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Optimal Binary Signaling for a Two Sensor Gaussian MAC Network

Luca Sardellitti, Glen Takahara, Fady Alajaji

TL;DR

The paper addresses distributed detection of a binary event with two sensors over a Gaussian MAC, deriving an optimal one-dimensional asymmetric constellation under per-sensor power constraints. By analyzing MAP decision boundaries and partitioning parameter space into three cases, it shows how power should be allocated—sometimes not exhaustively—to minimize error probability, with Case III illustrating the counterintuitive benefit of partially using the noisier sensor's power. The results demonstrate that the MAC-based constellation design outperforms orthogonal signaling in most scenarios and provide high-SNR insights and extensive numerical validations. These findings offer practical guidelines for energy-efficient, high-performance sensor networks and lay groundwork for extending to larger sensor sets and dynamic pairing.

Abstract

We consider a two sensor distributed detection system transmitting a binary non-uniform source over a Gaussian multiple access channel (MAC). We model the network via binary sensors whose outputs are generated by binary symmetric channels of different noise levels. We prove an optimal one dimensional constellation design under individual sensor power constraints which minimizes the error probability of detecting the source. Three distinct cases arise for this optimization based on the parameters in the problem setup. In the most notable case (Case III), the optimal signaling design is to not necessarily use all of the power allocated to the more noisy sensor (with less correlation to the source). We compare the error performance of the optimal one dimensional constellation to orthogonal signaling. The results show that the optimal one dimensional constellation achieves lower error probability than using orthogonal channels.

Optimal Binary Signaling for a Two Sensor Gaussian MAC Network

TL;DR

The paper addresses distributed detection of a binary event with two sensors over a Gaussian MAC, deriving an optimal one-dimensional asymmetric constellation under per-sensor power constraints. By analyzing MAP decision boundaries and partitioning parameter space into three cases, it shows how power should be allocated—sometimes not exhaustively—to minimize error probability, with Case III illustrating the counterintuitive benefit of partially using the noisier sensor's power. The results demonstrate that the MAC-based constellation design outperforms orthogonal signaling in most scenarios and provide high-SNR insights and extensive numerical validations. These findings offer practical guidelines for energy-efficient, high-performance sensor networks and lay groundwork for extending to larger sensor sets and dynamic pairing.

Abstract

We consider a two sensor distributed detection system transmitting a binary non-uniform source over a Gaussian multiple access channel (MAC). We model the network via binary sensors whose outputs are generated by binary symmetric channels of different noise levels. We prove an optimal one dimensional constellation design under individual sensor power constraints which minimizes the error probability of detecting the source. Three distinct cases arise for this optimization based on the parameters in the problem setup. In the most notable case (Case III), the optimal signaling design is to not necessarily use all of the power allocated to the more noisy sensor (with less correlation to the source). We compare the error performance of the optimal one dimensional constellation to orthogonal signaling. The results show that the optimal one dimensional constellation achieves lower error probability than using orthogonal channels.
Paper Structure (25 sections, 16 theorems, 89 equations, 8 figures, 2 tables)

This paper contains 25 sections, 16 theorems, 89 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. System Model
  3. Source and Sensors
  4. Channel Model
  5. Maximum-a-Posteriori Detection
  6. Summary of Main Results
  7. Proof of Main Results
  8. Decision Boundaries
  9. Case I: $0 \leq p_1 \leq \frac{\epsilon_1\epsilon_2}{1-\epsilon_1-\epsilon_2+2\epsilon_1\epsilon_2}$
  10. Case II: $\frac{\epsilon_1\epsilon_2}{1-\epsilon_1-\epsilon_2+2\epsilon_1\epsilon_2} < p_1 \leq \frac{\epsilon_1-\epsilon_1\epsilon_2}{\epsilon_1+\epsilon_2-2\epsilon_1\epsilon_2}$
  11. Case III: $\frac{\epsilon_1-\epsilon_1\epsilon_2}{\epsilon_1+\epsilon_2-2\epsilon_1\epsilon_2} < p_1 \leq 0.5$
  12. High SNR Behaviour
  13. Case II
  14. Case III
  15. Numerical and Simulation Results
  16. ...and 10 more sections

Key Result

Theorem 1

For any combination of binary constellations ${\cal C} = {\cal C}_1 + {\cal C}_2$, there exists a constellation pattern ${\cal C}^* = {\cal C}_1^* + {\cal C}_2^*$ which has equal error probability, equal or better power consumption, with the composition ${\cal C}_i^* = \{-\sqrt{\frac{p_1}{p_0}}P_i,\

Figures (8)

  • Figure 1: The roots of \ref{['xBoundExponential']}, $x_1$, $x_2$ and $x_3$, as a function of $P_2$ in Case III ($p_1 = 0.4$, $\epsilon_1 = 0.01$, $\epsilon_2 = 0.05$, $N_0 = 1$, $P_1 = 1$).
  • Figure 2: Theoretical and simulated error probability in Case III ($p_1 = 0.45, \epsilon_1 = 0.01, \epsilon_2 = 0.05, P_1= 1, N_0=1$).
  • Figure 3: Error probability as a function of SNR in Case II ($p_1 = 0.3, \epsilon_1 = 0.1, \epsilon_2 = 0.15, P_1^{\text{max}} = 1, P_2^{\text{max}} = 1$).
  • Figure 4: Error probability as a function of SNR in Case III ($p_1 = 0.4, \epsilon_1 = 0.01, \epsilon_2 = 0.05, P_1^{\text{max}} = 1, P_2^{\text{max}} = 2$).
  • Figure 5: Case type regions for different values of $p_1$.
  • ...and 3 more figures

Theorems & Definitions (16)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Theorem 8
  • Proposition 9
  • Proposition 10
  • ...and 6 more