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

## TL;DR

This paper introduces a novel optimization method on the complex circle manifold for MIMO radar beampattern design, effectively handling non-convex constant modulus constraints and improving performance over existing methods.

## Contribution

It develops a new projection, descent, and retraction (PDR) algorithm for direct optimization on the complex circle manifold, with analytical guarantees and enhanced robustness.

## Key findings

- PDR outperforms existing wideband beampattern design methods.
- The algorithm guarantees monotonic cost reduction and convergence to local minima.
- Orthogonality and robustness to target mismatch are successfully incorporated.

## Abstract

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.