What Determines the Diversity Order of Linear Equalizers?

Diversity techniques are important in combating the deleterious effects of channel fading. To enjoy diversity, both transmitter and receiver have to be designed appropriately. Many existing designs have exploited different flavors of diversity by employing maximum likelihood (ML) or near-ML detectors at the receiver, which is of high complexity. On the other hand, empirical results have shown that linear equalizers (LEs) offer inferior performance but come with low complexity. Unfortunately, the diversity collected by general LEs has not been quantified, which reduces the attraction of LEs in theoretical and practical aspects. In this paper, we reveal a fundamental yet simple condition that determines whether LEs can achieve the same diversity order as an ML equalizer does. The condition is based on the orthogonal deficiency (od) of channel matrix. Based on the distribution of the channel od, we propose a framework of hybrid equalizers which guarantees maximum diversity while trading off complexity for coding gains. The theoretical findings are verified by simulation results.