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A Structured Family of Grassmannian Constellations via Geodesic Mapping for MIMO Noncoherent Communications

Álvaro Pendás-Recondo, Enrique Pendás-Recondo

TL;DR

This paper addresses noncoherent MIMO signaling over Rayleigh block-fading channels by designing a structured family of Grassmannian constellations using geodesic mappings on the complex Grassmann manifold. The method fixes an initial point $\mathbf{U}=\mathbf{I}_M$ and employs $2M^2$ diametral tangent vectors from Weyl–Heisenberg matrices to generate up to $L=4M^2$ constellation points, each row of the transmitted matrix containing only one nonzero entry, enabling a single active transmit antenna per slot and reducing ML detector complexity by a factor of $M$. The design achieves error performance comparable to state-of-the-art unstructured constellations within the target spectral-efficiency range of $0.25$–$1$ bps/Hz, while also simplifying bit labeling and offering hardware benefits (one DAC and one PA per slot). The work demonstrates that the proposed constellation can outperform or match existing approaches under practical constraints and highlights potential applicability to SIMO/uplink IoT scenarios and URLLC-like short-packet regimes where CSI is costly or unreliable. Overall, the geodesic-based Grassmannian design provides a balanced tradeoff between performance, complexity, and hardware simplicity for noncoherent MIMO deployments.

Abstract

This work presents a novel structured family of Grassmannian constellations for multiple-input multiple-output (MIMO) noncoherent communications over Rayleigh block-fading channels, where neither the transmitter nor the receiver has channel state information (CSI). The proposed constellation design is built upon the geodesic curves of the Grassmann manifold, thereby exploiting its underlying geometric structure. The resulting solution is limited in spectral efficiency (with a maximum constellation size of $4M^2$ points, where $M$ is the number of transmit antennas), targeting a rate in the range of $0.25$-$1$ bps/Hz. However, all space-time matrices resulting from this design exhibit the remarkable property of having a single nonzero entry per row, meaning that only one transmit antenna is active per time slot. This property significantly reduces hardware complexity and implementation cost, while also lowering power consumption, as only a single power amplifier is required for transmission. Furthermore, within the constellation size limits, the proposed design achieves error performance comparable to state-of-the-art optimization-based unstructured designs, as validated through symbol error rate (SER) numerical results. It also enables simple yet effective bit labeling, confirmed by comparisons of bit error rate (BER) and SER, and reduces the computational complexity of the maximum-likelihood (ML) detector for Grassmannian constellations by a factor of $M$.

A Structured Family of Grassmannian Constellations via Geodesic Mapping for MIMO Noncoherent Communications

TL;DR

This paper addresses noncoherent MIMO signaling over Rayleigh block-fading channels by designing a structured family of Grassmannian constellations using geodesic mappings on the complex Grassmann manifold. The method fixes an initial point and employs diametral tangent vectors from Weyl–Heisenberg matrices to generate up to constellation points, each row of the transmitted matrix containing only one nonzero entry, enabling a single active transmit antenna per slot and reducing ML detector complexity by a factor of . The design achieves error performance comparable to state-of-the-art unstructured constellations within the target spectral-efficiency range of – bps/Hz, while also simplifying bit labeling and offering hardware benefits (one DAC and one PA per slot). The work demonstrates that the proposed constellation can outperform or match existing approaches under practical constraints and highlights potential applicability to SIMO/uplink IoT scenarios and URLLC-like short-packet regimes where CSI is costly or unreliable. Overall, the geodesic-based Grassmannian design provides a balanced tradeoff between performance, complexity, and hardware simplicity for noncoherent MIMO deployments.

Abstract

This work presents a novel structured family of Grassmannian constellations for multiple-input multiple-output (MIMO) noncoherent communications over Rayleigh block-fading channels, where neither the transmitter nor the receiver has channel state information (CSI). The proposed constellation design is built upon the geodesic curves of the Grassmann manifold, thereby exploiting its underlying geometric structure. The resulting solution is limited in spectral efficiency (with a maximum constellation size of points, where is the number of transmit antennas), targeting a rate in the range of - bps/Hz. However, all space-time matrices resulting from this design exhibit the remarkable property of having a single nonzero entry per row, meaning that only one transmit antenna is active per time slot. This property significantly reduces hardware complexity and implementation cost, while also lowering power consumption, as only a single power amplifier is required for transmission. Furthermore, within the constellation size limits, the proposed design achieves error performance comparable to state-of-the-art optimization-based unstructured designs, as validated through symbol error rate (SER) numerical results. It also enables simple yet effective bit labeling, confirmed by comparisons of bit error rate (BER) and SER, and reduces the computational complexity of the maximum-likelihood (ML) detector for Grassmannian constellations by a factor of .
Paper Structure (22 sections, 3 theorems, 27 equations, 7 figures, 4 tables, 1 algorithm)

This paper contains 22 sections, 3 theorems, 27 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation: Coherent versus Noncoherent Schemes and Unitary Space-Time Modulation
  3. Related Works: Constellation Designs
  4. Contributions
  5. System Model
  6. System Model
  7. Grassmmannian Signaling Application
  8. Metrics and Error Probability
  9. Constellation Design
  10. Geodesics and Diametral Sets
  11. Algorithm Description
  12. Bit Labeling
  13. Receiver Computational Complexity
  14. Numerical Results
  15. SER Performance
  16. ...and 7 more sections

Key Result

Theorem 1

Let $[\mathbf{U}] \in \textup{Gr}_{\mathbb{C}}(2M,M)$ be a point and $\mathbf{\Delta} \in T_{[\mathbf{U}]}\textup{Gr}_{\mathbb{C}}(2M,M)$ a vector of the form where $\tilde{\mathbf{U}} \in \textup{St}_{\mathbb{C}}(M,M)$ and $\tilde{\mathbf\Delta} \in \mathbb{C}^{M\times M}$. If $\sqrt{M}\tilde{\mathbf{\Delta}} \in U(M)$, then $\mathbf{\Delta}$ is a diametral vector and the geodesic $\gamma_{\math

Figures (7)

  • Figure 1: Constellation metrics versus geodesic mapping parameter, $x$, for Case (ii) and Case (iv) of Algorithm \ref{['algc']} with $T=4$ and $M=2$. For the UB, the dotted line indicates the value of $x$ that achieves its minimum when $N=2$. A total of $5000$ values of $x$ are shown.
  • Figure 2: SER results for $L=4$ constellation points and different values of $T$, $M$, and $N$.
  • Figure 3: SER results across different values of $T$ and $M$ for fixed values of receive antennas and spectral efficiency.
  • Figure 4: SER results for all possible values of $L$ and fixed values of $T$, $M$, and $N$. Considering UB instead of DP for geodesic mapping in Algorithm \ref{['algc']} is denoted as (UB).
  • Figure 5: Evaluation of bit labeling performance, comparing SER and BER results for equal values of $T$, $M$, $N$, and $L$.
  • ...and 2 more figures

Theorems & Definitions (6)

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