Little need be said of the importance and ubiquity of the ordinary
Fourier transform in many areas of science and engineering.
As a generalization of the ordinary Fourier transform, the fractional
Fourier transform is only richer in theory and more flexible in
applications--but not more costly in implementation.
This book provides a comprehensive and widely accessible account of
the transform covering both theory and applications.
Properties and applications of the ordinary Fourier transform are
special cases of those of the fractional Fourier transform.
The ordinary frequency domain is a special case of the continuum of
fractional Fourier domains, which are intimately related to
time-frequency representations such as the Wigner distribution.
In every area in which Fourier transforms and frequency-domain
concepts are used, there exists the potential for generalization and
improvement by using the fractional transform.
So far applications of the transform have been studied mostly in the
areas of optics and wave propagation, and signal analysis and
processing. They include Fourier optics and optical
information processing, beam propagation and spherical mirror
resonators (lasers), optical
systems and lens design, quantum optics, statistical optics, beam
synthesis and shaping, optical and quantum wavefield reconstruction
and phase retrieval, perspective projections, flexible and
cost-effective time- or space-variant filtering with applications in
signal and image restoration and reconstruction, signal extraction,
signal and system synthesis, correlation, detection, pattern
recognition, phase retrieval, radar, tomography, multiplexing, data
compression, linear FM detection. study of time-frequency
representations, differential equations, and harmonic motion.
Many of these applications are discussed at length in this book.
By consolidating knowledge on the transform and illustrating how
it has been applied in diverse contexts, the book aims to make
possible the discovery of new applications in other areas as well.
The considerable amount of background material is an important feature
of the book which is of interest in its own right, as self-contained
expositions or by presenting certain less encountered perspectives and
results. This material includes an introduction to time-frequency
analysis emphasizing the Wigner distribution and ambiguity
function, and canonical transforms. The chapters on optics complement
introductory texts on Fourier optics, dealing with optical systems in
phase space (the space-frequency plane) in terms of canonical
transforms. Matrix algebra is employed in a unified manner for both
wave and geometrical optical perspectives, leading to many important
and fundamental results, such as those on general Fourier transform
planes and optical invariants.
- Of interest to graduate and senior undergraduate students,
academicians, researchers, and professionals in
branches of mathematics, science, and engineering where Fourier
transforms and related concepts are used. A partial list of these
areas is operator theory, harmonic analysis and integral transforms,
linear algebra, group representation theory, phase-space methods,
time- and space-frequency representations, transform theory and techniques,
signal analysis and processing, wave propagation, and many areas of optics.
- Unifies knowledge from the mathematics, optics, and signal processing
literature in a manner accessible to a broad audience.
- Includes a comprehensive bibliography.
- Discussion of optics completely segregated for readers with
no interest or background in optics.
- May be used for self-study or in courses on the fractional Fourier
transform and time-frequency analysis and their applications in optics
and/or signal processing, advanced signal processing, advanced Fourier
optics or information optics emphasizing phase-space concepts and the
Ozaktas is the recipient of the 1998 ICO International Prize in
Optics (jointly with D. Mendlovic) and the
1999 Scientific and Technical Research Council of Turkey Science
Award for his contributions to the subject.