Introduction to Orthogonal Transforms

Abstract

A systematic, unified treatment of orthogonal transform methods for signal processing, data analysis and communications, this book guides the reader from mathematical theory to problem solving in practice. It examines each transform method in depth, emphasizing the common mathematical principles and essential properties of each method in terms of signal decorrelation and energy compaction. The different forms of Fourier transform, as well as the Laplace, Z-, Walsh–Hadamard, Slant, Haar, Karhunen–Loève and wavelet transforms, are all covered, with discussion of how each transform method can be applied to real-world experimental problems. Numerous practical examples and end-of-chapter problems, supported by online Matlab and C code and an instructor-only solutions manual, make this an ideal resource for students and practitioners alike.

On the Comparisons of Decorrelation Approaches for Non-Gaussian Neutral Vector Variables

“…1) Decorrelation Effect on Neutral Vector Variables: Vectors generated from a Dirichlet distribution are completely neutral. In order to illustrate the decorrelation effect of the PNT and PCA on neutral vector variables, we generated vectors from a given Dirichlet distribution with parameter α = [3,5,15,9,12,8,7, 20] T . PNT and PCA were applied to this generated data set, respectively.…”

“…Linear orthogonal transforms, including the renowned Fourier transform, discrete cosine transform and Karhunen-Loève transform, are not only able to decorrelate the elements of a vector variable to various extents, but also able to concentrate the "energy" (in terms of variance) of the vector in a small number of scalar variables obtained from the transformation [5]. Hence, linear orthogonal transforms are widely used to decorrelate a vector variable.…”

Statistical Treatment, Fourier and Modal Decomposition

“…Excellent references with a focus on deterministic signals are Oppenheim et al (1996); Ljung and Glad (1994); Hsu (2013); Hayes (2011), while Smith (2007b,a,c) provides a comprehensive treatment of Fourier analysis and filters. A gentle introduction to stochastic signals is provided by Kay (1993); Williamson (1999); Oppenheim (2015) while Wang (2009) provides an excellent review of orthogonal transforms.…”

Fields of an ultrashort tightly focused radially polarized laser pulse in a linear response plasma

“…A plausible choice for f k will be key to determining the appropriate initial axial profile f (ζ). The simplest would be a uniform distribution of wavenumbers, given by the square function [26,27]…”