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Optimal Beamforming for Uplink Covert Communication in MIMO GEO Satellite-Terrestrial Systems

Zewei Guo, Ranran Sun, Yulong Shen, Xiaohong Jiang

TL;DR

This work tackles uplink covert communication in a MIMO GEO satellite-terrestrial system consisting of Alice, Bob, and multiple wardens. It develops optimal beamforming (OB) and joint OB with antenna orientation (JO-BA) strategies under both perfect and imperfect channel state information (CSI), formulating convex SDR-based optimizations and employing Rodrigues' rotation formula for orientation search. The analysis provides theoretical DEP, TOP, and CR characterizations and delivers robust design procedures with semidefinite relaxation, alternating optimization, and 1-D searches, validated by extensive simulations. The results show that OB and especially JO-BA substantially improve covert rate, with larger antenna arrays and accurate CSI yielding the strongest gains, highlighting practical pathways for secure uplink satellite-terrestrial communications.

Abstract

This paper investigates the uplink covert communication in a multiple-input multiple-output (MIMO) satellite-terrestrial system consisting of an Earth station transmitter Alice, a geosynchronous Earth orbit (GEO) satellite receiver Bob, and multiple GEO satellite wardens around Bob, where each node in the system is equipped with an array of directional antennas. Based on beamforming and the default antenna orientation setting, we first propose a scheme for covert Alice-Bob uplink transmission. Under the perfect channel estimation scenario, we provide theoretical modeling for the system performance in terms of detection error probability (DEP), transmission outage probability (TOP) and covert rate (CR), and then explore the optimal beamforming (OB) design as well as the joint optimal beamforming and antenna orientation (JO-BA) design for CR maximization. We then extend our study to the imperfect channel estimation scenario, and conduct related performance modeling and OB/JO-BA designs for CR maximization. We also apply the techniques of semidefinite relaxation, alternating optimization, Rodrigues' rotation formula and 1-D search algorithm to develop efficient algorithms to solve the above optimization problems. Finally, extensive numerical results are presented to verify our theoretical results and to illustrate the efficiency of beamforming and antenna orientation design for supporting the uplink covert communication in MIMO GEO satellite-terrestrial systems.

Optimal Beamforming for Uplink Covert Communication in MIMO GEO Satellite-Terrestrial Systems

TL;DR

This work tackles uplink covert communication in a MIMO GEO satellite-terrestrial system consisting of Alice, Bob, and multiple wardens. It develops optimal beamforming (OB) and joint OB with antenna orientation (JO-BA) strategies under both perfect and imperfect channel state information (CSI), formulating convex SDR-based optimizations and employing Rodrigues' rotation formula for orientation search. The analysis provides theoretical DEP, TOP, and CR characterizations and delivers robust design procedures with semidefinite relaxation, alternating optimization, and 1-D searches, validated by extensive simulations. The results show that OB and especially JO-BA substantially improve covert rate, with larger antenna arrays and accurate CSI yielding the strongest gains, highlighting practical pathways for secure uplink satellite-terrestrial communications.

Abstract

This paper investigates the uplink covert communication in a multiple-input multiple-output (MIMO) satellite-terrestrial system consisting of an Earth station transmitter Alice, a geosynchronous Earth orbit (GEO) satellite receiver Bob, and multiple GEO satellite wardens around Bob, where each node in the system is equipped with an array of directional antennas. Based on beamforming and the default antenna orientation setting, we first propose a scheme for covert Alice-Bob uplink transmission. Under the perfect channel estimation scenario, we provide theoretical modeling for the system performance in terms of detection error probability (DEP), transmission outage probability (TOP) and covert rate (CR), and then explore the optimal beamforming (OB) design as well as the joint optimal beamforming and antenna orientation (JO-BA) design for CR maximization. We then extend our study to the imperfect channel estimation scenario, and conduct related performance modeling and OB/JO-BA designs for CR maximization. We also apply the techniques of semidefinite relaxation, alternating optimization, Rodrigues' rotation formula and 1-D search algorithm to develop efficient algorithms to solve the above optimization problems. Finally, extensive numerical results are presented to verify our theoretical results and to illustrate the efficiency of beamforming and antenna orientation design for supporting the uplink covert communication in MIMO GEO satellite-terrestrial systems.
Paper Structure (28 sections, 4 theorems, 79 equations, 10 figures, 1 table, 3 algorithms)

This paper contains 28 sections, 4 theorems, 79 equations, 10 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. System Model and Preliminaries
  3. System Model
  4. Channel Model
  5. Noise Uncertainty at Bob and Wardens
  6. Detection Model
  7. Covert Communication Scheme
  8. Performance Metric
  9. Covert Communication Scheme under Perfect Channel Estimation
  10. Covertness Requirement under Perfect Channel Estimation
  11. Reliability Requirement under Perfect Channel Estimation
  12. Beamforming Design under Perfect Channel Estimation
  13. Joint Beamforming and Antenna Orientation Design under Perfect Channel Estimation
  14. Covert Communication Scheme under Imperfect Channel Estimation
  15. Channel Estimation Error Model
  16. ...and 13 more sections

Key Result

Lemma 1

In the considered system, the CDF of $Z_j$ is lower-bounded by $F_{Z_j}(\tau_j)\geq \check{F}_{Z_j}(\tau_j)$. Here, $\check{F}_{Z_j}(\tau_j)$ is given by where $\Gamma(a,x)=\int_x^\infty t^{a-1}e^{-t}dt$ is the upper incomplete Gamma function, $\gamma(a,x)=\int_0^x t^{a-1}e^{-t}dt$ is the lower incomplete Gamma function, and $\mu_j(x)\triangleq{m_j \left(x-\hat{\sigma}_{\mathrm{w}_j}^{\mathrm{lb}

Figures (10)

  • Figure 1: System Model.
  • Figure 2: Geometrical relation between UPA of Alice and that of any satellite.
  • Figure 3: Detection error probability vs. detection threshold $\tau_j$.
  • Figure 4: Validation of minimum DEP under different parameter settings.
  • Figure 5: Maximum CR vs. number of the wardens $n_{\mathrm{w}}$.
  • ...and 5 more figures

Theorems & Definitions (8)

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