Ambiguity Function Shaping for Weak Target Detection: A Joint Design Approach for Unimodular Waveform and Mismatched Filter

Unimodular waveform design for ambiguity function (AF) shaping is a key technology in cognitive radar system. Existing methods are limited either by low degrees of freedom (DOF) under matched-filter schemes or by relaxation errors in joint waveform–filter design. To address these issues, we propose a relaxation-free optimization framework with high DOF for unimodular waveform and mismatched filter design, formulated as a nonconvex minimization under constant modulus (CMC) and complex sphere (CSC) constraints. Recognizing that CMC and CSC are naturally satisfied in corresponding Riemannian manifolds, a product manifold for ambiguity function shaping (PMAFS) method is proposed. The proposed approach transforms the problem into an unconstrained optimization over a product manifold and solves it via a parallel conjugate gradient algorithm. An exact-penalty extension, EP-PMAFS, is also developed for precise energy spectral density (ESD) control. Simulation results confirm that the proposed PM optimization framework achieves stronger detection performance for weak target and precise control of the waveform ESD.