The Multi-Way Relay Channel  includes a full data exchange model, in which each user receives messages of all other users, and the pairwise data exchange model, which consists of multiple two-way relay channels over which two users ( and ) exchange messages with the help of a common intermediate relay . In order to adapt to modern requirements, relaying schemes with high spectrum efficiency have recently attracted considerable attention [2, 3, 4].
An important two-way protocol category is called Multiple-Access Broadcast-Channel (MABC). In MABC decode-and-forward (DF) protocols, as in TW-Max-Min , transmission is organized in a prefixed schedule with two consecutive time slots. In the first time slot (MA phase), a selected relay receives and decodes the data simultaneously transmitted from two source nodes and physical layer network coding (PLNC) may be employed on the decoded data. In the second time slot (BC phase), the same relay forwards the decoded data to the two source nodes, which become destinations. Since all the channels are reciprocal (restricted to Time Division Multiplexing - TDM) and fixed during the two phases of the MABC protocol, the TW-Max-Min protocol  achieves a maximum diversity gain. On the other hand, by considering non reciprocal channels, the performance of relaying schemes may be improved by using a buffer-aided relaying protocol, where the relay may accumulate packets in a buffer, before transmitting to the destination nodes, as in the one-way Max-Link protocol, which selects in each time slot the more powerful channel among all the available source-relay (SR) and relay-destination (RD) channels (i.e., among channels) . For independent and identically distributed (i.i.d.) channels, Max-Link achieves a diversity gain of , where is the number of relays. Prior work has not considered multi-way protocols for multi-antenna systems or the use of a multi-way Max-Link (in which each pair of users has a particular buffer in the relays) or a central processor node.
In this work, we propose a multi-way Max-Link protocol for buffer-aided cooperative multi-antenna systems (MW-Max-Link) in non reciprocal channels. The proposed MW-Max-Link protocol  selects the best channels among pairs of users and achieves a diversity gain of . We also extend the MMD criterion  to multi-way systems for selection of relays in the proposed scheme and the existing TW-Max-Min (here adapted for multi-antenna systems) and carry out pairwise error probability (PEP), sum rate and computational complexity analyses.
Ii System Description
We consider a multi-antenna multi-way MABC relay scheme with pairs of users and half-duplex DF relays, ,…, operating in a spatial multiplexing mode. The users are equipped with antennas and each relay with antennas. A total of buffers are accessed by the selected relays for storing or extracting (each pair of users has a particular buffer established on demand in the relays), as shown in Fig.1. In the MA phase, a relay will be selected to receive simultaneously packets from a selected cluster (pair of users and ) and decode the data. Then, PLNC is employed on the decoded data and the resulting packets are stored in their particular buffers. In the BC phase, a relay will be selected to transmit packets from the particular buffer to the selected cluster. We remark that distributed space-time coding can also be examined in the described framework .
In each time slot, the total energy transmitted from each user to the relay selected for reception or from the relay selected for transmission to the selected cluster is the same and equal to
. The channel coefficients are drawn from mutually independent zero mean complex Gaussian random variables. The transmission is performed in data packets and the channels are constant for the duration of one packet and vary independently from one packet to the following. The order of the data packets is inserted in the preamble of each packet, so the original order is restored at the destination nodes. Pilot symbols for training and estimation of channel state information (CSI), and signaling for network coordination are also inserted in the preamble of the packet[12, 13]. A central processor node is responsible for deciding whether a cluster or the relay should transmit in a given time slot , through a feedback channel. This can be ensured by an appropriate signalling that provides global CSI at the central processor node . Furthermore, we assume that each relay only has information about its and channels. The use of a unique central processor node reduces its complexity, since a single central node is responsible for deciding which node will transmit (rather than all destination nodes being responsible together).
Ii-B System Model
At the MA phase of multi-way MABC DF systems, the received signal from the selected cluster (formed by and ) to the selected relay is formed by an vector given by
where represents the vector formed by symbols transmitted by and ( and ), is the matrix of and channels and represents the zero mean additive white complex Gaussian noise (AWGN) at the relay selected for reception.
Assuming synchronization, we employ the Maximum Likelihood (ML) receiver at the selected relay for reception:
where represents each of the possible transmitted symbols vector ( is the number of constellation symbols). The ML receiver computes an estimate of the vector of symbols transmitted by the users . Considering BPSK (), unit power symbols and , the estimated symbol vector may be , , or .
By employing PLNC (XOR), it is not necessary to store the packets transmitted by the selected cluster, but only the resulting packets (XOR outputs) with the information: "the bit transmitted by is different (or not) from the corresponding bit transmitted by ". Then, we employ the XOR: and store the resulting packets in the buffer.
At the BC phase, the signal transmitted from the selected relay and received at the selected cluster ( and ) is formed by an vector given by
where is the matrix of channels and represents the AWGN at or . At the selected cluster, we also employ the ML receiver which yields
where represents each possible vector formed by symbols. Then, at we compute the vector of symbols transmitted by by employing PLNC (XOR): = . The same reasoning is applied at to compute the vector of symbols transmitted by : = . Considering imperfect CSI when applying the ML receiver , the estimated channel matrix is assumed instead of in (2) and (4). Other suboptimal receivers such as linear, successive interference cancellation and decision feedback [15, 16, 17, 18] can also be considered.
Iii Proposed MW-Max-Link Relay Selection Scheme
The proposed MW-Max-Link scheme is modelled by the system shown in Fig. 1. This proposed scheme operates in two possible modes in each time slot: MA or BC. It is not necessary that a number of the buffer elements be filled with packets before the system starts its normal operation for this scheme to work properly and may be empty. Despite that, in this work, we consider that half of the buffer elements are filled in an initialization phase , before the scheme is used. The following subsections explain how MW-Max-Link works.
Iii-a Relay selection metric
In the first step, for each cluster formed by and , we compute the metric associated with the user-relay (UR) channels of each relay , for the MA mode:
where , and represent each possible vector formed by symbols and "" is different from "". This metric is computed for each of the (combination of in ) possibilities.
In the second step, we store the smallest metric (), for being critical, and thus each relay will have a minimum distance associated with its UR channels:
Then, in the third step, we perform ordering on and store the largest of these distances:
where . After finding for each cluster, we perform ordering and store the largest of these distances:
Then, we select the cluster and the relay that is associated with this distance to receive simultaneously packets from the selected cluster.
In the fourth step, for each cluster, we compute the metric associated with the channels of each relay , for the BC mode:
where , and represent each possible vector formed by symbols and "" is different from "". This metric is computed for each of the possibilities. In the fifth step, we find the minimum distance for each relay :
In the eighth step, after finding for each relay , we perform ordering and store the largest of these distances:
where . After finding for each cluster, we perform ordering and store the largest of these distances:
Then, we select the cluster and the relay that are associated with this distance to transmit simultaneously packets from the particular buffer to the selected cluster. Considering imperfect CSI, the estimated channel matrix is assumed, instead of in (5) and (9). We remark that other resource allocation techniques such as power allocation and relays [19, 20, 21, 22, 23] can also be considered.
Iii-B Comparison of metrics and choice of transmission mode
After computing all the metrics associated with the UR and RU channels and finding and , we compare these parameters and choose the transmission mode:
- If , we select "MA mode",
- Otherwise, we select "BC mode".
where . Thus, the probability of a relay being selected for transmission is close to the probability of a relay being selected for reception, and, consequently, the protocol works in a balanced way, even for asymmetric channels.
In this section, we first analyze the proposed MW-Max-Link in terms of PEP. Then, an approximated expression for the sum rate of the proposed protocol is derived and the complexity of the proposed and existing schemes are also presented.
Iv-a Pairwise Error Probability
The equations for , and may be simplified by making , where in (5) and (9). The PEP considers the error event when is transmitted and the detector computes an incorrect (where "" is different from ""), based on the received symbol . The PEP is given by
where is the power spectrum density of the AWGN. The PEP will have its maximum value for the minimum value of (PEP worst case). Thus, for cooperative transmissions, an approximated expression for computing the PEP worst case () in each time slot (regardless of whether it is an UR or RU channel) is given by
The extended MMD relay selection algorithm maximizes the metric and, consequently, minimizes the PEP worst case in the proposed MW-Max-Link scheme.
Iv-B Sum Rate
where the first and second terms in (16) represent the maximum rate at which can reliably decode the messages transmitted by the selected cluster ( and ) and at which the selected cluster can reliably decode the estimated messages transmitted by , respectively.
For the mutual information between and and , considering perfect CSI, we have
where represents a channel matrix and
where is the covariance matrix of the transmitted symbols. Note that the vectors are formed by independent and identically distributed (i.i.d.) symbols. The same reasoning can be applied to :
where and represents an channel matrix.
To compute the sum rate of the MW-Max-Link scheme, instead of (16), we consider an approximated expression for the sum rate in each time slot, depending on the kind of transmission. Then, in the case of a time slot selected for UR transmission, the approximated sum rate is given by
Moreover, in the case of a time slot selected for RU transmission, the approximated sum rate is given by
So, the average sum rate () of the MW-Max-Link scheme may be approximated by
where and represent the number of time slots selected for UR and RU transmissions, respectively.
Iv-C Computational Complexity
The complexity of the proposed MW-Max-Link, TW-Max-Link and the existing TW-Max-Min scheme (here adapted for multi-antenna systems) are associated with the complexity of the MMD protocol . The number of calculations of the metric for each channel matrix is given by
where in the case of , for the MA mode (), and in the case of and , for the BC mode (), is the number of different distances between the constellation symbols. If we have BPSK, , and QPSK, .
Table I shows the complexity of the proposed MW-Max-Link and the existing TW-Max-Min, for clusters, relays, antennas at the user nodes and antennas at the relays. Note that TW-Max-Link is a special case of MW-Max-Link, for a single two-way relay channel (). The complexity of MW-Max-Link is equal to the complexity of the adapted TW-Max-Min, multiplied by .
V Simulation Results
This section illustrates and discusses the simulation results of the proposed MW-Max-Link, the TW-Max-Link and the adapted TW-Max-Min , using the extended MMD relay selection criterion. The transmitted signals belong to BPSK constellations. The use of high order constellations as QPSK and 16-QAM was not included in this work but can be considered elsewhere. We tested the performance for different , but found that packets is sufficient to ensure a good performance in TW-Max-Link and MW-Max-Link. We also assume unit power channels ( ) and . The transmit signal-to-noise ratio SNR () ranges from 0 to 10 dB, where is the total energy transmitted by each user or the relay. The performances of the schemes were tested for packets, each containing 100 symbols. For imperfect CSI, the estimated channel matrix is assumed instead of : =+
, where the variance of thecoefficients is given by ( and ) .
Fig. 2 shows the BER performance of the MW-Max-Link for , TW-Max-Link and TW-Max-Min protocols, for , , BPSK, perfect and imperfect CSI ( and ). The performances of the MW-Max-Link are considerably better than those of TW-Max-Link and TW–Max-Min for the total range of SNR values tested, both for perfect and imperfect CSI.
shows the PEP and the Sum Rate performances, for BPSK and Gaussian distributed signals, respectively, of the MW-Max-Link (forand ), TW-Max-Link and TW-Max-Min protocols, for , and perfect CSI. The performances of the MW-Max-Link are very close, and considerably better than those of TW-Max-Link and TW–Max-Min for the total range of SNR values tested, as MW-Max-Link selects the best links among clusters and relays.
In this paper, we have presented a relay-selection strategy for multi-way cooperative multi-antenna systems that is aided by a central processor node, where a cluster formed by two users is selected to simultaneously transmit to each other with the help of relays. In particular, the proposed multi-way relay selection strategy selects the best link, exploiting the use of buffers and PLNC, that is called MW-Max-Link. The proposed MW-Max-Link was evaluated experimentally and outperformed the TW-Max-Link and the existing TW-Max-Min scheme. The use of a central processor node and buffers in the relays is presented as a promising relay selection technique and a framework for multi-way protocols.
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