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.

INTRODUCTION

We start this chapter with an intuitive derivation of the main result for Fourier integrals from the fundamental theorem of Fourier series. A mathematical rigorous treatment of the results obtained is postponed until chapter 7. In the present chapter the Fourier integral will thus play a minor role. First we will concentrate ourselves on the Fourier transform of a non-periodic function, which will be introduced in section 6.2, motivated by our intuitive derivation. After discussing the existence of the Fourier transform, a number of frequently used and often recurring examples are treated in section 6.3. In section 6.4 we prove some basic properties of Fourier transforms. Subsequently, the concept of a ‘rapidly decreasing function’ is discussed in section 6.5; in fact this is a preparation for the distribution theory of chapters 8 and 9. The chapter closes with the treatment of convolution and the convolution theorem for non-periodic functions.

LEARNING OBJECTIVES

After studying this chapter it is expected that you

1. - know the definition of the Fourier transform

2. - can calculate elementary Fourier transforms

3. - know and can apply the properties of the Fourier transform

4. - know the concept of rapidly decreasing function

5. - know the definition of convolution and know and can apply the convolution theorem.

An intuitive derivation

In the introduction we already mentioned that in order to make the basic formulas of the Fourier analysis of non-periodic function plausible, we use the theory of Fourier series.

INTRODUCTION

In the previous chapter we have seen that distributions form an extension of the familiar functions. Moreover, in most cases it is not very hard to imagine a distribution intuitively as a limit of a sequence of functions. Especially when introducing new operations for distributions (such as differentiation), such an intuitive representation can be very useful. In section 8.1 we applied this method to make it plausible that the Fourier transform of the delta function is the constant function 1, and also that the reciprocity property holds in this case.

The purpose of the present chapter is to develop a rigorous Fourier theory for distributions. Of course, the theory has to be set up in such a way that for functions we recover our previous results; this is because distributions are an extension of functions. This is why we will derive the definition of the Fourier transform of a distribution from a property of the Fourier transform of functions in section 9.1. Subsequently, we will determine the spectrum of a number of standard distributions. Of course, the delta function will be treated first.

In section 9.2 we concentrate on the properties of the Fourier transform of distributions. The reciprocity property for distributions is proven. We also treat the correspondence between differentiation and multiplication. Finally, we show that the shift properties also remain valid for distributions.

It is quite problematic to give a rigorous definition of the convolution product or to state (let alone prove) a convolution theorem.

INTRODUCTION

Many new concepts and theories in mathematics arise from the fact that one is confronted with problems that existing theories cannot solve. These problems may originate from mathematics itself, but often they arise elsewhere, such as in physics. Especially fundamental problems, sometimes remaining unsolved for years, decades or even centuries, have a very stimulating effect on the development of mathematics (and science in general). The Greeks, for example, tried to find a construction of a square having the same area as the unit circle. This problem is known as the ‘quadrature of the circle’ and remained unsolved for some two thousand years. Not until 1882 it was found that such a construction was impossible, and it was discovered that the area of the unit circle, hence the number π, was indeed a very special real number.

Many of the concepts which one day solved a very fundamental problem are now considered obvious. Even the concept of ‘function’ has one day been heavily debated, in particular relating to questions on the convergence of Fourier series. Problems arising in the context of the solutions of quadratic and cubic equations were solved by introducing the now so familiar complex numbers. As is well-known, the complex numbers form an extension of the set of real numbers.

In this chapter we will introduce new objects, the so-called ‘distributions’, which form an extension of the concept of function.

INTRODUCTION

In the first three sections of this chapter the number of properties of the Laplace transform will be extended even further. We start in section 13.1 with the treatment of the by now well-known convolution product. As for the Fourier transform, the convolution product is transformed into an ordinary product by the Laplace transform.

In section 13.2 we treat two theorems that have not been encountered earlier in the Fourier transform: the so-called initial and final value theorems for the Laplace transform. The initial value theorem relates the ‘initial value’ f(0+) of a function f(t) to the behaviour of the Laplace transform F(s) for s → ∞. Similarly, the final value theorem relates the ‘final value’ limt→∞f(t) to the behaviour of F(s) for s → 0. Hence, the function F(s) can provide information about the behaviour of the original function f(t) shortly after switching on (the value f(0+)) and ‘after a considerable amount of time’ (the value limt→∞f(t)).

In section 13.3 we will see how the Laplace transform of a periodic function can be determined. It will turn out that this is closely related to the Laplace transform of the function which arises when we limit the periodic function to one period.

In order to determine the Laplace transform of a periodic function, it is not necessary to turn to the theory of distributions. This is in contrast to the Fourier transform (see section 9.1.2).

INTRODUCTION

The Fourier transform is one of the most important tools in the study of the transfer of signals in control and communication systems. In chapter 1 we have already discussed signals and systems in general terms. Now that we have the Fourier integral available, and are familiar with the delta function and other distributions, we are able to get a better understanding of the transfer of signals in linear time-invariant systems. The Fourier integral plays an important role in continuous-time systems which, moreover, are linear and time-invariant. These have been introduced in chapter 1 and will be denoted here by LTC-systems for short, just as in chapter 5.

Systems can be described by giving the relation between the input u(t) and the corresponding output or response y(t). This can be done in several ways. For example, by a description in the time domain (in such a description the variable t occurs), or by a description in the frequency domain. The latter means that a relation is given between the spectra (the Fourier transforms) U(ω) and Y(ω) of, respectively, the input u(t) and the response y(t).

In section 10.1 we will see that for LTC-systems the relation between u(t) and y(t) can be expressed in the time domain by means of a convolution product. Here the response h(t) to the unit pulse, or delta function, δ(t) plays a central role.

INTRODUCTION

Applications of Fourier series can be found in numerous places in the natural sciences as well as in mathematics itself. In this chapter we confine ourselves to two kinds of applications, to be treated in sections 5.1 and 5.2. Section 5.1 explains how Fourier series can be used to determine the response of a linear time-invariant system to a periodic input. In section 5.2 we discuss the applications of Fourier series in solving partial differential equations, which often occur when physical processes, such as heat conduction or a vibrating string, are described mathematically.

The frequency response, introduced in chapter 1 using the response to the periodic time-harmonic signal eiωt with frequency ω, plays a central role in the calculation of the response of a linear time-invariant system to an arbitrary periodic signal. Specifically, a Fourier series shows how a periodic signal can be written as a superposition of time-harmonic signals with frequencies being an integer multiple of the fundamental frequency. By applying the so-called superposition rule for linear time-invariant systems, one can then easily find the Fourier series of the output. This is because the sequence of Fourier coefficients, or the line spectrum, of the output arises from the line spectrum of the input by a multiplication by the frequency response at the integer multiples of the fundamental frequency.

For stable systems which can be described by ordinary differential equations, which is almost any linear time-invariant system occurring in practice, we will see that the frequency response can easily be derived from the differential equation.

This book arose from the development of a course on Fourier and Laplace transforms for the Open University of the Netherlands. Originally it was the intention to get a suitable course by revising part of the book Analysis and numerical analysis, part 3 in the series Mathematics for higher education by R. van Asselt et al. (in Dutch). However, the revision turned out to be so thorough that in fact a completely new book was created. We are grateful that Educaboek was willing to publish the original Dutch edition of the book besides the existing series.

In writing this book, the authors were led by a twofold objective:

1. - the ‘didactical structure’ should be such that the book is suitable for those who want to learn this material through self-study or distance teaching, without damaging its usefulness for classroom use;

2. - the material should be of interest to those who want to apply the Fourier and Laplace transforms as well as to those who appreciate a mathematically sound treatment of the theory.

We assume that the reader has a mathematical background comparable to an undergraduate student in one of the technical sciences. In particular we assume a basic understanding and skill in differential and integral calculus. Some familiarity with complex numbers and series is also presumed, although chapter 2 provides an opportunity to refresh this subject.

The material in this book is subdivided into parts. Each part consists of a number of coherent chapters covering a specific part of the field of Fourier and Laplace transforms.

INTRODUCTION TO PART 5

In the previous chapters we have seen how the Fourier transform of a continuous-time signal can be calculated using tables and properties. However, it is not always possible to apply these methods. The reason could be that we only know the continuous-time signal for a limited number of moments in time, or simply that the Fourier integral cannot be determined analytically. Starting from a limited set of data, one then usually has to rely on numerical methods in order to determine Fourier transforms or spectra. To turn a numerical method into a manageable tool for a user, it is first transformed into an algorithm, which can then be processed by a digital computer. The user then has a program at his/her disposal to calculate spectra or Fourier transforms. Calculating the spectrum of a continuous-time signal using a computer program can be considered as signal processing. When an algorithm for such a program is studied in more detail, then one notices that almost all calculations are implemented in terms of numbers, or sequences of numbers. In fact, the continuous-time signal is first transformed into a sequence of numbers (we will call this a discrete-time signal) representing the function values, and subsequently this sequence is processed by the algorithm. One then calls this digital signal processing. It is clear that because of the finite computing time available, and the limited memory capacity of a digital computer, the spectrum can only be determined for a finite number of frequencies, and is seldom exact.

Newton's laws of motion

Kinetics is concerned with the relation between force and motion. The basis of this science is the laws of motion formulated by Isaac Newton (1642–1727) in his Principia, published in 1687. The essence of these laws is as follows.

1. First law. If the resultant force acting on a particle is zero, the acceleration of the particle will also be zero, i.e. it will continue in its state of rest or of constant velocity in a straight line.

2. Second law. If the resultant force acting on a particle is non-zero, the particle will accelerate in the direction of the force with a magnitude proportional to the magnitude of the force and inversely proportional to the mass of the particle.

3. Third law. To every action there is an equal and opposite reaction. For instance, if particle A collides with particle B, the reaction from B on A will be equal and opposite to the collision force from A on B.

The laws have been stated here with regard to the motion of a particle. This is convenient for the later development of the equations of motion of a body involving rotation as well as translation. However, in many cases the laws apply equally well to the motion of a body as to a particle. Indeed, Newton introduced the laws with regard to bodies rather than particles.

In order to perform calculations using Newton's laws of motion, it is necessary to specify units of measurement.