Linear Systems, Fourier Transforms, and Optics

This book is a textbook on linear systems, Fourier analysis, diffraction theory, and image formation. It is not a textbook on Fourier optics, but was intended to helps students with the basics before attempting that subject. This book might also be helpful to students that are studying linear systems theory or image processing alone and need an additional reference. There are problems at the end of each chapter, and the problems include both numerical calculations and derivations. No solutions to the problems are included. Numerous examples are shown with complete steps. Some examples are numerical, and many are not. Minus the optical material, I had already seen the rest of the material in the book before I used it, so perhaps I am not the best judge of how complete a textbook it was, but to me it seemed very complete and clear. Unlike many similar textbooks, the author did not assume much about the reader's background other than the Calculus, differential equations, and linear algebra that you would expect any graduate engineering student to have already mastered. I definitely recommend going through it or having access to it before you enroll in a class on Fourier optics. Chapter 2 presents an elementary review of various properties and classes of mathematical functons, as well as a description of the manner in which these functions represent physical quantities. Chapter 3 introduces a number of special functions that are of great use in later chapters. In particular the rectangle function, the sinc function, the delta function, and the comb function are very useful. Also, several special functions of two variables are described. In Chapter 4 the fundamentals of harmonic analysis are explored as well as how various arbitrary functions may be represented by linear combinations of other more elementary functions. Chapter 5 discusses the physical systems in term of linear operators, and the notions of linearity and shift invariance are introduced. Next, the impulse response function, the transfer function, and the eigenfunctions associated with linear shift-invariant systems are discussed. Chapter 6 is devoted to studies of the convolution, cross-correlation, and autocorrelation operations. The properties of these operations are explored in considerable depth. The fact that the output of a linear shift-invariant system is given by the convolution of the input with the impulse response of the system is derived and explored. In Chapter 7 the properties of the Fourier transformation is investigated, as well as the importance of this transform in the analysis of linear shift-invariant systems. In chapter 8 the characteristics of various types of linear filters are described. Their applications in various types of signal processing and recovery is discussed. Also discussed is the matched-filter problem and the various interpretations of the sampling theory. Chapter 9 extends the previous material on one-dimensional systems to two dimensions. In

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.

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 develo

Jack Gaskill and his book is the most practical book on this subject. His examples and explainations are straightforeword and organized.