Multidimensional Digital Signal Processing

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Author(s): Dan E. Dudgeon, Russell M. Mersereau
Series: Prentice-Hall Signal Processing
Publisher: Prentice Hall
Year: 1984

Language: English
Pages: 406

Contents......Page 1
Introduction......Page 7
Ch1 Multidimensional Signals & Systems......Page 11
1.1 Two-Dimensional Discrete Signals......Page 12
1.1.1 Some Special Sequences......Page 13
1.1.2 Separable Sequences......Page 14
1.1.4 Periodic Sequences......Page 15
1.2.1 Fundamental Operations on Multidimensional Signals......Page 18
1.2.3 Shift-Invariant Systems......Page 20
1.2.4 Linear Shift-Invariant Systems......Page 21
1.2.5 Cascade & Parallel Connections of Systems......Page 26
1.2.6 Separable Systems......Page 28
1.2.8 Regions of Support......Page 29
1.2.9 Vector Input-Output Systems......Page 31
1.3.1 Frequency Responses of a 2-D LSI Systems......Page 32
1.3.2 Determining Impulse Responses from Frequency Responses......Page 35
1.3.3 Multidimensional Fourier Transform......Page 37
1.3.4 Other Properties of 2-D Fourier Transform......Page 39
1.4.1 Periodic Sampling with Rectangular Geometry......Page 42
1.4.2 Periodic Sampling with Arbitrary Sampling Geometries......Page 45
1.4.3 Comparison of Rectangular & Hexagonal Sampling......Page 50
1.5 Processing Continuous Signals with Discrete Systems......Page 53
1.5.1 Relationship between System Input & Output Signals......Page 54
1.5.2 System Frequency Responses......Page 55
1.5.3 Alternative Definition of Fourier Transform for Discrete Signals......Page 56
Problems......Page 57
References......Page 64
Ch2 Discrete Fourier Analysis of Multidimensional Signals......Page 66
2.1 Discrete Fourier Series Representation of Rectangularly Periodic Sequences......Page 67
2.2.1 Definitions......Page 69
2.2.2 Properties of Discrete Fourier Transform......Page 73
2.2.3 Circular Convolution......Page 76
2.3 Calculation of Discrete Fourier Transform......Page 80
2.3.2 Row-Column Decompositions......Page 81
2.3.3 Vector-Radix Fast Fourier Transform......Page 82
2.3.4 Computational Considerations in DFT Calculations......Page 87
2.4.1 DFT Relations for General Periodically Sampled Signals......Page 93
2.4.2 FFT Algorithms for General Periodically Sampled Signals......Page 96
2.4.3 Some Special Cases......Page 102
2.5 Interrelationship between M-Dimensional & 1-Dimensional DFTs......Page 106
2.5.1 Slice DFT......Page 107
2.5.2 Good's Prime Factor Algorithm for Decomposing 1-D DFT......Page 109
Problems......Page 112
References......Page 117
3.1 FIR Filters......Page 118
3.2.1 Direct Convolution......Page 119
3.2.2 DFT Implementations of FIR Filters......Page 120
3.2.3 Block Convolution......Page 122
3.3.1 Description of the Method......Page 124
3.3.2 Choosing Window Function......Page 125
3.3.3 Design Example......Page 126
3.3.4 Image Processing Example......Page 130
3.4 Optimal FIR Filter Design......Page 132
3.4.1 Least Square Designs......Page 134
3.4.2 Design of Zero-Phase Equiripple FIR Filters......Page 136
3.5.1 Cascaded FIR Filters......Page 138
3.5.2 Parallel FIR Filters......Page 140
3.5.3 Design of FIR Filters using Transformations......Page 143
3.5.4 Implementing Filters Designed using Transformations......Page 150
3.5.5 Filters with Small Generating Kernels......Page 154
3.6 FIR Filters for Hexagonally Sampling Signals......Page 155
3.6.2 Design of Hexagonal FIR Filters......Page 156
Problems......Page 160
References......Page 166
Ch4 Multidimensional Recursive Systems......Page 168
4.1.1 Realizing LSI Systems using Difference Equations......Page 169
4.1.2 Recursive Computability......Page 170
4.1.3 Boundary Conditions......Page 174
4.1.4 Ordering Computation of Output Samples......Page 177
4.2.1 Transfer Function......Page 180
4.2.2 z-Transform......Page 181
4.2.3 Properties of 3-D z-Transform......Page 186
4.2.4 Transfer Functions of Systems Specified by Difference Equations......Page 188
4.2.5 Inverse z-Transform......Page 192
4.2.6 Two-Dimensional Flowgraphs......Page 193
4.3 Stability of Recursive Systems......Page 195
4.3.1 Stability Theorems......Page 196
4.3.2 Stability Testing......Page 199
4.3.3 Effect of Numerator Polynomial on Stability......Page 202
4.3.4 Multidimensional Stability Theorems......Page 203
4.4.1 Definition of Complex Cepstrum......Page 204
4.4.2 Existence of Complex Cepstrum......Page 205
4.4.3 Causality, Minimum Phase, & Complex Cepstrum......Page 207
4.4.4 Spectral Factorization......Page 208
4.4.5 Computing 2-D Complex Cepstrum......Page 211
Problems......Page 214
References......Page 221
5.1 Classical 3-D IIR Filter Implementations......Page 224
5.1.1 Direct Form Implementations......Page 225
5.1.2 Cascade & Parallel Implementations......Page 227
5.2 Iterative Implementations for 2-D IIR Filters......Page 230
5.2.1 Basic Iterative Implementation......Page 231
5.2.2 Generalizations of Iterative Implementation......Page 234
5.2.3 Truncation, Boundary Conditions, & Signal Constraints......Page 236
5.3.1 Circuit Elements & their Realizations......Page 240
5.3.2 Minimizing Number of Shift Operators......Page 244
5.3.3 State-Variable Realizations......Page 246
5.4 Space-Domain Design Techniques......Page 250
5.4.1 Shanks' Method......Page 252
5.4.2 Descent Methods for Space-Domain Design......Page 254
5.4.3 iterative Prefiltering Design Method......Page 256
5.5.1 General Minimization Procedures......Page 259
5.5.3 Magnitude Design with Stability Constraint......Page 261
5.5.4 Zero-Phase IIR Frequency-Domain Design Methods......Page 262
5.5.5 Frequency Transformations......Page 265
5.6.1 Cascade Designs......Page 268
5.6.2 Separable Denominator Designs......Page 269
5.6.3 Lattice Structures......Page 272
5.7.1 Cepstral Stabilization......Page 282
5.7.2 Shaw's Stabilization Technique......Page 283
Problems......Page 285
References......Page 292
Ch6 Processing Signals Carried by Propagating Waves......Page 295
6.1.1 Elemental Signals......Page 296
6.1.2 Filtering in Wavenumber-Frequency Space......Page 297
6.2.1 Weighted Delay-and-Sum Beamformer......Page 299
6.2.2 Array Pattern......Page 300
6.2.3 Example of Array Pattern......Page 303
6.2.4 Effect of Receiver Weighting Function......Page 305
6.2.5 Filter-and-Sum Beamforming......Page 306
6.2.6 Frequency-Domain Beamforming......Page 307
6.3 Discrete-Time Beamforming......Page 309
6.3.1 Time-Domain Beamforming for Discrete-Time Signals......Page 310
6.3.2 Interpolation Beamforming......Page 313
6.3.3 Frequency-Domain Beamforming for Discrete-Time Signals......Page 315
6.4 Further Considerations for Array Processing Applications......Page 317
6.4.1 Analysis of Narrowband Beamformer......Page 318
6.5 Multidimensional Spectral Estimation......Page 321
6.5.1 Classical Spectral Estimation......Page 322
6.5.2 High-Resolution Spectral Estimation......Page 327
6.5.3 All-Pole Spectral Modeling......Page 331
6.5.4 Maximum Entropy Spectral Estimation......Page 337
6.5.5 Extendibility......Page 344
Problems......Page 345
References......Page 351
Ch7 Inverse Problems......Page 354
7.1 Constrained Iterative Signal Restoration......Page 355
7.1.1 Iterative Procedures for Constrained Deconvolution......Page 356
7.1.2 iterative Procedures for Signal Extrapolation......Page 360
7.1.3 Reconstructions from Phase or Magnitude......Page 362
7.2 Seismic-Wave Migration......Page 365
7.3.1 Projections......Page 369
7.3.2 Projection-Slice Theorem......Page 372
7.3.3 Discretiration of Reconstruction Problem......Page 373
7.3.4 Fourier-Domain Reconstruction Algorithms......Page 375
7.3.5 Convolution / Back-Projection Algorithm......Page 379
7.3.7 Fan-Beam Reconstructions......Page 382
7.4 Projection of Discrete Signals......Page 385
Problems......Page 389
References......Page 395
Index......Page 397