Constrained Riemannian Manifold Optimization for the Simultaneous Shaping of Ambiguity Function and Transmit Beampattern

Designing the transmit waveforms with prescribed ambiguity functions (AFs) and beampatterns while adhering to the constant modulus (CM) constraint is pivotal for the forthcoming cognitive multiple-input multiple-output (MIMO) radar systems. This study delves into the AF shaping quandary within the MIMO radar framework, considering the joint constraints of waveform unimodality and desired beampattern. The established model explores higher dimensions to realize the waveform design in range–Doppler and spatial dimensions, to improve the possibility of separating target and interference. Specifically, we first formulate the waveform design problem as a jointly constrained quartic problem, with the aim of minimizing the response values corresponding to the different range–Doppler bins within the defined compound AF. Leveraging the geometric properties of CM constraint, we further transform the jointly constrained problem in the Euclidean space into a single-constraint optimization problem in the Riemannian space. Then, the Riemannian augmented Lagrangian method (RALM) is proposed to iteratively search for the optimal waveform. Subsequently, we conduct numerical experiments to validate the efficacy of the RALM algorithm. In addition, we implemented the designed waveforms in hardware systems to analyze the effects induced by nonlinear instruments.