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Fourier transform is one of the most widely used transforms for the analysis and design of signals and systems in several fields of science and engineering. The primary objective of writing this book is to present the discrete Fourier transform theory, practically efficient algorithms, and basic applications using a down-to-earth approach. The computation of discrete cosine transform and discrete Walsh-Hadamard transforms are also described.
The book is addressed to senior undergraduate and graduate students in engineering, computer science, mathematics, physics, and other areas who study the discrete transforms in their course work or research. This book can be used as a textbook for courses on Fourier analysis and as a supplementary textbook for courses such as digital signal processing, digital image processing, digital communications engineering, and vibration analysis. The second group to whom this book is addressed is the professionals in industry and research laboratories involved in the design of general- and special-purpose signal processors, and in the hardware and software applications of the discrete transforms in various areas of engineering and science. For these professionals, this book will be useful for self study and as a reference book.
As the discrete transforms are used in several fields by users with different mathematical backgrounds, I have put considerable effort to make things simpler by providing physical explanations in terms of real signals, and through examples, figures, signal-flow graphs, and flow charts so that the reader can understand the theory and algorithms fully with minimum effort. Along with other forms of description, the reader can easily understand that the mathematical version presents the same information in a more abstract and compact form. In addition, I have deliberately made an attempt to present the material quite explicitly and describing only practically more useful methods and algorithms in very simple terms.
With the arrival of more and more new computers, the user needs a deep understanding of the algorithms and the architecture of the computer used to achieve an efficient implementation of algorithms for a given application. By going through the mathematical derivations, signal-flow graphs, flow charts, and the numerical examples presented in this book, the reader can get the necessary understanding of the algorithms. Large number of exercises are given, analytical and programming, that will further consolidate the readers' confidence. Answers to selected analytical exercises marked * are given at the end of the book. Answers are given to all the programming exercises on the Internet at www.wspc.com/others/software/4610/. Important terms and expressions are defined in the glossary. A list of abbreviations is also given. For readers with little or no prior knowledge of discrete Fourier analysis, it is recommended that they read the chapters in the given order.
Introduction
The Discrete Sinusoid
The Discrete Fourier Transform
Properties of the DFT
Fundamentals of the PM DFT Algorithms
The u X 1 PM DFT Algorithms
The 2 x 2 PM DFT Algorithms
DFT Algorithms for Real Data - I
DFT Algorithms for Real Data - II
Two-Dimensional Discrete Fourier Transform
Aliasing and Other Effects
The Continuous-Time Fourier Series
The Continuous-Time Fourier Transform
Convolution and Correlation
Discrete Cosine Transform
Discrete Walsh-Hadamard Transform
A The Complex Numbers
B The Measure of Computational Complexity
C The Bit-Reversal Algorithm
D Prime-Factor DFT Algorithm
E Testing of Programs
F Useful Mathematical Formulas