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2D Coprime Pilots for Delay-Doppler Sensing in OFDM-ISAC Systems
Guanping Shang, Zeyan Zhuang, Anzheng Tang, Shenghui Song, Chi-Ying Tsui
TL;DR
Sparse pilots can reduce communication overhead, but jointly optimizing unambiguous sensing region and accuracy remains difficult without a unified metric and flexible design framework. The paper uses SMI, derives a 2D coprime condition, and applies AGIR; numerical results report advantages over existing designs in sensing region and accuracy.
Problem
Sparse-pilot designs lack a unified metric and framework for jointly optimizing unambiguous sensing region and delay-Doppler estimation accuracy.
Source: A. Background · p. 1, B. Related Works and Motivations · p. 1Method
The paper uses SMI as a unified metric, derives a necessary-and-sufficient 2D coprime condition, and proposes AGIR to optimize accuracy under that condition.
Source: C. Contributions · p. 2, B. Related Works and Motivations · p. 2Results
The proposed pilot patterns outperform existing designs in unambiguous sensing region and sensing accuracy.
Source: §VI · p. 11Takeaways & Limitations
The 2D coprime condition provides structural flexibility for optimizing sensing accuracy across different delay-Doppler parameter distributions.
Source: §VI · p. 11
Abstract
from arXiv · showhide
Integrated Sensing and Communication (ISAC) is envisioned to endow future 6G systems with seamless sensing capabilities. To support efficient sensing with minimum communication overhead, sparse pilots embedded within communication frames have emerged as a promising solution. Along this line of research, existing studies have achieved engaging results in maximizing the unambiguous sensing region. However, jointly maximizing the sensing region and sensing accuracy remains challenging due to the lack of a unified performance metric and an effective pilot design framework. This paper jointly optimizes the unambiguous sensing region and sensing accuracy for estimating delay-Doppler (DD) parameters in Orthogonal Frequency Division Multiplexing (OFDM)-ISAC systems, where sensing mutual information (SMI) is adopted as a unified performance metric to characterize the overall sensing capability. Specifically, the joint optimization is formulated as an SMI maximization problem by systematically resolving sensing ambiguity and optimizing sensing accuracy. In particular, based on the generalized Bezout identity, we derive a 2D (timefrequency) coprime condition, which, as far as the authors know, is the first necessary and sufficient condition to achieve the unique estimation of DD parameters in the literature. Under this unambiguous condition, we further propose an Adam-Guided Iterative Refinement (AGIR) algorithm to optimize the sensing accuracy. Numerical results demonstrate the advantage of the proposed framework over existing designs, owing to the freedom offered by the 2D coprime condition in optimizing the sensing accuracy.
I. INTRODUCTION
Sparse pilots limit communication overhead but impair delay-Doppler sensing, and prior designs lack a unified way to optimize sensing region and accuracy. The paper uses SMI, a 2D coprime condition, and AGIR to address this joint design problem in OFDM-ISAC.
A. Background
- Sparse pilots reduce sensing-resource use but can shrink the unambiguous delay-Doppler region and degrade parameter-estimation accuracy.The introduction identifies joint optimization of sensing region and accuracy as the design need.Source: A. Background · p. 1
B. Related Works and Motivations
- Prior work addressed ambiguity, but separate region and accuracy metrics and rigid pilot constraints leave joint optimization unresolved.The paper identifies SMI as a unified metric and proposes a more flexible pilot-design framework.Source: B. Related Works and Motivations · pp. 1–2
C. Contributions
- The paper uses SMI to measure both sensing region and accuracy, derives a necessary-and-sufficient 2D coprime condition, and proposes AGIR to optimize accuracy.The condition is based on the generalized Bézout identity; the BCRB serves as AGIR’s tractable accuracy surrogate.Source: C. Contributions · p. 2
D. Paper Structure and Notations
- The paper is organized around the system model, SMI analysis and problem formulation, 2D coprime condition and AGIR, numerical results, and conclusion.The notation section defines matrix and operator conventions used throughout.Source: D. Paper Structure and Notations · p. 2
II. SYSTEM MODEL
- The modeled system uses a dual-functional transmitter and sensing receiver for communication and sensing, with target-reflected echoes processed for parameter estimation.The system is bistatic and uses periodic sensing pilots embedded in OFDM frames.Source: §II · p. 2
A. Sensing Model
- The sensing model represents each OFDM frame as Bt × Bf resource blocks and uses an identical pilot pattern across blocks, indexed by subcarrier and symbol.The pilot set in one block contains P pilot symbols; delay and Doppler are the target parameters, and residual inter-carrier interference is included in the noise term.Source: A. Sensing Model · pp. 2–3
B. Sensing Region and Accuracy with Sparse Sampling
Sparse sampling creates both ambiguity in the delay-Doppler region and local estimation uncertainty. The section motivates SMI as a unified metric for the joint sensing problem.
B. Sensing Region and Accuracy with Sparse Sampling
- Sparse pilots limit observable delay and Doppler regions, creating indistinguishable parameter subregions and degrading local estimation accuracy.The paper describes ambiguity and local uncertainty as separate effects of sparse sampling.Source: B. Sensing Region and Accuracy with… · p. 3, Figure 2 · p. 3
III. SMI ANALYSIS AND PROBLEM FORMULATION
- Traditional metrics assess sensing region or accuracy separately, so the paper introduces SMI to measure their coupled contribution to overall sensing capability.SMI is the mutual information between received pilots YRS and delay-Doppler parameters θ.Source: B. Sensing Region and Accuracy with… · p. 3, §III · p. 3, A. Sensing Mutual Information · p. 3
A. Sensing Mutual Information
- The pilot-design problem maximizes SMI for a representative resource block subject to the available sensing-pilot count.The constraint is P = NRS.Source: A. Sensing Mutual Information · p. 4
B. The Unified Metric for Sensing Region and Accuracy
SMI captures both ambiguity and uncertainty within the unambiguous region. The paper therefore resolves ambiguity first, then optimizes sensing accuracy under the unambiguous constraint.
B. The Unified Metric for Sensing Region and Accuracy
- SMI decomposes into a penalty from the ambiguity set and uncertainty within the unambiguous region, so it reflects both sensing region and accuracy.The ambiguity set comprises alternative parameters with the same received-pilot likelihood as the ground truth.Source: B. The Unified Metric for Sensing… · p. 4
C. Problem Formulation
- As SNR rises, SMI approaches the entropy within the unambiguous region, but ambiguity leaves an irreducible entropy loss.Resolving ambiguity removes that loss; improving accuracy then allows SMI to approach its maximum.Source: C. Problem Formulation · p. 4, Figure 3 · p. 4
- The formulation first enforces an empty ambiguity set, then maximizes sensing accuracy under that unambiguous constraint.The objective U(P) measures accuracy as the inverse of local uncertainty.Source: C. Problem Formulation · p. 4
IV. COPRIME PILOTS FOR DELAY-DOPPLER SENSING
The framework resolves delay-Doppler ambiguity through intra-block pilot structure, then optimizes sensing accuracy under the resulting unambiguous design. It characterizes ambiguity with integer equations and pairwise pilot-index differences.
IV. COPRIME PILOTS FOR DELAY-DOPPLER SENSING
- The framework derives a 2D coprime condition for unambiguous sensing and proposes AGIR to optimize sensing accuracy under that condition.The section first characterizes ambiguity through integer equations for delay-Doppler parameters.Source: §IV · pp. 4–5, A. Coprime Condition for Resolving… · p. 5
1) Ambiguity Regions:
- Periodic resource-block sampling leaves multiple equivalent delay-Doppler locations, indexed by ambiguity integers, despite unique fractional components within the minimal unambiguous region.The ambiguity integers specify which sub-region contains the target.Source: 1) Ambiguity Regions · p. 5
2) Congruence Equations for Solving Ambiguity Integers:
- Intra-block pilot phases provide additional information for resolving the delay and Doppler ambiguity integers and attaining the maximum unambiguous sensing region.Pairwise phase differences of distinct pilots yield residual phases, which are organized as congruence equations using the pilot-index difference matrix.Source: 2) Congruence Equations for Solving… · p. 5
3) 2D Coprime Condition:
A generalized Bézout identity yields a necessary and sufficient 2D coprime condition for uniquely recovering ambiguity integers. Compared with orthogonal coprime designs, this condition permits more flexible pilot placement for accuracy optimization.
3) 2D Coprime Condition:
- The 2D coprime condition uniquely recovers ambiguity integers if and only if a subset of pilot difference vectors has cross-determinants and periodicity terms with greatest common divisor 1.The generalized Bézout identity establishes the condition; the resulting integer combinations synthesize the finest symbol and subcarrier resolution grid.Source: 3) 2D Coprime Condition · p. 6
4) Comparison with Orthogonal Coprime:
- Orthogonal coprime designs impose a sufficient but unnecessary condition on the two axes, whereas the 2D condition allows more pilot-design freedom for sensing-accuracy optimization.The orthogonal design uses coprime intervals across subcarriers and symbols; CRT-based designs instead use coprime sampling intervals across the frame in time or frequency.Source: 4) Comparison with Orthogonal Coprime · p. 6
B. An Insight about Sparse Sampling
The ambiguity function reveals how sparse sampling creates aliased responses and how pilot patterns shape delay-Doppler resolution. For sensing-accuracy optimization, the paper uses a log-determinant BCRB surrogate for SMI and derives the bound from the Bayesian Fisher information matrix.
B. An Insight about Sparse Sampling
- Sparse sampling creates aliased ambiguity-function lobes with responses identical to the target's main lobe, while lobe resolution reflects sensing accuracy.Coupled sampling produces an elliptical contour; its principal axes indicate the finest and coarsest resolution, and the 2D coprime ridge yields better accuracy for correlated delay-Doppler parameters than the orthogonal-coprime axes.Source: B. An Insight about Sparse Sampling · pp. 6–7
C. Sensing Accuracy Evaluation
- Because SMI lacks a closed-form expression, the paper uses the log-determinant BCRB as a tractable accuracy surrogate that is asymptotically tight as SNR increases.The BCRB lower-bounds estimation error and is defined through the inverse Bayesian Fisher information matrix.Source: C. Sensing Accuracy Evaluation · p. 7
- The likelihood Fisher information is calculated from retained received-pilot observations under a zero-mean complex Gaussian propagation-coefficient model independent of the delay-Doppler parameters.The noise is modeled as complex Gaussian; the likelihood Fisher information entries depend on centered second-order statistics of intra-block pilot indices, not on the parameter values.Source: C. Sensing Accuracy Evaluation · p. 7
D. Sensing Accuracy Optimization
The sensing-accuracy problem is a non-convex integer program under coprime and guard-interval constraints. AGIR combines continuous Adam-guided exploration with projection and discrete refinement to optimize pilot patterns.
D. Sensing Accuracy Optimization
- Coprime and integer constraints on block scales and guard intervals make the optimization problem a non-convex integer program.The problem is reformulated as BCRB optimization subject to pilot overhead and 2D coprime constraints.Source: D. Sensing Accuracy Optimization · p. 7
- AGIR relaxes discrete constraints into continuous space so objective gradients guide the pilot pattern toward a promising neighborhood.It projects the explored solution to valid integer patterns, then locally refines pilot indices to recover performance lost in projection.Source: D. Sensing Accuracy Optimization · p. 8
- AGIR alternates Adam exploration, projection onto valid patterns, and local discrete search to return an optimized coprime pilot pattern.Projection enforces valid-pattern constraints; discrete refinement searches local integer neighborhoods while holding other pilots fixed.Source: D. Sensing Accuracy Optimization · p. 8
- AGIR has complexity O(T_epoch · P · (N_cN_s)), matching Greedy Search’s linear complexity while avoiding suboptimal traps through continuous gradient guidance.Exhaustive Search has exponential complexity O((N_cN_s)^P).Source: D. Sensing Accuracy Optimization · p. 8
V. SIMULATION RESULTS
The simulations compare pilot patterns using ambiguity-function behavior and sensing mutual information across parameter distributions and operating conditions. The 2D Coprime pattern performs better than Orthogonal Coprime for joint Gaussian delay-Doppler parameters, while patterns with unresolved ambiguity incur SMI losses.
A. Comparison of AFs for different patterns
- Full Sampling gives a sharp, sidelobe-free ambiguity response at high overhead; Uniform Sampling produces identical lobes that prevent identifying true target parameters.The four sparse patterns use the same number of pilots, NRS = 4.Source: A. Comparison of AFs for different… · p. 8, Figure 8 · p. 9
- Orthogonal Coprime has sidelobes below its main lobe, whereas 2D Non-Coprime has identical lobes that cause sensing ambiguity.The Orthogonal Coprime pattern is suitable only for independent DD parameters.Source: A. Comparison of AFs for different… · pp. 8–9
B. SMI Performance Comparison
- The SMI comparisons cover SNR, pilot overhead, and resource-block size; the baseline evaluation uses NRS = 4 in an 8 × 8 block.SMI is evaluated using discrete Monte Carlo simulation.Source: B. SMI Performance Comparison · p. 9, Figure 10 · p. 10
1) SMI versus SNR:
- Numerical results favor the proposed framework over baseline designs; under joint Gaussian DD parameters, 2D Coprime outperforms Orthogonal Coprime.The comparison is shown in Fig. 9b.Source: Figure 9 · p. 9, 1) SMI versus SNR · p. 9
- Uniform Sampling loses ln(4 × 4) ≈ 2.77 nats of SMI as it shrinks the unambiguous region to 1/4 for both delay and Doppler.In the independent, uniform DD case, coprime patterns reach optimal SMI at high SNR.Source: 1) SMI versus SNR · p. 9, Figure 9 · p. 9
- Patterns that violate the coprime constraint consistently suffer severe SMI loss, supporting ambiguity as a dominant factor that sets an SMI upper limit.At low SNR, sacrificing sensing region for accuracy may yield only a marginal SMI increase, so the design resolves ambiguity before optimizing accuracy.Source: 1) SMI versus SNR · pp. 9–10
2) SMI versus Sensing Overhead:
The 2D Coprime design combines maximum unambiguous sensing with accuracy close to the D-optimal pattern, while outperforming Orthogonal Coprime in the reported comparisons. Its flexibility helps adapt to varying delay-Doppler distributions, though larger resource blocks increase computational complexity.
2) SMI versus Sensing Overhead:
- The 2D Coprime pattern achieves the highest SMI across pilot overheads, while D-optimal has much lower SMI because it fails to resolve sensing ambiguity.Larger resource blocks reach maximum SMI at considerably lower pilot overhead, at the expense of increased computational complexity.Source: 2) SMI versus Sensing Overhead · p. 10
1) Sensing Accuracy versus SNR:
- The 2D Coprime pattern achieves significantly lower MSE than Orthogonal Coprime and closely approaches D-optimal accuracy while retaining the maximum unambiguous sensing region.Fig. 11 reports delay and Doppler MSE versus SNR; D-optimal MSE is evaluated only locally within its unambiguous region.Source: 1) Sensing Accuracy versus SNR · p. 10, Figure 11 · p. 10
2) The Structural Flexibility of 2D Coprime:
- As the DD covariance becomes more anisotropic, 2D Coprime adapts its equivalent FIM to align with the covariance comparably to D-optimal, unlike rigid Orthogonal Coprime.Fig. 12 visualizes the alignment as condition number κ increases from 1.0 to 64.0.Source: 2) The Structural Flexibility of 2D… · p. 10, Figure 12 · p. 11
- Across condition numbers κ, 2D Coprime accuracy approaches the D-optimal pattern, while Orthogonal Coprime incurs substantial MSE loss from its rigid structure.The κ sweep evaluates robustness across DD parameter covariances.Source: 2) The Structural Flexibility of 2D… · pp. 10–11
- As the covariance rotation angle ϕ varies, all patterns show periodic MSE fluctuations and severe degradation at some mismatch angles, while 2D Coprime remains near the D-optimal limit.The reported degradation around 90°–150° is attributed to resource-block sensing blind spots; Fig. 13 depicts robustness versus κ and ϕ.Source: 2) The Structural Flexibility of 2D… · p. 11, Figure 13 · p. 11
VI. CONCLUSION
- The proposed pilot patterns outperform existing designs in unambiguous sensing region and sensing accuracy, with flexibility to optimize for different delay-Doppler distributions.The maximum unambiguous sensing region is strictly guaranteed.Source: §VI · p. 11
APPENDIX A PROOF OF THEOREM 1
The proof recasts unique recovery of the ambiguity integers as an integer-span question. The generalized Bézout and Smith normal form criteria yield a gcd condition for the lattice to span Z2.
APPENDIX A PROOF OF THEOREM 1
- Unique recovery of the ambiguity integers is equivalent to whether the linear system admits integer solutions isolating the target variables.For qτ, the proof seeks the unit vector [1, 0]T in the integer span.Source: Appendix A · p. 11
- The generalized Bézout identity requires the integer lattice generated by the constituent vectors to cover Z2.The proof characterizes this lattice with an augmented integer matrix.Source: Appendix A · p. 11
3) Periodicity-periodicity minor:
- The lattice spans Z2, and thus uniquely resolves the ambiguity integers, if and only if the gcd of all 2 × 2 minors of the augmented matrix is 1.The proof identifies this criterion using the Smith normal form.Source: Appendix A · p. 11, 3) Periodicity-periodicity minor · p. 12
APPENDIX B SMI COMPUTATION
The SMI computation separates conditional and marginal entropy calculations. It models received pilots over a finite delay-Doppler grid and estimates the marginal entropy numerically using Monte Carlo samples.
APPENDIX B SMI COMPUTATION
- SMI is computed by separately evaluating conditional entropy h(yRS|θ) and marginal entropy h(yRS), since direct calculation is intractable.The numerical method computes the conditional entropy and estimates the marginal entropy by Monte Carlo.Source: Appendix B · p. 12
1) Conditonal Distribution:
- The computation assumes delay and Doppler parameters lie on finite grids, with the possible channel states occurring with equal probability.The total number of states is T = Lτ × Lν, each with probability 1/T.Source: 1) Conditonal Distribution · p. 12
- For each grid state, the procedure constructs and stores the steering-vector-based pilot signal quantities, then generates Monte Carlo samples for marginal-entropy estimation.The received signal is modeled as complex Gaussian conditional on the DD parameters.Source: 1) Conditonal Distribution · p. 12
2) Conditional Entropy Calculation:
The appendix evaluates the conditional and marginal entropy components used to compute SMI. Because the marginal-entropy expectation has no closed form, it is approximated with Monte Carlo sampling.
2) Conditional Entropy Calculation:
- The conditional entropy h(yRS|θ) is computed as one component of the SMI calculation.The appendix introduces its calculation separately from the marginal entropy.Source: 2) Conditional Entropy Calculation · p. 12
3) Marginal Entropy approximation:
- The marginal entropy requires an expectation over the received-signal density, and the expectation has no closed form.The marginal PDF is used in this expectation.Source: 3) Marginal Entropy approximation · pp. 12–13
- Monte Carlo sampling approximates the marginal entropy using Cholesky-generated samples, while Log-Sum-Exp accumulates the marginal PDF in the logarithmic domain to prevent underflow.SMI is computed as the difference between marginal and conditional entropies.Source: 3) Marginal Entropy approximation · p. 13