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Developing algorithms for multi-dimensional Fourier transforms,
this book presents results that yield highly efficient code on a
variety of vector and parallel computers. By emphasising the
unified basis for the many approaches to both one-dimensional and
multidimensional Fourier transforms, this book not only clarifies
the fundamental similarities, but also shows how to exploit the
differences in optimising implementations. It will thus be of great
interest not only to applied mathematicians and computer
scientists, but also to seismologists, high-energy physicists,
crystallographers, and electrical engineers working on signal and
image processing.
The aim of this work is to present several topics in time-frequency
analysis as subjects in abelian group theory. The algebraic point
of view pre dominates as questions of convergence are not
considered. Our approach emphasizes the unifying role played by
group structures on the development of theory and algorithms. This
book consists of two main parts. The first treats Weyl-Heisenberg
representations over finite abelian groups and the second deals
with mul tirate filter structures over free abelian groups of
finite rank. In both, the methods are dimensionless and
coordinate-free and apply to one and multidimensional problems. The
selection of topics is not motivated by mathematical necessity but
rather by simplicity. We could have developed Weyl-Heisenberg
theory over free abelian groups of finite rank or more generally
developed both topics over locally compact abelian groups. However,
except for having to dis cuss conditions for convergence, Haar
measures, and other standard topics from analysis the underlying
structures would essentially be the same. A re cent collection of
papers 17] provides an excellent review of time-frequency analysis
over locally compact abelian groups. A further reason for limiting
the scope of generality is that our results can be immediately
applied to the design of algorithms and codes for time frequency
processing."
This graduate-level text provides a language for understanding,
unifying, and implementing a wide variety of algorithms for digital
signal processing - in particular, to provide rules and procedures
that can simplify or even automate the task of writing code for the
newest parallel and vector machines. It thus bridges the gap
between digital signal processing algorithms and their
implementation on a variety of computing platforms. The
mathematical concept of tensor product is a recurring theme
throughout the book, since these formulations highlight the data
flow, which is especially important on supercomputers. Because of
their importance in many applications, much of the discussion
centres on algorithms related to the finite Fourier transform and
to multiplicative FFT algorithms.
The aim of this work is to present several topics in time-frequency
analysis as subjects in abelian group theory. The algebraic point
of view pre dominates as questions of convergence are not
considered. Our approach emphasizes the unifying role played by
group structures on the development of theory and algorithms. This
book consists of two main parts. The first treats Weyl-Heisenberg
representations over finite abelian groups and the second deals
with mul tirate filter structures over free abelian groups of
finite rank. In both, the methods are dimensionless and
coordinate-free and apply to one and multidimensional problems. The
selection of topics is not motivated by mathematical necessity but
rather by simplicity. We could have developed Weyl-Heisenberg
theory over free abelian groups of finite rank or more generally
developed both topics over locally compact abelian groups. However,
except for having to dis cuss conditions for convergence, Haar
measures, and other standard topics from analysis the underlying
structures would essentially be the same. A re cent collection of
papers 17] provides an excellent review of time-frequency analysis
over locally compact abelian groups. A further reason for limiting
the scope of generality is that our results can be immediately
applied to the design of algorithms and codes for time frequency
processing."
This graduate-level text provides a language for understanding, unifying, and implementing a wide variety of algorithms for digital signal processing - in particular, to provide rules and procedures that can simplify or even automate the task of writing code for the newest parallel and vector machines. It thus bridges the gap between digital signal processing algorithms and their implementation on a variety of computing platforms. The mathematical concept of tensor product is a recurring theme throughout the book, since these formulations highlight the data flow, which is especially important on supercomputers. Because of their importance in many applications, much of the discussion centres on algorithms related to the finite Fourier transform and to multiplicative FFT algorithms.
This book develops theory and algorithms leading to systematic
waveform design in time-frequency space. The key tool employed in
the work is the Zak transform, which provides a two-dimensional
image for sequences, the Fourier transform, convolution, and
correlation, and allows for the design of sequences directly in Zak
space. Application areas covered include pulse radars and sonars,
multibeam radar and sonar imaging systems, remote dielectric
material identification, and code division multiple-access
communication systems. This is an excellent reference text for
graduate students, researchers, and engineers in radar, sonar, and
communication systems.
Developing algorithms for multi-dimensional Fourier transforms, this book presents results that yield highly efficient code on a variety of vector and parallel computers. By emphasising the unified basis for the many approaches to both one-dimensional and multidimensional Fourier transforms, this book not only clarifies the fundamental similarities, but also shows how to exploit the differences in optimising implementations. It will thus be of great interest not only to applied mathematicians and computer scientists, but also to seismologists, high-energy physicists, crystallographers, and electrical engineers working on signal and image processing.
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