On Uplink-Downlink Duality of Multi-Hop MIMO Relay Channel

For two-hop amplify-and-forward (AF) multiple-input multiple-output (MIMO) relay systems, the uplink-downlink duality has been recently investigated. In this paper, we establish the duality between uplink and downlink multi-hop AF-MIMO relay channels with any number of hops and any number of antennas at each node, which is a further generalization of several previously established results. We show that in the downlink relay system, signal-to-interference-noise ratios (SINRs) identical to those in the uplink relay system, and vice versa, can be achieved by two approaches. First, with the same total network transmission power constraint, one simply applies Hermitian transposed uplink relay amplifying matrices at relay nodes in the downlink system. Second, with transmission power constraint at each node of the relay network, one can use scaled and Hermitian transposed uplink relay amplifying matrices in the downlink system, with scaling factors obtained by switching power constraints at different nodes of the uplink system. As an application of the uplink-downlink duality, we propose an optimal design of the source precoding matrix and relay amplifying matrices for multi-hop MIMO relay system with a dirty paper coding (DPC) transmitter at the source node.