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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
Dual-functional MIMO radar-communication waveform design must minimize downlink multi-user interference while meeting radar-specific constraints. The paper proposes omnidirectional, directional, weighted, and constant-modulus designs; numerical results show the proposed methods outperform zero-forcing precoding while maintaining radar and communications performance.
Problem
The design challenge is to minimize downlink multi-user interference in dual-functional radar-communication systems while satisfying radar-specific constraints.
Source: §I · p. 2Method
The paper develops closed-form omnidirectional and directional designs, a weighted trade-off optimization, and a branch-and-bound algorithm for constant-modulus waveforms with similarity constraints.
Source: §I · p. 2, C. Upper-bound and Lower-bound… · p. 7Results
Numerical results show all proposed methods outperform zero-forcing precoding while guaranteeing radar and communication performance; the first three designs have computational costs at the same level as zero-forcing.
Source: C. Complexity Analysis · p. 4, §VII · p. 11Takeaways & Limitations
The proposed waveforms can be used for both target detection and downlink communications, with the weighted design offering a flexible radar-communications performance trade-off.
Source: §I · p. 2
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 · II. SYSTEM MODEL · A. Communication Model
The paper develops dual-functional MIMO radar-communication waveform designs that minimize downlink multi-user interference under radar constraints. Its system model links this objective to downlink sum-rate and assumes shared radar-communication waveforms and a fixed, perfectly estimated channel during each frame.
I. INTRODUCTION
- Radar-communication spectrum sharing is motivated by projected growth in connected devices and aims to reuse spectrum occupied by radar applications –.The introduction presents spectrum sharing as a way to use scarce spectrum more efficiently.Source: §I · p. 1
- Dual-functional RadCom combines radar detection and communications, avoiding the control-center coordination and side-information exchanges required by earlier coexistence schemes.Earlier waveform approaches could also limit performance or data rate, while the cited cellular-downlink beamforming work did not address constant-modulus signals.Source: §I · p. 2
- Closed-form omnidirectional and directional waveform designs and a low-complexity global weighted optimization enable trade-offs between radar and communication performance.The designs minimize downlink multi-user interference under radar-specific constraints.Source: §I · p. 2
- The constant-modulus and similarity-constrained design uses branch-and-bound for a globally optimal solution and outperforms the conventional SQR algorithm.Prior constant-modulus approaches, including SQR, provide only local optimality guarantees.Source: §I · p. 2
II. SYSTEM MODEL
- The system uses a uniform linear array with N antennas to serve K single-antenna downlink users while detecting radar targets simultaneously.The transmitted probing waveforms and communication symbols operate in the same dual-functional MIMO system.Source: §II · p. 3
A. Communication Model
- The transmitted signal matrix serves as both the radar waveform and communication signal; the channel is flat Rayleigh fading, constant per frame, and perfectly estimated from pilots.The communication frame has length L, and the received-symbol model includes Gaussian noise.Source: A. Communication Model · p. 3
- For a fixed-energy constellation, minimizing total multi-user interference maximizes the users’ achievable sum-rate.The passage identifies the second received-signal term as MUI and links MUI energy to downlink sum-rate.Source: A. Communication Model · p. 3
B. Radar Model · III. CLOSED-FORM WAVEFORM DESIGN FOR GIVEN RADAR BEAMPATTERNS · A. Omnidirectional Beampattern Design
The section formulates dual-functional waveform design as minimizing multi-user interference under radar-specific constraints and gives an SVD-based global solution for the omnidirectional case.
B. Radar Model
- Uncorrelated waveforms give MIMO radar higher degrees of freedom than traditional phased-array radar, and beampattern design can be treated as probing-signal covariance design.This work focuses on designing the dual-functional waveform matrix X and its spatial covariance.Source: B. Radar Model · p. 3
- The radar model assumes L ≥ N to ensure RX is positive-definite and defines the transmit array steering vector using adjacent-antenna spacing normalized by wavelength.Source: B. Radar Model · p. 3
- The RadCom waveform optimization is framed as minimizing PMUI subject to MIMO radar-specific constraints.Source: B. Radar Model · p. 3
III. CLOSED-FORM WAVEFORM DESIGN FOR GIVEN RADAR BEAMPATTERNS
- The section considers omnidirectional beampattern design for initial MIMO radar probing, followed by directional design aimed at directions of interest.Source: §III · p. 3
A. Omnidirectional Beampattern Design
- The omnidirectional design minimizes MUI energy subject to an orthogonal waveform matrix, whose covariance is the identity; its non-convex Stiefel-manifold problem is an Orthogonal Procrustes problem with a closed-form global SVD solution.The solution uses the SVD of H^H S and the rectangular identity matrix I_N×L.Source: A. Omnidirectional Beampattern Design · p. 4
B. Directional Beampattern Design · C. Complexity Analysis
The directional design transforms MUI minimization under a desired radar covariance into an equivalent orthogonal Procrustes problem and obtains a globally optimal solution. Its computational cost is of the same order of magnitude as conventional zero-forcing precoding.
B. Directional Beampattern Design
- The design starts from a desired Hermitian positive semidefinite covariance matrix Rd with trace equal to transmit power PT.The formulation minimizes MUI while matching the specified radar covariance.Source: B. Directional Beampattern Design · p. 4
- A Cholesky factorization rewrites the covariance constraint; the derivation assumes Rd is positive-definite so the lower-triangular factor F is invertible.Source: B. Directional Beampattern Design · p. 4
- The directional design obtains a globally optimal solution by converting the covariance-constrained problem into an orthogonal Procrustes problem and using an SVD-based construction.The original solution is recovered from the SVD result.Source: B. Directional Beampattern Design · p. 4
C. Complexity Analysis
- The omnidirectional design uses two matrix multiplications and one SVD.Source: C. Complexity Analysis · p. 4
- The directional design uses one Cholesky decomposition, four matrix multiplications, and one SVD.Source: C. Complexity Analysis · p. 4
- Its stated complexity is O(2NL^2 + N^2L + NKL + N^3 + N^2K).Source: C. Complexity Analysis · pp. 4–5
- The proposed closed-form approaches have computational costs of the same order of magnitude as conventional zero-forcing precoding.The ZF baseline uses one pseudoinverse of H and a matrix multiplication between the precoder and transmitted symbols.Source: C. Complexity Analysis · p. 4
IV. TRADE-OFF BETWEEN RADAR AND COMMUNICATION PERFORMANCE · A. Problem Formulation · B. Low-complexity Algorithm
Strict radar-beampattern equality can leave communication performance poor, especially for ill-conditioned channel covariance matrices, motivating a design that tolerates beampattern mismatch. The proposed trade-off formulation preserves the power budget and is solved globally by a low-complexity search.
IV. TRADE-OFF BETWEEN RADAR AND COMMUNICATION PERFORMANCE
- Strict beampattern equality can guarantee optimal radar performance while leaving communication performance poor, particularly with ill-conditioned channel covariance matrices.The trade-off design therefore allows tolerable mismatch between designed and desired radar beampatterns.Source: §IV · p. 4
A. Problem Formulation
- The trade-off formulation weights radar and communication performance by ρ, 0 ≤ ρ ≤ 1, and enforces equality in the power budget.It begins from the optimal solution X0 of problem (8) or (10); the equality constraint reflects radar’s practical need to transmit at maximum available power.Source: A. Problem Formulation · p. 4
- The formulation is a non-convex QCQP whose SDR is tight and globally optimal, but its large number of variables makes SDR computationally inefficient.The tightness follows from having only one quadratic constraint.Source: A. Problem Formulation · p. 5
B. Low-complexity Algorithm
- With Hermitian Q, the reformulated problem is a matrix trust-region subproblem with zero duality gap.The strong-duality result is attributed to.Source: B. Low-complexity Algorithm · p. 5
- The algorithm uses the TRS optimality conditions to obtain Xopt from the optimal dual variable λopt.The conditions include stationarity and primal and dual feasibility; the solution uses the Moore-Penrose pseudoinverse.Source: B. Low-complexity Algorithm · p. 5
- A unique λopt is found by line search on a strictly decreasing, convex function over λ ≥ −λmin, using eigenvalue decomposition to reduce each iteration to diagonal-matrix inversion.The paper gives Golden-section search as an example and computes Xopt = (Q + λoptIN)†G.Source: B. Low-complexity Algorithm · p. 5
C. Complexity Analysis
Algorithm 1’s complexity is dominated by matrix multiplications, the pseudoinverse, and eigenvalue decomposition; its total cost is stated to be of the same order as communication-only ZF precoding.
C. Complexity Analysis
- O(log(1/ε0)) iterations suffice for the Golden-section search to find an ε0-solution, and its per-iteration one-dimensional function evaluation is generally omitted from the complexity analysis.The analysis therefore treats Algorithm 1’s complexity as dominated by its matrix operations and decompositions.Source: C. Complexity Analysis · p. 5
- The matrix multiplications in Algorithm 1 involve complexity O(N^2L + NKL + N^3 + N^2K).Table I summarizes the computational costs of the three proposed waveform design approaches.Source: C. Complexity Analysis · p. 5
- O(N^2L + NKL + 3N^3 + N^2K) is Algorithm 1’s total complexity, which shares the same order of magnitude as communication-only ZF precoding.The analysis says matrix multiplications, the pseudoinverse, and eigenvalue decomposition dominate the cost.Source: C. Complexity Analysis · p. 5
V. CONSTANT MODULUS WAVEFORM DESIGN · A. Problem Formulation
This section formulates constant-modulus RadCom waveform design to minimize communication MUI energy, addressing a limitation of the preceding total-power-constrained designs. Its problem structure enables column-wise solution, while the per-entry feasible region is non-convex, motivating a branch-and-bound approach.
V. CONSTANT MODULUS WAVEFORM DESIGN
- The design minimizes communication MUI energy subject to a constant-modulus constraint, unlike the preceding total-power-constrained designs, which do not guarantee constant-modulus signals.Source: §V · p. 5
A. Problem Formulation
- The similarity constraint limits deviation from a constant-modulus benchmark radar matrix X0, such as a chirp signal, with η specifying the tolerable difference.Source: A. Problem Formulation · p. 6
- The objective is separable, allowing the formulation to be simplified using a normalized vector variable.Source: A. Problem Formulation · p. 6
- The full problem can be solved by concurrently solving the corresponding problem for each column of X.Source: A. Problem Formulation · p. 6
- Because x and x0 have unit modulus, the parameter ε lies in the range 0 ≤ ε ≤ 2, and the similarity constraint can be rewritten accordingly.Source: A. Problem Formulation · p. 6
- Each entry’s feasible region is an arc on the unit circle, making the problem non-convex and generally NP-hard; the paper therefore considers a branch-and-bound algorithm for global optimization.Source: A. Problem Formulation · p. 6
B. The Branch-and-bound Framework
The branch-and-bound (BnB) framework partitions the feasible region into subproblems, updates lower and upper bounds, and repeats until their gap closes. Finite convergence requires appropriate branching, subdivision, and bounding conditions.
B. The Branch-and-bound Framework
- BnB repeatedly branches on the subproblem with the smallest lower-bound, computes bounds for its two children, and updates the global bounds until their gap is at most the tolerance.Optional pruning limits the stored problem set without affecting the procedure’s effectiveness.Source: B. The Branch-and-bound Framework · pp. 6–7
- The worst-case BnB complexity is exponential in N, but tight bounds can identify and prune unqualified branches to accelerate the algorithm.The optional pruning step saves memory and does not affect the procedure’s effectiveness.Source: B. The Branch-and-bound Framework · pp. 6–7
- Finite convergence requires branching on the smallest lower-bound, exhaustive subdivision whose maximum subregion length tends to zero, and bounds consistent with branching.Under these conditions, the upper-bound approaches the original problem’s optimal value as the subregions shrink.Source: B. The Branch-and-bound Framework · p. 7
- Both basic and adaptive rectangular subdivision satisfy the exhaustive-subdivision condition, while simulations show that adaptive subdivision converges faster.Each rule equally divides the selected arc while leaving the other arcs unchanged.Source: B. The Branch-and-bound Framework · p. 7
C. Upper-bound and Lower-bound Acquisition
This section develops lower- and upper-bound acquisition methods for accelerating branch-and-bound, including convex relaxation, projection, and gradient-based alternatives.
Lower-bound acquisition
- The lower bound is computed by convexly relaxing (32) over each entry’s convex hull, yielding a QCQP solvable with numerical tools such as CVX.Source: C. Upper-bound and Lower-bound… · p. 7
Upper-bound acquisition
- Projecting each relaxed solution entry onto its corresponding arc gives a feasible upper-bound candidate.For a tighter bound, the projected candidate initializes a non-convex QCQP solved locally by fmincon; descent guarantees a smaller objective value than the initial point. For QP-UB, the gradient iteration projects onto the arc with PR1 rather than using interpolation.Source: C. Upper-bound and Lower-bound… · p. 8
Accelerated gradient projection
- Accelerated gradient-projection methods use a convex-hull projector and, for least squares, a step size equal to the reciprocal of the maximum eigenvalue of H̃ᴴH̃.The scheme’s interpolation applies only to convex feasible regions; the QP-UB variant instead uses PR1 to project onto the feasible arc.Source: C. Upper-bound and Lower-bound… · p. 8
Computational cost
- Gradient-based methods cost O(NK) per iteration and are far more efficient for a fixed iteration number.Source: C. Upper-bound and Lower-bound… · p. 8
D. Convergence Analysis and Worst-case Complexity
The analysis shows that Algorithm 2’s upper and lower bounds become uniformly tight as subdivisions shrink, supporting finite convergence to a value arbitrarily close to optimal. It also characterizes BRS iteration complexity and reports that simulations usually terminate in few iterations despite exponential worst-case costs.
D. Convergence Analysis and Worst-case Complexity
- As φ_max or d_max approaches zero, UB−LB becomes uniformly arbitrarily small, establishing consistent bounds for Algorithm 2’s subdivision rules.For every δ > 0, the lemma gives thresholds η1 and η2 such that φ_max ≤ η1 or d_max ≤ η2 implies UB−LB ≤ δ.Source: D. Convergence Analysis and Worst-case… · pp. 8–9
- Algorithm 2 converges in finitely many iterations to a value arbitrarily close to fopt.The proof links finite convergence to the consistency of the bounds under the two subdivision rules.Source: D. Convergence Analysis and Worst-case… · p. 9
- For BRS, Theorem 2 bounds iterations to reach a δ-optimal solution; BnB-ARS and BnB-BRS have exponential worst-case costs in N, but simulations usually terminate in few iterations due to tight bounds.The supplied theorem statement omits the iteration-bound expression, so its specific value cannot be reported.Source: D. Convergence Analysis and Worst-case… · p. 9
VI. NUMERICAL RESULTS
The numerical section evaluates the proposed waveform designs using simulations with a 16-antenna ULA and unit-power QPSK communication symbols.
VI. NUMERICAL RESULTS
- Simulations use P_T = 1, standard complex Gaussian channel entries, a 16-element half-wavelength-spaced ULA, and unit-power QPSK symbols.Each channel entry h_i,j follows CN(0, 1), and each symbol-matrix entry has power 1.Source: §VI · p. 9
A. Dual-functional Waveform Design with Given Radar Beampatterns · B. Dual-functional Constant Modulus Waveform Design · VII. CONCLUSION
The paper presents dual-functional waveform designs that balance downlink communications with radar performance, including fixed-beampattern designs and a globally optimized constant-modulus design. Simulations report improved communication performance over ZF or SQR-BS while preserving radar constraints or trading some radar performance for higher rates.
A. Dual-functional Waveform Design with Given Radar Beampatterns
- Strict omnidirectional and directional designs outperform ZF, while the strict directional design matches its desired beampattern and ρ = 0.1 trade-off designs raise sum-rate with only slight radar-pattern loss.The trade-off sum-rates approach the zero-MUI AWGN capacity.Source: Figure 3 · p. 9, A. Dual-functional Waveform Design… · pp. 9–10, Figure 4 · p. 9
- Communication rate trades off against radar detection probability for omnidirectional designs and beampattern MSE for directional designs; at fixed detection probability, fewer users yield higher achievable rate.The reported interpretation is that increasing degrees of freedom can further minimize multi-user-interference energy.Source: A. Dual-functional Waveform Design… · p. 10, Figure 5 · p. 10, Figure 6 · p. 10
B. Dual-functional Constant Modulus Waveform Design
- BnB-ARS converges faster than BnB-BRS, and both subdivision rules converge in finitely many iterations, but BnB is needed to confirm global optimality for the non-convex problem.The passage notes that local algorithms such as a QCQP solver or gradient projection can be iterated to reach the optimal value, while BnB certifies it.Source: B. Dual-functional Constant Modulus… · p. 10, Figure 7 · p. 10
- BnB significantly outperforms SQR-BS, approaches the QP-LB convex-relaxation bound, and can approach AWGN capacity at sufficiently large similarity tolerance ε.The paper attributes the advantage over SQR-BS to BnB obtaining the global optimum while SQR-BS yields local minima.Source: B. Dual-functional Constant Modulus… · pp. 10–11, Figure 8 · p. 11, §VII · p. 12
- Constant-modulus designs trade communication sum-rate against radar waveform similarity and pulse-compression performance; BnB and SQR-BS yield nearly identical pulse-compression results under the same similarity constraint.Pulse compression uses the first-antenna waveform and FFT-IFFT with a Taylor window to reduce sidelobes.Source: B. Dual-functional Constant Modulus… · p. 11, Figure 9 · p. 11
VII. CONCLUSION
- The constant-modulus method uses a branch-and-bound framework with gradient-projection algorithms to obtain upper and lower bounds for global optimization under similarity constraints.The conclusion reports that this algorithm considerably outperforms SQR-BS by obtaining the global optimum.Source: §VII · pp. 11–12
- The three fixed-beampattern and weighted designs have computational costs at the level of communication-only ZF precoding while providing radar and communication performance.The conclusion also reports that the weighted design can significantly improve communication performance with slight radar-performance loss.Source: §VII · p. 11
APPENDIX DERIVATION OF THE PROJECTOR PR2
The appendix derives PR2 by partitioning the complex plane according to the angle interval and selecting the nearest point in M1. For φ ≥ π, the projector is unchanged except for the definition of f1.
APPENDIX DERIVATION OF THE PROJECTOR PR2
- For an open angle below π, the complex plane is divided into five regions using bounds l and u, with A = exp(jl), B = exp(ju), and midpoint T = (A + B)/2.The projection seeks a nearest point in M1 and leaves points already in M1 unchanged.Source: APPENDIX DERIVATION OF THE PROJECTOR PR2 · p. 12
- The piecewise projector returns A for M2, B for M3, the perpendicular foot on AB for the specified M4 region, and otherwise the normalized input X/|X|.Points in M1 project to themselves; the M4 foot XT lies on AB and satisfies XXT ⊥ AB.Source: APPENDIX DERIVATION OF THE PROJECTOR PR2 · p. 12
- For φ ≥ π, the projector remains the same, but f1(X) receives a different definition.The supplied passage does not include that alternate definition.Source: APPENDIX DERIVATION OF THE PROJECTOR PR2 · p. 12