Introduction to Shannon Sampling and Interpolation Theory

by Robert J. II Marks

Overview

Much of that which is ordinal is modeled as analog. Most computational engines on the other hand are dig- ital. Transforming from analog to digital is straightforward: we simply sample. Regaining the original signal from these samples or assessing the information lost in the sampling process are the fundamental questions addressed by sampling and interpolation theory. This book deals with understanding, generalizing, and extending the cardinal series of Shannon sampling theory. The fundamental form of this series states, remarkably, that a bandlimited signal is uniquely specified by its sufficiently close equally spaced samples. The contents of this book evolved from a set of lecture notes prepared for a graduate survey course on Shannon sampling and interpolation theory. The course was taught at the Department of Electrical Engineering at the University of Washington, Seattle. Each of the seven chapters in this book includes a list of references specific to that chapter. A sequel to this book will contain an extensive bibliography on the subject. The author has also opted to include solutions to selected exercises in the Appendix.

Product Details

ISBN-13:	9781461397106
Publisher:	Springer New York
Publication date:	12/14/2011
Series:	Springer Texts in Electrical Engineering
Edition description:	Softcover reprint of the original 1st ed. 1991
Pages:	324
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1 Introduction.- 1.1 The Cardinal Series.- 1.2 History.- 2 Fundamentals of Fourier Analysis and Shastic Processes.- 2.1 Signal Classes.- 2.2 The Fourier Transform.- 2.3 Shastic Processes.- 2.4 Exercises.- 3 The Cardinal Series.- 3.1 Interpretation.- 3.2 Proofs.- 3.3 Properties.- 3.4 Application to Spectra Containing Distributions.- 3.5 Application to Bandlimited Shastic Processes.- 3.6 Exercises.- 4 Generalizations of the Sampling Theorem.- 4.1 Generalized Interpolation Functions.- 4.2 Papoulis’ Generalization.- 4.3 Derivative Interpolation.- 4.4 A Relation Between the Taylor and Cardinal Series.- 4.5 Sampling Trigonometric Polynomials.- 4.6 Sampling Theory for Bandpass Functions.- 4.7 A Summary of Sampling Theorems for Directly Sampled Signals.- 4.8 Lagrangian Interpolation.- 4.9 Kramer’s Generalization.- 4.10 Exercises.- 5 Sources of Error.- 5.1 Effects of Additive Data Noise.- 5.2 Jitter.- 5.3 Truncation Error.- 5.4 Exercises.- 6 The Sampling Theorem in Higher Dimensions.- 6.1 Multidimensional Fourier Analysis.- 6.2 The Multidimensional Sampling Theorem.- 6.3 Restoring Lost Samples.- 6.4 Periodic Sample Decimation and Restoration.- 6.5 Raster Sampling.- 6.6 Exercises.- 7 Continuous Sampling.- 7.1 Interpolation From Periodic Continuous Samples.- 7.2 Prolate Spheroidal Wave Functions.- 7.3 The Papoulis-Gerchberg Algorithm.- 7.4 Exercises.