Approximation of the Wigner Distribution for Dynamical Systems Governed by Differential Equations
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pproximation of the Wigner Distribution for Dynamical Systems Governed by Differential Equations Lorenzo Galleani Dipartimento di Elettronica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy Email: galleani@cicoria.polito.it
Leon Cohen City University of New York, 695 Park Avenue New York, NY 10021, USA Email: leon.cohen@hunter.cuny.edu Received 31 July 2001 and in revised form 12 October 2001 A conceptually new approximation method to study the time-frequency properties of dynamical systems characterized by linear ordinary differential equations is presented. We bypass solving the differential equation governing the motion by writing the exact Wigner distribution corresponding to the solution of the differential equation. The resulting equation is a partial differential equation in time and frequency. We then show how it lends itself to effective approximation methods because in the time frequency plane there is a high degree of localization of the signal. Numerical examples are given and compared to exact solutions. Keywords and phrases: differential equations, Wigner distribution, time-frequency analysis.
1. INTRODUCTION Many dynamical systems are governed by an equation of motion that is an ordinary differential equation with a known driving function. In particular, by an
dn x dn−1 x dx + an−1 + · · · + a1 + a0 x = f (t), (1) n dt dt n−1 dt
where f (t) is the driving force and x(t) the state function. Often, one wants to study the time-frequency properties of the solution. That would be done by solving (1) and putting the solution into a time-frequency distribution such as the Wigner distribution. However, only in rare cases one is able to solve (1) exactly, and even in those cases f (t) must be necessarily of a simple form (constant, sinusoid, polynomial, etc.). Alternatively, one can attempt to solve the differential equations approximately and substitute the solution into the Wigner distribution. However this is generally problematic because of the many possible regimes and we point out that even in the relatively simple case of the so-called gliding tone problem (to be discussed in Section 3) approximate solutions are quite involved. Since the introduction of time-frequency methods, it has
been realized that signals which may be complicated as a function of time or frequency are often simple in the timefrequency plane. We have developed an approach that takes advantage of this in a direct way. Our procedure is as follows. In contrast to the standard methods where one solves the differential equation and then uses a time-frequency distribution, for example the Wigner distribution, to ascertain the time-frequency properties of the solution, we show that one can obtain a differential equation for the Wigner distribution of the solution and hence bypass the necessity for solving (1). That is, if the Wigner distribution is defined by W (t, ω) =
1 2π
1 1 x ∗ t − τ x t + τ e−jτω dτ, 2 2
(2)
we obtain an exact equation of motion for W (t, ω) directly and show that one can app
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