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A Numerical Approach for Designing Unitary Space Time Codes with Large Diversity

Guangyue Han, Joachim Rosenthal

TL;DR

This work tackles unitary space-time code design for non-coherent Rayleigh channels by introducing a diversity-centric framework that encompasses both diversity product (high-SNR) and diversity sum (low-SNR). It develops a numerical design pipeline using the complex Stiefel manifold and Cayley transformation to parameterize unitary constellations, and leverages Simulated Annealing and Genetic Algorithms to optimize both unconstrained and algebraically structured constellations across dimensions and rates. Key findings show that optimizing diversity sum can yield substantial gains at low SNR, while diversity product optimization remains crucial at high SNR; numerically derived constellations achieve large diversity sums and competitive products, illustrating the practical value of the approach for various MIMO configurations. The methods enable scalable design for arbitrary dimensions and rates, offering a flexible toolkit for non-coherent MIMO communications and highlighting future directions in the geometry of unitary groups and Stiefel manifolds.

Abstract

A numerical approach to design unitary constellation for any dimension and any transmission rate under non-coherent Rayleigh flat fading channel.

A Numerical Approach for Designing Unitary Space Time Codes with Large Diversity

TL;DR

This work tackles unitary space-time code design for non-coherent Rayleigh channels by introducing a diversity-centric framework that encompasses both diversity product (high-SNR) and diversity sum (low-SNR). It develops a numerical design pipeline using the complex Stiefel manifold and Cayley transformation to parameterize unitary constellations, and leverages Simulated Annealing and Genetic Algorithms to optimize both unconstrained and algebraically structured constellations across dimensions and rates. Key findings show that optimizing diversity sum can yield substantial gains at low SNR, while diversity product optimization remains crucial at high SNR; numerically derived constellations achieve large diversity sums and competitive products, illustrating the practical value of the approach for various MIMO configurations. The methods enable scalable design for arbitrary dimensions and rates, offering a flexible toolkit for non-coherent MIMO communications and highlighting future directions in the geometry of unitary groups and Stiefel manifolds.

Abstract

A numerical approach to design unitary constellation for any dimension and any transmission rate under non-coherent Rayleigh flat fading channel.

Paper Structure

This paper contains 22 sections, 9 theorems, 96 equations, 6 figures.

Table of Contents

  1. Introduction and Model
  2. The Diversity Function, the Diversity Sum and the Diversity Product
  3. Design criterion for high SNR
  4. Design criterion for low SNR channel
  5. Three illustrative examples
  6. Orthogonal Design:
  7. Unitary Representation of $SL_2(\mathbb{F}_5)$:
  8. Numerically Derived Constellation:
  9. Numerical Design of Unitary Constellations with Good Diversity
  10. The complex Stiefel manifold
  11. Cayley transformation
  12. Simulated Annealing Algorithm (SA)
  13. Genetic Algorithm (GA)
  14. Constellations with algebraic structure
  15. Numerical Results of Constellation Design
  16. ...and 7 more sections

Key Result

Theorem 3.2

$\mathcal{S}_{T,M}$ is a smooth, real and compact sub-manifold of $\mathbb{C}^{MT}=\mathbb{R}^{2MT}$ of real dimension $2TM-M^2$.

Figures (6)

  • Figure 1: Diversity function $\mathcal{D}(\mathcal{V},\rho)$ and exact diversity function $\mathcal{D}_e(\mathcal{V},\rho)$ of a fully isotropic constellation.
  • Figure 2: Diversity function $\mathcal{D}(\mathcal{V},\rho)$ and exact diversity function for the group constellation $SL_2(\mathbb{F}_5)$.
  • Figure 3: Simulations of three constellations having sizes $T=4$, $M=2$ and $L=120$ respectively $L=121$.
  • Figure 4: Optimized constellation by exact diversity function versus constellation with optimal diversity product and sum. From the graph, one can see that the two constellation almost have the same performance, except there are minor deviations at certain interval. Actually it can be verified that the elements in the second constellation are similar to (up to one unitary matrix) those in the first constellation, so it has also optimal diversity product and diversity sum. For more details about $2$ dimensional optimal constellation with $3$ elements, one can look at Appendix \ref{['three']}.
  • Figure 5: Performance of different dimensional constellations with the same rate
  • ...and 1 more figures

Theorems & Definitions (26)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 3.1
  • Theorem 3.2
  • Example 3.3
  • Example 3.4
  • Corollary 3.5
  • Lemma 3.6
  • ...and 16 more