Ambiguity Function Shaping via Manifold Optimization Embedding With Momentum

The design of the constant-modulus waveform for Doppler ambiguity function shaping in slow-time domain is the key technology in cognitive radar systems. The design problem is the non-convex fourth-order optimization problem with constant-modulus constraints on the waveform. The existing methods mainly solve the problem by relaxing the objective function, but the relaxation will inevitably lead to relaxation error. We noticed that the original problem is a natural fourth-order unconstrained problem on the Complex Circle Manifold (CCM), and its optimization process is equivalent to a particle moving on a high-dimensional plane. Based on this feature, we propose the Manifold Optimization Embedding with Momentum (MOEM) method without relaxation. We first derive the momentum update strategy, then adaptively update the descent step size, and finally optimize on the manifold to ensure the monotonicity and convergence of the objective function. Compared with the existing state-of-the-art methods, the proposed method improves the Signal-to-Interference Ratio (SIR) by 65 dB with less computational cost.