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Sparse Grassmannian Design for Noncoherent Codes via Schubert Cell Decomposition

Joe Asano, Yuto Hama, Hiroki Iimori, Chandan Pradhan, Szabolcs Malomsoky, Naoki Ishikawa

TL;DR

The paper tackles noncoherent MIMO signaling by introducing sparse Grassmannian constellations that exploit Schubert-cell sparsity to reduce detection complexity while maintaining performance. It provides a rank-aware PEP bound to correctly characterize SER under rank-deficient configurations and derives a closed-form AMI lower bound that leads to a practical design metric. The proposed MCPD-oriented design, combined with Schubert-cell decomposition and pattern lifting, yields sparse constellations that outperform conventional Grassmannian designs and approach optimal performance at high SNR, with linear-time/space complexity in the number of codewords. The approach is demonstrated through theoretical insights and a detailed example at (T,M)=(4,2), showing improved SER/AMI and reduced complexity, with scalable behavior for moderate cardinalities. This framework offers a viable path for robust, low-complexity noncoherent signaling in large-MIMO IoT and beyond-5G/6G systems.

Abstract

In this paper, we propose a method for designing sparse Grassmannian codes for noncoherent multiple-input multiple-output systems. Conventional pairwise error probability formulations under uncorrelated Rayleigh fading channels fail to account for rank deficiency induced by sparse configurations. We revise these formulations to handle such cases in a unified manner. Furthermore, we derive a closed-form metric that effectively maximizes the noncoherent average mutual information (AMI) at a given signal-to-noise ratio. We focus on the fact that the Schubert cell decomposition of the Grassmann manifold provides a mathematically sparse property, and establish design criteria for sparse noncoherent codes based on our analyses. In numerical results, the proposed sparse noncoherent codes outperform conventional methods in terms of both symbol error rate and AMI, and asymptotically approach the performance of the optimal Grassmannian constellations in the high-signal-to-noise ratio regime. Moreover, they reduce the time and space complexity, which does not scale with the number of transmit antennas.

Sparse Grassmannian Design for Noncoherent Codes via Schubert Cell Decomposition

TL;DR

The paper tackles noncoherent MIMO signaling by introducing sparse Grassmannian constellations that exploit Schubert-cell sparsity to reduce detection complexity while maintaining performance. It provides a rank-aware PEP bound to correctly characterize SER under rank-deficient configurations and derives a closed-form AMI lower bound that leads to a practical design metric. The proposed MCPD-oriented design, combined with Schubert-cell decomposition and pattern lifting, yields sparse constellations that outperform conventional Grassmannian designs and approach optimal performance at high SNR, with linear-time/space complexity in the number of codewords. The approach is demonstrated through theoretical insights and a detailed example at (T,M)=(4,2), showing improved SER/AMI and reduced complexity, with scalable behavior for moderate cardinalities. This framework offers a viable path for robust, low-complexity noncoherent signaling in large-MIMO IoT and beyond-5G/6G systems.

Abstract

In this paper, we propose a method for designing sparse Grassmannian codes for noncoherent multiple-input multiple-output systems. Conventional pairwise error probability formulations under uncorrelated Rayleigh fading channels fail to account for rank deficiency induced by sparse configurations. We revise these formulations to handle such cases in a unified manner. Furthermore, we derive a closed-form metric that effectively maximizes the noncoherent average mutual information (AMI) at a given signal-to-noise ratio. We focus on the fact that the Schubert cell decomposition of the Grassmann manifold provides a mathematically sparse property, and establish design criteria for sparse noncoherent codes based on our analyses. In numerical results, the proposed sparse noncoherent codes outperform conventional methods in terms of both symbol error rate and AMI, and asymptotically approach the performance of the optimal Grassmannian constellations in the high-signal-to-noise ratio regime. Moreover, they reduce the time and space complexity, which does not scale with the number of transmit antennas.
Paper Structure (28 sections, 5 theorems, 69 equations, 8 figures, 2 tables)

This paper contains 28 sections, 5 theorems, 69 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Grassmannian Constellation
  3. Definition bendokat2024grassmann
  4. Conventional Grassmannian Constellation
  5. Manopt
  6. Exp-Map kammoun2003newkammoun2007noncoherent
  7. System Model
  8. Mathematical Analysis
  9. PEP Analysis for General Dense and Sparse Codes
  10. Closed-Form Lower Bound for AMI
  11. Proposed Design Method
  12. General Design Method
  13. Schubert Cell Decomposition konishi2022novelhansen2007schubert
  14. Lifting Schubert Cells onto the Complex Grassmann
  15. Optimization
  16. ...and 13 more sections

Key Result

Theorem 1

When the principal angles between $\mathbf{X}_i$ and $\mathbf{X}_j$ are defined as eq:p_angle, the lower bound of the AMI given in ineq:AMI can be expressed as where with and

Figures (8)

  • Figure 1: Variations in $\lambda^\star$ and $\kappa(\lambda^\star)$ with respect to SNR when $T/M=2$.
  • Figure 2: SER comparison with $(T,M)=(4,2)$ and $|\mathcal{X}|=4$.
  • Figure 3: SER comparison with $(T,M)=(6,2)$ and $|\mathcal{X}|=64$.
  • Figure 4: SER comparison with $(T,M)=(6,3)$ and $|\mathcal{X}|=64$.
  • Figure 5: AMI comparison with $(T,M)=(4,2)$ and $|\mathcal{X}|=4$.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Lemma 1
  • Lemma 2