Fast Fourier Transform

In a hypothetical conversation between a trader in interest-rate derivatives and a quantitative analyst, Brigo and Mercurio (2001) let the trader answer about the pros and cons of short rate models: ”… we should be careful in thinking market models are the final and complete solution to all problems in interest rate models … and who knows, maybe short rate models will come back one day…”

This book describes how a key signal/image processing algorithm – that of the fast Hartley transform (FHT) or, via a simple conversion routine between their outputs, of the real‑data version of the ubiquitous fast Fourier transform (FFT) – might best be formulated to facilitate computationally-efficient solutions.

. . . that is what learning is. You suddenly understand something you've un­ derstood all your life, but in a new way. Various transforms have been widely used in diverse applications of science, engineering and technology. New transforms are emerging to solve many problems, which may have been left unsolved in the past, or newly created by modern science or technologies.

This book presents a comprehensive overview of Fourier transforms and the treatment of superconductivity under the vision of the electromagnetic properties of superconductors

This book has two main objectives, the first of which is to extend the power of numerical Fourier analysis and to show by means of theoretical examples and numerous concrete applications that when computing discrete Fourier transforms of periodic and non periodic functions, the usual kernel matrix of the Fourier transform, the discrete Fourier transform (DFT), should be replaced by another kernel matrix, the eXtended Fourier transform (XFT), since the XFT matrix appears as a convergent quadrature of a more general transform, the fractional Fourier transform.

Following an introduction to the basis of the fast Fourier transform (FFT), this book focuses on the implementation details on FFT for parallel computers. FFT is an efficient implementation of the discrete Fourier transform (DFT), and is widely used for many applications in engineering, science, and mathematics. Presenting many algorithms in pseudo-code and a complexity analysis, this book offers a valuable reference guide for graduate students, engineers, and scientists in the field who wish to apply FFT to large-scale problems.