Piecewise Polynomial Phase-Coded Waveform Design for Enhanced Doppler Tolerance

Phase-coded waveforms, an important class of radar signals, exhibit low peak sidelobe level (PSL) but lack sufficient Doppler tolerance. Inspired by the excellent Doppler properties of chirp-like signals, researchers have developed Piecewise Polynomial Phase-Coded Waveforms (PPPCW), which model phase using quadratic polynomials within sections, employing different coefficients across intervals. This structure combines the Doppler tolerance of chirp-like signals with the enhanced correlation performance of piecewise design, enabling joint optimization of PSL and Doppler tolerance. However, when segment count increases beyond two, existing models fail to maintain the periodic Doppler tolerance characteristic of chirp-like waveforms, causing this desirable property to rapidly deteriorate with increasing frequency offset. In this paper, we overcome this limitation through systematic analysis of the relationships between polynomial coefficients and Doppler tolerance, leading to an optimization model that integrates tolerance metrics with coefficient constraints to preserve Doppler performance regardless of segment count. By approximating PSL using the Log-Sum-Exp (LSE) function and employing automatic differentiation techniques derived from neural networks, we effectively solve this model via gradient descent methods. Experimental results demonstrate that our proposed model yields waveforms with both low PSL and periodic Doppler tolerance regardless of segment count. Notably, the model readily extends to Multiple-Input MultipleOutput (MIMO) radar applications through simple orthogonality constraints.