Quadratic Time-Frequency Representations

Abstract

In this chapter we present some of the highlights from the theory of the Wigner distribution and other quadratic time-frequency representations. Such time-frequency representations were investigated early in quantum mechanics by E. Wigner [253] with the goal of finding joint probability distributions for the position and momentum variables. In signal analysis they were first introduced by E. Ville [245] for the description of stochastic signals. Later quadratic time-frequency representations became immensely popular in engineering through a series of influential articles by Claasen and Mecklenbräuker [45–47]. They became an essential tool in signal analysis, and the Wigner distribution in particular was believed to be the ideal mathematical description of the time-frequency behavior of signals. However, the non-linearity of these time-frequency representations makes the numerical treatment of signals difficult and often impractical. Owing to the recent boom of fast numerical algorithms in wavelet theory and in time-frequency analysis, quadratic time-frequency representations seem to have lost some of their importance in applications. They remain a beautiful and fascinating mathematical object, and understanding the Wigner distribution is vital for the analysis of pseudodifferential operators in Chapter 14.