I Introduction
In wireless networks, multipath propagation is a channel propagation phenomenon that affects the transmission of signals and can be mitigated through the use of cooperative diversity[1, 2, 3]. In cooperative communications with multiple relays, where a number of relays help a source in transmitting data packets to a destination, by receiving, processing (decoding) and forwarding these packets, relay selection schemes are key because of their high performance [4, 5, 6]. As cooperative communication can improve the throughput and extend the coverage of wireless communications systems, the task of relay selection serves as a building block to realize it. In this context, relay schemes have been included in recent/future wireless standards such as Long Term Evolution (LTE) Advanced [7, 8] and 5G standards [9].
Ia Prior and Related Work
In conventional relaying, using half duplex (HD) and decodeandforward protocols, transmission is usually organized in a prefixed schedule with two successive time slots. In the first time slot, the relay receives and decodes the data transmitted from the source, and in the second time slot the relay forwards the decoded data to the destination. Single relay selection schemes use the same relay for reception and transmission, so they are not able to simultaneously exploit the best available sourcerelay (SR) and relaydestination (RD) channels. The two most common schemes are bottleneck based and maximum harmonic mean based best relay selection (BRS)
[4]. The performance of relaying schemes can be improved if the link with the highest power is used in each time slot. This can be achieved via a bufferaided relaying protocol, where the relay can accumulate packets in its buffer, before transmitting. The use of buffers provides an improved performance and new degrees of freedom for system design
[7, 10]. However, it suffers from additional delay that must be well managed for delaysensitive applications. Bufferaided relaying protocols require not only the acquisition of channel state information (CSI), but control of the buffer status. Some possible applications of bufferaided relaying are: vehicular, cellular, and sensor networks [7]. In MaxMax Relay Selection (MMRS) [4], in the first time slot, the relay selected for reception can store the received packets in its buffer and forward them at a later time when selected for transmission. In the second time slot, the relay selected for transmission can transmit the first packet in the queue of its buffer, which was received from the source earlier. MMRS assumes infinite buffer sizes. To overcome this limitation, in [4] a hybrid relay selection (HRS) scheme, that is a combination of BRS and MMRS, was proposed. Although MMRS and HRS improve the throughput and/or SNR gain as compared to BRS, their diversity gain is limited to N (the quantity of relays). This can be improved by combining adaptive link selection with MMRS, which results in the MaxLink [11] protocol.The main idea of MaxLink is to select in each time slot the strongest link among all the available SR and RD links (i.e., among links) for transmission [12]. For independent and identically distributed (i.i.d.) links and no delay constraints, MaxLink achieves a diversity gain of , which is twice the diversity gain of BRS and MMRS. MaxLink has been extended in [13] to account for direct sourcedestination (SD) connectivity, which provides resiliency in low transmit SNR conditions [12]. In [14, 15, 16, 17], some bufferaided relay selection protocols improve the MaxLink performance by: reducing the average packet delay, maintaining a good diversity gain, and/or achieving full diversity gain with a smaller buffer size compared to MaxLink. In summary, the previous schemes (MMRS, HRS and MaxLink) only use bufferaided relay selection for cooperative singleantenna systems.
IB Contributions
In this work, we examine bufferaided relay selection for cooperative multipleantenna systems. In particular, we combine the concept of switching with the concept of selecting the best link used by the MaxLink protocol for cooperative multipleantenna systems, which results in the proposed Switched MaxLink protocol. We also introduce the MMD criterion for selection of relays in the proposed scheme, which is based on the ML criterion and the PEP. The advantage of the MMD algorithm is that it maximizes the minimum value of the PEP argument (PEP worst case). Simulations illustrate the performance of the proposed relay selection techniques. This paper is structured as follows. Section II describes the system model and the main assumptions made. Section III presents the proposed Switched MaxLink relay selection protocol and compares the proposed MMD criterion to the existing QN, in terms of PEP and complexity. Section IV illustrates and discusses the simulation results whereas Section V gives the concluding remarks.
Ii System Description
We consider a multipleantenna relay network with one source node, , one destination node, , and halfduplex decodeandforward (DF) relays, ,…,. Each relay is equipped with a buffer, whose size is packets and each node is equipped with antennas, and the transmission is organized in time slots [4]. This configuration is considered for simplicity. The considered system is shown in Fig. 1.
Iia Assumptions
In cooperative transmissions two time slots are needed to transmit data packets from the source to the destination, so the energy transmitted in direct transmission (from the source to the destination) is twice the energy transmitted in the cooperative transmission, from the source to the relay selected for reception () or from the relay selected for transmission to the destination (), . For this reason, the energy transmitted from each antenna in cooperative transmissions is and the energy transmitted from each antenna in direct transmissions is .
We consider that the channel coefficients are mutually independent zero mean complex Gaussian random variables (Rayleigh fading). Moreover, we assume that the transmission is organized in data packets and the channels are constant for the duration of one packet and vary independently from one packet to the next. The information about the order of the data packets is contained in the preamble of each packet, so the original order is restored at the destination node. Other information such as signaling for network coordination and pilot symbols for training and knowledge of the channel (CSI) are also inserted in the preamble of the packet. We consider perfect and imperfect CSI. Furthermore, we assume that the relays do not communicate with each other.
We also assume that the destination is the central node, being responsible for deciding whether the source or the relay should transmit in a given time slot . The central node has a perfect channel and buffer state information, so it may run the algorithm in each time slot and select the relay for transmission or reception through an errorfree feedback channel. This assumption can be ensured by an appropriate signalling that provides global channel state information (CSI) at the destination node [11]. Furthermore, we assume that the source has no CSI and each relay has only information about its SR channel and buffer status.
IiB System Model
The received signal from the source to the destination is organized in an vector given by
(1) 
where represents the total energy of the symbols transmitted from the source, represents the vector formed by symbols sent by the antennas of the source (a symbol of each packet). The quantity represents the matrix of links and denotes the zero mean additive white complex Gaussian noise (AWGN) at the destination receiver.
The received signal from the source to the selected relay is organized in an vector given by
(2) 
where refers to the selected relay for reception, is the matrix of links and represents the AWGN at the relay selected for reception.
The signal transmitted from the selected relay and received at the destination is structured in an vector given by
(3) 
where represents the total energy of the decoded symbols transmitted from the relay selected for transmission , is the vector formed by previously decoded symbols in the relay selected for reception and stored in its buffer and now transmitted by and is the matrix of links.
Assuming perfect CSI, at the relays, we employ the maximum likelihood (ML) receiver [5]:
(4) 
where represents each possible vector formed by symbols. As an example, if we have BPSK (number of constellation symbols ), unit power symbols and
, the estimated symbol vector
may be , , or .At the destination, we also resort to the ML receiver which depending on the transmission ( or ) yields
(5) 
(6) 
The ML receiver of the DF relay looks for an estimate of the vector of symbols transmitted by the source , comparing the quadratic norm between the output and the term multiplied by , that represents each of the possible transmitted symbols vector . We compute the symbol vector which is the optimal solution for the ML rule. The same reasoning is applied to the ML receiver at the destination. Other detection techniques can also be employed [22, 23, 24, 25, 26, 27, 28, 66, 61, 34, 31, 30, 32, spa2, 34, 35, 36, 37, 38, 41, 40, 41, 42, 43, 21, 44].
Iii Proposed Switched MaxLink Relay Selection Protocol
In this section, we detail the proposed Switched MaxLink relay selection protocol for cooperative multipleantenna systems. The proposed Switched MaxLink scheme can be implemented by making use of a network with one source node, , one destination node, , and halfduplex DF relays, ,…,. Each relay is equipped with a buffer, whose size is packets, and each node is equipped with antennas, resulting in a number of SR channels (links) for reception, RD links for transmission and SD links, as illustrated in Fig. 1. This scheme selects the best relay for reception () or the best relay for transmission () between relays (the best set of links among sets or the best set of links among sets). Similarly to the scheme proposed in [19], the MMD relay selection criterion (incorporated in Switched MaxLink), is based on the ML criterion. However, the metrics calculated by MMD are different from those of the scheme in [19], which leads to considerably better performance. MMD is also based on the worst case of the PEP and chooses the relay that has the highest minimum distance. So, it requires calculating the distance between the possible vectors of transmitted symbols.
For Switched MaxLink to work properly, it is not necessary that a certain number of buffer elements be filled with data before the system starts its normal operation. The buffers may be empty. Despite of that, in this work, for security, we considered that half of the buffer elements are filled in an initialization phase [4], by allowing the source to transmit a number of packets to the relays, before Switched MaxLink is used. During this initialization phase the relays do not transmit and the source transmits to the relay with the best set of SR links among the available relays.
In each time slot, the proposed Switched MaxLink Protocol may operate in two possible modes ("Direct Transmission" or "MaxLink"), with three options: a) work in "Direct Transmission" mode, by sending a quantity of packets directly to ; b) work in "MaxLink" mode, by sending a quantity of packets to and these packets are stored in its buffer; c) work in "MaxLink" mode, by forwarding a quantity of packets from its buffer to . Table 1 shows the Switched MaxLink pseudocode and the following subsections explain how this protocol works.
1: Calculate the metric 
; 
2: Find the minimum distance  
3: Calculate the metric 
; 
4: Find the minimum distance  
5: Perform ordering on and 
6: Find the maximum minimum distance (considering the buffer status) 
; 
7: Calculate the metric 
8: Find the minimum distance  
9: Select the transmission mode 
Operate in "Direct transmission mode"; 
Operate in "MaxLink mode" ; 
Iiia Calculation of relay selection metric
In the first step we calculate the metric related to the channels of each relay in MaxLink mode:
(7) 
where "" is different from "", and represent each possible vector formed by symbols.
This metric is calculated for each of the (combination of in ) possibilities. As an example, if and , we have possibilities. Then, we store the information related to the smallest metric (), for being critical (a bottleneck) in terms of performance, and thus each relay will have a minimum distance associated with its SR channels.
In the second step we calculate the metric related to the channels of each relay :
(8) 
where "" is different from "". This metric is calculated for each one of the possibilities. Then, we store the information related to the minimum distance (), and thus each relay will have a minimum distance associated with its RD channels.
In the third step, after calculating the metrics and for each of the relays, as described previously, we look for the largest value of the minimum distance:
(9) 
where "" is the index of each relay (). Therefore, we select the relay that is associated with this , considering its buffer status. This relay will be selected for reception (if its buffer is not full) or transmission (if its buffer is not empty), depending on this metric is associated with the SR or RD channels, respectively.
IiiB Calculation of the metric for direct transmission
In this step we calculate the metric related to the channels for the direct transmission mode:
(10) 
where "" is different from "". This metric is calculated for each of the possibilities. Then, we store the information related to the minimum distance (), associated with SD channels.
IiiC Comparison of metrics and choice of transmission mode
After calculating all the metrics associated to the SR and RD channels, finding and calculating the metrics associated to the SD channels and finding , we compare these parameters and select the transmission mode:
 If , we select "Direct transmission mode",
 Otherwise, we select "MaxLink mode".
If we do not consider the possibility of a direct SD connectivity ("Direct Transmission mode"), considering only the cooperative SRRD connectivity ("MaxLink mode"), we have another scheme, called "MMDMaxLink", instead of the proposed "Switched MaxLink" scheme.
IiiD Pairwise Error Probability
We can simplify the equations in (7), (8) and (10), making , where =, for and , or = 2 , for . We know that the PEP considers the error event when is transmitted and the detector computes an incorrect (where "" is different from ""), based on the received symbol [20]. The PEP is given by
(11) 
where is the AWGN noise spectrum density.
We may consider that the PEP will have its maximum value for the minimum value of (worst case of the PEP). So, for the worst case of the PEP (), in direct SD transmissions, in each time slot, we have
(12) 
However, for cooperative SRRD transmissions, an approximated expression for computing the worst case of the PEP in each time slot (regardless of whether it is an SR or RD link) is given by
(13) 
The advantage of the MMD algorithm compared to QN is that MMD maximizes the metric and QN does not take it into account. The QN algorithm is based only on the total power of these links (as the traditional MaxLink). Its metric is related to the quadratic norm of each matrix , and the matrix selected by this criterion is: . Even though the QN criterion selects the relay that has the largest quadratic norm of the channel coefficients matrices, the minimum value of the PEP argument associated with , selected by the QN criterion, may be not as high as the minimum value of the PEP argument associated with , selected by the MMD criterion. So, we have
(14) 
where is the PEP for the worst case in the MMD criterion and is the PEP for the worst case in the QN criterion.
IiiE Complexity
As we have seen, the metric may be calculated for each of the possibilities. However, it is not necessary to calculate all of them. We may generalize the total number of calculations of the metric , needed by the MMD criterion, for each matrix :
(15) 
where is the total number of different distances between the constellation symbols. If we have BPSK, , and QPSK, .
Operations/Criterion  Maximum Minimum Distance  Quadratic Norm 

additions  
multiplications 
Table II shows the complexity of the MMD and QN criteria for a number of relays, antennas, considering only the cooperative transmission (not considering the direct transmission mode), and the constellation type.
Fig. 2 shows the complexity of the MMD and QN criteria, for (a source, 3 relays and a destination), and BPSK. By the analysis of this result, it is observed that the complexity of the MMD criterion with is not so higher than the complexity of the QN criterion. If we increase the number of antennas to (or more) in each node, the complexity of MMD criterion becomes considerably higher than the complexity of QN criterion.
Iv Simulation Results
This section illustrates and discusses the simulation results of the proposed "Switched MaxLink", the "MMDMaxLink", the "conventional MIMO" (direct transmission, without relaying) and the MaxLink with the QN criterion ("QNMaxLink"). QNMaxLink with a single antenna is equal to the traditional MaxLink. We assume that the transmitted signals belong to BPSK or QPSK constellations. The 16QAM constellation was not included in this work because of its higher complexity. Each relay is equipped with a buffer whose size is packets. Note that we tested the performance for different but found that is sufficient to ensure a good performance. We also assume unit power channels ( ), and (total energy transmitted). The transmit signaltonoise ratio SNR () ranges from 0 to 12 dB and the performances of the transmission schemes were tested for 20000 packets, each containing 100 symbols.
Fig. 3 shows the PEP performance of the MMDMaxLink and QNMaxLink protocols, for , = 3, 5 and 10, BPSK and perfect CSI. By the analysis of this result, it is observed, as expected, that for multiple antennas the performance of the MMDMaxLink scheme is much better than the performance of QNMaxLink for the total range of SNR tested. When we increase , the MMDMaxLink has its performance improved. The same does not happen to QNMaxLink, as the QN criterion does not take the metric into account.
Fig. 4 shows the Switched MaxLink, the MMDMaxLink, the QNMaxLink and the conventional MIMO (direct transmission) BER performance comparison for , , BPSK and perfect CSI. We notice that the performance of the MMDMaxLink scheme is worse than the performance of the conventional MIMO scheme for a SNR less than 2 dB. Nevertheless, the performance of the proposed Switched MaxLink scheme is better than the performance of the conventional MIMO for a wide range of SNR values. It is observed, as expected, that the performance of the proposed Switched MaxLink scheme is better than the performance of the MMDMaxLink scheme, as well as its resiliency in low transmit SNR conditions.
Fig. 5 shows the Switched MaxLink, the MMDMaxLink and the conventional MIMO BER performance comparison for , , QPSK and perfect CSI (the QNMaxLink was not considered as its performance is worse than the performance of the proposed protocol). The performance of the MMDMaxLink scheme is worse than the performance of the conventional MIMO scheme for a SNR less than 6 dB. Nevertheless, the performance of the proposed Switched MaxLink scheme is better than the performance of the conventional MIMO for a wide range of SNR values. It is observed that the performance of the proposed Switched MaxLink scheme is better than the performance of the MMDMaxLink scheme, as well as its resiliency in low transmit SNR conditions.
Fig. 6 shows the Switched MaxLink, the MMDMaxLink and the conventional MIMO BER performance comparison for , , BPSK and imperfect channel knowledge (, and ). As in the case of perfect channel knowledge, the performance of the proposed Switched MaxLink scheme is better than the performance of the conventional MIMO for a wide range of SNR values. It is observed that the performance of the proposed Switched MaxLink scheme is still better than the performance of the MMDMaxLink scheme, as well as its resiliency in low transmit SNR conditions.
V Conclusions
In this paper we have presented the benefits of using buffers and multiple antennas for the design of halfduplex decodeandforward relaying protocols in cooperative communication systems, by using the MMD relay selection criterion, based on the ML criterion and the PEP. Moreover, a new cooperative protocol using multiple antennas that combines the concept of switching and the concept of selection of the best link used by MaxLink and incorporates the MMD selection criterion has been proposed. The proposed Switched MaxLink was evaluated experimentally and outperformed the conventional direct transmission and the existing QN MaxLink scheme.
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 [79] Z. Yang, R. C. de Lamare and X. Li, “L1Regularized STAP Algorithms With a Generalized Sidelobe Canceler Architecture for Airborne Radar," IEEE Transactions on Signal Processing, vol.60, no.2, pp.674686, Feb. 2012.
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