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Integrated Communication and Binary State Detection Under Unequal Error Constraints

Daewon Seo, Sung Hoon Lim

TL;DR

This work develops an information-theoretic framework for integrated sensing and communication where binary state detection is performed with unequal false-alarm and missed-detection constraints alongside a data-rate objective. It derives a complete three-way tradeoff for a fixed state using the cumulant-generating function of the log-likelihood ratio, and a ROC-based region for i.i.d. state variations, employing constant-composition codes and a mixture-based detector to achieve the optimal boundaries. The results are supported by analytical bounds and numerical examples (binary and Gaussian/Vector Gaussian channels), and they reveal how the input distribution and a tunable likelihood-ratio threshold govern the rate-exponent frontier. The findings offer constructive strategies (including a waterfilling approach) and practical insights for designing ISAC systems that reliably balance communication performance with two distinct sensing-error constraints, with potential extensions to multi-target and multi-user settings.

Abstract

This work considers a problem of integrated sensing and communication (ISAC) in which the goal of sensing is to detect a binary state. Unlike most approaches that minimize the total detection error probability, in our work, we disaggregate the error probability into false alarm and missed detection probabilities and investigate their information-theoretic three-way tradeoff including communication data rate. We consider a broadcast channel that consists of a transmitter, a communication receiver, and a detector where the receiver's and the detector's channels are affected by an unknown binary state. We consider and present results on two different state-dependent models. In the first setting, the state is fixed throughout the entire transmission, for which we fully characterize the optimal three-way tradeoff between the coding rate for communication and the two possibly nonidentical error exponents for sensing in the asymptotic regime. The achievability and converse proofs rely on the analysis of the cumulant-generating function of the log-likelihood ratio. In the second setting, the state changes every symbol in an independently and identically distributed (i.i.d.) manner, for which we characterize the optimal tradeoff region based on the analysis of the receiver operating characteristic (ROC) curves.

Integrated Communication and Binary State Detection Under Unequal Error Constraints

TL;DR

This work develops an information-theoretic framework for integrated sensing and communication where binary state detection is performed with unequal false-alarm and missed-detection constraints alongside a data-rate objective. It derives a complete three-way tradeoff for a fixed state using the cumulant-generating function of the log-likelihood ratio, and a ROC-based region for i.i.d. state variations, employing constant-composition codes and a mixture-based detector to achieve the optimal boundaries. The results are supported by analytical bounds and numerical examples (binary and Gaussian/Vector Gaussian channels), and they reveal how the input distribution and a tunable likelihood-ratio threshold govern the rate-exponent frontier. The findings offer constructive strategies (including a waterfilling approach) and practical insights for designing ISAC systems that reliably balance communication performance with two distinct sensing-error constraints, with potential extensions to multi-target and multi-user settings.

Abstract

This work considers a problem of integrated sensing and communication (ISAC) in which the goal of sensing is to detect a binary state. Unlike most approaches that minimize the total detection error probability, in our work, we disaggregate the error probability into false alarm and missed detection probabilities and investigate their information-theoretic three-way tradeoff including communication data rate. We consider a broadcast channel that consists of a transmitter, a communication receiver, and a detector where the receiver's and the detector's channels are affected by an unknown binary state. We consider and present results on two different state-dependent models. In the first setting, the state is fixed throughout the entire transmission, for which we fully characterize the optimal three-way tradeoff between the coding rate for communication and the two possibly nonidentical error exponents for sensing in the asymptotic regime. The achievability and converse proofs rely on the analysis of the cumulant-generating function of the log-likelihood ratio. In the second setting, the state changes every symbol in an independently and identically distributed (i.i.d.) manner, for which we characterize the optimal tradeoff region based on the analysis of the receiver operating characteristic (ROC) curves.

Paper Structure

This paper contains 18 sections, 6 theorems, 121 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main contribution
  4. Related works
  5. Organization and notation
  6. Problem Statement
  7. Statement for a Fixed State
  8. Statement for i.i.d. States
  9. Tradeoff for a fixed state
  10. Preliminaries
  11. Achievability Proof
  12. Converse proof
  13. Numerical Examples
  14. Tradeoff for i.i.d. States
  15. Proof of Thm. \ref{['thm:iid_state']}
  16. ...and 3 more sections

Key Result

Theorem 1

The rate-exponent region $\mathcal{R}$ is the set of tuples $(R, E_{\sf FA}, E_{\sf MD})$ such that for some $p_X(x)$ and $u \in (0,1)$ where Moreover, the detection performance corresponding to each $u$ can be obtained by the log-likelihood ratio test such that where $\tau = D(p_{u} \| p_0 | p_X) - D(p_{u} \| p_1 | p_X)$.

Figures (6)

  • Figure 1: Channel model for the joint communication and binary state detection problem.
  • Figure 2: Tradeoff boundary points of $\mathcal{R}$ for the binary multiplicative state channel with $p=0.1, q=0.2$. Units are bits.
  • Figure 3: Projection of $\mathcal{R}$ on the $E_{\sf MD}$-$E_{\sf FA}$ plane for the Gaussian example with $P=1, P=2$ and $P=4$. We set $h=0.5$.
  • Figure 4: Achievable points $(R, E_{{\sf FA}}, E_{{\sf MD}})$ for the vector Gaussian channel with $P=10$ and channel gains given in \ref{['eq:mimo_channel_gains']}.
  • Figure 5: Binary example assuming $\gamma_1 = \gamma_2 = 0.2$. In (b), the maximal $P_{\sf D}$ achieved at the highest data rate (i.e., channel capacity) is indicated by circle, corresponding to the input distribution $p_X(1) = 0.4824$.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Corollary 1: Joudeh--Willems2022
  • Corollary 2
  • Theorem 2
  • ...and 2 more