Ambiguity function shaping via quartic descent on the complex circle manifold

Controlling the range-Doppler response, i.e. Ambiguity Function (AF) continues to be of great interest in cognitive radar. The design problem is known to be a nonconvex quartic function of the transmit radar waveform. This AF shaping problem becomes even more challenging in the presence of practical constraints on the transmit waveform such as the Constant Modulus Constraint (CMC). Most existing approaches address the aforementioned challenges by suitably modifying or relaxing the design cost function and/or CMC. In a departure from such methods, we develop a solution that involves direct optimization over the non-convex complex circle manifold, i.e. the CMC set. We derive a new update strategy (Quartic-Gradient-Descent (QGD)) that computes an exact gradient of the quartic cost and invokes principles of optimization over manifolds towards an iterative procedure with guarantees of monotonic cost function decrease and convergence. Experimentally, QGD can outperform state of the art approaches for shaping the ambiguity function under CMC while being computationally less expensive.