Achievable Rate-Sensing Region in ISAC

• Achievable Rate-Sensing Region is the set of jointly attainable communication rates and sensing metrics in ISAC systems, defining the tradeoff between throughput and sensing accuracy.
• It is characterized using metrics like mutual information, CRB, and distortion, with formulations based on finite blocklength, MIMO models, and hybrid architectures.
• The design leverages joint waveform optimization, advanced beamforming, and dynamic feedback to maximize performance under practical hardware and resource limitations.

An achievable rate-sensing region defines the set of jointly attainable communication rates and sensing performance metrics for integrated sensing and communication (ISAC) systems, given resource constraints and system architectures. The operational boundaries of this region formalize the tradeoff between maximizing information throughput and minimizing sensing error/distortion, under unified waveforms, joint waveform designs, advanced beamforming, or feedback protocols. The region can be quantified in terms of mutual information, Cramér–Rao bound (CRB), detection-error exponent, and distortion constraints, with Pareto frontier characterizations spanning finite blocklength, MIMO architectures, near-field models, hybrid analog–digital hardware, and multi-user/multi-target scenarios.

1. Formal Definition and Mathematical Framework

The achievable rate-sensing region encompasses pairs (or tuples) of communication rate $R$ and sensing metric $Q$ (e.g., CRB, distortion, error exponent) that can be simultaneously achieved by an ISAC system, subject to resource constraints such as transmit power, coding complexity, and architecture. For a blocklength $n$, under error probability $\varepsilon$ and distortion $D$, the region is: $$\mathcal{R} = \left\{ (R, Q): \exists~\text{codebook, waveform, or beamformer such that } \operatorname{Rate} \ge R,~\operatorname{Sensing} \le Q \right\}$$ Specific characterizations include:

• Finite Blocklength DMC ISAC (Rate–Distortion–Error): For discrete memoryless state-dependent channels (Nikbakht et al., 28 Jan 2024)

$$R \leq \max_{P_X: \mathbb{E}[d(S, \hat s^*(X,Z))] \leq D} I(X ; Y) - \sqrt{\frac{V}{n}}Q^{-1}(\varepsilon) + O\left(\frac{\log n}{n}\right)$$

with distortion-constrained input $P_X$ and channel dispersion $V$.

• MIMO CRB-rate ISAC: For simultaneous communications (rate $R$) and extended-target estimation (CRB) (Hua et al., 2022, Hua et al., 2022)

$$\max_{R_x \succeq 0: \mathrm{CRB}_{\text{metric}}(R_x) \le \Gamma,\, \operatorname{tr}(R_x) \le P} \log_2 \det(I + H R_x H^H/\sigma_c^2)$$

subject to CRB constraints in trace, maximum eigenvalue, or determinant forms.

2. Pareto Frontier: Sensing-Centric and Rate-Centric Optimization

The Pareto boundary of the achievable region is traced by sweeping constraints on one metric and optimizing the other:

• Sensing-centric: Minimize sensing CRB/distortion for a fixed rate requirement (communication constraint)
• Rate-centric: Maximize communication rate under fixed sensing accuracy constraints

For hybrid analog–digital beamforming architectures (fully-connected or partially-connected), this translates into optimization over codebooks, precoders, and beamformer weight matrices $\{\mathbf F, \mathbf W, \mathbf c\}$: $$\min_{\mathbf F, \mathbf W, \mathbf c}~\mathrm{Tr}[\mathrm{CRB}(\mathbf r, \bm\theta)]~\text{subject to}~R_k(\mathbf F,\mathbf W,\mathbf c)\geq R_{\mathrm{th}},~~\|\mathbf F\mathbf W\|^2 \leq P_{\max}$$ Solving these for all $R_{\mathrm{th}}$ produces the full boundary (Zhou et al., 17 Feb 2025).

3. Impact of System Architecture and Resource Allocation

The region is critically affected by architecture:

• Fully digital beamforming yields maximum flexibility (largest region), often unattainable due to RF hardware constraints.
• Fully-connected hybrid analog–digital (FC-HAD): Near-optimal region with reduced RF chains; strict performance loss scaled by disparity in chains.
• Partially-connected HAD: Smaller region, but efficient for severely constrained hardware.
• Multi-user/multi-target, rate-splitting multiple access (RSMA): Enhanced interference management widens the region, especially in near-field spherical wave settings (Zhou et al., 17 Feb 2025).

4. Role of Feedback, Randomness, and Coding Paradigms

Feedback, randomization, and identification coding dramatically alter the region shape:

• Variable-length coding with feedback: Feedback introduces a rate-exponent tradeoff, forcing a true Pareto frontier and excluding simultaneous maximization of rate and exponent (Papoutsidakis et al., 24 Jan 2025).
• Identification/sensing codes (JIDAS, randomized/deterministic): Achievable region expands strictly beyond separation-based time-sharing, with doubly exponential ID code size growth and explicit lower bounds (Zhao et al., 20 Aug 2024).
• Random signaling (e.g., uniform over Stiefel manifold) can optimize sensing at the cost of communication DoF; Gaussian signaling is optimal for rate but may degrade CRB (Xiong et al., 2022).

5. Region Bounding and Characterization Techniques

Achievable regions are typically bounded (inner/outer) and characterized:

• Analytic/Algorithmic Pareto tracing: SVD-based water-filling on composite or communication channels, augmented Lagrangian or block-coordinate descent in beamformer design (Hua et al., 2022, Hua et al., 2022, Zhou et al., 17 Feb 2025).
• CRB-rate region for pinching antenna systems: Rate profile optimization yields convex hulls containing conventional fixed-antenna system regions (Ouyang et al., 15 May 2025).
• Bounding Techniques: Use Cauchy–Schwarz and majorization inequalities for outer bounds, and time-sharing between sensing- and rate-optimal points for inner bounds.

Architecture	Maximal Region	Complexity
Full digital (FD)	Largest	Highest
Fully-connected HAD	Near-maximal	Moderate
Partially-connected	Reduced	Lowest
JIDAS/RSMA	Strictly largest vs. separation	Varies

6. Geometry, Resource Splitting, and Comparative Insights

• ISAC region supersedes time/frequency separation: Convex rectangle—maximal joint rates—whereas separation yields piecewise linear tradeoff curves or staircases (Ouyang et al., 2022, Chang et al., 2022).
• Subspace tradeoff (ST): Overlap between comm. and sensing subspaces in transmit covariance shapes the frontier (Xiong et al., 2022).
• Deterministic-random tradeoff (DRT): Sensing prefers deterministic signals, comms prefers randomness; codomain overlap governs DoF loss on either side.

7. Representative Models and Quantitative Examples

• Finite blocklength DMC ISAC: Achievable rate shrinks with higher sensing fidelity (smaller $D$) and finite $n$ backoff.
• MIMO ISAC with extended target: For $\mathbf S$ transmit covariance, rate and trace-CRB are simultaneously optimized via dual constraints. In full-rank settings, box-shaped regions emerge; rank deficiency expands the region.
• Near-field multi-target sensing/RSMA: RSMA-FC-near approaches FD-near performance with only $N_f\sim8$ RF chains, strictly outperforms SDMA and far-field ISAC (Zhou et al., 17 Feb 2025).
• PASS pinching-antenna ISAC: Pareto profile design and element-wise alternating optimization achieve rate regions larger than any fixed-antenna baseline (Ouyang et al., 15 May 2025).

8. Comparative Region Analysis and Design Guidelines

• Time-sharing is strictly suboptimal compared to joint designs; region boundaries for ISAC are convex and reach simultaneous optima, while separation is linear and strictly contained within ISAC regions (Nikbakht et al., 28 Jan 2024, Ouyang et al., 2022).
• Hybrid designs (two-stage, penalty-dual-decomposition, identification coding) recover most joint region performance at reduced complexity, with explicit region bounding.
• Choice of architecture and beamforming balances region size, hardware constraints, and computational tractability.

9. Fundamental Limits and Future Directions

• No scheme surpasses the joint region defined by fidelity-rate constraints for any feasible input law or waveform.
• Closed-loop sensing and joint decoding push the Pareto region boundary outward, outperforming successive or open-loop approaches (Chang et al., 2022).
• A plausible implication is that further advances in joint waveform/codebook design, dynamic feedback exploitation, and hardware-parsimonious hybrid analog-digital architectures will continue to expand the achievable rate-sensing region in practical ISAC deployments.