The time-frequency poles of a differential equation

A linear differential equation in the time domain can be transformed to an equation in the time-frequency domain of the Wigner distribution. Even though the Wigner distribution is a nonlinear transform, the obtained time-frequency equation is still linear. We define time-frequency poles as the roots of the characteristic equation associated to the time-frequency equation. By studying the nature of time-frequency poles we can better understand the properties of the time-frequency equation, and, more importantly, of the original equation in the time domain. We first obtain the connection between time-frequency poles and time poles, which are the roots of the characteristic equation associated to the equation in the time domain. This relationship can be used to characterize the stability of the time-frequency equation, to show that time-frequency poles are a function of frequency, and, consequently, to study their multiplicity with respect to frequency. Second, we prove that the time-frequency spectrum of the response of the time equation to nonstationary driving terms can be written with respect to time-frequency poles. Finally, we show how time-frequency poles control the shape of the time-frequency spectrum of the solution to the time domain equation around resonant frequencies. We discuss our results by using analytic examples.