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Dual-Functional Radar-Communication Waveform Design: A Symbol-Level Precoding Approach

Rang Liu, Ming Li, Qian Liu, A. Lee Swindlehurst

arXiv:2108.05043v1eess.SP

TL;DR

DFRC waveform design must jointly support radar sensing and multi-user communications despite conflicting requirements and limited block-level degrees of freedom. This paper uses symbol-level precoding to optimize instantaneous dual-use waveforms under communication and power constraints, with two non-convex optimization algorithms. Simulations report more accurate angle estimation, better target detection with limited collected signals, and lower communication SER than conventional block-level precoding.

  • Problem

    DFRC waveform design must balance conflicting radar and communication requirements, while conventional block-level precoding offers limited degrees of freedom and can distort beampatterns with few samples.

  • Method

    The paper optimizes symbol-dependent transmit vectors to minimize beampattern error under communication QoS and constant-modulus constraints, using PDD-MM-BCD and ALM-RBFGS algorithms.

  • Results

    Simulations show more accurate angle estimation, better target detection with limited collected signals, and lower multi-user communication SER than conventional block-level precoding.

  • Takeaways & Limitations

    Symbol-level precoding provides a dual-use DFRC waveform design that improves reported radar sensing and multi-user communication performance.

Abstract

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Dual-functional radar-communication (DFRC) systems can simultaneously perform both radar and communication functionalities using the same hardware platform and spectrum resource. In this paper, we consider multi-input multi-output (MIMO) DFRC systems and focus on transmit beamforming designs to provide both radar sensing and multi-user communications. Unlike conventional block-level precoding techniques, we propose to use the recently emerged symbol-level precoding approach in DFRC systems, which provides additional degrees of freedom (DoFs) that guarantee preferable instantaneous transmit beampatterns for radar sensing and achieve better communication performance. In particular, the squared error between the designed and desired beampatterns is minimized subject to the quality-of-service (QoS) requirements of the communication users and the constant-modulus power constraint. Two efficient algorithms are developed to solve this non-convex problem on both the Euclidean and Riemannian spaces. The first algorithm employs penalty dual decomposition (PDD), majorization-minimization (MM), and block coordinate descent (BCD) methods to convert the original optimization problem into two solvable sub-problems, and iteratively solves them using efficient algorithms. The second algorithm provides a much faster solution at the price of a slight performance loss, first transforming the original problem into Riemannian space, and then utilizing the augmented Lagrangian method (ALM) to obtain an unconstrained problem that is subsequently solved via a Riemannian Broyden-Fletcher-Goldfarb-Shanno (RBFGS) algorithm. Extensive simulations verify the distinct advantages of the proposed symbol-level precoding designs in both radar sensing and multi-user communications.

I. INTRODUCTION

DFRC systems share hardware and spectrum for radar sensing and communications, but their conflicting requirements make waveform design challenging. This paper introduces symbol-level precoding for MIMO DFRC and develops two algorithms for the resulting non-convex design problem.

  • Motivation: DFRC uses a shared transmitter for radar and communications, reducing platform complexity while requiring careful balancing of conflicting functional requirements.MIMO architectures add waveform diversity for target detection, beamforming gains, and spatial multiplexing.
  • Limitations of prior designs: Conventional block-level precoding has limited waveform degrees of freedom and may distort average beampatterns when only a few radar-pulse samples are available.Its degrees of freedom are limited by the number of users, while covariance-based beampatterns rely on sufficiently many waveform samples.
  • Symbol-level precoding: Symbol-level precoding optimizes each instantaneous transmit vector from the current symbols, enabling per-slot beampattern control and constructive use of multi-user interference.The approach converts harmful interference into useful components for communication QoS.
  • Novelty: Unlike prior RCC applications, the proposed DFRC design jointly realizes radar sensing and communications through a dual-use waveform.The paper identifies this DFRC symbol-level formulation as previously uninvestigated.
  • Problem formulation: The waveform minimizes squared beampattern error subject to communication QoS and constant-modulus power constraints.The system serves multiple single-antenna users while detecting targets from several directions.
  • Algorithms: Two solvers are proposed: PDD-MM-BCD targets the non-convex problem through solvable subproblems, while ALM-RBFGS offers a faster solution with slight performance loss.Extensive simulations evaluate the designs for both radar sensing and multi-user communications.

B. Multi-user Communication Performance Metric

The communication metric is based on how far each received noise-free symbol lies from its nearest decision boundary. The radar metric evaluates instantaneous beampattern agreement with a desired pattern, addressing the limitations of covariance-based average-beampattern design.

  • Communication QoS: Constructive interference turns multi-user interference into helpful signal components that push received symbols farther from decision boundaries.The resulting distance supports more robust symbol detection under noise.
  • Communication QoS: The communication QoS constraint requires each received signal to remain at least a preset minimum distance from its closest decision boundary.For the k-th user, β_k specifies the minimum QoS requirement.
  • Modulation extensions: The constructive-region formulation extends beyond PSK to QAM and arbitrary modulation formats when symbol decision regions are convex.For QAM, different constellation-point types produce different convex regions expressible through linear inequalities.
  • Radar sensing metric: Radar beampattern design directs transmit power toward potential targets and compares the resulting pattern with a desired pattern using squared error.Stronger target echoes support more accurate parameter estimation, while reduced off-target power suppresses clutter.
  • Radar sensing metric: Covariance-based block-level design estimates an average beampattern from second-order symbol statistics, which can distort performance with few waveform samples.Symbol-level precoding instead optimizes the instantaneous transmit beampattern to address this limitation.

D. Problem Formulation

The paper formulates symbol-level transmit-vector design as minimizing beampattern mismatch while satisfying communication QoS and constant-modulus power constraints. The resulting problem is nonconvex because of its quartic objective and constant-modulus constraint.

  • The symbol-level transmit vector minimizes squared error between designed and desired beampatterns under CI-based communication QoS and transmit-power constraints.
  • Each transmit antenna uses a constant-modulus constraint at every symbol time, supporting high power efficiency and low peak-to-average power ratio.
  • The formulation removes the time-slot index for conciseness before expressing the optimization problem.
  • The objective is first minimized over the auxiliary scaling variable because the original formulation is quadratic in that variable.
  • The compact formulation is nonconvex due to its quartic objective and constant-modulus constraint, motivating specialized iterative algorithms.

III. PROPOSED PDD-MM-BCD ALGORITHM

The PDD-MM-BCD approach separates the difficult waveform design into tractable iterative subproblems. It introduces an auxiliary variable, penalizes coupling constraints, majorizes the quartic objective, and applies block coordinate descent.

  • PDD-MM transformation: An auxiliary variable decouples the convex and constant-modulus constraints before applying PDD to the resulting formulation.
  • PDD-MM transformation: PDD uses an augmented Lagrangian framework with inner BCD updates and outer penalty or dual-variable updates.
  • MM transformation: MM replaces the complicated nonconvex objective with a locally tractable surrogate that upper-bounds it at each iteration.
  • MM transformation: Second-order Taylor expansions and amplitude constraints convert quadratic terms into linear upper bounds, simplifying optimization and reducing computational complexity.
  • BCD procedure: The resulting surrogate problem is solved by alternating updates of x and v using a two-block BCD procedure.

B. BCD Algorithm

The BCD stage alternates updates of the auxiliary variable and waveform vector. Phase alignment solves the auxiliary update, while the waveform update is handled through a real-valued Lagrangian-dual formulation and derivative-free search.

  • Update v: With fixed x, the auxiliary-variable subproblem is solved in closed form through phase alignment.
  • Update x: With fixed v, the x-update is convex but is solved through its Lagrangian dual to avoid the high cost of repeated interior-point iterations.
  • Update x: The x-update is transformed into an equivalent real-valued problem before constructing its Lagrangian dual function.
  • Update x: The Lagrangian dual is efficiently searched with the derivative-free Hooke-Jeeves Pattern Search algorithm because derivative calculations are costly.
  • Update x: After obtaining the dual solution, the waveform solution and the original x-subproblem solution are reconstructed.

C. Summary and Analysis

The proposed PDD-MM-BCD method is implemented as a nested iterative algorithm with explicit convergence control and complexity analysis. Its complexity can be favorable to block-level precoding for BPSK and QPSK in small systems.

  • Algorithm 1 initializes the primal, auxiliary, penalty, and dual variables, then alternates inner updates until convergence.
  • The inner loop updates x and v until the objective converges, while the outer loop updates penalty and dual variables until the equality constraint is approximately satisfied.
  • Convergence analysis: The inner-loop objective is non-increasing because MM generates non-increasing sequences, and the objective is lower-bounded.
  • Convergence analysis: The PDD-based algorithm has guaranteed convergence under the stated analysis.
  • Computational complexity: For BPSK and QPSK in small-scale systems, the proposed symbol-level scheme can be more efficient than the conventional block-level scheme.

IV. EFFICIENT ALM-RBFGS ALGORITHM

The ALM-RBFGS algorithm reduces the computational burden of symbol-level precoding for large-scale systems by reformulating the constant-modulus problem on a Riemannian space and solving penalized constraints with RBFGS.

  • The algorithm targets the high computational cost of the earlier MM-based method as the number of users increases.The MM-based method requires considerable iterations, making its complexity unaffordable for larger systems.
  • RBFGS is then used to efficiently solve the unconstrained augmented-Lagrangian problem.The method is designed as a faster alternative for applying symbol-level precoding to large-scale systems.
  • The constant-modulus constraint is represented as a complex circle manifold, allowing the problem to be reformulated in Riemannian space.This reformulation preserves the constant-modulus structure while enabling manifold-based optimization.
  • The augmented Lagrangian method penalizes the communication inequality constraints and converts the reformulated problem into an unconstrained Riemannian optimization problem.The penalty parameter and dual variables are updated iteratively.

B. RBFGS Algorithm

The RBFGS inner solver performs manifold-aware quasi-Newton updates while the outer ALM procedure updates dual variables and penalty parameters until convergence.

  • RBFGS Algorithm: RBFGS determines search directions from the Riemannian gradient and a Hessian approximation, then updates the solution with Armijo line search.The variable update uses a retraction so iterates remain on the complex circle manifold.
  • RBFGS Algorithm: The Hessian approximation is updated iteratively, while vector transport maps tangent-space vectors between successive manifold points.Vector transport is required because vectors in different tangent spaces cannot be added directly.
  • RBFGS Algorithm: The RBFGS iterations continue until convergence to a locally optimal solution of the augmented Lagrangian problem.These iterations form the inner loop summarized in Algorithm 2.
  • ALM Updates: The ALM outer loop updates dual variables and the penalty parameter using constraint violations, with safeguards based on maximum limits.The penalty parameter increases only when constraint violations shrink sufficiently fast; otherwise it remains unchanged.
  • Algorithm 2: Algorithm 2 initializes feasible manifold points and alternates inner RBFGS optimization with outer ALM updates until the stopping threshold is met.The procedure accepts system and algorithm parameters as inputs and returns x⋆ and α⋆.

D. Summary and Analysis

The complete ALM-RBFGS procedure uses warm-started manifold optimization and has lower theoretical and observed computational cost than PDD-MM-BCD, making it more practical for large-scale systems.

  • Summary: Because RBFGS is a local-search method, the authors recommend warm-starting it with a solution from a scenario without communication services.That initialization problem can itself be solved using RBFGS.
  • Summary: The solution, penalty parameter, and dual variables are iteratively updated until convergence.This completes the outer ALM procedure after initialization and inner optimization.
  • Computational Complexity Analysis: The ALM-RBFGS algorithm has lower theoretical computational complexity than the PDD-MM-BCD algorithm.The complexity includes RBFGS updates and updates of the dual variables and penalty parameter.
  • Computational Complexity Analysis: The numerical results show that ALM-RBFGS requires fewer iterations and less execution time, supporting its use in large-scale systems.The paper describes this as a more practical implementation for large-scale systems.

V. EXTENSIONS FOR DOPPLER PROCESSING

The paper’s spatial, negligible-Doppler formulation does not directly cover rapidly moving targets, so it discusses extensions based on space-time waveform similarity or block-level super-symbol precoding.

  • Scope and Limitation: The proposed approach focuses on spatial radar properties and assumes Doppler effects are negligible.Rapidly moving targets require temporal waveform characteristics to be considered.
  • Possible Extensions: A waveform-similarity extension would compare the transmitted waveform with a reference waveform having desirable space-time correlation properties.The resulting optimization can be divided into sub-problems resembling the paper’s original formulation.
  • Possible Extensions: A second extension would jointly prec ode the entire space-time waveform using radar objectives such as angle-Doppler spectrum or ambiguity-function matching.Communication constraints would apply to symbols over the entire block, producing a super-symbol decoded as a block.
  • Scope and Limitation: The proposed extensions are outside the paper’s scope but illustrate that symbol-level precoding can be generalized to DFRC systems.The paper presents these directions as possible approaches rather than completed methods.
  • Simulation Context: The simulations evaluate proposed symbol-level designs against block-level DFRC precoding using specified communication, antenna, target, and beampattern settings.Figure 3 compares instantaneous beampatterns for the two approaches.
  • Simulation Context: In Figure 3, black lines denote the radar-only benchmark, colored lines denote time-slot beampatterns, and red square-marked lines denote their average.The comparison concerns instantaneous transmit beampattern behavior across time slots.

A. Instantaneous Transmit Beampattern

The proposed symbol-level precoding designs improve instantaneous radar beampattern quality and target sensing performance while maintaining stronger multi-user communication performance than block-level approaches. Increasing communication QoS requirements and user count creates a radar–communication trade-off, while the proposed algorithms converge effectively.

  • Instantaneous beampattern: Symbol-level precoding keeps instantaneous beampatterns clustered around the radar-only benchmark, unlike the dramatic fluctuations of block-level precoding.Its average beampattern is also better than that of block-level precoding.
  • Beampattern MSE: Transmit beampattern MSE increases with larger Γ and Ku, revealing a trade-off between radar sensing and multi-user communications.Fig. 4 compares Ku = 3 using solid lines with Ku = 4 using dashed lines.
  • Target angle estimation: The proposed algorithms provide lower angle-estimation RMSE than and, especially with few collected samples such as N = 15.Accuracy improves for all schemes as the number of collected samples increases.
  • Target detection: At false alarm probability 10^-5, PDD-MM-BCD and ALM-RBFGS achieve much higher detection probability than the two block-level approaches.Detection performance is evaluated using ROC curves with detection probability plotted against false alarm probability.
  • Communication performance: Symbol-level approaches achieve lower SER than block-level methods, while ALM-RBFGS trades slight SER loss against substantial computational complexity reduction relative to PDD-MM-BCD.ALM-RBFGS still outperforms block-level precoding in SER, although communication-constraint violations remain in its augmented-Lagrangian optimization.

C. Comparisons of Convergence and Complexity

Both proposed algorithms converge, while ALM-RBFGS offers substantially lower computational complexity than PDD-MM-BCD with a reported performance trade-off.

  • Convergence: The PDD-MM-BCD inner loop converges monotonically within a limited number of iterations, with fewer iterations required after the initial loop.
  • Convergence: The ALM-RBFGS algorithm converges rapidly in both its RBFGS inner loop and ALM outer loop.
  • Complexity: The ALM-RBFGS algorithm has much lower total computational complexity than PDD-MM-BCD.
  • Complexity: ALM-RBFGS requires only about 2% of the PDD-MM-BCD execution time for obtaining the precoded transmit vector.
  • Complexity: Execution time increases with the number of users because additional constraints produce more variables and iterations.
  • Complexity: Parallel computation can precalculate possible transmitted signals and further reduce execution time in practical implementations.
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