INTRODUCTION

In this chapter we present an overview of the necessary basic knowledge that will be assumed as mathematical prerequisite in the chapters to follow. It is presupposed that the reader already has previous knowledge of the subject matter in this chapter. However, it is advisable to read this chapter thoroughly, and not only because one may discover, and fill in, possible gaps in mathematical knowledge. This is because in Fourier and Laplace transforms one uses the complex numbers quite extensively; in general the functions that occur are complex-valued, sequences and series are sequences and series of complex numbers or of complex-valued functions, and power series are in general complex power series. In introductory courses one usually restricts the treatment of these subjects to real numbers and real functions. This will not be the case in the present chapter. Complex numbers will play a prominent role.

In section 2.1 the principal properties of the complex numbers are discussed, as well as the significance of the complex numbers for the zeros of polynomials. In section 2.2 partial fraction expansions are treated, which is a technique to convert a rational function into a sum of simple fractions. Section 2.3 contains a short treatment of differential and integral calculus for complex-valued functions, that is, functions which are defined on the real numbers, but whose function values may indeed be complex numbers. One will find, however, that the differential and integral calculus for complex-valued functions hardly differs from the calculus of real-valued functions.

INTRODUCTION

The greater part of this chapter consists of section 14.1 on linear time-invariant continuous-time systems (LTC-systems). The Laplace transform is very well suited for the study of causal LTC-systems where switch-on phenomena occur as well: at time t = 0 ‘a switch is thrown’ and a process starts, while prior to time t = 0 the system was at rest. The input u(t) will thus be a causal signal and since the system is causal, the output y(t) will be causal as well. Applying the Laplace transform is then quite natural, especially since the Laplace transform exists for a large class of inputs u(t) as an ordinary integral in a certain half-plane Re s > ρ. This is in contrast to the Fourier transform, where distributions are needed more often. For the Laplace transform we can usually restrict the distribution theory to the delta functions δ(t − a) with a ≥ 0 (and their derivatives). As in chapter 10, the response h(t) to the delta function δ(t) again plays an important role. The Laplace transform H(s) of the impulse response is called the transfer function or system function. An LTC-system is then described in the s-domain by the simple relationship Y(s) = H(s)U(s), where Y(s) and U(s) are the Laplace transforms of, respectively, the output y(t) and the input u(t) (compare this with (10.6)).

INTRODUCTION

In this chapter we give a brief introduction to the theory of complex functions. In section 11.1 some well-known examples of complex functions are treated, in particular functions that play a role in the Laplace transform. In sections 11.2 and 11.3 continuity and differentiability of complex functions are examined. It will turn out that both the definition and the rules for continuity and differentiability are almost exactly the same as for real functions. Still, complex differentiability is surprisingly different from real differentiability. In the final section we will briefly go into this matter and treat the so-called Cauchy–Riemann equations. The more profound properties of complex functions cannot be treated in the context of this book.

LEARNING OBJECTIVES

After studying this chapter it is expected that you

1. - know the definition of a complex function and know the standard functions zn, ez, sin z and cos z

2. - can split complex functions into a real and an imaginary part

3. - know the concepts of continuity and differentiability for complex functions

4. - know the concept of analytic function

5. - can determine the derivative of a complex function.

Definition and examples

The previous parts of this book dealt almost exclusively with functions that were defined on ℝ and could have values in ℂ. In this part we will be considering functions that are defined on ℂ (and can have values in ℂ).

INTRODUCTION TO PART 2

The Fourier series that we will encounter in this part are a tool to analyse numerous problems in mathematics, in the natural sciences and in engineering. For this it is essential that periodic functions can be written as sums of infinitely many sine and cosine functions of different frequencies. Such sums are called Fourier series.

In chapter 3 we will examine how, for a given periodic function, a Fourier series can be obtained, and which properties it possesses. In chapter 4 the conditions will be established under which the Fourier series give an exact representation of the periodic functions. In the final chapter the theory of the Fourier series is used to analyse the behaviour of systems, as defined in chapter 1, and to solve differential equations. The description of the heat distribution in objects and of the vibrations of strings are among the oldest applications from which the theory of Fourier series has arisen. Together with the Fourier integrals for non-periodic functions from part 3, this theory as a whole is referred to as Fourier analysis.

Jean-Baptiste Joseph Fourier (1768 – 1830) was born in Auxerre, France, as the son of a tailor. He was educated by Benedictine monks at a school where, after finishing his studies, he became a mathematics teacher himself. In 1794 he went to Paris, where he became mathematics teacher at the Ecole Normale.

INTRODUCTION

Fourier and Laplace transforms provide a technique to solve differential equations which frequently occur when translating a physical problem into a mathematical model. Examples are the vibrating string and the problem of heat conduction. These will be discussed in chapters 5, 10 and 14.

Besides solving differential equations, Fourier and Laplace transforms are important tools in analyzing signals and the transfer of signals by systems. Hence, the Fourier and Laplace transforms play a predominant role in the theory of signals and systems. In the present chapter we will introduce those parts of the theory of signals and systems that are crucial to the application of the Fourier and Laplace transforms. In chapters 5, 10, 14 and 19 we will then show how the Fourier and Laplace transforms are utilized.

Signals and systems are introduced in section 1.1 and then classified in sections 1.2 and 1.3, which means that on the basis of a number of properties they will be divided into certain classes that are relevant to applications. The fundamental signals are the sinusoidal signals (i.e. sine-shaped signals) and the time-harmonic signals. Time-harmonic signals are complex-valued functions (the values of these functions are complex numbers) which contain only one frequency. These constitute the fundamental building blocks of the Fourier and Laplace transforms.

The most important properties of systems, treated in section 1.3, are linearity and time-invariance. It is these two properties that turn Fourier and Laplace transforms into an attractive tool.

Fourier and Laplace transforms are examples of mathematical operations which can play an important role in the analysis of mathematical models for problems originating from a broad spectrum of fields. These transforms are certainly not new, but the strong development of digital computers has given a new impulse to both the applications and the theory. The first applications actually appeared in astronomy, prior to the publication in 1822 of the famous book Théorie analytique de la chaleur by Joseph Fourier (1768 – 1830). In astronomy, sums of sine and cosine functions were already used as a tool to describe periodic phenomena. However, in Fourier's time one came to the surprising conclusion that the Fourier theory could also be applied to non-periodic phenomena, such as the important physical problem of heat conduction. Fundamental for this was the discovery that an arbitrary function could be represented as a superposition of sine and cosine functions, hence, of simple periodic functions. This also reflects the essential starting point of the various Fourier and Laplace transforms: to represent functions or signals as a sum or an integral of simple functions or signals. The information thus obtained turns out to be of great importance for several applications. In electrical networks, for example, the sinusoidal voltages or currents are important, since these can be used to describe the operation of such a network in a convenient way.

INTRODUCTION

Applications of discrete transforms can mainly be found in the processing of discrete signals in discrete-time systems. In chapter 1 we have already discussed such systems in general terms. Since we now have certain discrete transforms available, we are able to get a better understanding of the discrete systems. Hence, in the present chapter we we will focus on a further analysis of the discrete-time systems.

In systems theory we distinguish inputs and the corresponding outputs or responses of the system. For discrete-time systems these signals are discrete-time signals. A system can be described by giving the relation that exists between the inputs and the outputs. This can be done in several ways. For example, by describing the relation in the n-domain, or in the z-domain, or, just as important, by describing it in the frequency or ω-domain. In the latter case we have the relationship between the spectra of the input and outputs in mind.

Discrete transforms play a special role in linear time-invariant discrete systems, similar to the role played by the Fourier integral in continuous-time systems (see chapter 10). Linear time-invariant systems have already been introduced in chapter 1 (see section 1.3.2). Discrete-time systems that are linear and time-invariant will henceforth be called LTD-systems for short. In section 19.1 we will see that for an LTD-system the relationship between an input and the corresponding output can be described in the n-domain by means of a convolution product.

INTRODUCTION

From parts 2 and 3 it is obvious that Fourier series and Fourier integrals play an important role in the analysis of continuous-time signals. In many cases we are forced to calculate the Fourier coefficients or the Fourier integral on the basis of a given sampling of the signal. We are therefore interested in a transformation that will transform a discrete-time signal, in this case a sampling, directly into the frequency domain. In general, such transformations are called discrete transforms. A particularly important discrete transform is the so-called discrete Fourier transform, abbreviated as DFT, and it will be the central theme of the present chapter. It arises naturally if one approximates the Fourier coefficients of a periodic continuous-time signal numerically using the trapezoidal rule. This is the subject of the first section of this chapter, which also introduces the DFT as a transform defined for periodic discrete-time signals. In the next section we introduce the inverse DFT in the so-called fundamental theorem of the discrete Fourier transform, which very much resembles the fundamental theorem of Fourier series. In the remaining sections, all kinds of properties of the DFT are treated, and again we will encounter many similarities with Fourier series and Fourier integrals. For example, one can formulate a Parseval theorem for periodic discrete-time signals. Following the introduction of the so-called cyclical convolution we can derive convolution theorems, which look very similar to the ones derived for continuous-time signals.

INTRODUCTION

Many phenomena in the applications of the natural and engineering sciences are periodic in nature. Examples are the vibrations of strings, springs and other objects, rotating parts in machines, the movement of the planets around the sun, the tides of the sea, the movement of a pendulum in a clock, the voltages and currents in electrical networks, electromagnetic signals emitted by transmittters in satellites, light signals transmitted through glassfibers, etc. Seemingly, all these systems operate in complicated ways; the phenomena that can be observed often behave in an erratic way. In many cases, however, they do show some kind of repetition. In order to analyse these systems, one can make use of elementary periodic functions or signals from mathematics, the sine and cosine functions. For many systems, the response or behaviour can be completely calculated or measured, by exposing them to influences or inputs given by these elementary functions. When, moreover, these systems are linear, then one can also calculate the response to a linear combination of such influences, since this will result in the same linear combination of responses.

Hence, for the study of the aforementioned phenomena, two matters are of importance.

On the one hand one should look at how systems behave under influences that can be described by elementary mathematical functions. Such an analysis will in general require specific knowledge of the system being studied.

INTRODUCTION

In chapter 1 signals were divided into continuous-time and discrete-time signals. Ever since, we have almost exclusively discussed continuous-time signals. This chapter, being the first chapter of part 5, can be considered as sort of a transition from the continuous-time to the discrete-time signals. In section 15.1 we first introduce a number of important discrete-time signals, which are very similar to well-known continuous-time signals like the unit pulse or delta function. Subsequently, we pay special attention in section 15.2 to the transformation of a continuous-time signal into a discrete-time signal (sampling) and vice versa (reconstruction), leading to the formulation and the proof of the so-called sampling theorem in section 15.3. The sampling theorem gives a lower bound (the so-called Nyquist frequency) for the sampling frequency such that a given continuous-time signal can be transformed into a discrete-time signal without loss of information. We close with the treatment of the so-called aliasing problem in section 15.4. This problem arises when a continuous-time signal is transformed into a discrete-time signal using a sampling frequency which is too low.

LEARNING OBJECTIVES

After studying this chapter it is expected that you

1. - can describe discrete-time signals using unit pulses

2. - can describe periodic discrete-time signals using periodic unit pulses

3. - can explain the meaning of the terms sampling, sampling period and sampling frequency

4. - can explain the sampling theorem and can apply it

5. - can understand the reconstruction formula for band-limited signals

6. - can describe the consequences of the aliasing problem.

INTRODUCTION

Chapter 3 has been a first introduction to Fourier series. These series can be associated with periodic functions. We also noted in chapter 3 that if the function satisfies certain conditions, the Fourier series converges to the periodic function. What these specific conditions should be has not been analysed in chapter 3. The conditions that will be imposed in this book imply that the function should be piecewise smooth. In this chapter we will prove that a Fourier series of a piecewise smooth periodic function converges pointwise to the periodic function. We stress here that this condition is sufficient: when it holds, the series is pointwise convergent. This condition does not cover all cases of pointwise convergence and is thus not necessary for convergence.

In the first section of this chapter we derive a number of properties of Fourier coefficients that will be useful in the second section, where we prove the fundamental theorem. In the fundamental theorem we prove that for a piecewise smooth periodic function the Fourier series converges to the function. In the third section we then derive some further properties of Fourier series: product and convolution, Parseval's theorem (which has applications in the analysis of systems and signals), and integration and differentiation of Fourier series. We end this chapter with the treatment of Gibbs' phenomenon, which describes the convergence behaviour of the Fourier series at a jump discontinuity.