Transforms and Fast Algorithms for Signal Analysis and Representations

This book is a comprehensive presentation of recent results and developments on several widely used transforms and their fast algorithms. In many cases, new options are provided for improved or new fast algorithms, some of which are not well known in the digital signal processing community. The book is suitable as a textbook for senior undergraduate and graduate courses in digital signal processing. It may also serve as an excellent self-study reference for electrical engineers and applied mathematicians whose work is related to the fields of electronics, signal processing, image and speech processing, or digital design and communication.

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. . . 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. Various meth­ ods have been continuously reported to improve the implementation of these transforms. Early developments of fast algorithms for discrete transforms have significantly stimulated the advance of digital signal processing technologies. More than 40 years after fast Fourier transform algorithms became known, several discrete transforms, including the discrete Hart­ ley transform and discrete cosine transform, were proposed and widely used for numerous applications. Although they all are related to the discrete Fourier transform, different fast algorithms and their implementations have to be separately developed to minimize compu­ tational complexity and implementation costs. In spite of the tremendous increase in the speed of computers or processors, the demands for higher processing throughout seemingly never ends. Fast algorithms have become more important than ever for modern applications to become a reality. Many new algorithms recently reported in the literature have led to important improvements upon a number of issues, which will be addressed in this book. Some discrete transforms are not suitable for signals that have time-varying frequency components. Although several approaches are available for such applications, various inher­ ent problems still remain unsolved.

Contenu
1 Introduction.- 1.1 Discrete linear transforms.- 1.2 Fast algorithms.- 1.3 New transforms.- 1.4 Organization of the book.- References.- 2 Polynomial Transforms and Their Fast Algorithms.- 2.1 Basic number theory.- 2.2 Basic polynomial theory.- 2.3 1D polynomial transform.- 2.4 Fast polynomial transform.- 2.5 MD polynomial transform and fast algorithm.- 2.6 Chapter summary.- References.- 3 Fast Fourier Transform Algorithms.- 3.1 Introduction.- 3.2 Radix-2 and split-radix algorithms.- 3.3 Generalized split-radix algorithm.- 3.4 Prime factor algorithms.- 3.5 Generalized 2D split-radix algorithms.- 3.6 Fast algorithms for generalized DFT.- 3.7 Polynomial transform algorithms for MD DFT.- 3.8 Chapter summary.- 4 Fast Algorithms for 1D Discrete Hartley Transform.- 4.1 Introduction.- 4.2 Split-radix algorithms.- 4.3 Generalized split-radix algorithms.- 4.4 Radix-2 algorithms for type-II, -III and -IV DHTs.- 4.5 Prime factor algorithms.- 4.6 Radix-q algorithms.- 4.7 Fast algorithms using type-I DHT.- 4.8 Chapter summary.- 5 Fast Algorithms for MD Discrete Hartley Transform.- 5.1 Introduction.- 5.2 Split-radix algorithms for 2D type-I DHT.- 5.3 Fast algorithms for 2D type-II, -III and -IV DHTs.- 5.4 Fast algorithms based on type-I DHT.- 5.5 PT-based radix-2 algorithm for MD type-I DHT.- 5.6 PT-based radix-2 algorithm for MD type-II DHT.- 5.7 PT-based radix-q algorithm for MD type-I DHT.- 5.8 PT-based radix-q algorithm for MD type-II DHT.- 5.9 Chapter summary.- References.- 6 Fast Algorithms for 1D Discrete Cosine Transform.- 6.1 Introduction.- 6.2 Radix-2 algorithms.- 6.3 Prime factor algorithms.- 6.4 Radix-q algorithms.- 6.5 Fast algorithms based on type-I bCT.- 6.6 Chapter summary.- 7 Fast Algorithms for MD Discrete Cosine Transform.- 7.1 Introduction.- 7.2 Algorithms for 2Dtype-I, -II and -III DCTs.- 7.3 Prime factor algorithm for MD DCT.- 7.4 PT-based radix-2 algorithm for MD type-II DCT.- 7.5 PT-based radix-2 algorithm for MD type-III DCT.- 7.6 PT-based radix-q algorithm for MD type-II DCT.- 7.7 PT-based radix-q algorithm for MD type-III DCT.- 7.8 Chapter summary.- 8 Integer Transforms and Fast Algorithms.- 8.1 Introduction.- 8.2 Preliminaries.- 8.3 Integer DCT and fast algorithms.- 8.4 Integer DHT and fast algorithms.- 8.5 MD Integer DCT and fast algorithms.- 8.6 MD Integer DHT and fast algorithms.- 8.7 Chapter summary.- References.- 9 New Methods of Time-Frequency Analysis.- 9.1 Introduction.- 9.2 Preliminaries.- 9.3 Harmonic transform.- 9.4 Tomographic time-frequency transform.- 9.5 Chapter summary.- References.