Fourier Analysis

Roger Ceschi is the Director at Esigetel and Deputy Director General at Groupe Efrei in Paris, France. He is also Professor at several Chinese and South African universities. His research focuses on signal theory and stochastic processes.Jean-Luc Gautier retired as University Professor at ENSEA in Cergy, France in 2014. His research focused on the design of microwave circuits and the architecture of digital communications systems.

• Preface xiChapter 1. Fourier Series  11.1. Theoretical background 11.1.1. Orthogonal functions  11.1.2. Fourier Series 31.1.3. Periodic functions  51.1.4. Properties of Fourier series 61.1.5. Discrete spectra. Power distribution 81.2. Exercises  91.2.1. Exercise 1.1. Examples of decomposition calculations  101.2.2. Exercise 1.2  111.2.3. Exercise 1.3  121.2.4. Exercise 1.4  121.2.5. Exercise 1.5  121.2.6. Exercise 1.6. Decomposing rectangular functions 131.2.7. Exercise 1.7. Translation and composition of functions  141.2.8. Exercise 1.8. Time derivation of a function 151.2.9. Exercise 1.9. Time integration of functions 151.2.10. Exercise 1.10  151.2.11. Exercise 1.11. Applications in electronic circuits 161.3. Solutions to the exercises  171.3.1. Exercise 1.1. Examples of decomposition calculations  171.3.2. Exercise 1.2  251.3.3. Exercise 1.3  261.3.4. Exercice 1.4  261.3.5. Exercise 1.5  271.3.6. Exercise 1.6 271.3.7. Exercise 1.7. Translation and composition of functions  291.3.8. Exercise 1.8. Time derivation of functions  311.3.9. Exercise 1.9. Time integration of functions  321.3.10. Exercise 1.10 321.3.11. Exercise 1.11 35Chapter 2. Fourier Transform  392.1. Theoretical background  392.1.1. Fourier transform 392.1.2. Properties of the Fourier transform  422.1.3. Singular functions 462.1.4. Fourier transform of common functions  512.1.5. Calculating Fourier transforms using the Dirac impulse method  532.1.6. Fourier transform of periodic functions  542.1.7. Energy density 542.1.8. Upper limits to the Fourier transform 552.2. Exercises  562.2.1. Exercise 2.1  562.2.2. Exercise 2.2  572.2.3. Exercise 2.3  582.2.4. Exercise 2.4  592.2.5. Exercise 2.5  592.2.6. Exercise 2.6  592.2.7. Exercise 2.7  602.2.8. Exercise 2.8  602.2.9. Exercise 2.9  612.2.10. Exercise 2.10 622.2.11. Exercise 2.11 622.2.12. Exercise 2.12 632.2.13. Exercise 2.13 632.2.14. Exercise 2.14 642.2.15. Exercise 2.15 642.2.16. Exercise 2.16 652.2.17. Exercise 2.17 662.3. Solutions to the exercises 672.3.1. Exercise 2.1  672.3.2. Exercise 2.2  682.3.3. Exercise 2.3  742.3.4. Exercise 2.4  742.3.5. Exercise 2.5  762.3.6. Exercise 2.6  762.3.7. Exercise 2.7  772.3.8. Exercise 2.8  792.3.9. Exercise 2.9  822.3.10. Exercise 2.10  852.3.11 Exercise 2.11 862.3.12 Exercise 2.12 882.3.13 Exercise 2.13 912.3.14 Exercise 2.14 912.3.15 Exercice 2.15  922.3.16 Exercise 2.16 942.3.17 Exercise 2.17 95Chapter 3. Laplace Transform 973.1. Theoretical background 973.1.1. Definition 973.1.2. Existence of the Laplace transform  983.1.3. Properties of the Laplace transform  983.1.4. Final value and initial value theorems  1023.1.5. Determining reverse transforms  1023.1.6. Approximation methods  1053.1.7. Laplace transform and differential equations  1073.1.8. Table of common Laplace transforms  1083.1.9. Transient state and steady state  1103.2. Exercise instruction  1113.2.1. Exercise 3.1  1113.2.2. Exercise 3.2  1113.2.3. Exercise 3.3  1123.2.4. Exercise 3.4  1123.2.5. Exercise 3.5  1123.2.6. Exercise 3.6  1133.2.7. Exercise 3.7  1133.2.8. Exercise 3.8  1153.2.9. Exercise 3.9  1153.2.10. Exercise 3.10  1153.3. Solutions to the exercises  1163.3.1. Exercise 3.1  1163.3.2. Exercise 3.2  1173.3.3. Exercise 3.3  1213.3.4. Exercise 3.4  1223.3.5. Exercise 3.5  1303.3.6. Exercise 3.6  1313.3.7. Exercise 3.7  1323.3.8. Exercise 3.8  1363.3.9. Exercise 3.9  1383.3.10. Exercise 3.10 139Chapter 4. Integrals and Convolution Product  1434.1. Theoretical background  1434.1.1. Analyzing linear systems using convolution integrals 1434.1.2. Convolution properties  1444.1.3. Graphical interpretation of the convolution product 1454.1.4. Convolution of a function using a unit impulse 1454.1.5. Step response from a system  1474.1.6. Eigenfunction of a convolution operator 1484.2. Exercises  1494.2.1. Exercise 4.1  1494.2.2. Exercise 4.2  1504.2.3. Exercise 4.3  1504.2.4. Exercise 4.4  1514.2.5. Exercise 4.5  1514.2.6. Exercise 4.6  1524.3. Solutions to the exercises 1534.3.1. Exercise 4.1  1534.3.2. Exercise 4.2  1564.3.3. Exercise 4.3  1604.3.4. Exercise 4.4  1634.3.5. Exercise 4.5  1644.3.6. Exercise 4.6  165Chapter 5. Correlation 1695.1. Theoretical background  1695.1.1. Comparing signals  1695.1.2. Correlation function 1705.1.3. Properties of correlation functions 1725.1.4. Energy of a signal 1765.2. Exercises  1775.2.1. Exercise 5.1  1775.2.2. Exercise 5.2  1785.2.3. Exercise 5.3  1785.2.4. Exercise 5.4  1785.2.5. Exercice 5.5  1795.2.6. Exercice 5.6  1795.2.7. Exercise 5.7  1795.2.8. Exercice 5.8  1805.2.9. Exercise 5.9  1805.2.10. Exercise 5.10  1815.2.11. Exercise 5.11  1815.2.12. Exercise 5.12  1825.2.13. Exercise 5.13  1825.2.14. Exercise 5.14  1835.3. Solutions to the exercises  1835.3.1. Exercise 5.1  1835.3.2. Exercice 5.2  1885.3.3. Exercise 5.3  1915.3.4. Exercice 5.4  1925.3.5. Exercise 5.5  1935.3.6. Exercise 5.6  1965.3.7. Exercise 5.7  1975.3.8. Exercise 5.8  2015.3.9. Exercise 5.9  2045.3.10. Exercise 5.10  2055.3.11 Exercise 5.11 2065.3.12 Exercise 5.12 2075.3.13 Exercise 5.13 2085.3.14 Exercise 5.14 209Chapter 6. Signal Sampling 2136.1. Theoretical background 2136.1.1. Sampling principle  2136.1.2. Ideal sampling  2146.1.3. Finite width sampling  2186.1.4. Sample and hold (S/H) sampling 2216.2. Exercises  2256.2.1. Exercise 6.1  2256.2.2. Exercise 6.2  2256.2.3. Exercise 6.3  2266.2.4. Exercise 6.4  2266.2.5. Exercise 6.5  2266.2.6. Exercise 5.6  2276.2.7. Exercise 6.7  2276.2.8. Exercice 6.8  2286.3. Solutions to the exercises 2296.3.1. Exercise 6.1  2296.3.2. Exercise 6.2  2296.3.3. Exercise 6.3  2336.3.4. Exercice 6.4  2356.3.5. Exercise 6.5  2366.3.6. Exercise 6.6  2386.3.7. Exercise 6.7  2406.3.8. Exercise 6.8  242Bibliography  245Index 247