Optimality of diagonalization of multi-hop MIMO relays

For a two-hop linear non-regenerative multiple-input multiple-output (MIMO) relay system where the direct link between source and destination is negligible, the optimal design of the source and relay matrices has been recently established for a broad class of objective functions. The optimal source and relay matrices jointly diagonalize the MIMO relay system into a set of parallel scalar channels. In this paper, we show that this diagonalization is also optimal for a multihop MIMO relay system with any number of hops, which is a further generalization of several previously established results. Specifically, for Schur-concave objective functions, the optimal source precoding matrix, the optimal relay amplifying matrices and the optimal receiving matrix jointly diagonalize the multihop MIMO relay channel. And for Schur-convex objectives, such joint diagonalization along with a rotation of the source precoding matrix is also shown to be optimal. We also analyze the system performance when each node has the same transmission power budget and the same asymptotically large number of antennas. The asymptotic analysis shows a good agreement with numerical results under a finite number of antennas.

I. Introduction

IT is well-known that multiple-input multiple-output (MIMO) wireless communication techniques enhance system reliability and increase system capacity. To efficiently exploit the multi-antenna hardware, an important issue in MIMO system design is to optimize the source precoding matrix [1], [2]. A general framework of optimizing the source precoding matrix has been developed in [2] by using the majorization theory [3]. It has been shown that the optimal source precoding matrix and the optimal receiving matrix diagonalize the MIMO source-destination channel for Schur concave objective functions. And for Schur-convex objectives, the MIMO channel is also diagonalized by the optimal source matrix and the optimal receiving matrix except for a special rotation matrix at the source node.