On the Fundamental Limits of Integrated Sensing and Communications Under Logarithmic Loss

TL;DR

This work develops a unified information-theoretic framework for integrated sensing and communications (ISAC) under logarithmic loss, encompassing both monostatic and bistatic sensing. By deriving tight lower and upper bounds on the capacity-distortion function $C(D)$ and identifying matching conditions under channel degradation, it clarifies when sensing and communication decouple and when they remain coupled. The paper provides explicit single-letter characterizations for binary-symmetric and Gaussian ISAC models (including a state-splitting approach and entropy-power-type arguments), offering deep insights into waveform design and fundamental limits. Overall, the results establish a rigorous foundation for ISAC under soft sensing and log-loss criteria, with open questions on non-degraded channels and efficient numerical evaluation.

Abstract

We study a unified information-theoretic framework for integrated sensing and communications (ISAC), applicable to both monostatic and bistatic sensing scenarios. Special attention is given to the case where the sensing receiver (Rx) is required to produce a "soft" estimate of the state sequence, with logarithmic loss serving as the performance metric. We derive lower and upper bounds on the capacity-distortion function, which delineates the fundamental tradeoff between communication rate and sensing distortion. These bounds coincide when the channel between the ISAC transmitter (Tx) and the communication Rx is degraded with respect to the channel between the ISAC Tx and the sensing Rx, or vice versa. Furthermore, we provide a complete characterization of the capacity-distortion function for an ISAC system that simultaneously transmits information over a binary-symmetric channel and senses additive Bernoulli states through another binary-symmetric channel. The Gaussian counterpart of this problem is also explored, which, together with a state-splitting trick, fully determines the capacity-distortion-power function under the squared error distortion measure.

Figures (4)

• Figure 1: Bistatic sensing (including monostatic sensing as a degenerate case).
• Figure 2: Plots of $C_{\mathrm{B}}(D)$ under sybmol-wise logarithmic loss with $\beta_2=0.2$ and $\beta_S=0.1$ are shown for $\beta_1=0.3$, $\beta_1=0.24$, and $\beta_1=0.18$, corresponding to cases 1), 2) and 3) in Theorem \ref{['thm:binary']}, respectively. The plots for $\beta_1=0.3$ and $\beta_1=0.18$ also apply to sequence-wise logarithmic loss, while the plot for $\beta_1=0.24$ serves as a lower bound on $C_{\mathrm{B}}(D)$ under sequence-wise logarithmic loss.
• Figure 3: Plots of $C_{\mathrm{G}}(D,P)$ with $P=1$, $N_2=2$, and $N_S=1$ are shown for $N_1=3.5$, $N_1=2.5$, ad $N_1=1.5$, corresponding to cases 1), 2) and 3) in Theorem \ref{['thm:Gaussian1']}, respectively. They apply to both sequence-wise and symbol-wise logarithmic loss. The plots for $N_1=3.5$ and $N_1=1.5$ depict the exact values of $C_{\mathrm{G}}(D,P)$, while the plot for $N_1=2.5$ serves as a lower bound on $C_{\mathrm{G}}(D,P)$.
• Figure 4: Plots of $C'_{\mathrm{G}}(D,P)$ under the squared error distortion measure with $P=1$, $N_2=2$, and $N_S=1$ are shown for $N_1=3.5$, $N_1=2.5$, and $N_1=1.5$. Here, $N_1=3.5$ corresponds to case 1), while both $N_1=2.5$ and $N_1=1.5$ correspond to case 2) in Theorem \ref{['thm:Gaussian2']}. Different from its logarithmic loss counterpart, the plot for $N_1=2.5$ depicts the exact valus of $C'_{\mathrm{G}}(D,P)$, rather than just a lower bound.

Theorems & Definitions (15)

• Definition 1
• Remark 1
• Theorem 1
• Remark 2
• Theorem 2
• Remark 3
• Corollary 1
• Corollary 2
• Remark 4
• Remark 5