Ambiguity function shaping for cognitive radar via complex quartic optimization

In this paper, we propose a cognitive approach to design phase-only modulated waveforms sharing a desired range-Doppler response. The idea is to minimize the average value of the ambiguity function of the transmitted signal over some range-Doppler bins, which are identified exploiting a plurality of knowledge sources. From a technical point of view, this is tantamount to optimizing a real and homogeneous complex quartic order polynomial with a constant modulus constraint on each optimization variable. After proving some interesting properties of the considered problem, we devise a polynomial-time waveform optimization procedure based on the Maximum Block Improvement (MBI) method and the theory of conjugate-partial-symmetric/conjugate-super-symmetric fourth order tensors. At the analysis stage, we assess the performance of the proposed technique showing its capability to properly shape the range-Doppler response of the transmitted waveform.