Chapter 2 - Heuristic Formulation of Time-Frequency Distributions

Abstract

This chapter discusses heuristic formulation of time frequency distributions (TFD). The chapter examines a variety of ad hoc approaches to the problem, namely the Wigner-Ville distribution, localized forms of the Fourier Transform, filter banks, Page's instantaneous power spectrum, and related energy densities. The chapter also illustrates how all these distributions are related to the Wigner-Ville distribution. Constructing a quadratic TFD from the analytic associate of a given real signal, rather than from the signal itself, avoids spurious terms caused by interference between positive-frequency and negative-frequency components. Every TFD derived heuristically is quadratic in the signal; that is, if the signal is scaled by a factor k, the TFD is scaled by a factor k 2. This is to be expected because each TFD is related to some sort of energy density; the signal has been assumed to be an effort variable or a flow variable and power is proportional to the product of an effort variable and the corresponding flow variable, hence to the square of the effort variable.