relay controls which stations are connected to which other stations. Each beamforming matrix realizes a permutation among the stations. By scheduling a set of different switch matrices, full connectivity among the stations can be established. 

Prior work that investigated the set-up of multiple stations exchanging data via a relay includes [2], [3], [4], and [5]. Ref. [2] studied “pairwise data exchange”, in w hich stations form pairs, and two stations in a pair exchange data with each other only. Specifically for pai rwise data exchange, if station i transmits to station j , then station j transmits to station i as well. In [2], MIMO relays with different forward strategies were considered. The bounds on the symmetric capacity were presented. Ref. [3] also studied pairwise data exchange, but the relay adopts the decode-and-forward strategy only. The diversity-multiplexing tradeoffs under reciprocal and non-reciprocal channels were analyzed. Both [2] and [3] studied the case in which a station communicates with one other station only. In a general setting, a station could have data for more than one station. In this paper, we focus on a uniform traffic s etting in which the amounts of traffic from station i to station j are the same for all i, j ∈ { 1 , · · · , N } , i ̸ = j . We refer to meeting such a uniform traffic requirement as “fair switching”. Fair switching is r ealized by scheduling a set of switch matrices. To the best of our knowledge, the framework of fair switching has not been considered in the existing literature. 

Refs. [4] and [5] investigated the case of full data exchange, in which all stations want to broadcast their data to all other stations$^{1}$. Data transmissions in [4] and [5] can be summarized as follows: in the first slot, all stations transmit to the relay simultaneousl y; the first slot is followed by multiple slots for downlink transmissions; in each downlink slot, the relay multiplies the signal received in the first time slot by a different beamforming matrix, such that at the end of all downlink slots, all stations receive the broadcast data from all other stations. By contrast, the framework investigated in this paper is more general in that it can accommodate the pure unicast case, the mixed unicast-multicast case, as well as the pure broadcast case as in [4] and [5]. In particular, a station i can have M$_{i}$ data streams, and each station j ̸ = i is a target receiver of one of the M$_{i}$ streams. 

In our framework, the MIMO relay serves as a general switch that switches traffic among the stations. We focus on the use of beamforming at the relay and linear detection to realize different connectivity patterns among the stations. Each beamforming matrix realizes a permutation connectivity among the stations. By scheduling a set of switch matrices, the MIMO switching system can realize any general 

$^{1}$Note that full data exchange is also discussed in [2]. But they consider a single-antenna relay. 

