Quartic Gradient Descent for Tractable Radar Slow-Time Ambiguity Function Shaping

We consider the problem of minimizing the disturbance power at the output of the matched filter in a single antenna cognitive radar set-up. The aforementioned disturbance power can be shown to be an expectation of the slow-time ambiguity function (STAF) of the transmitted waveform over range-Doppler bins of interest. The design problem is known to yield a nonconvex quartic function of the transmit radar waveform. This STAF 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 the CMC. In a departure from such methods, we develop a solution that involves direct optimization over the nonconvex 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 toward 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 the CMC while being computationally less expensive.