Time-Frequency Wigner Distribution Approach to Differential Equations - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 121

4

Time-Frequency Wigner Distribution
Approach to Differential Equations

Lorenzo Galleani and Leon Cohen

CONTENTS
4.1

.........................................................

Introduction 121

4.2 Harmonic Oscillator with Chirp Driving Force ..................... 122

4.3 Approximation Method ............................................. 126

.................................................

4.4 Stochastic Processes 129

4.5 Partial Differential Equations ....................................... 133

4.6 Conclusion ........................................................... 140

Acknowledgment .......................................................... 140

Appendix 4.1: Notations and Definitions ................................. 140

Appendix 4.2: Ordinary Differential Equations ........................... 142

Appendix 4.3: Exact Solution to the Gliding Tone Problem ............... 143

Appendix 4.4: Partial Differential Equations .............................. 144

Appendix 4.5: Derivation of the Wigner Equation of Motion
for Diffusion

................................................ 146

Appendix 4.6: Green's Function for the Wigner Distribution ............. 147

References .................................................................. 149

4.1 Introduction

In the field of signal analysis, time-frequency distributions have historically
been used as a means of analyzing signals for their time-varying spectra. 6 , 7 , 11
In physics, however, these distributions have been used to understand the
solution of the Schr odinger equation, which is a partial differential equa-
tion. The idea is to obtain the equation of motion for the Wigner distribution
corresponding to the solution of the Schr odinger equation. The basic rea-
sons for doing so is that one gains considerable insight into the nature of
the solution, and that it leads to new analysis and approximation methods.
Wigner, Moyal, Kirkwood, and many others made significant contributions