Error exponents for two-hop Gaussian multiple source-destination relay channels

. RESEARCH PAPER. SCIENCE CHINA Information Sciences February 0 Vol. 55 No. : 38 359 doi: 0.007/s3-0-36- Error exponents for two-hop Gaussian multiple source-destination relay channels DENG PanLiang,3,LIUYuanAn,3,XIEGang,3, WANG XiaoYu,3, TANG BiHua,3 &SUNJianFeng,3 School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing 00876, China; School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, Beijing 00876, China; 3 Key Laboratory of Universal Wireless Communication, Ministry of Education, Beijing 00876, China Received July 6, 00; accepted September, 00; published online July 30, 0 Abstract We investigate the two-hop multiple source-destination relay channels using the rate splitting transmission scheme where each source can split its message into private and public parts. We determine the system error probability via integrated exponent function under amplify-and-forward AF and decode-and-forward DF relay strategies. The most significant difference between AF and DF system error probability evaluations lies in a minimized cut-set bound of transmission rate under the DF strategy. There are many cases among transmission rate intervals for different system parameters e.g. transmit power and it is extremely complex to derive the system error probability for the DF strategy. We obtain a relatively simple result, which unifies various cases by a few expressions. Numerical results show that the system error probability decreases with the increase of the public message. Moreover, in order to draw deep insight into the reliability requirement of each source node in this network, we provide the error exponent region EER for different source nodes to show the trade-off of error probability among source nodes. Keywords error exponent, error exponent region, rate splitting, multiple source-destination relay channel Citation Deng P L, Liu Y A, Xie G, et al. Error exponents for two-hop Gaussian multiple source-destination relay channels. Sci China Inf Sci, 0, 55: 38 359, doi: 0.007/s3-0-36- Introduction In recent years multi-hop networks including relay networks, communication with feedback, discrete memoryless networks and so on have spurred significant interest among communication engineers and information theorists []. Relaying is a powerful technique for improving the throughput and reliability of communication networks. Thus, relay channel has become an important research topic both in academic and industry fields. For example, the IEEE 80. Task Group, the IEEE 80.6 Task Group and the 3rd Generation Partnership Project 3GPP are all working on standardization of the relay scheme [ ]. Actually, the information-theoretic analysis of the relay channel was initiated 0 years ago. In the early 970s, Meulen first introduced the relay channel [5], and then Cover and EI Gamal exploited two coding Corresponding author email: dengpanliangbupt@gmail.com c Science China Press and Springer-Verlag Berlin Heidelberg 0 info.scichina.com www.springerlink.com

Deng P L, et al. Sci China Inf Sci February 0 Vol. 55 No. 39 schemes for the relay channel [6]. Recently, Kramer generalized those two strategies to the multisource and multidestination relay channels and derived lower bounds of the corresponding achievable rates [7]. They all have investigated the capacity region with different strategies. However, practical communication designers are more concerned about the relation among the communication reliability, codeword length and transmission rate. Error exponent can characterize this fundamental trade-off from the informationtheoretic view, which was firstly pointed out in a point-to-point channel by Shannon [8,9]. Gallager s information theory book characterizes this systemically and completely [0]. Some scholars extended it to multiple access channels MAC and broadcast channels BC [ 3]. Etkin presented achievable error exponent for interference channels IFC [] based on the previous single user and MAC results. Hien Quoc Ngo derived the random coding error exponents for amplify-and-forward AF two-way relay channels [5]. Zhang compared two multi-hop strategies for delay in terms of error exponent [6]. In this paper, we study the trade-off between the communication reliability and transmission rate for half-duplex two-hop Gaussian multiple source-destination relay channels. This scenario is a kind of alternative technology for the future wireless network [ ], where multiple sources need to multicast their messages to a relay station, and relay station broadcasts those source messages to their respective destination in wireless environments. In order to simplify the analysis, we consider the smallest multiple source-destination relay channel, which is consisting of two transmitters, two receivers and one relay. We study two relay strategies in this scenario: amplify-and-forward AF, and decode-and-forward DF. Under both relay strategies, we obtain an achievable rate region using the rate splitting transmission scheme, which was formerly proposed to derive the inner bound of interference channels [7]. In the rate splitting approach, each source can split its message into private and public parts, and the public one can be decoded by all destinations. When all of the messages are public, each destination can decode all the messages and the network becomes an interference-free system. In contrast, when all of the messages are private, each destination can only decode its own messages and treat other ones as interference. The relay node in the AF strategy only amplifies the received signals and broadcasts them to the destinations, and the bound of system error probability is equal to the error probability bound in the broadcast phase. We derive the error probability bound via integrated exponent function when relay is working on the AF strategy, where the Gaussian input distribution is considered. The significant difference between the DF and AF strategies is whether relay can decode all the received source messages and encode the messages to destinations. Under the DF strategy, there is a minimized cut-set bound of the transmission rate between relay decoding messages phase and broadcasting messages phase. Because error probability bound is a piecewise function where the expressions of error exponent are different over different transmission rate intervals, there are many cases among transmission rate intervals for different system parameters e.g. transmit power and it is extremely complex to derive the reliability bound for the DF strategy. Fortunately, we obtain a relatively simple result, which unifies various cases by a few expressions. It is worth noting that the system error probability bound is not sufficient enough for us to draw deep insight into the requirement of each source in this network. Therefore, we derive error exponent regions EER for the relay network model. The EER was introduced in [3] for Gaussian broadcast and multiple access channels. In an EER, each source-destination pair can achieve a tradeoff of error exponents for a given achievable rate pair. There are two basic approaches to analyze the error exponent problem. One is the Gallager s approach [0], the other is the Csiszar and Korner s approach. Csiszar and Korner s approach [8] is called the method of types, which belongs to the large deviations theory [9]. It is more intuitive and straightforward in theory. However, the approach of Csiszar and Korner needs to solve the optimal probability distribution, which is difficult to compute in general. This paper extends the Gallager s approach to the two-hop multiple source-destination relay channel to get the computable bound analytically. The rest of the paper is organized as follows. In Section, we describe the system model. In Section 3, two relay strategies and rate splitting scheme are considered. We derive system error probability, system error exponent and EER under smallest multiple source-destination relay channels. In Section, numerical examples are provided. In Section 5, we conclude the paper.

350 Deng P L, et al. Sci China Inf Sci February 0 Vol. 55 No. Destination Source S Destination D D Relay R Destination 3 S D D Source s Destination d Figure Multiple source-destination relay channels. System model Consider a scenario with multi-source, multi-destination and one relay, as is shown in Figure. This model has s source nodes with i {,,...,s}, d destination nodes with j {,,...,d} and one relay node denoted by R. Each source node wishes to send a message W i {,,...,M i } over {,,..., NRi } to the corresponding destination via a relay node, where N is the block length. The relay node works on half-duplex mode, and it needs two time slots to complete information transmission. In the first slot, each source node encodes its message into input signal X i and transmits it to the relay. In the second slot, the relay broadcasts X R to the destination Y j. Before describing system model in detail, we define two error exponent functions. Let E r R denote the random coding exponent with relay node working on half-duplex mode. Define the function as where E r R = max {E 0R ρr}, px, 0 ρ { { +ρ } E 0 R = ln p Xe P rx p Y X +ρ dx} dy, Y X p X is the distribution of input, p Y X is the transition probability. P is the variance of random variable X. The coefficient in is due to the relay working on half-duplex mode. Let E ex R denote the expurgated exponent with relay node working on half-duplex mode, which is defined as E ex R = max {E xr ρr}, 3 px, ρ> where E x R = ρ ln p X p X e r X +X { } σ ρ p Y X p Y X dy dx dx. X X Y Let E R be the integrated exponent based on aforementioned two kinds of exponent functions, which is given by ER =max{e r R,E ex R}. 5 After defining those exponent functions, we describe the system model as the following equations: in the first slot, the system is a MAC. The received signal at the relay is given by Y R = s X i + Z R, 6 i=

Deng P L, et al. Sci China Inf Sci February 0 Vol. 55 No. 35 where Z R is Gaussian random variance with zero mean and variance σr. The transmit power constraint on source node i is N N n= E[ Xi n ] P i. Let p slot e,sys N,R,...,R s denote the smallest system error probability of block decoding in the first slot, where R,...,R s are the rates of source nodes,,...,s, respectively. The corresponding error exponent is defined as Esys slot R,...,R s = Δ lim N logpslot e,sys N. destination is described by In the second slot, the system is a BC. The received signal at each Y j = X R + Z j, 7 where Z j is Gaussian random variance with zero mean and variance σj. The relay broadcasts X R = fy R to the destinations with power constraint N N n= E[ XR n ] P R. Suppose that decoding function of the j th destination is φ j : Yj N { },..., NRj. Let p slot e,j N,R j denote the smallest system error probability of block decoding, where the message is transmitted from the relay to destination j. Through the union bound we can get the system error probability p slot e,sys N,R,...,R d d j= pslot e,j N,R j. The error exponent in this slot is defined as Esys slot R,...,R d = Δ lim N logpslot e,sys N,R,...,R d N. For the whole two-hop network we define system error probability as p total e,sys N,{R,...,R s }, {R,...,R d } Δ = p slot e,sys N,R,...,R s +p slot e,sys N,R,...,R d. 8 And the error exponent for the system is defined as Esys total {R,...,R s }, {R,...,R d } = Δ lim log p total e,sys N,{R,...,R s }, {R,...,R d }. 9 N N 3 Error exponents for two-hop Gaussian two-source, two-destination relay channels In this section, we consider the simplest two-hop Gaussian multiple source-destination relay channels with two sources, two destinations and one relay. Each source can split its message into private and public parts, and each destination can decode all of the public messages but the private information from the corresponding source. Two schemes for relay are investigated, which are amplify-and-forward and decode-and-forward. Under those two relay strategies we aim to derive the system error probability performance upper bounds. Also, we present EER that describes the achievable error exponent region at a given rate pair. 3. Error exponents for the amplify-and-forward strategy In this part, we first analyze error probability bounds for the amplify-and-forward relay strategy. In the first slot, the relay node receives signal from two sources. Y R = X + X + Z R, 0 where X = X + X, X = X + X, X and X denote the public messages of each source, X and X denote the private messages. The transmit power of public message can be denoted by η i P i, where η i is the power allocation parameter over [0, ], and the rest power η i P i is allocated to private message, i {0, }. In the second slot, the relay node amplifies received signal directly, which is given by where α satisfies the relay power constraint α X R = αy R, P R P + P + σr.

35 Deng P L, et al. Sci China Inf Sci February 0 Vol. 55 No. Y R is the received signal in the first slot, and the received signal of two destinations satisfies the following equation: Y j = α X + X + Z R +Z j = α X + X + X + X + Z R +Z j. 3 Each destination can decode all of the public messages and its corresponding private messages. Thus, there is only one kind of interference coming from other source private message. We treat it as noise in the following analysis. Now, we assume that input distribution is Gaussian. If we choose an arbitrary input distribution, it is difficult to get the closed expressions about the system error performances. However, computable error probability bounds can be derived by assuming that the input distribution is Gaussian. Gallager also chose the Gaussian input distribution to investigate the system error performances of the point-to-point channel [0]. It is worth noting that there were many examples based on the Gaussian assumption in communication and signal processing research fields [5, 0]. Then, the input probability density function and the transition probability at the second slot can be written as p αx i = exp X i, πα P i P i p Y j αx i = π α η i c P i c + α σr + exp Y j αx i σ α η i c P i c + α σ j R + σ j where, i c is i s complement, j {0, }., 5 Theorem. For the two-hop Gaussian two-source, two-destination relay channels with the AF strategy using the Gaussian input distribution, the system error probability is given by p total AF e,sys E total AF sys exp { N min { E AF R,E AF R }}, 6 min { E AF R,E AF where, i = j {, }, leta = + θr,i + +θ r,i, B = R }, 7 e Rj θr,j e erj Rj θ r,j, θ r,i = θ x,i = α P i α η i c P i c + α σ R + σ j, 8 E AF R =max { Er, AF,Eex,} AF, 9 E AF R =max { Er, AF ex,},eaf. 0 Integrated exponent Ej AF R j is based on the value of random coding exponent and expurgated exponent. For different transmission rate internals, random coding exponent expression satisfies the following two cases: when ln A R j ln + θ r,i E AF r,j R j = +e R j θr,i e Rj B e Rj { + ln e Rj B + θ r,i e R j } ; when R j < ln A Er,j AF R j= A + θ r,i R j + ln A θ r,i + ln A, and expurgated exponent expression is expressed by when R j < ln + +θ r,i Ex,j AF R j = θ x,i e Rj. 3

Deng P L, et al. Sci China Inf Sci February 0 Vol. 55 No. 353 Proof. For the AF scheme, the relay only amplifies the signal so that there is no error in the first slot, and then we have p slot e,sys N,R,R = 0. So the total system error probability is equal to the second slot error probability. In the second slot, the system is a BC. For BC, we have the union bound p total AF e,sys From the definition of error probability, we can get p total AF e,sys p slot AF e, + p slot AF e,. exp { N min { E AF R,E AF R }}. Consequently, we only need to derive the expression of integrated exponent function, which is related with random coding function and expurgated exponent function. Thus, the proof of primal problem is completed. First, we derive random coding exponent function. Substituting distributed function and 5 into to calculate the maximization value through deriving with respect to ρ, we can obtain the relationship among transmission rate, β and exponent function R j = ln β, β = + θr,i +ρ + + E AF r,j R j= + β θ r,i + + ln β θr,i ρθ +ρ r,i +ρ β θ r,j α β β θ r,i β β θr,i 5, 6 α β β θ r,i + θ r,i β, where, θ r,i and θ r,j satisfy 8. From 6, we can see that β increases with the decrease of ρ. Sowhenρ =0,β is maximized. When ρ =,β is minimizing, then we have β min = + θ r,i + so, from 5, the rate can satisfy the interval ln + θ r,i + 7 β max =+θ r,i, 8 +θ r,i +θ r,i, 9 R j ln + θ r,i, 30 for the rate is less than the left-hand side of 30, we have to choose ρ =toget Er,j AF R j = ln β min + β min + θ r,i + { ln β min θ } r,i R j. 3 By substituting 5 into 7, then is satisfied in Theorem. Similarly, by substituting 9 into 3, is satisfied. Also, we can get the expurgated exponent function as 9 and 0 in the same manner as the random coding function. The derivation is omitted here. Remark. According to 8, when η i c = we mean that the network is an interference-free system. Because all of the messages are public messages, each destination can decode them, although all the destinations only concern their intended messages. When η i c = 0, each destination receives the interference from other sources, meaning that there is only private message in each transmission message.

35 Deng P L, et al. Sci China Inf Sci February 0 Vol. 55 No. Remark. Theorem provides a theoretical framework for a system reliability upper bound function through integrated exponent. The system error probability can be arbitrarily small with the increase of codeword length. It is worth noting that the system error probability is not sufficient enough for us to draw deep insight into the reliability requirement of each user in the multi-terminal network. One approach is to investigate the EER. For any fixed rate pair, there is an EER, describing the error exponent tradeoff between different terminals. In the following theorem, we give the EER in error bound for two-hop Gaussian two-source, twodestination relay channels, which can provide the trade-off of error exponent among terminals by considering the different rate pairs. Theorem. For the two-hop Gaussian two-source, two-destination relay channels with the AF strategy using the Gaussian input distribution, an inner bound for error exponent region is given by EER AF R,R = { E AF,E AF : E AF R E AF R,E AF R E AF R }, 3 where, E AF R ande AF R satisfy 9 and 0, respectively. Proof. Based on the random coding and expurgated argument, we can show the error probabilities for each source-destination pair p total AF e, and p total AF e, using single-user decoding that satisfies p total AF e,k exp N { max { E AF r,k,e AF ex,k}}, 33 where, k {, }, Er, AF, EAF ex,, EAF r,, EAF ex, have been derived in Theorem. Therefore, the achievable error exponents using single-user decoding can be obtained by E AF R = max { } Er, AF,EAF ex, and E AF R =max { Er, ex,} AF,EAF, respectively. Since source-destination pair chooses single-user decoding, the maximum of the error exponents is achievable. Remark 3. Based on Theorem, we can design the two source-destination relay networks such that each source-destination pair can achieve a tradeoff of error exponent in an EER. And we only consider the single-user decoding schemes in this paper, actually any other encoding and decoding scheme can be investigated such as superposition encoding, single-code encoding, joint ML decoding and so on. Also, this result can be straightforwardly extended to general multiple source-destination relay networks. 3. Error exponents for the decode-and-forward strategy In this part, we consider decode-and-forward strategy for the two sources, two destinations with one relay case. In the first slot, the relay node receives signal and decodes it from two sources. In the second slot, the relay node broadcasts the decoding signal, and each destination decodes its own private message and public message, which is given by Y j = X + X + Z j = ωp R η U + ωp R η U + ωp R η U + ωp R η U + Z j, 3 where, ωp R is the transmission power of X, ω is the power allocation parameter of relay, 0 ω and U, U, U, U are independent zero-mean, unit-variance Gaussian random variables, U, U denote the private messages and U, U denote the public messages. η i is the power allocation parameter described in AF schemes. Also, we assume that input distribution is Gaussian, and the input probability density function at the first slot is followed by p Xi = exp X i π Pi P. 35 i The transition probability at the second slot is p Y j Xi = π η i c P exp Yj X i i c + σj η i c P i c + σj, 36

Deng P L, et al. Sci China Inf Sci February 0 Vol. 55 No. 355 where, P = ωp R, P = ω P R. Theorem 3. For the two-hop Gaussian two source-destination relay channels with DF strategy of using the Gaussian input distribution, the system error probability is given by p total DF e,sys { 5exp N min { E DF slot R,E DF slot R, E DF slot, R + R,E DF slot R,E DF slot R }}, 37 E total DF sys where, i = j {, }, ϕ r,i = Pi σ R min { E DF slot R,E DF slot R,E, DF slot R + R, E DF slot R,E DF slot R }, 38 D R i = D R j = E DF slot j E DF slot j, ϕ r, = P+P, ϕ σr r,i j = P i η i c P,andlet i c +σj e Ri ϕ r,i +eri e Ri ϕ r,i, e Rj ϕ r,i j +erj e Rj ϕ r,i j, R j =max { E DF slot r,j R j,e DF slot ex,j R j }, 39 R j =max { E DF slot r,j R j,e DF slot ex,j R j }, 0 E, DF slot R + R =max { Er, DF slot R + R,Er, DF slot R + R }. Similar to the AF scheme, the integrated exponent of the DF scheme is composed of both random coding exponent and expurgated exponent in different slots. There is a cut-set bound from the network system rate flow view, where each source-destination pair can achieve minimized transmission rate during the two slots. In the first slot, relay node receives the message from each source and decodes them using the joint maximum likelihood ML decoding. In the second slot, the relay node allocates its power ωp R to transmit message and the destination decodes the message using the single-user decoding, where destination can decode all of the public messages and corresponding private messages. The expressions of random coding exponent are divided into those parts: when ln + ϕr,i + + ϕ r,i Ri min { ln + ϕ r,i, ln + ϕ r,i j }, E DF slot r,i R i = D R { i +e R i ϕr,i + + e Ri e Ri ln e Ri + ϕ r,i e R i } D R i ; when R i < min { ln + ϕ r,i j, ln + } ϕr,i + + ϕ r,i, Er,i DF slot R i = ϕ r,i + ϕ r,i + when ln big + ϕr, R + R min + E DF slot r, R,R = + ln + ϕ r,i + ln + + ϕ r, min { ln + ϕ r,, ln + ϕ r, } +min { ln + ϕ r,, ln + ϕ r, } + ϕ r,i + ϕ r,i +e R +R ϕ r, D R + R e R+R e R+R R i + ; 3, ln + ϕ r,,

356 Deng P L, et al. Sci China Inf Sci February 0 Vol. 55 No. { + ln e R+R + ϕ r, e R +R } D R + R ; {{ { min when R + R < min ln + ϕ r,, ln + ϕ r, } } +min { ln + ϕ r,, ln + ϕ r, } Er, DF slot R,R = ϕ r, + ϕ r, + ln + + ln + ϕ r, + ϕ r,, ln + ϕr, + + + ϕ r, + ϕ r, }, R R + ; 5 when ln + ϕr,i j + + ϕ r,i j Rj min { ln + ϕ r,i, ln + ϕ r,i j }, { +e E DF slot R j ϕr,j r,j R j = D R j + e Rj e Rj ln e Rj + ϕ r,j e R j } D R j ; 6 when R j < min { ln + ϕ r,i, ln + } ϕr,i j + + ϕ r,i j, Er,j DF slot R j = ϕ r,i j + ln + ϕ r,i j + + ϕ r,i j + ln + + ϕ r,i j + ϕ r,i j R j +. 7 According to the expurgated exponent, the expressions can be rewritten as when R i < min { ln + ϕ r,i j, ln + } + ϕ r,i, E DF slot x,i R i = ϕ r,i e Ri ; 8 when R + R < min when R j < min { ln + {{ min { ln + ϕ r,, ln + ϕ r, } +min { ln + ϕ r,, ln + ϕ r, } Ex, DF slot R,R = ϕ r, + ϕ r,i j, ln + ϕ r,i }, E DF slot x,j R j = ϕ r,i j }, ln + } + ϕ r,, e R+R ; 9 e Rj. 50 Proof. For the DF scheme, the total system error probability is equal to the sum of the first slot error probability and the second slot error probability. Then, we can convert the original problem into deriving the error probability of two slots. According to those two slots error probability, the proof is the same as that given in Theorem and is omitted here. Remark. The biggest difference between DF strategy and AF strategy is that the relay node decodes the message from each source. From the max-flow min-cut theorem [], the achievable rate of two-hop Gaussian two source-destination relay channels with DF strategy is the minimized rate between MAC slot and BC slot. By setting up different transmit power among each source and relay, the error exponents are different in Theorem 3. The uniform result in Theorem 3 is more complicated than the relay node using the AF scheme.

Deng P L, et al. Sci China Inf Sci February 0 Vol. 55 No. 357 Remark 5. According to Theorems and 3, the result only shows the two sources and two destinations scenario. However, we can extend them to general multiple source-destination scenarios in a straightforward way. Also, those results can be extended to fading channels such as Nakagami-m fading channels, which has been investigated in different scenarios [,3]. From Remark, we know that the system error probability is not sufficient to characterize the reliability requirement of each user in the multi-terminal network. We show the EER in error bound for two-hop Gaussian two-source, two-destination relay channels using DF strategy, which can provide the tradeoff of error exponent among terminals by considering the different rate pairs. Theorem. For the two-hop Gaussian two-source, two-destination relay channel with DF strategy of using the Gaussian input distribution, the system inner bound for EER is given by E DF,E DF : E DF E DF R EER DF R,R = E DF E DF R, 5 where, E DF R = min { E DF slot R,E, DF slot R + R, E DF slot R }, E, DF slot R + R, E DF R = min { E DF slot R,E, DF slot R + R,E DF slot R },E DF slot j R j, E DF slot j R j have been defined by 39, respectively. Proof. The proof follows the way of joint ML decoding in the first slot and single-user decoding in the second slot. In the following proof, we define k {, }. In the first slot, where the system is MAC, the total error probabilities for sources and using joint ML decoding can be union bounded by p total DF slot e,k p DF slot e,k + p DF slot e, exp NE DF slot k R k +exp NE, DF slot R +R. 5 In the second slot, where the system is BC, the error probabilities for Destinations and using single-user decoding can be upper-bounded by p total DF slot e,k = p DF slot e,k exp NE DF slot k R k. 53 Thus, the whole error probabilities for source-destination and source-destination can be upperbounded by p total DF e,k 3exp { N min { E DF slot k R k,e, DF slot R + R,E DF slot k R k }}, 5 Then, the achievable error exponents using joint ML decoding in the first slot and single-user decoding in the second slot are E DF k R k =min { E DF slot k R k,e, DF slot R + R,E k R k }, 55 Since source-destination pair chooses joint ML decoding in the first slot and single-user decoding in the second slot, the maximum of the error exponents is achievable. Numerical results In this section, we show the preceding results via following numerical examples. Figure gives system error exponent versus the transmission rate of source-destination pair for R = 0. under the AF relay scheme when public message allocation parameters η = η =0, 0.3, 0.5, 0.7,. Also, Figure 3 is obtained under the DF relay strategies for R =0.0. It can be seen from the Figures and 3 that the system error exponent increases with the number of public messages at the transmission rate of the each source-destination pair below the capacity. It indicates that the system error probability decreases with increasing public message. The reason is that all source-destination pairs can decode the public message freely, which means that increasing the public message is equivalent to decreasing the interference from other source-destination pairs and improving the SNR.

358 Deng P L, et al. Sci China Inf Sci February 0 Vol. 55 No. System error exponent...0 0.8 0.6 0. 0. η =η =0 η =η =0.3 η =η =0.5 η =η =0.7 η =η = 0 0 0. 0. 0.3 0. 0.5 0.6 0.7 R nats/s/hz for fixed R =0. nats/s/hz System error exponent.0.8.6...0 0.8 0.6 0. 0. η =η =0.3 η =η =0 η =η =0.5 η =η =0.7 η =η = 0 0 0. 0. 0.3 0. 0.5 0.6 0.7 0.8 R nats/s/hz for fixed R =0. nats/s/hz Figure System error exponent versus R using the AF relay strategy for P = 70, P = 50, P R =30and σ = σ = σ R = when fixed rate R =0., η = η = 0, 0.3, 0.5, 0.7,. Figure 3 System error exponent versus R using the DF relay strategy for P = 70, P = 50, P R = 30, ω =0.6 and σ = σ = σ R = when fixed rate R =0.0, η = η =0, 0.3, 0.5, 0.7,. E 0.8 0.6 0. 0. 0.0 0.08 0.06 0.0 0.0 0 0 0.0 0.0 0.06 0.08 0.0 0. 0. Figure Error exponent region using the AF relay strategy for P = 70, P = 50, P R = 30 and σ = σ = σ R = when fixed rate pair is equal to 0.5, 0. dashed, 0.3, 0.05 dash-dotted, 0.9, 0.08 solid, respectively. E E 0.35 0.30 0.5 0.0 0.5 0.0 0.05 0 0 0.05 0.0 0.5 0.0 0.5 0.30 0.35 0.0 0.5 Figure 5 Error exponent region using the DF relay strategy for P = 70, P = 50, P R = 30, ω = 0.6 and σ = σ = σ R = when fixed rate pair is equal to 0.5, 0. dashed, 0.3, 0.05dash-dotted, 0.9, 0.08 solid, respectively. E Figures and 5 describe achievable EER under the AF and DF relay strategies, respectively. In both figures, the dashed curve is the boundary of the achievable EER for given rate pair 0.5, 0., the dashdotted curve is the boundary of the achievable EER for given rate pair 0.3, 0.05, and solid curve is the boundary of the achievable EER for given rate pair 0.9, 0.08. The main goals of these two figures are to show that the boundaries of EER are different for different fixed transmission rate pairs in multiple source-destination relay channels, and different source-destination pairs can be compromised in an EER. 5 Conclusions In this paper, we obtain a system error probability via integrated exponent function under AF and DF relay strategies. Also, a rate splitting transmission scheme is applied to derive this error probability where each source can split its message into private and public parts. Numerical results show that the system error probability decreases with the increase of public message. The most significant difference between

Deng P L, et al. Sci China Inf Sci February 0 Vol. 55 No. 359 AF and DF system error probability evaluation lies in a minimized cut-set bound of transmission rate under the DF strategy. So, there are many cases among transmission rate intervals for different system parameters e.g. transmit power and it is extremely complex to derive the system error probability for the DF strategy. We obtain a relatively simple result, which unifies various cases by a few of expressions. Moreover, in order to draw deep insight into the reliability requirement of each source in this network, we study the EER for different source nodes to show the trade-off of error probability among source nodes. Acknowledgements This work was supported in part by Sino-Swedish IMT-Advanced Cooperation Project Grant No. 008DFA780, Canada-China Scientific and Technological Cooperation Grant No. 00DFA30, National Natural Science Foundation of China Grant Nos. 6080033, 6087390, National High Technology Research and Development Program of China Grant No. 008AA0Z and Fundamental Research Funds for the Central Universities Grant Nos. 009RC0308, G7009. We are grateful to anonymous reviewers and M.S. Hong Chen in Huawei Technologies Company for helpful comment and writing improvement. References Gamal A E, Young H K. Lecture notes on network information theory. http://arxiv.org/abs/00.30. 00 IEEE. IEEE Standard 80.: Wireless Local Area Networks. http://grouper.ieee.org/groups/80//. 006 3 IEEE. IEEE Standard 80.6j/D7: Air Interface for Fixed and Mobile Broadband Wireless Access Systems, Multihop Relay Specification. 007 3GPP. 3GPP TR36.93, V8.0.0: Requirements for further advancements for E-UTRA LTE-Advanced, 008 5 Van der Meulen E C. Three-terminal communication channels. Adv Appl Prob, 97, 3: 0 5 6 Cover T M, Gamal A E. Capacity theorems for the relay channel. IEEE Trans Inf Theory, 979, 5: 57 58 7 Kramer G, Gastpar M, Gupta P. Cooperative strategies and capacity theorems for relay networks. IEEE Trans Inf Theory, 005, 5: 3037 3063 8 Shannon C E. Probability of error for optimal codes in a Gaussian channel. J Bell Syst Tech, 959, 38: 6 656 9 Shannon C E, Gallager R G, Berlekamp E R. Lower bounds to error probability for coding on discrete memoryless channels part I. Inf Contr, 967, 0: 65 03 0 Gallager R G. Information Theory and Reliable Communication. New York: Wiley, 968 Gallager R G. A perspective on multiaccess channels. IEEE Trans Inf Theory, 985, 3: Korner J, Sgarro A. Universal attainable error exponents for broadcast channels with degraded message sets. IEEE Trans Inf Theory, 980, 6: 670 679 3 Weng L, Pradhan S S, Anastasopoulos A. Error exponent regions for Gaussian broadcast and multiple-access channels. IEEE Trans Inf Theory, 008, 5: 99 9 Etkin R, Merhav N, Ordentlich E. Error exponents of optimum decoding for the interference channel. In: Proceedings of IEEE International Symposium on Information Theory, Toronto, 008. 53 57 5 Ngo H Q, Quek T Q S, Shin H D. Amplify-and-forward two-way relay channels: Error exponents. In: Proceedings of IEEE International Symposium on Information Theory, Seoul, 009. 08 03 6 Zhang W Y, Mitra U. Multihopping strategies: An error-exponent comparison. In: Proceedings of IEEE International Symposium on Information Theory, Nice, 007. 5 7 Han T S, Kobayashi K. A new achievable rate region for the interference channel. IEEE Trans Inf Theory, 98, 7: 9 60 8 Csiszar I, Korner J. Information Theory: Coding Theorems for Discrete Memoryless Systems. Budapest: Akademiai Kiado, 98 9 Sanov I N. On the probability of large deviations of random variables. Sel Trans Math Statist, 96, : 3 0 Wang J Z, Milstein L B. CDMA overlay situations for microcellular mobile communications. IEEE Trans Commun, 995, 3: 603 6 Frod L R, Fulkerson D R. Maximal flow through a network. J Canadian Math, 956, 8: 399 0 Wang J Z, Chen J. Performance of wideband CDMA systems with complex spreading and imperfect channel estimation. IEEE J Sel Areas Commun, 00, 9: 5 63 3 Ngo H Q, Quek T Q S, Shin H D. Random coding error exponent for dual-hop Nakagami-m fading channels with amplify-and-forward relaying. IEEE Commun Lett, 009, 3: 83 85