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Towards Dual-functional Radar-Communication Systems: Optimal Waveform Design
Fan Liu, Longfei Zhou, Christos Masouros, Ang Li, Wu Luo, Athina Petropulu
TL;DR
The paper addresses waveform design for a dual-functional MIMO RadCom system that simultaneously detects radar targets and serves downlink users while minimizing multi-user interference. It develops closed-form beampattern designs, a weighted radar–communication trade-off algorithm, and a globally optimal branch-and-bound method for constant modulus waveforms. The proposed approaches have ZF-comparable computational cost and outperform ZF in the reported numerical evaluations.
Problem
Global algorithms for constant modulus waveform design remain widely unexplored, while dual-functional systems require waveforms satisfying both radar and communication objectives.
Method
The paper formulates omnidirectional and directional beampattern designs, weighted radar–communication optimization, and constant modulus design using closed-form, low-complexity, and branch-and-bound algorithms.
Results
The proposed designs have computational costs comparable in order to ZF precoding, and numerical results show performance gains over ZF while maintaining radar and communication performance.
Takeaways & Limitations
Dual-functional waveform designs can jointly support target detection and downlink communications, with a flexible radar–communication trade-off and globally optimal constant modulus solutions.
Abstract
from arXiv · showhide
We focus on a dual-functional multi-input-multi-output (MIMO) radar-communication (RadCom) system, where a single transmitter communicates with downlink cellular users and detects radar targets simultaneously. Several design criteria are considered for minimizing the downlink multi-user interference. First, we consider both the omnidirectional and directional beampattern design problems, where the closed-form globally optimal solutions are obtained. Based on these waveforms, we further consider a weighted optimization to enable a flexible trade-off between radar and communications performance and introduce a low-complexity algorithm. The computational costs of the above three designs are shown to be similar to the conventional zero-forcing (ZF) precoding. Moreover, to address the more practical constant modulus waveform design problem, we propose a branch-and-bound algorithm that obtains a globally optimal solution and derive its worst-case complexity as a function of the maximum iteration number. Finally, we assess the effectiveness of the proposed waveform design approaches by numerical results.
I. INTRODUCTION
The paper develops dual-functional RadCom waveform designs that simultaneously support radar sensing and downlink communications while minimizing multi-user interference. It addresses radar beampattern constraints, radar–communication trade-offs, practical waveform constraints, and computational complexity.
- Motivation: Spectrum sharing motivates dual-functional systems that combine radar and communications within shared frequency resources.The paper frames spectrum scarcity and simultaneous radar–communication operation as the motivating context.
- Waveform design: The proposed designs minimize downlink multi-user interference while supporting omnidirectional and directional radar beampatterns.The omnidirectional and directional formulations have closed-form globally optimal solutions.
- Waveform design: A weighted optimization enables a flexible trade-off between radar and communication performance through a low-complexity global algorithm.The weighting factor determines the relative emphasis on radar and communication performance.
- Practical constraints: Constant modulus waveform design is addressed with a branch-and-bound algorithm that obtains globally optimal solutions rather than only local optima.Constant modulus signals can avoid distortion from low-cost nonlinear power amplifiers and support energy-efficient transmission.
- Complexity and evaluation: The proposed closed-form waveform approaches have computational costs of the same order of magnitude as conventional communication-only ZF precoding.The paper also derives computational complexity analytically and evaluates the designs numerically.
- System model: The system uses one transmitted signal matrix as a dual-functional waveform for both radar sensing and downlink communication.The model includes a ULA with N antennas serving K single-antenna users while detecting radar targets.
B. Radar Model
The radar model designs a dual-functional waveform matrix that minimizes downlink multi-user interference under radar-specific beampattern constraints. It considers omnidirectional probing with orthogonal waveforms and directional probing toward selected directions.
- The design minimizes downlink multi-user interference under MIMO radar-specific constraints.
- Directional Beampattern Design: Directional design targets specified directions through a desired radar covariance matrix and its Cholesky factorization.
- Omnidirectional Beampattern Design: Omnidirectional probing requires an orthogonal waveform matrix whose covariance is proportional to the identity matrix.
- Omnidirectional Beampattern Design: The omnidirectional problem is a non-convex Orthogonal Procrustes problem with a closed-form globally optimal solution obtained by SVD.
B. Directional Beampattern Design
The directional beampattern problem matches a desired radar covariance while minimizing communication interference. Cholesky factorization and SVD yield a globally optimal solution with computational cost comparable in order to zero-forcing precoding.
- The directional design minimizes multi-user interference while enforcing a desired Hermitian positive semidefinite radar covariance matrix.
- Assuming positive definiteness, Cholesky factorization converts the covariance constraint into an equivalent form suitable for Orthogonal Procrustes optimization.
- The transformed problem is an Orthogonal Procrustes problem whose globally optimal solution is obtained through SVD and mapped back to the original waveform design.
- The proposed closed-form approaches have the same order of computational complexity as conventional communication-only zero-forcing precoding.
IV. TRADE-OFF BETWEEN RADAR AND COMMUNICATION PERFORMANCE
Strict radar constraints can degrade communication performance, motivating a weighted trade-off formulation. The resulting non-convex QCQP has a tight semidefinite relaxation, while a lower-complexity eigenvalue-search algorithm attains the global optimum.
- Strict equality constraints guarantee optimal radar performance but can cause serious communication loss, especially for ill-conditioned communication channels.
- The trade-off formulation uses a weighting factor ρ between 0 and 1 to balance radar and communication performance while enforcing the total-power equality constraint.
- The weighted objective combines communication mismatch with deviation from a radar waveform obtained from the earlier designs.
- The formulation becomes a non-convex QCQP whose semidefinite relaxation is tight because it has one quadratic constraint, yielding a globally optimal solution.
- The low-complexity algorithm computes the global optimum using eigenvalue decomposition, Golden-section search for λopt, and a pseudoinverse-based solution.
C. Complexity Analysis
The low-complexity trade-off algorithm is dominated by matrix operations, with total complexity sharing the same order of magnitude as communication-only zero-forcing precoding.
- Golden-section search requires O(log(1/ε0)) iterations for an ε0-solution, but its one-dimensional evaluations are generally negligible in total cost.
- The dominant operations are matrix multiplications, the Moore-Penrose pseudoinverse, and eigenvalue decomposition.
- The total complexity is O(N^2L + NKL + 3N^3 + N^2K).
- The algorithm and the communication-only ZF precoder share the same order of computational complexity.
V. CONSTANT MODULUS WAVEFORM DESIGN
The paper formulates constant-modulus RadCom waveform design as minimizing communication MUI energy while constraining similarity to a benchmark radar waveform. The resulting problem is generally non-convex and NP-hard, motivating a global branch-and-bound approach.
- Constant-modulus design minimizes communication MUI energy under a constant-modulus constraint.
- The similarity constraint limits the vectorized difference between the designed waveform and a constant-modulus benchmark radar signal.The benchmark may use chirp signals, and η controls the tolerable difference.
- The objective is separable, so the matrix problem can be solved concurrently for each column.
- The reformulated constant-modulus problem is generally non-convex and NP-hard.
B. The Branch-and-bound Framework
The branch-and-bound framework partitions the feasible region, computes bounds for subproblems, and iteratively branches, prunes, and updates global bounds until convergence. Basic and adaptive rectangular subdivision both satisfy the required convergence conditions, with adaptive subdivision observed to converge faster.
- Branch-and-bound partitions the feasible region and maintains lower and upper bounds for each subproblem.Iterations continue until the upper-bound and lower-bound difference approaches zero.
- Each iteration branches the subproblem with the smallest lower-bound, subdivides its region, and updates the global bounds.
- Finite convergence requires bounding-improving branching, exhaustive subdivision, and bounds consistent with branching.
- Both basic rectangular subdivision and adaptive rectangular subdivision satisfy exhaustive subdivision, while adaptive subdivision converges faster in simulations.
C. Upper-bound and Lower-bound Acquisition
The algorithm obtains lower bounds through convex-hull relaxation and upper bounds through projection of relaxed solutions onto feasible arcs, with optional refinement and gradient-based acceleration. Gradient-based methods have per-iteration cost O(NK).
- The lower-bound problem uses a convex relaxation whose per-entry feasible set is represented by a circular-segment convex hull.
- A feasible upper-bound solution is obtained by projecting each relaxed entry onto its corresponding feasible arc.
- The upper bound can be tightened by locally solving a non-convex QCQP initialized with the projected relaxed solution.
- Gradient-projection methods cost O(NK) per iteration, substantially reducing fixed-iteration costs relative to QCQP-based methods.
D. Convergence Analysis and Worst-case Complexity
The proposed branch-and-bound algorithm converges finitely to an arbitrarily close value of the optimum, and a worst-case iteration bound is derived for basic rectangular subdivision. Despite exponential worst-case order in N, simulations report small termination iteration counts in most cases.
- The upper-bound and lower-bound difference converges uniformly to zero as the maximum subregion angle or diameter decreases.
- Algorithm 2 converges in a finite number of iterations to a value arbitrarily close to the optimum.
- For basic rectangular subdivision, Theorem 2 gives a worst-case iteration bound for reaching a δ-optimal solution.
- Both BnB variants have exponential worst-case complexity in N, although tight bounds usually permit termination after few iterations in simulations.
VI. NUMERICAL RESULTS
The numerical evaluation uses fixed transmit power, standardized channel statistics, a 16-antenna half-wavelength ULA, and unit-power QPSK symbols.
- PT = 1 is used throughout the simulations.
- Each channel-matrix entry follows a standard Complex Gaussian distribution, hi,j ∼CN (0, 1).
- The simulations use N = 16 antennas arranged as a ULA with half-wavelength adjacent-antenna spacing.
- Communication users employ the unit-power QPSK alphabet, so every symbol-matrix entry has power 1.
A. Dual-functional Waveform Design with Given Radar Beampatterns
The paper develops waveform designs that reduce downlink multi-user interference while satisfying omnidirectional or directional radar-beampattern requirements, then extends them to radar-communication trade-offs and constant-modulus constraints.
- Given Radar Beampatterns: The proposed omnidirectional and directional strict waveform designs outperform communication-only ZF precoding while retaining comparable computational costs.The strict designs produce the desired directional beampattern exactly; their computational costs remain at the ZF level.
- Given Radar Beampatterns: A weighting factor ρ = 0.1 substantially increases sum-rates toward the AWGN capacity, while radar beampatterns experience only slight performance loss.This weighted design provides a flexible radar-communications trade-off rather than enforcing strict radar equality constraints.
- Radar-Communication Trade-offs: The omnidirectional and directional designs exhibit trade-offs between communication rate and radar performance, measured by detection probability and beampattern MSE, respectively.For fixed detection probability, achievable rate increases as the number of users decreases, consistent with reduced multi-user-interference energy from additional degrees of freedom.
- Constant-Modulus Waveform Design: The branch-and-bound algorithm converges in a finite number of iterations for the constant-modulus waveform problem under the tested subdivision rules.The convergence experiment uses N = 16, K = 4, and ε = 1, comparing ARS and BRS subdivision rules.
- Constant-Modulus Waveform Design: The proposed constant-modulus branch-and-bound design outperforms SQR-BS by obtaining the global optimum and approaches AWGN capacity when similarity tolerance ε is sufficiently large.Its communication-rate performance is close to the convex-relaxation bound, while pulse-compression results remain nearly the same as SQR-BS under the same waveform-similarity constraint.
APPENDIX DERIVATION OF THE PROJECTOR PR2
The appendix derives the projector PR2 for a circular-segment feasible region by partitioning the complex plane and selecting the nearest feasible point according to the region containing X.
- Projector Construction: The circular segment is parameterized by endpoint angles l and u, with A = exp(jl), B = exp(ju), and midpoint T = (A + B) / 2.The derivation distinguishes cases where the open angle φ is less than or greater than π.
- Projector Construction: For X in the feasible region M1, the projection is X itself; for X in M2 or M3, the nearest projection is endpoint A or B, respectively.These cases preserve feasible points and assign points in the adjacent regions to the corresponding segment endpoints.
- Projector Construction: For X in M4, the method projects X onto line AB and uses the resulting point as the projection, with normalization applied where specified.The supporting lines OA and OB are defined through f2(X) and f3(X), while the projector is assembled piecewise.