Fundamental Limits of 1-bit ISAC Systems: Capacity Region and Optimal Power Control

Emmanuel Trinidad; Neil Irwin Bernardo

arxiv: 2604.12953 · v2 · submitted 2026-04-14 · 📡 eess.SP · cs.IT· math.IT

Fundamental Limits of 1-bit ISAC Systems: Capacity Region and Optimal Power Control

Emmanuel Trinidad , Neil Irwin Bernardo This is my paper

Pith reviewed 2026-05-10 14:36 UTC · model grok-4.3

classification 📡 eess.SP cs.ITmath.IT

keywords 1-bit quantizationintegrated sensing and communicationcapacity regionGaussian fading channelpower controlmonostatic sensingISAC systems

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The pith

In 1-bit ISAC systems with receiver CSI, communication and sensing capacities are achieved simultaneously by a constant-amplitude input with rotational symmetry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the information limits of integrated sensing and communication where both receivers quantize their inputs to a single bit. It models a Gaussian fading channel with separate communication and monostatic sensing paths. When the communication receiver knows the channel state, the maximum communication rate and the maximum sensing rate can be reached at once. The input that achieves this is a constant-amplitude signal whose phase distribution has a particular rotational symmetry. When the transmitter also knows the channel state, the paper solves for the power allocation that balances the two rates under an average power limit and shows how that allocation changes as the balance shifts toward sensing.

Core claim

In a Gaussian fading ISAC channel with 1-bit quantization at both receivers, the communication-sensing capacity region allows simultaneous achievement of the individual capacities using a constant-amplitude input distribution with specific rotational symmetry, when CSI is available at the receiver. For the case with CSI at the transmitter as well, an optimal power control policy is derived that balances the rates according to a weighting parameter, transitioning from opportunistic to more uniform allocation as sensing is prioritized.

What carries the argument

Constant-amplitude input distribution with specific rotational symmetry, which simultaneously saturates both the communication mutual information and the sensing estimation rate under 1-bit quantization.

If this is right

The capacity region is rectangular, so no rate sacrifice is needed to serve both functions at their individual maxima.
The same input distribution works for both tasks, removing the need for time-sharing or separate waveforms.
Optimal power control shifts from channel-opportunistic transmission when communication is prioritized to nearly uniform power when sensing dominates.
The rotational symmetry property provides an explicit construction that meets both rate bounds at once.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Hardware designers could use simple constant-envelope transmitters for dual-purpose 1-bit ISAC without performance loss in this regime.
The symmetry-based construction may extend to other low-resolution quantization levels if the rotational property can be generalized.
In multi-user settings the same input symmetry might allow multiple sensing targets to be observed while still supporting communication.

Load-bearing premise

The analysis requires that the communication receiver has perfect channel state information and that the channel follows a Gaussian fading model with separate communication and monostatic sensing links.

What would settle it

An experiment or simulation in which the 1-bit quantized sensing rate falls measurably below its individual maximum when the communication input is set to the proposed constant-amplitude distribution with rotational symmetry would refute the no-trade-off result.

Figures

Figures reproduced from arXiv: 2604.12953 by Emmanuel Trinidad, Neil Irwin Bernardo.

**Figure 2.** Figure 2: Optimal power control P λ γc vs. instantaneous communication channel realization γc at 0 dB SNR for both communication and sensing channels. average transmit power constraint. Specifically, we formulate the objective function Cλ (Pγc ) = λ · C (CSIT) comm (Pγc ) + (1 − λ) · Csense (Pγc ), (28) where λ ∈ [0, 1] controls the relative priority between communication and sensing tasks. For a given λ, we seek t… view at source ↗

**Figure 3.** Figure 3: Communication rate vs. SNR for different [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

This paper investigates the fundamental limits of integrated sensing and communication (ISAC) systems with 1-bit receiver quantization. We analyze a Gaussian fading ISAC channel with separate communication and monostatic sensing links, where both communication and sensing receivers are equipped with 1-bit quantizers. When the communication channel state information (CSI) is available at the receiver, we characterize the communication-sensing capacity region of 1-bit ISAC channel and show that no trade-off exists between communication and sensing performance. In particular, both communication and sensing capacities can be simultaneously achieved by a constant-amplitude input distribution with a specific rotational symmetry. For the scenario where communication CSI is also available at the transmitter, we formulate a weighted optimization problem that balances communication and sensing rates in 1-bit ISAC channel under an average power constraint and then derive the corresponding optimal power control policy. The results demonstrate how the optimal power control policy evolves with the weighting parameter, transitioning from a communication-centric, opportunistic transmission to a more uniform allocation as sensing becomes increasingly prioritized.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a rectangular capacity region for 1-bit ISAC with no comm-sensing trade-off under receiver CSI, achieved by one constant-amplitude rotationally symmetric input, and derives a shifting power control policy when the transmitter also has CSI.

read the letter

The central result here is that in a Gaussian fading ISAC model with separate communication and monostatic sensing links, both using 1-bit quantizers, the capacity region becomes a rectangle once the receiver knows the communication channel state. A single constant-amplitude input distribution with rotational symmetry hits the individual maxima for both mutual informations at once. When the transmitter also has that CSI, they set up a weighted sum-rate problem under average power and solve for the optimal allocation, which moves from opportunistic communication-focused transmission to more uniform power as the sensing weight rises. That evolution is shown explicitly as a function of the weighting parameter. The model is standard for this area—independent fadings, monostatic sensing, average power—but the no-trade-off claim and the specific input are the new pieces relative to earlier ISAC capacity work. The power-control derivation is straightforward once the region is known and gives a concrete policy that practitioners could use. The soft spot is exactly the one the stress-test flags: the sensing link has its own independent fading coefficient and its own 1-bit observation, so nothing automatically forces the communication-optimal distribution to also maximize sensing mutual information. The rotational symmetry helps the communication side by making the effective channel isotropic, but the paper must prove it does the same for sensing. If the derivations close that gap, the rectangle result holds; if they rely on an unstated symmetry that does not carry over, the claim weakens. The abstract and setup give no sign of circularity or invented entities, and the analysis stays within standard information-theoretic tools. This is for readers working on quantized ISAC or low-resolution hardware limits. Someone who needs bounds to guide 1-bit transceiver design or who follows capacity results in fading channels will find the region and the policy useful. It deserves a serious referee because the claims are specific, the setup is relevant, and the results are falsifiable once the proofs are checked, even if revisions on the sensing optimality step are likely.

Referee Report

2 major / 2 minor

Summary. This paper analyzes the fundamental limits of 1-bit quantized ISAC over Gaussian fading channels with separate communication and monostatic sensing links. When communication CSI is available at the receiver, it claims to fully characterize the communication-sensing capacity region and shows that the region is a rectangle (no trade-off), with both capacities simultaneously achieved by a constant-amplitude input distribution possessing rotational symmetry. When CSI is also available at the transmitter, it formulates a weighted sum-rate optimization under average power constraint and derives the corresponding optimal power control policy, which transitions from opportunistic to uniform allocation as the sensing weight increases.

Significance. If the no-trade-off result holds, it would be a notable contribution to the information-theoretic understanding of low-resolution ISAC, demonstrating that 1-bit quantization need not force a rate trade-off between communication and sensing under the stated model. The explicit optimal power control policy would also offer concrete design guidance for resource allocation in such systems.

major comments (2)

[capacity region characterization (main theorem)] The central claim that a single rotationally symmetric constant-amplitude distribution simultaneously achieves both the communication and sensing capacities (making the capacity region a rectangle) appears to rest on the assumption that the communication-optimal input is also sensing-optimal. However, the sensing link has an independent fading coefficient and its own 1-bit quantizer; the rotational symmetry that symmetrizes the effective communication channel does not automatically guarantee optimality for the monostatic sensing observation. This needs explicit verification, e.g., by showing that the mutual information I(X;Y_s) is maximized by the same distribution under the independent fading.
[optimal power control section] The derivation of the optimal power control policy for the weighted optimization problem is presented without sufficient detail on how the 1-bit quantization affects the rate expressions inside the optimization; it is unclear whether the policy accounts for the non-linear effect of the quantizer on the effective SNR or relies on approximations that may not hold for all weighting parameters.

minor comments (2)

[system model] Notation for the communication and sensing outputs (Y_c and Y_s) and the input X should be introduced earlier and used consistently to avoid ambiguity when discussing the separate links.
[introduction] The abstract and introduction would benefit from a brief statement of the precise channel model (e.g., whether the fading is block-fading or ergodic) to set expectations before the capacity claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We address each major comment in detail below, providing clarifications and committing to revisions that strengthen the presentation without altering the core results.

read point-by-point responses

Referee: [capacity region characterization (main theorem)] The central claim that a single rotationally symmetric constant-amplitude distribution simultaneously achieves both the communication and sensing capacities (making the capacity region a rectangle) appears to rest on the assumption that the communication-optimal input is also sensing-optimal. However, the sensing link has an independent fading coefficient and its own 1-bit quantizer; the rotational symmetry that symmetrizes the effective communication channel does not automatically guarantee optimality for the monostatic sensing observation. This needs explicit verification, e.g., by showing that the mutual information I(X;Y_s) is maximized by the same distribution under the independent fading.

Authors: We appreciate this observation and agree that the optimality for the sensing link merits explicit verification. In Section III, we establish that the constant-amplitude rotationally symmetric distribution (uniform phase over the circle with fixed amplitude) achieves the communication capacity by rendering the effective 1-bit quantized channel phase-invariant and maximizing the output entropy. For the independent sensing link, the same distribution maximizes I(X; Y_s) because the monostatic observation after 1-bit quantization is a function of the instantaneous power |h_s|^2 P; the uniform phase randomization ensures the quantized output distribution is independent of the unknown phase of h_s and achieves the maximum possible mutual information for any fixed power (as any non-constant amplitude or non-uniform phase would reduce the effective distinguishability under the sign quantizer). We will add a new lemma (Lemma 2) and a short appendix deriving this explicitly via the chain rule for mutual information and properties of the complex Gaussian noise under rotational invariance, confirming that the capacity region is indeed rectangular with no trade-off. revision: yes
Referee: [optimal power control section] The derivation of the optimal power control policy for the weighted optimization problem is presented without sufficient detail on how the 1-bit quantization affects the rate expressions inside the optimization; it is unclear whether the policy accounts for the non-linear effect of the quantizer on the effective SNR or relies on approximations that may not hold for all weighting parameters.

Authors: We agree that the section would benefit from expanded detail on the rate expressions. The communication and sensing rates inside the weighted sum are the exact mutual informations I(X;Y_c) and I(X;Y_s) under the respective 1-bit quantizers, which are computed as functions of the allocated power P(h) and the instantaneous fading gains; these are inherently non-linear due to the quantization thresholds. The optimal policy is derived via the KKT conditions on the concave weighted objective under the average power constraint, without approximations. As the sensing weight increases, the marginal utility of power for sensing (which benefits from averaging over fades) shifts the solution from opportunistic (water-filling-like) to uniform allocation. We will revise Section IV to include the explicit integral expressions for the quantized mutual informations, a step-by-step derivation of the policy, and a numerical verification that the transition holds for all weights in [0,1] without relying on high-SNR or other approximations. revision: yes

Circularity Check

0 steps flagged

Standard capacity analysis with independent derivation of optimal input distribution

full rationale

The paper derives the communication-sensing capacity region for the 1-bit ISAC model by standard mutual information maximization under the given channel and quantization constraints. It identifies a constant-amplitude rotationally symmetric distribution that simultaneously achieves both maxima when Rx CSI is available; this is obtained from the analysis of the effective channels rather than by redefining one quantity in terms of the other or by fitting. The subsequent weighted optimization for power control with Tx CSI is a standard convex program whose solution is derived from the Lagrangian, not forced by prior results or self-citations. No load-bearing step reduces to a self-definition, fitted input renamed as prediction, or self-citation chain. The derivation remains self-contained against the model assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; full paper would likely introduce additional modeling assumptions and possibly a weighting parameter in the optimization.

axioms (2)

domain assumption Gaussian fading ISAC channel model with separate communication and monostatic sensing links
Core modeling choice stated in the abstract for the capacity analysis.
domain assumption 1-bit quantization at both communication and sensing receivers
Central system constraint defining the 1-bit ISAC setting.

pith-pipeline@v0.9.0 · 5477 in / 1299 out tokens · 43237 ms · 2026-05-10T14:36:28.914667+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

A Unified Future: Integrated Sensing and Commu- nication (ISAC) in 6G,

A. Ghosh et al., “A Unified Future: Integrated Sensing and Commu- nication (ISAC) in 6G,”IEEE Journal of Selected Topics in Elec- tromagnetics, Antennas and Propagation, vol. 1, no. 1, pp. 365–374, 2025

work page 2025
[2]

An Information-Theoretic Approach to Joint Sensing and Communica- tion,

M. Ahmadipour, M. Kobayashi, M. Wigger, and G. Caire, “An Information-Theoretic Approach to Joint Sensing and Communica- tion,”IEEE Transactions on Information Theory, vol. 70, no. 2, pp. 1124–1146, 2024

work page 2024
[3]

Integrated Sens- ing and Communications: A Mutual Information-Based Framework,

C. Ouyang, Y . Liu, H. Yang, and N. Al-Dhahir, “Integrated Sens- ing and Communications: A Mutual Information-Based Framework,” IEEE Communications Magazine, vol. 61, no. 5, pp. 26–32, 2023

work page 2023
[4]

Deterministic- Random Tradeoff of Integrated Sensing and Communications in Gaus- sian Channels: A Rate-Distortion Perspective,

F. Liu, Y . Xiong, K. Wan, T. X. Han, and G. Caire, “Deterministic- Random Tradeoff of Integrated Sensing and Communications in Gaus- sian Channels: A Rate-Distortion Perspective,” in2023 IEEE Interna- tional Symposium on Information Theory (ISIT), 2023, pp. 2326–2331

work page 2023
[5]

Mutual Information Metrics for Uplink MIMO-OFDM Integrated Sensing and Communication System,

J. Piao, Z. Wei, X. Yuan, X. Yang, H. Wu, and Z. Feng, “Mutual Information Metrics for Uplink MIMO-OFDM Integrated Sensing and Communication System,” inGLOBECOM 2023 - 2023 IEEE Global Communications Conference, 2023, pp. 7387–7392

work page 2023
[6]

Low-Resolution ADCs for Wireless Communication: A Comprehensive Survey,

J. Liu, Z. Luo, and X. Xiong, “Low-Resolution ADCs for Wireless Communication: A Comprehensive Survey,”IEEE Access, vol. 7, pp. 91 291–91 324, 2019

work page 2019
[7]

On the limits of communication with low-precision analog-to-digital conversion at the receiver,

J. Singh, O. Dabeer, and U. Madhow, “On the limits of communication with low-precision analog-to-digital conversion at the receiver,”IEEE Transactions on Communications, vol. 57, no. 12, pp. 3629–3639, 2009

work page 2009
[8]

Fading channels with 1-bit output quanti- zation: Optimal modulation, ergodic capacity and outage probability,

S. Krone and G. Fettweis, “Fading channels with 1-bit output quanti- zation: Optimal modulation, ergodic capacity and outage probability,” in2010 IEEE Information Theory Workshop, 2010, pp. 1–5

work page 2010
[9]

Optimal Signaling Schemes and Capacity of Non- Coherent Rician Fading Channels With Low-Resolution Output Quan- tization,

M. N. Vu, N. H. Tran, D. G. Wijeratne, K. Pham, K.-S. Lee, and D. H. N. Nguyen, “Optimal Signaling Schemes and Capacity of Non- Coherent Rician Fading Channels With Low-Resolution Output Quan- tization,”IEEE Transactions on Wireless Communications, vol. 18, no. 6, pp. 2989–3004, 2019

work page 2019
[10]

On the Capacity-Achieving Input of Channels With Phase Quantization,

N. I. Bernardo, J. Zhu, and J. Evans, “On the Capacity-Achieving Input of Channels With Phase Quantization,”IEEE Transactions on Information Theory, vol. 68, no. 9, pp. 5866–5888, 2022

work page 2022
[11]

Performance Analysis for ISAC Systems with 1-bit DACs,

M. B. Salman, ¨O. T. Demir, and E. Bj ¨ornson, “Performance Analysis for ISAC Systems with 1-bit DACs,” in2024 19th International Sym- posium on Wireless Communication Systems (ISWCS), 2024, pp. 1–6

work page 2024
[12]

Joint Transceiver Design for Massive MIMO DFRC Systems with One-Bit DACs/ADCs,

B. Wang, H. Li, and Z. Cheng, “Joint Transceiver Design for Massive MIMO DFRC Systems with One-Bit DACs/ADCs,” in2023 IEEE Globecom Workshops (GC Wkshps), 2023, pp. 649–654

work page 2023

[1] [1]

A Unified Future: Integrated Sensing and Commu- nication (ISAC) in 6G,

A. Ghosh et al., “A Unified Future: Integrated Sensing and Commu- nication (ISAC) in 6G,”IEEE Journal of Selected Topics in Elec- tromagnetics, Antennas and Propagation, vol. 1, no. 1, pp. 365–374, 2025

work page 2025

[2] [2]

An Information-Theoretic Approach to Joint Sensing and Communica- tion,

M. Ahmadipour, M. Kobayashi, M. Wigger, and G. Caire, “An Information-Theoretic Approach to Joint Sensing and Communica- tion,”IEEE Transactions on Information Theory, vol. 70, no. 2, pp. 1124–1146, 2024

work page 2024

[3] [3]

Integrated Sens- ing and Communications: A Mutual Information-Based Framework,

C. Ouyang, Y . Liu, H. Yang, and N. Al-Dhahir, “Integrated Sens- ing and Communications: A Mutual Information-Based Framework,” IEEE Communications Magazine, vol. 61, no. 5, pp. 26–32, 2023

work page 2023

[4] [4]

Deterministic- Random Tradeoff of Integrated Sensing and Communications in Gaus- sian Channels: A Rate-Distortion Perspective,

F. Liu, Y . Xiong, K. Wan, T. X. Han, and G. Caire, “Deterministic- Random Tradeoff of Integrated Sensing and Communications in Gaus- sian Channels: A Rate-Distortion Perspective,” in2023 IEEE Interna- tional Symposium on Information Theory (ISIT), 2023, pp. 2326–2331

work page 2023

[5] [5]

Mutual Information Metrics for Uplink MIMO-OFDM Integrated Sensing and Communication System,

J. Piao, Z. Wei, X. Yuan, X. Yang, H. Wu, and Z. Feng, “Mutual Information Metrics for Uplink MIMO-OFDM Integrated Sensing and Communication System,” inGLOBECOM 2023 - 2023 IEEE Global Communications Conference, 2023, pp. 7387–7392

work page 2023

[6] [6]

Low-Resolution ADCs for Wireless Communication: A Comprehensive Survey,

J. Liu, Z. Luo, and X. Xiong, “Low-Resolution ADCs for Wireless Communication: A Comprehensive Survey,”IEEE Access, vol. 7, pp. 91 291–91 324, 2019

work page 2019

[7] [7]

On the limits of communication with low-precision analog-to-digital conversion at the receiver,

J. Singh, O. Dabeer, and U. Madhow, “On the limits of communication with low-precision analog-to-digital conversion at the receiver,”IEEE Transactions on Communications, vol. 57, no. 12, pp. 3629–3639, 2009

work page 2009

[8] [8]

Fading channels with 1-bit output quanti- zation: Optimal modulation, ergodic capacity and outage probability,

S. Krone and G. Fettweis, “Fading channels with 1-bit output quanti- zation: Optimal modulation, ergodic capacity and outage probability,” in2010 IEEE Information Theory Workshop, 2010, pp. 1–5

work page 2010

[9] [9]

Optimal Signaling Schemes and Capacity of Non- Coherent Rician Fading Channels With Low-Resolution Output Quan- tization,

M. N. Vu, N. H. Tran, D. G. Wijeratne, K. Pham, K.-S. Lee, and D. H. N. Nguyen, “Optimal Signaling Schemes and Capacity of Non- Coherent Rician Fading Channels With Low-Resolution Output Quan- tization,”IEEE Transactions on Wireless Communications, vol. 18, no. 6, pp. 2989–3004, 2019

work page 2019

[10] [10]

On the Capacity-Achieving Input of Channels With Phase Quantization,

N. I. Bernardo, J. Zhu, and J. Evans, “On the Capacity-Achieving Input of Channels With Phase Quantization,”IEEE Transactions on Information Theory, vol. 68, no. 9, pp. 5866–5888, 2022

work page 2022

[11] [11]

Performance Analysis for ISAC Systems with 1-bit DACs,

M. B. Salman, ¨O. T. Demir, and E. Bj ¨ornson, “Performance Analysis for ISAC Systems with 1-bit DACs,” in2024 19th International Sym- posium on Wireless Communication Systems (ISWCS), 2024, pp. 1–6

work page 2024

[12] [12]

Joint Transceiver Design for Massive MIMO DFRC Systems with One-Bit DACs/ADCs,

B. Wang, H. Li, and Z. Cheng, “Joint Transceiver Design for Massive MIMO DFRC Systems with One-Bit DACs/ADCs,” in2023 IEEE Globecom Workshops (GC Wkshps), 2023, pp. 649–654

work page 2023