Advanced Topics in Shannon Sampling and Interpolation Theory

1 Gabor’s Signal Expansion and Its Relation to Sampling of the Sliding-Window Spectrum.- 1.1 Introduction.- 1.2 Sliding-Window Spectrum.- 1.3 Sampling Theorem for the Sliding-Window Spectrum.- 1.4 Examples of Window Functions.- 1.5 Gabor’s Signal Expansion.- 1.6 Examples of Elementary 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.- 2.5 Superresolving Pupils.- 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.- 3.4 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.- 3.8 Computational Complexity of the Two Formulations.- 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.- 4.3 Spectral Analysis of Nonuniform Samples and Signal Recovery.- 4.4 Discussion on Reconstruction Methods.- 5 Linear Prediction by Samples from the Past.- 5.1 Preliminaries.- 5.2 Prediction of Deterministic Signals.- 5.3 Prediction of Random Signals.- 6 Polar, Spiral, and Generalized Sampling and Interpolation.- 6.1 Introduction.- 6.2 Sampling in Polar Coordinates.- 6.3 SpiralSampling.- 6.4 Reconstruction from Non-Uniform Samples by Convex Projections.- 6.5 Experimental Results.- 6.6 Conclusions.- Appendix A.- Appendix B.- 7 Error Analysis in Application of Generalizations of the Sampling Theorem.- Foreword: Welcomed General Sources for the Sampling Theorems.- 7.1 Introduction — Sampling Theorems.- 7.2 Error Bounds of the Present Extension of the Sampling Theorem.- 7.3 Applications.- Appendix A.- A.1 Analysis of Gibbs’ Phenomena.