Resource Allocation via Linear Programming for Multi-Source, Multi-Relay Wireless Networks

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1 Resource Aocation via Linear Programming for Muti-Source, Muti-Reay Wireess Networs Nariman Farsad and Andrew W Ecford Dept of Computer Science and Engineering, Yor University 4700 Keee Street, Toronto, Ontario, Canada M3J 1P3 E-mai: nariman@cseyoruca, aecford@yoruca Abstract In a cooperative wireess networ, there may be many potentia reays within radio range of a source; simiary, there may be many potentia sources seeing to use reays Aocating these resources is a non-trivia optimization probem In this paper, fractiona cooperation is considered, where each potentia reay ony aocates a fraction of its resources to reaying It is shown that inear programming can be used to optimay aocate resources in muti-source, muti-reay networs, where the reays use a demoduate-and-forward (F) strategy, and where the transmissions are protected by owdensity parity-chec (LDPC) codes Compared with existing optimization schemes, this method is particuary suitabe for very arge networs with numerous sources and reays Simuation resuts are presented to demonstrate the performance of this scheme I INTRODUCTION In wireess networs, spatia distribution of nodes generay resuts in independent fading on different ins This spatia distribution can be expoited in cooperative diversity 1, 2, where each node can assist its neighbours in transmitting information to a data sin In its simpest form, a cooperative system is a reay system consisting of three nodes: a source, a reay, and a destination The reay can use various cooperative schemes such as decode-and-forward (DF) 3 and demoduate-and-forward (F) 4, 5, to assist the source in transmitting its information bits to the destination F is particuary appicabe to devices with reduced computationa abiities, such as in wireess sensor networs (WSNs) In most wireess networs, a source node is typicay in radio range of mutipe reays Ecford et a 6 introduced fractiona cooperation as a ow-compexity cooperative scheme for such muti-reay systems, to be used in conjunction with F Using this scheme, a arge number of reays retransmit a sma fraction of the source s transmission bits, so that the reaying cost is spread over a arge number of reays instead of one thus, each reay contributes as much as it can, whie reserving resources for the transmission of its own information A ey chaenge in fractiona cooperation is resource aocation, in which the system determines what fraction must be seected for retransmission by each reay This chaenge is exacerbated in systems with mutipe sources, a of which are competing for the same fractiona resource at the reays In This wor was financiay supported by a Discovery Grant from the Natura Sciences and Engineering Research Counci this paper, we pose optima resource aocation in fractiona cooperation as a convex optimization probem, ensuring successfu transmission whie minimizing energy consumption In particuar, we use inear programming to minimize the number of transmission bits subject to the constraint that decoding at the destination is successfu This constraint is derived using extrinsic information transfer (EXIT) charts 7 Our approach is party inspired by 8, in which inear programming was used in combination with EXIT charts in order to optimize LDPC degree sequences Further codeoptimization wor for reay channes was given in 9 11, though the current paper concerns resource aocation rather than code optimization Reated wor on resource aocation was done in 12 However, in that paper, the union bound was used for anaysis (rather than EXIT charts), which ed to a nonconvex optimization probem The current paper is a significant improvement on that wor: by stating the optimization probem in terms of inear programming, one is guaranteed to efficienty find a goba optimum, regardess of the number of sources and reays Furthermore, we give simuation resuts showing the performance of our approach, which is especiay suitabe for arge (many-source, manyreay) networs Athough we focus on F cooperation, our methods can aso be appied to DF, which we wi discuss in future wor The rest of this paper is organized as foows In section II, we present a muti-source, muti-reay system mode that represents a wireess networ empoying fractiona cooperation In section III, we formuate a inear programming mode that wi ensure successfu transmission whie optimizing the transmission power In section IV, we verify the vaidity of our inear programming mode by presenting simuation resuts A Reay mode II SYSTEM MODEL The reay mode that we use is an extension of the fractiona cooperation mode proposed in 6: we consider s sources, r reays, and a singe destination The r reays are shared amongst a s sources (ie each singe source has r reays that assists in its transmission), as shown in Figure 1 Each source measures a phenomenon, encodes it using LDPC codes, and broadcasts the encoded codeword to the r reays, as we as the destination Using F, the jth reay maes hard decisions, seects a sma fraction ɛ j (where, in genera, ɛ i ɛ j for

2 R 1 B oduate-and-forward In F, a reay first demoduates the signa received from a source This process can be formuated as S 1 S 2 S s Fig 1 R 2 R r Muti-Source, Muti-reay mode D z (Si,Rj) = φ 1 (y (Si,Rj) ), (4) where z (Si,Rj) is the resuts of hard decisions (demoduation) for the jth reay assisting ith source Each reay then seects a fraction of the demoduated signa, re-encodes it using error correcting codes and retransmits to the destination The vector b (Si,Rj) represents the demoduated bit positions seected for retransmission to the destination: if b (Si,Rj) =1, then the th bit is reayed; if b (Si,Rj) =0, then the th bit is not reayed Therefore, the demoduated sequence resuting from the jth reay assisting ith source is avaiabe at the destination as i j), re-encodes them using LDPC codes, and transmits the resuting codeword to the destination The destination wi then decode each source s information bits using the received signa from the r reays, as we as the source itsef Each source has a ength-n information sequence to transfer to the destination represented by x (Si) = x (Si) 1,x (Si) 2,, x n (Si), where x (Si) {0, 1} and S i represents the ith source Each information sequence is encoded by an LDPC code for each source Let R 1,R 2,,R s be the code rates at each source Therefore the codeword ready for transmission at the ith source is represented by z (Si) = z (Si) 1,z (Si) 2,, z c (Si) i, where c i = n/r i is the ength of the codeword As shown in Figure 1, there are sr source to reay (S-R), s source to destination (S-D), and r reay to destination (R- D) ins We assume these communication ins use binary phase shift eying (BPSK) for data moduation We define the function φ : {0, 1} {+1, 1} as the moduation function where 0 is mapped to a +1 and 1 is mapped to -1 With sight abuse of the inverse notation, the demoduation function is defined as φ 1 (y) = { 0 if y 0 1 otherwise The S-D ins are therefore given by, (1) y (Si,D) = φ(z (Si) )+n (Si,D), (2) where S i corresponds to the ith source and n (Si,D) is AWGN with variance σ(s 2 i,d) The channe SNRs for each of the s S-D ins are represented by γ (Si,D) =1/(2σ(S 2 ) i,d) The S-R ins are aso given by y (Si,Rj) = φ(z (Si) )+n (Si,Rj), (3) where S i and R j correspond to the ith source and the jth reay respectivey and n (Si,Rj) is AWGN with variance σ 2 (S i,r j) Therefore, a the S-R ins can be represented by sr channe SNRs, γ (Si,R j) =1/(2σ 2 (S i,r j) ) y (Si,Rj,D) F = b (Si,Rj) φ(z (Si,Rj) ), (5) where is eement-wise mutipication of vectors, z (Si,Rj) is given by equation (4), and y (Si,Rj,D) F represents the resuts of demoduations avaiabe at the destination The eements of y (Si,Rj,D) F can tae three possibe vaues: +1 (representing a demoduated 0 bit), 1 (representing a demoduated 1 bit), and 0 (representing a position that is not seected for reaying) The channe LLR for the S-D in is cacuated as (Si,D) =2y (Si,D) /σ 2 (S i,d) =4γ (S i,d)y (Si,D), (6) and for the S-Rs in as (Si,Rj,D) F = y (Si,Rj,D) F og 1, (7) where is the probabiity of demoduation error at the jth reay assisting ith source given by = 1 2 erfc ( γ(si,r j)) (8) Consequenty, the message LLR input to the iterative LDPC decoder of the ith source can be cacuated as (Si) F = (Si,D) + j=1 (Si,Rj,D) F (9) In F any type of code (such as RA codes or irreguar LDPC codes) can be used For simpicity we assume that using powerfu and capacity approaching codes such as RA 14, and irreguar LDPC 15 codes, over R-D ins, can resut in perfect recovery of demoduated bits at the destination, at rates cose to capacity Thus, for any F system, we wi assume that a capacity-approaching code is used in the R- D in, and is decoded without error However, for a fixed number of information bits, the tota number of transmitted bits over the R-D in wi depend on the capacity, which is a function of the in SNR, written γ Ri,D for the ith reay

3 III LINEAR PROGRAMMING MODEL In this section we present a inear programming mode that wi minimize the number of transmission bits of a mutisource muti-reay system, described in the previous section, subject to the constraint of successfu transmission In the next section we wi confirm these methods by presenting simuations A Key Assumptions and Definitions We consider the muti-reay, muti-source system, expained in section II, with r reays and s sources For each source, we have r reays, and therefore r source-reay (S-R) and reaydestination (R-D) ins (as we as a singe source-destination (S-D) in) Without oss of generaity, we assume that the a-zero codeword is transmitted by each source, which is equivaent to the a-(+1) channe codeword We assume a ins are independent AWGN channes, represented with their respective channe SNR Define a vector of ength s, γ SD, as γ SD = γ (S1,D) γ (S2,D) γ (Ss,D) T (10) As expained in section II, for simpicity, we assume that for F the R-D ins are perfect We define ɛ (Si,R j) as the fraction of ith source transmission that was forwarded by the jth reay We define a vector of ength sr, ɛ, as ɛ = T ɛ (1,1) ɛ (1,r) ɛ (2,1) ɛ (2,r) ɛ (s,1) ɛ (s,r) (11) where the eements represent the fractions that are seected for retransmission by each reay for each source In 8 it was shown that messages passed in the LDPC decoder can be (approximatey) represented using a singe parameter in that paper, message error probabiity was used, but in this paper, we use the mean of the LLR messages Therefore, for LDPC codes, an approximate minimum channe LLR mean required for successfu decoding, written m min, can be cacuated using simuations and EXIT chart anaysis Hence, we can assume that if the channe LLR mean, m, that is input to the iterative decoding agorithm, satisfies m m min, the iterative decoding process is assumed to be successfu Since for each source we can have a different minimum channe LLR mean requirements, we define a vector of ength s, m min, as vector with s eements that are T m min = m {1} min m {2} min m {s} min, (12) where m {i} min is the minimum channe LLR mean threshod for the ith source B Successfu Transmission Requirements As described in the previous section, in order to ensure successfu transmission and decoding at the destination, the mean of the input channe LLR shoud be greater than a minimum, m min Therefore, we need to cacuate the mean of the input channe LLRs In this section we present a theorem for cacuation of the input LLR mean to the decoder for F Theorem 1: For the system described in section II, assuming that the reays use F, the channe mean that is input to the iterative decoder for the ith source, S i, is given by m (F ) i = 2γ (Si,D) + j=0 ɛ (Si,R j)(1 2 ) og 1, where γ (Si,D) is the channe SNR between the ith source and the destination, is the probabiity of hard decision error at the reay given by equation (8) and ɛ (Si,R j) the fraction seected by each reay Proof: When the reays use F the input LLR to the iterative decoding agorithm at the destination wi consist of summation of, (Si,D) and (Si,Rj) as shown in equation (9) in section II For the singe S-D in the channe LLR mean is cacuated as m (Si,D) =2γ (Si,D), (13) where γ (Si,D) is the channe SNR of the S-D in for the ith source This is the first term in Theorem 1 Since we have assumed that the R-D ins are perfect, (Si,Rj) represents the LLR of hard decisions at the jth reay From equation (7) we now that the channe LLR mean for (Si,Rj) depends on the probabiity that an error occurs when maing hard decisions at the jth reay retransmitting ith source signa If for exampe we assume y (Si,Rj) (the resuts of hard decisions mapped to +1 and -1 instead of 0 and 1) is a-(+1) the channe LLR mean is given by m (Si,Rj) 1 = og, (14) where is cacuated according to equation (8) In genera since according to our assumption the a-zero codeword was transmitted by the source, assuming the ength of the codeword is m, the expected number of +1s in the demoduated sequence is (1 )m whie the expected number of -1s in the sequence is m Therefore, the LLR average is cacuated as (1 )m p(si,rj) m 1 og = m (15) (1 2 ) og 1 Since the reay wi ony forward, ɛ (Si,R j) fraction of the demoduated sequence which is equivaent to repacing the unseected positions with zero, the channe LLR mean for the jth reay is given by m (Si,Rj) = ɛ (Si,R j)(1 2 ) og 1 (16)

4 Finay, since a the channe LLRs from the sources and reays have a symmetric Gaussian distribution, the mean of their sum is the sum of a the channe LLR means of S-D and R-D ins; That is for the ith source S i m (F ) i = m (Si,D) + j=0 m (Si,Rj) (17) Coroary For the system described in section II assuming that the reays use F the transmission and the decoding process is successfu whenever m (F ) i m min (18) This coroary foows directy from Theorem 1 and the EXIT chart anaysis C Energy Minimization In the previous section we discussed the criteria for successfu transmission and decoding at the destination for F based on input channe LLR means In this section we present a inear programming mode that minimizes the number of transmission bits (ie transmission power), subject to the constraint that the decoding is successfu In this section we derive a inear programming mode that achieves this tas In our mode the objective is to minimize the number of transmitted bits with the constraint of successfu decoding at the destination, where decision variabes are ɛ (Si,R j), the forwarding fractions of reays If F is used by the reays, and the ith source has a codeword of ength m i to transmit to the destination, the objective function is given by f(ɛ) = s m i + i=0 s i=0 j=0 ɛ (Si,R j)m i R i,j, (19) Since in section II we assumed the R-D ins are perfect through the use of powerfu capacity approaching codes, we can repace R i,j with the capacity of the corresponding channe We can aso omit the constant terms since they have no effect on the optimization Therefore, the objective function can be simpified to s ɛ (Si,R f(ɛ) = j)m i C(γ, (20) (Rj,D)) i=0 j=0 where C(γ (Rj,D)) is the channe capacity between jth reay and the destination To derive the constraints for F, we define a variabe g (Si,Rj) as g (Si,R j) = (1 2 ) og 1, (21) where the term on the right side is derived in equation (15), and represents the R-D in channe LLR mean before fractiona seection at the reays An s (sr) matrix, G SR, is defined such that the rows of the matrix represent each source and the coumns represent sr S-R channes The coumns are isted in the order of (S 1,R 1 ) (S 1,R r )(S 2,R 1 ) (S 2,R r ) (S s,r r ), (22) which represents the reays 1 through r forwarding for the first source, and then for the second source, and so on For the ith, row the ony nonzero eements are coumns (S i,r 1 ) to (S i,r r ), where the vaues are the SNRs of corresponding S-R channes Furthermore, et g i,sr =g (Si,R 1),g (Si,R 2),,g (Si,R r) (23) represent the row vector of S-R channe SNRs from source S i to a r reays Matrix G SR is then given by g 1,SR 0 r 0 r 0 r g 2,SR 0 r, (24) 0 r 0 r g s,sr where 0 r is a row vector of r zerosthe constraints can be derived using Theorem 1 and matrix G SR as G SR ɛ m min 2γ SD, (25) where ɛ, m min, and γ SD are given by equations (11), (12), and (10), respectivey Further constraints are required on ɛ to obtain a meaningfu resut, namey that 0 ɛ (Si,R j) ɛ (S i,r j), (26) where ɛ (S 1 (Setting i,r j) ɛ (S i,r j) < 1 impies that there are imits on the resources reay R j is prepared to commit to source S i ) Thus, the inear program is described by the inear objective function in (20) and inear constraints in (25)-(26) IV EXPERIMENTAL RESULTS In this section, through simuation, we show the good practica performance of the estimated soution To do this we wi present two sets of simuations, one to prove that the concept wors through a simpe exampe, and another to show the method gives usefu and accurate resuts in a reaistic scenario For a of our simuations we use a (3,6) reguar LDPC code to encode the data, and therefore a sources have the same codeword ength Using EXIT chart anaysis of the (3,6) reguar LDPC code, we have cacuated the convergence threshod for channe LLR mean as m i =253 A Simpe Exampe For our simpe exampe we consider a system with 2 reays and a source as we as a system with 50 reays and a source The system is considered simpe since we assume that a the ins in the system have the same SNR Therefore, the objective function is a minimum when a the ɛ (Si,R j) are the same For our two reay, singe source system, we assume that the normaized SNR on a the ins is 15dB Using our inear programming mode the fraction to be forwarded by the two reays for F is ɛ (F ) =02926 Our 50 reay, singe source mode, is adjusted in such a way to have

5 Error Rate Error Rate FER(1S2R,10K) BER(1S2R,10K) FER(1S2R,100K) BER(1S2R,100K) FER(1S50R,10K) BER(1S50R,10K) ε Fig 2 F frame error rate and bit error rate Avg FER(5S50R,10K) Avg BER(5S50R,10K) Vaue added to non zero εs Fig 3 F and muti-source frame error rate and bit error rate the same threshod vaues Hence, for F the normaized SNR on a the ins is 1032dB Figure 2, shows the resuting BER and FER for F with the frame size of 10 and 100 Since our inear programming mode reies on exit chart anaysis it wi be more accurate as the frame size or number of reays increase This effect is captured in these graphs and it can be seen that for the case of 50 reays the curve drops cose to the threshod ɛ B Reaistic Exampe For our reaistic exampe, we consider a F cooperative scheme with 5 sources and 50 reays where the channe SNRs on each in is randomy seected from a Gaussian distribution with mean 9dB and variance 1dB Aso, we set ɛ (S i,r j) =025 instead of 1, which taes into account the fact that each node may be unwiing to aocate its entire resources to cooperation Since the SNRs on each in are different, the probem becomes highy non-trivia By running our inear programming mode we can cacuate the vaues of ɛ (Si,R j) Figure 3 represents the resut of simuating such a system The x-axis represents the vaue that is added to non-zero ɛ (Si,R j) (since some reays might not be seected to forward any information for a particuar source) that were cacuated using the inear programming mode The y-axis represents the error rate in terms of BER and FER We average the BER and FER of the five source to achieve one singe curve As we can see from the graph both curves drop quicy after the threshod vaue REFERENCES 1 A Sendonaris, E Erip, and B Aazhang, User cooperation diversitypart I: system description, IEEE Trans Commun, vo 51, pp , Nov A Sendonaris, E Erip, and B Aazhang, User cooperation diversity, Part II: Impementation aspects and performance anaysis, IEEE Trans Commun, vo 51, pp , Nov A Nosratinia, T Hunter, and A Hedayat, Cooperative communication in wireess networs, IEEE Commun Mag, vo 42, no 10, pp 68 73, October D Chen and J N Laneman, Moduation and demoduation for cooperative diversity in wireess systems, IEEE Trans on Wireess Commun, vo 5, no 7, pp , Ju J P K Chu and R S Adve, Impementation of co-operative diversity using message-passing in wireess sensor networs, in Proc IEEE Gobecom, St Louis, MO, pp , Dec A W Ecford, J P K Chu, and R S Adve, Low compexity and fractiona coded cooperation for wireess networs, IEEE Trans Wireess Commun, vo 7, no 5, pp , May S ten Brin, Convergence of iterative decoding, Eectron Lett, vo 35, no 10, pp , May M Ardaani and F R Kschischang, A more accurate one-dimensiona anaysis and design of LDPC codes, IEEE Trans Commun, vo 52, no 12, pp , Dec P Razaghi and W Yu, Biayer ow-density parity-chec codes for decode-and-forward in reay channes, IEEE Trans Inform Theory, vo 53, no 10, pp , Oct R Thobaben, On Distributed Codes with Noisy Reays, in Proc Asiomar Conference on Signas, Systems, and Computers, Pacific Grove, CA, USA, Oct J Hu and T M Duman, LDPC codes over ergodic and non-ergodic reay channes, in Proc 44th Annua Aerton Conference on Communications, Contro, and Computing, Monticeo, IL, USA, Sep J P K Chu, A W Ecford, and R S Adve, Optimization for fractiona cooperation in mutipe-source mutipe-reay systems, in Proc IEEE Internationa Conference on Communications, Dresden, Germany, Jun T J Richardson and R Urbane, The capacity of ow-density paritychec codes under message-passing decoding, IEEE Trans Inform Theory, pp , Feb D Divsaar, H Jin, and R J McEiece, Coding theorems for Turboie codes, in Proc 36th Annua Aerton Conference on Communications, Contro, and Computing, Monticeo, IL, USA, pp , Sept T J Richardson, M Shoroahi, and R Urbane, Design of capacityapproaching irreguar ow-density parity-chec codes, IEEE Trans Inform Theory, pp , Feb 2001