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Cramér-Rao Bound Optimization for Joint Radar-Communication Design

Fan Liu, Ya-Feng Liu, Ang Li, Christos Masouros, Yonina C. Eldar

arXiv:2101.12530v1eess.SP

TL;DR

The paper addresses joint radar sensing and multi-user communication by minimizing CRB-based target-estimation error under per-user SINR constraints. It develops closed-form single-user solutions and SDR-based multi-user solutions, with globally optimal beamformers reachable and improved estimation performance over benchmark designs.

  • Problem

    Joint radar-communication beamforming must optimize target estimation for point and extended targets while satisfying individual communication-user SINR constraints.

  • Method

    The paper uses CRB-based optimization, derives closed-form single-user designs, and applies semidefinite relaxation with rank-one recovery for multi-user beamforming.

  • Results

    The proposed methods reach globally optimal solutions and significantly outperform beampattern-approximation benchmark designs in target-estimation performance.

  • Takeaways & Limitations

    CRB-Min beamforming provides a joint radar-communication design approach that improves target estimation while maintaining prescribed multi-user communication QoS.

Abstract

from arXiv · show

In this paper, we propose multi-input multi-output (MIMO) beamforming designs towards joint radar sensing and multi-user communications. We employ the Cramér-Rao bound (CRB) as a performance metric of target estimation, under both point and extended target scenarios. We then propose minimizing the CRB of radar sensing while guaranteeing a pre-defined level of signal-to-interference-plus-noise ratio (SINR) for each communication user. For the single-user scenario, we derive a closed form for the optimal solution for both cases of point and extended targets. For the multi-user scenario, we show that both problems can be relaxed into semidefinite programming by using the semidefinite relaxation approach, and prove that the global optimum can always be obtained. Finally, we demonstrate numerically that the globally optimal solutions are reachable via the proposed methods, which provide significant gains in target estimation performance over state-of-the-art benchmarks.

I. INTRODUCTION

DFRC systems integrate radar sensing and communications, but existing designs typically optimize transmitter-side proxies rather than target-estimation performance directly. This paper develops CRB-minimizing MU-MIMO beamforming under communication SINR and power constraints for point and extended targets.

  • Motivation: DFRC jointly supports target sensing and wireless communications through shared spectrum, hardware, and signal processing.The integration is motivated by applications requiring high data rates, low latency, localization, and support for many devices.
  • Research gap: Existing DFRC strategies typically shape waveforms or beampatterns to imply estimation capability instead of directly optimizing a receiver-side target-estimation metric.The paper identifies explicit target-estimation optimization as the gap addressed by its framework.
  • Evaluation: Numerical results show that the proposed beamformers significantly outperform beampattern-approximation DFRC designs by reducing target-estimation errors.The simulations validate that the proposed methods reach the globally optimal solutions and improve estimation performance over benchmark techniques.
  • Proposed framework: The paper proposes CRB-based MU-MIMO beamforming for point and extended target estimation while guaranteeing per-user SINR and a transmit power budget.The CRB measures target-estimation performance, while per-user SINR measures communication quality of service.
  • Single-user design: For a single communication user, the paper derives optimal closed-form solutions for both point-target and extended-target beamforming problems.The single-user analysis characterizes the structure of both optimization problems before deriving their solutions.
  • Multi-user design: For multiple users, semidefinite relaxation produces solvable formulations, and the paper proves that rank-one globally optimal solutions can be obtained for both target models.The extended-target case admits direct closed-form extraction of the rank-one optimum from the relaxed solution.

C. Performance Metrics for the Extended Target Case

For extended targets, transmitting only communication streams leaves the target response matrix unidentifiable, so dedicated probing streams are added to restore full-rank sensing. The resulting CRB is achievable by maximum-likelihood estimation while probing signals also interfere with communications.

  • Identifiability: Transmitting only K signal streams gives insufficient degrees of freedom to recover the rank-Nt target response matrix G, making the Fisher information matrix singular.The model assumes K < Nt, so the transmitted signal matrix is rank-deficient.
  • Sensing design: Dedicated probing streams orthogonal to communication streams augment the beamformer and make the transmitted signal matrix full rank Nt.The first K beamformer columns carry user data, while the auxiliary columns support probing.
  • Performance metric: The extended-target CRB is achievable with maximum-likelihood estimation because the estimation problem is linear under i.i.d. Gaussian noise.In this setting, the MSE of the maximum-likelihood estimator equals the CRB.
  • Joint constraints: Dedicated probing signals interfere with communication users because they contain no useful information, so radar interference enters each SINR denominator.The beamforming optimization therefore includes both the CRB objective and communication SINR constraints.

˙AH ˙ARX

In the single-user point-target case, the CRB minimization is recast in terms of radiation toward the target direction. The optimal design is then characterized through a closed-form solution.

  • Single-user reduction: In the single-user case, minimizing the CRB reduces to maximizing radiation power at the target angle θ.This reduction motivates the subsequent lemma and theorem for the optimal beamforming solution.
  • Closed-form design: The paper derives a closed-form optimal solution for the resulting single-user problem.Theorem 1 states the optimal solution after applying Lemma 1.

C. Semidefinite Relaxation for the Multi-User Case

For multi-user beamforming, the non-convex CRB objectives are transformed into semidefinite programs and relaxed by dropping rank constraints. Under the stated rank condition, the relaxation yields globally optimal rank-one solutions.

  • Convex reformulation: The CRB objective is non-convex in RX because of its fractional structure, but minimizing it is equivalently formulated as a semidefinite program.This convex reformulation is established by Proposition 1.
  • Semidefinite relaxation: Dropping rank constraints on the beamforming matrices produces a standard SDP that can be solved numerically.The relaxed formulation is obtained through the classical semidefinite relaxation technique.
  • Point-target multi-user case: Under the condition that H[A, ˙A] has full column rank, every optimal beamforming matrix satisfies rank(Wk) = 1.The theorem assumes feasibility of the relaxed problem.
  • Global optimality: For random channels, the full-column-rank condition almost always holds for K ≥ 2, so solving the SDR generally obtains the globally optimal beamformers.The paper relates this condition to the communication and radar target channels being uncorrelated.
  • Extended-target case: The extended-target problem is likewise relaxed to a convex form, with a closed-form single-user solution that automatically satisfies the rank-one constraint.Theorem 3 provides the solution, and the resulting W1 has rank one.

C. Rank-One Optimal Solution of (35) in the Multi-User Case

The extended-target multi-user relaxation may produce high-rank solutions, motivating an exact rank-one extraction procedure. The construction preserves the covariance, objective, transmit power, useful signal power, interference, and SINR.

  • Rank issue: The convex relaxation may yield high-rank solutions, and common eigenvalue or Gaussian-randomization approximations are not guaranteed to be accurate.The matrix inverse in the objective can make eigenvalue-based approximation inaccurate.
  • Exact extraction: The paper proposes extracting the exact rank-one optimum directly from the convex-relaxation solution.The procedure is constructive rather than an approximation.
  • Covariance preservation: Replacing each relaxed communication matrix with a rank-one matrix and transferring the difference to the auxiliary radar covariance preserves the total covariance.Because RX remains unchanged, the objective value and transmit power also remain unchanged.
  • Communication feasibility: The extracted beamformers preserve each user’s useful signal power and interference, so the resulting SINRs remain unchanged.The communication beamformer is obtained from the extracted rank-one components, while the auxiliary beamformer accounts for the residual covariance.

V. NUMERICAL RESULTS

The numerical studies validate the proposed CRB-minimizing beamforming designs for point and extended targets under communication SINR constraints. Across the tested settings, the proposed methods reach optimal solutions and improve target-estimation performance over benchmark designs.

  • Simulation setup: The simulations use a DFRC base station with N_t = 16 transmit antennas, N_r = 20 receive antennas, a 30dBm power budget, and frame length L = 30.For point targets, θ = 0°; for extended targets, the response matrix has i.i.d. zero-mean, unit-variance Gaussian entries, and MSE measures estimation performance.
  • Single-user results: Closed-form and numerical solutions match well in the single-user point- and extended-target cases, while increasing required SINR raises the CRB and MSE.The single-user results compare root-CRB for target angle estimation and MSE for target-response estimation.
  • Point target, multiple users: With K = 4 users and 15dB SINR thresholds, the proposed CRB-Min beamformer focuses the highest power toward the target angle among the compared designs.The comparison includes the beampattern approximation designs from and; all three methods focus their mainlobes toward 0°.
  • Point target, multiple users: The proposed CRB-Min method outperforms both beampattern approximation designs in target-estimation RMSE, especially at low radar SNR.The CRB lowerbounds RMSE and becomes tight in the high-SNR regime, where the MLE can achieve it.
  • Point target, multiple users: Increasing user SINR generally raises the CRB, while the proposed technique remains superior to both benchmark beampattern methods in the radar-communication tradeoff.For a smaller number of users, the CRB remains low despite increasing user SINR.
  • Extended target, multiple users: For extended targets, the globally optimal rank-one solution outperforms eigenvalue-decomposition rank-one approximation, whose MSE-SINR tradeoff is not monotonically increasing.As user number increases, estimation worsens; with 10dB SINR, the MSE variation remains within 1dB.

APPENDIX A PROOF OF LEMMA 1

The lemma proof reduces the optimal single-user beamformer to the subspace spanned by the target and user channels. It also establishes that the full transmit-power budget is used at the optimum.

  • Subspace reduction: The optimal beamformer can be decomposed into components in span {a, h_1} and its null space, but only the former contributes to the SINR.The null-space component can therefore be removed while improving the objective without violating the constraints.
  • Power utilization: The transmit-power budget is fully exploited at the optimum because scaling any feasible underpowered beamformer to P_T increases the objective while preserving feasibility.The proof considers an optimal solution with power below P_T and constructs a higher-objective feasible solution.
  • Closed-form construction: When the SINR constraint is inactive, the solution allocates all power along the target-channel direction a.When the SINR constraint is active, the beamformer is represented in the reduced span and its coefficient magnitudes satisfy |x_1|^2 + |x_2|^2 = P_T.
  • Closed-form construction: The objective is maximized by choosing x_1 and x_2 phases opposite to the corresponding channel inner products, yielding the closed-form expressions in (28).This phase alignment completes the proof for the active-SINR case.

˙AH ˙ARX

The proof uses a Schur-complement condition to rewrite the original problem as an equivalent semidefinite program.

  • SDP reformulation: A Schur-complement condition transforms problem (54) into the semidefinite-program formulation (29).The reformulation completes the proof of the stated equivalence.

APPENDIX D PROOF OF THEOREM 2

The theorem proof introduces dual variables for the linear and semidefinite constraints of problem (32) and applies complementary conditions at optimality.

  • Dual formulation: Problem (32) has K + 1 linear constraints and K + 1 semidefinite constraints, with corresponding dual variables and positive-semidefinite matrices.Assuming primal and dual optimality, the proof invokes the resulting complementary conditions.

˙AH ˙ARX

The proof establishes rank-one optimal solutions by partitioning users according to dual variables and analyzing three possible cases. Under a full-column-rank condition on H̄A, all cases yield rank-one solutions.

  • Assumption: The eigenvalue argument assumes Nt ≠ Nr, a condition stated to hold generally for MIMO radar when Nr is chosen larger than Nt.Under this condition, F has rank two and distinct nonzero eigenvalues λ1 and λ2.
  • Case analysis: The analysis partitions users into K1 with µk > 0 and K2 with µk = 0, covering all possible dual-variable patterns.This partition supports the subsequent three-case rank analysis.
  • Case I: |K1| = 0: When |K1| = 0, λ1 > λ2 implies rank(Zk) = Nt − 1 and rank(Wk) = 1 for every user.With all SINR constraints ineffective, the singularity of Zk determines the rank of Wk.
  • Case II: |K1| = 1: When |K1| = 1, rank-one optimal solutions can always be attained by replacing the solution while preserving objective value and feasibility.The construction uses the nullspace of Zk and conditions ensuring unchanged objective and feasible constraints.
  • Case III: |K1| ≥ 2: If |K1| ≥ 2 and D has full column rank, then all dual variables are positive and rank(Wk) = 1 for every user.The lemma first establishes µT INt − F̄ ≻ 0; otherwise, users in K2 would force Wk = 0 and violate feasibility.
  • Theorem conclusion: If H̄A has full column rank, solving problem (32) always yields rank-one solutions, completing the proof of Theorem 2.This conclusion combines the rank results from all three cases.

APPENDIX E PROOF OF THEOREM 3

The appendix derives a closed-form optimum using an equivalent formulation, Lagrangian dual variables, and KKT conditions. The analysis shows that the power budget is reached and distinguishes inactive and active SINR constraints.

  • Equivalent formulation: The optimization problem is reformulated equivalently before deriving its closed-form optimal solution.Lemma 2 is used in this reformulation and subsequent solution derivation.
  • KKT derivation: The Lagrangian introduces dual variables ω, µ, and ηi, whose KKT conditions characterize the optimum.The stationarity conditions are written for the first coordinate and for i = 2, 3, ..., Nt.
  • KKT consequences: The conditions imply ηi = 0 and equal diagonal terms λii = µ for i = 2, 3, ..., Nt.The result follows because λii > 0 for every i.
  • Power constraint: The power budget is always reached, giving Σ_i=1^Nt λii = PT.This follows from µ > 0 in the KKT conditions.
  • SINR cases: When ω = 0, the SINR constraint is inactive; when ω > 0, it is active and λ11 = Γ1σ2.The two cases determine the corresponding closed-form solution branches.
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