A multiple input multiple output (MIMO) two-way relay channel is considered, where two sources want to exchange messages with each other using multiple relay nodes, and both the sources and relay nodes are equipped with multiple antennas. Both the sources are assumed to have equal number of antennas and have perfect channel state information (CSI) for all the channels of the MIMO two-way relay channel, whereas, each relay node is either assumed to have CSI for its transmit and receive channel (the coherent case) or no CSI for any of the channels (the non-coherent case). The main results in this paper are on the scaling behavior of the capacity region of the MIMO two-way relay channel with increasing number of relay nodes. In the coherent case, the capacity region of the MIMO two-way relay channel is shown to scale linearly with the number of antennas at source nodes and logarithmically with the number of relay nodes. In the non-coherent case, the capacity region is shown to scale linearly with the number of antennas at the source nodes and logarithmically with the signal to noise ratio.
Relay channels are the most basic building block for cooperative and multihop communication in wireless networks. In a relay channel, one or more nodes, without data of their own to transmit, help a source destination pair communicate. The origins of the relay channel -as a three terminal communication channel -go back to Van der Meulen [1]. Despite the passage of time, the capacity of even the most arXiv:0706.2906v3 [cs.IT] 7 Apr 2008 basic relay channels is still unknown. Nonetheless, bounds derived in [1], [2] show that using a relay, it is possible to increase the reliable rate of data transfer between the source and the destination.
Motivated by the capacity improvements obtained by using multiple antennas at the source and the destination for point-to-point channels [29], recently, there has been a significant research focus on finding the capacity of the multiple input multiple output (MIMO) relay channel, where the source, the destination, and the relay may have multiple antennas [3], [10], [11]. The capacity of the MIMO relay channel was first studied in [3], [13], where upper and lower bounds on the capacity of the MIMO relay channel are derived for the deterministic and the Gaussian fading channel. Improved lower bounds for the MIMO relay channel with Gaussian fading channel were provided by [10], where message splitting and superposition coding are used at the transmitter to improve the bounds provided in [3]. In [3], [10] only full-duplex relays (can transmit and receive at the same time) were considered. Upper and lower bounds on the capacity for the more practical Gaussian MIMO relay channel with half-duplex relays, where the relays cannot transmit and receive at the same time, were developed in [11]. The bounds in [3], [10], [11] indicate that with relays there is a potential capacity gain to be leveraged by using multiple antennas.
In [1]- [3], [10], [11] only a single source destination pair is considered with a single relay node.
For a practical wireless network setting, where there are multiple source destination pairs, the concept of cooperative communication has been recently proposed [4]- [7], where different users in the network cooperate by taking turns relaying each others data. Thanks to the spatial separation between users, cooperation between users provides a means to obtain and exploit spatial diversity gain, called cooperative diversity gain, which increases the achievable data rate between each source and its destination. Several different protocols have been proposed to exploit the cooperative diversity gain, e.g. amplify and forward (AF) [4]- [7], [13], decode and forward (DF) [15], [18], with half-duplex [14], and full-duplex assumptions [16].
Prior work on the relay channel mostly considers one-way communication, i.e. a source wants to send data to a destination. In most networks, however, the destination also has some data to send to the source, e.g. packet acknowledgements from the destination to the source, downlink and uplink in cellular networks. Consequently, there has been interest in the two-way relay channel, where the bidirectional nature of communication is taken into account [21]- [24]. The two-way relay channel was studied in [22],
where upper and lower bounds on the capacity region were derived for a general discrete memoryless channel.
The MIMO two-way relay channel was introduced in [21], where two terminals T 1 and T 2 want to large, K → ∞ with probability 1. Thus, we characterize the scaling behavior of the capacity region of the MIMO two-way relay channel as the number of relay nodes grow large. Our approach is similar to the asymptotic (in the number of relays) capacity formulation of [12], [19], [20].
Our system model and the key assumptions are as follows. We assume that two terminals T 1 and T 2 want to communicate with each other via K relay nodes. None of the relays have any data of their own and only facilitate communication between T 1 and T 2 . Both T 1 and T 2 are equipped with M antennas, while all the K relays have N antennas each. We consider a two-phase communication protocol, where in the any given time slot, for the first α, α ∈ [0, 1] fraction of the time slot, both T 1 and T 2 transmit simultaneously and all the relays receive. In the rest 1 -α fraction of the time slot, all the relays simultaneously transmit and both T 1 and T 2 receive the signal transmitted by all relays. We assume that there is no direct path between T 1 and T 2 and that T 1 , T 2 and all the nodes (T 1 , T 2 and all relay nodes) can only operate in half-duplex mode. No direct path assumption is reasonable for the case when relay nodes are used for coverage improvement and the signal strength on the direct path is very weak. The half-duplex assumption is made since full-duplex nodes are difficult to realize in practice. We assume that both T 1 and T 2 have perfect CSI for all the channels of the MIMO two-way relay channel in the receive mode. This could be e
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