A corrected wideband ambiguity function for robust biosonar strategy analysis

The wideband ambiguity function (WAF) is a foundational tool for analyzing sonar signals, as it jointly characterizes time delay and Doppler-induced time stretching. However, when applied to nonstationary biological signals, this function is sensitive to temporal misalignment and integration-window effects, which can bias estimates of sonar performance. This paper proposes a corrected formulation of the WAF that compensates for Doppler-induced temporal shifts by centering the analysis around the energy centroid of the emitted signal. The corrected wideband ambiguity function (CWAF) preserves the intrinsic resolution properties imposed by signal bandwidth and duration, while ensuring invariance to time translation and integration time within the analysis window. This method is applied to field recordings of echolocation during foraging and depredation, produced by bats (Pipistrellus kuhlii) and sperm whales (Physeter macrocephalus). The results reveal contrasting biosonar strategies: bats dynamically modulate spectral and temporal parameters across foraging phases, whereas sperm whales emit spectrally stable, Doppler tolerant pulses and primarily adjust inter-click intervals and output levels. Additionally, analyzing the resolution of pulses P0 and P1 lends support to the hypothesis of a three-dimensional sperm whale biosonar model. Overall, the CWAF provides a robust framework for linking biosonar signal design, estimator theory, and animal behavior.