1 Resource Aocation via Linear Programming for Fractiona Cooperation Nariman Farsad and Andrew W Ecford Abstract In this etter, resource aocation is considered for arge muti-source, muti-reay networs empoying fractiona cooperation, in which each potentia reay ony aocates a fraction of its resources to reaying Using a Gaussian approximation, it is shown that the optimization can be posed as a inear program, where the reays use a demoduate-and-forward (F) strategy, and where the transmissions are protected ow-density paritychec (LDPC) codes This is usefu since existing optimization schemes for this probem are nonconvex 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 In most wireess networs, a source node is typicay in radio range of mutipe reays Fractiona cooperation [6] is a ow-compexity cooperative scheme for such muti-reay systems, often used in conjunction with F (though it can aso be used with DF) Using this scheme, a arge number of reays forward a sma fraction of the source s transmission bits, so that the reaying cost is spread over a arge number of reays This scheme has good diversity order properties in fading channes A ey chaenge in fractiona cooperation is resource aocation, in which the system determines what fraction must be seected for transmission 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 ight of this chaenge, this etter maes two concusions: Resource aocation in mutipe-source, mutipe-reay fractiona cooperation networs can be posed as an instance of inear programming Our optimization minimizes the number of transmission bits (ie, energy), subject to the constraint that decoding at the destination is successfu Our approach is particuary usefu since Submitted to IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, September 27, 2010; revised Apri 2, 2011 and October 7, 2011; accepted January 20, 2012 Materia in this etter was presented in part at the IEEE Internationa Conference on Communications, Cape Town, South Africa, 2010 This wor was financiay supported a Discovery Grant from the Natura Sciences and Engineering Research Counci The authors are with the Department of Computer Science and Engineering, Yor University, 4700 Keee Street, Toronto, Ontario, Canada M3J 1P3 E-mais: nariman@cseyoruca, aecford@yoruca efficient optimization agorithms are not nown to exist for this probem; an earier approach, based on the union bound, resuted in nonconvex optimization [7] The optima resource aocations found our method are not, in genera, equivaent to seection: we give exampes in which mutipe sources and reays are assigned some fraction that is equa to neither its maximum nor minimum possibe contribution This suggests that there is not aways a best reay for a source using fractiona cooperation with F Our optimization strategy is simiar to the use of inear programming to optimize ow-density parity-chec (LDPC) codes [8]: a Gaussian approximation is used, which mae the objective and constraints a inear Reated wor aso incudes [9] [11], where extrinsic information transfer (EXIT) chart techniques [12] were used for code optimization for reays, not resource aocation A Channe Mode II SYSTEM MODEL Our system uses F aong with fractiona cooperation (the reader is directed to [6] for compete detais) In fractiona cooperation, instead of forwarding the entire source s transmission sequence the reay nodes seect a random portion of the source s transmission bits for reaying Because of this random seection, some of the source s transmission bits wi not be received at the destination The source empoys powerfu error correcting codes to encodes its transmission bits, there ensuring successfu reconstruction of its symbos at the destination despite the missing bits Furthermore, F is used for its ow compexity Consider s sources, r reays, and a singe destination, as in Figure 1 The r reays are shared amongst a s sources Let S i and R j represent the ith source and jth reay, respectivey Each source has a ength-n information sequence to transfer to the destination represented x (Si) = [x (Si) 1,x (Si) 2,,x n (Si) ], where x (Si) {0, 1} Each source encodes its information sequence using an LDPC code Let ρ 1,ρ 2,,ρ s be the code rates at each source Therefore, the codeword ready for transmission at the ith source is represented z (Si) = [z (Si) 1,z (Si) 2,,z m (Si) i ], where m i = n/ρ i is the ength of the codeword We assume a ins are independent additive white Gaussian noise (AWGN) channes, represented with their respective channe signa-to-noise ratio (SNR) (The channes are AWGN since the channe state information is assumed to be nown, but we wi consider cases where the SNR is random and arises from Rayeigh-distributed channe ampitudes) There are s
2 S 1 S 2 Fig 1 S s R 1 R 2 R r Muti-Source, Muti-reay mode source-to-destination (S-D) ins, r reay-to-destination (R-D) ins, and sr source-to-reay (S-R) 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 The S-D ins are therefore given y (Si,D) = φ(z (Si) )+n (Si,D), (1) where 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 γ (Si,D) = 1/(2σ(S 2 i,d) ) The S-R ins are aso given D y (Si,Rj) = φ(z (Si) )+n (Si,Rj), (2) wheres i and R j correspond to the ith source and the jth reay respectivey and n (Si,Rj) is AWGN with variance σ(s 2 i,r j) Therefore, a the S-R ins can be represented sr channe SNRs, γ (Si,R j) = 1/(2σ(S 2 ) i,r j) In F, a reay first demoduates the signa received from a source With sight abuse of the inverse notation, the demoduation function is defined as φ 1 : R {0,1}, where { φ 1 0 if y 0 (y) =, (3) 1 otherwise Thus, the F process can be formuated as 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 transmits to the destination The vector b (Si,Rj) represents the demoduated bit positions seected for transmission 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 The vector b (Si,Rj), has a Hamming weight of m i ǫ (i,j), where ǫ (i,j) is the fraction to be forwarded the jth reay for the ithe source Thus, the reaying fraction ǫ (Si,R j) is defined as mi i=1 ǫ (Si,R j) = b(si,rj) (5) m i This random seection of bits is simiar to puncturing codes, and can be carried out using pseudorandom number generators Finay, transmission to the destination is encoded using a powerfu and capacity-approaching error-correcting code We wi consider two cases in the seque: first, we assume that decoding is successfu at the destination, with an energy cost equa to the in capacity; and second, we use a particuar punctured systematic repeat-accumuate (PSRA) code [6], with possibe decoding faiures If the code is decoded successfuy, the demoduated sequence resuting from the jth reay assisting ith source is avaiabe at the destination as y (Si,Rj,D) F = b (Si,Rj) φ(z (Si,Rj) ), (6) where is eement-wise mutipication of vectors, z (Si,Rj) is given equation (4), and y (Si,Rj,D) represents the resuts of demoduations avaiabe at the destination The eements of can tae three possibe vaues: +1 (representing a demoduated 0 bit), 1 (representing a demoduated 1 bit), and 0 (representing an unseected bit, ain to an erasure) y (Si,Rj,D) F B Fractiona Cooperation and Protoco Here we present a brief and simpe protoco for fractiona cooperation (A simiar protoco, minus the optimization, was impemented in hardware in [15]) 1) Each in is an orthogona channe The channe state information (CSI) of a the ins are nown to the destination, and the destination performs the optimization For each S-R in, the reay estimates the SNR γ (Si,R j) and informs the destination; for the S-D and each R-D in, the destination estimates the SNR The destination performs the optimization and transmits the required fractions ǫ (Si,R j), for each source S i, and the required R-D code rates, to every reay R j (This exchange is infrequent and invoves itte overhead if the environment is static or sow-moving) 2) Each source encodes its information using LDPC codes, and broadcasts the encoded codeword to the r reays, as we as the destination 3) (F) For each source S i, reay R j observes m i bits The reay demoduates symbos from S i, without decoding the underying codewords 4) (Fractiona Cooperation) For each source S i, R j seects m i ǫ (Si,R j) of the demoduated bits for reaying (noting that ǫ (Si,R j) is possi zero) We assume the reays use a pseudorandom number generator to randomy seect the symbos to be forwarded, with seed transmitted to the destination Note that, aside from nowedge of the seed at the destination, there is no coordination of the seection among reays themseves 5) The jth reay re-encodes the m i ǫ (Si,R j) bits using error correcting codes, at a rate specified the destination, and transmits the resuting codeword to the destination The destination then decodes each source s information
3 bits using the received signa from the r reays, and finay the source itsef III LINEAR PROGRAMMING MODEL In this section we present a inear programming mode that minimizes the number of transmission bits of a muti-source muti-reay system, described in the previous section, subject to the constraint of successfu transmission We consider the muti-reay, muti-source system, expained in section II, with r reays and s sources As is typica for anaysis of LDPC decoding, we assume that the sources transmit the a-zero codeword (ie, the a-(+1) channe codeword) For codes with asymptoticay ong boc ength, the convergence of LDPC decoding is soey dependent on the distribution of the channe og-ieihood ratio (LLR) [13], given for a bit x x = ogf Y (y x = 0)/f Y (y x = 1), where y represents a observations of x avaiabe to the receiver Letting γ (Si,D) represent the SNR on the ith S-D in, the vector of channe LLRs for a symbos on the S-D ins is cacuated as (Si,D) = 2y (Si,D) /σ 2 (S i,d) = 4γ (S i,d)y (Si,D), (7) and for the S-R ins as (Si,Rj,D) F = y (Si,Rj,D) F og [ ] 1, (8) where is the probabiity of demoduation error between the ith source and jth reay, given = 1 2 erfc( γ(si,r j)) (9) LLRs of independent observations of a symbo, such as those passed aong different reay ins, are additive 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 (10) As is assumed in the EXIT chart iterature, in (10) we assume that the distribution of (Si) F can be approximated the Gaussian distribution (This assumption is reasonabe if the number of component messages r is sufficienty arge, but high accuracy is not essentia here: anaysis of LDPC decoding under this assumption is nown to be robust even if the true distribution is quite far from Gaussian; cf [8]) Furthermore, a property of Gaussian-distributed LLRs is that the distribution is symmetric [14], with variance equa to twice the mean Thus, we may use the mean of the LLR messages to represent their distribution, and hence to determine whether LDPC coding converges or not: this is the ey observation which aows us to use inear programming Assuming a symmetric Gaussian-distributed LLR, a minimum channe LLR mean is required for successfu decoding, written m min This vaue can be cacuated using EXIT chart anaysis, density evoution, or simuation Let m represent the mean of the channe LLR messages; then the constraint m m min (11) is (approximatey) sufficient to ensure successfu decoding This mean may be different for each source, so we wi write m (i) min to represent the minimum LLR for the ith source The foowing resut gives the cacuation of the input LLR mean to the decoder for F Proposition 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 m (F) i =2γ (Si,D)+ (12) ǫ (Si,R j)(1 2 )og 1, j=0 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 equation (9) andǫ (Si,R j) the fraction seected each reay Proof: Taing the expected vaue of (10), since we have assumed that the transmission bits over R-D ins are decoded successfuy, we have m (F) i = m (Si,D) + r j=0 m(si,rj) For the singe S-D in the channe LLR mean is cacuated as m (Si,D) = 2γ (Si,D), where γ (Si,D) is the channe SNR of the S-D in for the ith source, which gives the first term Now, depends on the crossover probabiity on the (S i,r j ) in, as we as the probabiity of seection ǫ (Si,R j) If a bits are reayed, = (1 2 )og 1 (13) In fractiona cooperation, unseected positions have zero LLR (ie an erasure) Thus, = ǫ (Si,R j)(1 2 )og 1, (14) and the proposition foows We are now ready to set up the inear program In our mode the objective is to minimize the number of transmitted bits (equivaent to minimizing energy) with the constraint of successfu decoding at the destination The objective variabes are ǫ (Si,R j), the forwarding fractions of reays Define the objective vector of ength sr, ǫ, as ǫ = Ț ǫ (1,1) ǫ (1,r) ǫ (2,1) ǫ (2,r) ǫ (s,1) ǫ (s,r) (15) representing the fractions that are seected for transmission each reay for each source The objective function gives the tota energy consumption of the system If F is used the reays, and theith source has a codeword of ength m i to transmit to the destination, the tota energy consists of the energy required for the sources to transmit (proportiona to m i for the ith source), pus the energy committed the reays to forward a fraction of each source (for the ith source and jth reay, proportiona to ǫ (Si,R j)m i /r i,j, where r i,j is the rate of the jth reay s code)
4 Thus, the objective function is given s s ǫ (Si,R f(ǫ) = m i + j)m i (16) r i,j i=0 Since in Section II we assumed that powerfu capacity approaching codes are used over the R-D ins to ensure successfu decoding at the destination, we can repace r i,j with the capacity of the corresponding channe We can aso omit terms not in ǫ since they have no effect on the optimization Therefore, the objective function becomes f(ǫ) = s ǫ (Si,R j)m i C(γ (Rj,D)), (17) where C(γ (Rj,D)) is the channe capacity between jth reay and the destination Note that f(ǫ) is inear in ǫ To derive the constraints for F, we define a variabe g (Si,Rj) as [ g (Si,R j) = (1 2 )og ] 1, (18) where the term on the right side is derived in equation (13), 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 s rows of the matrix represent the sources, and the sr coumns represent the 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 ), 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 g (Si,R 1) to g (Si,R r) respectivey Therefore, using the row vector g i,sr = [g (Si,R 1),g (Si,R 2),,g (Si,R r)], the matrix G SR is then given g 1,SR 0 r 0 r 0 r g 2,SR 0 r G SR =, (19) 0 r 0 r g s,sr [m (1) min m (2) min where 0 r is a row vector of r zeros Let m min = m (s) min ] T represent the vector of minimum LLR means to ensure successfu decoding Aso et γ SD = [γ (S1,D) γ (S2,D) γ (Ss,D)] T represent the vector of S-D channe SNRs Then, from Proposition 1, the successfu decoding constraint may be stated in terms of ǫ as G SR ǫ m min 2γ SD, (20) Further constraints are required on ǫ to obtain a meaningfu resut, namey that 0 ǫ (Si,R j) ǫ (S i,r j), (21) where ǫ (S i,r j) 1 Additiona constraints may aso be added, depending on the appication Given the objective function in (17) and constraints in (20)-(21), the inear program may be stated competey as foows: Error Rate 10 0 10 1 10 2 10 3 10 4 FER (EXIT 252) FER (Density 259) BER (EXIT 252) BER (Density 259) 10 5 0 0005 001 0015 002 0025 003 Vaue added to non zero εs Fig 2 Average frame error rate (FER) and bit error rate (BER) for 5 source, 50 reay system versus the vaue added to the non-zero fractions FER and BER are averaged over a 5 sources A sources use reguar (3,6) LDPC code with codeword ength of 10,000 The normaized SNR of a ins are derived from Rayeigh distributed random variabe Both EXIT chart threshod (m min = 252) and density evoution threshod (m min = 259) are considered Minimize subject to f(ǫ) = s ǫ (Si,R j)m i C(γ (Rj,D)) (22) G SR ǫ m min 2γ SD ; ǫ 0; ǫ ǫ, (23) where 0 is an a-zero vector the same ength as ǫ, and ǫ is the vector of ǫ (S i,r j) corresponding to ǫ IV RESULTS For a of our simuations we use a (3,6) reguar LDPC code at every source as we as a simiar codeword ength To iustrate the sensitivity of our method to an accurate vaue of m min, we present resuts deriving m min from both EXIT charts and density evoution: we have m min = 252 and m min = 259 for EXIT charts and density evoution, respectivey The codeword ength is 10,000 bits, and reported bit error rate (BERs) / frame error rates (FERs) are averaged over a sources For our first set of simuations, we consider a cooperative scheme with 5 sources and 50 reays, and we assume the reays transmission bits are perfecty decoded at the destination We use a channe where every in has an independent Rayeigh-distributed signa strength; thus, the channe SNRs γ on a the ins are independent, identicay χ 2 -distributed random variabes with two degrees of freedom For these resuts, we use f Γ (γ) = 1/01265exp( γ/01265), giving an average SNR of 898dB Aso, we setǫ (S i,r j) = 025 instead of 1 (ie, each reay may be unwiing to forward a the bits it receives), and add an extra constraint as 5 i=1 ǫ (S i,r j) 1 (ie, the reay s maximum reaying commitment does not exceed the equivaent of a singe source) Figure 2 shows the
5 TABLE I VALUES OF ǫ BASED ON EXIT CHART AND DENSITY EVOLUTION THRESHOLDS Reay S 1 (EXIT, Dens) S 2 (EXIT, Dens) S 3 (EXIT, Dens) S 4 (EXIT, Dens) S 5 (EXIT, Dens) R 4 (0 0 ) (0 025 ) (0 0 ) (025 025 ) (00980 025) R 16 (00984 00963 ) (025 025) (00430 01537) (025 025 ) (025 025) R 20 (0 025 ) (0 0 ) (025 025) (0 0 ) (0 00057) R 26 (025 025 ) (025 025 ) (025 01568) (0 00932 ) (025 025) R 32 (0 00229) (025 025 ) (0 0 ) (025 025 ) (0 0 ) R 38 (025 025 ) (0 02447 ) (025 025) (025 025 ) (025 00053) 10 0 mode We see that the BER and FER waterfas start right at the predicted threshod Error Rate 10 1 10 2 10 3 10 4 FER (EXIT 252) FER (Density 259) BER (EXIT 252) BER (Density 259) 10 5 0 001 002 003 004 005 006 007 Vaue added to non zero εs Fig 3 Average frame error rate (FER) and bit error rate (BER) for 2 source, 7 reay system versus the vaue added to the non-zero fractions R-D ins transmission symbos are encoded using PSRA codes with rates seected based on the channe SNR FER and BER are averaged over the 2 sources A sources use reguar (3,6) LDPC code with codeword ength of 10,000 The normaized SNR of a ins are derived from Rayeigh distributed random variabe Both EXIT chart threshod (m min = 252) and density evoution threshod (m min = 259) are considered average BER and FER versus the vaue that is added to nonzero ǫ (Si,R j) that were obtained using the inear programming mode Tabe I iustrates the fractions cacuated our inear programming mode for the 5 source, 50 reay system considered in Figure 2 We observe that not a the reays are forwarding the maximum fraction of 025; for instance, the density evoution resuts for sources 1, 3, and 5 show fractions being spit among mutipe reays Thus, in genera, the optima strategy is not equivaent to reay seection For our second set of simuations, we consider a cooperative scheme with 2 sources and 7 reays Here, we use PSRA codes over reaistic R-D ins, where the code rates are seected based on the density evoution threshod of the PSRA code for a given channe SNR For these resuts, we use f Γ (γ) = exp( γ), giving an average SNR of 0dB Aso, we set ǫ (S i,r j) = 025, and add an extra constraint as 5 i=1 ǫ (S i,r j) 04 (ie, the reay s maximum reaying commitment does not exceed the equivaent of a 04 source) Figure 3 shows the average frame error rate (FER) and bit error rate (BER) of the system versus the vaue that is added to nonzero ǫ (Si,R j) that were obtained using the inear programming REFERENCES [1] A Sendonaris, E Erip, and B Aazhang, User cooperation diversitypart I: system description, IEEE Trans Commun, vo 51, pp 1927 1938, Nov 2003 [2] A Sendonaris, E Erip, and B Aazhang, User cooperation diversity, Part II: Impementation aspects and performance anaysis, IEEE Trans Commun, vo 51, pp 1939 1948, Nov 2003 [3] A Nosratinia, T Hunter, and A Hedayat, Cooperative communication in wireess networs, IEEE Commun Mag, vo 42, no 10, pp 68 73, October 2004 [4] 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 1785 1794, Ju 2006 [5] 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 1167 1171, Dec 2005 [6] 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 1917 1929, May 2008 [7] 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 2009 [8] 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 2106 2114, Dec 2004 [9] 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 3723 3739, Oct 2007 [10] R Thobaben, On Distributed Codes with Noisy Reays, in Proc Asiomar Conference on Signas, Systems, and Computers, Pacific Grove, CA, USA, Oct 2008 [11] 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 2006 [12] S ten Brin, Convergence of iterative decoding, Eectron Lett, vo 35, no 10, pp 806 808, May 1999 [13] T J Richardson and R Urbane, The capacity of ow-density paritychec codes under message-passing decoding, IEEE Trans Inform Theory, pp 599 618, Feb 2001 [14] T J Richardson, M Shoroahi, and R Urbane, Design of capacityapproaching irreguar ow-density parity-chec codes, IEEE Trans Inform Theory, pp 619 637, Feb 2001 [15] A Cace, N Farsad, and A W Ecford, An experimenta study of fractiona cooperation in wireess mesh networs, in Proc 22nd Annua IEEE Symposium on Persona Indoor Mobie Radio Communications (PIMRC), Toronto, ON, pp 990 994, 2011