Digital Signal Processing

PREFACE .................................................................................... xi
INTRODUCTION ............................................................................... 1
1 DISCRETE-TIME SIGNALS AND SYSTEMS ........................................................ 6
1.0 Introduction ........................................................................... 6
1.1 Discrete-Time Signals-Sequences ........................................................ 8
1.2 Linear Shift-Invariant Systems ......................................................... 11
1.3 Stability and Causality ................................................................ 13
1.4 Linear Constant-Coefficient Difference Equations ....................................... 16
1.5 Frequency-Domain Representation of Discrete-Time Systems and Signals ................... 18
1.6 Some Symmetry Properties of the Fourier Transform ...................................... 24
1.7 Sampling of Continuous-Time Signals .................................................... 26
1.8 Two-Dimensional Sequences and Systems .................................................. 30
Summary .................................................................................... 34
Problems ................................................................................... 35
2 THE Z-TRANSFORM .......................................................................... 45
2.0 Introduction ........................................................................... 45
2.1 z-Transform ............................................................................ 45
2.2 Inverse z-Transform .................................................................... 52
2.3 z-Transform Theorems and Properties .................................................... 58
2.4 System Function ........................................................................ 67
2.5 Two-Dimensional Z-Transform ............................................................ 73
Summary .................................................................................... 77
Problems ................................................................................... 78
3 THE DISCRETE FOURIER TRANSFORM ........................................................... 87
3.0 Introduction ........................................................................... 87
3.1 Representation of Periodic Sequences-The Discrete Fourier Series ....................... 88
3.2 Properties of the Discrete Fourier Series .............................................. 91
3.3 Summary of Properties of the DFS Representation of Periodic Sequences .................. 95
3.4 Sampling the z-Transform ............................................................... 96
3.5 Fourier Representation of Finite-Duration Sequences - The Discrete Fourier Transform ... 99
3.6 Properties of the Discrete Fourier Transform ........................................... 101
3.7 Summary of Properties of the Discrete Fourier Transform ................................ 110
3.8 Linear Convolution Using the Discrete Fourier Transform ................................ 110
3.9 Two-Dimensional Discrete Fourier Transform ............................................. 115 
Summary .................................................................................... 121
Problems ................................................................................... 121
4 FLOW GRAPH AND MATRIX REPRESENTATION OF DIGITAL FILTERS .................................. 136
4.0 Introduction ........................................................................... 136
4.1 Signal Flow Graph Representation of Digital Networks ................................... 137
4.2 Matrix Representation of Digital Networks .............................................. 143
4.3 Basic Network Structures for HR Systems ................................................ 148
4.4 Transposed Forms ....................................................................... 153
4.5 Basic Network Structures for FIR Systems ............................................... 155
4.6 Parameter Quantization Effects ......................................................... 165
4.7 Tellegen's Theorem for Digital Filters and Its Applications ............................ 173
Summary .................................................................................... 181
Problems ................................................................................... 182
5 DIGITAL FILTER DESIGN TECHNIQUES ......................................................... 195
5.0 Introduction ........................................................................... 195
5.1 Design of HR Digital Filters from Analog Filters ....................................... 197
5.2 Design Examples: Analog-Digital Transformation ......................................... 211
5.3 Computer-Aided Design of HR Digital Filters ............................................ 230
5.4 Properties of FIR Digital Filters ...................................................... 237
5.5 Design of FIR Filters Using Windows .................................................... 239
5.6 Computer-Aided Design of FIR Filters ................................................... 250
5.7 A Comparison of HR and FIR Digital Filters ............................................. 268
Summary .................................................................................... 269
Problems ................................................................................... 271
6 COMPUTATION OF THE DISCRETE FOURIER TRANSFORM ............................................ 284
6.0 Introduction ........................................................................... 284
6.1 Goertzel Algorithm ..................................................................... 287
6.2 Decimation-in-Time FFT Algorithms ...................................................... 291
6.3 Decimation-in-Frequency FFT Algorithms ................................................. 302
6.4 FFT Algorithms for N a Composite Number ................................................ 307
6.5 General Computational Considerations in FFT Algorithms ................................. 315
6.6 Chirp Z-Transform Algorithm ............................................................ 321
Summary .................................................................................... 326
Problems ................................................................................... 328
7 DISCRETE HILBERT TRANSFORMS .............................................................. 337
7.0 Introduction ........................................................................... 337
7.1 Real- and Imaginary-part Sufficiency for Causal Sequences .............................. 339
7.2 Minimum-Phase Condition ................................................................ 345
7.3 Hilbert Transform Relations for the DFT ................................................ 353
7.4 Hilbert Transform Relations for Complex Sequences ...................................... 358
Summary .................................................................................... 365
Problems ................................................................................... 367
8 DISCRETE RANDOM SIGNALS .................................................................. 376
8.0 Introduction ........................................................................... 376
8.1 A Discrete-Time Random Process ......................................................... 377
8.2 Averages ............................................................................... 382
8.3 Spectrum Representations of Infinite-Energy Signals .................................... 388
8.4 Response of Linear Systems to Random Signals ........................................... 391
Summary .................................................................................... 395
Problems ................................................................................... 395
9 EFFECTS OF FINITE REGISTER LENGTH IN DIGITAL SIGNAL PROCESSING ........................... 404
9.0 Introduction ........................................................................... 404
9.1 Effect of Number Representation on Quantization ........................................ 406
9.2 Quantization in Sampling Analog Signals ................................................ 413
9.3 Finite-Register-Length Effects in Realizations of IIR Digital Filters .................. 418
9.4 Finite-Register-Length Effects in Realizations of FIR Digital Filters .................. 438
9.5 Effects of Finite Register Length in Discrete Fourier Transform Computations ........... 444
Summary .................................................................................... 462
Problems ................................................................................... 464
10 HOMOMORPHIC SIGNAL PROCESSING ........................................................... 480
10.0 Introduction .......................................................................... 480
10.1 Generalized Superposition ............................................................. 481
10.2 Multiplicative Homomorphic System ..................................................... 484
10.3 Homomorphic Image Processing .......................................................... 487
10.4 Homomorphic Systems for Convolution ................................................... 490
10.5 Properties of the Complex Cepstrum .................................................... 500
10.6 Computational Realizations of the Characteristic System D* ............................ 507
10.7 Applications of Homomorphic Deconvolution ............................................. 511
Summary .................................................................................... 527
Problems ................................................................................... 529
11 POWER SPECTRUM ESTIMATION ............................................................... 532
11.0 Introduction .......................................................................... 532
11.1 Basic Principles of Estimation Theory ................................................. 533
11.2 Estimates of the Autocovariance ....................................................... 539
11.3 The Periodogram as an Estimate of the Power Spectrum .................................. 541
11.4 Smoothed Spectrum Estimators .......................................................... 548
11.5 Estimates of the Cross Covariance and Cross Spectrum .................................. 554
11.6 Application of the FFT in Spectrum Estimation ......................................... 555
11.7 Example of Spectrum Estimation ........................................................ 562
Summary .................................................................................... 571
Problems ................................................................................... 571
INDEX ...................................................................................... 577