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ISAC with Co-Prime Arrays: Virtual-Aperture Sensing and uplink downlink communications

Jing Zhang, Yuxiao Liu, Jiayi Sun, Junliang Ye, Derrick Wing Kwan Ng

arXiv:2609.01979v1cs.IT

TL;DR

ISAC UAV networks face limited physical aperture and residual self-interference during full-duplex sensing, motivating improved shared hardware and resource use. The paper embeds a sparse CPA in a ULA grid for sensing, reuses remaining positions for TDD communication, analyzes CRB and sampling tradeoffs, and jointly optimizes sensing and communication designs. The CPA offers stronger asymptotic sensing performance than a partitioned ULA, while simulations show consistent gains over considered baselines.

  • Problem

    ISAC UAV networks are constrained by physical antenna aperture and residual self-interference in full-duplex sensing.

  • Method

    The paper embeds a sparse CPA in a ULA grid for FD sensing, reuses remaining antenna positions for TDD communication, and jointly optimizes sensing covariance, downlink precoding, and uplink receive beamforming.

  • Results

    The CPA provides stronger asymptotic sensing performance than the partitioned ULA in single-target and nondegenerate multi-target scenarios, with consistent simulation gains over baselines.

  • Takeaways & Limitations

    The shared aperture combines CPA virtual-aperture sensing with bidirectional communication, while sensing gains generally require additional temporal snapshots under the same physical aperture.

Abstract

from arXiv · show

Integrated sensing and communication (ISAC) enables simultaneous communication and environmental sensing in unmanned aerial vehicle (UAV) networks, but its performance is constrained by the physical antenna aperture and residual self-interference (SI) in full-duplex (FD) sensing. To address these issues, we propose a shared-aperture ISAC architecture in which a sparse co-prime array (CPA) is embedded in a uniform linear array (ULA) grid for FD sensing, while the remaining antenna positions support time-division duplexing (TDD) communication. We characterize the sensing performance through an order-wise Cramer-Rao bound (CRB) analysis, showing that the CPA achieves a stronger asymptotic sensing gain than the partitioned ULA benchmark in both single-target and nondegenerate multi-target scenarios. We further reveal a space-time sampling tradeoff under the same physical aperture. Based on the proposed architecture, we formulate a non-convex joint resource allocation problem that maximizes the weighted downlink-uplink sum rate by jointly designing the sensing transmit covariance, downlink precoder, and uplink receive beamformers under sensing accuracy, BS transmit-power, communication QoS, and residual SI constraints. An alternating-optimization-based algorithm is developed. Simulations demonstrate consistent performance gains over the considered baselines and confirm the complementary benefits of the CPA virtual aperture and sensing covariance optimization.

I. INTRODUCTION

The paper proposes a shared-aperture ISAC architecture combining CPA-based full-duplex sensing with TDD communication on one physical aperture. It analyzes sensing–sampling tradeoffs and jointly optimizes sensing and communication resources under accuracy, power, QoS, and residual-SI constraints.

  • Architecture: The shared-aperture design embeds a sparse CPA in a common ULA grid and reuses remaining antenna positions for TDD communication.The architecture supports continuous FD sensing alongside downlink transmission and uplink reception without a dedicated communication array.
  • Sensing analysis: The order-wise CRB analysis shows stronger asymptotic CPA sensing performance than a partitioned ULA in single-target and nondegenerate multi-target scenarios.The comparison provides theoretical support for CRB improvement from the enlarged virtual aperture.
  • Sensing analysis: Under a fixed physical aperture, CPA spatial sparsity trades fewer sensing elements for generally more temporal snapshots to compensate estimation-accuracy loss relative to a same-aperture ULA.This establishes a space–time sampling tradeoff.
  • Resource allocation: The joint resource-allocation problem maximizes weighted downlink–uplink sum rate by optimizing sensing, downlink, and uplink designs under CRB, power, QoS, and residual-SI constraints.The AO-based solution combines fractional programming, closed-form uplink beamforming, convex downlink optimization, and semidefinite programming for sensing covariance.
  • Sensing architecture: With M_s=M_1+M_2−1 physical sensing positions, the CPA synthesizes M_1M_2 distinct virtual positions and enlarges the sensing aperture.The virtual aperture improves angular resolution without requiring a fully populated sensing array.

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The signal model describes sensing echoes and communication signals sharing a full-duplex, co-located platform, with residual self-interference and aggregate communication leakage included at the receivers.

  • Communication signal: Communication transceiver antennas use linear downlink precoding and uplink receive beamforming for the TDD communication links.Downlink data symbols and uplink symbols are transmitted or detected through the communication antenna set.
  • Sensing signal: The BS transmits sensing waveforms through an M_1-element subarray and receives target echoes through an M_2-element subarray.Targets are modeled as far-field point targets characterized by their angles and sensing steering vectors.
  • Sensing signal: Residual sensing self-interference is modeled as H_SI,s x_s(n) after imperfect cancellation, capturing leakage from the sensing transmit subarray into its receive subarray.The residual channel represents effects including imperfect estimation, hardware impairments, nonlinear distortion, and multipath leakage.
  • Interference model: The sensing receiver also accounts for aggregate leakage from uplink and downlink communication signals through a covariance R_ell,s.Sensing receiver noise is modeled as circularly symmetric complex Gaussian noise.
  • Communication signal: Concurrent sensing transmission introduces residual SI into uplink reception, alongside uplink signals and receiver noise.The uplink model includes the sensing-to-communication residual-interference channel H_SI,c x_s(n).

D. System Performance

The system evaluates downlink and uplink communication rates through SINR-based expressions and combines them using fixed directional weights into a weighted sum-rate objective.

  • Downlink rate: The downlink SINR of each communication UAV determines the downlink sum rate.The rate expression follows from the downlink received-signal model.
  • Uplink rate: The uplink SINR of each communication UAV determines the corresponding uplink sum rate.The expression uses the uplink received-signal and receive-beamforming model.
  • Weighted objective: R_sum=α_DLR_DL+α_ULR_UL combines downlink and uplink rates using fixed nonnegative weights.The weights balance the two communication directions.

2) CRB-Based Sensing Performance:

The CPA sensing model uses a virtual array manifold and an equivalent Fisher information matrix after eliminating nuisance reflection parameters. Its CRB analysis shows stronger asymptotic sensing behavior than the partitioned ULA, alongside a snapshot tradeoff under equal physical aperture.

  • CRB formulation: The sensing model collects L received snapshots and represents the CPA response through a virtual array manifold of dimension M1M2.The effective interference-plus-noise vector includes communication leakage, residual self-interference, and sensing receiver noise.
  • CRB formulation: The angle-related CRB is obtained from the equivalent FIM after eliminating complex target reflection coefficients as nuisance parameters via a Schur complement.The full parameter vector contains target angles and the real and imaginary parts of the reflection coefficients.
  • Order-wise comparison: The CPA produces a narrower normalized correlation mainlobe than the partitioned ULA for the same sensing transmit and receive antenna counts.The transmit-target and target-receive phase paths create the relevant virtual-aperture response for angular discrimination.
  • Order-wise comparison: For equal sensing-element counts and fixed per-transmit-antenna power, the CPA has faster asymptotic CRB decay than the partitioned ULA in the single-target analysis.The comparison attributes this advantage to the enlarged virtual aperture created by the CPA geometry.
  • Order-wise comparison: The multi-target CRB trace also receives an order-wise CPA analysis for fixed nondegenerate target placement and fixed per-transmit-antenna power.The analysis uses the full Fisher information matrix after eliminating nuisance reflection parameters.
  • Space-time tradeoff: Under the same physical aperture, the CPA generally needs more snapshots than the ULA to attain the same spatial-frequency CRB.The additional snapshots compensate for estimation-accuracy loss caused by sparse spatial sampling despite the CPA’s larger effective aperture.
  • Space-time tradeoff: When ρ = 0.5, the balanced transmit-receive allocation yields lower CRB than ρ = 0.2 because it maximizes M1M2 and enriches the virtual array.The resulting larger effective aperture improves angular resolution.

B. Problem Formulation

The paper formulates a joint resource-allocation problem for TDD communication and FD sensing. It maximizes weighted downlink-uplink sum rate while enforcing sensing, power, QoS, and residual-SI requirements.

  • Objective: The objective jointly designs sensing, downlink, and uplink variables to maximize the weighted communication sum rate.The optimization variables are the sensing precoder, downlink communication precoder, and uplink receive beamformers.
  • Constraints: The constraints enforce angle-related sensing accuracy, total BS transmit power, residual SI limits, and downlink and uplink QoS.The sensing waveform and downlink communication signals share the BS transmit-power budget.
  • Non-convexity: The problem is non-convex because the communication rates, fractional SINR constraints, and CRB constraint are coupled through the sensing and communication designs.The sensing precoder affects downlink interference, uplink residual SI, and sensing accuracy.

IV. PROBLEM SOLUTION

The solution reformulates the communication rates using fractional programming and then applies alternating optimization. The algorithm updates auxiliary variables, communication beamformers, and sensing covariance iteratively until convergence.

  • Solution strategy: Fractional programming provides an equivalent reformulation of the downlink and uplink rate terms.This reformulation enables tractable updates within the iterative procedure.
  • Solution strategy: The alternating-optimization algorithm successively updates FP auxiliary variables, communication beamformers, and sensing transmit covariance until convergence.The updates separate the coupled design variables into successive optimization steps.

A. FP-Based Rate Reformulation

The FP-based reformulation yields alternating updates for auxiliary variables, uplink receivers, downlink precoders, and sensing covariance. The downlink subproblem becomes a convex quadratically constrained problem under fixed auxiliary variables.

  • FP reformulation: The weighted downlink-uplink sum rate is rewritten in an equivalent FP form using auxiliary variables.The optimal auxiliary variables have closed-form updates for fixed physical design variables.
  • Alternating optimization: The overall AO procedure alternates updates of FP variables, communication beamformers, and sensing covariance.This combines closed-form uplink updates with tractable downlink and sensing-design subproblems.
  • Communication updates: For fixed sensing precoding, the uplink receive beamformers are updated in closed form before optimizing the downlink precoder.The receive update maximizes the uplink SINR and preserves the uplink QoS condition for a feasible iterate.
  • Communication updates: The downlink SINR constraints are recast using a phase choice and second-order-cone representation.The desired downlink signal can be chosen real and nonnegative without loss of optimality.
  • Communication updates: For fixed FP auxiliary variables, the downlink subproblem is convex because its objective is concave and its power and QoS constraints define a convex feasible set.It can therefore be solved with standard convex optimization tools.

C. Sensing Transmit Design Optimization with Fixed Communication Beamformers

With communication beamformers fixed, the sensing transmit design is optimized through covariance reformulation, convex relaxation, and integration into an alternating-optimization procedure.

  • Sensing transmit design: The sensing transmit design subproblem updates the sensing covariance while accounting for communication interference, residual SI, sensing accuracy, and transmit-power constraints.The transformed weighted sum-rate objective reduces to minimizing weighted sensing-interference and residual-SI terms after omitting common and independent terms.
  • Sensing transmit design: Algorithm 1 successively updates auxiliary variables, uplink and downlink communication beamformers, and sensing covariance until convergence.The per-iteration cost is dominated by communication-beamformer updates and the SDP sensing-covariance subproblem.
  • Sensing transmit design: The CRB constraint is conservatively reformulated using an auxiliary matrix Tθ, with Eθ selecting angle-related parameters from the complete parameter vector.
  • Sensing transmit design: Dropping the rank constraint on the lifted sensing covariance yields a relaxed positive-semidefinite sensing-design problem.The relaxation replaces the implicit rank(Rs) ≤ ds condition with Rs ⪰ 0.
  • Sensing transmit design: The relaxed sensing subproblem is an SDP that can be efficiently solved using standard convex optimization tools.
  • Sensing transmit design: An exact sensing precoder is recovered from a compact eigenvalue decomposition when the relaxed covariance has rank at most ds; otherwise, Gaussian randomization produces a feasible rank-ds precoder.This recovery is incorporated into the overall AO procedure.

c + CSOCP + CSDP]

The overall per-iteration complexity combines beamformer updates with SOCP and SDP subproblems, with the SDP typically dominating computational cost.

  • The overall complexity scales with the AO iteration count and the costs of uplink beamforming, SOCP, and SDP updates.Typically, the SDP subproblem P3 dominates the computational cost.

V. SIMULATION RESULTS

Simulations evaluate the shared-aperture ISAC design across power, sensing-accuracy, residual-SI, field-of-view, and target-count conditions against several array and sensing baselines. The proposed CPA design consistently offers favorable sum-rate and feasibility behavior, supported by both virtual-aperture geometry and optimized sensing covariance.

  • Simulation setup: The study benchmarks the proposed CPA design against partitioned ULA, isotropic CPA, and minimum-redundancy array configurations under matched antenna resources.The baselines compare sensing accuracy, communication throughput, and hardware-related array tradeoffs.
  • Convergence: Algorithm 1 rapidly increases system sum rate during the first few iterations before reaching a stable value across tested power and antenna configurations.At the same transmit-power level, M = 17 achieves higher sum rate than M = 10.
  • Power and CRB threshold: The proposed CPA scheme achieves the highest sum rate across the entire maximum-power range and consistently outperforms the MRA baseline.Relaxing the CRB threshold from ΓCRB = 2 × 10^-6 rad2 to ΓCRB = 3 × 10^-6 rad2 places the latter curves higher at equal transmit power.
  • Power and CRB threshold: As the CRB threshold increases, sum rate improves because stricter sensing accuracy requirements consume more communication resources; some schemes are infeasible at tight thresholds.The proposed CPA with M = 17 maintains a nearly constant sum rate across the tested threshold range, while M = 10 remains feasible under relatively stringent thresholds.
  • Residual SI: Sum rate decreases monotonically as residual SI increases, while a looser ΓCRB = 5 × 10^-6 rad2 threshold yields higher sum rate at the same SI level.The decrease is attributed to stronger sensing-transmission leakage increasing interference at sensing-receive and communication antennas.
  • Field of view and target count: Under increasing sensing field of view or target count, the CPA remains more feasible and higher-performing than same-size baselines, whereas the partitioned ULA degrades most and can become infeasible.The results associate this robustness with the CPA’s larger virtual aperture and, beyond geometry alone, optimized sensing covariance.
  • Overall findings: The conclusion reports consistent baseline gains and favorable communication performance under increasingly demanding sensing and interference conditions.It also identifies a space-time sampling tradeoff: sparse spatial sampling generally requires additional temporal snapshots under the same physical aperture.

APPENDIX A PROOF OF THEOREM 1

The appendix derives the single-target CPA sensing CRB by evaluating Fisher-information terms for the virtual manifold and compares its leading-order behavior with a partitioned ULA benchmark.

  • Single-target CRB derivation: For a single target, the sensing mean is differentiated with respect to spatial frequency and complex reflectivity to form the relevant Fisher-information blocks.The nuisance parameters are βR and βI, and a Schur complement removes their effect when obtaining the angle-related information.
  • Asymptotic assumptions: The asymptotic analysis assumes fixed per-sensing-antenna power and a full-rank, uniformly well-conditioned sensing covariance as the array grows.Under these assumptions, tr(Rs) = Θ(M1), so total sensing transmit power grows linearly with the number of sensing transmit antennas.
  • Single-target CRB derivation: The CPA transmit and receive steering vectors use co-prime spatial spacings, producing a corresponding virtual manifold for the sensing model.
  • Single-target CRB derivation: The projected Fisher-information term is evaluated through three inner products involving the virtual manifold and its derivative.The derivation uses the mixed-product property of the Kronecker product.
  • ULA comparison: The partitioned ULA benchmark is analyzed using the same asymptotic sequence, with analogous projected-manifold expressions leading to its order-wise comparison.

APPENDIX B PROOF OF THEOREM 2

Appendix B proves the multi-target CPA CRB scaling by showing that, under nondegenerate target spacing and well-conditioned sensing covariance, diagonal FIM terms dominate cross-target terms after Schur reduction. The same order-wise argument is then applied to the partitioned ULA benchmark.

  • CPA multi-target analysis: The multi-target parameter vector contains all target spatial frequencies and reflection-coefficient components, enabling a block-partitioned mean-based FIM analysis.The spatial-frequency CRB is obtained using block-matrix inversion and Schur-complement reduction.
  • CPA multi-target analysis: For fixed sensing-target count, full-rank uniformly well-conditioned sensing covariance changes quadratic-form scalings only by multiplicative constants as the sensing-array size grows.Fixed per-antenna sensing power gives tr(R_s) = Θ(M_1), while eigenvalues remain uniformly bounded above and below.
  • CPA multi-target analysis: Nondegenerate target spacing keeps pairwise spatial-frequency differences away from CPA aliasing singularities, so inter-target oscillatory sums do not add coherently.Consequently, all inter-target cross terms are asymptotically weaker than their corresponding diagonal terms.
  • CPA multi-target analysis: Each diagonal entry of the reduced spatial-frequency FIM inherits the single-target projected-term scaling after nuisance reflection coefficients are eliminated.Cross-target contributions entering a diagonal are lower-order, so the leading order is preserved.
  • CPA multi-target analysis: Schur-complement corrections cannot increase the order of off-diagonal terms, leaving them lower-order than the diagonal principal terms.The reduced FIM is therefore diagonally dominated for sufficiently large sensing-array size, and its inverse follows the inverse order of those diagonal terms.
  • Partitioned ULA benchmark: The partitioned ULA benchmark follows the same proof pattern when target placement stays away from ULA degeneracy points, yielding the stated inverse-order CRB scaling.Its off-diagonal entries remain two orders weaker than the diagonal entries, and combining the benchmark bounds completes the theorem proof.
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