Uniform Asymptotics of the Pseudo Wigner-Ville Distribution for Nonlinear Chirps

Abstract

The analysis of non stationary signals in complex physical systems often relies on Time Frequency distributions. Among these, the Pseudo Wigner Ville Distribution (PWVD) stands out for its superior resolution but is mathematically challenging due to its inherent quadratic nonlinearity. This nonlinearity generates complex interference artifacts and cross terms in the phase space, potentially obscuring the physical features of the signal, particularly for nonlinear chirps. In this work, we establish a mathematically grounded framework for the PWVD for general windowed nonlinear chirps. By leveraging the theory of oscillatory integrals with coalescing stationary points, we derive a uniform asymptotic expansion that bridges the gap between heuristic signal processing and semiclassical geometric approaches (Berry's chord construction). The resulting closed form representation, expressed in terms of symmetric incomplete Airy functions, provides a unified description of the nonlinear transform's behavior, regularizing the transition across the instantaneous frequency caustics. While the framework is general, we show its power on two illustrative examples: the high precision nonlinear chirps of coalescing binaries in gravitational-wave astronomy and radar nonlinear chirps for pulse compression applications. The analytical results successfully predict the structure of interference patterns and quantify the systematic bias in peak based frequency estimation. Therefore, this study establishes a systematic bridge between nonlinear mathematical analysis and precision experimental physics, validating the PWVD as a robust tool for detailed source characterization in high signal to noise regimes.