Advanced time-frequency signal and system analysis

Abstract

This chapter extends Part I by presenting additional advanced key principles underlying the use of time-frequency (t,f) methods. The topic is covered in 11 focused sections.

Section 4.1 describes the relationships between quadratic TFDs and time-scale methods, as an extension of Section 2.3.3; it further motivates the use of quadratic TFDs in most situations and not just in exploratory signal analysis. Then, the issue of cross-terms generation and localization is described in detail. The quadratic superposition principle explains the mechanism generating the cross-terms and the subsequent trade-off between cross-term reduction and increased localization as well as (t,f) resolution (Section 4.2). This is followed by an examination of the covariance property of TFDs for important signal transformations like (t,f) shifts or scaling (Section 4.3). Another significant aspect of (t,f) methods is that the (t,f) uncertainty relations determine the issue of lower bounds in achievable (t,f) resolution (Section 4.4). Using methods such as coordinate change methods, we can also define joint distributions of other variables than t and f that may be better suited for specific applications (Section 4.5). To estimate signal parameters directly from the (t,f) plane, formulations of measures such as spread measures are then provided (Section 4.6). Then, (t,f) methods are used to describe linear time-varying input-output relationships (Section 4.7). The relationships between (t,f) methods such as the WVD and the fractional FT is described using the Radon-Wigner transform (Section 4.8). Then, the next sections focus on (1) a (t,f) perspective of MIMO dynamical systems (Section 4.9) and (2) Teager-Kaiser operators in (t,f) analysis (Section 4.10). This is followed by the presentation of the Gabor spectrogram in Section 4.11, which relates the properties of the WVD to energy atoms. The chapter ends with a presentation of the empirical mode decomposition and Hilbert Spectrum.