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Optimal Transmit Beamforming for Integrated Sensing and Communication

Haocheng Hua, Jie Xu, Tony Xiao Han

arXiv:2104.11871v4cs.IT

TL;DR

The paper asks how downlink ISAC can jointly support multiuser communication and radar sensing despite dedicated-radar interference and beampattern-design trade-offs. It formulates receiver-aware beamforming problems and solves them globally using SDR tightness proofs. Minimum weighted beampattern gain maximization improves sensing with lower complexity, and Type-II receivers provide better sensing than Type-I and conventional designs.

  • Problem

    Dedicated radar signals add sensing degrees of freedom but can interfere with communication, while conventional SINR-constrained beampattern matching has sensing trade-offs.

  • Method

    The paper jointly designs information and dedicated radar beamforming for Type-I and Type-II receivers under SINR and transmit-power constraints, using SDR with tightness proofs.

  • Results

    Minimum weighted beampattern gain maximization yields enhanced sensing and significantly lower execution time than beampattern matching, while Type-II receivers outperform Type-I and conventional designs.

  • Takeaways & Limitations

    Known-radar-signal interference cancellation and dedicated radar signaling can improve ISAC sensing, with radar signals always beneficial for Type-II receivers but unnecessary for Type-I receivers in special LOS channels.

Abstract

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This paper studies the transmit beamforming in a downlink integrated sensing and communication (ISAC) system, where a base station (BS) equipped with a uniform linear array (ULA) sends combined information-bearing and dedicated radar signals to simultaneously perform downlink multiuser communication and radar target sensing. Under this setup, we maximize the radar sensing performance (in terms of minimizing the beampattern matching errors or maximizing the minimum weighted beampattern gains), subject to the communication users' minimum signal-to-interference-plus-noise ratio (SINR) requirements and the BS's transmit power constraints. In particular, we consider two types of communication receivers, namely Type-I and Type-II receivers, which do not have and do have the capability of cancelling the interference from the {\emph{a-priori}} known dedicated radar signals, respectively. Under both Type-I and Type-II receivers, the beampattern matching and minimum weighted beampattern gain maximization problems are globally optimally solved via applying the semidefinite relaxation (SDR) technique together with the rigorous proof of the tightness of SDR for both Type-I and Type-II receivers under the two design criteria. It is shown that at the optimality, radar signals are not required with Type-I receivers under some specific conditions, while radar signals are always needed to enhance the performance with Type-II receivers. Numerical results show that the minimum weighted beampattern gain maximization leads to significantly higher beampattern gains at the worst-case sensing angles with a much lower computational complexity than the beampattern matching design. We show that by exploiting the capability of canceling the interference caused by the radar signals, the case with Type-II receivers results in better sensing performance than that with Type-I receivers and other conventional designs.

I. INTRODUCTION

The paper develops downlink ISAC beamforming that jointly supports multiuser communication and radar sensing, while addressing radar-signal interference and sensing-design limitations. It proposes receiver-aware formulations, globally optimal SDR-based solutions, and numerical comparisons of sensing performance and complexity.

  • Motivation: ISAC jointly uses wireless infrastructures for radar sensing and communication, exploiting multiple-antenna spatial degrees of freedom for multi-target and multi-user operation.The paper motivates transmit waveform and beamforming design as essential for simultaneous sensing and communication.
  • Research gap: Dedicated radar signals provide sensing degrees of freedom but can introduce harmful communication interference, motivating exploitation of their a-priori known sequences.The paper identifies interference cancellation as an opportunity not previously investigated in the cited literature.
  • Receiver designs: Type-II receivers cancel known radar-signal interference before decoding, whereas Type-I receivers lack this capability.The receiver distinction is central to the paper’s joint beamforming analysis.
  • Design criteria: The paper formulates beampattern matching and minimum weighted beampattern gain maximization under users’ minimum SINR constraints.The second criterion targets improved sensing at angles of interest while differing from conventional beampattern matching.
  • Optimization: SDR yields globally optimal solutions for the four receiver-and-criterion combinations because the corresponding relaxations are rigorously proved tight.The four problems combine Type-I and Type-II receivers with the two sensing objectives.
  • Beamforming insights: Dedicated radar signals are always beneficial with Type-II receivers, whereas Type-I receivers need none in the special LOS communication-channel case.The Type-I exception occurs when information-signal degrees of freedom are sufficient for ISAC.
  • Results: Numerical results report better sensing performance and lower execution time for minimum weighted beampattern gain maximization, while Type-II receivers outperform Type-I and conventional designs.The comparisons are reported under both sensing design criteria.

III. SINR-CONSTRAINED BEAMPATTERN MATCHING

The section jointly optimizes information beamforming and dedicated radar-signal covariance to minimize beampattern mismatch under BS power and user SINR constraints.

  • Problem formulation: Information beamforming vectors and the dedicated radar covariance matrix are jointly optimized to minimize matching error to a desired transmit beampattern.The optimization includes the BS sum-power constraint and minimum SINR constraints for communication users.

A. Problem Formulation

The formulation defines desired-beampattern matching under SINR and power constraints, then applies SDR to obtain globally optimal solutions for the receiver-specific problems.

  • A. Problem Formulation: The desired beampattern specifies the target transmit-power distribution across M sensing angles.For target tracking, it may be nonzero at angles containing potential targets and zero elsewhere.
  • A. Problem Formulation: The matching error compares the transmit beampattern with a scaled desired beampattern using steering vectors and a scaling coefficient α.The coefficient adjusts the transmit-pattern scale to improve matching.
  • A. Problem Formulation: Type-I and Type-II formulations minimize beampattern matching error subject to each receiver’s minimum SINR requirement Γ_i.The SINR thresholds are nonnegative and indexed by communication receiver.
  • A. Problem Formulation: The problems impose BS power constraints, including equality power use when all available transmit power is intended to maximize radar performance.This equality constraint is motivated by target tracking or detection objectives.
  • B. Optimal Solution via SDR: Introducing covariance variables T_k=t_k t_k^H reformulates the beamforming problems but retains positive-semidefinite and rank-one constraints.The rank-one constraints preserve their interpretation as individual information beamformers.
  • B. Optimal Solution via SDR: Relaxing the rank-one constraints produces convex QSDPs solvable by convex optimization software, with rank-one recovery determining exact optimality.Gaussian randomization is otherwise needed to construct generally sub-optimal rank-one solutions.
  • B. Optimal Solution via SDR: Propositions establish that SDR1 and SDR2 each possess globally optimal solutions, so the relaxed formulations can solve the original problems when rank-one solutions are recovered.The proposition proofs are supplied or referenced in the paper’s appendices.
  • B. Optimal Solution via SDR: The Type-II problem has an equal or smaller optimal matching error than Type-I because its feasible region is enlarged.Every Type-I feasible solution remains feasible for Type-II, but the converse need not hold.

C. Analytic Performance Comparison versus Corresponding Design without Radar Signals

This section compares joint beamforming with and without dedicated radar signals for Type-I and Type-II receivers, including the special case of LOS communication channels. It shows when radar signals improve sensing and when Type-I receivers do not require them.

  • Dedicated radar signals generally reduce beampattern matching errors for both receiver types by exploiting additional spatial degrees of freedom.The comparison is made against corresponding designs without radar signals.
  • Type-II receivers outperform Type-I receivers in beampattern matching because they can cancel interference from dedicated radar signals.
  • For LOS communication channels, Type-I receivers achieve the same optimal matching error with radar signals as without them.The information-bearing signals alone provide sufficient degrees of freedom for ISAC in this special case.

IV. SINR-CONSTRAINED MINIMUM WEIGHTED BEAMPATTERN GAIN MAXIMIZATION

The paper introduces minimum weighted beampattern gain maximization as an alternative sensing criterion to conventional beampattern matching. The criterion targets the weakest weighted gain across desired sensing angles while satisfying communication SINR constraints.

  • Minimum weighted beampattern gain maximization increases the weakest weighted sensing gain at interested angles while satisfying individual users' SINR constraints.
  • Compared with beampattern matching, the proposed criterion provides enhanced sensing performance at interested angles with significantly lower execution time.

A. Problem Formulation

The formulation maximizes the minimum weighted beampattern gain over desired sensing angles under communication SINR constraints for Type-I and Type-II receivers. Although the resulting problems are non-convex, SDR produces globally optimal solutions because the relaxations are tight.

  • A. Problem Formulation: The design considers a quantized set of interested sensing angles, with angle selection determined by tasks such as target detection or tracking.
  • A. Problem Formulation: Problems P4 and P5 maximize the minimum weighted beampattern gain variable t for Type-I and Type-II receivers, respectively.
  • A. Problem Formulation: The weight η_θ represents the beampattern gain weight at each interested sensing angle and characterizes transmit power density over space.
  • B. Optimal Solution via SDR: Both formulations remain non-convex because the transmit covariance matrices T_k are constrained to be positive semidefinite and rank one.
  • B. Optimal Solution via SDR: Type-II receivers always achieve equal or greater minimum weighted beampattern gains than Type-I receivers because radar-interference cancellation enlarges the feasible region.
  • B. Optimal Solution via SDR: Relaxing the rank-one constraints yields separable semidefinite programs that can be solved optimally by CVX.
  • B. Optimal Solution via SDR: The SDRs for both Type-I and Type-II formulations are tight, so globally optimal solutions satisfying the original rank-one constraints can be constructed.

C. Analytic Performance Comparison versus Corresponding Design without Radar Signals

The analysis compares minimum weighted beampattern gain designs with and without dedicated radar signals, showing that radar signals generally improve sensing and Type-II receivers outperform Type-I receivers. Under LOS channels, however, Type-I receivers can achieve the same gain without radar signals.

  • Corresponding design without radar signals: The analysis benchmarks Type-I and Type-II minimum weighted beampattern gain designs against corresponding designs without dedicated radar signals.The no-radar benchmark is formed by setting R_d = 0 and relaxing rank-one constraints in the semidefinite formulation.
  • General channel conditions: Dedicated radar signals generally improve minimum weighted beampattern gain for both receiver types by exploiting the full spatial degrees of freedom.The comparison identifies higher minimum weighted beampattern gain when radar signals are added.
  • General channel conditions: Type-II receivers outperform Type-I receivers because they can cancel interference caused by the dedicated radar signals.This comparison applies to the minimum weighted beampattern gain criterion under the analyzed channel conditions.
  • LOS communication channels: With LOS communication channels, Type-I receivers achieve the same minimum weighted beampattern gain with or without dedicated radar signals.The result implies that information-signal degrees of freedom are sufficient for ISAC in this special case.

D. Complexity Analysis

The paper contrasts the computational formulations of beampattern matching and minimum weighted beampattern gain maximization, then evaluates the proposed designs numerically under both receiver types and channel conditions. The numerical results show lower matching error and beampatterns close to radar-only sensing for Type-II receivers, while LOS Type-I designs can avoid radar signals.

  • Complexity analysis: Beampattern matching formulations are QSDPs with worst-case interior-point complexity O(K^6.5N^6.5 log(1/ϵ)).The complexity is stated for solution accuracy ϵ.
  • Complexity analysis: Minimum weighted beampattern gain formulations are SSDPs with lower worst-case interior-point complexity than the QSDP formulations.The supplied complexity expression for SSDPs is truncated, but the paper explicitly contrasts it with the QSDP complexity.
  • SINR-constrained beampattern matching: Under both Rayleigh fading and LOS channels, Type-II receivers achieve much lower beampattern matching error than the other evaluated designs.The matching-error curves average results over 200 random channel realizations.
  • SINR-constrained beampattern matching: At Γ = 20 dB, Type-II beampatterns are close to the radar-only upper bound under both channel conditions.For LOS channels, only the Type-II design closely resembles radar-only sensing around the interested region near 60°.
  • SINR-constrained beampattern matching: Under LOS channels, Type-I designs use zero radar-signal beampattern and reuse information signals, whereas Type-II designs allocate radar gain to the interested sensing regions.The decomposition of the achieved beampatterns shows the distinct roles of information and radar signals.
  • SINR-constrained beampattern matching: Beampattern gains outside sensing regions reflect a trade-off: communication SINR requires gain there, while sensing does not desire it.Users near −10° and 10° receive relatively smaller gains than the user near −60° in the sensing region.

B. SINR-Constrained Minimum Weighted Beampattern Gain Maximization

The max-min design targets worst-case weighted beampattern gains at selected sensing angles under SINR constraints, outperforming beampattern matching in sensing performance and execution time. Type-II receivers further improve sensing by cancelling radar-signal interference, while the design remains strongest across tested channel conditions and user loads.

  • The max-min design selects angles with positive desired gains and maximizes the minimum weighted beampattern gain over those sensing angles.The angles of interest are denoted Θ, with unit weights ηθ = 1 in the simulation.
  • Max-min designs achieve much higher worst-case beampattern gains at angles of interest than beampattern matching designs under Rayleigh fading and LOS channels.The comparison averages results over 200 random channel realizations and varies the SINR threshold Γ.
  • With Γ ≥ 10 dB, Type-II receivers perform much better than other schemes because they cancel interference caused by dedicated radar signals.The Type-II advantage is reported under both design criteria, especially at high SINR thresholds.
  • At Γ = 20 dB, the Type-II max-min design produces more balanced beampatterns than the other designs under both channel conditions.Under LOS channels, the Type-I max-min pattern is distorted but still has a minimum gain above 0.1, compared with less than 0.04 for beampattern matching.
  • As the number of users increases, the minimum weighted beampattern gain decreases, while the Type-II max-min design outperforms all other approaches under both channel conditions.The results use Γ = 20 dB, N = 8, and averages over 200 Rayleigh and LOS channel realizations.
  • The paper globally solves the four non-convex beamforming problems by applying semidefinite relaxation and proving that the resulting relaxations are always tight.The four problems combine two receiver types with beampattern matching and minimum weighted beampattern gain maximization.

APPENDIX

The appendix constructs rank-one information covariance matrices from an SDR solution while preserving feasibility, objective value, power constraints, and communication-user SINR constraints.

  • APPENDIX: An SDR solution is transformed into rank-one positive semidefinite matrices while retaining the objective value and satisfying the power constraint with equality.The construction establishes the required covariance-matrix structure for the tightness proof.
  • APPENDIX: The reconstructed radar covariance remains positive semidefinite because it combines a positive semidefinite original covariance with positive semidefinite matrix sums.
  • APPENDIX: Reformulating the SINR constraints shows that the reconstructed information and radar covariances remain feasible for all communication users.

B. Proof of Proposition 3

The proof shows that an optimal solution of SDR1 yields a feasible and optimal solution of SDR3, establishing the claimed relationship between the two relaxed problems.

  • B. Proof of Proposition 3: The constructed variables are feasible for SDR3 because they remain positive semidefinite and satisfy the constraints inherited from the feasible SDR1 solution.
  • B. Proof of Proposition 3: For any feasible SDR3 solution, the proof establishes an objective bound showing that the constructed SDR3 solution is optimal.

C. Proof of Proposition 4

The proof reconstructs rank-one transmit covariance matrices from a high-rank optimum while preserving the sensing objective, SINR performance, and sum-power feasibility.

  • C. Proof of Proposition 4: Each high-rank optimal covariance is decomposed and reconstructed as a rank-one solution without changing the received SINR across sensing angles.The Riesz-Fejér theorem supplies a vector representation independent of the angle variable.
  • C. Proof of Proposition 4: The reconstructed rank-one matrices satisfy the BS sum-power constraints.
  • C. Proof of Proposition 4: The reconstructed rank-one solution achieves the same objective-function value as the original high-rank optimum.
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