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Joint Subcarrier and Power Allocation for Localization-Oriented Multi-Carrier Underwater ISAC Systems
Wei Huang, Sitong Liu, Christos Masouros, Hao Zhang, T. Aaron Gulliver, Kaitao Meng
TL;DR
Frequency-selective underwater channels give subcarriers uneven localization capabilities, making multi-anchor resource allocation challenging under communication constraints. The paper derives localization-information metrics and a distance-frequency matching rule, then uses them in a greedy allocation scheme. Simulations report improved localization accuracy over conventional schemes while maintaining communication performance.
Problem
Frequency-dependent underwater attenuation, ambient noise, and channel selectivity give subcarriers heterogeneous ranging capabilities, complicating localization-oriented allocation across cooperating anchors.
Source: §I · p. 1Method
The paper derives closed-form CRBs and a prior-averaged subcarrier-level localization-information metric, then formulates communication-constrained A-optimal allocation with a greedy algorithm and power refinement.
Source: §I · p. 2, §VI · p. 13Results
Simulations demonstrate improved localization accuracy over conventional schemes while maintaining communication performance.
Source: §VI · p. 13Takeaways & Limitations
The derived distance-frequency matching rule provides a theoretical basis for balancing localization information across subcarriers in localization-oriented allocation.
Source: §VI · p. 13
Abstract
from arXiv · showhide
Multi-carrier underwater integrated sensing and communication (ISAC) realizes concurrent acoustic communication and localization, yet the frequency selectivity of the acoustic channel creates uneven positioning capabilities across subcarriers. This paper proposes a localization-oriented subcarrier and power allocation scheme for multi-anchor underwater ISAC systems. We build a unified orthogonal subcarrier multiplexing framework and derive closed-form Cramer-Rao bounds (CRBs) under frequency-selective channels with colored noise. We derive a subcarrier-level localization information contribution from the time of arrival Fisher information, which enables localization-oriented resource allocation. We derive several theoretical properties regarding the distance and frequency dependence of the localization information metric, and establish a distance-frequency matching principle. These results provide a theoretical foundation for localization-oriented resource allocation and guide the design of a low-complexity greedy algorithm that improves positioning accuracy. Specifically, the algorithm iteratively allocates subcarriers to minimize the residual positioning-error objective, thereby improving positioning accuracy while satisfying communication quality-of-service (QoS) constraints. Simulations verify that our method yields much lower positioning error bound than uniform, random and high-frequency-first subcarrier allocation in symmetric or asymmetric topologies while guaranteeing communication QoS, achieving superior communication-localization tradeoffs.
I. Introduction
Underwater multi-carrier ISAC shares acoustic resources for communication and localization, but frequency-dependent propagation gives subcarriers different ranging capabilities and complicates multi-anchor allocation. The paper develops a localization-oriented framework, derives a distance-frequency matching rule, and models the signal using orthogonal subcarriers and frequency-selective channels.
I. Introduction
- Frequency-dependent attenuation, ambient noise, and channel selectivity give subcarriers heterogeneous ranging capabilities, while multi-anchor positioning also depends on anchor geometry and cooperation.Most existing resource-allocation schemes prioritize communication, and localization-oriented allocation for multi-anchor underwater ISAC remains largely unexplored.Source: §I · pp. 1–2
- The framework jointly treats orthogonal subcarrier allocation, power allocation, communication performance, and localization accuracy under communication constraints.It derives closed-form TOA and localization CRBs for frequency-selective channels with colored noise, then uses a subcarrier-level localization-information metric to formulate a communication-constrained A-optimal allocation problem.Source: §I · p. 2
- High-frequency subcarriers should be assigned to close anchors and low-frequency subcarriers to far anchors, according to the distance-dependent frequency-ordering theorem.This matching rule quantifies the joint influence of propagation distance and carrier frequency on localization information.Source: §I · p. 2
A. Multi-carrier Underwater Signal Model
- Each anchor uses known pilots for TOA estimation and data during a separate interval, with anchors assigned disjoint subcarrier sets.The acoustic channel is frequency-selective across the system bandwidth but locally flat within each narrowband subcarrier; received signals also include colored ambient noise and external interference.Source: A. Multi-carrier Underwater Signal Model · pp. 2–3
B. Communication Rate
Communication rate depends jointly on subcarrier assignment, power, channel gain, and noise-plus-interference, creating a tradeoff with localization. Localization uses TOA estimates from anchor signals, with delays tied to acoustic propagation paths rather than simply to Euclidean distance.
B. Communication Rate
- Communication-oriented allocation may favor subcarriers with high channel gain and low noise-plus-interference, which need not match the localization-oriented allocation.Power and assignment jointly determine data rate; the per-anchor rate must satisfy Rm ≥ Rmin.Source: B. Communication Rate · p. 3
C. TOA and Localization Model
- TOA estimates provide the observations for positioning, but their accuracy depends on the individual delay estimates and anchor geometry.The received signals are affected by noise, interference, and frequency-selective channel attenuation.Source: C. TOA and Localization Model · p. 3, 1) TOA Measurement · p. 4
1) TOA Measurement:
- TOA is determined by the actual acoustic propagation path, which may be curved by sound-speed refraction, and ray tracing converts the estimated delay into propagation distance.The path is calculated using the sound-speed profile, anchor positions, and AUV depth.Source: 1) TOA Measurement · p. 4
2) TOA-to-Range Transformation:
The positioning model converts TOA into range using an equivalent sound speed and represents the unknown AUV position relative to known anchors. Its Fisher-information analysis accounts for frequency-dependent channel response and colored disturbance when assessing delay-estimation accuracy.
2) TOA-to-Range Transformation:
- The TOA-to-range transformation expresses one-way propagation distance as dm = ceff,mτm and uses ray tracing under a known layered sound-speed profile.The ray path is represented by segments whose sound speeds and depth intervals determine the propagation distance.Source: 2) TOA-to-Range Transformation · p. 4
3) Geometric Position Model:
- The geometric model uses known anchor positions and a nominal AUV position with Gaussian uncertainty rather than treating the nominal estimate as the true position.The uncertainty parameter σp characterizes the position uncertainty.Source: 3) Geometric Position Model · p. 4
1) Fisher Information Matrix:
- Whitening accounts for the frequency-varying disturbance PSD in the TOA Fisher-information analysis, whose information contribution depends on channel, disturbance, and signal spectra across allocated subcarriers.The resulting Fisher information yields a closed-form expression for propagation-delay information.Source: 1) Fisher Information Matrix · pp. 4–5
2) Closed-form TOA CRB:
The section derives a conditional TOA CRB and uses it to define localization information at the subcarrier level. It relates the accumulated ranging information of each anchor to positioning accuracy and geometry.
2) Closed-form TOA CRB:
- The conditional TOA CRB is the reciprocal of the TOA Fisher information when channel amplitudes and phases are known.The subcarrier-level metric uses a locally constant disturbance PSD and a flat-spectrum approximation.Source: 2) Closed-form TOA CRB · p. 5
- Localization information measures the positioning contribution of a subcarrier assigned to an anchor at a candidate AUV position.The metric is introduced to connect frequency allocation directly to localization performance.Source: 2) Closed-form TOA CRB · p. 5
- Positioning accuracy depends jointly on the accumulated TOA information from all anchors and their geometric configuration.The conditional TOA CRB depends on candidate AUV position through propagation distance and channel attenuation.Source: 2) Closed-form TOA CRB · p. 5
B. Position FIM, CRB and Metric
This section projects ranging information into the position domain and defines a Bayesian positioning error bound. Its information-matrix structure also motivates sequential anchor–subcarrier allocation.
B. Position FIM, CRB and Metric
- Independent anchors’ conditional measurement Fisher information matrices combine to form the information matrix for a candidate position.The position-domain information is obtained by projecting ranging information through the geometric observation model.Source: B. Position FIM, CRB and Metric · p. 6
- The Bayesian positioning error bound depends on localization information accumulated across anchors and their spatial geometry.The BPEB lower-bounds Bayesian root-mean-square position error averaged over prior position uncertainty and measurement observations.Source: B. Position FIM, CRB and Metric · p. 6
- Each anchor–subcarrier pair contributes a prior-averaged information matrix, motivating greedy selection by reduction in the Bayesian A-optimal objective.The prior-averaged matrices characterize how frequency and power allocations affect Bayesian positioning performance.Source: B. Position FIM, CRB and Metric · p. 6
C. Problem Formulation
The formulation minimizes Bayesian positioning error while enforcing communication QoS and resource constraints. It accounts for uncertainty in the nominal AUV position used for allocation.
C. Problem Formulation
- The objective minimizes Bayesian positioning error subject to communication QoS, orthogonal assignment, equal subcarrier allocation, and per-anchor power constraints.The formulation also requires non-negative power and binary assignment variables.Source: C. Problem Formulation · p. 7
- Resource allocation uses a prior distribution around the nominal AUV position rather than treating that position as true.The nominal position may come from a previous localization result, dead reckoning, or trajectory prediction; σ_p characterizes its uncertainty.Source: C. Problem Formulation · p. 7
IV. Localization-Oriented Joint Resource Allocation
The localization information metric decreases with propagation distance, while its frequency dependence reflects competing effects. Frequency improves localization information only when delay-resolution gain dominates absorption and disturbance-spectrum variation.
1) Distance Dependence:
- Localization information decreases monotonically with propagation distance, making distance a key ordering factor for allocation.The paper identifies this distance dependence as a guideline for distance-aware allocation strategies.Source: 1) Distance Dependence · p. 7
2) Frequency Dependence:
- The frequency dependence has an optimal carrier frequency that maximizes localization information for a given distance.The proposition defines the optimum as the frequency maximizing λ(f, d).Source: 2) Frequency Dependence · p. 7
- Higher carrier frequency improves localization information only when delay-resolution gain outweighs absorption loss and disturbance-spectrum variation.The frequency derivative reflects the balance among these effects.Source: 2) Frequency Dependence · p. 7
3) Frequency Ordering Principle:
The frequency-ordering result shows how relative localization information varies with frequency for anchors at different distances. It favors matching higher frequencies to nearer anchors and lower frequencies to farther ones.
3) Frequency Ordering Principle:
- The ordering provides a theoretical basis for matching frequency resources to anchors at different propagation distances.Source: 3) Frequency Ordering Principle · p. 7
- For anchors with d_i < d_j, the ratio of their localization information increases monotonically with frequency.This is the distance-dependent ordering property established by Theorem 1.Source: 3) Frequency Ordering Principle · p. 7
- Higher-frequency subcarriers provide a larger relative localization benefit to nearer anchors, while lower-frequency subcarriers are relatively more favorable for distant anchors.Source: 3) Frequency Ordering Principle · p. 8
B. Greedy Subcarrier Allocation with Local Power Refinement
The paper uses greedy subcarrier assignment followed by local power refinement to address a communication-constrained positioning objective. The allocation is evaluated by simulation and has polynomial-order complexity when sample and dimension counts are fixed.
B. Greedy Subcarrier Allocation with Local Power Refinement
- At each iteration, the greedy method selects the feasible anchor–subcarrier pair that most reduces the Bayesian A-optimal objective, then locally refines power under communication and power constraints.The subcarrier-assignment stage uses a reference equal-power allocation and prior-averaged information matrices.Source: B. Greedy Subcarrier Allocation with… · p. 8
C. Computational Complexity Analysis
- With D and N_s small and fixed, the overall complexity reduces to O(MK^2 + N_pK), the same polynomial order as the nominal-position implementation.The complexity analysis includes prior-information computation, greedy candidate evaluation, and local power refinement.Source: C. Computational Complexity Analysis · p. 8
A. Simulation Setup
- Simulations compare the A-optimal strategy with continuous and interleaved uniform, random, and high-frequency-first allocation using BPEB and effective communication throughput.Results generally average 100 independent Monte Carlo trials.Source: A. Simulation Setup · p. 8
B. Validation of Localization Information Properties
The simulations examine localization-information trends and joint allocation across frequency, distance, power, anchor geometry, and communication requirements. The proposed scheme has the lowest median BPEB across the tested anchor configurations, while stricter communication requirements can raise positioning error.
1) Frequency Effect:
- Localization information first rises and then falls with frequency as delay resolution competes with absorption loss, and its optimal frequency shifts lower as distance increases.Source: 1) Frequency Effect · p. 8, Figure 2 · p. 9, C. Performance of Joint Resource… · p. 9
2) Distance Effect:
- Localization information decreases with distance at all tested frequencies, with higher-frequency subcarriers degrading faster because of stronger acoustic attenuation.Source: 2) Distance Effect · p. 9
3) Power Effect:
- Localization information increases almost linearly with transmit power, indicating that power allocation should account jointly for propagation distance and carrier frequency.Source: 3) Power Effect · p. 9
1) Prior-Averaged Information Distribution:
- Under asymmetric geometry, uniform allocation yields unequal anchor contributions, whereas Bayesian allocation compensates for weak links using prior-averaged information matrices.Both schemes provide balanced information under symmetric geometry.Source: Figure 3 · p. 9, 1) Prior-Averaged Information… · p. 9
2) BPEB and Communication Feasibility:
- The proposed scheme achieves the lowest median BPEB under all three tested anchor configurations and reduces BPEB relative to four benchmarks for most position realizations.The comparison covers uniform, random, and high-frequency-first allocation schemes; the advantage does not necessarily grow monotonically with anchor count because instantaneous geometry and channel conditions also matter.Source: 2) BPEB and Communication Feasibility · pp. 9–10, Figure 4 · p. 10
- At low data-rate requirements, BPEB remains nearly unchanged; stricter requirements raise BPEB, and nine-subcarrier cases accommodate higher rates but have higher BPEB than corresponding six-subcarrier cases.Source: 2) BPEB and Communication Feasibility · p. 10, Figure 5 · p. 11
- Higher required frame-averaged throughput reduces available pilot time, and the shorter pilots reduce ranging information and increase simulated localization RMSE.The plotted communication rates are data-phase rates; effective frame-averaged rates equal the prescribed requirements under the adopted time-split rule.Source: 2) BPEB and Communication Feasibility · pp. 10–11, Figure 5 · p. 11
3) Practical Evaluation:
Across anchor layouts, the evaluation reports favorable positioning accuracy from adaptive resource allocation, close tracking between practical localization error and the theoretical BPEB, and communication and computational costs tied to localization resources.
3) Practical Evaluation:
- BPEB maps show favorable localization accuracy across the service area, with allocation gains particularly noticeable for asymmetric anchors, where allocation helps compensate for geometric imbalance.The maps cover three- and four-anchor symmetric layouts and a five-anchor asymmetric layout, with AUV depth fixed.Source: 3) Practical Evaluation · p. 11, Figure 6 · p. 12
- Low-frequency subcarriers tend to go to distant anchors and high-frequency subcarriers to nearby anchors in the symmetric three-anchor topology.The reported pattern is attributed to frequency-dependent propagation loss and acoustic absorption.Source: 3) Practical Evaluation · p. 11, Figure 7 · p. 12
- The matched-filter localization error follows the BPEB trend as received SNR rises, but remains above the bound across the SNR range.The gap is attributed mainly to geometric approximation, TOA discretization, and position-estimation nonlinearity.Source: 3) Practical Evaluation · pp. 11–12, Figure 8 · p. 12
D. Communication Performance Evaluation
- Effective communication throughput decreases approximately linearly as localization pilot duration increases, while higher received SNR yields greater throughput at the same duration.The reduction comes from less time for data transmission, not deterioration in channel quality.Source: Figure 8 · p. 12, D. Communication Performance Evaluation · p. 12
E. Computational Complexity Analysis
- The proposed method costs more computation than heuristic baselines, but its average runtime stays below 10 ms with 12 subcarriers.High-frequency-first has the lowest complexity because it uses sorting without iterative optimization.Source: Table II · p. 13, E. Computational Complexity Analysis · p. 13
VI. Conclusion
- The paper formulates allocation as A-optimal optimization with communication constraints and proposes a low-complexity greedy algorithm with power refinement.The conclusion reports improved localization accuracy over conventional schemes while maintaining communication performance.Source: §VI · p. 13