Sampling Theory in Fourier and Signal Analysis: Foundations
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Product details:
- Publisher OUP Oxford
- Date of Publication 30 May 1996
- Number of Volumes laminated boards
- ISBN 9780198596998
- Binding Hardback
- No. of pages236 pages
- Size 242x161x19 mm
- Weight 489 g
- Language English
- Illustrations figures 0
Categories
Short description:
This book is about the equivalence of signal functions with their sets of values taken at discrete points. Beginning with an introduction to the main ideas, and background material on Fourier analysis and Hilbert spaces amd their bases, the book covers a wide variety of topics including: sampling of Bernstein and Paley-Wiener spaces; Kramer's Lemma and its application to eigenvalue problems; contour integral methods including a proof of the equivalence of the sampling theory; the Poisson summation formula and Cauchy's integral formula; optimal regular, irregular, multi-channel, multi-band and multi dimensional sampling, and Campbell's generalized sampling theorem.
MoreLong description:
Containing important new material unavailable previously in book form, this book covers a wide variety of topics which will be great interest to applied mathematicians and engineers. Introducing the main ideas, background material is provided on Fourier analysis, Hilbert spaces, and their bases, before the book moves on to discuss more complex topics and their applications.
...the text is written by use of LATEX and its beautiful graphics reveal the power and the advantages of this system.
Table of Contents:
An introduction to sampling theory
General introduction
Introduction - continued
The seventeenth to the mid twentieth century - a brief review
Interpolation and sampling from the seventeenth century to the mid twentieth century - a brief review
Introduction - concluding remarks
Background in Fourier analysis
The Fourier Series
The Fourier transform
Poisson's summation formula
Tempered distributions - some basic facts
Hilbert spaces, bases and frames
Bases for Banach and Hilbert spaces
Riesz bases and unconditional bases
Frames
Reproducing kernel Hilbert spaces
Direct sums of Hilbert spaces
Sampling and reproducing kernels
Finite sampling
A general setting for finite sampling
Sampling on the sphere
From finite to infinite sampling series
The change to infinite sampling series
The Theorem of Hinsen and Kloösters
Bernstein and Paley-Weiner spaces
Convolution and the cardinal series
Sampling and entire functions of polynomial growth
Paley-Weiner spaces
The cardinal series for Paley-Weiner spaces
The space ReH1
The ordinary Paley-Weiner space and its reproducing kernel
A convergence principle for general Paley-Weiner spaces
More about Paley-Weiner spaces
Paley-Weiner theorems - a review
Bases for Paley-Weiner spaces
Operators on the Paley-Weiner space
Oscillatory properties of Paley-Weiner functions
Kramer's lemma
Kramer's Lemma
The Walsh sampling therem
Contour integral methods
The Paley-Weiner theorem
Some formulae of analysis and their equivalence
A general sampling theorem
Ireggular sampling
Sets of stable sampling, of interpolation and of uniqueness
Irregular sampling at minimal rate
Frames and over-sampling
Errors and aliasing
Errors
The time jitter error
The aliasing error
Multi-channel sampling
Single channel sampling
Two channels
Multi-band sampling
Regular sampling
An algorithm for the optimal regular sampling rate
Selectively tiled band regions
Harmonic signals
Band-ass sampling
Multi-dimensional sampling
Remarks on multi-dimensional Fourier analysis
The rectangular case
Regular multi-dimensional sampling
Sampling and eigenvalue problems
Preliminary facts
Direct and inverse Sturm-Liouville problems
Further types of eigenvalue problem - some examples
Campbell's generalised sampling theorem
L.L. Campbell's generalisation of the sampling theorem
Band-limited functions
Non band-limited functions - an example
Modelling, uncertainty and stable sampling
Remarks on signal modelling
Energy concentration
Prolate Spheroidal Wave functions
The uncertainty principle of signal theory
The Nyquist-Landau minimal sampling rate
