Robert J. Marks I1 Editor

Advanced Topics in Shannon Sampling and Interpolat ion Theory

With 91 Figures

Springer-Verlag New York Berlin Heidelberg London Paris

Springer Texts in Electrical Engineering

Multivariable Feedback Systems F.M. Callier/C.A. Desoer

Linear Programming M. Sakarovitch

Introduction to Random Processes E. Wong

Stochastic Processes in Engineering Systems E. WongIB. Hajek

Introduction to Probability J.B. Thomas

Elements of Detection and Signal Design C.L. Weber

An Introduction to Communication Theory and Systems J.B. Thomas

Signal Detection in Non-Gaussian Noise S.A. Kassam

An Introduction to Signal Detection and Estimation H.V. Poor

Introduction to Shannon Sampling and Interpolation Theory R.J. Marks I1

Random Point Processes in Time and Space, 2nd Edition D.L. Snyder/M.I. Miller

Linear System Theory F.M. Callier1C.A. Desoer

Advanced Topics in Shannon Sampling and Interpolation Theory R.J. Marks I1 (ed.)

Springer Texts in Electrical Engineering

Multivariable Feedback Systems F.M. Callier/C.A. Desoer

Linear Programming M. Sakarovitch

Introduction to Random Processes E. Wong

Stochastic Processes in Engineering Systems E. WongIB. Hajek

Introduction to Probability J.B. Thomas

Elements of Detection and Signal Design C.L. Weber

An Introduction to Communication Theory and Systems J.B. Thomas

Signal Detection in Non-Gaussian Noise S.A. Kassam

An Introduction to Signal Detection and Estimation H.V. Poor

Introduction to Shannon Sampling and Interpolation Theory R.J. Marks I1

Random Point Processes in Time and Space, 2nd Edition D.L. Snyder/M.I. Miller

Linear System Theory F.M. Callier1C.A. Desoer

Advanced Topics in Shannon Sampling and Interpolation Theory R.J. Marks I1 (ed.)

Contents

Acknowledgments v

Contributors xiii

1 Gabor's Signal Expansion and Its Relation to Sampling of the Sliding-Window Spectrum

. . . . . . . . . . . . . . . . . . . . . . . . . . . Introductioii 1 . . . . . . . . . . . . . . . . . . . Sliding-Il~iildo~~~ Spectruin 4 . . . . . . . . . . . . . . . . . . . 1.2.1 Ii~versionForili~~las 5

. . . . . . . . . . . . . . . 1.2.2 Space and Frequency Shift 6 1.2.3 Some 111tegra.l~ Coilceriling the Sliding-IYindow

. . . . . . . . . . . . . . . . . . . . . . . . Spectruin 6 . . . . . . . . . . . . . . . . . 1.2.4 Discrete-Time Signals 6

Sainpling Theorein for the Sliding-IYindo~v Spectruin . . . . 7 . . . . . . . . . . . . . . . . . 1.3.1 Discrete-Time Sigilals 11

. . . . . . . . . . . . . . . . Examples of IVindow Fuilctioils 12 . . . . . . . . . . . . . . 1.4.1 Gaussiai~I4~ind01~rF~illctioll 14

. . . . . . . . . . . . . . . . . 1.4.2 Discrete-Time Signals 15 . . . . . . . . . . . . . . . . . . . Gabor's Sigilal Expailsioil 17

. . . . . . . . . . . . . . . . . 1.5.1 Discrete-Time Sigilals 21 . . . . . . . . . . . . . . . Exainples of Eleineiltary Sigilals 21

. . . . . . . . . . . . . . . . 1.6.1 Rect Eleiiieiltary Signal 22

. . . . . . . . . . . . . . . . 1.6.2 Siilc Eleineiltary Signal 22 . . . . . . . . . . . . . . 1.6.3 Gaussian Elementary Signal 22

. . . . . . . . . . . . . . . . . 1.6.4 Discrete-Time Signals 25 . . . . . . . . . . . . . . . . Degrees of Freedom of a Signal 27

0pt.ical Geileratioil of Gabor's Expailsion Coefficients for . . . . . . . . . . . . . . . . . . . . . . . . Rastered Signals 31

. . . . . . . . . . . . . . . . . . . . . . . . . . . Coilclusioil 34

2 Sampling in Optics 37

Franc0 Gori

viii Contents

. . . . . . . . . . . . . . . . . . . . . 2.2 Historical Backgrouild 39

. . . . . . . . . . . . . . . . . . . . . 2.3 The von Laue Analysis 40 . . . . . . . . . . . . . . . 2.4 Degrees of Reedoin of a.11 Iinage 42

. . . . . . . . . . . . . 2.4.1 Use of the Sampling Theorem 44 . . . . . . . . . . . . . . . . . . . . 2.4.2 SomeObjectioils 47

. . . . . . . . . . . . . 2.4.3 The Eigeizfuilction Techiiique 49 . . . . . . . . . . . . . . . . 2.4.4 The Gerchberg Method 55

. . . . . . . . . . . . . . . . . . . . . 2.5 Superresolviilg Pupils 59 . . . . . . . . . . . . . . . . 2.5.1 Singu1a.r Va.lue Aila.lysis 61

. . . . . . . . . . . . . . . . . . . 2.5.2 Illcoherent 1ina.ging 64 . . . . . . . . . . . . . . . . . . 2.5.3 Survey of Extensions 66

. . . . . . . . . . . . . . . . . . . . . . . . 2.6 Fresnel Sai~~pling 67 . . . . . . . . . . . . . . . . . . . . . 2.7 Exponential Sainpliilg 70

. . . . . . . . . . . . . . . . . . . 2.8 Partially Coherent Fields 76 . . . . . . . . . . . . . . . . . . . . . . . 2.9 Optical Processing 78

. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Conclusioil 83

3 A Multidimensional Extension of Papoulis' Generalized Sampling Expansion with the Application in Minimum Density Sampling

PART I: A ~IC'LTIDIAIENSIONAL EXTENSION OF PAPOULIS' GENERALIZED SA~IPLING EXPANSION 85

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Iiltroductioil 85 . . . . . . . . . . . . . . . . . . . . . . . 3.2 GSE Forillulatioil 86

. . . . . . . . . . . . . . . . . . . . . . . . . 3.3 hl-D Extensioil 89 . . . . . . . . . . . . . . . . 3.3.1 hl-D Sainpliilg Theoreill 89 . . . . . . . . . . . . . . . . 3.3.2 hl-D GSE Fori~~ulation 92

. . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Examples 95 . . . . . . . . . . . . . . . . . . . 3.4 Extensioil Generalization 100

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion 102

PART 11: SAMPLING ~:IULTIDIMENSIONAL BAND-LI~IITED FUNCTIONS AT ~ ~ Z I N I A . I C R . I DENSITIES 102

. . . . . . . . . . . . . . . . . . . . 3.6 Sa.~nple Ii~terdepenclency 103 . . . . . . . . 3.7 Sa.inpling Density Reductioil Usiiig M-D GSE 104

. . . . . . . . . . . . . . . . . 3.7.1 Sainpling D e i n t i o i 107 3.7.2 A Secoild Forinulatioii for Sainpliilg Deciination . . 110

. . . . 3.8 Computationa.1 Coinplexity of the Two Forillulatioils 114 . . . . . . . . . 3.8.1 Grain-Scllinidt Searchiilg Algorithin 115

. . . . . . . . . . . . . . 3.9 Sampling at the hiiiliinuin Density 117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Discussion 118

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Coilclusioil 119

Contents ix

4- Nonuniform Sampling

Farokl~ Alar vast i

. . . . . . . . . . . . . . . . . . . . 4.1 Preliilliiiary Discussions . . . . . . . . . . . 4.2 General Nonulliforin Sainpling Theorems

4.2.1 La.gra.nge 1nterpola.tioil . . . . . . . . . . . . . . . . . 4.2.2 1nterpola.tion from Noiluiliforin Sailiples of a Sigila.1

and Its Deriva.tives . . . . . . . . . . . . . . . . . . . 4.2.3 Nonuniform Sainpling for Nonband-Limited Sigilals

. . . . . . . . . . . . . . . . . . . 4.2.4 Jittered Sainpliilg . . . . . . . . . . . . . . . . . . . . . 4.2.5 Past Sainpling

. . . 4.2.6 Stability of Nonuniforin Sainpling Interpolatioil 4.2.7 1nterpola.tion Viewed a.s a . Time Va.rying System . .

. . . . . . . . . . . . . . . . . . . 4.2.8 Randoin Sainpliilg 4.3 Spectral Ana.lysis of Noiluiliforili Sainples and

Signal Recovery . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Exteilsioil of the Pa.rseval Relatioilship to

. . . . . . . . . . . . . . . . . . Noi~uiiiform Sa.lnples 4.3.2 Estill~a.t.iug the Spectruin of Nolluiliforill Sa.inples 4.3.3 Spectra.1 Aaalysis of Raildoin Sa.inpling . . . . . . .

4.4 Discussion on R.econstructioa hiethods . . . . . . . . . . . . 4.4.1 Signa.1 R.ec0ver.y Through Non-Linear Methods . . . 4.4.2 1tera.tive R.iethoc1s for Signal Recovery . . . . . . . .

5 Linear Prediction by Samples from the Past

P.L. Butzer and R.L. Stem

. . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Preliilliilaries 5.2 Prediction of Deterillillistic Signals . . . . . . . . . . . . . .

5.2.1 General Resl-11ts . . . . . . . . . . . . . . . . . . . . . 5.2.2 Specific Predictlioil Suins . . . . . . . . . . . . . . . . 5.2.3 An Inverse Result . . . . . . . . . . . . . . . . . . . 5.2.4 Prediction of Derivatives f ( S ) 1)y Samples of f . . . . 5.2.5 Round-Off and Time Jitter Errors . . . . . . . . . .

5.3 Prediction of Randoin Signals . . . . . . . . . . . . . . . . . 5.3.1 Contiilr~ous and Differentiable Stocllastic Processes 5.3.2 Predictioil of Weak Sense . Statioilary

Stochastic Processes . . . . . . . . . . . . . . . . . .

6 Polar. Spiral. and Generalized Sampling and Interpolat ion

Henry Stark

6.1 Iiltroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Sampling in Polar Coordina.tes . . . . . . . . . . . . . . . .

6.2.1 Sainpliilg of Periodic F~~nctions . . . . . . . . . . . .

Con tents

6.2.2 A Forinula for 1nterpola.tiag froin Sainples on a . . . . . . . . . . . . . . . . . Uniforin Polar Lattice 188

. . . . 6.2.3 Applicatioils in Computer Tomography (CT) 189 . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Spiral Sainpling 192

. . . . . . . . . . . 6.3.1 Linear Spiral Sainpliilg Tlieorein 192 6.3.2 Reconstruction froin Samples on Expanding Spirals 197

6.4 Reconstructioil from Non-Uaiforill Sainples by . . . . . . . . . . . . . . . . . . . . . . . Convex Projections 199

6.4.1 The Method of Projections onto . . . . . . . . . . . . . . . . . . . . . . . Convex Sets 199

. . . . . . . . . . 6.4.2 Iterative Recoilstructioil by POCS 200 . . . . . . . . . . . . . . . . . . . . . . 6.5 Experimental Results 204

. . . . . 6.5.1 Reconst.ruction of One-Diillensiona.1 Sigilals 204 . . . . . . . . . . . . . . . 6.5.2 Recoilstructioil of Iillages 205

. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Coi~clusioi~s 206

. . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A 213 A.1 Deriva.tion of Projectioils oilto Coilvex Sets Ci . . . 213

. . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B 217 B.l Derivation of the Projectioil onto tlie Set

. . . . . . . . . . . . . . . . . . . . . . . Co r niCi 217

7 Error Analysis in Application of Generalizations of the Sampling Theorem 219

A bdul J . Je1.l.i

FOREWORD: I'ITELCO~IED GENERAL SOURCES FOR THE S A ~ ~ P L I N G THEOREM 219

. . . . . . . . . . . . . . 7.1 1ntroduct.ion-Sampling Theoreins 220 7.1.1 Tile Shannon Sampling Theoreill-A Brief

. . . . . . . . . . . . . . . Introduction aild History 220 . . . 7.1.2 The Generalized Transforin Sampling Theorell1 224

7.1.3 Systein Interpretation of the Sainpliiig Theoreins 228 7.1.4 Self-Truncating Sainpliiig Series for Better

. . . . . . . . . . . . . . . Truncation Error Bound 233 7.1.5 A New Iinpulse Train-The Extended Poissoil Suin

. . . . . . . . . . . . . . . . . . . . . . . . Formula 239 7.2 Error Bounds of the Present Extensioil of the

. . . . . . . . . . . . . . . . . . . . . . . Sampling Theorem 249 . . . . . . . . . . . . . . . 7.2.1 The Aliasing Error Bouild 249

. . . . . . . . . . . . . 7.2.2 The Truncation Error Bound 254 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Applications 258

7.3.1 Optics-Ii~tegral Ecluatioils Representation for . . . . . . . . . . . . . . . . . . . Circular Aperture 259

7.3.2 The Cibbs' Phenoi~lena of the General Orthogonal . . . . . . . . . . . Expansion-A Possible Renledy 261

Contents xi

. . . . . . . . . . . . . . 7.3.3 Boundary-Value Problems 272 . . . 7.3.4 Other Applicatiolls and Suggested Exteilsioils 274

. . . . . . . . . . . . . . . . . . . . . . . . . . . Appeildix A 287 . . . . . . . . . . . . A.l Ailalysis of Gibbs' Phenomei~a 287

Bibliography 299

Index 357

Contributors

Mart in J. Bast iaans Technisclle Uiliversi teit Eindhoven, Faculteit Elek- trotechniek, Post bus 51 3, 5600 h.1 B Eindho~ren, Net herlands.

P. L. Butzer Lehrstuhl A fiir hlatheillatik, RIVTH Aaclieil Teilipler- graben, D-5100 Aachen, Geri11aiiy.

Kwan F. Cheung Departilleilt of Electrical and Electroilic Engineering, The Hoilg Koilg Uiliversity of Science and Technology, Clear IVay Bay, Kotvloon, Hong Kong.

Fkanco Gori Dipartiineilto di Fisica - UniversitSt di Roina "La, Sapienza" Piazza.le Aldo AIoro. 2, I - 00185 Rome, Italy.

Abdul J. Jerri Department of h~fathema.tics and Computer Scieilce, Clarksoil University. Potsda.m. NY 13699. USA.

Robert J. Marks I1 Departillent of Electrical Engineering, Uiliversity of Ilrashington FT-10, Seattle? 'IVA 98195: USA.

Farokh Marvasti Departineilt of Electroilic and Electrical Engineeriag, King's College London, University of London, Straild, Lolldoll M'C2R 2LS, England.

Henry Stark Departineilt of Electrical and Coinputer Engineering, Illi- nois Ii~stitut~e of Technology. 3301 South Dearborn? Chicago, IL 60616, USA.

R. L. Stens Lehrstuhl A fiir hlathematik, RjVTH Aachen Templer- graben, D-5100 Aachen, Gerinai~y.