Non-Orthogonal Multiple Access combined with Random Linear Network Coded Cooperation
aa r X i v : . [ c s . I T ] J un Non-Orthogonal Multiple Access combined withRandom Linear Network Coded Cooperation
Amjad Saeed Khan,
Student Member, IEEE, and Ioannis Chatzigeorgiou,
Senior Member, IEEE
Abstract —This letter considers two groups of source nodes.Each group transmits packets to its own designated destinationnode over single-hop links and via a cluster of relay nodesshared by both groups. In an effort to boost reliability withoutsacrificing throughput, a scheme is proposed, whereby packetsat the relay nodes are combined using two methods; packetsdelivered by different groups are mixed using non-orthogonalmultiple access principles, while packets originating from thesame group are mixed using random linear network coding. Ananalytical framework that characterizes the performance of theproposed scheme is developed, compared to simulation resultsand benchmarked against a counterpart scheme that is based onorthogonal multiple access.
Index Terms —Network coding, non-orthogonal multiple access,sparse random matrices, decoding probability, throughput.
I. I
NTRODUCTION
Random Linear Network Coding (RLNC) is a scheme thatallows an intermediate node to combine and forward the dataof multiple users in a single transmission, and can effectivelyimprove network capacity [1]. RLNC has the inherent capabil-ity to achieve spatial diversity. For example, it has been shownin [2] that network coding can improve the diversity gain ofnetworks that either contain distributed antenna systems orsupport cooperative relaying. Furthermore, RLNC can improveboth the throughput [1] and the latency in a network [3] byreducing the number of distinct transmissions.The benefits of network coding have made it an attractivesolution for challenges encountered in existing and futurecommunication systems. For instance, it has been shown in [4]that by modifying the IEEE 802.11g frame structure, networkcoding combined with Orthogonal Frequency Division Mul-tiplexing (OFDM) can significantly improve throughput. Theimportance of network-coded cooperation has been demon-strated in [5] and implemented in [6] with Orthogonal Fre-quency Multiple Access (OFDMA). Recently, Non-OrthogonalMultiple Access (NOMA) has been recognised as a promisingmultiple access technique for 5G mobile networks [7], [8].It has been shown in [9], [10] that combining NOMA withOFDM can improve the spectral efficiency and accommodatemore users than the conventional OFDMA-based systems.Moreover, the usefulness of RLNC for downlink NOMA-based transmissions has been studied in [11].This letter considers network-coded cooperation in aNOMA-based scenario with two groups of source nodes. Eachgroup communicates with a different destination node viamultiple relay nodes. To the best of our knowledge, this work
A. S. Khan and I. Chatzigeorgiou are with the School of Computing andCommunications, Lancaster University, Lancaster, United Kingdom (e-mail: { a.khan9, i.chatzigeorgiou } @lancaster.ac.uk). represents the first attempt to characterise the performanceof NOMA-based RLNC cooperation. The main contributionsof our work can be summarized as follows: (i) we proposea framework which integrates the benefits of NOMA-basedmultiplexing and RLNC-based cooperative relaying; (ii) usingthe fundamentals of RLNC and uplink/downlink NOMA, wederive closed-form expressions for the network performance,in terms of the decoding probability at each node, and thesystem throughput; (iii) we validate the accuracy of the derivedexpressions through simulations and we investigate the impactof the system parameters on the network performance andthroughput. II. S YSTEM M ODEL
Consider a network with two source groups, two des-tination nodes and N commonly shared relay nodes r , r , . . . , r N . Each source group G k contains K source nodes s ( k )1 , s ( k )2 , . . . , s ( k ) K for k = 1 , . The packets transmitted bysource nodes in G k are meant to be received by destination d k ,either directly or via relay nodes. The acceptable transmissionrate for G is R ∗ and for G is R ∗ . Without loss of generality,we assume that all source nodes in G require a comparativelyhigh quality of service with R ∗ < R ∗ . In practice, G couldbe a group of devices (e.g., sensors) associated to high riskapplications that need to be connected quickly with low datarate, and G could be a group of devices related to lowrisk applications that can afford opportunistic connectivity. Allnodes operate in half duplex mode. The links connecting thenodes are modeled as quasi-static Rayleigh fading channels.The channel gain between nodes i and j is represented by h ij ,which is a zero-mean circularly symmetric complex Gaussianrandom variable with variance σ ij .Before the communication process is initiated, source nodesfrom the two groups are paired according to their indices,such that s (1) i in group G is paired with s (2) i in G . Onlypaired nodes are allowed to transmit simultaneously over thesame frequency band. The simultaneous transmission of twonodes exploits the principle of superposition coding, whichis a key component of NOMA. Node pairing in NOMAhas been recently proposed for 3GPP Long Term EvolutionAdvanced (LTE-A) [12]. Source nodes in different pairstransmit over orthogonal frequency bands, and therefore canbe recovered independently. This approach is also known asOFDM-NOMA [9] but, for the sake of brevity, we shallsimply refer it to as NOMA. We consider the worst casescenario, in which both source groups contain an equal (i.e., K ) number of source nodes, such that relay nodes alwaysreceive superimposed signals. The proposed communicationprocess is divided into the broadcast phase and the relay phase. During the broadcast phase , each source node broadcastsa packet in the form of an information-bearing signal to therelay and destination nodes. The signals transmitted by the i th source pair (cid:0) s (1) i , s (2) i (cid:1) , and received by a relay node r j anddestination nodes d and d , are respectively given by z i r j = p α P s h s (1) i r j ˜ x i + p α P s h s (2) i r j ˜ y i + w i r j ,z i d = p α P s h s (1) i d ˜ x i + w i d ,z i d = p α P s h s (2) i d ˜ y i + w i d , where P s is the total transmission power by the source pair, α and α are the fractions of P s transmitted by s (1) i and s (2) i ,respectively, with α + α = 1 , and { ˜ x i , ˜ y i } represent themodulated signals of data packets { x i , y i } . The additive whiteGaussian noise components at the relay and destination nodesare represented by w i r j and w i d k , respectively. All relay nodesemploy Successive Interference Cancellation (SIC) to recoverthe transmitted signals, and then disjointly demodulate andstore the correctly received packets.During the relay phase , a relay node r j employs RLNC onthe successfully received and stored data packets of groups G and G , and generates coded packets m (1) j and m (2) j , respec-tively, given by m (1) j = P Ki =1 c (1) i,j x i and m (2) j = P Ki =1 c (2) i,j y i ,where, c ( k ) i,j represents the coding coefficients over a finitefield F q of size q . The value of a coefficient is zero if areceived packet contains irrecoverable errors; otherwise, thevalue of that coefficient is selected uniformly at random fromthe remaining q − elements of F q . The probability massfunction of c ( k ) i,j is given as Pr (cid:0) c ( k ) i,j = t (cid:1) = ǫ s ( k ) i r j , for t = 0 , − ǫ s ( k ) i r j q − , for t ∈ F q \ { } , (1)where ≤ ǫ s ( k ) i r j ≤ is the outage probability of thelink connecting the source node s ( k ) i with the relay node r j .The closed form expression of ǫ s ( k ) i r j will be presented inSection III. This type of RLNC at the relay nodes is known as sparse RLNC, where the sparsity level is determined by theoutage probability ǫ s ( k ) i r j [13], [14].Each relay node, instead of transmitting two separatenetwork-coded signals (one for each destination), generatesa signal that is the superposition of the two network-codedsignals and broadcasts it to both destinations. Relay transmis-sions are orthogonal, either in time or in frequency. The su-perimposed signal transmitted by relay r j can be expressed as ( √ P r β ˜ m (1) j + √ P r β ˜ m (2) j ) , where P r is the total transmittedpower, and β , β denote the power allocation coefficients,such that β + β = 1 with β > β in order to satisfy thequality of service requirement [15]. Thus, the received signalat destination d k is given as ˆ z j d k = h r j d k ( p P r β ˜ m (1) j + p P r β ˜ m (2) j ) + ˆ w j d k where ˆ w j d k is the Gaussian noise component. Each destinationnode employs SIC in order to separate the superimposedsignals and retrieve the relevant coded packets. Destination d k will recover the data packets of source group G k if it collects K linearly independent packets directly from that source groupand via the relay nodes.III. A CHIEVABLE RATE AND LINK OUTAGE PROBABILITY
This section describes the achievable transmission rate ofsource-to-destination, source-to-relay and relay-to-destinationlinks. An outage occurs when the achievable rate is lessthan the target rate of transmission. Therefore, the outageprobability of each link can be expressed in terms of thecorresponding achievable rate and the target rate.Let us first consider the broadcast phase, during whichsignals arrive at each destination node directly from therespective source group. The achievable rate of the s ( k ) i d k link,which originates from group G k , can be obtained as R s ( k ) i d k = B s log (cid:0) P s α k | h s ( k ) i d k | B s N (cid:1) (2)where k ∈ { , } , i ∈ { , , . . . , K } , N represents the noisepower and B s denotes the bandwidth of the frequency bandallocated to each source pair for simultaneous transmissions,as discussed in Section II. The outage probability of the s ( k ) i d k link can be derived if we combine expression (2) withthe cumulative distribution function of Rayleigh fading [16,eq. (7.6)], which gives ǫ s ( k ) i d k = Pr( R s ( k ) i d k ≤ R ∗ k ) = 1 − exp( − τ k ρ s α k σ ( k ) i d k ) where ρ s = P s B s N and τ k = 2 R ∗ k /B s − . The achievable rateof the link between one of the nodes of a source pair and arelay node r j depends on the channel conditions of both linksthat connect the nodes of the source pair with r j . For example,assume that α | h s (1) i r j | > α | h s (2) i r j | . In that case, SIC at therelay node r j will first recover the signal of the node from G and treat the other signal as interference. Thus, the achievablerate of a link between s ( k ) i and r j can be expressed as [17] R s (1) i r j = B s log (cid:0) α | h s (1) i r j | α | h s (2) i r j | + 1 /ρ s (cid:1) (3) R s (2) i r j = B s log (cid:0) ρ s α | h s (2) i r j | (cid:1) . (4)The outage probability of a link between s ( k ) i and r j can beobtained as ǫ s (1) i r j = Pr( R s ( k ) i r j < R ∗ k ) , thus ǫ s (1) i r j = 1 − α σ (1) i r j τ α σ (2) i r j + α σ (1) i r j exp( − τ ρ s α σ (1) i r j ) ǫ s (2) i r j = 1 − Pr (cid:2) ( R s (1) i r j > R ∗ ) ∩ ( R s (2) i r j > R ∗ ) (cid:3) = 1 − α σ (1) i r j τ α σ (2) i r j + α σ (1) i r j exp( − τ ( τ + 1) ρ s α σ (1) i r j − τ ρ s α σ (2) i r j ) . During the relay phase, the destination node d can onlysuccessfully recover the coded signals corresponding to sourcegroup G , when R r j d > R ∗ provided that R r j d > R ∗ . On the other hand, the destination d can recover the coded signalsof G , when R r j d > R ∗ . The achievable rates are given as R r j d = B s log (cid:0) β | h r j d | β | h r j d | + 1 /ρ r (cid:1) (5) R r j d = B s log (cid:0) ρ r β | h r j d | (cid:1) (6)where B s is the bandwidth allocated to each pair of relays,and ρ r = P r B s N . It is assumed that β ≥ τ β , otherwise theoutage probability is always one [7]. The outage probabilityof links r j d and r j d can be respectively obtained as ǫ r j d = Pr( β | h r j d | β | h r j d | + 1 /ρ r ≤ τ )= 1 − exp( − τ ( ρ r β − τ ρ r β ) σ j d ) ,ǫ r j d = 1 − Pr( β | h r j d | β | h r j d | + 1 /ρ r > τ , ρ r β | h r j d | > τ )= 1 − exp( − ρ r σ j d max( τ β − τ β , τ β )) . OMA-based Benchmark scheme
In this letter, we consider conventional OFDMA as thebenchmark Orthogonal Multiple Access (OMA) scheme. Ac-cording to this scheme, all nodes s ( k ) i and r j transmit overorthogonal frequency bands. As a result, likewise (2), theachievable rates of source-to-relay and source-to-destinationlinks during the broadcast phase, and the relay-to-destinationlinks during the relay phase can be respectively obtained as R s ( k ) i u = B s P s α k | h s ( k ) i u | . B s N ) ,R r j d k = B s P r β k | h r j d k | . B s N ) where u ∈ { r j , d k } . The factor / is due to the fact that,unlike NOMA, each sub-band is now further split between twotransmitting nodes. Note that, using the achievable rates, wecan derive the outage probabilities. These results can be furtherextended to RLNC-based analysis, which will be presented inthe next section, and can be used as benchmarks against theproposed NOMA-based scheme.In the remainder of the letter, we assume that links con-necting co-located transmitting nodes with receiving nodes arestatistically similar, hence ǫ r j d k = ǫ rd k , ǫ s ( k ) i r j = ǫ s ( k ) r and ǫ s ( k ) i d k = ǫ s ( k ) d k for all valid values of i , j and k .IV. D ECODING PROBABILITY AND A NALYSIS
This section analyses the system performance in terms ofthe probability of a destination node successfully recoveringthe packets of all nodes in the corresponding source group.Furthermore, the system throughput is derived as a functionof the number of packet transmissions.The destination node d k can recover the packets of allsource nodes in group G k if and only if it collects packets thatyield K degrees of freedoms (dofs). Note that dofs at a destina-tion node represent successfully received linearly independent packets, which can be either source packets delivered duringthe broadcast phase, or coded packets transmitted during therelay phase. According to [14, eq. (5)] and [14, eq. (8)], theprobability that the N ≥ K coded packets, which have beentransmitted by the N relay nodes, will yield K dofs can bebounded as follows: P ′ ( K, N, ǫ s ( k ) r , q ) ≥ max n K Y i =1 (cid:0) − Γ N − i +1max (cid:1) , − K X w =1 (cid:18) Kw (cid:19) ×× ( q − w − (cid:2) q − + (1 − q − ) (cid:0) − − ǫ s ( k ) r − q − (cid:1) w (cid:3) N o (7)where Γ max = max (cid:8) ǫ s ( k ) r , − ǫ s ( k ) r q − (cid:9) .In order to formulate the decoding probability at eachdestination node, let us assume that the destination d k suc-cessfully received m packets, given that K + N packets weretransmitted, i.e., K source packets during the broadcast phaseand N coded packets during the relay phase. If we denoteby f ℓ ( N T , ǫ ) the probability mass function of the binomialdistribution, that is, f ℓ ( N T , ǫ ) = (cid:18) N T ℓ (cid:19) ǫ N T − ℓ (1 − ǫ ) ℓ (8)then the probability that h of the m packets are source packetsand the remaining m − h are coded packets is given by P h/m ( ǫ s ( k ) d k , ǫ rd k ) = f h ( K, ǫ s ( k ) d k ) f m − h ( N, ǫ rd k ) . (9)The contribution of the h recovered source packets to the m − h coded packets can be removed, so that the m − h coded packets become linear combinations of the remaining K − h source packets only. Thus, at this point of the decodingprocess, the destination node d k can successfully recover theremaining data packets if and only if the modified m − h coded packets yield K − h dofs. By employing (7), (9) andthe law of total probability, the overall decoding probabilityat the destination d k can be expressed as P d k ( K, N ) = N + K X m = K K X h = h min P h/m ( ǫ s ( k ) d k , ǫ rd k ) P ′ ( K − h, m − h, ǫ s ( k ) r , q ) (10)where h min = max(0 , m − N ) .Note that retransmissions are not allowed in case of packetfailures during the broadcast phase or the relay phase. There-fore, by modifying the expression of the end-to-end throughputin [18], the average system throughput can be defined as η = KK + max { E d ( N ) , E d ( N ) } (11)where E d k ( N ) is the average number of relay nodes neededby each destination node d k to recover the entire source group G k , and can be calculated using [19] E d k ( N ) = N − N − X v =0 P d k ( K, v ) . (12)Moreover, by following (12), the average number of re-lays required for both destinations to decode the pack-ets of the respective source groups can be repre-sented as E T ( N ) = N − P N − v =0 P joint ( K, v ) , where P joint ( K, v ) = P d ( K, v ) P d ( K, v ) . SNR ¯ ρ (dB) D ec o d i n g P r o b a b ili t y NOMA-RLNC (d )NOMA-RLNC (d )OMA-RLNC (d )OMA-RLNC (d )Simulation Figure 1: Simulation results and performance com-parison between NOMA-RLNC and OMA-RLNC,when K = 20 , N = 10 and q = 4 . Number of Relays ( N ) J o i n t D ec o d i n g P r o b a b ili t y P j o i n t NOMA-RLNC ( q = 2)NOMA-RLNC ( q = 4)NOMA-RLNC ( q = 64)Simulation Figure 2: Effect of the field size q and the numberof relay nodes N on the joint decoding probability,when K = 20 .
15 20 25 300510152025 A v e r ag e N u m b e r o f r e l a y s E T ( N ) SNR ¯ ρ (dB) NOMA-RLNCOMA-RLNC
Figure 3: Comparison between the two schemesin terms of the required average number of relaynodes and the SNR when K = 20 and q = 4 . V. N
UMERICAL R ESULTS
In this section, the accuracy of the derived analytical boundin (7), when used in combination with the decoding probabilityin (10), is verified through simulations. In the consideredsystem setup, the bandwidth of each sub-band is normalizedto 1, i.e., B s = 1 . The source nodes and relay nodes havebeen positioned such that σ (1) d = 0 . , σ (2) d = 0 . , σ (1) r = 2 . , σ (2) r = 1 , σ = 1 . and σ = 1 . .We set α = 0 . and α = 0 . , while exhaustive search hasbeen used to identify the values of β and β that maximizethe joint decoding probability mentioned in Section IV. Theaverage system SNR is set equal to ρ s = ρ r = ¯ ρ and, unlessotherwise stated, we consider R ∗ = 1 , R ∗ = 1 . .Fig. 1 shows the decoding probabilities P d and P d at thetwo destination nodes in terms of the system SNR. The figureclearly demonstrates the tightness of the analytical curve tothe simulation results. The decoding probability P d is greaterthan P d because node d supports a lower target rate thannode d , and d is allocated more power than d to ensurethat the quality of service requirements are met. As expected,NOMA-RLNC outperforms OMA-RLNC because each sourcenode in NOMA-RLNC benefits from being allocated twice thebandwidth that is allocated in OMA-RLNC.Fig. 2 shows the joint decoding probability, for differentvalues of field size q , as a function of the number of relays.The analytical bound is close to the simulation results for q = 2 and becomes tighter for greater values of q . A significantgain in performance can be observed when the field sizeincreases from q = 2 to q = 4 . However, the increase in gainis markedly smaller when q further increases from to .This is because the certainty of linear independence betweencoded packets increases with the field size and approaches thehighest possible degree even for relatively small values of q .We stress that the computational complexity of the decoder atthe destination nodes also depends on the value of q . Thus, thechoice of the field size over which RLNC is performed resultsin a trade-off between complexity and performance gain.Fig. 3 illustrates the relationship between the system SNRand the average number of relays required for the decodingof the source packets of both source groups by the respectivedestination nodes. The curves establish the diversity advantageoffered by the combination of NOMA with RLNC as opposedto OMA with RLNC. For a fixed value of SNR, OMA-RLNC clearly needs more relays for cooperation than NOMA-RLNC.Alternatively, OMA-RLNC can achieve the same performanceas NOMA-RLNC at the expense of a higher SNR.Fig. 4 presents the system throughput as a function ofthe system SNR, for different target rates. The performancegap between NOMA-RLNC and OMA-RLNC is evident. Weobserve that, for a fixed SNR value, when the target rateincreases from R ∗ = 1 . to R ∗ = 2 , the outage probabil-ity increases and, therefore, the system throughput reduces.Interestingly, an increase in the target rate also increases theperformance gap between NOMA-RLNC and OMA-RNC,that is, the throughput degradation of NOMA-RLNC is lesssevere than that of OMA-RLNC. An intuitive reason for thisobservation is that the / spectral loss in OMA dominatesthe system throughput.
15 20 25 300.40.50.60.70.80.91
SNR ¯ ρ (dB) S y s t e m T h r o u g hpu t η NOMA−RLNC (R =1, R =1.5)OMA−RLNC (R =1, R =1.5)NOMA−RLNC (R =1, R =2.0)OMA−RLNC (R =1, R =2.0) Figure 4: Effect of target rates on the system throughput against the systemSNR, when K = 20 and q = 4 . VI. C
ONCLUSIONS
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