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.

Introduction

The principal objective of this chapter is to present some of the essential basic mathematical background that is employed in later sections. A good deal of this material should be straightforward for a well trained undergraduate mathematics or physics major; however, there are a few slightly more advanced topics that are treated concisely. For these topics, collateral reading in a standard text is highly recommended. Some suggestions of where to start additional reading are provided in the end–notes. The mathematically talented reader could bypass most of this chapter and skip to the derivations in Chapter 3.

Almost all the mathematical notation employed can be found in the List of symbols; please consult that list for the definition or for the first use of a particular symbol. Some common notational devices are reviewed first, and this is followed by a concise description of some of the more important mathematical tools, such as Fourier analysis, complex variable theory, and the basics of integration theory, i.e. topics that are central to later developments. Further extensions of some of these tools are given later as needed.

Introduction

The developments of the previous chapter are continued in this chapter. The principal focus is on the reflectance and the energy loss function. Knowledge of the reflectance and phase allows the real and imaginary parts of the complex refractive index to be evaluated, and hence the real and imaginary parts of the complex dielectric constant can be determined. Measurement of the reflectance and calculation of the phase provides a convenient route to a number of optical properties.

The procedure for dealing with the formulation of dispersion relations for the reflectance and phase has some features that are different from the cases of the dielectric constant and the refractive index. This is tied directly to the fact that the real and imaginary parts of the complex reflectivity depend upon both the reflectance and the phase. To uncouple these quantities requires consideration of the logarithm of the complex reflectivity, which introduces some additional issues into the discussion. This also has a direct bearing on the types of sum rules that can be derived.

Dispersion relations for the normal-incident reflectance and phase

The topic of obtaining dispersion relations for the normal-incident reflectivity is examined in this section. Consideration is restricted to the normal-incidence case, and it will be assumed that light is impinging on the material surface from a vacuum. The reflectivity, R(ω), is the ratio of the reflected to the incident light intensities.

Systems

This chapter is concerned with setting up the foundations that allow the connection between causality and analyticity to be established. The interplay between these two topics and the Hilbert transform is also treated. The material of this chapter lays the basis for many of the applications discussed in the following chapters.

Consider the arrangement in Figure 17.1, where the input to the system is denoted by i, and r designates the output response. The input and output could be of the same nature, for example a voltage, or very different variables, for example a voltage input, with the output being a physical displacement of a mass. As an example of Figure 17.1, consider the input to be the driving force acting on a simple oscillator arrangement with a mass hanging from a fixed point by a spring. The output is the displacement of the mass. The input has some dependence on time, and likewise the output response. Henceforth, the temporal dependence is made explicit. Often the input and output response are continuous functions of the time variable, but this is not a requirement. In later sections, the focus will include consideration of step function and impulse inputs.

The term “system” refers to a device capable of converting an input into some output response. The output response should be totally characterized by the system input and the characteristics of the system.