Forty-Fifth Annual Allerton Conference Allerton House, UIUC, Illinois, USA September 26-28, 2007
WeC2.3
Joint Physical Layer Coding and Network Coding for Bi-Directional Relaying Krishna Narayanan, Makesh Pravin Wilson and Alex Sprintson Abstract— We consider the problem of two transmitters wishing to exchange information through a relay in the middle. The channels between the transmitters and the relay are assumed to be synchronized, average power constrained additive white Gaussian noise channels with a real input with signal-tonoise ratio (SNR) of snr. An upper bound on the capacity is 12 log(1 + snr) bits per transmitter per use of the mediumaccess phase and broadcast phase of the bi-directional relay channel. We show that using lattice codes and lattice decoding, we can obtain a rate of 12 log( 12 + snr) bits per transmitter, which is essentially optimal at high SNR’s. The main idea is to decode the sum of the codewords modulo a lattice at the relay followed by a broadcast phase which performs Slepian-Wolf coding with structured codes. For asymptotically low SNR’s, jointly decoding the two transmissions at the relay (MAC channel) is shown to be optimal. We conjecture that if the two transmitters use identical lattices, we can achieve a rate of only 1 log( 12 + snr) even with maximum likelihood decoding. The 2 proposed scheme can be thought of as a joint physical layer, network layer code which outperforms the recently proposed analog network coding scheme.
I. I NTRODUCTION , S YSTEM M ODEL AND P ROBLEM STATEMENT
We consider the bi-directional relaying problem where two users try to exchange information with each other through a relay in the middle. More specifically, we study a simple 3-node linear Gaussian network as shown in Fig. 1. Node A and node B wish to exchange information between each other through the relay node R, however, nodes A and B cannot communicate with each other directly. Let uA ∈ {0, 1}k and uB ∈ {0, 1}k , be the information vector at nodes A and B (vectors are denoted by bold face letters such as u throughout the paper). The information is assumed to be encoded in to vectors (codewords) xA ∈ Rn and xB ∈ Rn at nodes A and B, respectively, and transmitted. We assume that communication takes place in two phases - a multiple access (MAC) phase and a broadcast phase. a) MAC phase: During the MAC phase, nodes A and B transmit xA and xB in n uses of an AWGN channel to the relay. Further, it is assumed that the two transmissions are perfectly synchronized. Hence, the received signal at the relay is yR ∈ Rn which is given by yR = xA + xB + n where the components of n are independent identically distributed (i.i.d) Gaussian random variables with zero mean This work was supported by the National Science Foundation under grant CCR-0515296. The authors are with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843, USA krn,makesh,
[email protected]
RelayNodeR
yR=xA +xB +n xA
MACphase
NodeA
xR
xB
NodeB
xR
RelayNodeR Broadcastphase NodeB
NodeA
yA =xR +nB
yA =xR +nA
Fig. 1.
System Model with 3 Nodes
and variance σ 2 . Further, it is assumed that there is an average power constraint of P at both nodes and, hence, E[||xA ||2 ] ≤ P and E[||xB ||2 ] ≤ P . b) Broadcast phase: During the broadcast phase, the relay node transmits xR ∈ Rn in n uses of an AWGN broadcast channel to both nodes A and B. It is assumed that the average transmit power at the relay node is also constrained to P and that the noise variance at the two nodes is σ 2 . It is assumed that communication in the MAC and broadcast phases are orthogonal. For example, this could be in two separate frequency bands (or in two different time slots) and, hence, the MAC phase and broadcast phase does not interfere. We simply think of this as 2n dimensions being available for communication out of which n dimensions are allocated to the MAC phase and n dimensions are allocated to the broadcast phase. Also note that the signal-to-noise ratio for all transmissions is snr = P/σ 2 and, hence, we refer to this as simply the SNR without having to distinguish between the SNR’s of the different phases. We are mainly interested in good encoding and decoding schemes that maximize the amount of information (maximize k/n) that can be exchanged reliably (such that the probability of error can be made arbitrarily small in the limit of n → ∞) reliably. We refer to the maximum value of k/n that can be reliably exchanged with a given scheme as the exchange rate for that scheme. The exchange capacity is then the supremum of all such rates over the encoding schemes.
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WeC2.3 II. M AIN R ESULTS AND COMMENTS We mainly consider the case when the MAC and broadcast phase are both restricted to using exactly n uses of the channel each. For this case, the main results in this paper are • An upper bound on the constrained (to the MAC and broadcast phase using n-uses of the AWGN channel orthogonally) exchange capacity is 12 log(1 + σP2 ). • An exchange rate of 1 P 1 + 2 RLattice = log 2 2 σ can be obtained using the lattice encoding and decoding scheme in Section V-B and an exchange rate of 2P 1 RJD = log 1 + 2 4 σ
•
•
can be obtained using the joint decoding scheme in Section V-C. It can be seen that the lattice decoding scheme becomes nearly optimal at high SNR’s whereas the joint decoding scheme of Section V-C is nearly optimal at low SNRs. Clearly, any rate of the form βRJD + (1 − β)RLattice can be obtained for any 0 ≤ β ≤ 1 by time sharing between the two schemes. This outperforms the recently proposed analog network coding idea in [3]. We conjecture that even with maximum likelihood decoding, only a rate of 12 log 12 + σP2 can be obtained with the lattice coding scheme of Section V-B. III. R ELATED P RIOR W ORK
Recently, there has been a lot of work on coding for the bidirectional relay problem [1]-[7]. In [2], Katti et al., showed the usefulness of network coding for this problem. Although they do not consider the physical layer explicitly in this work, the natural extension of their solution to our problem would work as follows. The 2n orthogonal dimensions available for signalling would be split in to three slots with 2n/3 dimensions each. In the first time slot, uA is encoded using an optimal channel code for the AWGN channel in to xA and transmitted from node A. Similarly, in the second time slot uB is encoded in to xB and transmitted from node B. At the relay, uA and uB can be decoded and then the relay form uR = uA ⊕ uB and encodes uR in to xR using an optimal code for the Gaussian channel and transmitted from the relay. The two nodes decode uR and then since they have uA and uB , they can obtain uB and uA at the nodes A and B, respectively. Here, the physical layer and network layer are completely separated and coding (or mixing of the information) is performed only at the network layer. In the system model considered in Fig. 1, the physical layer naturally performs mixing of the signals from the two transmitter. Can we take advantage of this? Such schemes can be called joint physical layer coding and network coding solution. One such a scheme called analog network coding was recently proposed in [3]. In this case, the MAC phase
and broadcast phase uses n uses of the AWGN and are orthogonal to each other. Gaussian code books are used at the transmitters to encode uA into xA and uB into xB , respectively. Analog network coding is an amplify (rather scale) and forward scheme where the received signal at the relay during the MAC phase yR , is scaled to satisfy the power constraint and transmitted during the broadcast phase, i.e., xR = 2P P+σ2 yR . It can be seen that this scheme can P/σ 2 1 achieve an exchange capacity of 2 log 1 + 3P +σ2 , which P
is higher than that achievable with the pure network coding scheme in [2] for high SNRs. The schemes proposed here can be thought of as decode and forward schemes which outperform the amplify and forward scheme in [3]. In a very recent independent work [6], it is conjectured that an exchange of 12 log 1 + σP2 can be achieved, however, no explicit scheme is given. The scheme in this paper is a constructive scheme that performs close to the rate conjectured in [6]. IV. A N O PTIMAL T RANSMISSION S CHEME FOR THE BSC C HANNEL To motive our proposed scheme in Section V-B, we first consider a system where the physical layer channels are all binary symmetric channels. i.e., xA ∈ {0, 1}n , xB ∈ {0, 1}n and the signal received at the relay is yR = xA ⊕ xB ⊕ e, where ⊕ denotes binary addition and e is an error sequence whose components are 0 or 1 with probability q and 1 − q respectively and are i.i.d. Similarly, during the broadcast phase also let the channel be a BSC channel with crossover probability q. In this case, an upper bound on the exchange capacity can be seen to be 1−H(q) since this is the maximum information that can flow to any of the nodes from the relay. This can be achieved using the following coding scheme. The two nodes use identical capacity achieving binary linear codes of rate 1 − H(q). Consider again the received signal at the relay given by yR = xA ⊕ xB ⊕ e
(1)
Notice that since xA and xB are codewords of the same linear code, xA ⊕ xB is also a valid codeword from the same code which achieves capacity over a BSC channel with crossover probability q. Hence, the relay can decode xA ⊕xB and transmit this during the broadcast phase. The nodes A and B can also decode xA ⊕ xB and since they have xA and xB , they can obtain xB and xA , respectively. This scheme achieves an exchange rate of 1−H(q) and, is hence, optimal. c) Random codes versus structured codes: It is quite interesting to note that if random codes were used instead of linear codes, xA ⊕ xB cannot be decoded at the relay. The linearity (or group structure) of the code is exploited to make xA ⊕ xB decodable at the relay and, hence, structured codes with a group structure outperform random codes for this problem. Examples of schemes were structured codes outperform random codes have been given in Korner and
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WeC2.3 Marton [8] and more recently by Nazer and Gastpar in [9], [10]. V. B I -D IRECTION R ELAYING WITH G AUSSIAN LINKS NodeA
Let us now return to the problem outlined in Section I, where the channels between the nodes and the relay are AWGN channels.
RelayNodeV
A. Upper Bound on the Exchange Rate
tA
We restrict our attention to schemes where the MAC phase and broadcast phase are both orthogonal to each other and use n channel uses (or dimensions). With this restriction, a simple upper bound on the exchange rate can be obtained as follows. Consider a cut between the relay node and node A. The maximum amount of information that can flow to either of the nodes from the relay is 12 log 1 + σP2 . Hence, using the max-flow min-cut theorem, the exchange capacity is upper bounded by Cex,ub = 12 log 1 + σP2 . We now consider coding schemes and analyze their performance.
NodeB
t
tB
tA
t = (tA + tB )mod
B. Nested Lattice Based Coding Scheme
Fig. 2. Lattice code based scheme showing the transmitted signals tA , tB and the decoded signal at the relay t
The result in Section IV for the BSC channel shows that codes with a group structure (linear codes) enable decoding of a linear combination (or, sum) codewords at the relay. This motivates the use of lattice codes for the Gaussian channel since lattices have a similar group structure with respect to real vector addition. We begin with some preliminaries about lattices [12],[14]. A n-dimensional lattice Λ is a subgroup of Rn under normal vector addition. This implies that if λ1 , λ2 ∈ Λ, then λ1 + λ2 ∈ Λ. We can next define a quantizer QΛ . For any x ∈ Rn , QΛ (x) is defined as the λ ∈ Λ that is closest to x. The fundamental Voronoi region V(Λ) is defined as V(Λ) = {x : QΛ (x) = 0}. Next we define the mod operation as (x mod Λ) = x − QΛ (x). This can be interpreted as the error in quantizing an x to the closest point in the lattice Λ. In this work, we are interested in nested lattices. Formally, we can say the lattice Λ (the coarse lattice) is nested in the lattice Λ1 (the fine lattice) if Λ ⊆ Λ1 [13]. Let the fundamental Voronoi regions of the lattices, Λ and Λ1 be V(Λ) and V(Λ1 ) and the volumes be V (Λ) and V (Λ1 ) respectively, i.e., V (Λ) = V(Λ) dx, and V (Λ1 ) = V(Λ1 ) dx. The second moment of the coarse and fine lattices are 1 1 2 σ (Λ) = ||x||2 dx V (Λ) n V(Λ) 1 1 2 ||x||2 dx σ (Λ1 ) = V (Λ1 ) n V(Λ1 )
[15],[16],[11]. More recently, nested lattices have been shown to achieve the capacity of the single user AWGN channel under lattice decoding [12], [14]. The main idea here is to use the coarse lattice as a shaping region and the lattice points from the fine lattice contained within the basic voronoi region of the coarse lattice as the codewords. The existence of good nested lattices that achieve capacity has been shown in [12]. We now extend the idea in [12] for the bi-directional relaying problem. The encoding and decoding operations during the MAC and broadcast phase are explained below. A general schematic is also shown in Fig. 2. 1) MAC Phase: Let there be k information bits in the information vector uA and uB and, hence, the exchange rate is R = k/n. At node A, the information vector uA is mapped on to a fine lattice point tA ∈ {Λ1 ∩ V}, i.e., the set of all fine lattice points in the basic voronoi region of the coarse lattice is taken to be the code. An identical code is used at node B and the information vector uB is mapped onto the codeword tB ∈ {Λ1 ∩ V}. We then generate dither vectors dA and dB which are randomly generated n length vectors uniformly distributed over V. The dither vectors are mutually independent of each other and are known at both the relay node and the nodes A and B. Now node A and node B form the transmitted signal xA and xB as follows
The normalized second moment of the lattice G(Λ) = σ 2 (Λ)/V (Λ)1/n . The existence of nested lattices for which G(Λ) ≈ 1/(2πe) and G(Λ1 ) ≈ 1/(2πe) has been shown in [13], [14]. The number of lattice points of the fine lattice in the basic voronoi
of the coarse lattice is given by the region (Λ) nesting ratio VV(Λ . 1) Lattice codes can be used to achieve capacity on the single user AWGN channel under maximum likelihood
xA = (tA − dA )
mod Λ
(2)
xB = (tB − dB )
mod Λ
(3)
By choosing an appropriate coarse lattice with second moment P , the transmit power constraint will be satisfied at both nodes. Assuming perfect synchronization the relay node receives
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WeC2.3 y given by yR = xA + xB + n
(4)
where n is the noise vector whose components have variance σ2 . The main idea is that the relay decodes t = (tA + tB )
mod Λ
from the received signal yR as follows. The decoder at the relay node forms ˆ t = (αy + dA + dB ) mod Λ (α will be determined later) and finds lattice point in the fine lattice that is closest to ˆ t. Using the distributive property of the mod operation, ˆ t can be written as, ˆ t =
(αy + dA + dB )
mod Λ
2) Broadcast Phase: In the broadcast phase, the relay node transmits the index of t using a capacity achieving code for the AWGN channel. Since the capacity of the AWGN is 12 log 1 + σP2 which is higher than RLattice in (7) and, hence, the index (or, equivalently, t) can be obtained at the nodes A and B. Since node A already has uA and, hence, tA , it needs to recover uB or, equivalently, tB , from t, tA . This can be done as follows. Node A computes (t − tA ) mod Λ, which can be written as (t − tA ) mod Λ = ((tA + tB ) mod Λ − tA ) mod Λ = tB mod Λ = tB (8)
Similarly, tA can also be obtained in node B by taking (t − tB )Λmod Λ. = (xA + xB + dA + dB + αn − (1 − α)(xA + xB )) mod Hence, an effective rate of Rex = 12 log 12 + σP2 can be = ((tA − dA ) mod Λ + (tB − dB ) mod Λ + obtained using nested lattices with lattice decoding. We can dA + dB + αn − (1 − α)(xA + xB )) mod Λ see athigh SNR’s, this scheme approaches the upper bound = ((tA + tB ) mod Λ + αn − (1 − α)(xA + xB )) mod12 log Λ 1 + σP2 and is hence, nearly optimal. This = (t + αn − (1 − α)(xA + xB )) mod Λ (5) scheme can be interpreted as a Slepian-Wolf coding scheme using nested lattices, i.e., the relay wishes to convey Due to the group structure of the lattice, t is a lattice tA + tB to node A, where some side information (namely, point in the fine lattice (more precisely, t ∈ {Λ1 ∩ V(Λ)}). tA ) is available. The broadcast phase of the proposed scheme Since tA and tB are uniformly distributed over lattice points can be thought of using nested lattices for solving this in {Λ1 ∩ V(Λ)}, t is also uniformly distributed over all the Slepian-Wolf coding problem. lattice points in {Λ1 ∩V(Λ)}. Further, note that tA and tB are This scheme can also be thought of as a decode and independent of n, xA and xB and, hence, we can define an forward scheme where the relay decodes a function of tA equivalent noise term as neq = αn − (1 − α)(xA + xB ) such and tB , namely t = (tA + tB ) mod Λ and forwards this that t and neq are independent of each other. The second to the nodes. Notice however, that the relay will not know 2 2 2 2 moment of Neq is given by σeq = E[Neq ] = α σ + (1 − either tA or tB exactly, it only knows t. 2 α) 2P . Since the nested lattice code we have used is a capacity 2 We can now choose α to minimize E[neq ] and the resultachieving code for the AWGN channel, one does not have 2P 2 ing optimum values of α and σeq are αopt = 2P +σ2 and to encode t again. The relay can simply transmit the t to the 2P σ 2 2 2 σeq,opt = 2P +σ 2 . We choose a fine lattice with σ (Λ1 ) = nodes A and B and t can be decoded at the nodes A and 2 σeq,opt + , for an arbitrarily small . Using the results in B in the presence of noise at the nodes A and B. Notice [12] and [14] it can be shown that that E[||t||2 ] ≤ P and, hence, the power constraint will be satisfied at the relay node. t) = t} → 0 as n → ∞ P r{QΛ1 (ˆ The proposed lattice code based scheme can be seen to be and, hence, we can guarantee that t can be decoded with somewhat similar to the scheme proposed independently by probability of error that can be made arbitrarily small as Nazer and Gastpar [10] for a related but different problem. n → ∞. The number of lattice points of the fine lattice in Λ1 ∩ V(Λ) is then C. Joint Decoding Based Scheme 2 2/n σ (Λ) V (Λ) While the aforementioned scheme is nearly optimal at high = |Λ1 ∩ V(Λ)| = V (Λ1 ) σ 2 (Λ1 ) SNR’s, the performance of this scheme at low SNR’s is 2 very poor. In fact, for P/σ 2 < 1/2, the scheme does not For → 0, σ 2 (Λ1 ) = σeq,opt and, hence, even provide a non-zero rate. In this regime, we can use 2/n 2/n 2P + σ 2 1 P any coding scheme which is optimal for the multiple access + 2 |Λ1 ∩ V(Λ)| = = (6) 2 channel and perform joint decoding at the relay such that 2σ 2 σ uA and uB can be decoded. Then, the relay can encode Hence, we can use a rate of uA ⊕ uB and transmit to the nodes. Any coding scheme that 1 P 1 1 RLattice = log2 |Λ1 ∩ V(Λ)| = log + (7) is optimal for the MAC channel can be used in the MAC n 2 2 σ2 phase. A simple scheme is time sharing (although it is not at each of the nodes and (tA + tB ) mod Λ can be decoded the only one) where nodes A and B transmit with powers 2P for a duration of n/2 channel uses each but they do not at the relay. =
(α(xA + xB ) + αn + dA + dB )
mod Λ
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WeC2.3 interfere with one another. In this case, a rate of 2P 1 RJD = log 1 + 2 4 σ
(9)
can be obtained. It can be seen that this is optimal at asymptotically low SNRs, since log(1 + snr) ≈ snr for snr → 0. The performance of these schemes are shown in Fig. 3 where the upper bound and the achievable rates with these proposed schemes are shown. The achievable rate with the analog network coding scheme is also shown. It can be seen that our schemes outperform analog network coding. 2 1.8 1.6
Exchange Rate
1.4 Lattice based scheme 1.2 1 Upper bound 0.8 0.6 0.4
Joint decoding Analog network coding
0.2 0 −10
Fig. 3.
−5
0 SNR in dB
5
10
Achievable Exchange Rates for the Proposed Schemes
Clearly, we can time share between the lattice based scheme and this scheme in order to obtain rates of the form Rex = βRJD + (1 − β)RLattice .
(10)
In both the lattice based scheme and the joint decoding based scheme, if the restriction to use n channel uses during the MAC phase and n during the broadcast phase is removed, i.e., only the total number of uses is constrained to be 2n, better schemes can be easily designed. Similarly, different power sharing may also lead to better schemes. VI. E XTENSION TO MULTIPLE HOPS We can extend the above results to multiple hops. We can again show that rate of 12 log 12 + σP2 is achievable using structured coding even in the multiple hop scenario. It should be noted that the advantage of this scheme over the amplify and forward scheme [3] becomes more pronounced in the multi-hop case, since at each stage for the amplify and forward scheme, the channel noise is amplified and hence the amplify and forward scheme will suffer a huge rate loss as the number of hops increase. The relay nodes and the nodes A and B can transmit only to the two nearest nodes. During a single transmission slot (n uses of the channel), a node can either listen or transmit. It can not do both simultaneously. We explain our structured coding scheme using a simple
example of a 3-relay network. The different transitions are shown in the table given below. Here node A and node B have data that need to be exchanged between each other. Each node has a stream of packets. Node A has packets named uA,1 , uA,2 , . . . and node B has packets named uB,1 , uB,2 , . . .. In the first transmission slot the nodes A, R2 , B transmit. Nodes A, B transmit vectors xA,1 and xB,1 , respectively using our proposed lattice coding scheme. At the beginning of transmission, the node R2 no data to transmit in the first transmission slot. The node R1 and R2 decode to xA,1 mod Λ and xB,1 mod Λ, respectively. During the second transmission slot the nodes R1 and nodes R3 transmit, while the other nodes remain silent. This can be seen as shown in the table above. So, in each stage the nodes transmit and the listening nodes decode to a lattice point in the fine lattice in the Voronoi region of the coarse lattice. In every second transmission slot a new packet is transmitted to the relay nodes by the nodes A and B as can be seen from the table. During slots 2, 4, 6 nodes A, B transmits new packets into the relay channel. From this example, we can see that at the 4th slot the node A and B decode xB,1 and xA,2 respectively. This is because the node A receives (xA,1 + xA,2 + xB,1 ) mod Λ during the 4th transmission and, hence, since xA,1 , xA,2 are already known at the node A, the node A can decode to xB,1 using modulo operation. The same argument holds for node B also. From every two transmission from this stage a new packet can be decoded at each node. This shows that for sufficiently large number packets we can approach 12 log( 12 + σP2 ). We can easily prove that the same argument holds for a general say L relay nodes in between. This is because by our coding scheme we can see in every two slots a new packet is sent out from the nodes A,B. After an initial 2L transmission slot delay, in every two slots the relay nodes receives a new packet from the other nodes. Here, we mean that every two slots the relay node decodes to a lattice point which is a linear function of a new packet. Hence, at the decoding stage at the nodes B and A, we can decode after every 2 slots since only one variable is unknown, since only one new packet(or function of new packet) moves from one node to the other. Hence, we can still achieve the rate of 1 1 P 2 log( 2 + σ 2 ). Moreover, the functions in each stage are bounded and, hence, we can always perform the decoding at the receiver nodes. VII. P ERFORMANCE UNDER M AXIMUM L IKELIHOOD D ECODING In Section V-B, we have assumed lattice decoding. Can we achieve higher rates if maximum likelihood (ML) decoding is used with lattice codes? We conjecture that the upper bound 12 log 1 + σP2 cannot be achieved even with ML decoding. Our conjecture is based on the idea that if xA and xB are lattice points in the intersection of a lattice with √ a thin spherical shell of radius P , then the lattice points x √A +xB will with high probability lie in a thin shell of radius 2P . In order to guarantee successful decoding with a noise variance of σ 2 , the volume of the voronoi region needs to
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WeC2.3 TABLE I TABLE S HOWING S EQUENCE OF PACKETS IN THE M ULTIHOP C ASE Slot 1 2 3 4 5 6
Node A Transmits xA,1 Remains Silent Transmits xA,2 Decodes xB,1 Transmits xA,3 Decodes xB,2
Node R1 ( mod Λ) xA,1 Transmits xA,2 + xA,1 + xB,1 Transmits 2(xA,1 + xB,1 ) +xA,2 + xB,2 + xA,3 Transmits
Node R2 ( mod Λ) Transmits xA,1 + xB,1 Transmits 2(xA,1 + xB,1 ) + xA,2 + xB,2 Transmits 4(xA,1 + xb,1 ) + 2(xA,2 + xB,2 ) + xA,3 + xB,3
This allow us to pack only 12 log 12 + σP2 voronoi √ regions within a sphere of radius P , obtaining only the same performance with lattice decoding! Our current work focusses on resolving this issue. σ 2 2P 2P +σ 2 .
VIII. C ONCLUSION We considered joint physical layer and network layer coding for the bi-directional relay problem where two nodes wish to exchange information through AWGN channels. Under the restrictive model of the MAC and broadcast phase using n channel uses separately, we showed upper bounds on the exchange capacity and constructive schemes based on lattices that is nearly optimal at high SNR’s. At low SNR’s joint decoding based schemes (optimal coding schemes for the MAC channel) are nearly optimal. These schemes outperform the recently proposed analog network coding. Interestingly, our result shows that structured codes such as lattice codes outperform random codes for such networking problems. We also showed extensions of this scheme to a network with many relay nodes, where the advantages of the proposed scheme over simple amplify and forward will be higher than in single relay case.
Node R3 ( mod Λ) xB,1 Transmits xA,1 + xB,1 + xB,2 Transmits 2(xA,1 + xB,1 ) +xA,2 + xB,2 + xB Transmits
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