Heuristic Formulation of Time-Frequency Distributions

Abstract

Having established the basic signal formulations in the first chapter, we now turn to the problem of representing signals in a joint time-frequency domain. Given an analytic signal z(t) obtained from a real signal s(t), we seek to construct a time-frequency distribution z(t, f) to represent precisely the energy, temporal and spectral characteristics of the signal. We choose the symbol z in the expectation that the TFD will represent an “energy density of z” in the (t, f) plane. We would also like the constant-t cross-section of z(t, f) to be some sort of “instantaneous spectrum” at time t. In this chapter we examine a variety of ad hoc approaches to the problem, namely the Wigner-Ville distribution (Section 2.1), a time-varying power spectral density called the Wigner-Ville Spectrum (2.2), localized forms of the Fourier Transform (2.3), filter banks (2.4), Page’s instantaneous power spectrum (2.5), and related energy densities (2.6). Finally (in Section 2.7), we show how all these distributions are related to the first-mentioned Wigner-Ville distribution, thus setting the scene for the more systematic treatment in the next chapter. The various distributions are illustrated using a linear FM asymptotic signal. The linear FM signal [Eq. (1.1.5)] is regarded as the most basic test signal for TFDs because it is the simplest example of a signal whose frequency content varies with time. It is clearly monocomponent, and is asymptotic if its BT product is large. The minimum requirement for a useful TFD is that it clearly shows the IF law of an asymptotic linear FM signal, giving a reasonable concentration of energy about the IF law (which, for an asymptotic signal, is equivalent to the TD law).