درس‌هایی در مورد تبدیل فوریه و کاربردهای آن (متون کارشناسی محض و کاربردی) (متون کارشناسی محض و کاربردی، ۳۳)

This book is derived from lecture notes for a course on Fourier analysis for engineering and science students at the advanced undergraduate or beginning graduate level. Beyond teaching specific topics and techniques all of which are important in many areas of engineering and science the author's goal is to help engineering and science students cultivate more advanced mathematical know-how and increase confidence in learning and using mathematics, as well as appreciate the coherence of the subject. He promises the readers a little magic on every page. The section headings are all recognizable to mathematicians, but the arrangement and emphasis are directed toward students from other disciplines. The material also serves as a foundation for advanced courses in signal processing and imaging. There are over 200 problems, many of which are oriented to applications, and a number use standard software. An unusual feature for courses meant for engineers is a more detailed and accessible treatment of distributions and the generalized Fourier transform. There is also more coverage of higher-dimensional phenomena than is found in most books at this level. Cover Title page Preface Thanks Chapter 1. Fourier Series 1.1. Choices: Welcome Aboard 1.2. Periodic Phenomena 1.3. It All Adds Up 1.4. Two Examples and a Warning 1.5. The Math, Part 1: A Convergence Result 1.6. Fourier Series in Action 1.7. The Math, Part 2: Orthogonality and Square Integrable Functions 1.8. Appendix: Notes on the Convergence of Fourier Series 1.9. Appendix: The Cauchy-Schwarz Inequality Problems and Further Results Chapter 2. Fourier Transform 2.1. A First Look at the Fourier Transform 2.2. Getting to Know Your Fourier Transform 2.3. Getting to Know Your Fourier Transform, Better 2.4. Different Definitions of the Fourier Transform, and What Happens to the Formulas Problems and Further Results Chapter 3. Convolution 3.1. A * Is Born 3.2. What Is Convolution, Really? 3.3. Properties of Convolution: It’s a Lot Like Multiplication 3.4. Convolution in Action I: A Little Bit on Filtering 3.5. Convolution in Action II: Differential Equations 3.6. Convolution in Action III: The Central Limit Theorem 3.7. Heisenberg’s Inequality Problems and Further Results Chapter 4. Distributions and Their Fourier Transforms 4.1. The Day of Reckoning 4.2. The Best Functions for Fourier Transforms: Rapidly Decreasing Functions 4.3. A Very Little on Integrals 4.4. Distributions 4.5. Defining Distributions 4.6. Fluxions Finis: The End of Differential Calculus 4.7. Convolutions and the Convolution Theorem 4.8. Appendix: Windowing, Convolution, and Smoothing 4.9. Epilog and Some References Problems and Further Results Chapter 5. δ Hard at Work 5.1. Filters, Redux 5.2. Diffraction: Sincs Live and in Pure Color 5.3. X-Ray Diffraction 5.4. The \boldmath\shah-Function on Its Own 5.5. Periodic Distributions and Fourier Series 5.6. A Formula for δ Applied to a Function, and a Mention of Pullbacks 5.7. Cutting Off a \dl 5.8. Appendix: How Special Is \shah? Problems and Further Results Chapter 6. Sampling and Interpolation 6.1. Sampling sines and the Idea of a Bandlimited Signal 6.2. Sampling and Interpolation for Bandlimited Signals 6.3. Undersampling and Aliasing 6.4. Finite Sampling for a Bandlimited Periodic Signal 6.5. Appendix: Timelimited vs. Bandlimited Signals 6.6. Appendix: Linear Interpolation via Convolution 6.7. Appendix: Lagrange Interpolation Problems and Further Results Chapter 7. Discrete Fourier Transform 7.1. The Modern World 7.2. From Continuous to Discrete 7.3. The Discrete Fourier Transform 7.4. Notations and Conventions 1 7.5. Two Grids, Reciprocally Related 7.6. Getting to Know Your Discrete Fourier Transform 7.7. Notations and Conventions 2 7.8. Getting to Know Your DFT, Better 7.9. The Discrete Rect and Its DFT 7.10. Discrete Sampling and Interpolation 7.11. The FFT Algorithm Problems and Further Results Chapter 8. Linear Time-Invariant Systems 8.1. We Are All Systemizers Now 8.2. Linear Systems 8.3. Examples 8.4. Cascading Linear Systems 8.5. The Impulse Response, or the Deepest Fact in the Theory of Distributions Is Well Known to All Electrical Engineers 8.6. Linear Time-Invariant (LTI) Systems 8.7. The Fourier Transform and LTI Systems 8.8. Causality 8.9. The Hilbert Transform 8.10. Filters Finis 8.11. A Tribute: The Linear Millennium Problems and Further Results Chapter 9. n -Dimensional Fourier Transform 9.1. Space, the Final Frontier 9.2. Getting to Know Your Higher-Dimensional Fourier Transform 9.3. A Little \dl Now, More Later 9.4. Higher-Dimensional Fourier Series 9.5. \boldmath\shah, Lattices, Crystals, and Sampling 9.6. The Higher-Dimensional DFT 9.7. Naked to the Bone 9.8. Appendix: Line Impulses 9.9. Appendix: Pullback of a Distribution Problems and Further Results Appendix A. A List of Mathematical Topics that Are Fair Game Appendix B. Complex Numbers and Complex Exponentials B.1. Complex Numbers B.2. The Complex Exponential and Euler’s Formula B.3. Further Applications of Euler’s Formula Problems and Further Results Appendix C. Geometric Sums Problems and Further Results Index Back Cover This book is derived from lecture notes for a course on Fourier analysis for engineering and science students at the advanced undergraduate or beginning graduate level. Beyond teaching specific topics and techniques—all of which are important in many areas of engineering and science—the author's goal is to help engineering and science students cultivate more advanced mathematical know-how and increase confidence in learning and using mathematics, as well as appreciate the coherence of the subject. He promises the readers a little magic on every page. The section headings are all recognizable to mathematicians, but the arrangement and emphasis are directed toward students from other disciplines. The material also serves as a foundation for advanced courses in signal processing and imaging. There are over 200 problems, many of which are oriented to applications, and a number use standard software. An unusual feature for courses meant for engineers is a more detailed and accessible treatment of distributions and the generalized Fourier transform. There is also more coverage of higher-dimensional phenomena than is found in most books at this level. A thoroughly enjoyable yet careful mathematical perspective of the underlying concepts and many applications of modern signal analysis. —Les Atlas, University of Washington Osgood leads his readers from the basics to the more sophisticated parts of applicable Fourier analysis with a lively style, a light touch on the technicalities, and an eye toward communications engineering. This book should be a great resource for students of mathematics, physics, and engineering alike. —Gerald B. Folland, University of Washington Fourier analysis with a swing in its step. —Tom Körner, University of Cambridge