Theory of Quadratic TFDs

Abstract

The quadratic time-frequency distributions (TFDs) introduced in the last chapter represent the majority of the methods used in practical applications that deal with non-stationary signals. In this chapter, which completes the introductory tutorial, we show that these particular quadratic TFDs belong to a general class of TFDs whose design follows a common procedure, and whose properties are governed by common laws. This quadratic class may be considered as the class of smoothed Wigner-Ville distributions (WVDs), where the “smoothing”is described in the (t, f) domain by convolution with a “time-frequency kernel” function (t, f), and in other domains by multiplication and/or convolution with various transforms of (t, f). The generalized approach allows the definition of new TFDs that are better adapted to particular signal types, using a simple and systematic procedure as opposed to the ad hoc methods of Chapter 2. The first section (3.1) extends Section 2.1 by enumerating in detail the key properties and limitations of theWVD. Thus it motivates the introduction of general quadratic TFDs (Section 3.2) and prepares for the discussion of their properties (Section 3.3). In Section 3.2, 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. One of these, namely the Doppler-lag ( , ) domain, leads to the definition of the “ambiguity function” and allows the smoothing of the WVD to be understood as a filtering operation. In Section 3.3, the list of properties of the WVD is supplemented by mentioning some desirable TFD properties not shared by the WVD. The various TFD properties are then expressed in terms of constraints on the kernel, so that TFD design is reduced to kernel design. Three tables are provided showing the kernel properties equivalent to various TFD properties, the kernels of numerous popular TFDs in the various two-dimensional domains, and the properties of those same TFDs.