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

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combines the signals received from the previous relays andalong with that received from the source.For a general scheme m-cooperation, (1 ≤ m ≤ N + 1),each receiving node decodes the information after combiningthe signals received from the previous m transmitting nodes.Fig. 1 shows a wireless multihop network consisting of asource node s, N relays, and a destination node d, whichis operating under m-cooperation scenario. The cooperationprotocol has N +1 phases. In Phase 1, the source transmits theinformation, and the received signal at the destination (nodenumber N + 1) and the relays can be modeled asbe written as)−γ m∑th N 0P−γ th N 0P i =σi−1,i 2 ln(1 − ρ 0) . (4) Pr {γ rec < γ th } = α i(1 − e i σh 2 i .i=1(10)Now, assume a connection from the source node to the destinationvia N intermediate nodes. For decoding the messagereliably, the outage probability must be less than the desiredend-to-end outage probability ρ max . The probability of correctreception isN+1∏{Pn |h n−1,n | 2 } ( )N+1∏ −γ th N 0P c (N)= Pr≥γ th = expNn=10 Pn=1 n σ 2 .n−1,n(5)Thus, in the multihop case, a target outage probability ρ 0 =1− N+1√ 1 − ρ max is required at each hop. Since ln(1−ρ max ) =y 0,i = √ P 0 h 0,i s + v i , for i = 1, 2, . . . , N + 1, (1)(N + 1) ln(1 − ρ 0 ), the total power in this case, and hence,the non-cooperative multihop link cost is given byN+1∑N+1∑ −γ th N 0 (N + 1)P T (non-coop) = C(n−1, n) =σn=1n=1 n−1,n 2 ln(1 − ρ max) ,(6)where C(n−1, n) is the point-to-point link cost when the (n−1)th node transmits to the nth node, which is given in (4).B. Cooperative Multihop Link CostIn this case, a set of multiple nodes Tx n ={tx n,1 , tx n,2 , . . . , tx n,m } cooperate to transmit the same informationto a single receiver node rx n . Assuming coherentdetection at the receiving node, the signals simply add up atthe receiver, and acceptable decoding is possible as long as thereceived SNR becomes larger than γ th . Using (2), the receivedSNR at the receiving node can be written as γ rec = ∑ mi=1 γ i,i=max(1,n−m)where γ i = P i|h n−i,n | 2N 0. For the notational simplicity, weskipped the index n. Our objective is to find the minimumvalue of cost function C(Tx n , n) = ∑ mi=1 P i such that theoutage probability at the receiving node become less than thetarget value ρ 0 . In this case, the probability of outage can becalculated as Pr { ∑ mi=1 γ i < γ th }.Now, we are going to derive a tractable outage probabilityformula at the receiving node rxIII. OUTAGE PROBABILITY-BASED LINK COSTFORMULATIONn . For calculatingPr {γ rec < γ th }, we first derive the moment generating function(MGF) of the random variable γ rec . Since γ i ’s are independentexponential random variables, the MGF of γ rec , i.e.,M rec (−s) = E{e −sγ rec}, can be written asm∏ 1M rec (−s) =. (7)i=1 1 + P iσh 2 s iN 0Using partial fraction expansion, and by assuming that all linkshave different variances, (7) can be decomposed intom∑ α iM rec (−s) =, (8)i=1 1 + P iσh 2 s iN 0where≥ γ th . That ism∏ P i σh 2 α i =i{Pi |h i−1,i | 2 } ( )Pj=1 i σh 2 i− P j σh 2 . (9)j−γ th N 0j≠iPr≥ γ th = expN 0 P i σi−1,i2 . (3) Since each term in the summation in (8) is correspondingto the MGF of an exponential distribution, Pr {γ rec < γ th } canwhere P 0 is the average total transmitted symbol energy ofthe source, since we assume the information bearing symbolss’s have zero-mean and unit variance, v i is complex zeromeanwhite Gaussian noise at the ith receiving node. Thechannel coefficients h i,j , i = 0, 1, . . . , N, j = 1, 2 . . . , N + 1,are complex Gaussian random variables with zero-mean andvariances σi,j 2 . In Phase 2, relay nodes are sorted based ontheir received SNR, such that relay number 1 has the highestreceived SNR. Generally, in Phase n, 2 ≤ n ≤ N + 1, theprevious min{m, n} nodes are transmitting their signal towardthe next node. Similar to [4], we assume that transmitters areable to adjust their phases in such a way that the receivedsignal at the nth receiving node in Phase n isy n = √ n−1∑ √P 0 |h 0,n | u(m−n) s + Pi |h i,n | ŝ i +v n , (2)where the function u(x) = 1, when x ≥ 0, and otherwise iszero and the symbol ŝ i is the re-encoded symbol at the ithrelay.In this section, our objective is to find the optimal powerallocation required for successful transmission from a set oftransmitting nodes to a receiving node.A. Non-Cooperative Multihop Link CostFirst, we consider the simplest case where only one node istransmitting within a time slot to a single target node. Now, weinvestigate how the transmitter node should decide the valueof its transmit power P i to satisfy the target SNR, γ th , at thedestination with a target outage probability of ρ 0 . We considerthat the receiver can correctly decode the source data wheneverSNR d = Pi|hi−1,i|2N 0Therefore, the required transmit power can be calculated as
Page 1: Minimum Power Allocation for Cooper
Page 5: Eergy Savings10.90.80.70.60.5ρ 0=1

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

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