Linear Canonical Transforms

This book provides a clear and accessible introduction to the essential mathematical foundations of linear canonical transforms from a signals and systems perspective. Substantial attention is devoted to how these transforms relate to optical systems and wave propagation. There is extensive coverage of sampling theory and fast algorithms for numerically approximating the family of transforms. Chapters on topics ranging from digital holography to speckle metrology provide a window on the wide range of applications. This volume will serve as a reference for researchers in the fields of image and signal processing, wave propagation, optical information processing and holography, optical system design and modeling, and quantum optics. It will be of use to graduate students in physics and engineering, as well as for scientists in other areas seeking to learn more about this important yet relatively unfamiliar class of integral transformations.

Although it is straightforward to determine the relationship between the in-focus image and the object of a simple optical system such as a lens, it is far more challenging to compute the input/output relationships of general first-order and astigmatic optical systems. Such optical systems are known as quadratic-phase systems (QPS) and they include the Fresnel propagation in free space, propagation in graded-index media, passage through thin lenses, and arbitrary concatenations of any number of these, including anamorphic, astigmatic, nonorthogonal elements. Such computation is accomplished by representing the physical system with a general mathematical framework of integrations against kernels and then distilling the entire system into one input-output relationship that can be represented by a linear integral transform. The underlying mathematical integral transforms can be applied to a wider field of signal processing where they are known as the linear canonical transform (LCT) of a signal. Conventional numerical integration methods have a computational complexity of O(N^2) where N is the space-bandwidth product of the sampling scheme, e.g. the number of pixels in the field for an optical system. The algorithms described here yield a complexity of only O(Nlog N). The key is the use of different decompositions (or factorizations) of a given input/output relationship into simpler ones. Instead of following the general physical subparts in cascaded systems and computing input-output relations separately, these algorithms use the simplest possible decompositions to represent the entire system in terms of least possible number of steps. The algorithms are Fast Fourier Transform (FFT) based methods and the only essential deviation from exactness arises from approximating a continuous Fourier transform (FT) with the discrete Fourier transform (DFT). Thus the algorithms work with a performance similar to that of the fast Fourier transform algorithm in computing the Fourier transform, both in terms of speed and accuracy. Unlike conventional techniques these algorithms also track and control the space-bandwidth products, in order to achieve information that is theoretically sufficient but not wastefully redundant.

The Linear Canonical Transform (LCT) is a mathematical operation that generalizes the Fourier transform, fractional Fourier transform, and other similar transforms. It describes the transformation of signals or functions in time and frequency domains in a linear, continuous, and reversible manner. The LCT can be expressed using a 4x4 matrix, which allows for efficient computation and manipulation. The broadened scope of the LCT encompasses various applications in signal processing, optics, image processing, and quantum mechanics. Its versatility lies in its ability to handle a wide range of signal types, including non-stationary, chirp-like, and spatially varying signals. Additionally, the LCT provides a unified framework for analyzing and processing signals with different degrees of chirpiness or time-frequency characteristics. Applications of the LCT include signal modulation, filtering, feature extraction, pattern recognition, and image reconstruction. It has found particular utility in optical systems for beam shaping, propagation, and manipulation. Furthermore, the LCT has been applied in quantum mechanics to study wave-particle duality and quantum information processing.

Two dramatically different philosophical approaches to classical mechanics were proposed during the 17th - 18th centuries. Newton developed his vectorial formulation that uses time-dependent differential equations of motion to relate vector observables like force and rate of change of momentum. Euler, Lagrange, Hamilton, and Jacobi, developed powerful alternative variational formulations based on the assumption that nature follows the principle of least action. These variational formulations now play a pivotal role in science and engineering.This book introduces variational principles and their application to classical mechanics. The relative merits of the intuitive Newtonian vectorial formulation, and the more powerful variational formulations are compared. Applications to a wide variety of topics illustrate the intellectual beauty, remarkable power, and broad scope provided by use of variational principles in physics.The second edition adds discussion of the use of variational principles applied to the following topics:(1) Systems subject to initial boundary conditions(2) The hierarchy of related formulations based on action, Lagrangian, Hamiltonian, and equations of motion, to systems that involve symmetries.(3) Non-conservative systems.(4) Variable-mass systems.(5) The General Theory of Relativity.Douglas Cline is a Professor of Physics in the Department of Physics and Astronomy, University of Rochester, Rochester, New York.

Undergraduate-level introduction to linear algebra and matrix theory. Explores matrices and linear systems, vector spaces, determinants, spectral decomposition, Jordan canonical form, much more. Over 375 problems. Selected answers. 1972 edition.

Special relativity and quantum mechanics, formulated early in the twentieth century, are the two most important scientific languages and are likely to remain so for many years to come. In the 1920's, when quantum mechanics was developed, the most pressing theoretical problem was how to make it consistent with special relativity. In the 1980's, this is still the most pressing problem. The only difference is that the situation is more urgent now than before, because of the significant quantity of experimental data which need to be explained in terms of both quantum mechanics and special relativity. In unifying the concepts and algorithms of quantum mechanics and special relativity, it is important to realize that the underlying scientific language for both disciplines is that of group theory. The role of group theory in quantum mechanics is well known. The same is true for special relativity. Therefore, the most effective approach to the problem of unifying these two important theories is to develop a group theory which can accommodate both special relativity and quantum mechanics. As is well known, Eugene P. Wigner is one of the pioneers in developing group theoretical approaches to relativistic quantum mechanics. His 1939 paper on the inhomogeneous Lorentz group laid the foundation for this important research line. It is generally agreed that this paper was somewhat ahead of its time in 1939, and that contemporary physicists must continue to make real efforts to appreciate fully the content of this classic work.

Wavelet Transforms: Kith and Kin serves as an introduction to contemporary aspects of time-frequency analysis encompassing the theories of Fourier transforms, wavelet transforms and their respective offshoots. This book is the first of its kind totally devoted to the treatment of continuous signals and it systematically encompasses the theory of Fourier transforms, wavelet transforms, geometrical wavelet transforms and their ramifications. The authors intend to motivate and stimulate interest among mathematicians, computer scientists, engineers and physical, chemical and biological scientists. The text is written from the ground up with target readers being senior undergraduate and first-year graduate students and it can serve as a reference for professionals in mathematics, engineering and applied sciences. Features Flexibility in the book’s organization enables instructors to select chapters appropriate to courses of different lengths, emphasis and levels of difficulty Self-contained, the text provides an impetus to the contemporary developments in the signal processing aspects of wavelet theory at the forefront of research A large number of worked-out examples are included Every major concept is presented with explanations, limitations and subsequent developments, with emphasis on applications in science and engineering A wide range of exercises are incoporated in varying levels from elementary to challenging so readers may develop both manipulative skills in theory wavelets and deeper insight Answers and hints for selected exercises appear at the end The origin of the theory of wavelet transforms dates back to the 1980s as an outcome of the intriguing efforts of mathematicians, physicists and engineers. Owing to the lucid mathematical framework and versatile applicability, the theory of wavelet transforms is now a nucleus of shared aspirations and ideas.

In the world of physical science, important physical quantities like sound, pressure, electrical current, voltage, and electromagnetic fields vary with time. Such quantities are labeled as signals/waveforms and include oral signals, optical signals, acoustic signals, biomedical signals, radar, and sonar. Time-frequency analysis is a vital aid in signal analysis, which is concerned with how the frequency of a function (or signal) behaves in time, and it has evolved into a widely recognized applied discipline of signal processing. This book discusses the Fourier transform (FT), which is one of the most valuable and widely used integral transforms that converts a signal from time versus amplitude to frequency versus amplitude. It is one of the oldest tools in the time-frequency analysis of signals. The book includes five chapters that discuss general Fourier transforms as well as new and novel transforms such as hybrid transforms, quadratic-phase Fourier transforms, fractional Fourier transforms, linear canonical transforms, and more.