Su Yuping (蘇玉萍)

(State Key Lab of ISN, Xidian University, Xi’an 710071, P.R.China)

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Diversity-multiplexing tradeoff of half-duplex multi-input multi-output two-way relay channel with decode-and-forward protocol①

Su Yuping (蘇玉萍)②

(State Key Lab of ISN, Xidian University, Xi’an 710071, P.R.China)

A multi-input multi-output (MIMO) separated two-way relay channel (STWRC) is considered, where two users exchange their messages via a relay node. When each link is quasi-static Rayleigh fading, the achievable diversity-multiplexing tradeoff (DMT) of the half-duplex STWRC is analyzed. Firstly, the achievable DMT of the STWRC with static decode-and-forward (DF) protocol is obtained. Then, a dynamic decode-and-forward (DDF) protocol for the STWRC is proposed, where the relay listening time varies dynamically with the channel qualities of the links between two users and the relay. Finally, the achievable DMT of the proposed DDF protocol is derived in a case-by-case manner. Numerical examples are also provided to verify the theoretical analysis of both protocols.

two-way relay channel (TWRC), half-duplex, diversity-multiplexing tradeoff (DMT), decode-and-forward (DF), outage probability

0 Introduction

Recently, relay techniques have attracted increasing attentions due to their ability of increasing reliability and throughput of the system. For slow fading relay channels, the diversity-multiplexing tradeoff (DMT)[1]is often analyzed to describe the tradeoff between the transmission rate and the reliability at high signal-to-noise ratio (SNR) regime. Existing works related to the DMT analysis mainly focus on two kinds of relay channels: one-way relay channels[2,3]and two-way relay channels (TWRCs)[4,5]. In the one-way relay channel, a source unidirectionally transmits messages to the destination via relay node(s). On the contrary, in the TWRC, two sources exchange their messages via relay node(s) and data flows bidirectionally.

According to whether a relay node can transmit and receive messages simultaneously, the relay channels can operate in the full-duplex mode or the half-duplex mode. For TWRC, existing works mainly focus on the DMT analysis of the full-duplex relay[4,5]. Specifically, for the separated two-way relay channel (STWRC) where two users cannot overhear each other directly, it is shown in Ref.[4] that the compress-and-forward (CF) strategy achieves the optimal DMT. Similarly, for TWRC where a direct link exist between two users, its optimal DMT is also obtained by using the CF protocol[5]. For multi-way relay channels, the optimal DMT for the full-duplex relay is studied in Ref.[6]. Recently, the finite-SNR DMT analysis of TWRC is also studied[7,8]. Besides, spatial channel pairing and beamforming for the multi-pair two-way relay networks are also investigated in Refs[9] and [10], respectively.

To the best of our knowledge, there's no work focusing on the DMT analysis of the half-duplex STWRC. As a result, the paper focuses on the achievable DMT of the STWRC with half-duplex relay. When each link is of frequency non-selective quasi-static Rayleigh fading, the achievable DMT of the multi-input multi-output (MIMO) STWRC with decode-and-forward (DF) protocol is analyzed. Firstly, the achievable DMT of the static DF protocol is obtained by scaling the DMT curve of the full-duplex case[4]. Then, a dynamic decode-and-forward (DDF) protocol is proposed, in which the listening time of the relay varies dynamically with the channel qualities of the links from two users to the relay. Finally, the achievable DMT of the proposed DDF protocol is derived in a case-by-case manner. Numerical examples are also provided to illustrate the achievable DMT performance for both protocols.

This paper is organized as follows. In Section 1, the channel model is introduced. In Section 2, the achievable DMT for both static and dynamic DF protocols is investigated. The paper is finally concluded in Section 3.

1 Channel model

The STWRC model, where two users exchange messages with the assistance of a single relay, is shown in Fig.1. User i, i∈A={1,2}, is equipped with Miantennas and the relay is equipped with N antennas. A channel such as (M1, N, M2)-STWRC system is refered. It is assumed the channel operates in the time-division duplex (TDD) mode and all the nodes are of half-duplex. Let the codeword of each user consisting of L symbols and t (t∈(0,1)) be the time fraction of the relay to listen. During the first tL symbol intervals, the relay only listens to the users transmission (listening phase) and during the remaining (1-t)L symbol intervals, the relay transmits its own codeword to two users (transmission phase). All the links in the STWRC are assumed to be frequency non-selective quasi-static fading and the codeword length L is assumed to be sufficiently long so that the error probability is dominated by the channel outage probability.

Fig.1 Channel model of separated two-way relay channel

During the listening phase, the received signal at the relay node is

During the transmission phase, the received signal at user i is

where Xi∈Mi×1,i∈A, and Xr∈N×1are the transmitted signal vectors at user i and the relay, respectively. Vectors Wr∈N×1and Wi∈Mi×1, i∈A, are the additive noise, whose entries are independent and identically distributed (i.i.d.) complex Gaussian CN(0,1) random variables. Matrices Hi∈N×Miand Gi∈Mi×N, i∈A, are the channel matrices with i.i.d. CN(0,1) entries. Due to the reciprocity of channel matrices in the TDD mode, we have, where (·)Hdenotes the matrix conjugate transpose. SNR is the average signal-to-noise ratio at each receive antenna.

2 Achievable DMT of MIMO STWRC

In this section, the achievable DMT of the MIMO STWRC with DF protocols is derived, including both static and dynamic cases.

Before proceeding, first some definitions are given as in Ref.[1]. a scheme is considered as a sequence of codes {C(SNR)}, where for each SNR, the corresponding code C(SNR) consists of 2LR1(SNR)×2LR2(SNR) codewords and the code rate for user i is Ri(SNR), i∈A.

For this sequence of codes, multiplexing gain of user i is riif

holds. Symbol “?” is used to denote exponential equality, i.e., the equality f(SNR)?SNRbto denote

2.1 DMT of MIMO STWRC with static DF protocol

For the static DF protocol, the time allocation between the listening phase and the transmission phase is fixed and independent of the channel realization. Such a protocol is referred to as DF with fixed time allocation (fDF). For the DF protocols, due to the decoding requirement of both messages at the relay, the DMT analysis for different diversity requirement becomes very difficult[11]. Thus, it is assumed that two users have the same diversity gain requirement d. The achievable DMT of the fDF protocol is given as follows.

Proposition 1: The achievable DMT of the half-duplex STWRC with fDF protocol is

(1)

Proof: The achievable DMT of a half-duplex STWRC with fDF protocol can be directly obtained by scaling the DMT curve of the full-duplex case[4]with time division coefficients.

When r1=r2=r, the achievable symmetric DMT is got as

(2)

Fig.2 Achievable symmetric DMT for a (4,2,3)-STWRC system

2.2 DDF Protocol for MIMO STWRC

In Ref.[13], the DDF protocol for the cooperative relay channel with single antenna is proposed. In DDF for the relay channel, the relay listens until the accumulated mutual information over the source-relay channel is sufficient for the transmission rate. Here, DDF is considered for the MIMO STWRC. During the listening phase, the message transmission from two users to the relay is a multiple-access channel (MAC), and its instantaneous capacity region is characterized as[12]

RS≤I(XS; Yr|XSc)IS,R, S?A

(3)

where RS=∑i∈SRi, XS={Xi:i∈S} and Scis the complement of S in A. To ensure that the relay can decode two users' messages successfully, accumulated mutual information tLIS,Rmust exceed LRSfor each S?A. Therefore, time fraction t is chosen as

(4)

where RS/IS,Ris the ratio between the real transmission rate and the corresponding mutual information (referred to as rate-to-mutual information ratio (RMR)).

In Eq.(4), each term (i.e.,RS/IS,R) in the bracket is a random variable that depends on the channel state between users and the relay. Time fraction t is the maximum of such three random variables. As a result, t is also a random variable depending on the channel state. Unfortunately, the probability density function (p.d.f) of each RMR is very complex and they are not independent with each other, so p.d.f of t is very difficult to obtain. As an alternative method,some simulation results of the percentage are given that t is equal to each RMR in Eq.(4). The simulation result is obtained in the following way. Given multiplexing gain r1, r2and the end step numstop. For each SNR, Rayleigh fading matrices H1, H2are generated to compute the values of three RMRs. At each SNR, that is done for numstoptimes and finally the percentage is computed when each RS/IS,Ris the maximum among the three RMRs.

For the (4, 2, 3) and (5, 8, 7)-STWRC system, the percentage of each RMR is illustrated when it is the maximum among the three RMRs in Fig.3～Fig.6. It is shown in Fig.3～Fig.5 that if r1≥r2when M1≥M2, the percentage of (R1+R2)/IA,Rwhen it is the maximum among the three approaches 1 with the increase of SNR. Intuitively, the real transmission rate

Fig.3 Percentage of each RMR when it is the maximum among the three RMRs for a (4, 2, 3)-STWRC system with r1=0.6, r2=0.4

Fig.4 Percentage of each RMR when it is the maximum among the three RMRs for a (4, 2, 3)-STWRC system with r1=0.1, r2=0.1

Fig.5 Percentage of each RMR when it is the maximum among the three RMRs for a (5, 8, 7)-STWRC system with r1=0.3, r2=0.6

Fig.6 Percentage of each RMR when it is the maximum among the three RMRs for a (5, 8, 7)-STWRC system with r1=0.5, r2=0.3

which has the form of rlog SNR increases faster with SNR than the corresponding instantaneous mutual information. Therefore, the term (R1+R2)/IA,Rincreases faster since the sum transmission rate has the largest multiplexing gain. If user’s real transmission rate is inversely proportional to its number of antenna, just as shown in Fig.6, time allocation fraction t is determined with a large probability by RMR which corresponds to higher transmission rate but smaller mutual information.

When t≥1, the channel is in outage during the MAC phase. When t<1, the relay decodes two users' messages and transmits them to both users. This message transmitting from the relay to two users is in fact a broadcast channel (BC) with receiver side information. For the BC phase transmission, if the transmission rate pair (R1, R2) does not lie in the corresponding achievable rate region, the channel is also in outage.

2.3 DMT of MIMO STWRC with DDF protocol

Since the p.d.f of t is difficult to obtain, a case-by-case method is developed to analyze the achievable DMT of the DDF relay protocol. The main result of this subsection is given in the following theorem.

Theorem 1: For the (M1, N, M2)-STWRC system with given multiplexing gain pair (r1, r2), the achievable DMT of the DDF protocol is given as follows.

Case 1: If R1/I1,R=max{RS/IS,R,S?A}, the achievable diversity gain satisfies

dDDF(r1,r2)=min{dN,M*(2r1),dN,M1(r1+r2)}

(5)

Case 2: If R2/I2,R=max{RS/IS,R,S?A}, the achievable diversity gain satisfies

dDDF(r1,r2)=min{dN,M*(2r2),dN,M2(r1+r2)}

(6)

Case 3: If (R1+R2)/IA,R=max{RS/IS,R,S?A}, the achievable diversity gain satisfies

dDDF(r1,r2)=dM*,N(2(r1+r2))

(7)

Proof: The proof is given in the Appendix.

Symmetric Tradeoff: Assuming that each user has symmetric multiplexing gain, i.e., R1=R2=R=rlogSNR, the tradeoff region can be further simplified.

Theorem 2: For the (M1, N, M2)-STWRC system and given common multiplexing gain r, the achievable diversity gain of DDF protocol is given as follows.

Case 1: If R/I1,R=max{|S|R/IS,R,S?A}or R/I2,R=max{|S|R/IS,R,S?A}, the achievable diversity gain satisfies

(8)

Case 2: If 2R/IA,R=max{|S|R/IS,R,S?A}, the achievable diversity gain satisfies

(9)

where |S| denotes the cardinality of set S.

Proof: This theorem can be easily proved by using the similar steps as the proof of Theorem 1 by replacing R1and R2with R=r logSNR.

For case 1, it can be seen that its achievable symmetric DMT is the same as the outer bound of the fixed time allocation scheme and thus the DDF protocol is superior to any fDF protocol in this case. For case 2, its achievable symmetric DMT is equal to that of fDF protocol when t=0.5. The achievable symmetric diversity gain of a (2, 2, 2)-STWRC with static and dynamic protocols is illustrated in Fig. 7. Just as the analysis above, it is shown that the achievable symmetric DMT of case 1 for DDF is better than that of the fDF protocol with various time allocations (t=0.2, 0.5, 0.8). Case 2 for DDF has the same symmetric DMT performance as the fDF protocol when t=0.5.

Fig.7 The achievable symmetric DMT of a half-duplex (2, 2, 2)-STWRC with static and dynamic protocols

3 Conclusion

The achievable DMT of MIMO STWRC with half-duplex relay is derived. Both static DF and DDF protocols are considered, which shows that the DDF protocol achieves better DMT performance than the static DF protocol for some cases. Besides, finding the optimal DMT performance of the half-duplex STWRC is still a challenge problem, which will be our future work.

Appendix: Proof of Theorem 1

In the DDF protocol, the achievable rate region of the BC phase is[14]

Ri≤I(Xr; YA{i})IR,A{i}, i∈A

(10)

(11)

(12)

Define

According to the outage events analysis in Section 2.2, the overall outage probability of the DDF protocol is upper bounded as

Pout≤P{t>1}+P{t<1∩R1>(1-t)IR,2} +P{t<1∩R2>(1-t)IR,1} ≤P{t>1}+P{R1>(1-t)IR,2} +P{R2>(1-t)IR,1}

(13)

In the following, the proof for each case is given.

where step (a) is due to that the constant before SNR can be ignored on the scale of DMT analysis [1] and step (b) follows from the DMT result for the MIMO point-to-point channel [1, Theorem 2].

The second term in Eq.(13) is computed as

where (c) follows from Lemma 3 in Ref.[2].

Similarly, the last term in (13) is computed as

Thus, the overall outage probability is upper bounded as

and the achievable diversity gain for this case satisfies

d≤min{dN,M*(2r1),dN,M1(r1+r2)}

Using the similar steps as for case 1, the achievable diversity gain is obtained as

d≤min{dN,M*(2r2),dN,M2(r1+r2)}

This condition implies that

For this case, t=(R1+R2)/IA,Ris chosen. The first term in Eq.(13) is easily computed as

P{t≥1}?SNR-dM1+M2,N(r1+r2)

The second term in Eq.(13) is computed as

where (f) is due to the fact that R1≤(R1+R2)I1,R/IA,Rfrom Eq.(14) and (g) follows from the fact that IA,R≥I1,Rsince logdet(·) is a monotonically increasing function in the cone of psd matrices. The last term is similarly computed as

By using the results obtained above, we have the achievable diversity gain for this case as

d≤dM*,N(2(r1+r2))

Combining the results for the three cases yields Theorem 1.

Reference

[ 1] Zheng L, Tse D. Diversity and multiplexing: a fundamental tradeoff in multiple antenna channels. IEEE Transactions on Information Theory, 2003, 49(4): 1073-1096

[ 2] Yuksel M, Erkip E. Multi-antenna cooperative wireless systems: a diversity multiplexing tradeoff perspective. IEEE Transactions on Information Theory, 2007, 53(10): 3371-3393

[ 3] Gündüz D, Khojastepour M A, Goldsmith A, et al. Multi-hop MIMO relay networks: diversity-multiplexing trade-off analysis. IEEE Transactions on Wireless Communication, 2010, 9(5): 1738-1747

[ 4] Gündüz D, Goldsmith A, Poor H V. MIMO two-way relay channel: diversity-multiplexing tradeoff analysis. In: Proceedings of the 42nd Asilomar Conference on Signals, Systems and Computers, Pacific Grove, USA, 2008. 1474-1478

[ 5] Vaze R, Heath R W. On the capacity and diversity-multiplexing tradeoff of the two-way relay channel. IEEE Transactions on Information Theory, 2011, 57(7): 4219-4234

[ 6] Su Y, Li Y. Optimal diversity-multiplexing tradeoff of MIMO multi-way relay channel. ETRI Journal, 2013, 35(5): 919-922

[ 7] Wang L, Cai Y, Yang W. On the finite-SNR DMT of two-way AF relaying with imperfect CSI, IEEE Wireless Communications Letters, 2012, 1(3): 161-164

[ 8] Lin X, Tao M, Xu Y, et al. Finite-SNR diversity-multiplexing tradeoff for two-way relay fading channel, IEEE Transactions on Vehicular Technology, 2013, 62(7): 3123-3136

[ 9] Shu F, Chen Y, You X H, et al. Low-complexity optimal spatial channel pairing for AF-based multi-pair two-way relay networks, Science China Information Sciences, 2014, 57(10): 1-10

[10] Shu F, Lu Y Z, Chen Y, et al. High-sum-rate beamformers for multi-pair two-way relay networks with amplify-and-forward relaying strategy, Science China Information Sciences, 2014, 57(2): 1-11

[11] Tse D, Viswanath P, Zheng L. Diversity-multiplexing tradeoff in multiple access channels. IEEE Transactions on Information Theory, 2004, 50(9): 1859-1874

[12] Cover T M, Thomas J A. Elements of Information Theory. New York: Wiley-Interscience, 1991. 389,445

[13] Azarian K, Gamal H E, Schniter P. On the achievable diversity-multiplexing tradeoff in half-duplex cooperative channels. IEEE Transactions on Information Theory, 2005, 51(12): 4152-4172

Su Yuping, born in 1988. She is now studying for a doctor’s degree in State Key Lab of ISN, Xidian University. She received her B.S. degree from Henan Normal University in 2009. Her research interests include cooperative users and relay systems for wireless communications.

10.3772/j.issn.1006-6748.2015.03.018

①Supported by the National Basic Research Program of China (No.2012CB316100) and National Natural Science Foundation of China (No. 61072064， 61301177).

②To whom correspondence should be addressed. E-mail: ypsuxidian@gmail.com Received on Apr. 14， 2014, Li Ying, Liu Yang