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

Xuehan Wang, Jinhong Yuan, Jintao Wang, Kehan Huang

TL;DR

This work tackles the problem of achieving full multipath diversity in time- and doubly dispersive channels for general modulation schemes. It proposes random constellation rotation as a lightweight mechanism and derives a set of rank-based criteria for both linearly precoded CP-OFDM and general linear modulation to guarantee maximum diversity, with random rotations guaranteeing diversity with probability 1. The key theoretical contributions include a sufficient condition for CP-OFDM systems and a necessary-and-sufficient condition for general modulation, plus extensions to doubly dispersive channels and a PEP-centered simplification to one-error-event analyses. Numerical results confirm the theory, showing substantial diversity gains for DFT-s-OFDM, OTFS/ODDM/AFDM-like schemes, and improved reliability for practical detectors, along with PAPR benefits. Overall, the approach provides a general, tractable framework for designing modulation schemes that harness full multipath diversity in realistic wireless channels.

Abstract

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.

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

TL;DR

This work tackles the problem of achieving full multipath diversity in time- and doubly dispersive channels for general modulation schemes. It proposes random constellation rotation as a lightweight mechanism and derives a set of rank-based criteria for both linearly precoded CP-OFDM and general linear modulation to guarantee maximum diversity, with random rotations guaranteeing diversity with probability 1. The key theoretical contributions include a sufficient condition for CP-OFDM systems and a necessary-and-sufficient condition for general modulation, plus extensions to doubly dispersive channels and a PEP-centered simplification to one-error-event analyses. Numerical results confirm the theory, showing substantial diversity gains for DFT-s-OFDM, OTFS/ODDM/AFDM-like schemes, and improved reliability for practical detectors, along with PAPR benefits. Overall, the approach provides a general, tractable framework for designing modulation schemes that harness full multipath diversity in realistic wireless channels.

Abstract

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.
Paper Structure (15 sections, 7 theorems, 24 equations, 5 figures, 1 algorithm)

This paper contains 15 sections, 7 theorems, 24 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. System Model
  4. Diversity
  5. Full Diversity Based on Linearly Precoded CP-OFDM Systems
  6. Sufficient Condition for Full Diversity in Time Dispersive Channels
  7. Extension and Discussions
  8. Full Diversity Based on General Linear Modulation Systems
  9. Sufficient and Necessary Condition for Full Diversity in Time Dispersive Channels
  10. Extension and Discussions
  11. Numerical Results
  12. Conclusions
  13. Proof of Theorem \ref{['th1_spread']}
  14. Proof of Lemma \ref{['lemma1_lowrank_finite']}
  15. Proof of Theorem \ref{['th2_modu']}

Key Result

Theorem 1

If $\tilde{\mathbf{\Psi}}$ satisfies $|\tilde{\mathbf{\Psi}}(p,q)|>0$ for $\forall p,q$, there are infinite choices for $\phi_{0},\phi_{1},\cdots,\phi_{M-1}$ to achieve full diversity in time dispersive channels. Meanwhile, if $\phi_{0},\phi_{1},\cdots,\phi_{M-1}$ are randomly generatedPlease note t

Figures (5)

  • Figure 1: BER against SNR under $M=4$ and time dispersive channels in \ref{['IO_delayOnly']} with $L=2$.
  • Figure 2: BER against SNR under $M=8$ and time dispersive channels in \ref{['IO_delayOnly']} with $L=4$.
  • Figure 3: CCDF evaluation with the phase rotation.
  • Figure 4: BER against SNR under $M=8$ and doubly dispersive channels in \ref{['IO_double']} with $K=1$ and $L=2$.
  • Figure 5: BER against SNR under practical detection and channels.

Theorems & Definitions (8)

  • Theorem 1
  • Remark 1
  • Corollary 1
  • Lemma 1
  • Theorem 2
  • Corollary 2
  • Corollary 3
  • Corollary 4