On the Capacity and Diversity-Multiplexing Tradeoff of the Two-Way Relay Channel

# On the Capacity and Diversity-Multiplexing Tradeoff of the Two-Way Relay Channel

Rahul Vaze and Robert W. Heath Jr.
The University of Texas at Austin
Department of Electrical and Computer Engineering
Wireless Networking and Communications Group
1 University Station C0803
Austin, TX 78712-0240
email: vaze@ece.utexas.edu, rheath@ece.utexas.edu
This work was funded by DARPA through IT-MANET grant no. W911NF-07-1-0028.
###### Abstract

This paper considers a multiple input multiple output (MIMO) two-way relay channel, where two nodes want to exchange data with each other using multiple relays. An iterative algorithm is proposed to achieve the optimal achievable rate region, when each relay employs an amplify and forward (AF) strategy. The iterative algorithm solves a power minimization problem at every step, subject to minimum signal-to-interference-and-noise ratio constraints, which is non-convex, however, for which the Karush Kuhn Tuker conditions are sufficient for optimality. The optimal AF strategy assumes global channel state information (CSI) at each relay. To simplify the CSI requirements, a simple amplify and forward strategy, called dual channel matching, is also proposed, that requires only local channel state information, and whose achievable rate region is close to that of the optimal AF strategy. In the asymptotic regime of large number of relays, we show that the achievable rate region of the dual channel matching and an upper bound differ by only a constant term and establish the capacity scaling law of the two-way relay channel. Relay strategies achieving optimal diversity-multiplexing tradeoff are also considered with a single relay node. A compress and forward strategy is shown to be optimal for achieving diversity multiplexing tradeoff for the full-duplex case, in general, and for the half-duplex case in some cases.

## I Introduction

We consider a multiple antenna two-way relay channel as shown in Fig. 1, where two nodes and want to exchange information with each other with the help of a relay node and all the nodes are equipped with one or more than one antenna. The two-way relay channel models the communication scenario where the destination terminal also has some data to send to source terminal e.g. downlink and uplink in cellular communication, or packet acknowledgments in a wireless network. The general discrete memoryless two-way relay channel was introduced in [1], and the multiple antenna two-way relay channel in [2]. In the literature, the two-way relay channel is also known by several other names, including the: bidirectional relay channel [3, 4, 5] and analog network coding [6].

A specific embodiment of a multiple antenna two-way relay channel that assumes half-duplex relays and the absence of a direct path between source and destination was proposed in [2]. An illustration is provided in Fig. 1. As shown in Fig. 1, in phase or the first time slot, both terminals and are scheduled to transmit simultaneously while the relay receives. In phase or the second time slot, the relay is scheduled to transmit while terminals and receive. The key idea with the two-way relay channel is that each terminal can cancel the interference (generated by its own transmission) from the signal it receives from the relay to recover the transmission from the other terminal. The idea is reminiscent of work in network coding [7], though note that here the coding is done in the analog domain, [6] rather than in digital domain [7]. In this paper we only consider multiple antenna two-way relay channel and for brevity, drop the prefix multiple antenna from here onwards.

There has been a growing interest in finding the capacity region of the two-way relay channel with a single relay node [8, 3, 4, 5, 9, 10, 11, 12, 13]. Achievable sum rate expressions (sum of the rates achievable from and links) have been derived in [2] and [5, 3, 4], for the half-duplex two-way relay channel, using amplify and forward (AF), decode and forward (DF) and compress and forward (CF) at the relay. It is shown that in a two-way relay channel, it is possible to remove the rate loss factor in spectral efficiency due to the half duplex assumption on the nodes. For a general full-duplex two-way relay channel with a single relay node (, and relay can transmit and receive at the same time) achievable rate regions are derived in [8] for AF, DF, and CF. For the AWGN two-way relay channel (no fading), using nested lattice coding and DF at the relay, the achievable rate region has been shown to be very close to the upper bound for all SNRs [10, 11]. Using the deterministic channel approach, the achievable rate region has been shown to be at most three bits away from the upper bound for the full-duplex two-way relay channel [9]. The capacity region of the two-way relay channel has also been studied in [12, 13], where in [13], it has been shown that in the low SNR regime the upper bound can be achieved by choosing a suitable relay mapping function, together with LDPC codes. The achievable rate region [3, 4, 9, 10, 11, 5, 8, 12, 13] does not meet the upper bound [14], in general. Consequently, the problem of finding the capacity region of the two-way relay channel is currently open.

The problem of finding the capacity region of the two-way relay channel becomes even more challenging when there are multiple relay nodes that can help and , and to the best of our knowledge has not been addressed in the literature. The problem becomes hard, because it is known that for the one-way relay channel with multiple relay nodes, DF does not work well [15], while the partial DF and distributed CF [15] lead to complicated achievable rate regions that are very hard to compute and analyze. The same conclusion holds true for the two-way relay channel; the only simple strategy that is well suited for multiple relay nodes is AF. With this motivation, in this paper we attempt to find the optimal relay beamformers that maximize the achievable rate region of the two-way relay channel with AF. For the one-way relay channel with multiple relays, optimal relay beamformers have been found [16], however, they are not known for the two-way relay channel.

For the case when both and have a single antenna, and each relay has an arbitrary number of antennas, we solve the problem of finding optimal relay beamformers by recasting it as an iterative power minimization algorithm. The iterative algorithm, at each step, solves a power minimization problem with minimum signal-to-interference-noise (SINR) constraints, for which satisfying the Karush Kuhn Tucker (KKT) conditions [17, 18] are sufficient for optimality. We consider both the sum power constraint across relays, as well as an individual relay power constraint. The optimal AF solution requires each relay to have channel state information (CSI) for all relays and leads to an achievable rate region that cannot be expressed in closed form.

For the case when each relay knows its own CSI, finding the optimal AF strategy is quite hard and intractable, even for the one-way relay channel case [16]. To remove the global CSI requirement, and to obtain a simple achievable rate region expression, next, we propose a simple AF strategy, called dual channel matching strategy, which works for any number of antennas at and . In dual channel matching, relay transmits the received signal multiplied with , if the channel between and relay is , between relay and is , between and relay is and between relay and is . Using dual channel matching, we lower bound the achievable rate region of the optimal AF strategy, which is unknown for more than one antenna at and , and bound the gap between the optimal AF strategy and the upper bound. The dual channel matching is quite simple to implement and its achievable rate region can be shown to be quite close (by simulation) to the optimal AF strategy, when and each have single antenna.

We upper bound the capacity region of the two-way relay channel using the cut-set bound [19] on the broadcast cut (), and , and the multiple access cut and (), over all possible two phase protocols (with different time allocation between first and second phase). We show that the gap between the upper and lower bound (dual channel matching) is quite small for small values of . In the limit , we show that the gap is constant with increasing , and thus establish the scaling law [20] of the capacity region of the two-way relay channel, which shows that bits can be transmitted from both and , simultaneously.

We also consider the problem of finding relay transmission strategies to achieve the optimal diversity multiplexing (DM)-tradeoff [21] of the two-way relay channel with a single relay node, in the presence of a direct path between and . The DM-tradeoff captures the maximum rate of fall of error probability with signal to noise ratio (SNR), when rate of transmission is increased as . The DM-tradeoff for the two-way relay channel is a two-dimensional region spanned by the , where and are the negatives of the exponent of the probability of error from and , respectively, when is transmitting at rate and at . The DM-tradeoff for the one-way relay channel has been studied in [22, 23, 24, 25], where notably in [25], it has been shown that the CF strategy achieves the DM-tradeoff for both the full-duplex as well as the half-duplex case. The DM-tradeoff of the two-way relay channel has been recently studied in [26], where upper and lower bounds are obtained on the DM-tradeoff which are shown to match for the case when each node has a single antenna.

We first consider the full-duplex two-way relay channel and show that a slightly modified version of the CF strategy [27] achieves the optimal DM-tradeoff. More importantly, we show that does not depend on () and the two-way relay channel can be decoupled into two one-way relay channels using the CF strategy. Then we consider the more interesting case of half-duplex nodes, where the achievable rate regions are protocol dependent. For the two-way relay channel it is not known which protocol achieves the highest possible rates [3, 4, 5]. We use a three phase protocol, where in phase one transmits to both the relay and , in phase two transmits to both the relay and and in phase three the relay transmits to and . This three phase protocol makes use all the direct links between different nodes in a two-way relay channel. For this three phase protocol, we propose a modified CF strategy and show that it can achieve the optimal DM-tradeoff in some cases. We conjecture that our strategy can also achieve the optimal DM-tradeoff in general, but we are yet to prove it.

Notation: The following notation is used in this paper. The superscripts represent the transpose and transpose conjugate. denotes a matrix, a vector and the element of . For a matrix by we mean . and denotes the determinant and trace of matrix , respectively. denotes the expectation. denotes the usual Euclidean norm of a vector and denotes the absolute value of a scalar. is a identity matrix. is the cardinality of set . We use the usual notation for if remains bounded, as . means is a circularly symmetric complex Gaussian random variable with zero mean and variance and means given , is a circularly symmetric complex Gaussian random variable with zero mean and variance . denotes the set of matrices with complex entries. denotes that the sequence of random variables converge to a random variable with probability . We use to denote equality with probability i.e. and is defined similarly. denotes the mutual information between and and the differential entropy of [19]. To define a variable we use the symbol .

Organization: The rest of the paper is organized as follows. In Section II, we describe the two-way relay channel system model, the protocol under consideration and the key assumptions. In Section III, we obtain the optimal AF strategy to maximize the achievable rate region of the two-way relay channel. In Section IV, we introduce a simple AF strategy, dual channel matching, and lower bound the achievable rate region of the optimal AF strategy of Section III. In Section V, we derive an upper bound on the capacity of the two-way relay channel capacity and compare it with the achievable rate region of the optimal AF strategy and dual channel matching. In Section VI, we show that the CF strategy can achieve the optimal DM-tradeoff for full-duplex two-way relay channel, in general, and in some cases for the half-duplex case. Final conclusions are made in Section VII.

## Ii System and Channel Model

In this section we describe the two-way relay channel system model under consideration, and then present the relevant signal and channel models.

### Ii-a System Model

For the first part of the paper Section III, IV, and V, we consider a wireless network where there are two terminals and who want to exchange information via relays, as shown in Fig. 2. The relays do not have any data of their own and only help and communicate. The relays are assumed to be located randomly and independently so that the channel coefficients between each relay and and are independent. We also assume that there is no direct path between and and that they can communicate only through the relays. This is a realistic assumption when relaying is used for coverage improvement in cellular systems, since at the cell edge the signal to noise ratio is extremely low for the direct path. In ad-hoc networks, it can be the case that two terminals want to communicate, but are out of each other’s transmission range.

We assume that both the terminals and have antennas and all the relays have antennas each. We further assume that both the terminals and all the relays can operate only in half-duplex mode (cannot transmit and receive at the same time). The communication protocol is summarized as follows [2]. In any given time slot, for the first fraction of time, called the transmit phase, both and are scheduled to transmit and all the relays receive a superposition of the signals transmitted from and . In the rest fraction of the time slot, called the receive phase, all the relays are scheduled to transmit simultaneously and both the terminals receive. Both and are assumed to have power constraint of , while for relays we assume two different power constraints, the sum power constraint where the sum of the power of all relays is or the individual power constraint where each relay has power constraint of .

For the second part of the paper, Section VI, we assume a two-way relay channel with a single relay node and the presence of a direct path between and as shown in Fig. 4. We assume that has antennas, has antennas, and the relay node has antennas.

### Ii-B Channel and Signal Model

Throughout this paper we assume that all the channels are frequency flat slow fading block fading channels, where in a block of time duration (called the coherence time), the channel coefficients remain constant and change independently from block to block. We assume that is more that the duration of time slot used by and to communicate with each other as described before. As shown in Fig. 3, let the forward channel between and the relay be and the backward channel between relay and be . Similarly let the forward channel between relay and be and the backward channel between and the relay be . For Section VI, where the direct path between and is considered, the channel between and is denoted by and in the reverse direction by . We assume that with independent and identically distributed (i.i.d.) entries.

For the first part of the paper Section III, IV and V, we consider the following signal model. The received signal at the relay is given by

 rk=√PMHkx1+√PMGrkx2+nk (1)

if and are the signals transmitted from and to be decoded at and respectively, with , is the power transmitted by and , respectively. The noise is the spatio-temporal white complex Gaussian noise independent across relays with . Relay processes its incoming signal to transmit a signal with (sum power constraint) or (individual power constraint) in the receive phase. The received signals and at terminal and , respectively, in the receive phase, are given by

 y1=K∑k=1Hrktk+z1, (2)
 y2=K∑k=1Gktk+z2, (3)

where and are spatio-temporal white complex Gaussian noise vectors with .

Throughout this paper we assume that both and perfectly know in the receive mode. To be precise, in the receive phase (i.e. when and receive signal from all the relays), and both know and . We also assume that no transmit CSI is available at and , i.e. in the transmit phase and have no information about what the realization of and is going to be when it transmits its signal to all the relays in the transmit phase, respectively.

In this paper we assume different CSI assumptions at the relay. For finding the optimal AF strategy (Section III) we assume that each relay knows for all . To reduce the CSI requirements next, we present a simple AF strategy in Section IV where we assume that relay only knows . In Section VI, we assume that the relay knows , as well as , the channel coefficient between and .

## Iii Optimal AF strategy for two-way relay channel

In this section we will find optimal relay beamformers that maximize the achievable rate region of the two-way relay channel with AF, when and have a single antenna each, . For simplicity of exposition, in this section we consider the case when each relay nodes has a single antenna, . Generalizations to are straightforward, and will be described later.

To start with, because of single antenna restriction, the channel between and relay is denoted by and between relay and denoted by . For the reverse direction the channel coefficients are the same as in forward direction but with an added superscript , e.g. channel coefficient between relay and is denoted by . With AF strategy, each relay node transmits the received signal multiplied with to both and . Thus, if and is the transmitted signal from and , respectively, then the received signal at , , and , is

 y1 = K∑k=1√Phrkwkgrkx2+√Phrkwkhkx1+hrkwknk+z1, y2 = K∑k=1√Pgkwkhkx1+√Pgkwkgrkx1+gkwknk+z2, (4)

where is noise added at relay and , and are added at and . Since and are known at and , respectively, their contribution can be removed from the received signal at and , respectively. Let the rate of transmission from to be and from to be , then from (4)

 R12 = log⎛⎜ ⎜⎝1+P(∣∣∑Kk=1gkwkhk∣∣2)1+∑Kk=1|gkwk|2⎞⎟ ⎟⎠, R21 = log⎛⎜ ⎜⎝1+P(∣∣∑Kk=1hrkwkgrk∣∣2)1+∑Kk=1|hrkwk|2⎞⎟ ⎟⎠.

Thus, the achievable rate region for the two-way relay channel with AF for a sum power constraint across all relays, i.e. is the set and for individual power constraint at each relay, i.e. is the set . Therefore, the problem is to find optimal ’s that achieve the boundary points of the region , for both the sum power constraint and an individual power constraint.

For the one-way relay channel, no communication from to , optimal ’s have been found in [16] to maximize . The solution of [16], provides an upper bound on individual rates and and is equivalent to solutions where or is greedily maximized disregarding the other. The problem in the two-way relay channel case is to find optimal ’s such that is maximized, for each , where , and . Towards that end, we use the rate profile method [28] to identify ’s that meet the boundary point of . Next, we only consider the sum power constraint across the relays. For individual power constraints the same procedure can be applied as pointed out later. Thus, the optimization problem can be formulated as follows.

Maximize
subject to
,
(5)

An equivalent problem to this problem is the following iterative power minimization problem subject to rate constraints,

Minimize
subject to ,
,
(6)

where at each iteration is changed to maximize the achievable rate, subject to power constraint. To be precise, if the value of at iteration is say and the solution to (6) is feasible (i.e. if ) 111For an individual power constraint the same can be done by checking at each iteration whether the obtained solution is feasible with individual power constraints or not., then is incremented in next iteration, otherwise decreased. Choice of the step size of increase or decrease determines the speed of convergence to the optimal rate , for which . One possible starting point for is times the maximum provided by [16] for one way relay channel. The step size can be chosen by bisection between the last feasible (initially ) and the last infeasible . Even though this equivalent problem provides a solution to (5) in a iterative manner, the problem (6) is in general non-convex, and not easy to solve. To overcome this limitation, we recast the problem (6) as a standard power minimization problem subject to signal-to-interference-noise ratio (SINR) [18], where the forwarded noise from each relay plays the role of interference. For a given and , the problem (6) is of the form

Minimize
subject to
.
(7)

This problem again is non-convex, however, it is of the form

Minimize f(x) ||ai(x)||2−|bi(x)|2≤0, ∀ i,
(8)

where is a convex function, is an affine function of and , by noting that if , or are less than zero or complex, then they can be scaled by appropriate phases to make them real and positive, without changing the objective function or the constraints 222An immediate consequence of this property is that the optimal solution does not change if all are scaled by , or all ’s are scaled by ..

For the problem (8), it has been shown in [18], that if the problem is strictly feasible, then KKT conditions [17] are necessary and sufficient to find the optimal solution. It is easy to see that the problem (7) is strictly feasible and therefore KKT conditions are sufficient for optimality. The Lagrangian of problem (7) is of the form

 L = wAw∗+λ1(wBw∗−1γ0|cwT|2+1)+λ2(wDw∗−1γ1|ewT|2+1),

where and

 A=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣P(|h1|2+|gr1|2)+10…00P(|h2|2+|gr2|2)+1…000⋱00…0P(|hK|2+|grK|2)+1⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,
 B=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣|g1|20…00|g2|2…000⋱00…0|gK|2⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦, D=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣|hr1|20…00|hr2|2…000⋱00…0|hrK|2⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,

and

Differentiating the Lagrangian yields

 (A+λ1B+λ2D−λ1γ0c∗c+λ2γ1e∗e)w=0,

and the optimal is found by solving for and using the constraints 333Clearly, the optimal lies in the null space of some matrix that is a function of , and and hence not unique..

Therefore, by recasting our original problem of obtaining the boundary points of to the power minimization problem with SINR constraints, we have shown that the optimal solution can be found in an efficient way. In Section V, we plot the achievable rate region of the optimal AF strategy and compare it with the lower bound obtained by using dual channel matching, and an upper bound.

Recall that we only considered a two-way relay channel, where each relay had a single antenna, . Extension to , is straightforward by replacing by , by , by and by , which are scalars as before, and the optimal solution to ’s can be found using the iterative power minimization algorithm (6).

Our algorithm to optimize the achievable region with AF is fairly simple, however, it assumes that each relay has CSI for all the relay nodes, and requires . Finding optimal relay beamformers where each relay has only its CSI, and , is rather complicated and has not been solved even for the one-way relay channel [16]. Another limitation of the optimal AF strategy is that the expression for the obtained rate region cannot be written down in close form, and therefore does not allow analytical tractability for comparison with an upper bound. To remove these restrictions, in the next section we propose a simple AF strategy, called dual channel matching, where each relay uses its own CSI, and for which the achievable rate region expression can be written down in a closed form. Since dual channel matching is in general, a suboptimal AF strategy, the achievable rate region of dual channel matching lower bounds the rate region of the optimal AF strategy, and allows to estimate the difference between the optimal AF strategy and the upper bound.

## Iv Dual Channel Matching Strategy

In this section we propose a simple AF strategy, called dual channel matching, and derive a lower bound on the achievable rate region for the two-way relay channel. With the dual channel matching strategy relay multiplies to the received signal and forwards it to and , where is the normalization constant to satisfy the power constraint. Dual channel matching tries to match both the channels which the data streams from to and to experience at each relay node. The motivation for this strategy is that for one-way relay channel (i.e. has no data for ) with one relay node, the optimal AF strategy is to multiply to the signal at the relay, where the singular value decomposition of is and is and is a diagonal matrix whose entries are chosen by waterfilling [29]. In dual channel matching the complex conjugates of the channels are used directly rather than the unitary matrices from the SVD of the channels [29]. This modification makes it easier to analyze the achievable rates for the two-way relay channel. Note that the dual channel matching is an extension of the listen and transmit strategy of [30] for the one-way relay channel, where each relay transmits the received signal after scaling it with the complex conjugates of the forward and backward channel coefficients.

Together with dual channel matching we restrict the signal transmitted from and , and , respectively, to be circularly symmetric complex Gaussian distributed with covariance matrix , to obtain a lower bound on the achievable rate region of two-way relay channel. Moreover, we use i.e. and transmit and receive for same amount of time. The achievable rates and using the dual channel matching can be computed as follows.

From (1), the received signal at the relay is given by

 rk=√PMHkx1+√PMGrkx2+nk.

Using dual channel matching as described above, at relay , is multiplied to the received signal so that the transmitted signal is given by

 tk=√βk(G∗kH∗k+Hr∗kGr∗k)rk

where is to ensure that 444This is for the sum power constraint. For an individual power constraint, is chosen such that for each .. With dual channel matching the received signal at is given by

 y2=K∑k=1Gktk+z. (9)

Expanding (9) we can write

 y = K∑k=1√PβkMGk(G∗kH∗k+Hr∗kGr∗k)HkAx1+K∑k=1√PβkMGk(G∗kH∗k+Hr∗kGr∗k)Grkx2 +K∑k=1√βkGk(G∗kH∗k+Hr∗kGr∗k)Bknk+z.

Since and all the channel coefficients are known at , the second term can be removed from the received signal at . Moreover, as described before is circularly symmetric complex Gaussian vector with covariance matrix , thus the achievable rate for to link is [31]

 R12=12I(x1;y2)=12logdet⎛⎝IM+AA∗(K∑k=1BkB∗k+IM)−1⎞⎠, (10)

since Similarly, we obtain the expression for ,

 R21=12logdet⎛⎝IM+CC∗(K∑k=1DkD∗k+IM)−1⎞⎠, (11)

where and This rate region expression obtained is analytically tractable and can be used to compare the loss between the optimal AF strategy and the upper bound. Another interesting question of interest is how does the achievable rate region behaves with . To answer that question, we turn to asymptotics and compute the rate region in the limit , in the next lemma.

###### Lemma 1

As grows large, ,

 limK→∞R12 =w.p.1 M2logK+O(1), limK→∞R21 =w.p.1 M2logK+O(1).

Proof: Consider

 2R12−logdetKIM = logdet⎛⎝IM+AA∗(K∑k=1BkB∗k+IM)−1⎞⎠−logdetKIM, = logdet⎛⎝1KIM+A√KA∗√K(K∑k=1BkB∗k+IM)−1⎞⎠.

To satisfy the sum power constraint, let 555Equal power allocation among relays., where is a constant such that

which is same for all . Then,

 A√K=√PPRc1M1KK∑k=1Gk(G∗kH∗k+Hr∗kGr∗k)Hk,

which by using strong law of large numbers, converges to,

 A√Kw.p.1−−−→√PPRc1MN2IM,

since , and . Same result holds true for . With ,

 K∑k=1BkB∗k=PRc11KK∑k=1(Gk(G∗kH∗k+Hr∗kGr∗k))(Gk(G∗kH∗k+Hr∗kGr∗k))∗,

which again using the strong law of large numbers converges to , for some finite , since are i.i.d. with finite variance. Thus, in the limit ,

 2R12−logdetKIM→Mlog(PPRN4c1M(PRθ+c1)),

and thus it follows that

 R12=w.p.1M2logK+O(1). (12)

Similarly we get the achievable rate on the to link as

 limK→∞R21=w.p.1M2logK+O(1). (13)
\QED

Discussion: In this section we introduced the dual channel matching AF strategy, and obtained a lower bound on the capacity region of the two-way relay channel. Dual channel matching is a simple AF strategy that requires local CSI, and as we will see in Section V, has achievable rate region very close to that of the optimal AF strategy (Section III) for . We also derived the asymptotic achievable rate region of the dual channel matching, by taking the limit , and using the law of large numbers. We showed, that in the asymptotic regime, both and scale as with increasing .

Next, we derive an upper bound on the capacity region of the two-way relay channel, and compare it with the achievable rate region of the dual channel matching.

## V Upper Bound on the Two-Way Relay Channel Capacity

In this section we upper bound the capacity region of the two-way relay channel using the cut-set bound [19] for the broadcast cut, and the multiple access cut. We assume a general two-phase protocol where for fraction of the time slot and transmit to all relays and the rest of the fraction of time slot all relays simultaneously transmit to both and . Note that to lower bound the capacity of the two-way relay channel using dual channel matching, we used which might be suboptimal. We prove later that for the asymptotic case of , is optimal.

The upper bound is derived as follows. We start by first separating and then