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On the Capacity Region of Additive-Multiplicative MAC with Heterogeneous Input Constraints

Qianqian Zhang, Ying-Chang Liang

Abstract

This paper characterizes the capacity region of a two-user additive-multiplicative multiple access channel (AM-MAC) under heterogeneous input constraints. This model captures the fundamental limits of symbiotic radio, where an active primary transmitter (PT) conveys information via active transmission subject to an average power constraint, while a passive backscatter device (BD) modulates signals through backscattering under a peak amplitude constraint. Our main results are threefold. Firstly, we prove that the sum-rate capacity equals the PT's point-to-point capacity, achieved when the PT employs Gaussian signaling and the BD acts as a pure reflector to assist the PT's transmission. Secondly, to achieve the BD's maximum achievable rate, the PT must adopt a constant-envelope signaling strategy, while the optimal BD distribution exhibits a concentric-circle structure with a uniform phase. Thirdly, for the remaining boundary points, we establish that the optimal PT signal consists of a continuous uniform phase and a discrete amplitude, whereas the optimal BD distribution is fully discrete. Finally, numerical results are provided to characterized the capacity region by solving a specialized nonlinear optimization problem. To demonstrate the practical implications, we also characterize an baseline rate pair and evaluate the overall performance of the AM-MAC.

On the Capacity Region of Additive-Multiplicative MAC with Heterogeneous Input Constraints

Abstract

This paper characterizes the capacity region of a two-user additive-multiplicative multiple access channel (AM-MAC) under heterogeneous input constraints. This model captures the fundamental limits of symbiotic radio, where an active primary transmitter (PT) conveys information via active transmission subject to an average power constraint, while a passive backscatter device (BD) modulates signals through backscattering under a peak amplitude constraint. Our main results are threefold. Firstly, we prove that the sum-rate capacity equals the PT's point-to-point capacity, achieved when the PT employs Gaussian signaling and the BD acts as a pure reflector to assist the PT's transmission. Secondly, to achieve the BD's maximum achievable rate, the PT must adopt a constant-envelope signaling strategy, while the optimal BD distribution exhibits a concentric-circle structure with a uniform phase. Thirdly, for the remaining boundary points, we establish that the optimal PT signal consists of a continuous uniform phase and a discrete amplitude, whereas the optimal BD distribution is fully discrete. Finally, numerical results are provided to characterized the capacity region by solving a specialized nonlinear optimization problem. To demonstrate the practical implications, we also characterize an baseline rate pair and evaluate the overall performance of the AM-MAC.

Paper Structure

This paper contains 12 sections, 8 theorems, 92 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The Main Results
  3. Converse Analysis
  4. Achievability
  5. Case for $\mu_1=\mu_2$
  6. Case for $\mu_1 >\mu_2$
  7. Case for $\mu_1 < \mu_2$
  8. Numerical Results
  9. Baseline Achievable Rate Pair
  10. Numerical Optimization for the Capacity Region Boundary
  11. Results
  12. Conclusions

Key Result

Theorem 1

When $\mu_1 = 0$ and $\mu_2 = 1$, the capacity-achieving distributions satisfy the following conditions: (1) The PT transmits constant-envelope signal with $|X_1|^2 = P$ almost surely; and (2) the BD transmits a signal characterized by a discrete amplitude and an independent, uniformly distributed p

Figures (8)

  • Figure 1: A two-user AM-MAC model with heterogeneous input constraints.
  • Figure 2: Capacity region for model \ref{['eq:system model']} with heterogeneous input constraints.
  • Figure 3: An example of the optimal input distribution with peak amplitude constraint.
  • Figure 4: Geometric illustration of the level sets of $\omega_2(X_2)$. The intersection of circles centered at the origin and $-a$ characterizes the isolated points of the optimal BD support $\mathcal{S}_{f_{X_2}^*}$.
  • Figure 5: Capacity region with $a=2$.
  • ...and 3 more figures

Theorems & Definitions (11)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Theorem 4
  • Remark 2
  • Lemma 2
  • Remark 3
  • Corollary J.1
  • ...and 1 more