In this paper, we consider a multicasting multipleinput multiple-output (MIMO) relay system where multiple transmitters multicast their own messages to a group of receivers over multiple hops, and all nodes are equipped with multiple antennas. Joint transmit and relay precoding design problem has been investigated for multicasting multiple data streams based on min-max mean-squared error (MSE) criterion. We aim at minimizing the maximal MSE of the signal waveform estimation among all receivers subjecting to power constraints at the transmitters and all the relay nodes. This problem is highly nonconvex with matrix variables and the exactly optimal solution is very hard to obtain. We develop an iterative algorithm to jointly optimize the transmitter, relay, and receiver matrices through solving convex subproblems. By exploiting the optimal structure of the relay precoding matrices, we then propose a low complexity solution for the problem under some mild approximation. In particular, we show that under (moderately) high signal-to-noise ratio (SNR) assumption, the min-max optimization problem can be solved using the semidefinite programming (SDP) technique. Numerical simulations demonstrate the effectiveness of the proposed algorithms.
In many practical communication systems, multiple users (transmitters) need to send their messages to a group of receivers simultaneously. The simplest way to send data to multiple receivers simultaneously is to transmit individual copies of the data to each receiver. However, this is highly inefficient, since multiple copies of the same data are sent from the source through one or more networks. Multicasting enables a single transmission to be received by multiple users, significantly reducing the required bandwidth. For example, in war fields, military troops need to share their current status, pass secret messages to the allied groups. In an online interactive gaming scenario, all the participants are interested to know the current status of their rivals. If the participants multicast their information, the interested users can receive all the messages simultaneously. Multicasting from multiple sources can also be used to support video conferencing and webcasts among multiple users.
The broadcasting nature of the wireless channel makes it naturally suitable for multicasting applications, since a single transmission may be simultaneously received by multiple users. Recently, wireless multicasting technology has attracted great research interest, due to the increasing demand for mobile applications such as streaming media, software updates, and location-based services involving group communications. Particular multicasting applications include live IP-TV, Internet radio, video conferencing and webcasts. However, wireless channel is subject to fading. By exploiting the spatial diversity, multi-antenna techniques can be applied to combat channel fading , . Next generation wireless communication standards such as WiMAX 802.16m and 3GPP LTE-Advanced have already included technologies which enable better multicasting solutions based on multi-antenna and beamforming techniques .
Due to its nonconvex nature, the problem of designing optimal beamforming vectors for multicasting is hard in general. The authors of  have designed transmit beamformers for physical layer multicasting using rank relaxations, where two design criteria were adopted, namely minimizing the transmit power subject to minimum received signal-to-noise ratio (SNR) at each of the intended receivers and a related max-min SNR problem subject to a transmit power constraint. It has been proven in  that both problems are NP-hard. Using lower complexity transmission schemes, the information theoretic capacity of the multi-antenna multicasting channel was studied in  with a particular focus on the scaling of the capacity and achievable rates as the number of antennas and/or users approaches infinity. The effect of channel spatial correlation on the multicasting capacity has been investigated in . The asymptotic capacity limits of multi-antenna multicasting channel have been studied in  based on antenna subset selection. The authors of  investigated transmit precoding design for multi-antenna multicasting systems where the channel state information (CSI) is obtained via limited feedback. The authors of  considered transmit covariance design for a secrecy rate maximization problem, where a multi-antenna transmitter delivers a confidential message to multiple singleantenna receivers in the presence of multiple multi-antenna eavesdroppers. In , the tight upper bound and the lower bound of the multicast secrecy rate are defined via convex approximation.
The works in - solved the max-min SNR/rate beamforming problems with the aid of semidefinite relaxation (SDR) and rank-one approximation. Note that the rankrelaxation technique is suboptimal in general. In -, a stochastic beamforming strategy is proposed for multi-antenna multicasting where the randomization is guided by SDR, but without the need of rank-one approximation. While the use of channel coding and the assumption of sufficiently long code lengths play a vital role in achieving the above result, a combination of transmit beamforming and the Alamouti spacetime code has also been considered in  which yields a rank-two generalization of the SDR-based beamforming. The fundamental limit of the max-min beamforming is that as the number of users increases to infinity, the achievable rate decreases to zero . To solve this problem, a joint beamforming and admission control approach has been developed in  and , where a subset of users is selected so that certain quality-of-service (QoS) requirements can be satisfied. An iterative transmit beamforming algorithm was proposed in  for multiple cochannel multicasting groups to minimize the total power consumed by the antenna array subjecting to signal-to-interference-plus-noise ratio (SINR) constraints at the receivers. However, the method used in  has a high computational complexity. The authors in  attempted to reduce the complexity of  by combining the concept of the iterative second-order cone programming (SOCP) with that of interior-point methods.
While the works in - investigated multicasting systems with single-antenna receivers, recently multi-antenna receivers have been considered - for multicasting systems since receiver beamforming can significantly improve the system performance. In particular, coordinated beamforming techniques have been investigated in , where a generalized form of block diagonalization has been proposed to make orthogonal transmissions to distinct multicasting groups with multi-antenna receivers. The scaling of the achievable rate with increasing number of users was investigated in  for multiple-input multiple-output (MIMO) multicasting where the transmission is coded at the application layer over a number of channel realizations. In , non-iterative nearly optimal transmit beamformers are designed for wireless link layer multicasting with real-valued channels, and for complexvalued channels an upper-bound on the multicasting rate is derived.
The above works - considered single-hop multicasting systems. However, as the transmitter-receiver distance increases, it becomes necessary to adopt relay nodes to efficiently combat the pathloss of wireless channel. Relay nodes are also essential to overcome the shadowing effect of wireless links in large urban areas with giant buildings and other obstacles, behind-the-hill areas, and so on. Hence, efforts are being made to design optimal beamformers for multicasting over more than one hops using relay nodes. A multi-group multicasting relay network has been considered in  and a distributed beamforming algorithm was proposed to minimize the total relay power where each node is equipped with a single antenna. The authors in  studied the lowerbound for the outage probability of cooperative multi-antenna multicasting schemes based on the amplify-and-forward (AF) strategy where the users are equipped with a single antenna. In , multicast scheduling with multiple sessions and multiple channels was investigated where the base station may multicast data in two sessions using MIMO simultaneously through the same channel and the users are allowed to cooperatively help each other on orthogonal channels. Thus, the scheme in  leads to a higher multicasting rate than single-session transmissions. Joint transmit and relay precoding design problems were investigated in - for a two-hop multicasting MIMO relay system where all nodes are equipped with multiple antennas. An iterative algorithm has been developed in  to jointly optimize the source, relay, and receiver matrices. In order to reduce the computational complexity of the iterative algorithm, a simplified algorithm has also been proposed in - for the two-hop multicasting system. Multicasting from multiple sources in a dual-hop MIMO relay system has been considered in .
In this paper, we consider multi-hop multicasting MIMO relay systems where multiple transmitters multicast their messages to a group of receivers with the aid of multiple relay nodes located in series. The transmitters, relay nodes, and receivers are all equipped with multiple antennas. To the best of our knowledge, such multicasting (from multiple sources) MIMO relay system has not been investigated in existing works. Note that our paper generalizes the multicasting scheme in - in two ways. Firstly, we consider multicasting from multiple sources instead of the single-transmitter multicasting in -. Secondly, we generalize the two-hop MIMO relay multicasting scheme to multi-hop systems with any number of hops. Such extension is important in the case of long source-destination distance where a two-hop relay is not sufficient and multi-hop relays are necessary to establish a reliable source-destination link. It is obvious that due to the introduction of multiple users and multiple relay nodes, the mean-squared error (MSE) matrix decomposition and hence the source and relay matrices optimization procedure become much more challenging than that for the single-transmitter two-hop system. For the sake of the implementation simplicity, we choose the AF relaying strategy at all relay nodes. We consider the joint transmit and relay precoding design problem based on the min-max MSE criterion. We aim at minimizing the maximal MSE of the signal waveform estimation among all receivers subjecting to power constraints at the transmitters and the relay nodes. The problem is highly nonconvex with matrix variables and the exactly optimal solution is very difficult to obtain. We develop an iterative algorithm to jointly optimize the transmitter, relay, and receiver matrices through solving convex subproblems. By exploiting the optimal structure of the relay precoding matrices, we propose a lowcomplexity solution to the problem under some mild approximation. We apply the same concept of high SNR assumption as in  in order to decompose the complicated original optimization problem into smaller easily solvable subproblems. In particular, we show that under (moderately) high SNR assumption, the problem can be solved using standard semidefinite programming (SDP) techniques. Numerical simulations demonstrate the effectiveness of the proposed algorithms. Note that the proposed algorithms support multicasting multiple data streams in contrast to the existing single data stream multicasting schemes -. In this paper, for notational convenience, we consider a narrow-band single-carrier system. However, our results can be straightforwardly generalized to each subcarrier of a broadband multi-carrier multicasting MIMO relay system.