Multidimensional Digital Signal Processing (Prentice-Hall Signal Processing Series)

CONTENTS......Page 2
INTRODUCTION......Page 8
1. MULTIDIMENSIONAL SIGNALS AND SYSTEMS......Page 12
1.1 Two-Dimensional Discrete Signals......Page 13
1.1.1 Some Special Sequences......Page 14
1.1.2 Separable Sequences......Page 15
1.1.4 Periodic Sequences......Page 16
1.2.1 Fundamental Operations on Muhidimensional Signals......Page 19
1.2.3 Shift-lnvmiant Systems......Page 21
1.2.4 Linear Shift-Invariant Systems......Page 22
1.2.5 Cascade and Parallel Connections of Systems......Page 27
1.2.6 Separable Syste1m......Page 29
1.2.8 Regions of Suppon......Page 30
1.2.9 Vector lnput-Outpllt Systems......Page 32
1.3.1 Frequency Response of a 2-D LSI System......Page 33
1.3.2 Determining rite Impulse Response from the Frequency Response......Page 36
1.3.3 Multidimensional Fourier Transform......Page 38
1.3.4 Other Properties of the 2-D Fourier Transform......Page 40
1.4.1 Periodic Sampling with Rectangular Geometry......Page 43
1.4.2 Periodic Sampling with Arbitrary Sampling Geometries......Page 46
1.4.3 Comparison of Rectangular and Hexagonal Sampling......Page 51
1.5 Processing Continuous Signals with Discrete Systems......Page 54
1.5.1 Relationship between the System Input and Output Signals......Page 55
1.5.2 System Frequency Response......Page 56
1.5.3 Alternative Definition of the Fourier Transform for Discrete Signals......Page 57
2. DISCRETE FOURIER ANALYSIS OF MULTIDIMENSIONAL SIGNALS......Page 67
2.1 Discrete Fourier Series Representation of Rectangularly Periodic Sequences......Page 68
2.2.1 Definitions......Page 70
2.2.2 Properties of the Discrete Fourier Transform......Page 74
2.2.3 Circular Convolution......Page 77
2.3 Calculation of the Discrete Fourier Transform......Page 81
2.3.2 Row-Column Decompositions......Page 82
2.3.3 Vector-Radix Fast Fourier Transform......Page 83
2.3.4 Computational Considerations in DFT Calculations......Page 88
2.4.1 DFT Relations for General Periodically Sampled Signals......Page 94
2.4.2 Fast Fourier Transform Algorithms for General Periodically Sampled Signals......Page 97
2.4.3 Some Special Cases......Page 103
2.5 Interrelationship between M-dimensional and One-Dimensional DFTs......Page 107
2.5.1 Slice DFT......Page 108
2.5.2 Good\'s Prime Factor Algorithm for Decomposing a 1-D DFT......Page 110
3.1 FIR Filters......Page 119
3.2.1 Direct Convolution......Page 120
3.2.2 Discrete Fourier Transform Implementations of FIR Filters......Page 121
3.2.3 Block Convolution......Page 123
3.3.1 Description of the Method......Page 125
3.3.2 Choosing the Window Function......Page 126
3.3.3 Design Example......Page 127
3.3.4 Image Processing Example......Page 131
3.4 Optimal Fl R Filter Design......Page 133
3.4.1 Least-Squares Designs......Page 135
3.4.2 Design of Zero-Phase Equiripple FIR Filters......Page 137
3.5.1 Cascaded FIR Filters......Page 139
3.5.2 Parallel FIR Filters......Page 141
3.5.3 Design of FIR Filters Using Transformations......Page 144
3.5.4 Implementing Filters Designed Using Transformations......Page 151
3.5.5 Filters with Small Generating Kernels......Page 155
3.6 FIR Filters for Hexagpnally Sampled Signals......Page 156
3.6.2 Design of Hexagonal FIR Filters......Page 157
4. MULTIDIMENSIONAL RECURSIVE SYSTEMS......Page 169
4.1.1 Realizing LSI Systems Using Difference Equations......Page 170
4.1.2 Recursive Computability......Page 171
1.1.3 Boundtzry Conditions......Page 175
4.1.4 Ordering the Computation of Output Samples......Page 178
4.2.1 Transfer Function......Page 181
4.2.2 The z-Transform......Page 182
4.2.3 Properties of the 2-D z-Transform......Page 187
4.2.4 Transfer Functions of Systems Specified by Difference Equations......Page 189
4.2.5 Inverse z-Transform......Page 193
4.2.6 Two-Dimensional Flowgraphs......Page 194
4.3 Stability of Recursive Systems......Page 196
4.3.1 Stability Theorems......Page 197
4.3.2 Stability Testing......Page 200
4.3.3 Effect of the Numerator Polynomial on Stability......Page 203
4.3.4 Multidimensional Stability Theorems......Page 204
4.4.1 Definition of the Complex Cepjtrum......Page 205
4.4.2 Existence of the Complex Cepstrum......Page 206
4.4.3 Causality  Minimum Phase  and the Complex Cepstrum......Page 208
4.4.4 Spectral Factorization......Page 209
4.4.5 Computing the 2-D Complex Cepstrum......Page 212
5.1 Classicai2-D IIR Filter Implementations......Page 225
5.1.1 Direct Form Implementations......Page 226
5.1.2 Cascade and Parallel Implementations......Page 228
5.2 Iterative lmplementMions for 2-D IIR Filters......Page 231
5.2.1 Basic Iterative Implementation......Page 232
5.2.2 Generalizations of the Iterative Implementation......Page 235
5.2.3 Truncation  Boundary Con:itions  and Signal Constraints......Page 237
5.3.1 Circuit Elements and Their Realization......Page 241
S.3.2 Minimizing the Number of Shift Operators......Page 245
5.3.3 State-Variable Realizations......Page 247
5.4 Space-Domafn Design Techniques......Page 251
5.4.1 Shanks\'s Method......Page 253
5.4.2 Descent Methodsfor Space-Domain Design......Page 255
5.4.3 lt.erative Pre.filtering Design Method......Page 257
5.5 Frequency-Domain Design Techniques......Page 260
5.7.1 Cepstral Stabilization......Page 0
5.5.3 Magnitude Design with a Stability Constraint......Page 262
5.5.4 Zero-Phase fiR Frequency-Domain Dtsign Methods......Page 263
5.6.1 Cascade Designs......Page 269
5.6.2 Separable Denominator Designs......Page 268
5.6.3 Lattice Structures......Page 273
5.7 Stabilization Techniques......Page 283
5.7.2 Shaw\'s Stabilization Technique......Page 284
6. PROCESSING SIGNALS CARRIED BY PROPAGATING WAVES......Page 296
6.1.1 Elemental Signals......Page 297
6.1.2 Filtering in Wavenumber-Frequency Space......Page 298
6.2.1 Weighted Delay-and-Sum Beamformer......Page 300
6.2.2 Array Pattern......Page 301
6.2.3 Example of an Array Pattern......Page 304
6.2.4 Effect of the Receiver Weighting Function......Page 306
6.2.5 Filter-and-Sum Beamforming......Page 307
6.2.6 Frequency-Domain Beamforming......Page 308
6.3 Discrete-Time Beamforming......Page 310
6.3.1 Time-Domain Beamformingfor Discrete-Time Signals......Page 311
6.3.2 Interpolation Beamforming......Page 314
6.3.3 Frequency-Domain Beamforming for Discrete-Time. Signals......Page 316
6.4 Further Considerations for Array Processing Applications......Page 318
6.4.1 Analysis of a Narrowband Beamformer......Page 319
6.5 Multidimensional Spectral Estimation......Page 322
6.5.1 Classical Spectral Estimation......Page 323
6.5.2 High-Resolution Spectral Estimation......Page 328
6.5.3 All-Pole Spectral Modeling......Page 332
6.5.4 Maximum Entropy Spectral Estimation......Page 338
6.5.5 Extendibility......Page 345
7. INVERSE PROBLEMS......Page 355
7.1 Constrained Iterative Signal Restoration......Page 356
7.1.1 Iterative Procedures for Constrained Deconvolution......Page 357
7.1.2 Iterative Procedures for Signal Extrapolation......Page 361
7.1.3 Reconstructions from Phase or Magnitude......Page 363
7.2 Seismic Wave Migration......Page 366
7.3.l Projections......Page 370
7.3.2 Projections-Slice Theorem......Page 373
7.3.3 Discretization of the Reconstruction Problem......Page 374
7.3.4 Fourier-Domain Reconstruction Algorithms......Page 376
7.3.5 Convolution/Back-Projection Algorithm......Page 380
7.3.7 Fan-Beam Reconstructions......Page 383
7.4 Projection of Discrete Signals......Page 386
INDEX......Page 398