Mathematics in Signal Processing

This volume provides an overview of the wide range of mathematical topics in signal processing. The three main contributions explore the theme of alternative algebras for signal processing, covering the already significant impact of multilinear algebra, and the potential applications of geometric algebra and Grobner bases. Other chapters cover a wide range of subjects from array processing and digital communications to wavelets and nonlinear signal processing. These, in turn, encompass a rich variety of mathematical techniques including Pade approximation, convex optimization and generalized eigenvalue decomposition. This intriguing blend of theory and application should appeal to a wide range of engineers and mathematicians.

Table of Contents:
From matrix to tensor: multilinear algebra and signal processing; The analysis of linear functions using geometric algebraan alternative to matrices and tensors; Rational approximation using Grobner bases: some numerical results; Grobner basis design of incomplete Chebyshev polynomials; Fractals, linear channels, and delay methods; Blind binary signal separation using joint diagonalization techniques; Adaptive algorithms for single user blind channel equalization; Assessing stability for adaptive filtering algorithms in signal processing; Constrained Kalman filter for least-squares estimation of time-varying beamforming weights; Using algorithmic engineering to derive a fast parallel weight extraction algorithm based on QRD recursive least squares; The simultaneous strict positive real problem: similarities with the simultaneous stability problem and applications in adaptive recursive algorithms; Asymptotic performance analysis of Yang's subspace tracking algorithm; Generalized prediction error approach to spectral analysis and filtering of irregularly-sampled data; Decaying exponentials and signal decomposition; Robust envelope-constrained filter design; On the use of positivity in reconstructing images from noisy data; Treatment of noise in positive reconstructions in linear inverse problems; Ill conditioning and gradient based optimization of multi-layer perceptrons; Online modelling of time series with resource allocating neural networks; Bayesian model selection for linear and non-linear time series using the Gibbs sampler; Bayesian image estimation from sparse raw MRI data; Polynomial models for 3D motion parameters and depth estimation in monocular computer vision; Symmetry and wavelet transforms for image data compression; Vanishing moments and biorthogonal wavelet systems; On sliding window schemes for discrete least-squares approximation by trigonometric polynomials; Jump detection by Pade approximation; Discrete simulation models for multidimensional systems based on functional transformations; Analytic performance evaluation of third-order statistics based estimators for Allpass identification; Detection of sleep apnoea using time-frequency analysis of heart rate variability; Robust estimation of oceanic background spectrum