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Rating: 
Summary: Well written and complete
Review: I consider Gaskill's book to be the best I've seen for advanced undergraduate and first-year graduate classes on linear systems. Gaskill approaches the subject in a clear and understandable style while dealing with the subject in a complete and quantitative manner. Though he does not eschew mathematical rigor by any means, the text is well written and logically formatted, making it refreshingly easy to follow what is, in other texts, a more difficult subject. Though I've filed Gaskill's book in my library alongside other dealing with optics, this is primarily a book on mathematics, but written more for engineers and scientists than for mathematicians. After a brief introduction, the author begins (in chapter 2) with a quick summary of mathematical concepts, including classes of functions, one and two-dimensional functions, complex numbers, phasors, and the scalar wave equation. The third chapter introduces useful functions (many of a discontinuous nature) that find application in modeling linear systems. These include step functions and the impulse function in both one and two dimensions. Development of these functions follows an intuitive path that reflects the way in which they are often used. The many figures are particularly useful in conveying concepts more effectively. Chapter four develops the theme of harmonic analysis by introducing the notion of orthogonal expansions and extending this development to the Fourier series, leading to development of the Fourier integral. The chapter finishes with some worked examples showing the spectra of simple functions. Chapter 7 seems a little out of place, since it deals with the Fourier transform, yet appears in the book several chapters later, after the author introduces the concepts of linear systems and the convolution. Though one of the shorter chapters, chapter five is pivotal, and develops the idea of mathematical operators and physical systems - with the crucial development of the impulse response. The application of the impulse response is extended by chapter 6, which develops the mathematics of convolution. For a linear, shift-invariant system the impulse response convolved with the input to the system gives the system's output. Chapter 8 pulls together the material in the previous chapters to mathematically describe the characteristics and applications of linear filters. Examples include amplitude filters, phase filters, combination amplitude and phase filters, and some interesting applications showing (for example) how to filter the noise from a signal of interest. All this development is strictly mathematical, with no real-world worked examples (except in the abstract). Nevertheless, this chapter is very useful and (in the author's style) easy to understand and follow. Chapter 9 deals with two-dimensional convolutions and the two-dimensional Fourier transform. This chapter is essentially an extension of the earlier one-dimensional developments in earlier chapters, but introduces some useful mathematical tools, including the convolution and Fourier transform in polar coordinates. The Hankel transform, developed in this chapter, is particularly useful for work in optics where many examples (laser beams, for instance) exhibit circular symmetry. In these examples the two-dimensional integrals may be greatly simplified by the Hankel transform to a one-dimensional form where (even in the absence of a closed-form equation) they are far more tractable. The chapter concludes with useful tables of common transforms. Chapter 10 leaves the almost purely mathematical forum of the previous chapters by introducing the subject of propagation and diffraction of optical waves. Gaskill first develops the mathematics of the optical waves and then derives the equations that show how these waves are diffracted. Not surprisingly, the diffraction fields are expressible in terms of the transforms developed earlier in the book. The chapter also describes the influence of optical lenses on the diffraction patterns and the very important subject of propagation of Gaussian beams (since many laser beams, and the fundamental mode in weakly guiding optical fibers have Gaussian profiles). Chapter 11 continues the optical theme by explaining image-forming systems. The student will be particularly enabled in this chapter if he or she has had prior exposure to the subject of diffraction and perhaps some exposure to the idea of image aberrations. The book ends with appendix 1, on special functions, and appendix 2, on elementary geometric optics. Each chapter has a list of references, and problems for the student, and the book has a complete index making it useful as a desk references book as well as a textbook for advanced undergraduate and first-year graduate coursework. Gaskill's book is mathematically intense, but the author's style and frequent use of figures makes the book surprisingly easy to read. Prerequisites for this book should include a couple of years of calculus, differential equations, and a smattering of linear algebra. Some exposure to concepts in optics, including diffraction and aberrations would also be helpful. Gaskill's book will be helpful far beyond optics, with applications in electrical engineering, mechanical engineering, digital image processing, or anywhere else that linear systems might be encountered.
Rating:
Summary: Well written and complete
Review: I consider Gaskill's book to be the best I've seen for advanced undergraduate and first-year graduate classes on linear systems. Gaskill approaches the subject in a clear and understandable style while dealing with the subject in a complete and quantitative manner. Though he does not eschew mathematical rigor by any means, the text is well written and logically formatted, making it refreshingly easy to follow what is, in other texts, a more difficult subject. Though I've filed Gaskill's book in my library alongside other dealing with optics, this is primarily a book on mathematics, but written more for engineers and scientists than for mathematicians. After a brief introduction, the author begins (in chapter 2) with a quick summary of mathematical concepts, including classes of functions, one and two-dimensional functions, complex numbers, phasors, and the scalar wave equation. The third chapter introduces useful functions (many of a discontinuous nature) that find application in modeling linear systems. These include step functions and the impulse function in both one and two dimensions. Development of these functions follows an intuitive path that reflects the way in which they are often used. The many figures are particularly useful in conveying concepts more effectively. Chapter four develops the theme of harmonic analysis by introducing the notion of orthogonal expansions and extending this development to the Fourier series, leading to development of the Fourier integral. The chapter finishes with some worked examples showing the spectra of simple functions. Chapter 7 seems a little out of place, since it deals with the Fourier transform, yet appears in the book several chapters later, after the author introduces the concepts of linear systems and the convolution. Though one of the shorter chapters, chapter five is pivotal, and develops the idea of mathematical operators and physical systems - with the crucial development of the impulse response. The application of the impulse response is extended by chapter 6, which develops the mathematics of convolution. For a linear, shift-invariant system the impulse response convolved with the input to the system gives the system's output. Chapter 8 pulls together the material in the previous chapters to mathematically describe the characteristics and applications of linear filters. Examples include amplitude filters, phase filters, combination amplitude and phase filters, and some interesting applications showing (for example) how to filter the noise from a signal of interest. All this development is strictly mathematical, with no real-world worked examples (except in the abstract). Nevertheless, this chapter is very useful and (in the author's style) easy to understand and follow. Chapter 9 deals with two-dimensional convolutions and the two-dimensional Fourier transform. This chapter is essentially an extension of the earlier one-dimensional developments in earlier chapters, but introduces some useful mathematical tools, including the convolution and Fourier transform in polar coordinates. The Hankel transform, developed in this chapter, is particularly useful for work in optics where many examples (laser beams, for instance) exhibit circular symmetry. In these examples the two-dimensional integrals may be greatly simplified by the Hankel transform to a one-dimensional form where (even in the absence of a closed-form equation) they are far more tractable. The chapter concludes with useful tables of common transforms. Chapter 10 leaves the almost purely mathematical forum of the previous chapters by introducing the subject of propagation and diffraction of optical waves. Gaskill first develops the mathematics of the optical waves and then derives the equations that show how these waves are diffracted. Not surprisingly, the diffraction fields are expressible in terms of the transforms developed earlier in the book. The chapter also describes the influence of optical lenses on the diffraction patterns and the very important subject of propagation of Gaussian beams (since many laser beams, and the fundamental mode in weakly guiding optical fibers have Gaussian profiles). Chapter 11 continues the optical theme by explaining image-forming systems. The student will be particularly enabled in this chapter if he or she has had prior exposure to the subject of diffraction and perhaps some exposure to the idea of image aberrations. The book ends with appendix 1, on special functions, and appendix 2, on elementary geometric optics. Each chapter has a list of references, and problems for the student, and the book has a complete index making it useful as a desk references book as well as a textbook for advanced undergraduate and first-year graduate coursework. Gaskill's book is mathematically intense, but the author's style and frequent use of figures makes the book surprisingly easy to read. Prerequisites for this book should include a couple of years of calculus, differential equations, and a smattering of linear algebra. Some exposure to concepts in optics, including diffraction and aberrations would also be helpful. Gaskill's book will be helpful far beyond optics, with applications in electrical engineering, mechanical engineering, digital image processing, or anywhere else that linear systems might be encountered.
Rating:
Summary: Lifesaver
Review: If you want to survive a first year graduate class on Fourier Optics, get this book. Gaskill is precise and comprehensive, presenting concepts incrementally with ample diagrams to illustrate all along the way. I've got Goodman and Bracewell on my shelf, but it's Gaskill's that's saving my life this semester.
Rating:
Summary: Lifesaver
Review: If you want to survive a first year graduate class on Fourier Optics, get this book. Gaskill is precise and comprehensive, presenting concepts incrementally with ample diagrams to illustrate all along the way. I've got Goodman and Bracewell on my shelf, but it's Gaskill's that's saving my life this semester.
Rating:
Summary: The best, practical book for this subject.
Review: Jack Gaskill and his book is the most practical book on this subject. His examples and explainations are straightforeword and organized.
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