 = 10 −7 . It can be observed that by increasingthe value of ρ 0 , more saving in the transmitted energy isobtainable. This is because of the fact that for the fixednetwork, as ρ 0 increases, the required received SNR alsogoes up, and the effect of cooperative diversity become moreprominent.REFERENCES[1] N. Vaidya, “Open problems in mobile ad-hoc networks,” in Keynote talkat the Workshop on Local Area Networks, (Tampa, Florida), Nov. 2001.[2] F. Li, K. Wu, and A. Lippman, “Energy-efficient cooperative routing inmulti-hop wireless ad hoc networks,” in Proc. IEEE Perform., Computing,and Commun. Conf. (IPCCC), pp. 215–222, 2006.[3] J. Boyer, D. D. Falconer, and H. Yanikomeroglu, “Multihop diversity inwireless relaying channels,” IEEE Trans. Commun., vol. 52, pp. 1820–1830, Oct. 2004.[4] A. E. Khandani, J. Abounadi, E. Modiano, and L. Zheng, “Cooperativerouting in static wireless networks,” IEEE Trans. Commun., pp. 2185–2192, Nov. 2007.[5] M. K. Simon and M.-S. Alouini, Digital Communication over FadingChannels: A Unified Approach to Performance Analysis. New York, NY:Wiley, 2005.[6] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, andProducts. San Diego, USA: Academic, 1996.[7] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge: CambridgeUniv. Press, 2004.
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