The Parameter Window Linear Canonical Transform: Properties and Energy Optimization

This paper investigates the fundamental properties of the parameter window linear canonical transform (PWLCT) and the problem of energy concentration in the time–frequency domain. Firstly, the mathematical definition of the PWLCT is established, and its fundamental properties are derived. Secondly, within the theoretical framework of the Bargmann transform, the optimization problem of time-frequency energy concentration is reformulated in Fock space, and an uncertainty principle for the PWLCT is derived. Finally, the time-frequency characteristics of Gaussian signals and nonlinear frequency modulation signals are analyzed through numerical experiments, and the Doppler characteristics of the flying heron radar echo are verified. Theoretical analysis shows that, under the normalized Fock-space formulation, disk-shaped regions provide sharp concentration bounds, and their inverse images correspond to stretched elliptical regions in the time-frequency plane. The chirp-modulated complex Gaussian window is introduced as a parameter-compatible analysis window that enables this Bargmann-Fock representation. Numerical experiments illustrate the practical concentration and tracking behavior of the proposed representation.