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Transmit MIMO Radar Beampattern Design Via Optimization on the Complex Circle Manifol

Khaled Alhujaili, Vishal Monga, Muralidhar Rangaswamy

arXiv:1904.07329v1eess.SP

TL;DR

MIMO radar beampattern design seeks desired-pattern fidelity under the difficult non-convex constant modulus constraint. The paper directly optimizes over the complex circle using PDR, proving monotonic improvement and convergence for quadratic costs. PDR outperforms competing methods in fidelity and computational cost, while its orthogonality extension supports robustness to target-direction mismatch.

  • Problem

    MIMO radar must design waveforms that match desired beampatterns despite the difficult, practically important non-convex constant modulus constraint.

  • Method

    The paper directly optimizes over the complex circle using a projection, descent, and retraction algorithm, extending the cost with an orthogonality penalty when needed.

  • Results

    PDR outperforms competing methods in deviation from the desired beampattern while requiring lower computational cost.

  • Takeaways & Limitations

    PDR provides a tractable constant-modulus beampattern-design framework with monotonic improvement, convergence guarantees, and an orthogonality extension associated with target-mismatch robustness.

Abstract

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The ability of Multiple-Input Multiple-Output (MIMO) radar systems to adapt waveforms across antennas allows flexibility in the transmit beampattern design. In cognitive radar, a popular cost function is to minimize the deviation against an idealized beampattern (which is arrived at with knowledge of the environment). The optimization of the transmit beampattern becomes particularly challenging in the presence of practical constraints on the transmit waveform. One of the hardest of such constraints is the non-convex constant modulus constraint, which has been the subject of much recent work. In a departure from most existing approaches, we develop a solution that involves direct optimization over the non-convex complex circle manifold. That is, we derive a new projection, descent, and retraction (PDR) update strategy that allows for monotonic cost function improvement while maintaining feasibility over the complex circle manifold (constant modulus set). For quadratic cost functions (as is the case with beampattern deviation), we provide analytical guarantees of monotonic cost function improvement along with proof of convergence to a local minima. We evaluate the proposed PDR algorithm against other candidate MIMO beampattern design methods and show that PDR can outperform competing wideband beampattern design methods while being computationally less expensive. Finally, orthogonality across antennas is incorporated in the PDR framework by adding a penalty term to the beampattern cost function. Enabled by orthogonal waveforms, robustness to target direction mismatch is also demonstrated.

I. INTRODUCTION

MIMO radar beampattern design must balance fidelity to a desired pattern with practical waveform constraints, especially the non-convex constant modulus constraint. The paper introduces PDR optimization on the complex circle, establishes guarantees for quadratic costs, and reports improved fidelity, lower computational cost, and robustness with orthogonal waveforms.

  • Motivation: The constant modulus constraint is practically important for radar amplifiers but makes transmit beampattern design difficult, motivating direct optimization methods.Existing indirect approaches approximate or relax the constraint, while direct methods can be computationally expensive.
  • Proposed approach: PDR projects the cost gradient onto the complex-circle tangent space, descends there, and retracts the result back to the feasible constant-modulus set.
  • Guarantees: For quadratic beampattern costs, the proposed updates monotonically decrease the cost and converge to a local minimum.
  • Orthogonal waveform design: PDR incorporates antenna orthogonality by adding a penalty term to the beampattern cost while optimizing over the complex circle manifold.
  • Validation: Simulations show better desired-beampattern fidelity at lower computational cost than competing methods, with orthogonality providing robustness to target-direction mismatch.

III. CONSTANT MODULUS CONSTRAINT AND OPTIMIZATION OVER MANIFOLDS

Under constant modulus constraints, the feasible waveform set is modeled as a complex circle manifold. The paper uses manifold optimization to develop a direct, feasible beampattern-design procedure.

  • Complex circle manifold: The feasible set is the product of L complex circles, with each waveform entry constrained to unit modulus.
  • Complex circle manifold: The complex circle manifold is defined as the set of vectors in C^L whose entries all have magnitude one.
  • Optimization strategy: The paper applies manifold optimization directly to this feasible set to avoid leaving the constant-modulus constraint during optimization.

A. Optimization over manifolds

Optimization over manifolds adapts classical line-search methods by projecting gradients onto tangent spaces, descending there, and retracting updates back onto the manifold.

  • A. Optimization over manifolds: Manifold optimization modifies classical line-search methods so updates remain compatible with a constraint manifold.The objective is defined on a manifold M, with a smooth real-valued cost function g(x).
  • A. Optimization over manifolds: The intrinsic gradient is obtained by projecting the Euclidean gradient onto the tangent space at the current point.This projection supplies a feasible search direction for descent on the manifold.
  • A. Optimization over manifolds: After tangent-space descent produces an off-manifold point, a retraction maps the update back onto the manifold.The update is first formed as x̄(k) = x(k) − βP_Tx(k)M(∇xg(x(k))), then retracted.

B. Complex circle manifold

The complex circle manifold is handled element-wise by relating each complex entry to a point on the real unit circle, where projection and retraction operators can be defined.

  • B. Complex circle manifold: Each feasible complex vector entry corresponds to a point on the real unit circle, enabling element-wise manifold operators.A feasible vector lies in the product of unit-circle components, with each component satisfying a unit-norm condition.
  • B. Complex circle manifold: Projection maps a vector component onto the tangent space of the unit circle at the corresponding feasible point.The tangent space consists of real two-dimensional vectors orthogonal to the point on the unit circle.
  • B. Complex circle manifold: Retraction maps an arbitrary component back to the unit circle, restoring feasibility after an update.The projection and retraction operators are illustrated in Figure 2.

C. Projection, Descent and Retraction (PDR) algorithm

PDR solves constant-modulus beampattern design by projecting a descent direction onto the complex-circle tangent space, updating there, and retracting to feasibility.

  • C. Projection, Descent and Retraction (PDR) algorithm: PDR alternates projection, tangent-space descent, and retraction so successive iterates remain on the complex circle manifold.The intermediate tangent-space point may violate constant modulus, but the next iterate is retracted back to the feasible set.
  • C. Projection, Descent and Retraction (PDR) algorithm: PDR applies this procedure to the quadratic beampattern cost using its gradient and a stopping threshold.Algorithm 1 evaluates the search direction, projects it, computes a tangent-space update, retracts it, and checks cost change.
  • C. Projection, Descent and Retraction (PDR) algorithm: PDR has per-iteration complexity O(L^2) for large L, dominated by the gradient computation.Projection and retraction each have O(L) complexity.
  • C. Projection, Descent and Retraction (PDR) algorithm: PDR has lower complexity than SDR and similar per-iteration complexity to ADMM and IA-CPC, while those methods require more iterations to match its performance.The comparison is reported for constant-modulus-constrained state-of-the-art methods in Table I.
  • C. Projection, Descent and Retraction (PDR) algorithm: The beampattern-design implementation alternates phase-related updates with PDR optimization over the waveform vector.Algorithm 2 invokes Algorithm 1 after updating auxiliary quantities used in practical beampattern design.

D. Convergence Analysis

For the complex-circle problem, suitable step-size and penalty conditions yield non-increasing costs and convergence, while orthogonality is encouraged through an added penalty term.

  • D. Convergence Analysis: With the stated step-size conditions, the PDR cost sequence is non-increasing and converges to a finite value because the cost is bounded below.The result follows from the descent and retraction lemmas together with nonnegativity of the original cost.
  • D. Convergence Analysis: The proposed gradient update preserves complex-circle feasibility while guaranteeing monotonic cost decrease and convergence for this structured problem.The paper cautions that these guarantees may not generalize to other non-convex manifolds.
  • D. Convergence Analysis: Orthogonality and constant modulus define an intersection that is difficult to optimize directly, so orthogonality is encouraged with a penalty term.The penalty is applied to the waveform matrix and optimized using the PDR procedure with a modified cost function.

IV. NUMERICAL RESULTS

Numerical simulations evaluate PDR for constant-modulus beampattern design across three desired-pattern cases and compare it with state-of-the-art methods. Across these cases, PDR achieves the closest designs to the unconstrained or desired patterns while offering a favorable complexity-performance trade-off.

  • Experimental setup: The evaluation covers three specifications: angular-only illumination, angle-frequency dependence, and restricted transmission bands for spectrally crowded scenarios.The simulations use a ULA MIMO radar with 10 transmit antennas and 32 time samples.
  • Case 1: 2.13 dB, 2.7 dB, and 5.5 dB gains over ADMM, SDR, and IA-CPC, respectively, are reported for PDR in Case 1.The gains refer to deviation from the desired beampattern, with lower deviation indicating better fidelity.
  • Case 1: In Case 1, PDR produces the beampattern closest to the unconstrained design, which serves as a performance bound.Figure 4 visualizes the unconstrained design and the competing constrained methods.
  • Case 2: In Case 2, PDR is closest to the unconstrained case with a 2 dB gap over the second-best method.The corresponding visualization also reports PDR as closer to the desired pattern than competing methods.
  • Case 3: PDR outperforms competing methods in deviation from the desired beampattern for Case 3 and is computationally bettered only by WBFIT overall.The reported comparison identifies PDR as offering the most favorable complexity-performance trade-off, with nearly 7 dB gain over WBFIT.

B. Joint CMC and orthogonality constraints

PDR is extended to jointly enforce constant modulus and waveform orthogonality through a penalty formulation. The resulting waveforms preserve close beampattern matching and exhibit robustness to target-direction mismatch.

  • Joint constraint performance: PDR achieves the closest desired-beampattern match under joint constant-modulus and orthogonality constraints, with SimulAnneal second best.The constrained feasible set is smaller, so deviation is higher than under constant modulus alone.
  • Joint constraint performance: At α = 100, PDR produces waveforms orthogonal for practical purposes, as indicated by the ISL measure.Increasing α places stronger emphasis on orthogonality in the penalized design.
  • Robustness to direction mismatch: Orthogonality is motivated by its increased virtual-array benefits and small transmit-receive gain loss under target-direction mismatch.The transmit-receive pattern measures response for a steered beam and a target at a potentially different angle.
  • Robustness to direction mismatch: PDR with α = 200 achieves direction-mismatch behavior comparable to LFM, while WBFIT, IA-CPC, and coherent transmission lose mainlobe strength.The comparison considers target offsets of 0°, 10°, and 20° from the transmit direction.
  • Scope: The paper focuses on transmit waveform design, leaving joint optimization of receive processing for future work.This bounds the scope of the reported transmit-receive-pattern results.

APPENDIX

The appendix develops the analytical machinery for PDR’s projected descent step on the complex circle manifold. It derives the tangent-space update and establishes a step-size condition supporting nonnegative cost improvement.

  • Projection and descent: The projected search direction is computed in the tangent space at the current constant-modulus iterate before applying a step β.The gradient is represented as w = 2Rx(k) − 2q, and the new projection formula supplies the tangent-space direction.
  • Quadratic analysis: The appendix establishes the quadratic cost representation on the tangent space using R = P + γI and the current tangent-space iterate.The proof uses the positive-semidefinite structure of the matrices involved.
  • Monotonicity condition: A nonnegative step β satisfying the positive-semidefinite condition on I − βΛ ensures nonnegative quadratic cost improvement.The bound depends on λR, the largest eigenvalue of the positive-definite matrix R.

B. Proof of Lemma 3.2

The proof of Lemma 3.2 analyzes the retraction from a tangent-space point back to the constant-modulus manifold. It uses matrix positivity and norm inequalities to bound the resulting cost-function difference.

  • Retraction: Retraction maps the tangent-space update, which is outside the manifold, to a constant-modulus iterate x(k+1) ∈ SL.The tangent-space input has entries with magnitude at least one, while the retracted output is feasible.
  • Cost bound: The resulting inequalities connect the retraction representation to a sufficient condition for the desired cost comparison.The proof introduces Ψ as a nonnegative diagonal matrix in the retraction-based representation.
  • Cost bound: The cost-difference proof relies on positive semidefiniteness of P + γI and Ψ, together with the constant-modulus identity for x(k+1).The argument bounds terms using the largest eigenvalues λΨ and λP and standard norm inequalities.

C. Proof of Lemma 3.3

The proof establishes that the iterative cost sequence is non-increasing and converges to a finite value because the cost is bounded below by zero.

  • The iterative cost sequence is non-increasing and converges to a finite value because f(x) ≥ 0 for all x.

D. The equivalence between Eq (42) and the retraction step

This subsection shows that Eq (42) and the retraction step are equivalent by rewriting the retracted vector through component magnitudes and substitution.

  • The derivation establishes equivalence between Eq (42) and Algorithm 1's retraction step.The proof starts from the retraction formula, solves for ¯x(k), and substitutes the resulting diagonal matrix expression.
  • Because each component magnitude satisfies |¯x_l(k)| ≥ 1, it is represented as 1 + ψ_l with ψ_l ≥ 0.
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