Unimodular waveform design for ambiguity function shaping with spectral constraint via a manifold-based exact penalty method

Unimodular waveform design for ambiguity function (AF) shaping is a key technology in radar system. Depending on whether the spectral compatibility of the waveform is considered or not, the problem can be categorized into two forms, the first without imposing spectral constraints and the second with spectral constraints. Both of them are complex quartic non-convex problems with constant modulus constraint (CMC), which are challenging to be solved. It is worth noting that the second one obtains more attention due to the improved spectral compatibility, but the involved inequality constraints further increase the difficulty of solving this problem. We note that the inequality constraints can be transformed into penalty terms of the objective function by constructing an exact penalty function. Additionally, the complex circle manifold (CCM) naturally satisfies the CMC, providing a suitable framework to address the resultant problem. Based on the above considerations, we propose a manifold-based exact penalty method (MEP). First, the inequality constraints are eliminated by constructing them as exact penalty terms of the objective function using an exact penalty function, which incorporates the constraints into the objective function. The resulting problem is then projected onto the CCM, transforming the problem into a complex quartic unconstrained optimization problem. Finally, a conjugate gradient (CG) method is derived to solve this unconstrained problem on the CCM. Simulation results confirm that compared to the methods in Wu et al. (2017), Alhujaili et al. (2020) and Zhang et al. (2023), the proposed method exhibits the following superior performance:(1) higher signal-to-interference ratios (SIR), (2) deeper notches and more precise control of the stopbands level of energy spectrum distribution (ESD), (3) stronger detection performance to weak targets.