Approximating the time-frequency output of a dynamical system for an arbitrary nonstationary input

We obtain the approximate analytic time-frequency spectrum of the output of a dynamical system when the input is an arbitrary finite-energy nonstationary signal. Our method is based on three steps. First, we transform the dynamical system to the time-frequency domain. Second, we approximate the time-frequency spectrum of the input as a sum of short duration sinusoids through a Fourier series expansion. Finally, we combine the time-frequency outputs corresponding to each individual short duration sinusoid, which are known in exact analytic form. An example shows that the proposed method requires a few terms only to obtain an approximate time-frequency output which is indistinguishable from the exact one. Furthermore, our method can clarify how dynamical systems process nonstationary signals. This processing mechanism is of fundamental interest since dynamical systems are a common model for real-world signals.