Mathematical aspects of signal processing (Q5890965)

The book deals with signal processing and associated mathematical techniques along four key aspects: Function representation of signals, generalized inverses, modal decomposition (in a broad sense, related to singular value decompositions) and optimization. All topics are illustrated with signal analysis examples. Here the guiding theme is an autoregressive signal model, equivalent to model the signal as a superposition of uniformly sampled complex damped exponentials. The task is, to estimate the frequencies and the remaining model parameters in various situations (equidistant vs. non-uniform sampling, noiseless vs. presence of noise).NEWLINENEWLINEThe chapter on function representations of (discretely sampled) signals covers topics from interpolation and (polynomial) approximation theory. The above mentioned estimation problem is tackled with Prony and related methods, polynomial interpolation is used to deal with non-uniform sampling.NEWLINENEWLINENEWLINEMatrix calculus resulting from inverse problems, i.e., solving \(A\cdot x = b\) for \(x\), is described in the chapter on generalized inverses. Again the techniques are illustrated in the context of the above-mentioned parameter estimation problem.NEWLINENEWLINEThe chapter on modal decomposition deals (among other techniques) with singular value decompositions allowing for robust and stable matrix inversions, also in the presence of noise. The methods are illustrated by a more sophisticated solution of the above-mentioned estimation problem via building Hankel matrices from the data and using them for frequency estimation via the Matrix-Pencil-Method.NEWLINENEWLINENEWLINESince all procedures involve the solution of optimization problems in suitable normed spaces, the last chapter deals with an overview on optimization techniques (with and without constraints).NEWLINENEWLINENEWLINEThe cover text states, that the textbook addresses advanced undergraduate or graduate students, which is definitely true. It is expected that the reader already has a suitable background in Fourier analysis and related harmonic expansions as well as in z-transform, transfer function etc. Thus for supplementary reading a text like [\textit{S. B. Damelin} and \textit{W. Miller jun.}, The mathematics of signal processing. Cambridge: Cambridge University Press (2012; Zbl 1257.94001)] is strongly recommended.