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On Outage-based Beamforming Design for Dual-Functional Radar-Communication 6G Systems
Ahmad Bazzi, Marwa Chafii
TL;DR
The paper addresses DFRC beamforming with imperfect CSI when one transmit signal serves communication users and target detection. It formulates outage-constrained radar-power maximization, derives tractable and sometimes closed-form solutions, and demonstrates communication–radar trade-offs and performance through simulations.
Problem
DFRC beamforming must jointly support communication users and target detection despite imperfect CSI and competing sensing–communication requirements.
Method
The authors maximize Bartlett radar output under probabilistic outage SINR constraints, using convex relaxations and rank techniques to obtain a tractable optimization problem.
Results
The relaxed problem is guaranteed to yield rank-one beamforming solutions, while simulations demonstrate radar–communication trade-offs and comparisons with robust beamforming methods.
Takeaways & Limitations
Closed-form analysis and simulations show how communication requirements, outage parameters, and channel–target correlation govern achievable radar performance under imperfect CSI.
Abstract
from arXiv · showhide
This article studies and derives beamforming design in a dual-functional radar-communication (DFRC) multiple-input-multiple-output system. We focus on a scenario, where the DFRC base station communicates with downlink communication users, with imperfect channel state information knowledge, and performs target detection, all via the same transmit signal. Through careful relaxation procedures, we arrive at a suitable and novel optimization problem, which maximizes the radar output power in the Bartlett sense, under probabilistic outage signal-to-interference-and-noise ratio constraints. Theoretical analysis proves optimality of the solution given by the relaxed version of the problem, as well as closed-form solutions in certain scenarios. Finally, the achieved performances and trade-offs of the proposed beamforming design are demonstrated through numerical simulations.
I. INTRODUCTION
The paper develops DFRC beamforming for imperfect CSI, addressing communication–radar trade-offs through outage-constrained optimization and numerical evaluation. It combines convex relaxations, closed-form single-user analysis, rank-one optimality, and simulations comparing sensing and communication performance.
- I. INTRODUCTION: The study targets unresolved joint sensing–communication trade-offs caused by shared hardware, spectrum, waveform, and imperfect-CSI constraints.
- B. Contributions and Insights: The proposed design maximizes Bartlett radar output under outage SINR constraints while accounting for imperfect CSI in DFRC beamforming.The stochastic constraints are converted into a tractable convex optimization problem using Bernstein-type inequalities, Schur complements, and rank relaxation.
- B. Contributions and Insights: Closed-form single-user solutions expose how outage parameters and steering-vector–channel correlation shape the radar–communication trade-off.The analysis identifies parameter dependencies that can tune the beamforming solution under imperfect CSI.
- B. Contributions and Insights: The relaxed optimization problem is proved to produce rank-one beamforming solutions, avoiding Gaussian randomization and principal-component post-processing.
- B. Contributions and Insights: Higher target SINR improves communication requirements but reduces the radar’s ability to form the desired look-direction pattern and reject sidelobes.
- B. Contributions and Insights: At unit transmit power, the proposed design achieves 4.78 bits/sec/Hz/user average sum-rate with approximately 0.99 detection probability, versus approximately 0.65 for state-of-the-art methods.
A. Communication System Model
The DFRC model uses one precoded transmit signal for downlink communication and radar sensing, with imperfect CSI represented by Gaussian channel errors. Communication quality is measured through SINR and outage constraints, while radar performance is tied to Bartlett power in the target look-direction.
- Signal and channel model: The same precoded transmit signal x[n] serves both communication users and radar sensing in a colocated DFRC system.The transmit symbols are independent and unit variance, and the beamforming matrix columns target individual users.
- Signal and channel model: Imperfect CSI is modeled as an estimated channel plus circularly symmetric complex Gaussian error with known covariance.For simplicity, the error covariance is assumed proportional to the identity matrix.
- Communication performance: The communication KPI is user SINR, with outage probability p_k specifying the allowed probability of failing to reach target γ_k.The outage constraint provides a probabilistic QoS requirement under random SINR.
- Radar performance: Radar performance is measured by Bartlett beamformer output power P(θ0) toward the target look-direction, which is linked to detection probability.The radar covariance follows from the received echo model, and increasing look-direction power increases detection probability.
III. BEAMFORMING DESIGN UNDER IMPERFECT CSI
The paper formulates joint DFRC beamforming as maximizing Bartlett radar output subject to probabilistic communication QoS constraints. It transforms the stochastic nonconvex problem through successive relaxations into a convex optimization problem solvable with standard convex solvers.
- Problem formulation: The joint design maximizes Bartlett beamformer output while satisfying communication-user QoS constraints under imperfect CSI.The objective and constraints jointly represent radar sensing and downlink communication requirements.
- Problem formulation: The initial problem is nonconvex because its communication constraints contain probability events involving random SINR.Directly deriving the exact SINR distribution would produce a highly nonlinear problem.
- Problem relaxation: Gaussian channel uncertainty is handled through normalization and a Bernstein-type inequality that replaces probabilistic constraints with deterministic bounds.The transformation introduces deterministic matrix and scalar conditions while preserving the intended outage-probability level through ε_k = −log(p_k).
- Problem relaxation: Successive slack-variable, second-order-cone, and Schur-complement transformations produce a tractable semidefinite formulation.The rank-one constraints remain the final source of nonconvexity before their relaxation.
- Final formulation: Dropping the rank-1 constraint yields the final convex optimization problem, which can be solved using a convex solver such as CVX.The formulation retains positive-semidefinite beamforming matrices and deterministic uncertainty constraints.
C. Rank-one beamforming
The paper addresses the practical need for rank-one beamforming vectors after semidefinite relaxation. It proves that the final convex problem itself admits rank-one optimal solutions, eliminating the need for approximation post-processing.
- Rank-one requirement: Practical beamforming requires matrices of the form W_k = w_k w_k^H, whereas relaxed solutions may otherwise require rank-one approximation.Common approximation methods include Gaussian randomization and principal component analysis.
- Rank-one optimality: Theorem 1 proves that every optimal solution of the final convex problem satisfies rank-one optimality for all users.The theorem establishes exactness of the relaxation within the stated optimization problem.
- Rank-one recovery: Because rank-one solutions are guaranteed, eigenvalue decomposition can directly recover the optimal beamforming vectors without Gaussian randomization or principal-component post-processing.The recovered vector is obtained from each optimal W_k.
D. Single-user closed-form solutions
For one communication user, the paper derives closed-form beamforming solutions that expose how channel–steering correlation, SINR threshold, and outage probability shape the radar–communication tradeoff. High correlation yields radar-oriented or matched-filter behavior, while outage restrictions alter the solution parameters.
- Closed-form solutions: The single-user case admits closed-form optimal beamforming solutions, including a formulation for vanishing uncertainty and a generalized form with nonzero outage parameters.The generalized solution reduces to the limiting result as ε approaches zero.
- Radar–communication tradeoff: When the channel and target steering vector are highly correlated, the optimal solution becomes the Bartlett beamformer and prioritizes the radar metric.The paper identifies a correlation threshold depending on system and SINR parameters.
- Radar–communication tradeoff: Increasing γ strengthens the channel–steering correlation required for the solution to become radar-oriented; with lower correlation, both vectors contribute.The SINR target therefore changes the balance between communication and radar objectives.
- Radar–communication tradeoff: At correlation ρ = 1, the optimal beamforming vector is a matched-filter solution, and feasibility is characterized by the maximum achievable average SINR.This is described as an extreme communication-perspective case.
- Outage effects: Tighter outage failure requirements increase Λ, with the paper summarizing its scaling as O(γ) and O(ε^(1/2)).The result identifies both the SINR target and outage parameter as direct controls of the tradeoff.
IV. SIMULATION RESULTS
Simulations use equal channel-error variances and outage parameters across users, with Monte Carlo averaging and a half-wavelength-spaced uniform linear array.
- Simulations assume equal channel-error variances and equal outage parameters across users, with σ∆ = 0.1 unless otherwise stated.Monte Carlo averaging is used throughout the simulations.
- Achievable-rate simulations vary γ, p, N, and K to evaluate communication performance under the stated setup.Figures 3 and 4 examine achievable rates as functions of γ and p, respectively.
- The simulations include a half-wavelength-spaced uniform linear array.
A. Influence of γ on the achievable rate
The simulations evaluate achievable rates and radar-communication trade-offs under imperfect CSI, showing how outage parameters, users, and radar metrics shape performance and feasibility.
- A. Influence of γ on the achievable rate: At γ = 2 dB with K = 2, the achieved total rate is 5.39 bits/sec/Hz despite imperfect CSI.At γ = 1 dB, the rate is close to the 2.17 bits/sec/Hz threshold; increasing γ or K produces a rising rate trend.
- B. Influence of p on the achievable rate: Increasing p degrades total achievable rate, with a breakout region around p = 0.1; additional degrees of freedom improve SINR through interference nulling.
- C. Feasibility rate: At 99% feasibility, γ = 3.8 dB is attainable for K = 2, while targets fall to 1.6 dB for K = 3 and 1.25 dB for K = 4.Increasing p enlarges the feasibility region but sacrifices achievable rate.
- D. ISMR-rate trade-off: For fixed ISMR^-1, average achievable rate increases with higher γ or fewer users, revealing a radar-rate trade-off.ISMR measures sidelobe energy relative to mainlobe energy, with specified mainlobe and sidelobe supports.
E. Probability of detection and rate trade-off
The proposed design exposes trade-offs among communication rate, radar detection, beampattern quality, clutter, and robustness, while outperforming comparison designs in key settings.
- Detection probability and rate: At 4.78 bits/sec/Hz/user, the proposed design achieves PD ≃0.99 versus ≃0.65 for.The advantage is especially pronounced as the number of users grows.
- Clutter impact: A 3 dB increase in S_C at a fixed 6 bits/sec/Hz/user raises PD by about 0.2 when S_C ≤ 7 dB.The clutter level directly changes the achieved detection-rate trade-off.
- Beampattern examples: Reducing γ by a factor of 3 lowers sidelobes by roughly 10 dB and improves look-direction accuracy by about 0.5° for N = 10 and K = 4.
- Minimum achievable rate: The proposed and Bernstein methods coincide for γ < 2.8, while the proposed method outperforms Bernstein for larger γ and Sphere bounding for γ > 5.6.The comparison concerns minimum achievable rate under the respective robust beamforming designs.
APPENDIX A RANK-1 OPTIMALITY
The appendix establishes that the relaxed convex problem can be solved through its dual and that its optimal beamforming matrices admit rank-one solutions.
- Because (P7) is convex with zero duality gap, its optimal solution can be obtained by solving the dual problem.
- The proof rules out an unbounded dual objective by showing that ζoptI + Ωopt_k cannot have negative eigenvalues.
- An eigenvalue decomposition constructs a rank-one contribution preserving the relevant bound, contradicting any optimal solution with rank greater than one.
- The resulting optimal beamforming matrices Wopt_k are rank-one.
APPENDIX B SINGLE USER CLOSED-FORM SOLUTION
The appendix introduces the proof strategy for the single-user closed-form solution, whose subsequent derivation proceeds through a three-part argument.
- The single-user closed-form solution is established using a proof organized into three parts.
A. Part 1: Optimal solution spans {a∗(θ0),h∗}
The single-user optimum lies in the subspace spanned by the target steering vector and the communication channel, because orthogonal components do not improve the objective.
- Part 1: Optimal solution spans {a*(θ0),h*}: The optimal beamformer belongs to span(a*(θ0), h*).The component orthogonal to this span contributes nothing to the objective and can be removed while preserving feasibility.
- Part 1: Optimal solution spans {a*(θ0),h*}: An orthonormal basis of this two-dimensional subspace expresses wopt as α∥a∥ + α⊥a⊥.
- Part 1: Optimal solution spans {a*(θ0),h*}: The objective increases with |α∥|^2 and does not depend on α⊥, so the optimum uses |α∥|^2 = 1 and |α⊥|^2 = 0 when feasible.
- Part 1: Optimal solution spans {a*(θ0),h*}: The feasibility condition is tied to the upper bound on single-user average SINR under a unit-power constraint.
C. Part 3: Optimal solution expressed in {h∥,h⊥}
The optimal beamforming vector is represented in the orthonormal basis {h∥, h⊥}, with coefficient phases aligned constructively and magnitudes selected under the problem constraints.
- The beamformer is expressed as wopt = α1h∥ + α2h⊥ after substituting the orthonormal representation into the optimization problem.
- The phases of α1 and α2 are adjusted so their contributions add constructively to the objective.This phase choice makes both complex terms contribute constructively.
- Constructive phase alignment implies jointly maximizing |α1| and |α2|, so the total-power constraint is active: |α1|^2 + |α2|^2 = 1.
- The optimal solution is obtained by setting |α1|^2 = ρ and |α2|^2 = 1 − ρ in the complementary region.The basis projections satisfy |aT(θ0)h∥|^2 ≤ Nρ and |aT(θ0)h⊥|^2 ≥ N(1 − ρ), which supports this magnitude allocation.
- Because h∥ and h⊥ form an orthonormal basis of span(a∗(θ0), h∗), their squared projections onto aT(θ0) sum to N.