# Handbook of Fourier Analysis & Its Applications

Fourier analysis has many scientific applications - in physics, number theory, combinatorics, signal processing, probability theory, statistics, option pricing, cryptography, acoustics, oceanography, optics and diffraction, geometry, and other areas. In signal processing and related fields, Fourier analysis is typically thought of as decomposing a signal into its component frequencies and their amplitudes. This practical, applications-based professional handbook comprehensively covers the theory and applications of Fourier Analysis, spanning topics from engineering mathematics, signal processing and related multidimensional transform theory, and quantum physics to elementary deterministic finance and even the foundations of western music theory. As a definitive text on Fourier Analysis, Handbook of Fourier Analysis and Its Applications is meant to replace several less comprehensive volumes on the subject, such as Processing of Multifimensional Signals by Alexandre Smirnov, Modern Sampling Theory by John J. Benedetto and Paulo J.S.G. Ferreira, Vector Space Projections by Henry Stark and Yongyi Yang and Fourier Analysis and Imaging by Ronald N. Bracewell. In addition to being primarily used as a professional handbook, it includes sample problems and their solutions at the end of each section and thus serves as a textbook for advanced undergraduate students and beginning graduate students in courses such as: Multidimensional Signals and Systems, Signal Analysis, Introduction to Shannon Sampling and Interpolation Theory, Random Variables and Stochastic Processes, and Signals and Linear Systems.

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### Contents

 1 Introduction 3 2 Fundamentals of Fourier Analysis 10 3 Fourier Analysis in Systems Theory 104 4 Fourier Transforms in Probability Random Variables and Stochastic Processes 151 5 The Sampling Theorem 217 6 Generalizations of the Sampling Theorem 242 7 Noise and Error Effects 288 8 Multidimensional Signal Analysis 326
 10 Signal Recovery 447 Alternating Projections Onto Convex Sets 495 12 Mathematical Morphology and Fourier Analysis on Time Scales 570 13 Applications 610 14 Appendices 660 15 References 680 Index 745 Copyright

 9 TimeFrequency Representations 411