Quartic Riemannian Trust Region Algorithm for Cognitive Radar Ambiguity Function Shaping

Waveform adaptation grants cognitive radar (CR) the ability to adapt to its environment, which requires an effective framework to synthesize waveforms sharing a desired ambiguity function (AF). In this letter, we propose a novel method for shaping the slow-time AF in order to adaptively suppress the interference power. The problem is formulated as minimizing the average value of the slow-time AF over some range Doppler bins spanned by the interference, which can be identified exploiting a plurality of knowledge sources. From a technical point of view, this is tantamount to optimizing a complex quartic-order polynomial with a constant modulus (CM) constraint on each optimization variable. To solve this problem, we proposed a quartic Riemannian trust region algorithm. This algorithm first transforms the optimization into an unconstrained one in a complex circle Riemannian manifold, then devises a new Riemannian trust region optimization algorithm that invokes Riemannian gradient and Hessian matrix to obtain an iterative solution with super-linear convergence rate and ability to escape potential saddle points. Simulation results demonstrated that our proposed algorithm outperformed state-of-the art approaches for AF shaping while being computationally less expensive.