Waveform optimization of bio-inspired cetacean whistle active sonar using the principle of stationary phase

Objective Underwater bio-inspired integrated sensing and communication (ISAC) systems require shared waveforms that simultaneously provide bio-inspired concealment and effective active detection performance. To address the waveform optimization problem for cetacean whistle-based active sonar, this study establishes a unified analytical framework based on the principle of stationary phase (PSP). The framework systematically reveals the causal relationship between instantaneous frequency characteristics and ambiguity function morphology. Based on this analysis, a multi-tone sinusoidal frequency-modulated waveform optimization method is proposed using a genetic algorithm with golden-ratio-based initialization.

Method First, with the wideband ambiguity function (WAF) as the primary analytical tool, asymptotic expressions of the WAF for six typical cetacean whistle waveforms (constant, concave, convex, up-sweep, down-sweep, and sinusoidal) are derived using the principle of stationary phase (PSP). This establishes the intrinsic mapping relationship of "number of monotonic intervals in the instantaneous frequency → number of effective stationary phase points → WAF morphology." This mapping provides a theoretical basis for waveform selection and identifies the sinusoidal whistle waveform as the optimal candidate. Second, to address the two inherent limitations of sinusoidal whistles—near-range sidelobes and periodic grating lobes—a genetic algorithm-based optimization framework for multi-tone sinusoidal FM waveforms is proposed, with golden-ratio initialization. Based on the stationary phase superposition theorem, the framework optimizes the specific values of multiple modulation frequency components while keeping the total frequency deviation (range resolution) and minimum modulation frequency (grating lobe constraint) fixed, thereby disrupting the periodic grating lobe structure and further suppressing Doppler sidelobes.

Results Both theoretical derivations and simulation results consistently demonstrate that the sinusoidal whistle waveform, owing to its periodic non-monotonic instantaneous frequency characteristics, contains the largest number of effective stationary phase points and achieves superior overall performance in terms of autocorrelation sidelobe suppression, ambiguity function concentration, reverberation resistance, and Doppler tolerance. After genetic algorithm optimization, a Doppler sidelobe suppression gain of 4–6 dB is achieved, and the grating lobe structure is significantly mitigated due to the incommensurability of the golden-ratio-based modulation frequencies.

Conclusion The proposed PSP-based waveform optimization method establishes a complete technical framework—from theoretical waveform selection to waveform optimization—for the engineering design of shared waveforms in bio-inspired underwater ISAC systems.