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NonRegenerative MultiAntenna MultiGroup MultiWay Relaying
EURASIP Journal on Wireless Communications and Networking volume 2011, Article number: 29 (2011)
Abstract
We consider nonregenerative multigroup multiway (MGMW) relaying. A halfduplex nonregenerative multiantenna relay station (RS) assists multiple communication groups. In each group, multiple halfduplex nodes exchange messages. In our proposal, the required number of communication phases is equal to the maximum number of nodes among the groups. In the first phase, all nodes transmit simultaneously to the RS. Assuming perfect channel state information is available at the RS, in the following broadcast (BC) phases the RS applies transceive beamforming to its received signal and transmits simultaneously to all nodes. We propose three BC strategies for the BC phases: unicasting, multicasting and hybrid uni/multicasting. For the multicasting strategy, network coding is applied to maintain the same number of communication phases as for the other strategies. We address transceive beamforming maximising the sum rate of nonregenerative MGMW relaying. Due to the high complexity of finding the optimum transceive beamforming maximising the sum rate, we design generalised low complexity transceive beamforming algorithms for all BC strategies: matched filter, zero forcing, minimisation of mean square error and BCstrategyaware transceive beamforming. It is shown that the sum rate performance of nonregenerative MGMW relaying depends both on the chosen BC strategies and the applied transceive beamforming at the RS.
Introduction
Twoway relaying is a spectrally efficient protocol to establish bidirectional communication between two halfduplex nodes via a halfduplex relay station (RS) [1–3]. It was shown in [1, 3] that twoway relaying outperforms the traditional oneway relaying due to its smaller number of communication resources. In twoway relaying, two communication phases are needed. The first phase is the multiple access (MAC) phase where the two communicating nodes send their data streams simultaneously to the RS. The second phase is the broadcast (BC) phase where the RS sends the processed signals simultaneously to both nodes. Consequently, the nodes need to cancel their selfinterference.
Regarding the signal processing at the RS, it can be either regenerative, cf. [1, 2] or nonregenerative, cf. [1, 3]. A regenerative RS regenerates (decodes and reencodes) the data streams of all nodes while a nonregenerative RS performs linear signal processing to the received signals and transmits the output to the nodes.
The use of multiple antennas can improve the spectral efficiency and/or the reliability of communication networks [4, 5]. For twoway relaying, a multiantenna RS that serves one bidirectional pair was considered in [6–8] for a regenerative RS and in [3, 9–11] for a nonregenerative RS. For the nonregenerative case, while [3, 9] assume multiantenna nodes, [10, 11] assume single antenna nodes. Their works consider optimal transceive beamforming at the RS maximising the sum rate as well as linear transceive beamforming based on Zero Forcing (ZF) [3, 9–11], Minimisation of Mean Square Error (MMSE) [3, 9, 10], Maximisation of Signal to Noise Ratio (MSNR) [3] and Matched Filter (MF) criteria [10, 11].
Multiuser twoway relaying, where an RS serves multiple bidirectional pairs, is treated in [12–14] for a regenerative RS and in [15, 16] for a nonregenerative RS. In [12], all bidirectional pairs are separated using Code Division Multiple Access. Every two nodes in a bidirectional pair have their own code which is different from the other pairs' codes. In contrast to [12], having multiple antennas at the RS and assuming perfect detection in MAC phase, in [13, 14] the separation of the pairs in the BC phase is done spatially using transmit beamforming employed at the RS. In [15], ZF and MMSE transceive beamforming for multiuser nonregenerative twoway relaying is designed to separate the nodes. In [16], blockdiagonalisationsingularvaluedecomposition (BDSVD) transceive beamforming is designed for separating the twoway pairs by extending BDSVD transmit beamforming proposed in [17].
In applications such as video conference and multiplayer gaming, multiple nodes exchange messages. In [18], the multiway relay channel is considered, where an RS assists multiple communication groups. In each group, each member node exchanges messages with other member nodes, but not with other nodes from the other groups. A fullduplex communication is assumed and time division is used to separate the multiway groups. The full duplex assumption, however, is not yet practical and halfduplex nodes and relays are more of practical importance [1, 19]. Therefore, communication protocols for multiway relaying for halfduplex nodes and a halfduplex RS are needed.
Multiway relaying protocols for one communication group, where a halfduplex multiantenna RS assists N halfduplex nodes to exchange messages, is proposed by the authors of this paper in [20] for a nonregenerative RS and in [21] for a regenerative RS. The required number of communication phases is only N, consisting of one MAC phase and N  1 BC phases. In [20, 21], in each BC phase, the RS sends N data streams to N nodes simultaneously. Thus, each node receives an intended data stream from a specific node, while seeing other data streams as interference. Nevertheless, the interference can be canceled by performing successive interference cancellation or by applying linear transceive beamforming, e.g., ZF, that nullifies the interference [20]. In [20], it was also shown that instead of transmitting N different data streams per BC phase, the RS can transmit one data stream simultaneously to all nodes in each BC phase. This data stream is a superposition of two different data streams. Consequently, each node has to perform self and knowninterference cancellation. Since in each BC phase the RS transmits only one superposed data stream to all nodes, there is no interstream interference and, thus, the performance is improved.
In this paper, we consider nonregenerative multigroup multiway (MGMW) relaying. A halfduplex multiantenna RS assists L multiway groups where each group consists of halfduplex single antenna nodes. We consider nonregenerative relaying where the RS performs transceive beamforming. Nonregenerative, compared to regenerative, has three advantages: no decoding error propagation, no delay due to decoding and deinterleaving, and transparency to the modulation and coding schemes that are used at the nodes [3].
In each lth multiway group, , , there are N_{ l } ≥ 2 nodes that exchange messages. In our proposal, the required number P of communication phases is equal to max _{ l }N_{ l } . Our work is a generalisation of some of the above mentioned publications. If L = 1 and N_{1} = 2, we have a nonregenerative twoway relaying as in [3, 10, 22]. If L > 1 and N_{ l } = 2, ∀l, , we have a nonregenerative multiuser twoway relaying as in [15, 16], and if L = 1 and N_{1} ≥ 2, we have a nonregenerative singlegroup multiway relaying as in [20].
We propose three BC strategies for the BC phases, namely, unicasting, multicasting and hybrid uni/multicasting. The proposed strategies are designed in such a way that the number of communication phases remains P = max _{ l }N_{ l } . We derive the sum rate expression for nonregenerative MGMW relaying with the proposed BC strategies for asymmetric and symmetric traffic. In asymmetric traffic all nodes in each group may communicate with different rate, while in symmetric traffic all nodes in each group communicate with the same rate.
We address the sum rate maximisation which requires optimum transceive beamforming. Due to the high complexity of finding the optimum transceive beamforming maximising the sum rate, we design generalised low complexity transceive beamforming algorithms for all proposed BC strategies, namely, ZF, MMSE, MF and BCstrategyaware (BCSA) transceive beamforming. BCSA transceive beamforming is designed by suppressing unwanted signals using either block diagonalisation (BD) [17] or regularised BD proposed in [23].
This paper is organised as follows. Section II explains the proposed broadcast strategies and the system model of nonregenerative MGMW relaying. Section III explains the sum rate expression. The transceive beamforming algorithms are explained in Section IV. The simulation results are given in Section V. Finally, Section VI provides the conclusion.
Notations
Boldface lower and upper case letters denote vectors and matrices, respectively, while normal letters denote scalar values. The superscripts (·)^{T}, (·)* and (·)^{H} stand for matrix or vector transpose, complex conjugate, and complex conjugate transpose, respectively. The operators mod _{ N } (x), E{X} and tr{X} denote the modulo N of x, the expectation and the trace of X, respectively, and denotes the circularly symmetric zeromean complex normal distribution with variance σ^{2}.
Broadcast Strategies And System Model
We consider L multiway communication groups. It is assumed that there are no direct links among the nodes and the MGMW communication can only be performed with the assistance of a halfduplex multiantenna RS with M antenna elements. In the l th group, , , there are N_{ l } nodes which exchange messages through an RS. For simplicity of notations, we consider the same number of nodes in all groups, i.e., N_{ l } = N_{mw}, . However, the extension to the case of different numbers of nodes in the groups is straightforward. The total number N of nodes in the network is .
Assuming that the RS already knows which nodes belong to which communication group, the RS makes the indexing of all nodes according to their group membership. Nodes in group one are indexed within the set {0,..., N_{1}  1}, nodes in group two are indexed within the set {N_{1},..., (N_{1} + N_{2})  1}, and so on. In general, it can be given as follows. The l th group consists of nodes Si_{ l } , , where is the set of node indices given by , with a_{ l } = (l  1)N_{mw}, b_{ l } = lN_{mw}  1. Each node only exchanges messages with the other nodes in its group and each node belongs only to one multiway group, i.e., , ∀l ≠ k and .
A. Broadcast Strategies
In this subsection, the broadcast strategies for nonregenerative MGMW relaying are described. The number P of communication phases to perform MGMW communication is given by the maximum number of nodes among all groups, i.e., P = max _{ l }N_{ l } = N_{mw}. In the first phase, the MAC phase, all nodes transmit simultaneously to the RS. In the following P  1 BC phases, the RS transmits to the nodes. Let p, , , denote the index of the BC phase. In p th phase, in group l, receiving node is intended to receive the data stream of transmitting node .
1) Unicasting Strategy
Using unicasting strategy, in each BC phase, the RS transmits different data streams to different nodes. Each data stream is intended only for one receiving node. Consequently, in each BC phase each node sees the other data streams transmitted by the RS to the other nodes as interference. The data stream transmitted from the RS to each particular node is changed in each BC phase, such that within P  1 BC phases, each node receives the data streams from all other nodes in its group.
The relationship of the parameters p, r_{ l } and t_{ l } is given by
Using such strategy, assuming each node knows its index and all other nodes' indices in its group, there is no signalling required in the network. The proposed unicasting strategy is a generalization of the work in [3, 10] for L = 1 and N_{1} = 2, in [15] for L > 1 and N_{ l } = 2, ∀l, and in [20] for L = 1 and N_{1} ≥ 2 with multiplexing transmission. Figure 1a shows an example of the unicasting strategy for MGMW relaying with L = 2 communication groups and N_{1} = N_{2} = 3 nodes.
2) Hybrid Uni/Multicasting Strategy
For each served group, one data stream is transmitted to one node exclusively (unicast transmission) and one data stream is transmitted to the other N_{ l }  1 nodes (multicast transmission). In each BC phase, the unicasted data stream is fixed and is transmitted to a different node in the group. Consequently, the multicasted data stream has to be changed in each BC phase to ensure that each node in each group receives all data streams of the other nodes in its group within P phases. Compared to the unicasting strategy, intragroup interference in each BC phase is reduced since only two data streams are transmitted simultaneously.
The procedure can be described as follows. For each group l, the RS chooses one data stream out of N_{ l } data streams. This data stream will be unicasted to different nodes in different BC phases. Therefore, in each BC phase, to ensure that each node receives the N_{ l }  1 data streams from the other N_{ l }  1 nodes, the multicasted data stream is the transmitted data stream of the node who will receive the unicasted data stream. In the following, we derive the mathematical formulation of the procedure for hybrid uni/multicasting.
In the l th group, given the index of the transmit node whose data stream is unicasted by the RS and the index , , of the transmit node whose data stream is multicasted, the relationship between r_{ l } , t_{ l } , and p is defined by
where
The relationships in (2) and (3) are defined after choosing the data stream to be unicasted for group l, , which remains the same in all N_{ l } 1 BC phases. In the p th phase, node , whose data stream is multicasted by the RS, receives the unicasted data stream from . The other nodes, r_{ l }, , receive the data stream from node which is multicasted by the RS to these N_{ l } 1 nodes. The multicasted data stream is changed in every BC phase as defined in Equation (3).
Using hybrid uni/multicasting strategy, the nodes need to know which data stream is unicasted and which data stream is multicasted by the RS in the p th phase. However, given (2) and (3), and by choosing the unicasted data stream from the node with the lowest index, that is, , there is no signalling effort needed. The RS is then multicasting the data streams in the P  1 BC phases starting from the lowest index in the set . In case of onepair twoway relaying and multiuser twoway relaying, the hybrid uni/multicasting strategy is the same as the unicasting strategy. The proposed hybrid uni/multicasting strategy is a generalisation of the work in [3, 10] for L = 1 and N_{1} = 2, and in [15] for L > 1 and N_{ l } = 2, ∀l. Figure 1b shows an example of the proposed hybrid uni/multicasting strategy for MGMW relaying when L = 2 and N_{1} = N_{2} = 3 nodes.
3) Multicasting Strategy
Using multicasting strategy, the RS transmits only one data stream for each served group in each BC phase. The RS transmits , i.e., the superposition of the data streams of nodes Sv_{ l } , , and Sw_{ l } , , in the l th group, to all N_{ l } nodes in group l. Prior to detection, each node has to cancel the self and knowninterference from each of the received data streams using the available side information. The side information can be its own transmitted data stream or a data stream which has been decoded in one of the previous BC phases. The general rule for selecting the two data streams for each group in each BC phase is that we have to ensure that the data stream of each node in each group is selected at least once.
The relationship between r_{ l } , t_{ l } , and p can be written as
Given the general rule, we may have several options to define the superposed data stream. However, each option will lead to different signaling requirement, since the RS has to inform the nodes about the indices v_{ l } and w_{ l } in each BC phase. In this work, we are interested in an option that does not need any signalling. Therefore, we extend the proposal in [20] to the case of MGMW relaying. We always choose v_{ l } = a_{ l } , and, consequently, w_{ l } is changed in each BC phase and is selected successively based on the relationship defined by w_{ l } = v_{ l } + p  1.
Using such relationships, node Sr_{ l } = a_{ l } always performs selfinterference cancellation, i.e., , to obtain all N_{ l }  1 data streams from other nodes t_{ l } = w_{ l } , . Regarding the other nodes , they have to be able to decode and, afterwards, use to perform knowninterference cancellation. Each node has to wait until its own data stream is superposed with , that is in the p th phase which leads to r_{ l } = (p + a_{ l } ) 1. In this corresponding p th phase, node r_{ l } = (p + a_{ l } )  1 performs selfinterference cancellation, i.e., to obtain . Afterwards, using , it performs knowninterference cancellation to obtain the other data stream from the other nodes t_{ l } = w_{ l } , received in the other BC phases. The proposed multicasting strategy is a generalisation of the work in [1, 22, 24] for L = 1 and N_{1} = 2, in [16] for L > 1 and N_{ l } = 2, ∀l, and in [20] for L = 1 and N_{1} ≥ 2 with analog network coding transmission. Figure 1c shows an example of MGMW relaying using the multicasting strategy when L = 2 and N_{1} = N_{2} = 3 nodes.
In this subsection, we have mathematically formulated the BC strategies. With the assumption of timeinvariant channels within P phases, for unicasting strategy, any other relationship that one may derive will lead to the same performance. The relationships given in Section IIA1 has an advantage that it does not require any signaling in the network. For hybrid uni/multicasting, one may also find other relationship than the relationships given in (2) and (3). However, with the timeinvariant channels assumption and given the same , the same performance will be obtained. In Section IIA2, we choose such that there will be no signaling in the network. This is suboptimum and one may improve the performance by exhaustively searching the best , ∀l. This will lead to a higher computational complexity and also requires signalling in the network since the RS needs to inform all nodes in group l about the chosen . Regarding multicasting strategy, in this work we propose v_{ l } = a_{ l } and w_{ l } = p + a_{ l } 1 which does not require any signaling in the network. This is suboptimum and one may improve the performance by exhaustively searching v_{ l } and w_{ l } which optimises the performance while fulfilling the general rule for multicasting strategy as described in Section IIA3. However, the computational complexity will be higher and there are signaling needed since the RS needs to inform the nodes in group l about v_{ l } and w_{ l } .
Note that all the relationships of parameters which are described in this subsection can also be directly applied for the case when the numbers of nodes are not equal in all groups. If the numbers of nodes are not equal in all groups, in each p th phase, the RS serves only the groups with N_{ l } ≥ p.
B. Generalised System Model
In this subsection, we explain the generalised system model for nonregenerative MGMW relaying which is valid for all BC strategies. In order to have nonregenerative MGMW relaying with a specific BC strategy, the relationship of p, r_{ l } , and t_{ l } as described in the previous subsection has to be set accordingly.
The overall channel matrix from the nodes to the RS is given by , with , , the channel vector between node Si and the RS. The channel coefficient h_{i, m},, , follows . The vector is equal to (x_{0},..., x_{N1})^{T}, with x_{ i } the transmit signal of node Si that follows . The AWGN noise vector at the RS is denoted as with z_{RSm}following . In this work, we assume that all nodes transmit with fixed and equal transmit power.
In the first phase, all nodes transmit simultaneously to the RS and the received signal at the RS is given by
Assuming reciprocal and timeinvariant channels in P phases, the downlink channel from the RS to the nodes is simply the transpose of the uplink channel H. In the p th phase, the RS performs transceive beamforming, denoted by matrix G^{p} , to the received signals and transmits to the nodes. Therefore, G^{p} has to be designed to ensure that the MGMW relaying is performed according to the chosen BC strategy. It is assumed that there is a transmit power constraint at the RS. The received signal vector of all nodes in the p th phase can be written as
where , with the noise at receiving node r_{ l } which follows . Accordingly, the received signal at node Sr_{ l } , , while receiving the data stream from node St_{ l } , , in the p th phase is given by
Sum Rate Expression
In this section, we derive the sum rate expression of nonregenerative MGMW relaying. We start by defining the signal to interference and noise ratio (SINR) for the BC strategies. The achievable sum rate of MGMW relaying for both asymmetric and symmetric traffic are explained afterwards. The achievable sum rate is the sum of the rates received at all nodes. Asymmetric traffic refers to the situation where we allow all nodes in the group to transmit with different rates. Each node transmits with a rate that ensures that in the following N_{ l }  1 consecutive BC phases, all N_{ l }  1 nodes in its group can decode its data stream correctly. Symmetric traffic is when all nodes in group l have to transmit simultaneously with the same rate that is defined by the lowest rate among all possible link combinations of receive and transmit node (r_{ l } , t_{ l } ) in group l.
A. Signal to Interference and Noise Ratio
It is assumed that x_{ i } , ∀i, , ∀m, and z_{ i } , ∀i, are all statistically independent. Therefore, given the received signal in (7), the SINR for the link between receiving node Sr_{ l } and transmitting node St_{ l } is given by
with the useful signal power at node Sr_{ l }
the RS's propagated noise power which appear at node Sr_{ l }
and the node Sr_{ l } 's noise power
The interference power at receiving node Sr_{ l } , is given by
with the same group interference power and the other group interference power. While depends on the applied BC strategy, does not depend on the BC strategy and it is given by
At each receiving node Sr_{ l } , includes the interference power caused by its own data stream and other data streams that have been decoded in the previous BC phases. These a priori known data streams can be canceled by each receiving node prior to detection by performing self and knowninterference cancellation. If self and knowninterference cancellation is performed, the remaining interference power which is not canceled by the receiving node, , is given by
with the interference power caused by the data streams which are a priori known by the receiving node Sr_{ l } and is canceled. With interference cancellation, the interference power in (12) can be rewritten as
In the following, we explain and for each BC strategy.
1) Unicasting
The same group interference power is given by
In every p th phase, node Sr_{ l } may perform interference cancellation. It subtracts the a priori known selfinterference as well as the a priori known same group otherstream interference from the previous BC phases. Once the nodes have decoded other nodes' data streams in the previous BC phases, they may use it to perform knowninterference cancellation in a similar fashion to selfinterference cancellation. Using interference cancellation, for unicasting strategy is given by
with the set of the nodes' indices whose data streams have been decoded by receiving node r_{ l } in the previous BC phases.
2) Hybrid uni/multicasting
The same group interference power can be decoupled into two parts. The first part is the interference caused by the unicasted or the multicasted data stream, denoted by . The second part is the interference caused by other data streams which can only appear at the receiving node r_{ l } if the transceive beamforming applied at the RS cannot fully suppress it. The same group interference power is given by
with the index of the transmitting node whose data stream is unicasted by the RS, the index of the transmitting node whose data stream is multicasted by the RS, and
the interference at the nodes which only can be either from the unicasted data stream (at N_{ l }  1 nodes which are intended to receive the multicasted data stream) or from the multicasted data stream (at the node which receives the unicasted data stream). Similar to the unicasting strategy, interference cancellation at the nodes can also be applied. For hybrid uni/multicasting transmission, is defined by
with the sets of nodes' indices whose data streams have been multicasted by the RS in the previous BC phases and
3) Multicasting
The same group interference power can be decoupled into two parts. The first part is the inherent interference within the superposed data stream which can only be either self or knowninterference, denoted by I_{sk}. The second part is the interference caused by other data streams which can only appear at the receiving node r_{ l } if the transceive beamforming applied at the RS cannot fully suppress it. The same group interference power is given by
with {v_{ l } , w_{ l } } the indices of the two nodes in group l whose data streams are superposed by the RS in the p th phase.
is the self or knowninterference power, which can only be either selfinterference power at nodes r_{ l } = w_{ l } and r_{ l } = v_{ l } given by
or knowninterference power at nodes r_{ l } ≠ w_{ l } ≠ v_{ l } given by
with the index of the knowninterference which can only be either w_{ l } or v_{ l } . As explained in Section IIA3, can be cancelled and, thus, . Moreover, once the nodes have decoded other nodes' data streams from the previous BC phases, they may use them to reduce the amount of interference in the second summand in (22). For the multicasting strategy, is defined by
with the set of the nodes' indices whose data streams have been decoded by receiving node r_{ l } in the previous BC phases.
B. Sum Rate for Asymmetric Traffic
Given the SINR as in (8), the information rate at receiving node r_{ l } when it receives from transmitting node t_{ l } in the p th phase is given by
Since in MGMW relaying there is only one MAC phase, the transmitting node t_{ l } has to ensure that its data stream can be decoded correctly by all N_{ l }  1 intended receiving nodes. Consequently, we have
which is the minimum rate among all receiving nodes r_{ l } in group l when they receive the data stream from a certain transmitting node t_{ l } . The achievable sum rate of nonregenerative MGMW relaying is given by
The factor N_{ l }  1 is since in group l there are N_{ l }  1 nodes that receive the same data stream from a certain transmitting node t_{ l } . The scaling factor is due to P channel uses for MGMW relaying.
One important note regarding (27) is that by taking the minimum, we ensure each node Si transmits x_{ i } with the rate that can be decoded correctly by all other nodes in its group. Thus, knowing x_{ i } , all other nodes in the group can use it to perform knowninterference cancellation in a similar fashion to their selfinterference cancellation.
C. Sum Rate for Symmetric Traffic
In certain scenarios, there may be a requirement to have a symmetric traffic between all nodes in group l. All nodes communicate with the same data rate defined by the minimum of ,∀t_{ l } , . The achievable sum rate for symmetric traffic for all BC strategies is given by
Transceive Beamforming
In this section, first, we formulate the optimisation problem of finding the optimum transceive beamforming maximising the sum rate. Afterwards, we explain the design of generalised low complexity transceive beamforming algorithms for all BC strategies. It is assumed that perfect channel state information is available at the RS whose number of antennas is higher than or equal to the total number of the nodes, i.e., M ≥ N.
A. Sum Rate Maximisation
The optimisation problem of finding the optimum transceive beamforming maximising the sum rate of nonregenerative MGMW relaying for asymmetric traffic can be written as
with , , and f(i, p) the receiving node index, which is a function of transmitting index i and BC phase index p, and depends on the applied BC strategy.
In this work, we assume that the transmit powers at the nodes are fixed and equal. In order to improve the sum rate, one could have the transmit powers at the nodes as variables to be optimised subject to a power constraint at each node. However, since there is only one MAC phase, one has to find the optimum transmit power at each node and, simulateneously, the transceive beamforming for all BC phases, i.e., G^{p} , ∀p, . This joint optimisation problem would further increase the computational effort.
The optimisation problem in (30) is nonconvex and it requires high computational complexity to find the global optimum solution. Thus, in the following, we propose generalised low complexity transceive beamforming algorithms for all proposed BC strategies.
As mentioned in Section IIB and as seen in (30), the transceive beamforming G^{p} depends on the BC strategy applied at the RS. In order to design generalised transceive beamforming for all BC strategies and to make the problem more tractable, we decouple G^{p} into transmit beamforming , BCstrategydefining permutation matrix Π^{p} and receive beamforming , such that .
In the following, we explain specially designed transceive beamforming for MGMW relaying. First, we explain the generalised linear transceive beamforming based on three different optimisation criteria, namely, MF, ZF, and MMSE. Afterwards, we explain the generalised BCstrategyaware (BCSA) transceive beamforming.
B. Linear Transceive Beamforming
In this subsection, we explain the design of three low complexity generalised linear transceive beamforming algorithms, namely, MF, ZF, and MMSE. Since we have only one MAC phase, the receive beamforming is computed only once, i.e., , . The BCstrategydefining permutation matrix Π^{p} defines the transmission from the RS according to the BC strategies. Table 1 shows Π^{p} for the example in Figure 1 for MF, ZF, and MMSE for all BC strategies. One important note is that, even though the derivation for the MF, ZF, and MMSE generalised transceive beamforming appears to be similar with the threestep transceive beamforming for twoway relaying in [9]; however, our generalised transceive beamforming algorithms have a different approach and are based on different motivations. In [9], the downlink (from the RS to the nodes) channel matrix is a permuted matrix of the uplink (from the nodes to the RS) channel matrix. Therefore, Π^{p} in [9] is a diagonal matrix with weighting factors in each of its diagonal elements. Such approach as in [9] is only suitable for unicasting strategy. Hence, our generalised transceive beamforming is a generalisation of the threestep transceive beamforming for twoway relaying in [9].
1) Matched Filter
Given the received signal at RS as in (5), the output of the receive filtering is given by
The MF optimisation problem for receive beamforming can be written as
The objective function in (32) can be written as
By taking the derivative of (33) with respect to G_{R} and setting it equal to zero, we have [see, e.g, [25, 26]]
The received signal at the nodes in (6) can now be rewritten as
with the transmitted signals from the RS in the p th phase. The MF optimization problem for transmit beamforming can be written as
Let and . Using the same steps as in [26] by deriving the Lagrangian function and solving the KarushKuhnTucker (KKT) conditions, we have
where is needed to fulfill the power constraint and given by
2) Zero Forcing
Given (31), the ZF optimisation problem for receive beamforming can be written as
where is the ZF constraint which implies that G_{R}H = I _{M}. Due to the ZF constraint, the objective function in (39) can be written as
Using the same steps as in [26] by deriving the Lagrangian function and solving the KKT conditions, we have [see, e.g., [25]]
Given (35), the ZF optimisation problem for transmit beamforming can be written as
where is the ZF constraint which implies that . Due to the ZF constraint, the objective function in (42) can be written as
Using the same steps as in [26] by deriving the Lagrangian function and solving the KKT conditions, we have
where is needed to fulfill the power constraint and given by
3) Minimisation of Mean Square Error
Given (31), the MMSE optimisation problem for receive beamforming can be written as
The objective function in (46) can be written as
By taking the derivative of (47) with respect to G_{R} and setting it equal to zero, we have [see, e.g., [25, 26]]
Given (35), the MMSE optimisation problem for transmit beamforming can be written as
where 1/β^{p} is introduced to modify the mean square error as in [27, 28]. Using the same steps as in [28] by deriving the Lagrangian function and solving the KKT conditions, we have
where is needed to fulfill the power constraint and given by
Finally, for MF, ZF, and MMSE the transceive beamforming is given by
where the subscript (·)_{algorithm} refers to either MF, ZF, or MMSE.
C. BroadcastStrategyAware Transceive Beamforming
In the following, we explain the design of BCSA transceive beamforming. Based on the chosen BC strategy, the RS separates the data streams which are going to be transmitted in the BC phase and transmits to the corresponding node or nodes. For unicasting strategy, the RS separates all data streams and transmits each data stream to each corresponding receiving node. For hybrid uni/multicasting, for each group, the RS separates the unicasted data stream from the other data streams and transmits it to the corresponding node whose data stream is multicasted. The RS also separates the multicasted data stream from the other data streams and transmits it to the remaining nodes in the corresponding group. For multicasting strategy, the RS separates the superposition of two data streams from the others and transmits the superposed data stream to all nodes in the group.
In order to compute the transceive beamforming, we first compute the equivalent channels for receive beamforming and transmit beamforming. The equivalent channels are needed to ensure that there will be no interstream interference received at the unintended receiving node or nodes. In order to find the equivalent channel, BD as proposed in [17] can be applied. Several works have considered BD for separation of data streams, e.g., [16, 20, 29, 30]. In this work, we also consider regularised BD (RBD) as proposed in [23]. RBD avoids the drawbacks of BD which has a quite poor performance if the subspaces of the users channel matrices overlap significantly [23].
Equivalent channel
Without loss of generality, in the following we omit the BC phase index p. Let and denote the channel matrix of the intended nodes and the channel matrix of the other unintended nodes, respectively, with the number of intended nodes. Both channel matrices are parts of the overall channel matrix, i.e., . Since the steps of computing the equivalent channel for receive beamforming and transmit beamforming are similar, we generally explain the methods for finding the equivalent channel using and . In order to relate them with the receive beamforming and transmit beamforming for BC strategies, we have to set and accordingly. Table 2 shows the corresponding and for all BC strategies. Given the singular value decomposition (SVD) of the unintended nodes' channels as
we compute the equivalent channel for the intended nodes . The equivalent channel is given by
where F_{null} is the nullspace matrix which can be computed either using BD or RBD.
Using BD, with denoting the rank of matrix . The BD approach can be used directly for receive and transmit beamforming, since it only deals with the channels without considering the noise. Using RBD, however, the equivalent channels for receive and transmit beamforming need to be computed differently. RBD for transmit beamforming has been derived in [23] and in this work, we provide the derivation of RBD for receive beamforming in the appendix. Using RBD, with for transmit beamforming [23] and for receive beamforming, see appendix.
Having , we can now compute the receive beamforming and transmit beamforming. In the following, when computing the receive beamforming, and relate to and as defined in Table 2 for receive beamforming, while when computing transmit beamforming, and relate to and as defined in Table 2 for transmit beamforming.
In this work, we consider signal processing algorithms which do not deal with interference since is free from unwanted data streams, namely, MF, SVD, and semidefinite relaxation (SDR) of maximising the minimum SNR. MF and SVD for singlepair twoway relaying have been investigated in [3, 10]. The BDMF and BDSDR have been designed in [20] for singlegroup multiway relaying for multicasting strategy. BDSVD has been designed in [16] only for multiuser twoway relaying with multicasting strategy. In this work, BCSA transceive beamforming is designed for nonregenerative MGMW relaying for all proposed BC strategies. Due to the requirement to make generalised BCSA transceive beamforming also suitable for nonregenerative MGMW relaying with multicasting strategy, it has a slight difference to [3, 10, 16]. Using BCSA for multicasting strategy, for each group l, the RS has to transmit one data stream, which is a superposition of two data streams, to N_{ l } nodes in the group where N_{ l } can be any number higher than two. For that reason, in the design of receive beamforming using MF and SVD, we have to do a superposition of two data streams. This makes the proposed BCSA not a direct generalisation of [3, 10, 16]. However, for cases of onepair twoway relaying and multiuser twoway relaying, if the superposition is not performed, BCSA is a generalisation of [3, 10, 16].
Matched filter The receive beamforming vector is given by
where a vector of ones of length . superposes (adds) twodata streams from two nodes in each group for multicasting strategy. can be seen as receive power loading where the modulus operator  ·  is assumed to be applied elementwise and the mean function returns the mean of a vector.
The transmit beamforming vector is given by
with the transmit power loading. is a vector of ones with size . For multicasting strategy, replicates the superposed data stream times.
Singular value decomposition Let the SVD of the equivalent channel be given by
The receive beamforming vector is given by
with the receive power loading.
The transmit beamforming vector is given by
with the transmit power loading.
Semidefinite Relaxation
Since in MGMW relaying all member nodes in each group exchange messages, we are also interested in a fair beamforming algorithm which aims at balancing the SNRs at the RS as well as at the receiving nodes in each group.
The SNR balancing problem for receive beamforming can be written as
with the set of intended nodes with cardinality equal to . is the index of a member node in and . The receive beamforming is given by
with the receive power loading.
Equation (60) is a nonconvex quadratically constrained quadratic program. A similar optimisation is also considered in [31]. It is proved to be NPhard in [31]. Nonetheless, it can be approximately solved using SDR techniques [31, 32]. We will not go further into this relaxation and the interested reader may find more detailed derivation in [20, 31, 32].
The SNR balancing problem for transmit beamforming can be written as
with . The transmit beamforming is given by
with the transmit power loading. Note that to compute the transmit beamforming with SDR, we assume that the information of the noise power at the nodes is available at the RS. Similar to (60), (62) can be approximately solved with semidefinite relaxation techniques using a solver such as SEDUMI [33].
In the following, we use again the BC phase index p to describe the BCSA transceive beamforming. For unicasting strategy, the receive beamforming matrix is given by
where , , is the receive beamforming as in (55), (58) or (61) given the equivalent channel of node t_{l}. The transmit beamforming matrix is given by
where , , is the transmit beamforming as in (56), (59), or (63) given the equivalent channel of node r_{l}.
For hybrid uni/multicasting strategy, the receive beamforming matrix is given by
where m , , and , are the receive beamforming as in (55), (58), or (61) given the equivalent channels of nodes and , respectively. The transmit beamforming matrix is given by
where , , and , , are the transmit beamforming as in (56), (59), or (63) given the equivalent channel of node and the equivalent channel of all other nodes in group l, , respectively.
For multicasting strategy, the receive beamforming matrix is given by
where , , is the receive beamforming as in (55), (58), or (61) given the equivalent channels of two nodes v_{ l } and w_{ l } whose data streams are superposed. The transmit beamforming matrix is given by
where , , is the transmit beamforming as in (56), (59), or (63) given the equivalent channels of all nodes in group l.
Finally, BCSA transceive beamforming is given by
where β^{p} is needed in order to satisfy the transmit power constraint at the RS, with
Note that Π^{p} is not the same for all BC strategies. For unicasting strategy, Π^{p} is the same as for MF, ZF, and MMSE, where an example for L = 2, N_{1} = N_{1} = 3 is given in Table 1. For hybrid uni/multicasting strategy, Π^{p} = I_{2L}and for multicasting strategy, Π^{p} = I _{L}.
Simulation Results
In this section, the sum rate performance is analysed based on simulation results. We set , , and E_{RS} = 1. The channel coefficients are i.i.d. , i.e., Rayleigh fading. Hence, the SNR value is given by .
We consider three scenarios. The first two scenarios are the wellknown scenarios, namely, onepair twoway relaying and multiuser twoway relaying. We consider both scenarios to show that the proposed BC strategies and the generalised transceive beamforming designed in this work are valid for both wellknown scenarios. The third scenario is the twogroup multiway case, where each group consists of three nodes.
A. First Scenario: L = 1 and N_{1} = 2
In onepair twoway relaying, unicasting and hybrid uni/multicasting are the same. Figure 2 shows the sum rate performance of onepair twoway relaying with MF, ZF, and MMSE transceive beamforming for both asymmetric traffic and symmetric traffic. Regarding symmetric traffic, to reduce the number of lines in the figure, we only plot the result for MF transceive beamforming. The approximation of maximum sum rate is also provided for two cases, i.e., with optimised and with fixed transmit power at the nodes. Both optimum transceive beamforming solutions maximising the sum rate were computed using fmincon from MATLAB to provide performance bounds for twoway relaying. We use the value of MMSE transceive beamforming as the initial value. In general, using MF, ZF and MMSE transceive beamforming, unicasting and hybrid uni/multicasting outperform multicasting strategy. A direct superposition of the output of receive beamforming for the multicasting strategy doubles the amount of the RS's filtered noise. Moreover, the RS transmit power is distributed within the superposed data stream and after the selfinterference cancellation, each node only receives half of the power. Since each node performs selfinterference cancellation, no interference appears at the nodes, and thus, for all BC strategies MF outperforms MMSE and ZF. At low SNR, MMSE converges to MF and in high SNR, ZF converges to MMSE. In this work, we assume fixed transmit power at all nodes and the performance of unicasting and hybrid uni/multicasting with MF is close to the approximation of maximum sum rate with fixed transmit power. If the nodes can optimise their transmit power, the sum rate is improved with a penalty of having higher computational complexity. It can also be seen that asymmetric traffic leads to a higher rate compared to symmetric traffic since the rate for symmetric traffic is defined by the weakest link among all available links. Therefore, in the following, we only consider asymmetric traffic.
Figure 3 shows the sum rate performance of twoway relaying with BCSA transceive beamforming. For multicasting strategy, since there is no separation needed both for receive beamforming and transmit beamforming, BD and RBD are the same. For unicasting and hybrid uni/multicasting strategies, BDMF, BDSVD, and BDSDR perform the same and they have similar performance to multicasting strategy with SVD. Different to multicasting strategy, for unicasting and hybrid uni/multicasting, since there is a stream separation both in receive beamforming and transmit beamforming, RBD improves the performance in low SNR region. In high SNR region, BD converges to RBD. For unicasting and hybrid uni/multicasting strategies, since the equivalent channels (which are free from interference) always correspond only to one intended node for both receive beamforming and transmit beamforming, MF, SVD, and SDR will always have the same performance. It can be seen that multicasting strategy with SDR performs best. Hence, having a suitable transceive beamforming, one can exploit the benefit of beamformingbased physical layer network coding for nonregenerative single group multiway relaying as proposed in [20].
B. Second Scenario: L = 2 and N_{1} = N_{2} = 2
Figure 4 shows the sum rate performance of multiuser twoway relaying with MF, ZF, and MMSE transmit beamforming. In this scenario, unicasting and hybrid uni/multicasting are the same and they outperform multicasting strategy. The reason is the same as in the case of onepair twoway relaying. Moreover, the direct superposition of the output of receive beamforming for multicasting strategy not only increases the amount of the RS's filtered noise but also increases the unwanted interference at the receiving nodes. For all strategies, MMSE performs best and in high SNR region, ZF converges to MMSE, while in low SNR region, MF converges to MMSE. Different to the case of onepair twoway relaying, in multiuser twoway relaying MF performs worse since it does not cancel the interference from other pairs which appears at each node. The transceive beamforming maximising the sum rate was computed using fmincon from MATLAB to provide a bound for multiuser twoway relaying. We use the value of MMSE tranceive beamforming as initial value. It can be clearly seen that if the transmit power at the nodes can be optimised, the sum rate can be improved at the expense of computational complexity.
Figure 5 shows the sum rate performance of multiuser twoway relaying with BCSA transceive beamforming. In general, RBD outperforms BD in low SNR region and BD converge to RBD in high SNR. Only for multicasting strategy, BDSVD outperforms RBDSVD for all SNR values and it has similar performance as unicasting and hybrid uni/multicasting with BDMF, BDSVD, and BDSDR. The gain of RBD compared to BD is obtained most for unicasting and hybrid uni/multicasting strategies, while for multicasting strategy (with MF and SDR), the gain is small. In medium to high SNR region, multicasting strategy with RBDSDR or BDSDR performs best. While for both unicasting and hybrid uni/multicasting strategies, the performance of MF, SVD, and SDR is the same, for multicasting strategy SDR always performs best followed by MF and SVD.
C. Third Scenario: L = 2 and N_{1} = N_{2} = 3
Figure 6 shows the sum rate performance of twogroup threeway relaying using MF, ZF, and MMSE. In general hybrid uni/multicasting performs best followed by unicasting and multicasting strategies. While hybrid uni/multicasting strategy with MMSE slightly ouperforms unicasting strategy with MMSE, both strategies have similar ZF performance. With MMSE, we find the trade off between the noise enhancement and the interference suppression. Since, hybrid uni/multicasting has smaller number of transmit data streams from the RS, it performs better than unicasting strategy both for MMSE and MF. ZF perfectly cancels the interference, and, thus, both unicasting and hybrid uni/multicasting perform similar. In general, for all strategies, ZF converges to MMSE in high SNR region and in low SNR region, MF converges to MMSE. It can be seen that multicasting strategy is outperformed by other strategies since it suffers from the increase of RS's filtered noise and the reduced received power at the nodes. This shows that analog network coding for nonregenerative MGMW relaying obtained by directly adding the output of receive beamforming (using MF, ZF, and MMSE receive beamforming) is not an efficient strategy and, thus, appropriate transceive beamforming is required.
Figs. 7 and 8 show the sum rate performance of twogroup threeway relaying using BCSA transceive beamforming with BD and RBD, respectively. Comparing both the figures, in general, RBD outperforms BD, and they converge in high SNR. Only when using SVD, for both hybrid uni/multicasting and multicasting strategies, RBDSVD performs worse than BDSVD and BDSVD does not converge to RBDSVD in high SNR region. In medium to high SNR, multicasting strategy outperforms other strategies when using SDR and MF. However, if SVD is applied, unicasting strategy performs best. For unicasting strategy, MF, SVD and SDR have similar performance.
Comparing Figures 6, 7, and 8, one can clearly see that BCSA transceive beamforming improves the sum rate performance compared to MF, ZF, and MMSE transceive beamforming, especially for multicasting strategy. For the multicasting strategy, only RBDSVD performs worse than ZF and MMSE. For the hybrid uni/multicasting strategy, BDMF performs similar to ZF, RBDMF performs similar to MMSE and both BDSDR and RBDSDR outperform MF, ZF, and MMSE. For the unicasting strategy, BDMF, BDSVD, and BDSDR perform similar to ZF, while RBDMF, RBDSVD, and RBDSDR perform similar to MMSE. The highest sum rate (especially in high SNR) is obtained by multicasting strategy with BCSA BDSDR and RBDSDR. Therefore, provided a suitable transceive beamforming is used which can exploit analog network coding, the sum rate of nonregenerative MGMW relaying can be improved.
Conclusion
In this paper, we consider nonregenerative MGMW relaying. A multiantenna RS assists L communication groups, where N_{ l } nodes in each group communicate to each other but not with other nodes in other groups. The number P of communication phases is equal to max_{ l }N_{ l } . Three BC strategies are proposed, namely, unicasting, hybrid uni/multicasting, and multicasting. We derive the sum rate expression for nonregenerative MGMW relaying for asymmetric and symmetric traffic. We address the optimum transceive beamforming maximising the sum rate of nonregenerative MGMW relaying. We design generalised low complexity suboptimum transceive beamforming for all BC strategies, namely, MF, ZF, MMSE, and BCSA transceive beamforming. It is shown that the performance of nonregenerative MGMW relaying depends on the BC strategy and the applied transceive beamforming. While multicasting strategy using BCSA SDR provides better sum rate performance compared to other strategies, however, if either MF, ZF, or MMSE are applied, multicasting strategy is outperformed by the other strategies.
Appendix
RBD for Receive Beamforming
For receive beamforming, the RS has to ensure that the interference from other users to the intended user i can be minimised while taking into consideration the appearance of noise at the RS. The matrix F_{Null} is designed to achieve the aim and, by rewriting the optimisation problem for transmit beamforming in [23] Equation (9) we have the optimisation problem for receive beamforming,
where β is a scaling factor needed to fulfill the nodes' transmit power constraint. In this work, we assume that all nodes transmit with fixed and equal unit power and, thus, the constraint can be written as
The objective function in (72), f(F _{i}), can be written as
Let the SVD of be given by

(74)
can be rewritten as
(76)
Let F_{ i } = F a_{i} F b _{ i } and let , the optimisation problem reduces to
where F a _{ i } needs to be positive definite in order to find a nontrivial solution [23]. Using the results from [23, 34], we have
with .
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Acknowledgements
The work of Aditya U. T. Amah is supported by the 'Excellence Initiative' of the German Federal and State Governments and the Graduate School of Computational Engineering, Technische Universität Darmstadt. The authors would like to thank the anonymous reviewers whose review comments are very helpful in improving this article.
Some parts of this paper have been presented at the IEEE ICASSP 2010, Dallas, USA, and IEEE WCNC 2010, Sydney, Australia.
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Amah, A.U.T., Klein, A. NonRegenerative MultiAntenna MultiGroup MultiWay Relaying. J Wireless Com Network 2011, 29 (2011). https://doi.org/10.1186/16871499201129
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Keywords
 Multiway relaying
 Nonregenerative
 Multiantenna
 Analog network coding
 Transceive beamforming