Transforms and Fast Algorithms for Signal Analysis and Representations

"This is perhaps the best text on transforms for signal processing since Nussbaumer's Fast Fourier Transform and Convolution Algorithms (Springer, 1982) and Elliott and Rao's, Fast Transforms: Algorithms, Analyses, Applications (Academic Press, 1982). Its nine chapters encompass almost all the knowledge needed to apply signal processing transforms successfully in practice. It is expected that the reader has had some exposure to transforms, so the first introductory chapter is very short; it mainly provides a sort of plan of things to come... The authors have chosen to provide proofs only to essential theorems, like the Chinese remainder theorem, a good decision for a reference book, but perhaps not so good for a textbook. However, this book does lean more towards serving as a professional reference than does the more academic Nussbaumer text. The lack of proofs felt by some readers is largely offset by an abundance of concrete examples... Chapter 8 is unique, in the sense that 'to the best of my knowledge' this material has never appeared in a book before. The chapter deals with integer transforms, which might be thought of as derived from the DCT and discrete sine transforms; these are gaining in popularity in various transform coding schemes for both video and audio... In conclusion, this book is a highly practical and very welcome addition to the collection of texts on signal processing techniques and applications." -Analog Dialogue -

• 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.