Fourier and Laplace Transforms

by R. J. Beerends , H. G. ter Morsche , J. C. van den Berg , E. M. van de Vrie

• ISBN13: 9780521806893
• ISBN10: 0521806895
• Format: Hardcover
• Copyright: 2003-08-18
• Publisher: Cambridge University Press

Summary

This textbook presents in a unified manner the fundamentals of both continuous and discrete versions of the Fourier and Laplace transforms. These transforms play an important role in the analysis of all kinds of physical phenomena. As a link between the various applications of these transforms the authors use the theory of signals and systems, as well as the theory of ordinary and partial differential equations. The book is divided into four major parts: periodic functions and Fourier series, non-periodic functions and the Fourier integral, switched-on signals and the Laplace transform, and finally the discrete versions of these transforms, in particular the Discrete Fourier Transform together with its fast implementation, and the z-transform. This textbook is designed for self-study. It includes many worked examples, together with more than 120 exercises, and will be of great value to undergraduates and graduate students in applied mathematics, electrical engineering, physics and computer science.

Table of Contents

Preface	ix
Introduction	1	(6)
Part 1 Applications and foundations
1 Signals and systems	7	(20)
1.1 Signals and systems	8	(3)
1.2 Classification of signals	11	(5)
1.3 Classification of systems	16	(11)
2 Mathematical prerequisites	27	(33)
2.1 Complex numbers, polynomials and rational functions	28	(7)
2.2 Partial fraction expansions	35	(4)
2.3 Complex-valued functions	39	(6)
2.4 Sequences and series	45	(6)
2.5 Power series	51	(9)
Part 2 Fourier series
3 Fourier series: definition and properties	60	(26)
3.1 Trigonometric polynomials and series	61	(4)
3.2 Definition of Fourier series	65	(6)
3.3 The spectrum of periodic functions	71	(1)
3.4 Fourier series for some standard functions	72	(4)
3.5 Properties of Fourier series	76	(4)
3.6 Fourier cosine and Fourier sine series	80	(6)
4 The fundamental theorem of Fourier series	86	(27)
4.1 Bessel's inequality and Riemann-Lebesgue lemma	86	(3)
4.2 The fundamental theorem	89	(6)
4.3 Further properties of Fourier series	95	(10)
4.4 The sine integral and Gibbs' phenomenon	105	(8)
5 Applications of Fourier series	113	(25)
5.1 Linear time-invariant systems with periodic input	114	(8)
5.2 Partial differential equations	122	(16)
Part 3 Fourier integrals and distributions
6 Fourier integrals. definition and properties	138	(26)
6.1 An intuitive derivation	138	(2)
6.2 The Fourier transform	140	(4)
6.3 Some standard Fourier transforms	144	(5)
6.4 Properties of the Fourier transform	149	(7)
6.5 Rapidly decreasing functions	156	(2)
6.6 Convolution	158	(6)
7 The fundamental theorem of the Fourier integral	164	(24)
7.1 The fundamental theorem	165	(7)
7.2 Consequences of the fundamental theorem	172	(9)
7.3 Poisson's summation formula*	181	(7)
8 Distributions	188	(20)
8.1 The problem of the delta function	189	(3)
8.2 Definition and examples of distributions	192	(5)
8.3 Derivatives of distributions	197	(6)
8.4 Multiplication and scaling of distributions	203	(5)
9 The Fourier transform of distributions	208	(21)
9.1 The Fourier transform of distributions: definition and examples	209	(8)
9.2 Properties of the Fonner transform	217	(4)
9.3 Conclusion	221	(8)
10 Applications of the Fourier integral	229	(24)
10.1 The impulse response	230	(4)
10.2 The frequency response	234	(5)
10.3 Causal stable systems and differential equations	239	(4)
10.4 Boundary and initial value problems for partial differential equations	243	(10)
Part 4 Laplace transforms
11 Complex functions	253	(14)
11.1 Definition and examples	253	(3)
11.2 Continuity	256	(3)
11.3 Differentiability	259	(4)
11.4 The Cauchy-Riemann equations*	263	(4)
12 The Laplace transform: definition and properties	267	(21)
12.1 Definition and existence of the Laplace transform	268	(7)
12.2 Linearity, shifting and scaling	275	(5)
12.3 Differentiation and integration	280	(8)
13 Further properties, distributions, and the fundamental theorem	288	(22)
13.1 Convolution	289	(2)
13.2 Initial and final value theorems	291	(3)
13.3 Periodic functions	294	(3)
13.4 Laplace transform of distributions	297	(6)
13.5 The inverse Laplace transform	303	(7)
14 Applications of the Laplace transform	310	(30)
14.1 Linear systems	311	(12)
14.2 Linear differential equations with constant coefficients	323	(4)
14.3 Systems of linear differential equations with constant coefficients	327	(3)
14.4 Partial differential equations	330	(10)
Part 5 Discrete transforms
15 Sampling of continuous-time signals	340	(16)
15.1 Discrete-time signals and sampling	340	(4)
15.2 Reconstruction of continuous-time signals	344	(3)
15.3 The sampling theorem	347	(4)
15.4 The aliasing problem*	351	(5)
16 The discrete Fourier transform	356	(19)
16.1 Introduction and definition of the discrete Fourier transform	356	(6)
16.2 Fundamental theorem of the discrete Fourier transform	362	(2)
16.3 Properties of the discrete Fourier transform	364	(4)
16.4 Cyclical convolution	368	(7)
17 The Fast Fourier Transform	375	(16)
17.1 The DFT as an operation on matrices	376	(4)
17.2 The N-point DFT with N = 2m	380	(3)
17.3 Applications	383	(8)
18 The z-transform	391	(21)
18.1 Definition and convergence of the z-transform	392	(4)
18.2 Properties of the z-transform	396	(4)
18.3 The inverse z-transform of rational functions	400	(4)
18.4 Convolution	404	(3)
18.5 Fourier transform of non-periodic discrete-time signals	407	(5)
19 Applications of discrete transforms	412	(17)
19.1 The impulse response	413	(6)
19.2 The transfer function and the frequency response	419	(5)
19.3 LTD-systems described by difference equations	424	(5)
Literature	429	(3)
Tables of transforms and properties	432	(12)
Index	444