Theory and design of high-resolution quadratic TFDs

Abstract

The quadratic time-frequency distributions (TFDs) introduced in the previous chapter represent the majority of traditional time-frequency ((t,f)) methods used in practical applications that deal with nonstationary signals. This key chapter completes the introductory tutorial and prepares the reader for the more advanced material presented in the following chapters. The core material shows that the particular quadratic TFDs introduced in Chapter 2 actually belong to a general quadratic class of TFDs whose design follows a common procedure, and whose properties are governed by common laws. This quadratic (or bilinear) class may be considered as the class of “smoothed” or filtered Wigner-Ville distributions (WVDs).

The “smoothing” or “filtering” is described in the (t,f) domain by a 2D convolution with a “(t,f) kernel” γ(t,f) and in other domains by multiplication and/or 1D convolution with various transforms of γ(t,f) such as kernel filter g(ν,τ). The generalized approach allows the definition of new TFDs that are better adapted to particular signal types, using a simple and systematic procedure. The first section reviews in detail the key properties and limitations of the WVD; it introduces the general quadratic TFDs (Section 3.2) and then discusses their properties. Then, using Fourier transforms from lag to frequency and from time to Doppler (frequency shift), the quadratic TFDs and their kernels are formulated in four different but related two-dimensional domains.

This leads to the definition of the “ambiguity function” and allows the smoothing of the WVD to be understood as a filtering operation.Section 3.3 lists some desirable TFD properties expressed in terms of constraints on the kernel, so that TFD design is reduced to kernel design.Section 3.4 deals with the question of TFD positivity and the corresponding condition on the kernel filter.