Decomposing the discrete Bai distribution function for efficient computation
Yushi Zheng, John J. Healy · 2025
Recently, numerous phase-space distributions have been proposed, among which the Bai distribution function (BDF) is independent from the others. It interpolates between the instantaneous auto-correlation function and the Wigner distribution function (WDF), and shows improved noise resilience when estimating the parameters of linear frequency modulated signals such as the Newton’s rings in optics. Previous studies have discretized the continuous BDF, analyzed its sampling requirement, and constructed a discrete BDF calculation procedure, which enables its use in simulations and experiments as the signals are digital in both scenarios. In this paper, we will propose a novel method for calculating the discrete BDF by decomposing it into a discrete WDF and a discrete linear canonical transform. We then analyze the sampling theorem of the proposed method, demonstrate its accuracy, and compare it with a previously published method. Our results show that each of the two methods exhibits its own strengths and limitations depending on the optical system used in different applications. It permits a more efficient simulation of the numerical BDF, which will be useful in applying it to the areas of, e.g., partial coherence, optical measurements based on Newton’s rings, or radar systems.