Multipair two-way relay systems have attracted significant attention in the research community, due to their inherent ability to overcome the halving of system sumrate (stemming from half-duplex operation), with essential doubling of sumrate compared to ordinary one-way relaying . Much of the research efforts regarding these systems were focused on developing signal processing algorithms at the relays that are tailored to fit a certain target objective, e.g. interference cancellation or sumrate maximization , . Recently, multipair two-way relaying systems with a large number of relays/relay antennas were considered [3, 4, 5, 6, 7, 8]. By employing a large-scale relay network, system performance is boosted via the channel hardening effect, thus either improving system sumrate or increasing coverage compared to relay systems that operate on a smaller scale.
Previous work on large-scale multipair two-way (LS–MTW) relay systems is limited to two extreme scenarios. In the first scenario, a large number of non-cooperating single-antenna relays processes the data in a fully decentralized fashion. Such a setup is described in , where individual single-antenna relays perform amplify–and–forward processing of the data on the bidirectional links, based only on their local CSI. No data or CSI is exchanged between the relays, and their sheer large number is relied upon to deliver satisfying performance. On the other end of the spectrum is the scenario where a single relay with a large number of antennas performs the uplink and downlink processing. Theoretical performance characterization for this setup was analyzed in [4, 5, 6, 7, 8], where the relay is assumed to employ simple linear processing (maximum ratio combining/transmission and zero forcing).
To the best of our knowledge, there is no prior analysis of LS–MTW relay networks with an arbitrary degree of cooperation among relays. By “degree of cooperation” we here refer to the number of relays that will exchange data and channel state information inside a closed group, with no exchange occurring between groups. The number of closely cooperating relays directly trades system performance against data exchange cost, and is therefore an important design parameter for practical implementations of LS–MTW networks. An illustration of a MTW network with grouped relays is shown in Fig. 1.
This work provides a comprehensive analysis of the effects of relay cooperation in an LS–MTW relay system where zero-forcing processing is used. We derive a lower bound for ergodic system sumrate that is tight at high SNR. Furthermore, we make use of this bound to analyze the behavior of system performance as the number of closely cooperating relays changes. Finally, we reflect on the choice of the degree of relay cooperation that maximizes the cost-effectiveness of cooperation.
Ii System setup
In this paper, we analyze the multi-pair bidirectional symmetric relay network, where relays serve to connect two separate groups of user units, as illustrated in Fig. 1. Each user from group A is connected in a pairwise fashion with a corresponding user in group B, with pairs formed in total. It is assumed that there is no direct link between the users in a pair, so the pairwise connections are established solely via the relays. Moreover, the information flow between the two users in a pair is assumed symmetric.
The exchange of information is split in two phases, uplink and downlink, observed from the perspective of the relays, which occur in alternating time slots. In the uplink phase, all user units simultaneously transmit to the relays. The relays process the received signal and send the processed information to the users in the downlink phase, with the direction of information flow swapped compared to the uplink phase (processed information that came on the uplink of channel is transmitted on the downlink of channel , and vice versa).
Furthermore, we assume that it is only the relays that have knowledge of the channels and
. In practice, the channels can be estimated in the uplink by transmission of orthogonal pilot sequences from the user units, and downlink channels are then automatically obtained assuming that radio channel reciprocity holds and that reciprocity calibration is performed at the relays. In this sense, the analyzed relay system is equivalent to collocated or distributed massive MIMO (MaMI) systems working in time division duplex (TDD), and channel estimation and reciprocity calibration methods developed for TDD MaMI readily apply[9, 10, 11]. However, for clarity of analysis, in this work we assume perfect channel knowledge and channel reciprocity.
The focus of investigation in this work is the impact of cooperation between the relays on the overall system performance. The level of inter-relay cooperation is the parameter that trades off network performance with cost of backhaul information exchange. To this end, we assume the following hierarchical structure of the relay system:
The relays are assumed to be divided in equally-sized groups, each containing relays. Inside the group, channel state information (CSI), symbols received in the uplink phase and symbols to be transmitted in the downlink phase are shared mutually among all relays. Moreover, the relays inside the group are assumed to be time and frequency synchronized. The data is congregated and processed for uplink and downlink in a central group processor (CGP). One of the relays can take on the role of the CGP, and is referred to as the group master (represented by in Fig. 1). Most importantly, no data or CSI information is exchanged between the groups, and each group performs data processing independently of others.
Groups (or equivalently, group masters) are assumed to be synchronized in time and frequency, and this is the only form of inter-group cooperation.
The two-tier hierarchy of cooperation enables us to cover the entire space of cooperative networks that lies in between the two extreme cases:
For we have the fully decentralized multipair relaying scenario, where single-antenna relays use their local CSI to process and relay the received data without exchanging any CSI or received data information with other relays. Such a scenario was analyzed in .
The case of represents the perfectly centralized relaying scenario where all CSI and received data is available at a central point that performs data processing. Usually this setup is cast in the form of one (massive) MIMO relay, as in .
In general, there are no constraints on the geographical distribution of relays in a group, which can be collocated or distributed. Likewise, the type of connections between the relays in a group is arbitrary and can be wireless or wired. We note, however, that a group of collocated users can be observed as a single MIMO relay with antennas. We also note that the stratification of relay cooperation enables the design of a layered and scalable synchronization protocol. Instead of synchronizing all the relays to a common beacon, synchronization can be done first on the group level and then among the group masters, resulting in a completely decentralized synchronization scheme. For the sake of simplicity, we assume that the intra- and inter-group synchronization is perfect, and leave the analysis of the impact of synchronization errors as a subject for future work.
Iii System model
We start the description of the system model by denoting with
the total number of relay groups. In each channel use, transmitted user symbols are represented by the
complex vectorwith covariance matrix . The user symbol vector can be represented as , where and are symbols transmitted from the left-hand-side and right-hand-side groups of users, illustrated in Figure 1, respectively.
Focusing on the th relay group, we build the system model step by step, following the uplink - downlink flow of information. First, we denote by
the composite uplink channel between the th relay group and all the users. Received signal vector at the th relay group is then
where is an zero-mean circularly symmetric complex Gaussian (ZMCSCG) vector of thermal noise with covariance matrix and is the uplink transmit power per user, assumed to be same for all users.
The received signal in the uplink is linearly filtered with to yield the estimates of user data symbols:
After uplink filtering, the downlink precoder is applied to symbol estimates, together with a scaling factor ensuring proper transmitted power. We assume that the user power allocation in the downlink is uniform.
The uplink/downlink linear processing can be compactly represented by a general complex gain matrix
Altogether, the transmit signal vector from the th relay group is
The power scaling coefficient is determined so that the transmitted power per group averaged over data and noise realizations equals . For the heavily restricted decentralized setup considered here, a practically implementable strategy of power allocation between relay groups is that all groups have the same transmit power, so . Overall, we have
which readily yields
We define the composite downlink channel between the th relay group and all the users as
The contribution of the th relay group to the received signal at the users, , is thus
The total received signal vector at the users is hence
where is the ZMCSCG vector of thermal noise at the users, with covariance .
For the benefit of further analysis, we define the uplink and downlink SNRs as
The overall system model (11) can be expanded for the received symbol at a particular user, say th user from group A. This reveals that the performance in the general case is limited by four distinct impairments: self-interference, interuser interference, precoded thermal noise at the relays and thermal noise at the users:
In the follow-up, we consider the case when zero-forcing is chosen as the linear processing scheme at individual relay groups and analyze system performance, averaged over channel realizations. The goal of the analysis is to determine the closed-form dependence of system performance (quantified by ergodic system sumrate) on relay group size .
Iv Ergodic system sumrate calculation with per-group zero-forcing
If zero-forcing (ZF) is chosen for linear processing, the uplink and downlink processing matrices at each relay group are calculated as
Back-to-back ZF processing will completely eliminate self- and interuser interference, leaving the precoded noise from relays and noise at the user terminals as sources of impairment. A strong requirement for total interference elimination is that .
In order to gain some insight in the connection between system performance (quantified by system sumrate) and system parameters and , we assume that is high. Typically, a high SNR would mean that the geographical distances between users and relays are small, and that the relays are used to boost the system sumrate (in contrast to e.g. a range extension scenario).
Under the high-SNR assumption and additionally assuming that and are well-conditioned, the influence of precoded thermal noise at the relays can be neglected, so the system model (11) simplifies to
A basis for performance evaluation is the per-user , defined for the th user in group A as the ratio of powers of the information signal part and impairments from (13), which, due to all interference being eliminated and the high-SNR assumption, becomes
Instantaneous per-user performance is characterized by user rate:
and overall system performance by ergodic system sumrate, calculated as
The factor of 2 accounts for the fact that the information flow in the system is symmetric, so members of the th user pair have the same information transmission capacity. The factor of , on the other hand, stems from half-duplex operation. From here, the benefit of the symmetric multi-pair setup compared with ordinary relaying schemes becomes clear: simultaneous and symmetric transmission from both user groups manages to (approximately) compensate for the halving of the capacity due to TDD splitting of uplink and downlink.
We proceed with calculating (18), and in the process, we make use of
Let be a vector of nonnegative random variables
be a vector of nonnegative random variables, with denoting realizations of . Then
Proof: The proof is given in Appendix A.
We employ Lemma 1, assuming in the process that without loss of generality, to obtain a lower bound on the ergodic information rate of the th user as
where, for sake of clarity, we introduce the ZF precoding scaling factor
Now we assume that and are iid Rayleigh fading channels with pathloss and shadowing, modeled as
The entries of matrices and
are iid ZMCSCG with unit variance, and the diagonal matrices
are used to model propagation losses and large-scale fading. In order for channel matrices to be decomposable as in (21), the propagation amplitude gain needs to be the same from user to all relays in group , which implies that the relays of that group are assumed to be collocated and experience the same large scale fading in relation to user . These conditions are readily satisfied if a relay group is implemented in form of a single MIMO relay with a compact form factor. Otherwise, they can be met by applying an appropriate relay grouping scheme, which is an interesting research problem in itself, but falls outside of the scope of this paper.
Using well-known results from random matrix theory and the identity , it can be shown that
For convenience of exposition, in the follow-up we will refer to and as power imbalance factor and array gain degradation factor, respectively.
V Analysis and discussion
In order to have a fair comparison between systems, we assume that the pathloss and shadowing power gains are normalized so that
which implies . Given the constraint (28), it is easy to show that power imbalance and array gain degradation factors are lower (respectively, upper) bounded as
where equality holds in the case . In other words, the lower bound on system sumrate from (27) is maximized when there are no pathloss/shadowing power imbalances between users. In practical system deployments, such imbalances will invariably exist and the resulting degradation of sumrate can be combatted by either performing waterfilling-based user power weighting in the downlink, or by employing advanced user scheduling techniques. Analysis of the effects of these approaches is beyond the scope of this work.
For a fair comparison between relay systems with differing and , we need to assume that the amount of transmit power allocated to the entire relay system is fixed, and we denote this power by . As mentioned previously, due to limited coordination, it is reasonable to assume that the total power allocated to relays is distributed equally among relay groups, so . By taking into account (29), we can write , which yields the lower bound on sumrate that allows for a fair comparison between different relay systems:
Now we can consider the case when . Even for a large number of user pairs, this case is feasible due to the fundamental assumption of a large number of relay units, . The array gain term from (30) then becomes
that is, the array gain becomes independent of . This insight is of fundamental importance for practical deployments of LS-MTW systems. What it implies is that, in the regime with a large number of relays , tight cooperation in information processing between all relays is not necessary. Instead, small, independent groups of tightly cooperating relays can be formed, and such a setup experiences only a marginal degradation of system sumrate compared to the case when all relays are cooperating. If we substitute the notion of a tightly cooperating relay group with a more specific notion of a MIMO relay, we can conclude that a single massive MIMO relay performing ZF can be substituted with several simpler and cheaper MIMO relays with smaller numbers of antennas, with a negligible reduction in system performance.
The presented observations are corroborated by simulations, results of which are presented in Fig. 2, where the lower bound (30) is compared to simulated system sumrate, averaged over channel realizations. It is assumed that the total transmit power in the system, which we denote by , is split between users and relays in two equal parts, so and . The results show an excellent match between the theoretical lower bound (30) and simulations. Moreover, it is clearly demonstrated how substituting one large group () with several smaller and independent groups of relays introduces only a slight degradation of sumrate (in the most extreme cases, sumrate degrades by to for the setups considered).
In order to gain deeper understanding of tradeoffs encountered in the design of LS-MTW systems, in addition to sumrate, we also need to take into account the cost of enabling cooperation between the relays. This cost, which we denote by , quantifies the resources spent (e.g. energy, bandwidth) or penalties in system performance incurred (e.g. latency) when CSI and uplink/downlink data are exchanged inside a relay group. In particular, we focus on resources that can be reused between groups. An example system setup would feature relays inside a group exchanging CSI and data with the CGP over dedicated short-range wireless links and employing frequency division multiplexing (FDM). With enough physical separation between individual groups, the short range of intra-group backhaul links would mean that the bandwidth dedicated for cooperation can be reused between groups. Moreover, the use of FDM implies that this bandwidth is proportional to the group size:
where is the bandwidth of the frequency slot allocated for one user-CGP link. As discussed previously, for , sumrate is independent of . Therefore, the cooperation efficiency of the described system,
increases for decreasing when is large.
Relative cooperation efficiency function is shown in Fig. 3, where but the constraint is relaxed. These results support the notion that using one large cooperating group is a suboptimal strategy from the point of view of cooperation efficiency, especially for low values of K.
We have analyzed a multipair two-way relay system with a large number of relays. The relays are assumed to form groups inside which data and channel state information is exchanged, and processing is done independently from other groups. Assuming that the groups perform zero-forcing and that the SNR is large, we derive a closed-form expression for a tight lower bound on the system sumrate. An asymptotic analysis of the bound shows that the sumrate is essentially independent from group sizewhen . This implies that one large group of cooperating users can be substituted with several smaller groups, with no significant impact on performance. We extend this result to take into account the efficiency of information exchange that supports relay cooperation. It is shown that using several smaller relay groups is more efficient than the use of a single large group, if the resource used for intra-group information exchange is reusable between groups.
The authors would like to thank the Intel Corporation for providing the funds for the research, which was conducted as a part of the SRC MSR-Intel Research Project P28832, “Coordination in Distributed Multi-User High-Performance Dense Networks”.
Appendix A: Proof of Lemma 1
By taking into account Jensen’s inequality
convex (concave), we can form a chain of (in)equalities
where inequality a) follows from convexity of on , so
Inequality b) follows from concavity of and from the fact that is monotonically decreasing, which yields
-  C. Wang, H. Chen, Q. Yin, A. Feng, and A. F. Molisch, “Multi-user two-way relay networks with distributed beamforming,” IEEE Transactions on Wireless Communications, vol. 10, no. 10, pp. 3460–3471, Oct 2011.
-  J. Zhang, F. Roemer, and M. Haardt, “Distributed beamforming for two-way relaying networks with individual power constraints,” in 2012 Conference Record of the Forty Sixth Asilomar Conference on Signals, Systems and Computers (ASILOMAR), Nov 2012, pp. 542–546.
-  H. Q. Ngo and E. G. Larsson, “Large-scale multipair two-way relay networks with distributed AF beamforming,” IEEE Communications Letters, vol. 17, no. 12, pp. 1–4, Dec 2013.
-  S. Jin, X. Liang, K. K. Wong, X. Gao, and Q. Zhu, “Ergodic rate analysis for multipair massive MIMO two-way relay networks,” IEEE Transactions on Wireless Communications, vol. 14, no. 3, pp. 1480–1491, Mar 2015.
-  T. V. T. Le and Y. H. Kim, “Power and spectral efficiency of multi-pair massive antenna relaying systems with zero-forcing relay beamforming,” IEEE Communications Letters, vol. 19, no. 2, pp. 243–246, Feb 2015.
-  Y. Dai and X. Dong, “Power allocation for multi-pair massive MIMO two-way AF relaying with linear processing,” IEEE Transactions on Wireless Communications, vol. 15, no. 9, pp. 5932–5946, Sept 2016.
-  M. Liu, J. Zhang, and P. Zhang, “Multipair two-way relay networks with very large antenna arrays,” in 2014 IEEE 80th Vehicular Technology Conference (VTC2014-Fall), Sept 2014, pp. 1–5.
-  C. Kong, C. Zhong, M. Matthaiou, E. Björnson, and Z. Zhang, “Multi-pair two-way AF relaying systems with massive arrays and imperfect CSI,” in 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Mar 2016, pp. 3651–3655.
-  E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” IEEE Communications Magazine, vol. 52, no. 2, pp. 186–195, Feb 2014.
-  J. Vieira, F. Rusek, O. Edfors, S. Malkowsky, L. Liu, and F. Tufvesson, “Reciprocity calibration for massive MIMO: Proposal, modeling, and validation,” IEEE Transactions on Wireless Communications, vol. 16, no. 5, pp. 3042–3056, May 2017.
-  R. Rogalin, O. Y. Bursalioglu, H. Papadopoulos, G. Caire, A. F. Molisch, A. Michaloliakos, V. Balan, and K. Psounis, “Scalable synchronization and reciprocity calibration for distributed multiuser MIMO,” IEEE Transactions on Wireless Communications, vol. 13, no. 4, pp. 1815–1831, Apr 2014.
-  A. M. Tulino and S. Verdu, Random Matrix Theory and Wireless Communications. Now Publishers, 2004.