Introduction

In this final chapter, some further applications of Hilbert transforms are considered. Several of these topics are of considerable significance and could easily be developed into rather lengthy chapters. The underlying theme of this chapter is that the applications all make use of the standard Hilbert transform, or one of its variants, as a substantial part of the technique or approach that is under discussion. Since problems from diverse areas are considered in this chapter, the reader is alerted to the fact that the same symbol may designate unrelated quantities in different sections.

Hilbert transform spectroscopy

The term Hilbert transform spectroscopy is used in the literature with two different meanings. Both techniques are discussed in this section. The first involves the use of the Josephson junction. Here the spectrum of the incident electromagnetic radiation on the junction is related to the Hilbert transform of the electrical response of the Josephson junction. The second development employs the conversion of a time domain signal into an absorption spectrum, using the Hilbert transform in a manner designed to try and improve spectral features, such as the identification of weak spectral lines and the minimization of spurious noise peaks.

The Josephson junction

The approach described in this section was proposed by Divin, Polyanskii, and Shul'man (1980), and called Hilbert transform spectroscopy by Divin et al. (1993). The terminology Hilbert spectroscopy has also been employed (Tarasov et al., 1995).