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.
Author(s): Tom J. Moir
Publisher: Springer
Year: 2021
Language: English
Pages: 466
City: Cham
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
