Digital Filters - Theory, Application And Design Of Modern Filters

A book on the various methods an engineer uses to separate the signal from the noise. As this is done most of the time by the use of a suitable filter, this book focuses on the understanding and design different types of filters, as there are discrete filters and linear filters which can be either deterministic filters or stochastic filters.
The first part of the book provides the reader with a general discussion of filters. All filters examined are linear time invariant and mostly finite in their impulse response. The reader is introduced to efficient ways of implementing these filters. The author describes multirate filters which change the sampling rate of the signal so that different techniques can be used to either permit the use the signal in different devices, or to compress it efficiently.
This wil be followed by a tretise on optimum filters. The modern approach to the design and use of filters covers advanced theories such optimization and prediction, which are thoroughly discussed on the example of Weiner filter which operate on stochastic signals.
The final part is devoted to application classes of adaptive filters which are used in diverse fields such as sonar, control systems, biomedical engineering and many others. The similarities in the use of the adaptive filters lead to four different ways the adaptive filter is used in any of these fields, and hence the author gives details on system identification, inverse modeling, prediction modeling, and interference canceling.
This book is suited for the practitioneer, and can
also serve as text for a graduate course. A collection of about 190 problems and projects is available online.

Rajiv J. Kapadia is a Professor in the Electrical Engineering Department at Minnesota State University in Mankato, USA. He obtained his academic degrees from the University of Bombay and from the University of Oklahoma. Dr. Kapadia is experienced in teaching in several different university systems as he has taught at the University of Queensland in Brisbane, Australia, and at the University of Mumbai, India. He spent most of his career teaching and consulting. Professor Kapadia has authored several scientific publications as well as two textbooks

1 Background and Introduction 1


1.1 Introduction 1


1.2 How is Digital Processing Done? 2


1.3 What is Filtering? 3


1.4 Linear Filters 6


1.5 Multirate Filters 8


1.6 A Classical Filtering Model 12


1.7 An Optimum Solution to the Classical Problem 14


1.8 Classes of Applications of Adaptive Filters 16


1.9 Chapter Summary 18


2 Discrete Time Signals and Systems 19


2.1 Introduction 19


2.2 Operations on Signals 22


2.3 Symmetry in Signals 27


2.4 Energy and Power Signals 29


2.5 The Concept of Frequency in Discrete Time Systems 30


2.6 Discrete Time Systems 34


2.7 Analysis of Shift-Invariant Linear System 39


2.8 The Convolution Sum 41


2.9 Systems Described by Difference Equations 44


2.10 Impulse Response to a System 51


2.11 Examples of Some Discrete Time Systems 53


2.12 Chapter Summary 57


3 Discrete Time Systems in the Frequency Domain 59


3.1 Introduction 59


3.2 Continuous Time Fourier Transform 59


3.3 Sampling an Analog Signal 61


3.4 Discrete Time Fourier Transform 62


3.5 Sampling a Continuous Time Signal 71


3.6 Discrete Fourier Transform 80


3.7 Properties of the DFT 89


3.8 Theorems of the DFT 93


3.9 DFT of Real Sequences 100


3.10 Convolution of Very Long Sequences 102


3.11 Chapter Summary 107


4 The Z-Transform 109


4.1 Introduction 109


4.2 Definition of the Z-Transform 109


4.3 The Inverse Z-Transform 115


4.4 Theorems and Properties of the Z-Transform 123


4.5 Application of Z-Transforms to Systems 129


4.6 Responses to Typical Pole–Zero Patterns 130


4.7 Introduction to Two-Dimensional Z-Transform 134


4.8 Chapter Summary 135


5 Discrete Filter Design Techniques 137


5.1 Introduction 137


5.2 Design of Analog Filters: A Review 138


5.3 Design of IIR Filters from Analog Filters 150


5.4 Design of FIR Filters 163


5.5 Design of Windows 179


5.6 FIR Filter Design Using Optimization Techniques 182


5.7 Chapter Summary 185


6 Computing the DFT 187


6.1 Introduction 187


6.2 Direct Computation of the DFT 188


6.3 The Goertzel Algorithm 190


6.4 Decimation in Time Algorithm 191


6.5 Decimation in Frequency Algorithm 197


6.6 Algorithm when N is a Composite Number 202


6.7 Computing the FFT of Only a Few Samples 204


6.8 The Chirp Z-Algorithm 207


6.9 Chapter Summary 213


7 Multirate Signal Processing and Devices 215


7.1 Introduction 215


7.2 Time Domain Characteristics of the Sampling Rate Alteration Devices 216


7.3 Frequency Domain Characteristics of the Sampling Rate Alteration Devices 219


7.4 Basic Sampling Rate Converters 227


7.5 Polyphase Decomposition 234


7.6 Computationally Efficient Interpolator and Decimator 236


7.7 Half Band and Nyquist Filters 243


7.8 Chapter Summary 245


8 Introduction to Stochastic Processes 247


8.1 Introduction 247


8.2 Types of Random Variables, Expected Value, and Moments 248


8.3 Correlation and Covariance 251


8.4 The Notion of the Stochastic Process 254


8.5 The Correlation Matrix 260


8.6 White Noise Process 263


8.7 Stochastic Process through a Linear Shift-Invariant Filter 263


8.8 Stochastic Models 266


8.9 Chapter Summary 273


9 Weiner Filters 276


9.1 Introduction 276


9.2 The Principle of Orthogonality 276


9.3 Weiner–Hopf Equations 279


9.4 Solution of the Weiner–Hopf Equations in the Time Domain 280


9.5 Solution of the Weiner–Hopf Equations in the Frequency Domain 284


9.6 Canonical Form of the Error Surface 295


9.7 Weiner Filters with Additional Constraints 297


9.8 Chapter Summary 299


10 Adaptive Filters 301


10.1 Introduction 301


10.2 Adaptive Direct Form FIR Filters 305


10.3 The Gradient Algorithm 306


10.4 Other Related Stochastic Gradient Algorithms 311


10.5 Properties of the Gradient Algorithms 314


10.6 The Recursive Least Squares Algorithm 321


10.7 Chapter Summary 332


Further Reading 334


Appendix A: Mathematical Identities 337


Appendix B: Transform Tables 339


B.1 Fourier Series 339


B.2 Fourier Transform 340


B.3 Laplace Transform 341


B.4 Z-Transform 342


B.5 Discrete Fourier Transform 344


Appendix C: Introduction to MATLAB 345


C.1 Introduction 345


C.2 Numbers and Data Representation 345


C.3 Control Flow 347


C.4 Special Operators and Predefined Variables 349


C.5 Drawing Plots in MATLAB 349


C.6 Some Special Commands Used in this Book 350


Index 353