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

Description

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. Front Matter....Pages i-ix Walsh Functions....Pages 1-15 Self-similarity in Walsh Functions....Pages 17-24 Computation of Farfield Diffraction Characteristics of Radial and Annular Walsh Filters on the Pupil of Axisymmetric Imaging Systems....Pages 25-46 Self-similarity in Transverse Intensity Distributions on the Farfield Plane of Self-similar Walsh Filters....Pages 47-58 Self-similarity in Axial Intensity Distributions of Self-similar Walsh Filters....Pages 59-71 Self-similarity in 3D Light Distributions Near the Focus of Self-similar Radial Walsh Filters....Pages 73-80 Concluding Remarks....Pages 81-82