Self-similarity in Walsh Functions and in the Farfield Diffraction Patterns of Radial Walsh Filters | springerprofessional.de

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Self-similarity in Walsh Functions and in the Farfield Diffraction Patterns of Radial Walsh Filters

Authors: Lakshminarayan Hazra, Pubali Mukherjee

Publisher: Springer Singapore

Book Series : SpringerBriefs in Applied Sciences and Technology

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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About this book

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 π phase, corresponding to +1 or -1 of the Walsh function value.

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Table of Contents

Frontmatter

Chapter 1. Walsh Functions

Abstract

With a brief description of the origin, basic characteristics and major practical applications of Walsh functions, different forms of Walsh functions in one and two dimensions are systematically developed from the orthogonality considerations. Approximation of a continuous function over a given domain by a finite set of Walsh functions provides a piecewise constant approximation of the function. The latter has important consequences in tackling practical problems by way of opening up new techniques for analysis and synthesis. Highlights of these aspects are provided along with a brief description of Walsh Block functions and Hadamard matrices.

Lakshminarayan Hazra, Pubali Mukherjee

Chapter 2. Self-similarity in Walsh Functions

Abstract

A structure which can be divided into smaller and smaller pieces, each piece being an exact replica of the entire structure is called self-similar. The set of Walsh functions can be classified into distinct self-similar groups and subgroups where members of each subgroup exhibit self-similarity. After a brief discussion on the generation of higher order Walsh functions from lower order Walsh functions by an alternating process, a scheme for classification of Walsh functions into self-similar groups and subgroups is presented. Self-similarity in radial and annular Walsh functions and the correspondence between Walsh filters and Walsh functions are also discussed.

Lakshminarayan Hazra, Pubali Mukherjee

Chapter 3. Computation of Farfield Diffraction Characteristics of Radial and Annular Walsh Filters on the Pupil of Axisymmetric Imaging Systems

Abstract

Pupil plane filtering by Walsh filters is a convenient technique for tailoring the intensity distribution of light near the focal plane of a rotationally symmetric imaging system. Walsh filters, derived from Walsh functions, form a set of orthogonal phase filters that take on values either 0 or π phase, corresponding to +1 or −1 values of the Walsh functions over prespecified annular regions of the circular filter. Order of these filters is given by the number of zero-crossings, or equivalently phase transitions within the domain over which the set is defined. In general, Walsh filters are binary phase zone plates, each of them demonstrating distinct focusing characteristics. With a backdrop of pupil plane filtering for image enhancement and a brief description of different types of zone plates for manipulation of axial intensity distribution, this chapter puts forward the inherent potential of Walsh filters in this context. The mathematical formulation utilized for computing the transverse and axial intensity distributions in and around the image/focal plane, when radial and annular Walsh filters are placed on the exit pupil of an axisymmetric imaging system, is presented.

Lakshminarayan Hazra, Pubali Mukherjee

Chapter 4. Self-similarity in Transverse Intensity Distributions on the Farfield Plane of Self-similar Walsh Filters

Abstract

The set of radial and annular Walsh filters can be classified into distinct self-similar groups and subgroups, where members of each subgroup possess self-similar structures or phase sequences. It has been observed that, the transverse intensity distributions in the farfield diffraction pattern of these self-similar radial and annular Walsh filters are also self-similar. In this chapter we report results of our investigations on the self-similarity in the intensity distributions on a transverse plane in the farfield diffraction patterns of the self-similar radial and annular Walsh filters.

Lakshminarayan Hazra, Pubali Mukherjee

Chapter 5. Self-similarity in Axial Intensity Distributions of Self-similar Walsh Filters

Abstract

The set of radial and annular Walsh filters can be classified into distinct self-similar groups and subgroups, where members of each subgroup possess self-similar structures or phase sequences. It has been observed that the axial intensity distributions around the focal/image plane are also self-similar when these self-similar radial and annular Walsh filters are used as pupil plane filters. In this chapter we report results of our investigations on the self-similarity in the intensity distributions around the focal/image plane when self-similar radial and annular Walsh filters are used as pupil filters.

Lakshminarayan Hazra, Pubali Mukherjee

Chapter 6. Self-similarity in 3D Light Distributions Near the Focus of Self-similar Radial Walsh Filters

Abstract

The three dimensional light distribution near the focus of rotationally symmetric imaging system may be tailored by the technique of pupil plane filtering using Walsh filters. The set of Walsh filters can be classified into distinct self-similar groups and subgroups, where members of each subgroup possess self-similar structures or phase sequences. The 3D light distribution around the focal/image plane portrays self-similarity which can be correlated to the structural self-similarity of Walsh filters themselves. In this chapter we report results of our investigations on the self-similarity in the three dimensional intensity distribution near the focus of the self-similar radial Walsh filters.

Lakshminarayan Hazra, Pubali Mukherjee

Chapter 7. Concluding Remarks

Abstract

The investigations reported in this monograph conclusively demonstrate that the three dimensional point spread function of radial and annular Walsh filters portray self-similarity which can be correlated to the self-similar structures of the diffracting apertures. Self-similarity and orthogonality of these filters may be harnessed to solve challenging problems for generation of prespecified 3D patterns. This chapter underscores the importance of the study reported in the monograph for tackling problems of wave optical engineering.

Lakshminarayan Hazra, Pubali Mukherjee

Title: Self-similarity in Walsh Functions and in the Farfield Diffraction Patterns of Radial Walsh Filters
Authors: Lakshminarayan Hazra
Pubali Mukherjee
Copyright Year: 2018
Publisher: Springer Singapore
Electronic ISBN: 978-981-10-2809-0
Print ISBN: 978-981-10-2808-3
DOI: https://doi.org/10.1007/978-981-10-2809-0