Performance Bounds for Bi-Directional Coded Cooperation Protocols

Transcription

1 1 Performance Bounds for Bi-Directional Coded Cooperation Protocols Sang Joon Kim, Patrick Mitran, and Vahid Tarokh arxiv:cs/ v4 [cs.it] 5 Jun 2008 Astract In coded i-directional cooperation, two nodes wish to exchange messages over a shared half-duplex channel with the help of a relay. In this paper, we derive performance ounds for this prolem for each of three decode-and-forward protocols. The first protocol is a two phase protocol where oth users simultaneously transmit during the first phase and the relay alone transmits during the second. In this protocol, our ounds are tight. The second protocol considers sequential transmissions from the two users followed y a transmission from the relay while the third protocol is a hyrid of the first two protocols and has four phases. In the latter two protocols the ounds are not identical. Numerical evaluation shows that in some cases of interest our ounds do not differ significantly. Finally, in the Gaussian case with path loss, we derive achievale rates and compare the relative merits of each protocol. This case is of interest in cellular systems. Surprisingly, we find that in some cases, the achievale rate region of the four phase protocol contains points that are outside the outer ounds of the other two protocols. Index Terms Cooperation, capacity ounds, performance ounds, i-directional communication, network coding. Sang Joon Kim and Vahid Tarokh are with the School of Engineering and Applied Sciences, Harvard University, Camridge, MA s: sangkim, vahid@fas.harvard.edu. Patrick Mitran is with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Canada. pmitran@uwaterloo.ca. This research is supported in part y NSF grant numer ACI and ARO MURI grant numer W911NF This work was supported in part y the Army Research Office,under the MURI award N0. N The views expressed in this paper are those of the authors alone and not of the sponsor. Octoer 9, 2018

2 2 I. INTRODUCTION Consider two users, denoted y a and, who wish to share independent messages over a shared channel. Traditionally, this prolem is known as the two-way channel [2], [10]. In many realistic roadcast environments, such as wireless communications, it is not unreasonale to assume the presence of a third node which may aid in the exchange of a and s messages. In particular, if a is a moile user and is a ase station, then we may suppose the presence of a relay station r to assist in the i-directional communication. Traditionally, without the presence of the relay station, communication etween nodes a and is performed in two steps: first a transmits its message to followed y similar transmission from to a (illustrated in Fig. 1.i). In the presence of relay node r, one might initially assume that four phases are needed (see Fig. 1.ii). However, y taking advantage of the shared wireless medium, it is known that the third and fourth transmissions may e comined (Fig. 1.iii) into a single transmission using, for example, ideas from network coding [1], [13]. In particular, if the messages of a and are w a and w respectively and elong to a group, then it is sufficient for the relay node to successfully transmit w a w simultaneously to a and. In [4], [5], such a three phase coded i-directional protocol is considered when the group is Z k 2, the inary operator is component-wise modulo 2 addition (i.e., exclusive or) and encoding is performed linearly to produce parity its. As each user transmits sequentially, each user is amenale to receive side-information from the opposite user during one of the first two phases. The works of [7] and [8] not only consider the three phase protocol, ut comine the first two phases into a single joint transmission y nodes a and followed y a single transmission y the relay which forwards its received signal (Fig. 1.iv). Coded i-directional cooperation may also e extended for the case of multiple relaying nodes [11], [12]. In [9], achievale rate regions are derived assuming full duplex capailities at all nodes. In this paper, we are interested in determining fundamental ounds on the performance of coded idirectional communications assuming various decode-and-forward protocols for half-duplex channels. In the case of a two phase protocol where oth users transmit simultaneously in the first phase followed y a transmission from the relay, we derive the exact performance 1. In the case of three or more phase protocols, we take into account any side information that a node may acquire when it is not transmitting and derive inner and outer ounds on the capacity regions. We find that a four phase hyrid protocol is sometimes strictly etter than the outerounds of two or three phase decode-and-forward protocols Octoer 9, 2018

3 3 Fig. 1. (i) Traditional approach, (ii) Naive four phase i-directional cooperation, (iii) coded roadcast three phase protocol, (iv) two phase protocol. previously introduced in the literature. This paper is structured as follows. In Section II, we define our notation and the protocols that we consider. In Section III, we derive performance ounds for the protocols while in Section IV, we numerically compute these ounds for fading Gaussian channels. A. Notation and Definitions II. PRELIMINARIES We first start with a somewhat more general formulation of the prolem. We consider an m node set, denoted as M := {1,2,...,m} (where := means defined as) for now, where node i has message W i,j that it wishes to send to node j. Each node i has channel input alphaet X i = X i { } and channel output alphaet Y i = Y i { }, where is a special symol distinct of those in X i and Y i and which denotes either no input or no output. In this paper, we assume that a node may not simultaneously transmit and receive at the same time. In particular, if node i selects X i =, then it receives Y i Y i and if X i X i, then necessarily Y i =, i.e., X i = iff Y i 2. Otherwise, the effect of one node remaining silent on the received variale at another node may e aritrary at this point. The channel is assumed discrete memoryless. In Section IV, we will e interested in the case X i = Y i = C { }, i M. The ojective of this paper is to determine achievale data rates and outer ounds on these for some particular cases. We use R i,j for the transmitted data rate of node i to node j, i.e., W i,j {0,..., 2 nri,j 1} := S i,j. 1 Similar results were independently derived in [6]. 2 Thus, FDM cannot e allowed as it violates the half-duplex constraint. Octoer 9, 2018

4 4 For a given protocol P, we denote y l 0 the relative time duration of the l th phase. Clearly, l l = 1. It is also convenient to denote the transmission at time k, 1 k n at node i y X k i, where the total duration of the protocol is n and X (l) i denotes the random variale with alphaet X i and input distriution p (l) (x i ) during phase l. Also, X k i corresponds to a transmission in the first phase if k 1 n, etc. We also define XS k := {Xk i i S}, the set of transmissions y all nodes in the set S at time k and similarly X (l) S := {X(l) i i S}, a set of random variales with channel input distriution p (l) (x S ) for phase l, where x S := {x i i S}. Lower case letters x i denote instances of the upper case X i which lie in the calligraphic alphaets X i. Boldface x i represents a vector indexed y time at node i. Finally, it is convenient to denote y x S := {x i i S}, a set of vectors indexed y time. Encoders are then given y functions X k i (W i,1,...,w i,m,y 1 i k 1,...,Y ), for k = 1,...,n and decoders y Ŵj,i(Y 1 i,...,y n i,w i,1,...,w i,m ). Given a lock size n, a set of encoders and decoders has associated error events E i,j := {W i,j Ŵi,j( )}, for decoding the message W i,j at node j at the end of the lock, and the corresponding encoders/decoders result in relative phase durations { l,n }, where the suscript n indicates that the phase duration depends on the choice of lock size (as they must e multiples of 1/n). A set of rates {R i,j } is said to e achievale for a protocol with phase durations { l }, if there exist encoders/decoders of lock length n = 1,2,... with P[E i,j ] 0 and l,n l as n l. An achievale rate region (resp. capacity region) is the closure of a set of (resp. all) achievale rate tuples for fixed { l }. i B. Basic Results In the next section, we will use a variation of the cut-set ound. We assume that all messages from different sources are independent, i.e., i j, W i,k and W j,l are independent k,l M. In contrast to [2], we relax the independent assumption from one source to different nodes, i.e., in our case W i,j and W i,k may not e independent. Given susets S,T M, we define W S,T := {W i,j i S,j T} and R S,T = lim n 1 n H(W S,T). Lemma 1: If in some network the information rates {R i,j } are achievale for a protocol P with relative durations { l }, then for every ǫ > 0 and all S {1,2,,m} = M R S,S c l l I(X (l) (l) S ;Y S X (l) c S,Q)+ǫ, c for a family of conditional distriutionsp (l) (x 1,x 2,...,x m q) and a discrete time-sharing random variale Q with distriution p(q). Furthermore, each p (l) (x 1,x 2,...,x m q)p(q) must satisfy the constraints of Octoer 9, 2018

5 5 phase l of protocol P. Proof: Replacing W (T) y W S,S c and W (Tc) y W Sc,M in (15.323) - (15.332) in [2], then all the steps in [2] still hold and we have n H(W S,S c) = H(W S,S c W Sc,M) I(XS;Y k S k c Xk S c)+nǫ n, where ǫ n 0 since i S,j S cp[e i,j] 0 and the distriutions p(x k 1,...,xk m,y1 k,...,yk m) are those induced y encoders for which P[E i,j ] 0 as n. DefiningQ 1,Q 2,... to e discrete random variales uniform over{1,...,n 1,n },{n 1,n +1,...,n 1,n +n 2,n },..., we thus have k=1 H(W S,S c) l n l,n I(X Ql Ql S ;YS X Ql c S,Q c l )+nǫ n, (2) Defining the discrete random variale Q := (Q 1,Q 2,...), then 1 n H(W S,S c) l l,n I(X (l) (l) S ;Y S X (l) c S,Q)+ǫ c n, (3) where X (l) S := X Ql S. Finally, since the distriutions p(l) (x 1,x 2,...,x m q)p(q) are those induced y encoders for which P[E i,j ] 0, if there is a constraint on the encoders (such as a power constraint), this constraint is also valid for the distriutions p (l) (x 1,x 2,...,x m q)p(q). C. Protocols In i-directional cooperation, two terminal nodes denoted a and exchange their messages. The messages to e transmitted are W a := W a,, W := W,a and the corresponding rates are R a := R a, and R := R,a. The two distinct messages W a and W are taken to e independent and uniformly distriuted in the set of {0,..., 2 nra 1} := S a and {0,..., 2 nr 1} := S, respectively. Then W a and W are oth memers of the additive group Z L, where L = max( 2 nra, 2 nr ). The simplest protocol for the i-directional channel, is that of Direct Transmission (DT) (Fig. 2.i). Here, since the channel is memoryless and ǫ > 0 is aritrary, the capacity region from Lemma 1 is : R a sup 1 I(X a ;Y = ) p (x a) R sup 2 I(X (2) (2) ;Y a X a (2) = ), p (2) (x ) where the distriutions are over the alphaets X a and X respectively. With a relay node r, we suggest three different decode-and-forward protocols, which we denote as Multiple Access Broadcast (MABC) protocol, Time Division Broadcast (TDBC) and Hyrid Broadcast Octoer 9, 2018

6 6 Fig. 2. Proposed protocol diagrams. Shaded areas denote transmission y the respective nodes. It is assumed that all nodes listen when not transmitting. (HBC). Then, the message from a (resp. ) to r is W a,r = W a (resp. W,r = W ) and the corresponding rate is R a,r = R a (resp. R,r = R ). Also, in our protocols, all phases are contiguous, i.e., they are performed consecutively and are not interleaved or re-ordered. 3 In the MABC protocol (Fig. 2.ii), terminal nodes a and transmit information simultaneously during phase 1 and the relay r transmits some function of the received signals during phase 2. With this scheme, we only divide the total time period into two regimes and neither node a nor node is ale to receive any meaningful side-information during the first phase due to the half-duplex constraint. In the TDBC protocol (Fig. 2.iii), only nodeatransmits during the first phase and only nodetransmits during the second phase. In phase 3, relay r performs a transmission ased on the received data from the first two phases. Here, node a attempts to recover the message W ased on oth the transmissions from node in the second phase and node r in the third phase. We denote the received signal at node a in the second phase as second phase side information. Likewise, node may also recover W a ased on first phase side information and the received signal at node during the third phase. Finally, we consider a Hyrid Broadcast (HBC) protocol (Fig. 2.iv) which is an amalgam of the MABC and TDBC protocols. In this scheme, there are 4 distinct transmissions, two of which result in side-information at a and. 3 If we relax the contiguous assumption, the achievale region could increase y cooperation etween interleaving phases. Octoer 9, 2018

7 7 III. PERFORMANCE BOUNDS A. MABC Protocol Theorem 2: The capacity region of the half-duplex i-directional relay channel with the MABC protocol is the closure of the set of all points (R a,r ) satisfying R a < min { 1 I(X a ;Y r,x r =,Q), 2 I(X r (2) ;Y (2) X a (2) = X (2) R < min { 1 I(X ;Y r X a,x r =,Q), 2 I(X r (2) ;Y a (2) X a (2) = X (2) R a +R < 1 I(X a,x ;Y r X r =,Q) over all joint distriutions p(q)p (x a q)p (x q)p (2) (x r q) with Q 5 over the alphaetx a X X r. Remark: If the relay is not required to decode oth messages, then the region aove is still achievale, and removing the constraint on the sum-rate R a +R yields an outer ound. Proof: Achievaility: Random code generation: For simplicity of exposition only, we take Q = 1 and therefore consider distriutions p (x a ), p (x ) and p (2) (x r ). First we generate random (n 1,n )- length sequences x a (w a ) with w a S a and x (w ) with w S, and (n 2,n )-length sequences x (2) r (w r ) with w r Z L where L = max( 2 nra, 2 nr ), according to p (x a ), p (x ) and p (2) (x r ) respectively. Encoding: During phase 1, encoders of node a and send the codewords x a (w a ) and x (w ) respectively. Relay r estimates ŵ a and ŵ after phase 1 using jointly typical decoding, then constructs w r = ŵ a ŵ in Z L and sends x (2) r (w r ) during phase 2. Decoding: a andestimate w and w a after phase 2 using jointly typical decoding. Sincew r = w a w and a knows w a, node a can reduce the numer of possile w r to 2 nr and likewise at node, the cardinality is 2 nra. Error analysis: For convenience of analysis, first define E (l) i,j as the error event at node j that node j attempts to decode w i at the end of phase l using jointly typical decoding. Let A (l) S,T represents the set of ǫ-weakly typical (x (l) S,y(l) T ) sequences of length n l,n according to the input distriutions employed in phase l. Also define the set of codewords x (l) S (w S) := {x (l) i (w i ) i S} and the events D (l) S,T (w S) := {(x (l) S (w S),y (l) T ) A(l) S,T }, where S and T are disjoint susets of nodes. P[E a, ] P[E a,r E,r E(2) r, ] (4) P[E a,r E,r ]+P[E(2) r, Ē a,r Ē,r ] (5) Octoer 9, 2018

8 8 Following the well-known MAC error analysis from (15.72) in [2]: P[E a,r E,r Also, ] P[ D {a,},{r} (w a,w )]+2 nra 2 n 1,n(I(X a ;Y r,x r = ) 3ǫ) + 2 nr 2 n 1,n(I(X ;Y r a,x r = ) 3ǫ) +2 n(ra+r) 2 n 1,n(I(X a,x ;Y r X r = ) 4ǫ) (6) P[E (2) r, Ē a,r Ē,r ] P[ D (2) {r},{} (w a w )]+P[ wa w a D (2) {r},{} ( w a w )] P[ D (2) {r},{} (w a w )]+2 nra 2 n 2,n(I(X(2) r ;Y (2) X a (2) =X (2) = ) 3ǫ) (7) Since ǫ > 0 is aritrary, with the conditions of Theorem 2 and the AEP property, we can make the right hand sides of (6) (7) tend to 0 as n. Similarly, P[E,a ] 0 as n. Converse: We use Lemma 1 to prove the converse part of Theorem 2. As we have 3 nodes, there are 6 cut-sets, S 1 = {a}, S 2 = {}, S 3 = {r}, S 4 = {a,}, S 5 = {a,r} and S 6 = {,r}, as well as two rates R a and R. The outer ound corresponding to S 1 is then R a 1 I(X a ;Y r,y r,x,q)+ 2I(X (2) a ;Y (2) r,y (2) X (2) r,x (2),Q)+ǫ (8) = 1 I(X a ;Y r,x r =,Q)+ǫ, (9) where (9) follows since in the MABC protocol, we must have We find the outer ounds of the other cut-sets in the same manner: Y a = Y = X r = (10) X (2) a = X (2) = Y (2) r =. (11) S 2 : R 1 I(X ;Y r X a,x r =,Q)+ǫ. (12) S 3 : N/A (13) S 4 : R a +R 1 I(X a,x ;Y r r =,Q)+ǫ, (14) S 5 : R a 2 I(X (2) r ;Y (2) X (2) a = X (2) =,Q)+ǫ, (15) S 6 : R 2 I(X (2) r ;Y (2) a X (2) a = X (2) =,Q)+ǫ. (16) Since ǫ > 0 is aritrary, together, (9), (12) (16) and the fact that the half-duplex nature of the channel constrains X a to e conditionally independent of X given Q yields the converse. By Fenchel-Bunt s theorem in [3], it is sufficient to restrict Q 5. Octoer 9, 2018

9 9 B. TDBC Protocol Theorem 3: An achievale region of the half-duplex i-directional relay channel with the TDBC protocol is the closure of the set of all points (R a,r ) satisfying R a < min { 1 I(X a ;Y r = X r =,Q), 1 I(X a ;Y = X r =,Q)+ 3 I(X (3) r ;Y (3) X (3) a = X (3) R < min { 2 I(X (2) (2) ;Y r X a (2) = X r (2) =,Q), 2 I(X (2) (2) ;Y a X a (2) = X r (2) =,Q)+ 3 I(X r (3) ;Y a (3) X a (3) = X (3) over all joint distriutions p(q)p (x a q)p (2) (x q)p (3) (x r q) with Q 4 over the alphaetx a X X r. Proof: Random code generation: First, we generate a partition of S a randomly y independently assigning every indexw a S a to a sets a,i, with a uniform distriution over the indicesi {0,..., 2 nra0 1}. We denote y s a (w a ) the index i of S a,i to which w a elongs and likewise, a partition for w S is similarly constructed. For simplicity of exposition, we take Q = 1. For any ǫ > 0 and distriutions p (x a ), p (2) (x ) and p (3) (x r ), we generate random (n 1,n )-length sequences x a (w a ) with w a S a, (n 2,n )-length sequencesx (2) (w ) with w S and(n 3,n )-length sequencesx (3) r (w r ) with w r Z L, L = 2 n max{ra0,r0}. Encoding: During phase 1 (resp. phase 2), the encoder at node a (resp. node ) sends the codeword x a (w a ) (resp x (2) (w )). Relay r estimates ŵ a and ŵ after phases 1 and 2 respectively. The relay then constructs w r = s a (ŵ a ) s (ŵ ) in Z L, and sends x (3) r (w r ) during phase 3. Decoding: Terminal nodes a and estimate the indices s (w ) and s a (w a ) after phase 3 from x (3) r and then decode w and w a if there exists a unique w S, s A (2) {},{a} and w a S a, sa A {a},{}. Error analysis: Define E (l) i,j as the error events from node i to node j assuming node j attempts to decode w i at the end of phase l using jointly typical decoding and s a or s if availale. Also we use the same definitions of A (l) S,T and D(l) S,T (w S) as in the proof of Theorem 2. Then : P[E a, ] P[E a,r E (2),r E(3) r, E(3) a, ] (17) P[E a,r]+p[e (2),r ]+P[E(3) r, Ē a,r Ē(2),r ]+P[E(3) a, Ē a,r Ē(2),r Ē(3) r, ]. (18) Octoer 9, 2018

10 10 Also P[E a,r] P[ D {a},{r} (w a)]+2 nra 2 n 1,n(I(X a ;Y r P[E (2) (2),r ] P[ D {},{r} (w )]+2 nr 2 n 2,n(I(X(2) ;Y r (2) X (2) =X r a =X r (2) = ) 3ǫ) = ) 3ǫ) (19) (20) P[E (3) r, Ē a,r Ē(2),r ] P[ D (3) {r},{} (s a(w a ) s (w ))]+P[ sa s a(w a)d (3) {r},{} ( s a s (w ))] P[ D (3) {r},{} (s a(w a ) s (w ))]+2 nra0 2 n 3,n(I(X(3) r ;Y (3) X (3) a =X (3) = ) 3ǫ) (21) P[E (3) a, Ē a,r Ē(2),r Ē(3) r, ] P[ D {a},{} (w a)]+p[ wa w a D {a},{} ( w a),s a (w a ) = s a ( w a )] P[ D {a},{} (w a)]+2 n(ra 1,nI(X a ;Y =X r = ) R a0+3ǫ) (22) Since ǫ > 0 is aritrary, with the proper choice of R a0, the conditions of Theorem 3 and the AEP property, we can make the right hand sides of (19) (22) vanish as n. Similarly, P[E,a ] 0 as n. By Fenchel-Bunt s theorem in [3], it is sufficient to restrict Q 4. Theorem 4: The capacity region of the i-directional relay channel with the TDBC protocol is outer ounded y the union of R a min{ 1 I(X a ;Y r,y = X r =,Q), 1 I(X a ;Y = X r =,Q)+ 3 I(X (3) r ;Y (3) X (3) a = X (3) =,Q)} R min{ 2 I(X (2) (2) ;Y r,y a (2) X a (2) = X r (2) =,Q), 2 I(X (2) (2) ;Y a X a (2) = X r (2) =,Q)+ 3 I(X r (3) ;Y a (3) X a (3) = X (3) =,Q)} R a +R 1 I(X a ;Y r = X r =,Q)+ 2 I(X (2) (2) ;Y r X a (2) = X r (2) =,Q) over all joint distriutions p(q)p (x a q)p (2) (x q) p (3) (x r q) with Q 5 over the alphaetx a X X r. Remark: If the relay is not required to decode oth messages, removing the constraint on the sum-rate R a +R yields an outer ound. Proof outline: The proof of Theorem 4 follows the same argument as in the proof of the converse part of Theorem 2. Octoer 9, 2018

11 11 C. HBC Protocol Theorem 5: An achievale region of the half-duplex i-directional relay channel with the HBC protocol is the closure of the set of all points (R a,r ) satisfying R a < min { 1 I(X a ;Y r = X r =,Q)+ 3 I(X a (3) ;Y r (3) X (3),X(3) r =,Q), 1 I(X a ;Y = X r =,Q)+ 4 I(X r (4) ;Y (4) X a (4) = X (4) R < min { 2 I(X (2) (2) ;Y r X a (2) = X (2) r =,Q)+ 3 I(X (3) (3) ;Y r X a (3),X r (3) =,Q), 2 I(X (2) (2) ;Y a X a (2) = X r (2) =,Q)+ 4 I(X r (4) ;Y a (4) X a (4) = X (4) R a +R < 1 I(X a ;Y r = X 3 I(X (3) a,x (3) (3) ;Y r X r (3) =,Q) r =,Q)+ 2 I(X (2) (2) ;Y r X a (2) = X r (2) =,Q)+ over the joint distriution p(q)p (x a q)p (2) (x q)p (3) (x a q)p (3) (x q) p (4) (x r q) over the alphaet X 2 a X 2 X r with Q 5. Proof outline: Generate random codewords x a (w a ), x (2) (w ), x (3) a (w a ), x (3) (w ). Relay r receives data from terminal nodes during phases 1 3, which is decoded y the relay using a MAC protocol to recover w a, w. Theorem 5 then follows the same argument as the proof of Theorem 3. Theorem 6: The capacity region of the i-directional relay channel with the HBC protocol is outer ounded y the union of R a min { 1 I(X a ;Y r,y = X r =,Q)+ 3 I(X a (3) ;Y r (3) X (3),X(3) r =,Q), 1 I(X a ;Y = X r =,Q)+ 4 I(X r (4) ;Y (4) X a (4) = X (4) R min { 2 I(X (2) (2) ;Y r,y a (2) X a (2) = X (2) r =,Q)+ 3 I(X (3) (3) ;Y r X a (3),X r (3) =,Q), 2 I(X (2) (2) ;Y a X a (2) = X r (2) =,Q)+ 4 I(X r (4) ;Y a (4) X a (4) = X (4) R a +R 1 I(X a ;Y r = X 3 I(X (3) a,x (3) (3) ;Y r X r (3) =,Q) r =,Q)+ 2 I(X (2) (2) ;Y r X a (2) = X r (2) =,Q)+ over all joint distriutions p(q)p (x a q)p (2) (x q)p (3) (x a,x q) p (4) (x r q) with Q 5 over the alphaet X 2 a X 2 X r. Remark: If the relay is not required to decode oth messages, then removing the constraint on the sum-rate R a +R in the region aove yields an outer ound. Proof outline: part of Theorem 2. The proof of Theorem 6 follows the same argument as the proof of the converse Octoer 9, 2018

12 12 IV. THE GAUSSIAN CASE In the following section, we apply the performance ounds derived in the previous section to the AWGN channel with pass loss. Definitions of codes, rate, and achievaility in the memoryless Gaussian channels are analogous to those of the discrete memoryless channels. If X a [k],x [k],x r [k] =, then the mathematical channel model is Y r [k] = g ar X a [k] +g r X [k] +Z r [k] and Y a [k] and Y [k] are given y similar expression in terms of g ar,g r and g a if only one node is silent. If X a [k] = X [k] = and X r [k], then Y a [k] = g ra X r [k] + Z a [k] and Y [k] = g r X r [k] + Z [k] and similar expressions hold if other pairs of nodes are silent, where the effective complex channel gain g ij etween nodes i and j comines oth quasi-static fading and path loss and the channels are reciprocal, i.e., g ij = g ji. For convenience, we define G ij := g ij 2, i.e. G ij incorporates path loss and fading effects on received power. Furthermore, we suppose the interesting case that G a G ar G r. Finally, we assume full Channel State Information (CSI) at all nodes (i.e. each node is fully aware of g a, g r and g ar ) and that each node has the same transmit power P for each phase, employs a complex Gaussian codeook and the noise is of unit power, additive, white Gaussian, complex and circularly symmetric. For convenience of analysis, we also define the function C(x) := log 2 (1+x). For a fading AWGN channel, we can optimize the i s for given channel mutual informations in order to maximize the achievale sum rate (R a + R ). First, we optimize the time periods in each protocol and compare the achievale sum rates otained to determine an optimal transmission strategy in terms of sum-rate in a given channel. For example, applying Theorem 3 to the fading AWGN channel, the optimization constraints for the TDBC protocol are 4 : R a min{ 1 C(PG ar ), 1 C(PG a )+ 3 C(PG r )} (23) R min{ 2 C(PG r ), 2 C(PG a )+ 3 C(PG ar )} (24) We have taken Q = 1 in the derivation of (23) and (24), since a Gaussian distriution simultaneously maximizes each mutual information term individually as each node is assumed to transmit with at most power P during each phase. Linear programming may then e used to find optimal time durations. The optimal sum rate corresponding to the inner ounds of the protocols is plotted in Fig. 3. As expected, the optimal sum rate of the HBC protocol is always greater than or equal to those of the other protocols since the MABC and TDBC protocols are special cases of the HBC protocol. Notaly, the sum rate of 4 The power constraint is satisfied almost surely as n in the random coding argument for Gaussian input distriutions with E[X 2 ] < P. Octoer 9, 2018

13 13 sum of achievale data rate (its/channel) DT(G a = 5 db) MABC TDBC(G a = 10 db) TDBC (G a = 5 db) HBC (G a = 10 db) HBC (G a = 5 db) G r (db) Fig. 3. Achievale sum rates of the protocols (P = 15 db, G ar = 0 db) the HBC protocol is strictly greater than the other cases in some regimes. This implies that the HBC protocol does not reduce to either of the MABC or TDBC protocols in general. In the MABC protocol, the performance region is known. However, in the other cases, there exists a gap etween the expressions. An achievale region of the 4 protocols and an outer ound for the TDBC protocol is plotted in Fig. 4 (in the low and the high SNR regime). As expected, in the low SNR regime, the MABC protocol dominates the TDBC protocol, while the latter is etter in the high SNR regime. It is difficult to compute the outer ound of the HBC protocol numerically since, as opposed to the TDBC case, it is not clear that jointly Gaussian distriutions are optimal due to the joint distriution p (3) (x a,x q) as well as the conditional mutual information terms in Theorem 6. For this reason, we do not numerically evaluate the outer ound. Notaly, some achievale HBC rate pairs are outside the outer ounds of the MABC and TDBC protocols. REFERENCES [1] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, Network information flow, IEEE Trans. Inform. Theory, vol. 46, pp , [2] T. Cover and J. Thomas, Elements of Information Theory, 2nd ed. New York:Wiley, [3] J. B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis. New York; Springer, [4] P. Larsson, N. Johansson, and K.-E. Sunell, Coded i-directional relaying, in the 5th Scandanavian Workshop on Wireless ad-hoc Networks, Stockholm, May [5], Coded i-directional relaying, in Proc. IEEE Veh. Technol. Conf. - Spring, 2006, pp Octoer 9, 2018

14 DT MABC TDBC HBC TDBC outer ound 0.5 R R a DT MABC TDBC HBC TDBC outer ound R R a Fig. 4. G a = 7 db) Achievale rate regions and outer ounds with P = 0 db (top) and P = 10 db (ottom) (G ar = 0 db, G r = 5 db, [6] T. J. Oechtering, C. Schnurr, I. Bjelakovic, and H. Boche, Achievale rate region of a two phase idirectional relay channel, in Proc. Conf. on Inf. Sci. and Sys., Baltimore, MD, Mar [7] P. Popovski and H. Yomo, The anti-packets can increase the achievale throughput of a wireless multi-hop network, in Proc. IEEE Int. Conf. Commun., 2006, pp [8], Bi-directional amplification of throughput in a wireless multi-hop network, in Proc. IEEE Veh. Technol. Conf. - Spring, 2006, pp [9] B. Rankov and A. Wittneen, Achievale rate regions for the two-way relay channel, in Proc. IEEE Int. Symp. Inform. Theory, Seattle, Jul. 2006, pp [10] C. E. Shannon, Two-way communications channels, in 4th Berkeley Symp. Math. Stat. Pro., Chicago, IL, Jun. 1961, pp [11] Y. Wu, P. A. Chou, and S.-Y. Kung, Information exchange in wireless networks with network coding and physical-layer Octoer 9, 2018

15 15 roadcast, Microsoft Research, Tech. Rep., Aug. 2004, MSR-TR [12], Information exchange in wireless networks with network coding and physical-layer roadcast, in Proc. Conf. on Inf. Sci. and Sys., Baltimore, MD, Mar [13] L. L. Xie, Network coding and random inning for multi-user channels, in the 10th Canadian Workshop on Information Theory, Edmonton, Alerta, Canada, Jun. 2007, pp Octoer 9, 2018