Advanced design and specifications of TFDs

Abstract

This chapter describes specific examples of design of time-frequency distributions (TFDs), as a complement to the material described in Chapters 2 and Section 3.1. This key time-frequency (t,f) topic is covered in 11 sections.

Ambiguity functions are traditionally used in radar and sonar. As 2D Fourier transforms of TFDs, they form the basis for TFD kernel design methodologies (Section 5.1). One of the first TFDs constructed on this basis is the exponential kernel distribution, a TFD designed for its reduced interference properties (Section 5.2). To better adapt to the signal under analysis and achieve higher resolution and concentration performance, we can design adaptive TFDs using optimization theory (Section 5.3). TFD design can also be adapted to a specific class of signals such as polynomial frequency modulated (FM) signals, leading to the formulation of polynomial Wigner-Ville distributions (WVDs) (Section 5.4). Such methods are related to higher-order spectra (HOS) (Section 5.5). Other TFDs can be designed by adaptation to signals with dispersive group delay (Section 5.6). A step-by-step methodology for the design of a specific TFD is provided using separable kernels (Section 5.7). Finally, the class of generalized marginal TFDs is related to the fractional Fourier transform, allowing further design possibilities (Section 5.8).Section 5.9 discusses fixed kernel high-resolution TFDs that improve performance by accounting for the directions of energy concentration of signal components in the (t,f) domain. Adaptive methods that estimate the direction of the smoothing kernel on a point-by-point basis are presented in Section 5.10.Section 5.11 uses linear (t,f) methods based on a modified version of the S-transform to estimate phase synchrony. Finally, Section 5.12 reviews the requirements for designing positive TFDs and their implications.