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Spie Press Book

Hadamard Transforms
Author(s): Sos S. Agaian; Hakob G. Sarukhanyan; Karen O. Egiazarian; Jaakko Astola
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Book Description

The Hadamard matrix and Hadamard transform are fundamental problem-solving tools in a wide spectrum of scientific disciplines and technologies, such as communication systems, signal and image processing (signal representation, coding, filtering, recognition, and watermarking), digital logic (Boolean function analysis and synthesis), and fault-tolerant system design. Hadamard Transforms intends to bring together different topics concerning current developments in Hadamard matrices, transforms, and their applications. Each chapter begins with the basics of the theory, progresses to more advanced topics, and then discusses cutting-edge implementation techniques. The book covers a wide range of problems related to these matrices/transforms, formulates open questions, and points the way to potential advancements.

Hadamard Transforms is suitable for a wide variety of audiences, including graduate students in electrical and computer engineering, mathematics, or computer science. Readers are not presumed to have a sophisticated mathematical background, but some mathematical background is helpful. This book will prepare readers for further exploration and will support aspiring researchers in the field.


Book Details

Date Published: 4 August 2011
Pages: 520
ISBN: 9780819486479
Volume: PM207

Table of Contents
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Preface
1. CLASSICAL HADAMARD MATRICES AND ARRAYS
1.1. Sylvester or Walsh-Hadamard matrices
1.2. Walsh-Paley Matrices
1.3. Walsh and Related Systems
1.3.1. Walsh system
1.3.2. Cal-Sal orthogonal system
1.3.3. The Haar system
1.4. Hadamard Matrices and Related Problems
1.5. Complex Hadamard Matrices
1.5.1. Complex Sylvester-Hadamard transform
1.5.2. Complex Walsh-Hadamard transform
1.5.3. Complex Paley-Hadamard transform
1.5.4. Complex Walsh transform
References
2. FAST CLASSICAL DISCRETE ORTHOGONAL TRANSFORMS
2.1. Matrix Based Fast DOTs Algorithms
2.2. Fast Walsh-Hadamard Transform
2.3. Fast Walsh-Paley Transform
2.4. Cal-Sal Fast Transform
2.5. Fast Complex Hadamard Transform
2.6. Fast Haar Transform
References
3. DISCRETE ORTHOGONAL TRANSFORMS AND HADAMARD MATRICES
3.1. Fast Discrete Orthogonal Transforms via Walsh-Hadamard Transform
3.2. Fast Fourier Transform Implementation
3.3. Fast Hartley Transform
3.4. Fast Cosine Transform
3.5. Fast Haar Transform
3.6. Integer Slant Transforms
3.6.1. Slant-Hadamard transforms
3.6.2. Parametric slant-Hadamard transform matrices
3.7. Construction of Sequential Integer Slant-Hadamard Transforms
3.7.1. Fast Algorithms
3.7.2. Examples of Slant Transform Matrices
3.8. Construction of the Iterative Parametric Slant-Haar Transform
References
4. "PLUG IN TEMPLATE" METHOD: WILLIAMSON-HADAMARD MATRICES
4.1. Williamson-Hadamard Matrices
4.2. Construction of Eight Williamson Matrices
4.3. Williamson Matrices from Regular Sequences
References
5. FAST WILLIAMSON-HADAMARD TRANSFORMS
5.1. Construction of Hadamard Matrices Using Williamson Matrices
5.2. Parametric Williamson Matrices and Block Representation of Williamson-Hadamard Matrices
5.3. Fast Block Williamson-Hadamard Transform
5.4. Multiplicative Theorem Based Williamson-Hadamard Matrices
5.5. Multiplicative Theorem Based Fast Williamson-Hadamard Transforms
5.6. Complexity and Comparison
5.6.1. Complexity of block-cyclic block-symmetric Williamson-Hadamard transform
5.6.2. Complexity of Hadamard transform from multiplicative theorem
References
6. SKEW WILLIAMSON-HADAMARD TRANSFORMS
6.1. Skew Hadamard Matrices
6.2. Skew-symmetric Williamson Matrices
6.3. Block Representation of Skew-Symmetric Williamson-Hadamard Matrices
6.4. Fast Block-Cyclic Skew-Symmetric Williamson-Hadamard Transform
6.5. Block-Cyclic Skew-Symmetric Fast Williamson-Hadamard Transform in Add/Shift Architectures
References
7. DECOMPOSITION OF HADAMARD MATRICES
7.1. Decomposition of Hadamard Matrices By (+1,-1) Vectors
7.2. Decomposition of Hadamard Matrices and Their Classification
7.3. Multiplicative Theorems of Orthogonal Arrays and Hadamard Matrices Construction
References
8. FAST HADAMARD TRANSFORMS FOR ARBITRARY ORDERS
8.1. Hadamard Matrix Construction Algorithms
8.2. Hadamard Matrix Vector Representation
8.3. Fast Hadamard Transform of Order n = 0(mod 4)
8.4. Fast Hadamard Transform via Four Vector Representation
8.5. Fast Hadamard Transform of Order n = 0(mod 4) on Shift/Add Architectures
8.6. Complexities of Developed Algorithms
8.6.1. Complexity of the general algorithm
8.6.2. Complexity of the general algorithm with shifts
References
9. ORTHOGONAL ARRAYS
9.1. Orthogonal Designs
9.2. Baumert-Hall Arrays
9.3. A-Matrices
9.4. Geothals-Seidel Arrays
9.5. Plotkin Arrays
9.6. Welch Arrays
References
10. HIGHER-DIMENSIONAL HADAMARD MATRICES
10.1. 3D Hadamard Matrices
10.2. 3D Williamson-HadamardMatrices
10.3. 3D Hadamard Matrices of Order 4n+2
10.4. Fast 3D Walsh Hadamard Transforms
10.5. Operations with Higher-Dimensional Complex Matrices
10.6. 3D Complex Hadamard Transforms
10.7. Construction of High-Dimensional Generalized Hadamard Matrices
References
11. EXTENDED HADAMARD MATRICES
11.1. Generalized Hadamard Matrices
11.1.1. Introduction and statement of problems
11.1.2. Some necessary conditions of generalized Hadamard matrices existence
11.1.3. Construction of generalized Hadamard matrices of new orders
11.1.4. Generalized Yang matrices and construction of generalized Hadamard matrices
11.2. Chrestenson Transform
11.2.1. The Rademacher-Walsh transforms
11.2.2. Chrestenson functions and matrices
11.3. Chrestenson Transforms Algorithms
11.3.1. Chrestenson transform of order 3n
11.3.2. Chrestenson transform of order 5n
11.4. Fast Generalized Haar Transforms
11.4.1. The generalized Haar functions
11.4.2. 2n-point Haar transform
11.4.3. 3n-Point generalized Haar transform
11.4.4. 4n-point generalized Haar transform
11.4.5. 5n-point generalized Haar transform
References
12. JACKET HADAMARD MATRICES
12.1. Introduction to Jacket Matrices
12.2. Weighted Sylvester-Hadamard Matrices
12.3. Parametric Reverse Jacket Matrices
12.4. Construction of Special Type Parametric Reverse Jacket Matrices
12.5. Fast Parametric Reverse Jacket Transform
12.5.1. Fast 4x4 parametric reverse jacket transform
12.5.2. Fast 8x8 parametric reverse jacket transform
References
13. APPLICATIONS OF HADAMARD MATRICES IN COOMMUNICATION SYSTEMS
13.1. Hadamard Matrices and Communication Systems
13.1.1. Overview of error-correcting codes
13.1.2. Levenshtein constructions
13.1.3. Uniquely decodable base codes
13.1.4. Shortened codes construction and application to data coding and decoding
13.2. Space-Time Codes From Hadamard Matrices
13.2.1. The general Wireless System Model
13.2.2. Orthogonal array and linear processing design
13.2.3. Design of space-time codes from Hadamard matrix
References
14. RANDOMIZATION OF DISCRETE ORTHOGONAL TRANSFORMS AND ENCRYPTION
14.1. Preliminaries
14.1.1. Matrix forms of DHT, DFT, DCT, and other DOTs
14.1.2. Cryptography
14.2. Randomization of Discrete Orthogonal Transforms
14.2.1. The theorem of randomizations of discrete orthogonal transforms
14.2.2. Discussions on the square matrices P and Q
14.2.3. Examples of randomized transform matrix Ms
14.2.4. Transform properties and features
14.2.5. Examples of randomized discrete orthogonal transforms
14.3. Encryption Applications
14.3.1. 1D data encryption
14.3.2. 2D data encryption and beyond
14.3.3. Examples of image encryption
14.3.3.1. Key space analysis
14.3.3.2. Confusion property
14.3.3.3. Diffusion property
APPENDIX
A.1. Elements of Matrix Theory
A.2. First Rows of Cyclic Symmetric Williamson Type Matrices of Order n, n = 3, 5, ..., 33, 37, 39, 41, 43, 49, 51, 55, 57, 61, 63
A.3. First Block-Rows of the Block-Cyclic Block-Symmetric Williamson-Hadamard Matrices of Order 4n, n = 3, 5, ..., 33, 37, 39, 41, 43, 49, 51, 55, 57, 61, 63
A.4. First Rows of Cyclic Skew-Symmetric Williamson Type Matrices of Order n, n = 3, 5, ..., 33, 35
A.5. First Block-Rows of Skew-Symmetric Block Williamson-Hadamard Matrices of Order 4n, n = 3, 5, ..., 33, 35

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