Sampling Theory in Fourier and Signal Analysis / Edition 1

by J. R. Higgins

Overview

With much material not previously found in book form, this book fills a gap by discussing the equivalence of signal functions with their sets of values taken at discreet points comprehensively and on a firm mathematical ground. The wide variety of topics begins with an introduction to the main ideas and background material on Fourier analysis and Hilbert spaces and their bases. Other chapters discuss 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. Mathematicians, physicists, and communications engineers will welcome the scope of information found here.

Product Details

ISBN-13:	9780198596998
Publisher:	Oxford University Press
Publication date:	08/01/1996
Series:	Oxford Science Publications
Pages:	236
Product dimensions:	9.56(w) x 6.42(h) x 0.76(d)

About the Author

Anglia Polytechnic University, Cambridge

Rheinisch-Westfaelische, Technische Hochschule, Aachen

Table of Contents

1. An introduction to sampling theory1.1. General introduction1.2. Introduction - continued1.3. The seventeenth to the mid twentieth century - a brief review1.4. Interpolation and sampling from the seventeenth century to the mid twentieth century - a brief review1.5. Introduction - concluding remarks2. Background in Fourier analysis2.1. The Fourier Series2.2. The Fourier transform2.3. Poisson's summation formula2.4. Tempered distributions - some basic facts3. Hilbert spaces, bases and frames3.1. Bases for Banach and Hilbert spaces3.2. Riesz bases and unconditional bases3.3. Frames3.4. Reproducing kernel Hilbert spaces3.5. Direct sums of Hilbert spaces3.6. Sampling and reproducing kernels4. Finite sampling4.1. A general setting for finite sampling4.2. Sampling on the sphere5. From finite to infinite sampling series5.1. The change to infinite sampling series5.2. The Theorem of Hinsen and Kloösters6. Bernstein and Paley-Weiner spaces6.1. Convolution and the cardinal series6.2. Sampling and entire functions of polynomial growth6.3. Paley-Weiner spaces6.4. The cardinal series for Paley-Weiner spaces6.5. The space ReH16.6. The ordinary Paley-Weiner space and its reproducing kernel6.7. A convergence principle for general Paley-Weiner spaces7. More about Paley-Weiner spaces7.1. Paley-Weiner theorems - a review7.2. Bases for Paley-Weiner spaces7.3. Operators on the Paley-Weiner space7.4. Oscillatory properties of Paley-Weiner functions8. Kramer's lemma8.1. Kramer's Lemma8.2. The Walsh sampling therem9. Contour integral methods9.1. The Paley-Weiner theorem9.2. Some formulae of analysis and their equivalence9.3. A general sampling theorem10. Irregular sampling10.1. Sets of stable sampling, of interpolation and of uniqueness10.2. Irregular sampling at minimal rate10.3. Frames and over-sampling11. Errors and aliasing11.1. Errors11.2. The time jitter error11.3. The aliasing error12. Multi-channel sampling12.1. Single channel sampling12.3. Two channels13. Multi-band sampling13.1. Regular sampling13.2 Optimal regular sampling. 13.3. An algorithm for the optimal regular sampling rate13.4. Selectively tiled band regions13.5. Harmonic signals13.6. Band-ass sampling14. Multi-dimensional sampling14.1. Remarks on multi-dimensional Fourier analysis14.2. The rectangular case14.3. Regular multi-dimensional sampling15. Sampling and eigenvalue problems15.1. Preliminary facts15.2. Direct and inverse Sturm-Liouville problems15.3. Further types of eigenvalue problem - some examples16. Campbell's generalised sampling theorem16.1. L.L. Campbell's generalisation of the sampling theorem16.2. Band-limited functions16.3. Non band-limited functions - an example17. Modelling, uncertainty and stable sampling17.1. Remarks on signal modelling17.2. Energy concentration17.3. Prolate Spheroidal Wave functions17.4. The uncertainty principle of signal theory17.5. The Nyquist-Landau minimal sampling rate1. An introduction to sampling theory1.1. General introduction1.2. Introduction - continued1.3. The seventeenth to the mid twentieth century - a brief review1.4. Interpolation and sampling from the seventeenth century to the mid twentieth century - a brief review1.5. Introduction - concluding remarks2. Background in Fourier analysis2.1. The Fourier Series2.2. The Fourier transform2.3. Poisson's summation formula2.4. Tempered distributions - some basic facts3. Hilbert spaces, bases and frames3.1. Bases for Banach and Hilbert spaces3.2. Riesz bases and unconditional bases3.3. Frames3.4. Reproducing kernel Hilbert spaces3.5. Direct sums of Hilbert spaces3.6. Sampling and reproducing kernels4. Finite sampling4.1. A general setting for finite sampling4.2. Sampling on the sphere5. From finite to infinite sampling series5.1. The change to infinite sampling series5.2. The Theorem of Hinsen and Kloösters6. Bernstein and Paley-Weiner spaces6.1. Convolution and the cardinal series6.2. Sampling and entire functions of polynomial growth6.3. Paley-Weiner spaces6.4. The cardinal series for Paley-Weiner spaces6.5. The space ReH16.6. The ordinary Paley-Weiner space and its reproducing kernel6.7. A convergence principle for general Paley-Weiner spaces7. More about Paley-Weiner spaces7.1. Paley-Weiner theorems - a review7.2. Bases for Paley-Weiner spaces7.3. Operators on the Paley-Weiner space7.4. Oscillatory properties of Paley-Weiner functions8. Kramer's lemma8.1. Kramer's Lemma8.2. The Walsh sampling therem9. Contour integral methods9.1. The Paley-Weiner theorem9.2. Some formulae of analysis and their equivalence9.3. A general sampling theorem10. Ireggular sampling10.1. Sets of stable sampling, of interpolation and of uniqueness10.2. Irregular sampling at minimal rate10.3. Frames and over-sampling11. Errors and aliasing11.1. Errors11.2. The time jitter error11.3. The aliasing error12. Multi-channel sampling12.1. Single channel sampling12.3. Two channels13. Multi-band sampling13.1. Regular sampling13.2 Optimal regular sampling. 13.3. An algorithm for the optimal regular sampling rate13.4. Selectively tiled band regions13.5. Harmonic signals13.6. Band-ass sampling14. Multi-dimensional sampling14.1. Remarks on multi-dimensional Fourier analysis14.2. The rectangular case14.3. Regular multi-dimensional sampling15. Sampling and eigenvalue problems15.1. Preliminary facts15.2. Direct and inverse Sturm-Liouville problems15.3. Further types of eigenvalue problem - some examples16. Campbell's generalised sampling theorem16.1. L.L. Campbell's generalisation of the sampling theorem16.2. Band-limited functions16.3. Non band-limited functions - an example17. Modelling, uncertainty and stable sampling17.1. Remarks on signal modelling17.2. Energy concentration17.3. Prolate Spheroidal Wave functions17.4. The uncertainty principle of signal theory17.5. The Nyquist-Landau minimal sampling rate