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The second volume of the two volumes book is dedicated to various
extensions and generalizations of Dyadic (Walsh) analysis and
related applications. Considered are dyadic derivatives on Vilenkin
groups and various other Abelian and finite non-Abelian groups.
Since some important results were developed in former Soviet Union
and China, we provide overviews of former work in these countries.
Further, we present translations of three papers that were
initially published in Chinese. The presentation continues with
chapters written by experts in the area presenting discussions of
applications of these results in specific tasks in the area of
signal processing and system theory. Efficient computing of related
differential operators on contemporary hardware, including graphics
processing units, is also considered, which makes the methods and
techniques of dyadic analysis and generalizations computationally
feasible. The volume 2 of the book ends with a chapter presenting
open problems pointed out by several experts in the area.
Dyadic (Walsh) analysis emerged as a new research area in applied
mathematics and engineering in early seventies within attempts to
provide answers to demands from practice related to application of
spectral analysis of different classes of signals, including audio,
video, sonar, and radar signals. In the meantime, it evolved in a
mature mathematical discipline with fundamental results and
important features providing basis for various applications. The
book will provide fundamentals of the area through reprinting
carefully selected earlier publications followed by overview of
recent results concerning particular subjects in the area written
by experts, most of them being founders of the field, and some of
their followers. In this way, this first volume of the two volume
book offers a rather complete coverage of the development of dyadic
Walsh analysis, and provides a deep insight into its mathematical
foundations necessary for consideration of generalizations and
applications that are the subject of the second volume. The
presented theory is quite sufficient to be a basis for further
research in the subject area as well as to be applied in solving
certain new problems or improving existing solutions for tasks in
the areas which motivated development of the dyadic analysis.
Decision diagrams (DDs) are data structures for efficient (time/space) representations of large discrete functions. In addition to their wide application in engineering practice, DDs are now a standard part of many CAD systems for logic design and a basis for severe signal processing algorithms. "Spectral Interpretation of Decision Diagrams" derives from attempts to classify and uniformly interpret DDs through spectral interpretation methods, relating them to different Fourier-series-like functional expressions for discrete functions and a group-theoretic approach to DD optimization. The book examines DDs found in literature and engineering practice and provides insights into relationships between DDs and different polynomial or spectral expressions for representation of discrete functions. In addition, it offers guidelines and criteria for selection of the most suitable representation in terms of space and time complexity. The work complements theory with numerous illustrative examples from practice. Moreover, the importance of DD representations to the verification and testing of arithmetic circuits is addressed, as well as problems related to various signal processing tasks.
Anyone who can interpret decision diagrams using the spectral
approach can advance both the utility and understanding of
classical DD techniques. This approach also provides a framework
for developing advanced solutions for digital design and a host of
other applications. Scientists, computer science and engineering
professionals, and researchers with an interest in the spectral
methods of representing discrete functions, as well as the
foundations of logic design, will find the book a clearly
explained, well-organized, and essential resource.
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