Minimal distance approach for studying multi-form MIMO precoders, application to finite-SNR DMT

The linear closed-loop MIMO precoding technique employs the channel state information (CSI) at both sides of the link to optimize various criteria such as the capacity, the mean square error, the signal to noise ratio (SNR), etc. Besides classical criteria such as capacity or bit error rate, the diversity-multiplexing trade-off (DMT) is now widely used to evaluate the performance of designed precoders. Indeed, it is known that a fundamental tradeoff between the spatial multiplexing and the diversity order exists. The first definition was given for asymptotic SNR, then was extended to finite values. The DMT was studied for open-loop scheme (Alamouti or V-BLAST) and we propose in this paper a method to obtain DMT of multi-form MIMO precoders. Although several multi-form solutions were found, to obtain their theoretical performance is still difficult. In order to tackle this challenge, we propose to investigate the minimal distance approach: starting from the probability density function of a square minimum distance, we obtain the outage probability and diversity-multiplexing trade-off (DMT) at operational SNR. We arbitrarily choose the max-d min precoder based on the maximization of a minimal distance using the CSI at the transmitter (closed-loop). This expression is validated by simulations and comparisons between different MIMO precoding schemes are performed. The method can be applied to others precoders and fading channels.