Achieving Full Multipath Diversity by Random Constellation Rotation: a Theoretical Perspective

Diversity is an essential concept associated with communication reliability in multipath channels since it determines the slope of bit error rate performance in the medium to high signal-to-noise ratio regions. However, most of the existing analytical frameworks were developed for specific modulation schemes while the efficient validation of full multipath diversity for general modulation schemes remains an open problem. To fill this research gap, we propose to utilize random constellation rotation to ease the conditions for full-diversity modulation designs. For linearly precoded cyclic-prefix orthogonal frequency division multiplexing (OFDM) systems, we prove that maximum multipath diversity can be attained as long as the spread matrix does not have zero entries, which is a sufficient but easily satisfied condition. Furthermore, we derive the sufficient and necessary condition for general modulation schemes, whose verification can be divided into validation tasks for each column of the modulation matrix. Based on the proposed conditions, maximum diversity order can be attained with the probability of 1 by enabling a randomly generated rotation pattern for both time and doubly dispersive channels. The theoretical analysis in this paper also demonstrates that the diversity evaluation can be concentrated on the pairwise error probability when the number of error symbols is one, which reduces the complexity of diversity-driven design and performance analysis for novel modulation schemes significantly in both time and doubly dispersive channels. Finally, numerical results for various modulation schemes confirm that the theoretical analysis holds in both time and doubly dispersive channels. Furthermore, when employing practical detectors, the random constellation rotation technique consistently enhance the transmission reliability for both coded and uncoded systems.