Introduction

Principal value integrals arise in a wide variety of applications, and in many cases it is not possible to evaluate such integrals in a simple closed analytic form. Consequently, there has been a significant investment of research effort devoted to the numerical evaluation of principal value integrals. The expression “numerical quadrature”, or frequently just the term “quadrature”, is used synonymously with numerical integration.

Some of the numerical integration approaches that have been developed for principal value integrals are outlined in this chapter. These range from rather simple schemes, which are sometimes quite effective, to approaches that yield fairly precise results and can be implemented in a high–speed calculation. Methods that are discussed include Maclaurin's formula, the trapezoidal rule, Simpson's formula, specialized Gaussian quadrature methods, and techniques involving Fourier transforms, including the fast Fourier transform, Fourier allied integral approaches, and methods based on conjugate Fourier series. Since a number of principal value integral problems arise in the context of transforming experimental data, some attention is devoted to the discretized nature of the data and how this can be handled.

Even if an analytic solution can be found for a particular principal value integral, numerical methods can be employed as a very useful check on the closed form result.

Some elementary transformations for Cauchy principal value integrals

Two straightforward, but potentially very useful, transformations that may be employed to simplify the evaluation of the Hilbert transform are discussed in this section.

Introduction

The focus of this chapter is the derivation of dispersion relations arising in magnetooptical applications and for natural optical activity. A further topic considered is the development of sum rules for some of the quantities that are experimentally accessible. These include the optical rotatory dispersion (commonly abbreviated as ORD in the literature), circular dichroism (CD), the magnetic analogs, the Faraday dispersion, and magnetic circular dichroism (MCD). These terms will be defined shortly.

A number of the ideas that are employed are very similar to those developed in the preceding two chapters. There are, however, some major differences that arise for the cases of magneto-optical and natural optical activity. The dispersion relations for magneto-optical and natural optical activity have a different structure compared with what was found for the typical case of other optical properties discussed in Chapters 19 and 20. The difference arises primarily from the form of the crossing symmetry relations for the refractive indices and absorption coefficients for left and right circularly polarized light. The reason for the absence of simple dispersion relations for the individual modes corresponding to the refractive indices for left and right polarization, and the corresponding absorption coefficients for right and left circularly polarized light, is also discussed.

My objective in this book is to present an elementary introduction to the theory of the Hilbert transform and a selection of applications where this transform is applied. The treatment is directed primarily at mathematically well prepared upper division undergraduates in physics and related sciences, as well as engineering, and first–year graduate students in these areas. Undergraduate students with a major in applied mathematics will find material of interest in this work.

I have attempted to make the treatment self–contained. To that end, I have collected a number of topics for review in Chapter 2. A reader with a good undergraduate mathematics background could possibly skip over much of this chapter. For others, it might serve as a highly condensed review of material used later in the text. The principal background mathematics assumed of the reader is a solid foundation in basic calculus, including introductory differential equations, a course in linear and abstract algebra, some exposure to operator theory basics, and an introductory knowledge of complex variables. Readers with a few deficiencies in these areas will find a number of recommendations for further reading at the end of Chapter 2. Some of the applications discussed require the reader to be familiar with basic electrodynamics.

My objective in this book is to present an elementary introduction to the theory of the Hilbert transform and a selection of applications where this transform is applied. The treatment is directed primarily at mathematically well prepared upper division undergraduates in physics and related sciences, as well as engineering, and first-year graduate students in these areas. Undergraduate students with a major in applied mathematics will find material of interest in this work.

I have attempted to make the treatment self-contained. To that end, I have collected a number of topics for review in Chapter 2. A reader with a good undergraduate mathematics background could possibly skip over much of this chapter. For others, it might serve as a highly condensed review of material used later in the text. The principal background mathematics assumed of the reader is a solid foundation in basic calculus, including introductory differential equations, a course in linear and abstract algebra, some exposure to operator theory basics, and an introductory knowledge of complex variables. Readers with a few deficiencies in these areas will find a number of recommendations for further reading at the end of Chapter 2. Some of the applications discussed require the reader to be familiar with basic electrodynamics.

Introductory ideas on signal processing

Signal processing plays a central role in a multitude of modern technologies. Think about those industries that are dependent on data communication, or on radar, to give just two examples, and the importance of signal processing becomes self-evident. The Hilbert transform plays a central role in a number of signal processing applications. Pioneering work on the application of Hilbert transforms to signal theory was carried out by Gabor (1946).

A notational alert to the reader is appropriate at the start of this chapter. In the following sections the standard Hilbert transform operator H, the Heaviside step function H(x), the Hermite polynomials Hn(x), the Hilbert transfer function H(ω), the fractional Hilbert transform Hα, and the fractional Hilbert transform filter Hp(ω) all appear, sometimes in close proximity, so the reader should pay careful attention to the particular symbols in use.

Broadly defined, a signal provides a means for transmission of information about a system. For the signals of interest in this chapter, it is assumed that a mathematical representation of the signal is known. There are two important types of signals. The first are the continuous or analog signals – sometimes referred to as continuoustime signals. Unless something explicitly to the contrary is indicated, it is assumed throughout this chapter that all signals of this group belong to the class L2(ℝ). In a number of places, this requirement can be generalized.

Some common integral transforms

Transform techniques have become familiar to recent generations of undergraduates in various areas of mathematics, science, and engineering. The principal integral transform that is perhaps best known is the Fourier transform. The jump from the time domain to the frequency domain is a characteristic feature of a number of important instrumental methods that are routinely employed in many university science departments and industrial laboratories. Fourier transform nuclear magnetic resonance spectroscopy (acronym FTNMR) and Fourier transform infrared spectroscopy (FTIR) are two extremely significant techniques where the Fourier transform methodology finds important application. Two transforms derived from the Fourier transform, the Fourier sine and Fourier cosine transforms, also find wide application. The Laplace transform is often encountered fairly early in the undergraduate mathematics curriculum, because of its utility in aiding the solution of certain types of elementary differential equations. The transforms that bear the names of Abel, Cauchy, Mellin, Hankel, Hartley, Hilbert, Radon, Stieltjes, and some more modern inventions, such as the wavelet transform, are much less well known, tending to be the working tools of specialists in various areas. The focus of this work is about the Hilbert transform. In the course of discussing the Hilbert transform, connections with some of the other transforms will be encountered, including the Fourier transform, the Fourier sine and Fourier cosine offspring, and the Hartley, Laplace, Stieltjes, Mellin, and Cauchy transforms. The Z–transform is studied as a prelude to a discussion of the discrete Hilbert transform.