Rudiments of Signal Processing and Systems

Description

This book is intended to be a little different from other books in its coverage. There are a great many digital signal processing (DSP) books and signals and systems books on the market. Since most undergraduate courses begin with signals and systems and then move on in later years to DSP, I felt a need to combine the two into one book that was concise yet not too overburdening. This means that students need only purchase one book instead of two and at the same time see the flow of knowledge from one subject into the next. Like the rudiments of music, it starts at the very beginning with some elementary knowledge and builds on it chapter by chapter to advanced work by chapter 15. I have been teaching now for 38 years and always think it necessary to credit the pioneers of the subjects we teach and ask the question “How did we get to this present stage in technological achievement”? Therefore, in Chapter 1 I have given a concise history trying to not sway too much away from the subject area. This is followed by the rudimentary theory in increasing complexity. It has already been taught successfully to a class at Auckland University of Technology New Zealand. Preface Contents 1 Introduction and Basic Signal Properties 1.1 Brief History of the Subject from a Personal Perspective 1.2 Basic Signals and Properties 1.3 Signal Operations 1.4 Signal Symmetry 1.4.1 Writing Any Signal as the Sum of Even and Odd Signals 1.5 Causal and Non-causal Signals 1.6 Signals of Special Importance References 2 Dynamic Systems Introduction 2.1 Definition of a Linear System 2.1.1 Linear System Definition 2.1.2 Examples of Linear and Nonlinear Systems 2.2 Definition of Time-Invariant System 2.2.1 Examples of Time-Invariance 2.3 Linear Time Invariant (LTI) Systems 2.4 Laplace Transforms 2.4.1 Laplace Transform of Common Signals. 2.4.2 Laplace Transform of a First Derivative 2.4.3 Laplace Transform of a Second Derivative 2.4.4 Laplace Transform of a System and Its Transfer-Function 2.4.5 Table of Laplace Transforms 2.4.6 Multiple Connected Systems 2.4.7 Example of Step Response, First-Order System 2.4.8 Higher Order Systems 2.5 Impulse Response Reference 3 Further Introductory Topics in Signals and Systems 3.1 Convolution Introduction 3.1.1 Examples of Convolution 3.2 Convolution by Parts (or Piecewise Convolution) 3.2.1 Example 1. Convolution by Parts 3.2.2 Example 2. Convolution by Parts 3.2.3 Example 3. Pulse of Duration 1 s Applied to an RC Network 3.3 Heaviside Piecewise Laplace Transform Method 3.3.1 Example of Heaviside Laplace Method for an RC Circuit 3.4 Laplace Transform of Convolution 3.5 Stability of LTI Systems 3.5.1 Introduction to Poles and Zeros of Transfer Functions 3.5.2 Final Value Theorem in Laplace Transforms Reference 4 Frequency-Domain Properties of Signals 4.1 Fourier Series for Periodic Waveforms 4.1.1 Example1. A Square Wave with Even Symmetry 4.1.2 Half-Range Fourier Series 4.1.3 Example 2. A Square Wave Being Neither Even nor Odd Symmetry 4.1.4 Example of an Even Triangular Waveform 4.1.5 Fourier Series of Rectified Cosine Waveform 4.2 Complex Fourier Series 4.2.1 Example: Square Wave in Complex Fourier Form 4.2.2 Complex Fourier Series of a Triangular Waveform 4.2.3 Complex Fourier Series of an Impulse Train 4.2.4 More on Square Waves 4.3 The Fourier Transform 4.3.1 The Ideal Filter 4.3.2 Transforms of Common Signals 4.3.3 Some Properties of the FT 4.3.4 Table of Common Fourier Transforms 4.4 Parseval’s Theorem 4.4.1 Example Using Parseval’s Theorem Reference 5 Sampling of Signals and Discrete Mathematical Methods 5.1 The Sampling Theorem 5.2 Zero-Order Hold (ZOH) 5.3 Aliasing of Signals and Prevention 5.3.1 Anti-Aliasing Filtering 5.4 The Z-Transform 5.4.1 Z-Transforms of Common Signals 5.4.2 Table of Unilateral Z-Transforms 5.4.3 Inverse Z-Transforms 5.4.4 The Bilateral Z-Transform 5.4.5 Discrete-Time Final Value Theorem 5.5 Finite Difference Equations 5.5.1 Steady-State Output of a Difference Equation References 6 Properties of Discrete-Time Systems and Signals 6.1 Stability of Discrete-Time Systems 6.2 Impulse Response and Convolution 6.2.1 Examples of Discrete-Time Convolution 6.2.2 Z-Transform of Discrete-Time Convolution 6.3 Frequency Response of Discrete-Time Systems 6.3.1 Example: Calculation of Frequency Response 6.3.2 Example: Frequency Response of a Finite-Impulse Response Filter (FIR Filter) 6.3.3 Design of a Simple Digital Notch Filter 6.4 The Discrete-Time Fourier Transform (DTFT) 6.4.1 DTFT of a Delayed Impulse by m Samples 6.4.2 DTFT of a Geometric Decay 6.4.3 DTFT of a Delayed Signal 6.4.4 DTFT of a Discrete-Time Pulse 6.4.5 DTFT Properties 7 A More Complete Picture 7.1 Link Between Various Methods 7.2 Conversion of G(s) to G(Z) 7.2.1 Example of Using the Bilinear Transform or Trapezoidal Integration 7.3 Frequency Warping of the Bilinear Transform 7.4 Impulse Invariance Method 7.5 Analogue and Digital Lowpass Butterworth IIR Filters 7.6 Using the Z-Transform Tables Directly 7.7 Pole-Zero Mapping or Matched Z-Transform Method 7.8 Implementing Difference Equations 7.9 The Discrete Fourier Transform (DFT) 7.9.1 DFT Example 1 7.9.2 DFT Example 2 7.9.3 DFT Example 3. Inverse DFT of an Impulse in the Frequency Domain 7.9.4 DFT of a Cosine mathcalF{ cos( 2nNko ) },n = 0,1,2 N - 1 7.9.5 Matrix Form of the DFT 7.10 The Fast Fourier Transform (FFT) 7.10.1 Spectral Leakage and Windowing References 8 FIR Filter Design 8.1 Definition of Linear Phase 8.2 Frequency Sampling Method of FIR Filter Design 8.2.1 Example: Design a Lowpass Filter with a Passband Frequency of One Quarter Sampling Frequency 8.2.2 Example: Design of a Bandpass Filter 8.2.3 Example. Design of an FIR Band Limited Differentiator 8.2.4 Z-Transform of IIR Frequency Sampling Filters 8.3 Interpolation Method of FIR Design 8.4 Outline of Optimal Design of FIR Filters References 9 State-Space Method of System Modelling 9.1 Motivation 9.2 Realizations 9.3 Solution of the State Equation 9.4 Poles of the State-Space System 9.5 State-Space Descriptions with Zeros 9.6 Controllability and Observability 9.7 Discrete-Time State-Space 9.8 Similarity Transformations 9.9 States as Measurable Signals 9.10 Conversion from Continuous to Discrete Time State-Space References 10 Toeplitz Convolution Matrix Method 10.1 Preliminaries 10.2 LTT Matrix Properties for Dynamic Systems 10.3 Inverse of a LTT Matrix Using Two FFTs References 11 FIR Wiener Filters and Random Signals 11.1 Motivation 11.2 Mathematical Preliminaries 11.2.1 Autocovariance or Autocorrelation 11.2.2 Autoregressive (AR) Time-Series Model 11.2.3 Moving Average Time-Series Model 11.2.4 Autoregressive Moving Average Time-Series Model 11.2.5 Colouring by Filtering White Noise 11.2.6 Contour Integration of Laurent Series 11.2.7 Vectors and Random Signals 11.2.8 Differentiation of Vector Products 11.3 The FIR Wiener Filtering Problem 11.3.1 FIR Wiener Filter Example with Additive White Noise 11.3.2 FIR Wiener Filter Improvements 11.4 The FIR Smoothing and Prediction Problem References 12 IIR Wiener Filtering, Smoothing and Prediction 12.1 Preliminaries and Statement of Problem 12.2 The IIR Wiener Filter 12.2.1 Wiener Filter Example 12.2.2 Innovations Form of the Wiener Filter 12.2.3 The Smoothing and Prediction Problems 12.3 Optimal Wiener IIR Filter, Smoother and Predictor for Additive Coloured Noise 12.3.1 Optimal Coloured Noise Smoother 12.3.2 Optimal Coloured Noise Predictor References 13 FIR Wiener Filters Using Lower Triangular Toeplitz Matrices 13.1 Preliminaries 13.2 LTT Model Description 13.3 The Estimation Problem 13.3.1 Estimation Error 13.3.2 Spectral Factorization of Toeplitz Matrices 13.3.3 Optimal Filter, Smoother or Predictor 13.4 The Noncausal Toeplitz Filtering Solution References 14 Adaptive Filters 14.1 Overview and Motivation 14.2 The Least-Mean Squares (LMS) Method 14.3 LMS for Wiener Estimators 14.4 Choice of Step Size and Normalized LMS 14.5 Adaptive Noise Cancellation 14.5.1 Example with Noisy Speech 14.6 Adaptive Noise Cancellation, Method 2 14.6.1 Example with Noisy Speech 14.7 Two Input Beamformer 14.8 The Symmetric Adaptive Decorrelator (SAD) 14.8.1 Example of Two Mixed Signals 14.9 Independent Component Analysis. (ICA), Some Brief Notes References 15 Other Common Recursive Estimation Methods 15.1 Motivation 15.2 The Recursive Least Squares (RLS) Method 15.3 The Kalman Filter 15.3.1 Kalman Filter Example 15.4 The Kalman Filter Used in System Identification 15.4.1 Illustrative Example 15.5 The LMS Algorithm with Optimal Gains 15.5.1 Newtons Method 15.5.2 White Driving Noise Case [11] 15.6 LMS with Coloured Driving Noise. Toeplitz Based LMS References