Optimal DMT of dynamic decode-and-forward protocol on a half-duplex relay channel with arbitrary number of antennas at each node

The diversity-multiplexing tradeoff (DMT) curve computed for a half-duplex (HD) relay network with arbitrary number of antennas at each node, operating in the dynamic decode-and-forward (DDF) [1] mode. The general case is considered where the source, relay and the destination have m, k and n antennas, respectively. Such a channel is denoted as a (m, k, n)-relay channel, hereafter. Computing the DMT of a (m, k, n)-relay channel requires the joint eigenvalue distribution of two specially correlated central Wishart random matrices. This distribution has been computed recently in [2] using spherical integrals, for the special case where m = n. In this paper, this distribution is computed using an alternate method which is more transparent (for the m = n case) and valid for arbitrary m, k and n. Using this result for the distribution, a complete characterization of the optimal DMT curves of a DDF protocol on a general (m, k, n)-relay channel (the special case of (n, k, n)-relay channel has been considered in [3]) is provided employing a numerical method and compared with the optimal DMTs of half-duplex static compress-and-forward (HD-SCF) and full-duplex decode-and-forward (FD-DF) protocols analyzed in [4]. Interestingly, simulation results show that, for some class of channel configurations such as (m, k, n > k)-relay channels, the HD-DDF and FD-DF protocols have identical optimal DMTs. For other channel configurations, the optimal diversity orders of the HD-DDF protocol are only marginally lower than those of the FD-DF protocol at high multiplexing gains whereas, at low multiplexing gains, the optimal diversity orders of both these protocols coincide. Our result also shows that for some channel configurations at low multiplexing gains the optimal diversity orders of HD-DDF protocol is greater than that of HD-SCF protocol, which was proved to attain the optimal DMT on a static HD relay channel([4]).