Our research applied tools from probability and communication theory to study four problems in wireless, summarised below. Main performance measures of interest were mutual information and diversity (high-SNR behaviour of error probability). The thesis is titled "Asymptotic Analysis of Wireless Systems with Rayleigh Fading" and can be found here.
1. Multiple Antenna Links at Low SNR: taking the block-fading multiple antenna channel model in which the channel is unknown to the transmitter and receiver, the low-SNR asymptotic mutual information has a quadratic leading term, contrasting with the known channel case where there is linear growth in the SNR. Under various signaling constraints (e.g. Gaussian modulation, unitary space-time modulation, peak constraints) this mutual information is maximized by using a single transmit antenna.
2. MIMO Link with Relays at High SNR: applying results from existing work on the Rayleigh product channel, we found the optimal diversity-multiplexing gain trade-off for an M by N multiple-antenna system with R single-antenna relay nodes that adopts a two-stage distributed space-time protocol. Previously only the maximum diversity (R.min(M,N)) and maximum multiplexing gain (min(M,N)) were known. We found that the optimal trade-off is the same as that of a MIMO link with R transmit and min(M,N) receive antennas. The optimal trade-off can be achieved by codes based on cyclic division algebras.
3. More than one source-destination pair at High SNR: initially we looked at a four-node network featuring two source-destination pairs. We showed how cooperation amongst the nodes (allowing transmitters and receivers to share information) can be introduced to increase diversity, but at the expense of a reduced rate. Two cooperative diversity schemes were considered and the trade-offs determined by outage probability calculations. The diversity results can be generalised from 2 to m source-destination pairs.
4. Source-destination pairs with relatys at High SNR: we introduced n relay nodes into the m source-destination pair channel considered before, with a view to improving the diversity-multiplexing gain trade-off. It was shown that at full rate (two channel uses) a diversity of n-m^2+2 can be obtained. If m^2>n we could show that by using some of the transmitters and receivers as relays, high-diversity schemes are also possible, but they would incur a rate penalty.