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Sensing with Random Signals: The Role of Time Sharing

This paper characterizes the capacity-sensing region for symbol-unaware integrated sensing and communications (ISAC) systems, demonstrating that explicit time sharing is strictly beneficial only when the single-mode frontier is convex due to sensing-side dominance in effective SNR distributions, whereas it is unnecessary when the frontier is already concave.

Original authors: Yi Geng, Wenyi Zhang

Published 2026-06-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yi Geng, Wenyi Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a radio operator trying to do two things at once with a single walkie-talkie: talk to a friend (Communication) and listen for echoes to map out the room (Sensing).

In many traditional setups, the person listening for echoes (the "sensing receiver") is also the person you are talking to. They know exactly what you said, so they can subtract your voice from the echo to see what's in the room. This is like having a script and knowing every line before you read it.

The Problem: The "Blind" Listener
This paper tackles a different, harder scenario: Passive or Bistatic Sensing. Imagine the person listening for echoes is a stranger standing in a different room. They know how you speak (the rules of the language, the volume, the frequency), but they do not know the specific words you are saying right now. To them, your voice is just part of the background noise.

The authors ask: How do we balance talking clearly and sensing the room well when the listener doesn't know our words?

The Core Discovery: The "Time-Sharing" Secret

The paper finds that the answer depends entirely on the quality of the connection for talking versus the quality of the connection for sensing. They use a concept called Time-Sharing, which is like a chef deciding whether to cook a perfect steak or a perfect salad, or to mix them together.

Here is the simple breakdown of their findings:

1. The "Smooth Curve" Scenario (Talking is King)

Imagine your connection for talking is much stronger than your connection for sensing.

The Metaphor: You are shouting through a megaphone to a friend, but whispering to the stranger in the other room.
The Result: The best strategy is to gradually change your voice. You don't need to switch modes abruptly. You can slowly shift from "random chatter" (sending data) to "sounding like a steady tone" (sending a pilot signal) without losing efficiency.
The Paper's Claim: In this case, the "frontier" (the limit of what you can achieve) is concave (curved like a bowl). You don't need to stop and switch strategies; a smooth mix is already perfect.

2. The "Bumpy Hill" Scenario (Sensing is King)

Imagine your connection for sensing is much stronger than your connection for talking.

The Metaphor: You are whispering to your friend, but the stranger in the other room has a super-sensitive microphone that hears everything clearly.
The Result: Here, mixing your voice halfway (half whisper, half tone) is actually wasteful. It's like trying to make a smoothie by blending a rock and a feather; you get a bad result.
The Paper's Claim: The "frontier" is convex (curved like a hill). To get the best results, you must Time-Share. You should spend 100% of your time shouting (pure sensing mode) and then switch to 100% whispering (pure data mode), rather than trying to do a weak version of both at the same time. The "optimal" path is a straight line connecting the two extremes, skipping the middle ground entirely.

3. The "Perfect Balance" Scenario

If the strength of the talking connection and the sensing connection are exactly equal, the result is a straight line. You can mix them in any proportion without penalty or gain.

The "Why" Behind the Magic

The authors used some heavy math to prove that the shape of this "efficiency curve" is determined by a stochastic ordering (a fancy way of saying "which signal is statistically stronger").

If the "Talking Signal" dominates: The curve is smooth. You can blend modes.
If the "Sensing Signal" dominates: The curve is bumpy. You must switch modes abruptly (Time-Sharing) to be efficient.

The SIMO Extension (Many Antennas)

The paper also looked at systems with multiple antennas (SIMO). They found that the same rule applies, but now it depends on two factors:

How many antennas you have on the talking side vs. the sensing side.
How strong the signal is for each antenna.

If the talking side has more antennas or stronger signals, you blend. If the sensing side has the advantage, you switch back and forth (Time-Share). If the advantages are mixed (e.g., more antennas on one side but weaker signal on the other), the math gets complicated, and the simple "switch or blend" rule might not apply cleanly.

Summary

In the world of "Symbol-Unaware" sensing (where the listener doesn't know your words), Time-Sharing isn't just a backup plan; it is sometimes the only way to be efficient.

If your talking channel is strong, blend your signals smoothly.
If your sensing channel is strong, switch between pure talking and pure sensing. Don't try to do both at once.

The paper proves that the decision to "switch" or "blend" is not a matter of engineering preference, but a fundamental law dictated by the relative strength of your signals.

Technical Summary: Sensing with Random Signals: The Role of Time Sharing

Problem Statement
This paper addresses the information-theoretic characterization of Integrated Sensing and Communications (ISAC) in settings where the sensing receiver is "symbol-unaware." In traditional monostatic or decision-aided ISAC formulations, the sensing receiver is assumed to know the transmitted waveform or symbol realization, allowing sensing performance to be measured by the conditional mutual information $I(S; V | X)$ . However, in passive, bistatic, or distributed scenarios, the sensing receiver may know the signaling rule but not the specific transmitted symbols. In such cases, the communication signal remains part of the sensing uncertainty, necessitating the use of the unconditioned mutual information $I(S; V)$ as the sensing metric. The central question is whether a single input distribution (a single signaling mode) is sufficient to characterize the optimal communication–sensing tradeoff under this metric, or if explicit time sharing is fundamentally required to convexify the achievable region.

Methodology
The authors model the system as a state-dependent memoryless channel where a message $W$ is transmitted over $n$ channel uses, generating communication output $Y$ and sensing output $V$ dependent on an i.i.d. state sequence $S$ . The sensing performance is quantified by the averaged symbol-by-symbol mutual information $D_n = \frac{1}{n}\sum I(S_i; V_i)$ , without conditioning on the input symbols.

The analysis proceeds in two main stages:

General Characterization: The authors derive the single-letter capacity–sensing region for finite-alphabet memoryless channels. They introduce an auxiliary time-sharing variable $Q$ to characterize the achievable region. Using the Fenchel–Eggleston strengthening of Carathéodory's theorem, they prove that the optimal upper boundary of the region is the upper concave envelope of the "single-mode frontier" (the set of points achievable by a single input distribution $P_X$ ).
Specific Model Analysis: The theoretical framework is applied to Rayleigh-fading Binary Phase Shift Keying (BPSK) and Single-Input Multiple-Output (SIMO) BPSK models. The authors analyze the curvature of the single-mode frontier $C_0(D)$ by examining the stochastic ordering of the effective Signal-to-Noise Ratio (SNR) distributions on the communication and sensing sides. They utilize the concept of Monotone Likelihood Ratio (MLR) ordering and tilted SNR densities to determine whether the frontier is concave, convex, or linear.

Key Contributions and Results

Capacity–Sensing Region Characterization: The paper establishes that for symbol-unaware ISAC, the achievable region is the downward closure of the convex hull of the single-mode frontier. Crucially, the optimal boundary is the upper concave envelope of the single-mode frontier. This implies that explicit time sharing is unnecessary only if the single-mode frontier is already concave; otherwise, time sharing is strictly beneficial and structurally necessary to achieve the optimal tradeoff.
Curvature Criterion for Rayleigh-Fading BPSK: For scalar Rayleigh-fading BPSK, the authors derive a precise condition determining the geometry of the frontier based on the average effective SNRs of the communication side ( $\bar{\gamma}_c$ $\overset{γ}{ˉ}_{c}$ ) and the sensing side ( $\bar{\gamma}_s$ $\overset{γ}{ˉ}_{s}$ ):
- Communication-Side Dominance ( $\bar{\gamma}_c > \bar{\gamma}_s$ ): The single-mode frontier is strictly concave. Time sharing provides no gain; the optimal strategy involves continuously biasing the input distribution from equiprobable data symbols to deterministic pilot-like symbols.
- Sensing-Side Dominance ( $\bar{\gamma}_c < \bar{\gamma}_s$ ): The single-mode frontier is strictly convex. Time sharing yields a strict gain, meaning the optimal boundary is the chord connecting the pure data mode and the pure pilot mode, rather than any intermediate biased mode.
- Equality ( $\bar{\gamma}_c = \bar{\gamma}_s$ ): The frontier is linear.
Extension to SIMO-BPSK: The results are extended to i.i.d. SIMO systems. The curvature is determined by the ordering of the post-combining SNR distributions. If the communication side has a larger number of antennas ( $N_c$ ) or a larger scale parameter ( $\theta_c$ ) compared to the sensing side, the frontier is concave. Conversely, if the sensing side dominates in both diversity and scale, the frontier is convex, necessitating time sharing. Mixed conditions may result in non-monotonic likelihood ratios, preventing a simple dominance rule.

Significance
The paper claims that under the symbol-unaware sensing metric $I(S; V)$ , time sharing is not merely a secondary design option or an implementation artifact but can be an intrinsic component of the optimal ISAC boundary. The findings explain the fundamental conditions under which a system should transition from continuous data-symbol transmission to a time-shared strategy involving pilot-like sensing modes. Specifically, when the sensing side's effective SNR dominates the communication side, the optimal strategy shifts away from intermediate biased inputs toward a strict alternation between pure communication and pure sensing modes. This work provides a rigorous geometric understanding of the tradeoffs in passive and distributed ISAC scenarios, distinguishing them from known-waveform formulations.