The Mathematics of Signal Processing

Abstract

Arising from courses taught by the authors, this largely self-contained treatment is ideal for mathematicians who are interested in applications or for students from applied fields who want to understand the mathematics behind their subject. Early chapters cover Fourier analysis, functional analysis, probability and linear algebra, all of which have been chosen to prepare the reader for the applications to come. The book includes rigorous proofs of core results in compressive sensing and wavelet convergence. Fundamental is the treatment of the linear system y=Φx in both finite and infinite dimensions. There are three possibilities: the system is determined, overdetermined or underdetermined, each with different aspects. The authors assume only basic familiarity with advanced calculus, linear algebra and matrix theory and modest familiarity with signal processing, so the book is accessible to students from the advanced undergraduate level. Many exercises are also included.

The effect of tRNA levels on decoding times of mRNA codons

“…w(t) = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$f(t)$\end{document} + \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$g(t)$\end{document}) is calculated as convolution between the two distributions (39): \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}\begin{eqnarray*} w(t) &=& f(t)*g(t)\nonumber \\ &=& \int\limits_{ - \infty }^t {f(\tau )g(t - } \tau )d\tau \quad \forall f,g:[ - \infty ,\infty ) \to \mathbb{R}.\end{eqnarray*}\end{document}…”

Time-domain numerical modeling of terahertz receivers based on photoconductive antennas

“…In addition, these results show that the receiver photoconductivity could be considered a good approximation of the impulsive response of the own receiver [47]. Thus convolving any input, for instance the electric field reaching the receiver, with the receiver photoconductivity we get a magnitude proportional to the detected intensity of current I THz t in the receiver.…”

Operational Risk Modelling in Insurance and Banking

“…In mathematics, convolution is a mathematical operation on two functions f and g , producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated (Bracewell, 1986;Damelin & Miller, 2011). Convolution is presented in the following manner:…”

“…However formula can be interpreted as a weighted average of the function ( ) f τ at the moment t where the weighting is given by ( ) g τ − simply shifted by amount t . As t changes, the weighting function emphasizes different part of the input function (Bracewell, 1986;Damelin & Miller, 2011).…”