The book explains the classification of a set of Walsh functions into distinct self-similar groups and subgroups, where the members of each subgroup possess distinct self-similar structures. The observations on self-similarity presented provide valuable clues to tackling the inverse problem of synthesis of phase filters. Self-similarity is observed in the far-field diffraction patterns of the corresponding self-similar filters.
Walsh functions form a closed set of orthogonal functions over a prespecified interval, each function taking merely one constant value (either +1 or 1) in each of a finite number of subintervals into which the entire interval is divided. The order of a Walsh function is equal to the number of zero crossings within the interval. Walsh functions are extensively used in communication theory and microwave engineering, as well as in the field of digital signal processing. Walsh filters, derived from the Walsh functions, have opened up new vistas. They take on values, either 0 or phase, corresponding to +1 or -1 of the Walsh function value.
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1 | (16) |
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1 | (1) |
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1.2 One Dimensional Walsh Functions |
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2 | (2) |
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1.3 Two Dimensional Walsh Functions |
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4 | (6) |
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1.3.1 Rectangular Walsh Functions |
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4 | (1) |
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1.3.2 Polar Walsh Functions |
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4 | (1) |
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1.3.3 Radial Walsh Functions |
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4 | (3) |
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1.3.4 Azimuthal Walsh Functions |
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7 | (1) |
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1.3.5 Annular Walsh Functions |
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8 | (2) |
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1.4 Walsh Block Functions and Hadamard Matrices |
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10 | (2) |
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1.5 Walsh Approximation of Functions |
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12 | (2) |
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1.6 Multiplication Rule for Walsh Functions |
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14 | (3) |
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15 | (2) |
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2 Self-similarity in Walsh Functions |
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17 | (8) |
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17 | (1) |
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2.2 Generation of Higher Order Walsh Functions from Lower Order Walsh Functions |
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18 | (2) |
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2.3 Classification of Walsh Functions of Various Orders in Self-similar Groups and Subgroups |
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20 | (2) |
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2.4 Self-similarity in Radial Walsh Functions |
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22 | (1) |
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2.5 Self-similarity in Annular Walsh Functions |
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22 | (1) |
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2.6 Radial and Annular Walsh Filters |
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22 | (1) |
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2.7 Self-similarity in Walsh Filters |
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23 | (2) |
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2.7.1 Self-similarity in Radial Walsh Filters |
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23 | (1) |
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2.7.2 Self-similarity in Annular Walsh Filters |
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23 | (1) |
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24 | (1) |
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3 Computation of Far-field Diffraction Characteristics of Radial and Annular Walsh Filters on the Pupil of Axisymmetric Imaging Systems |
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25 | (22) |
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25 | (6) |
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3.1.1 Pupil Plane Filtering and Walsh Filters |
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25 | (2) |
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3.1.2 Zone Plates and Walsh Filters |
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27 | (4) |
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3.2 Farfield Diffraction Characteristics of Radial and Annular Walsh Filters: Mathematical Formulation |
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31 | (1) |
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3.3 Radial Walsh Filters on the Exit Pupil |
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32 | (5) |
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3.3.1 Transverse Intensity Distribution on the Focal/Image Plane |
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32 | (2) |
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3.3.2 Intensity Distribution on a Transverse Plane Axially Shifted from the Focal/Image Plane |
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34 | (3) |
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3.4 Annular Walsh Filters |
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37 | (10) |
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3.4.1 Transverse Intensity Distribution |
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37 | (2) |
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3.4.2 Axial Intensity Distribution |
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39 | (2) |
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41 | (6) |
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4 Self-similarity in Transverse Intensity Distributions on the Farfield Plane of Self-similar Walsh Filters |
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47 | (12) |
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47 | (6) |
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4.2 Annular Walsh Filters |
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53 | (6) |
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58 | (1) |
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5 Self-similarity in Axial Intensity Distributions of Self-similar Walsh Filters |
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59 | (14) |
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5.1 Characteristics of Axial Intensity Distributions Around the Image Plane with Radial Walsh Filters on the Exit Pupil |
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59 | (5) |
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5.2 Characteristics of Axial Intensity Distribution Around the Image Plane with Annular Walsh Filters on the Exit Pupil |
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64 | (9) |
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71 | (2) |
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6 Self-similarity in 3D Light Distributions Near the Focus of Self-similar Radial Walsh Filters |
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73 | (8) |
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73 | (1) |
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6.2 Computation of Intensity Distributions on Transverse Planes in the Focal Region with Radial Walsh Filters on the Exit Pupil |
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74 | (2) |
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76 | (5) |
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80 | (1) |
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81 | (1) |
| References |
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A B.Sc. (Hons) Physics graduate and postgraduate of Applied Physics with a doctorate from the University of Calcutta, Kolkata. Prof. Lakshminarayan Hazra has over four decades of academic and industrial experience. He is an Emeritus Professor and Former Head of the Department of Applied Optics and Photonics at the University of Calcutta, Kolkata, India. His areas of professional specialization include lens design/optical system design, image formation & aberration theory, diffractive optics, and optical and photonic instrumentation. He is a Fellow of the Optical Society of America, and the International Society for Optics and Photonics (SPIE). He is the Editor-in-Chief of the archival journal, Journal of Optics, published by M/s Springer in collaboration with the Optical Society of India. He has published more than 150 journal articles and books.
Pubali Mukherjee holds B.Sc. (Hons.), M. Tech. and Ph.D. degrees, all from the University of Calcutta. Currently, she is an Assistant Professor in Electronics and Communication Engineering Department at the MCKV Institute of Engineering, Howrah, West Bengal, India. She has 10 years of teaching and 5 years of research experience. Her areas of interest include optical systems, image assessment criteria and diffraction pattern tailoring using phase filters and applications. She has published many papers in journals and conference proceedings.
