On signal processing with sampling errors: a worst case analysis
type: Item
instance of: scholarly article
title: On signal processing with sampling errors: a worst case analysis
zbMATH Open document ID: 0624.94007
DOI: 10.1007/BF00945408
author: Werner Krabs
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1987-01-01T00:00:00Z
review text: The starting point of this paper is a communication system by which a sequence \((s_ k)_{k\in {\mathbb{Z}}}\) of signals is modulated into a continuous signal of the form \[ y(t) = T\sum^{\infty}_{j=- \infty}s_ j h(t-jT),\quad t\in {\mathbb{R}}. \] The signals \(s_ k\) are assumed to be identically distributed real random variables with probability space (A,\({\mathfrak A},\mu)\) such that \[ \sum^{\infty}_{j=- \infty}| s_ j(a)| <\infty \quad for\quad \mu -almost\quad all\quad a\in A, \] E[s\({}_ k(\cdot)]=0\), \(E[s_ k(\cdot)^ 2]=1\) for all \(k\in {\mathbb{Z}}\), and \(E[s_ j(\cdot)s_ k(\cdot)]=0\) for all j,k\(\in {\mathbb{Z}}\) with \(j\neq k\) where \(E[S]=\int_{A}s(a)d\mu (a)\). The impulse response h is chosen as a real valued function is \(L^ 2({\mathbb{R}})\) such that its Fourier transform vanishes outside some interval [- \(\Omega\),\(\Omega\) ], \(\Omega >0\). If the modulated signal \((y(\cdot,t))_{t\in {\mathbb{R}}}\) is sampled in the form (y(\(\cdot,k\cdot T))\), \(k\in {\mathbb{Z}}\), then exact sampling is possible, i.e. \(y(a,k\cdot T)=s_ k(a)\) for all \(k\in {\mathbb{Z}}\) and \(\mu\)-almost all \(a\in A\), if, for instance, h is chosen as \(h_ n(t)=1/T\) for \(t=0\), \(h_ n(t)=(1/T)\sin (n\pi t/T)/(n\pi t/T)\) for \(t\neq 0\) and some \(n\in {\mathbb{N}}\) (in which case we have that \(\Omega =\Omega_ n=[- (n/2)\omega_ 0\), \((n/2)\omega_ 0]\) with \(\omega_ 0=2\pi /T).\) The paper is now concerned with the case in which y(\(\cdot\), \(k\cdot T+v(t))\) is sampled instead of y(\(\cdot\), \(k\cdot T)\) where \(v=v(t)\) is a random variable with probability space (\({\mathcal T}=[-\tau,\tau],{\mathfrak B},p)\) for some \(\tau\in [0,T/2]\), \({\mathfrak B}\) being the sigma algebra of Borel sets in \({\mathcal T}\) and p being a Borel measure on \({\mathfrak B}\) with p(\({\mathcal T})=1\). By v(t) a random sampling error is expressed whose measure space is assumed to be (\({\mathcal T},{\mathfrak B})\) and whose probability distribution is then gi…
Mathematics Subject Classification ID: 94A12
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 42A61
Mathematics Subject Classification ID: 94A14
zbMATH DE Number: 4014605
zbMATH Keywords: max-min-problem
zbMATH Keywords: communication system
zbMATH Keywords: modulated signal
zbMATH Keywords: sampling error
zbMATH Keywords: worst case analysis
zbMATH Keywords: mean square error
MaRDI profile type: Publication
cites work: Q5554487
cites work: On the determination of the worst sampling error in a communication system for pulse‐amplitude‐modulated signals by solving an approximation problem
cites work: Notes on Pólya's and Turán's hypotheses concerning Liouville's factor
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description: scientific article; zbMATH DE number 4014605