Advanced Topics in Shannon Sampling and Interpolation Theory

Advanced Topics in Shannon Sampling and Interpolation Theory is the second volume of a textbook on signal analysis solely devoted to the topic of sampling and restoration of continuous time signals and images. Sampling and reconstruction are fundamental problems in any field that deals with real-time signals or images, including communication engineering, image processing, seismology, speech recognition, and digital signal processing. This second volume includes contributions from leading researchers in the field on such topics as Gabor's signal expansion, sampling in optical image formation, linear prediction theory, polar and spiral sampling theory, interpolation from nonuniform samples, an extension of Papoulis's generalized sampling expansion to higher dimensions, and applications of sampling theory to optics and to time-frequency representations. The exhaustive bibliography on Shannon sampling theory will make this an invaluable research tool as well as an excellent text for students planning further research in the field.

1 Gabor’s Signal Expansion and Its Relation to Sampling of the Sliding-Window Spectrum.- 1.1 Introduction.- 1.2 Sliding-Window Spectrum.- 1.2.1 Inversion Formulas.- 1.2.2 Space and Frequency Shift.- 1.2.3 Some Integrals Concerning the Sliding-Window Spectrum.- 1.2.4 Discrete-Time Signals.- 1.3 Sampling Theorem for the Sliding-Window Spectrum.- 1.3.1 Discrete-Time Signals.- 1.4 Examples of Window Functions.- 1.4.1 Gaussian Window Function.- 1.4.2 Discrete-Time Signals.- 1.5 Gabor’s Signal Expansion.- 1.5.1 Discrete-Time Signals.- 1.6 Examples of Elementary Signals.- 1.6.1 Rect Elementary Signal.- 1.6.2 Sine Elementary Signal.- 1.6.3 Gaussian Elementary Signal.- 1.6.4 Discrete-Time Signals.- 1.7 Degrees of Freedom of a Signal.- 1.8 Optical Generation of Gabor’s Expansion Coefficients for Rastered Signals.- 1.9 Conclusion.- 2 Sampling in Optics.- 2.1 Introduction.- 2.2 Historical Background.- 2.3 The von Laue Analysis.- 2.4 Degrees of Freedom of an Image.- 2.4.1 Use of the Sampling Theorem.- 2.4.2 Some Objections.- 2.4.3 The Eigenfmiction Technique.- 2.4.4 The Gerchberg Method.- 2.5 Superresolving Pupils.- 2.5.1 Singular Value Analysis.- 2.5.2 Incoherent Imaging.- 2.5.3 Survey of Extensions.- 2.6 Fresnel SampHng.- 2.7 Exponential SampHng.- 2.8 Partially Coherent Fields.- 2.9 Optical Processing.- 2.10 Conclusion.- 3 A Multidimensional Extension of Papoulis’ Generalized Sampling Expansion with the Application in Minimum Density Sampling.- I: A Multidimensional Extension of Papoulis’ Generalized Sampling Expansion.- 3.1 Introduction.- 3.2 GSE Formulation.- 3.3 M-D Extension.- 3.3.1 M-D Sampling Theorem.- 3.3.2 M-D GSE Formulation.- 3.3.3 Examples.- 3.4 Extension Generalization.- 3.5 Conclusion.- II: Sampling Multidimensional Band-Limited Functions At Minimum Densities.- 3.6 Sample Interdependency.- 3.7 Sampling Density Reduction Using M-D GSE.- 3.7.1 Sampling Decimation.- 3.7.2 A Second Formulation for Sampling Decimation.- 3.8 Computational Complexity of the Two Formulations.- 3.8.1 Gram-Schmidt Searching Algorithm.- 3.9 Sampling at the Minimum Density.- 3.10 Discussion.- 3.11 Conclusion.- 4 Nonuniform Sampling.- 4.1 Preliminary Discussions.- 4.2 General Nonuniform Sampling Theorems.- 4.2.1 Lagrange Interpolation.- 4.2.2 Interpolation from Nonuniform Samples of a Signal and Its Derivatives.- 4.2.3 Nonuniform Sampling for Nonband-Limited Signals.- 4.2.4 Jittered Sampling.- 4.2.5 Past Sampling.- 4.2.6 Stability of Nonuniform Sampling Interpolation.- 4.2.7 Interpolation Viewed as a Time Varying System.- 4.2.8 Random Samphng.- 4.3 Spectral Analysis of Nonuniform Samples and Signal Recovery.- 4.3.1 Extension of the Parseval Relationship to Nonuniform Samples.- 4.3.2 Estimating the Spectrum of Nonuniform Samples.- 4.3.3 Spectral Analysis of Random Sampling.- 4.4 Discussion on Reconstruction Methods.- 4.4.1 Signal Recovery Through Non-Linear Methods.- 4.4.2 Iterative Methods for Signal Recovery.- 5 Linear Prediction by Samples from the Past.- 5.1 Preliminaries.- 5.2 Prediction of Deterministic Signals.- 5.2.1 General Results.- 5.2.2 Specific Prediction Sums.- 5.2.3 An Inverse Result.- 5.2.4 Prediction of Derivatives f(s) by Samples of.- 5.2.5 Round-OfF and Time Jitter Errors.- 5.3 Prediction of Random Signals.- 5.3.1 Continuous and Differentiable Stochastic Processes.- 5.3.2 Prediction of Weak Sense Stationary Stochastic Processes.- 6 Polar, Spiral, and Generalized Sampling and Interpolation.- 6.1 Introduction.- 6.2 Sampling in Polar Coordinates.- 6.2.1 Sampling of Periodic Functions.- 6.2.2 A Formula for Interpolating from Samples on a Uniform Polar Lattice.- 6.2.3 Applications in Computer Tomography (CT).- 6.3 Spiral Sampling.- 6.3.1 Linear Spiral Sampling Theorem.- 6.3.2 Reconstruction from Samples on Expanding Spirals.- 6.4 Reconstruction from Non-Uniform Samples by Convex Projections.- 6.4.1 The Method of Projections onto Convex Sets.- 6.4.2 Iterative Reconstruction by POCS.- 6.5 Experimental Results.- 6.5.1 Reconstruction of One-Dimensional Signals.- 6.5.2 Reconstruction of Images.- 6.6 Conclusions.- Appendix A.- A.1 Derivation of Projections onto Convex Sets Ci.- Appendix B.- B. 1 Derivation of the Projection onto the Set C0= ?iCi.- 7 Error Analysis in Application of Generalizations of the Sampling Theorem.- Foreword: Welcomed General Sources for the Sampling Theorems.- 7.1 Introduction — Sampling Theorems.- 7.1.1 The Shannon Sampling Theorem — A Brief Introduction and History.- 7.1.2 The Generalized Transform Sampling Theorem.- 7.1.3 System Interpretation of the Sampling Theorems.- 7.1.4 Self-Truncating Sampling Series for Better Truncation Error Bound.- 7.1.5 A New Impulse Train—The Extended Poisson Sum Formula.- 7.2 Error Bounds of the Present Extension of the Sampling Theorem.- 7.2.1 The Aliasing Error Bound.- 7.2.2 The Truncation Error Bound.- 7.3 Applications.- 7.3.1 Optics—Integral Equations Representation for Circular Aperture.- 7.3.2 The Gibbs’ Phenomena of the General Orthogonal Expansion—A Possible Remedy.- 7.3.3 Boundary-Value Problems.- 7.3.4 Other Apphcations and Suggested Extensions.- Appendix A.- A.1 Analysis of Gibbs’ Phenomena.