Performance Bounds for Bi-Directional Coded Cooperation Protocols

In coded bi-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 bounds for this problem for each of three protocols. The first protocol is a two phase protocol where both users simultaneously transmit during the first phase and the relay alone transmits during the second. In this protocol, our bounds are tight. The second protocol considers sequential transmissions from the two users followed by a transmission from the relay while the third protocol is a hybrid of the first two protocols and has four phases. In the latter two protocols the inner and outer bounds are not identical, and differ in a manner similar to the inner and outer bounds of Cover's relay channel. Numerical evaluation shows that at least in some cases of interest our bounds do not differ significantly. Finally, in the Gaussian case with path loss, we derive achievable rates and compare the relative merits of each protocol in various regimes. Surprisingly, we find that in some cases, the achievable rate region of the four phase protocol sometimes contains points that are outside the outer bounds of the other protocols.


I. INTRODUCTION
Consider two users, denoted by a and b, who wish to share independent messages over a shared channel.Traditionally, this problem is known as the two-way channel [2], [10].
In many realistic broadcast environments, such as wireless communications, it is not unreasonable to assume the presence of a third node which may aid in the exchange of a and b's messages.In particular, if a is a mobile user and b is a base station, then we may suppose the presence of a relay station r to assist in the bi-directional communication.
Traditionally, without the presence of the relay station, communication between nodes a and b is performed in two steps: first a transmits its message to b followed by similar transmission from b 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, by taking advantage of the shared wireless medium, it is known that the third and fourth transmissions may be combined (Fig. 1.iii) into a single transmission using, for example, ideas from network coding [1], [13].In particular, if the messages of a and b are w a and w b respectively and belong to a group, then it is sufficient for the relay node to successfully transmit w a ⊕ w b simultaneously to a and b.In [4], [5], such a three phase coded bi-directional protocol is considered when the group is Z k 2 , the binary operator is component-wise modulo 2 addition (i.e., exclusive or) and encoding is performed linearly to produce parity bits.As each user transmits sequentially, each user is amenable 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, but combine the first two phases into a single joint transmission by nodes a and b followed by a single transmission by the relay which forwards its received signal (Fig. 1.iv).Coded bi-directional cooperation may also be extended for the case of multiple relaying nodes [11], [12].In [9], achievable rate regions are derived assuming full duplex capabilities at all nodes.
In this paper, we are interested in determining fundamental bounds on the performance of coded bidirectional communications assuming various decode-and-forward protocols for half-duplex channels.In the case of a two phase protocol where both users transmit simultaneously in the first phase followed by 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 bounds on the capacity regions.We find that a four phase hybrid protocol is sometimes strictly better than the outerbounds of two or three phase decode-and-forward protocols 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 bounds for the protocols while in Section IV, we numerically compute these bounds for fading Gaussian channels.

A. Notation and Definitions
We first start with a somewhat more general formulation of the problem.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 alphabet X * i = X i ∪ {∅} and channel output alphabet Y * i = Y i ∪ {∅}, where ∅ is a special symbol 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 Otherwise, the effect of one node remaining silent on the received variable at another node may be arbitrary at this point.The channel is assumed discrete memoryless.In Section IV, we will be interested in the case The objective of this paper is to determine achievable data rates and outer bounds 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 .
For a given protocol P, we denote by ∆ 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 by X k i , where the total duration of the protocol is n and X (ℓ) i denotes the random variable with alphabet X * i and input distribution 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 X k S := {X k i |i ∈ S}, the set of transmissions by all nodes in the set S at time k and similarly X (ℓ) i |i ∈ S}, a set of random variables with channel input distribution 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 alphabets X * i .Boldface x i represents a vector indexed by time at node i.Finally, it is convenient to denote by x S := {x i |i ∈ S}, a set of vectors indexed by time.
Encoders are then given by functions ), for k = 1, . . ., n and de- ).Given a block 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 block, and the corresponding encoders/decoders result in relative phase durations {∆ ℓ,n }, where the subscript n indicates that the phase duration depends on the choice of block size (as they must be multiples of 1/n).A set of rates {R i,j } is said to be achievable for a protocol with phase durations {∆ ℓ }, if there exist encoders/decoders of block length n = 1, 2, . . .with P [E i,j ] → 0 and ∆ ℓ,n → ∆ ℓ as n → ∞ ∀ℓ.An achievable rate region (resp.capacity region) is the closure of a set of (resp.all) achievable rate tuples for fixed {∆ ℓ }.

B. Basic Results
In the next section, we will use a variation of the cut-set bound.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 be independent.Given subsets S, T ⊆ M, we define W S,T := {W i,j |i ∈ S, j ∈ T } and Lemma 1: If in some network the information rates {R i,j } are achievable for a protocol P with relative durations {∆ ℓ }, then for every ǫ > 0 and all S ⊂ {1, 2, for a family of conditional distributions p (ℓ) (x 1 , x 2 , . . ., x m |q) and a discrete time-sharing random variable Q with distribution p(q).Furthermore, each p (ℓ) (x 1 , x 2 , . . ., x m |q)p(q) must satisfy the constraints of phase ℓ of protocol P.
Proof: Replacing W (T ) by W S,S c and W (T c ) by W S c ,M in (15.323) -(15.332) in [2], then all the steps in [2] still hold and we have where ǫ n → 0 since i∈S,j∈S c P [E i,j ] → 0 and the distributions p(x k 1 , . . ., x k m , y k 1 , . . ., y k m ) are those induced by encoders for which . ., we thus have Defining the discrete random variable S := X Qℓ S .Finally, since the distributions p (ℓ) (x 1 , x 2 , . . ., x m |q)p(q) are those induced by 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 distributions p (ℓ) (x 1 , x 2 , . . ., x m |q)p(q).

C. Protocols
In bi-directional cooperation, two terminal nodes denoted a and b exchange their messages.The messages to be transmitted are W a := W a,b , W b := W b,a and the corresponding rates are R a := R a,b and R b := R b,a .The two distinct messages W a and W b are taken to be independent and uniformly distributed in the set of {0, . . ., ⌊2 nRa ⌋ − 1} := S a and {0, . . ., ⌊2 nRb ⌋ − 1} := S b , respectively.Then W a and W b are both members of the additive group Z L , where L = max(⌊2 nRa ⌋, ⌊2 nRb ⌋).
The simplest protocol for the bi-directional channel, is that of Direct Transmission (DT) (Fig. 2

.i).
Here, since the channel is memoryless and ǫ > 0 is arbitrary, the capacity region from Lemma 1 is : where the distributions are over the alphabets X a and X b 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 Hybrid Broadcast (HBC).Then, the message from a (resp.b) to r is W a,r = W a (resp.W b,r = W b ) and the corresponding rate is R a,r = R a (resp.R b,r = R b ).Also, in our protocols, all phases are contiguous, i.e., they are performed consecutively and are not interleaved or re-ordered. 3n the MABC protocol (Fig. 2.ii), terminal nodes a and b 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 b is able to receive any meaningful side-information during the first phase due to the half-duplex constraint.
In the TDBC protocol (Fig. 2.iii), only node a transmits during the first phase and only node b transmits during the second phase.In phase 3, relay r performs a transmission based on the received data from the first two phases.Here, node a attempts to recover the message W b based on both the transmissions from node b 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 b may also recover W a based on first phase side information and the received signal at node b during the third phase.
Finally, we consider a Hybrid 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 b.

A. MABC Protocol
Theorem 2: The capacity region of the half-duplex bi-directional relay channel with the MABC protocol is the closure of the set of all points (R a , R b ) satisfying over all joint distributions p(q)p (1) (x a |q)p (1) (x b |q)p (2) (x r |q) with |Q| ≤ 5 over the alphabet X a ×X b ×X r .
Remark: If the relay is not required to decode both messages, then the region above is still achievable, and removing the constraint on the sum-rate R a + R b yields an outer bound.
Proof: Achievability: Random code generation: For simplicity of exposition only, we take |Q| = 1 and therefore consider distributions p (1) a (w a ) with w a ∈ S a and x r (w r ) with w r ∈ Z L where L = max(⌊2 nRa ⌋, ⌊2 nRb ⌋), according to p (1) (x a ), p (1) (x b ) and p (2) (x r ) respectively.
Encoding: During phase 1, encoders of node a and b send the codewords x Decoding: a and b estimate wb and wa after phase 2 using jointly typical decoding.Since w r = w a ⊕w b and a knows w a , node a can reduce the number of possible w r to ⌊2 nRb ⌋ and likewise at node b, the cardinality is ⌊2 nRa ⌋.
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 = {b}, S 3 = {r}, S 4 = {a, b}, S 5 = {a, r} and S 6 = {b, r}, as well as two rates R a and R b .The outer bound corresponding to S 1 is then a ; Y r , Y b |X r , X a ; Y r , Y b |X r , X a ; Y r |X b , X where (9) follows since in the MABC protocol, we must have We find the outer bounds of the other cut-sets in the same manner: b ; Y r |X a , X S 3 : N/A (13) a , X b ; Y r |X r ; Y a |X (2) Since ǫ > 0 is arbitrary, together, ( 9), ( 12) -( 16) and the fact that the half-duplex nature of the channel constrains X a to be conditionally independent of X b given Q yields the converse.By Fenchel-Bunt's theorem in [3], it is sufficient to restrict |Q| ≤ 5. 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 between the expressions.An achievable region of the 4 protocols and an outer bound 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 better in the high SNR regime.
It is difficult to compute the outer bound of the HBC protocol numerically since, as opposed to the TDBC case, it is not clear that jointly Gaussian distributions are optimal due to the joint distribution p (3) (x a , x b |q) as well as the conditional mutual information terms in Theorem 6.For this reason, we do not numerically evaluate the outer bound.Notably, some achievable HBC rate pairs are outside the outer bounds of the MABC and TDBC protocols.

Fig. 2 .
Fig. 2. Proposed protocol diagrams.Shaded areas denote transmission by the respective nodes.It is assumed that all nodes listen when not transmitting.

( 1 )
b (w b ) with w b ∈ S b , and

( 1 )
a (w a ) and x (1) b (w b ) respectively.Relay r estimates ŵa and ŵb after phase 1 using jointly typical decoding, then constructs w r = ŵa ⊕ ŵb in Z L and sends x (2) r (w r ) during phase 2.

T
) sequences of length n • ∆ ℓ,n according to the input distributions employed in phase ℓ.Also define the set of codewords x (ℓ) S (w S ) := {x

i
(w i )|i ∈ S} and the events D (ℓ) S,T (w S ) := {(x }, where S and T are disjoint subsets of nodes.