Digital Filters Design for Signal and Image Processing

Dealing with digital filtering methods for 1-D and 2-D signals,
this book provides the theoretical background in signal processing,
covering topics such as the z-transform, Shannon sampling theorem
and fast Fourier transform. An entire chapter is devoted to the
design of time-continuous filters which provides a useful
preliminary step for analog-to-digital filter conversion.

Attention is also given to the main methods of designing finite
impulse response (FIR) and infinite impulse response (IIR) filters.
Bi-dimensional digital filtering (image filtering) is investigated
and a study on stability analysis, a very useful tool when
implementing IIR filters, is also carried out. As such, it will
provide a practical and useful guide to those engaged in signal
processing.

Auteur
Mohamed Najim has published several books, more than 220 technical papers and has taught courses in digital signal processing for more than 30 years.

Résumé
Dealing with digital filtering methods for 1-D and 2-D signals, this book provides the theoretical background in signal processing, covering topics such as the z-transform, Shannon sampling theorem and fast Fourier transform. An entire chapter is devoted to the design of time-continuous filters which provides a useful preliminary step for analog-to-digital filter conversion.
Attention is also given to the main methods of designing finite impulse response (FIR) and infinite impulse response (IIR) filters. Bi-dimensional digital filtering (image filtering) is investigated and a study on stability analysis, a very useful tool when implementing IIR filters, is also carried out. As such, it will provide a practical and useful guide to those engaged in signal processing.

Contenu

Introduction xiii

Chapter 1. Introduction to Signals and Systems 1
Yannick BERTHOUMIEU, Eric GRIVEL and Mohamed NAJIM

1.1. Introduction 1

1.2. Signals: categories, representations and characterizations 1

1.2.1. Definition of continuous-time and discrete-time signals 1

1.2.2. Deterministic and random signals 6

1.2.3. Periodic signals 8

1.2.4. Mean, energy and power 9

1.2.5. Autocorrelation function 12

1.3. Systems 15

1.4. Properties of discrete-time systems 16

1.4.1. Invariant linear systems 16

1.4.2. Impulse responses and convolution products 16

1.4.3. Causality 17

1.4.4. Interconnections of discrete-time systems 18

1.5. Bibliography 19

Chapter 2. Discrete System Analysis 21
Mohamed NAJIM and Eric GRIVEL

2.1. Introduction 21

2.2. The z-transform 21

2.2.1. Representations and summaries 21

2.2.2. Properties of the z-transform 28

2.2.2.1. Linearity 28

2.2.2.2. Advanced and delayed operators 29

2.2.2.3. Convolution 30

2.2.2.4. Changing the z-scale 31

2.2.2.5. Contrasted signal development 31

2.2.2.6. Derivation of the z-transform 31

2.2.2.7. The sum theorem 32

2.2.2.8. The final-value theorem 32

2.2.2.9. Complex conjugation 32

2.2.2.10. Parseval's theorem 33

2.2.3. Table of standard transform 33

2.3. The inverse z-transform 34

2.3.1. Introduction 34

2.3.2. Methods of determining inverse z-transforms 35

2.3.2.1. Cauchy's theorem: a case of complex variables 35

2.3.2.2. Development in rational fractions 37

2.3.2.3. Development by algebraic division of polynomials 38

2.4. Transfer functions and difference equations 39

2.4.1. The transfer function of a continuous system 39

2.4.2. Transfer functions of discrete systems 41

2.5. Z-transforms of the autocorrelation and intercorrelation functions 44

2.6. Stability 45

2.6.1. Bounded input, bounded output (BIBO) stability 46

2.6.2. Regions of convergence 46

2.6.2.1. Routh's criterion 48

2.6.2.2. Jury's criterion 49

Chapter 3. Frequential Characterization of Signals and Filters 51
Eric GRIVEL and Yannick BERTHOUMIEU

3.1. Introduction 51

3.2. The Fourier transform of continuous signals 51

3.2.1. Summary of the Fourier series decomposition of continuous signals 51

3.2.1.1. Decomposition of finite energy signals using an orthonormal base 51

3.2.1.2. Fourier series development of periodic signals 52

3.2.2. Fourier transforms and continuous signals 57

3.2.2.1. Representations 57

3.2.2.2. Properties 58

3.2.2.3. The duality theorem 59

3.2.2.4. The quick method of calculating the Fourier transform 59

3.2.2.5. The Wiener-Khintchine theorem 63

3.2.2.6. The Fourier transform of a Dirac comb 63

3.2.2.7. Another method of calculating the Fourier series development of a periodic signal 66

3.2.2.8. The Fourier series development and the Fourier transform 68

3.2.2.9. Applying the Fourier transform: Shannon's sampling theorem 75

3.3. The discrete Fourier transform (DFT) 78

3.3.1. Expressing the Fourier transform of a discrete sequence 78

3.3.2. Relations between the Laplace and Fourier z-transforms 80

3.3.3. The inverse Fourier transform 81

3.3.4. The discrete Fourier transform 82

3.4. The fast Fourier transform (FFT) 86

3.5. The fast Fourier transform for a time/frequency/energy representation of a non-stationary signal 90

3.6. Frequential characterization of a continuous-time system 91

3.6.1. First and second order filters 91

3.6.1.1. 1st order system 91

3.6.1.2. 2nd order system 93 3.7. Frequential characterization of discrete-tim...