1 BACKGROUND

1.1 Complex Numbers
1.1-1 A Historical Note
1.1-2 Algebra of Complex Numbers
1.2 Sinusoids and Exponentials
1.2-1 Addition of Sinusoids
1.2-2 Sinusoids in Terms of Exponentials
1.2-3 Monotonic Exponentials
1.2-4 The Exponentially Varying Sinusoid
1.3 Cramer's Rule
1.4 Partial Fraction Expansion
1.4-1 Method of Clearing Fractions
1.4-2 The Heaviside "Cover-Up" Method
1.4-3 Repeated Factors of Q(x)
1.4-4 A Combination of Heaviside "Cover-Up" and Clearing Fractions
1.4-6 Modified Partial Fractions
1.5 Vectors and Matrices
1.5-1 Some Definitions and Properties
1.5-2 Matrix Algebra
1.6 MATLAB: Elementary Operations
1.6-1 MATLAB Overview
1.6-2 Calculator Operations
1.6-3 Vector Operations
1.6-4 Simple Plotting
1.6-5 Element-by-Element Operations
1.6-6 Matrix Operations
1.6-7 Partial Fraction Expansions
1.7 Appendix: Useful Mathematical Formulas
1.7-1 Some Useful Constants
1.7-2 Complex Numbers
1.7-3 Sums
1.7-4 Taylor and Maclaurin Series
1.7-5 Power Series
1.7-6 Trigonometric Identities
1.7-7 Common Derivative Formulas
1.7-8 Indefinite Integrals
1.7-9 L'Hôpital's Rule
1.7-10 Solution of Quadratic and Cubic Equations
References
Problems

2 SIGNALS AND SYSTEMS
2.1 Size of a Signals
2.1-1 Signal Energy
2.1-2 Signal Power
2.2 Some Useful Signal Operations
2.2-1 Time Shifting
2.2-2 Time Scaling
2.2-3 Time Reversal
2.2-4 Combined Operations
2.3 Classification of Signals
2.3-1 Continuous-Time and Discrete-Time Signals
2.3-2 Analogue and Digital Signals
2.3-3 Periodic and Aperiodic Signals
2.3-4 Energy and Power Signals
2.3-5 Deterministic and Random Signals
2.4 Some Useful Signal Models
2.4-1 The Unit Step Function u(t)
2.4-2 The Unit Impulse Function ?(t)
2.4-3 The Exponential Function est
2.5 Even and Odd Functions
2.5-1 Some Properties of Even and Odd Functions
2.5-2 Even and Odd Components of a Signal
2.6 Systems and System Classification
2.6-1 Classification of Systems
2.6-2 Linear and Nonlinear Systems
2.6-3 Time-Invariant and Time-Varying Systems
2.6-4 Instantaneous and Dynamic Systems
2.6-5 Causal and Noncausal Systems
2.6-6 Continuous-Time and Discrete-Time Systems
2.6-7 Analogue and Digital Systems
2.6-8 Invertible and Noninvertible Systems
2.6-9 Stable and Unstable Systems
2.7 System Model: Input-Output Description
2.7-1 Electrical Systems
2.7-2 Mechanical Systems
2.7-3 Electromechanical Systems
2.8 Internal and External Descriptions of a System
2.8-1 Internal Description: The State-Space Description
2.9 MATLAB: Working with Functions
2.9-1 Anonymous Functions
2.9-2 Relational Operators and the Unit Step Functions
2.9-3 Visualising Operations on the Independent Variable
2.9-4 Numerical Integration and Estimating Signal Energy
2.10 Summary
References
Problems

3 TIME-DOMAIN ANALYSIS OF CONTINUOUS-TIME SYSTEMS
3.1 Introduction
3.2 System Response to Internal Conditions: The Zero-Input Response
3.2-1 Some Insights into the Zero-Input Behaviour of a System
3.3 The Unit Impulse Response h(t)
3.4 System Response to External Input: The Zero-State Response
3.4-1 The Convolution Integral
3.4-2 Graphical Understanding of Convolution Operation
3.4-3 Interconnected Systems
3.4-4 A Very Special Function for LTIC Systems:
The Everlasting Exponential est
3.4-5 Total Response
3.5 System Stability
3.5-1 External (BIBO) Stability
3.5-2 Internal (Asymptotic) Stability
3.5-3 Relationship Between BIBO and Asymptotic Stability
3.6 Intuitive Insights into System Behaviour
3.6-1 Dependence of System Behaviour on Characteristic Modes
3.6-2 Response Time of a System: The System Time Constant
3.6-3 Time Constant and Rise Time of a System
3.6-4 Time Constant and Filtering
3.6-5 Time Constant and Pulse Dispersion (Spreading)
3.6-6 Time Constant and Rate of Information Transmission
3.6-7 The Resonance Phenomenon
3.7 MATLAB: M-Files
3.7-1 Script M-Files
3.7-2 Function M-Files
3.7-3 For-Loops
3.7-4 Graphical Understanding of Convolution
3.8 Appendix: Determining the Impulse Response
3.9 Summary
References
Problems

4 TIME-DOMAIN ANALYSIS OF DISCRETE-TIME SYSTEMS
4.1 Introduction
4.1-1 Size of a Discrete-Time Signal
4.1-2 Useful Signal Operations
4.2 Some Useful Discrete-Time Signal Models
4.2-1 Discrete-Time Impulse Function ?[n]
4.2-2 Discrete-Time Unit Step Function u[n]
4.2-3 Discrete-Time Exponential ? n
4.2-4 Discrete-Time Sinusoid cos('04n+?)
4.2-5 Discrete-Time Complex Exponential ej'04n
4.3 Examples of Discrete-Time Systems
4.3-1 Classification of Discrete-Time Systems
4.4 Discrete-Time System Equations
4.4-1 Recursive (Iterative) Solution of Difference Equation
4.5 System Response to Internal Conditions: The Zero-Input Response
4.6 The Unit Impulse Response h[n]
4.6-1 The Closed-Form Solution of h[n]
4.7 System Response to External Input: The Zero-State Response
4.7-1 Graphical Procedure for the Convolution Sum
4.7-2 Interconnected Systems
4.7-3 Total Response
4.8 System Stability and Behaviour
4.8-1 External (BIBO) Stability
4.8-2 Internal (Asymptotic) Stability
4.8-3 Relationship Between BIBO and Asymptotic Stability
4.8-4 Intuitive Insights into System Behaviour
4.9 MATLAB: Discrete-Time Signals and Systems
4.9-1 Discrete-Time Functions and Stem Plots
4.9-2 System Responses Through Filtering
4.9-3 A Custom Filter Function
4.9-4 Discrete-Time Convolution
4.10 Appendix: Impulse Response for a Special Case
4.11 Summary
Problems

5 CONTINUOUS-TIME SYSTEM ANALYSIS USING THE
LAPLACE TRANSFORM
5.1 The Laplace Transform
5.2 Some Properties of the Laplace Transform
5.2-1 Time Shifting
5.2-2 Frequency Shifting
5.2-3 The Time-Differentiation Property
5.2-4 The Time-Integration Property
5.2-5 The Scaling Property
5.2-6 Time Convolution and Frequency Convolution
5.3 Solution of Differential and Integro-Differential Equations
5.3-1 Comments on Initial Conditions at 0? and at 0+
5.3-2 Zero-State Response
5.3-3 Stability
5.3-4 Inverse Systems
5.4 Analysis of Electrical Networks: The Transformed Network
5.4-1 Analysis of Active Circuits
5.5 Block Diagrams and System Realisations
5.5-1 Direct Form I Realisation
5.5-2 Direct Form II Realisation
5.5-3 Cascade and Parallel Realisations
5.5-4 Transposed Realisation
5.5-5 Using Operational Amplifiers for System Realisation
5.5-6 Application to Feedback and Controls
5.5-7 Analysis of a Simple Control System
5.6 Frequency Response of an LTIC System
5.6-1 Steady-State Response to Causal Sinusoidal Inputs
5.7 Bode Plots
5.7-1 Constant Ka1a2/b1b3
5.7-2 Pole (or Zero) at the Origin
5.7-3 First-Order Pole (or Zero)
5.7-4 Second-Order Pole (or Zero)
5.7-5 The Transfer Function from the Frequency Response
5.8 Filter Design by Placement of Poles and Zeros of H(s)
5.8-1 Dependence of Frequency Response on Poles and Zeros of H(s)
5.8-2 Lowpass Filters
5.8-3 Bandpass Filters
5.8-4 Notch (Bandstop) Filters
5.8-5 Practical Filters and Their Specifications
5.9 The Bilateral Laplace Transform
5.9-1 Properties of the Bilateral Laplace Transform
5.9-2 Using the Bilateral Transform for Linear System Analysis
5.10 MATLAB: Continuous-Time Filters
5.10-1 Frequency Response and Polynomial Evaluation
5.10-2 Butterworth Filters and the Find Command
5.10-3 Using Cascaded Second-Order Sections for Butterworth Filter Realisation
5.10-4 Chebyshev Filters
5.11 Summary
References
Problems

6 DISCRETE-TIME SYSTEM ANALYSIS USING THE Z-TRANSFORM
6.1 The z-Transform
6.1-1 Inverse Transform by Partial Fraction Expansion and Tables
6.1-2 Inverse z-Transform by Power Series Expansion
6.2 Some Properties of the z-Transform
6.2-1 Time-Shifting Properties
6.2-2 z-Domain Scaling Property (Multiplication by ? n)
6.2-3 z-Domain Differentiation Property (Multiplication by n)
6.2-4 Time-Reversal Property
6.2-5 Convolution Property
6.3 z-Transform Solution of Linear Difference Equations
6.3-1 Zero-State Response of LTID Systems: The Transfer Function
6.3-2 Stability
6.3-3 Inverse Systems
6.4 System Realisation
6.5 Frequency Response of Discrete-Time Systems
6.5-1 The Periodic Nature of Frequency Response
6.5-2 Aliasing and Sampling Rate
6.5-3 Frequency Response from Pole-Zero Locations
6.6 Digital Processing of Analogue Signals
6.7 The Bilateral z-Transform
6.7-1 Properties of the Bilateral z-Transform
6.7-2 Using the Bilateral z-Transform for Analysis of LTID Systems
6.7-3 Connecting the Laplace and z-Transforms
6.8 MATLAB: Discrete-Time IIR Filters
6.8-1 Frequency Response and Pole-Zero Plots
6.8-2 Transformation Basics
6.8-3 Transformation by First-Order Backward Difference
6.8-4 Bilinear Transformation
6.8-5 Bilinear Transformation with Prewarping
6.8-6 Example: Butterworth Filter Transformation
6.8-7 Problems Finding Polynomial Roots
6.8-8 Using Cascaded Second-Order Sections to Improve Design
6.9 Summary
References
Problems

7 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER SERIES
7.1 Periodic Signal Representation by Trigonometric Fourier Series
7.1-1 The Fourier Spectrum
7.1-2 The Effect of Symmetry
7.1-3 Determining the Fundamental Frequency and Period
7.2 Existence and Convergence of the Fourier Series
7.2-1 Convergence of a Series
7.2-2 The Role of Amplitude and Phase Spectra in Waveshaping
7.3 Exponential Fourier Series
7.3-1 Exponential Fourier Spectra
7.3-2 Parseval’s Theorem
7.3-3 Properties of the Fourier Series
7.4 LTIC System Response to Periodic Inputs
7.5 Generalised Fourier Series: Signals as Vectors
7.5-1 Component of a Vector
7.5-2 Signal Comparison and Component of a Signal
7.5-3 Extension to Complex Signals
7.5-4 Signal Representation by an Orthogonal Signal Set
7.6 MATLAB: Fourier Series Applications
7.6-1 Numerical Computation of Dn
7.6-2 Periodic Functions and the Gibbs Phenomenon
7.6-3 Optimisation and Phase Spectra
7.7 Summary
References
Problems

8 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER TRANSFORM
8.1 Aperiodic Signal Representation by the Fourier Integral
8.1-1 Physical Appreciation of the Fourier Transform
8.2 Transforms of Some Useful Functions
8.2-1 Connection Between the Fourier and Laplace Transforms
8.3 Some Properties of the Fourier Transform
8.4 Signal Transmission Through LTIC Systems
8.4-1 Signal Distortion During Transmission
8.4-2 Bandpass Systems and Group Delay
8.4-3 Ideal and Practical Filters
8.5 Signal Energy
8.6 Application to Communications: Amplitude Modulation
8.6-1 Double-Sideband, Suppressed-Carrier (DSB-SC) Modulation
8.6-2 Amplitude Modulation (AM)
8.6-3 Single-Sideband Modulation (SSB)
8.6-4 Frequency-Division Multiplexing
8.7 Data Truncation: Window Functions
8.7-1 Using Windows in Filter Design
8.8 MATLAB: Fourier Transform Topics
8.8-1 The Sinc Function and the Scaling Property
8.8-2 Parseval’s Theorem and Essential Bandwidth
8.8-3 Spectral Sampling
8.8-4 Kaiser Window Functions
8.9 Summary
References
Problems

9 SAMPLING: THE BRIDGE FROM CONTINUOUS TO DISCRETE
9.1 The Sampling Theorem
9.1-1 Practical Sampling
9.2 Signal Reconstruction
9.2-1 Practical Difficulties in Signal Reconstruction
9.2-2 Some Applications of the Sampling Theorem
9.3 Analogue-to-Digital (A/D) Conversion
9.4 Dual of Time Sampling: Spectral Sampling
9.5 Numerical Computation of the Fourier Transform: The Discrete
Fourier Transform
9.5-1 Some Properties of the DFT
9.5-2 Some Applications of the DFT
9.5-3 The Fast Fourier Transform (FFT)
9.6 MATLAB: The Discrete Fourier Transform
9.6-1 Computing the Discrete Fourier Transform
9.6-2 Improving the Picture with Zero Padding
9.6-3 Quantisation
9.7 Summary
References
Problems

10 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS
10.1 Discrete-Time Fourier Series (DTFS)
10.1-1 Periodic Signal Representation by Discrete-Time Fourier Series
10.1-2 Fourier Spectra of a Periodic Signal x[n]
10.2 Aperiodic Signal Representation by Fourier Integral
10.2-1 Nature of Fourier Spectra
10.2-2 Connection Between the DTFT and the z-Transform
10.3 Properties of the DTFT
10.4 LTI Discrete-Time System Analysis by DTFT
10.4-1 Distortionless Transmission
10.4-2 Ideal and Practical Filters
10.5 DTFT Connection with the CTFT
10.5-1 Use of DFT and FFT for Numerical Computation of the DTFT
10.5-2 Generalisation of the DTFT to the z-Transform
10.6 MATLAB: Working with the DTFS and the DTFT
10.6-1 Computing the Discrete-Time Fourier Series
10.6-2 Measuring Code Performance
10.6-3 FIR Filter Design by Frequency Sampling
10.7 Summary
Reference
Problems

11 STATE-SPACE ANALYSIS
11.1 Mathematical Preliminaries
11.1-1 Derivatives and Integrals of a Matrix
11.1-2 The Characteristic Equation of a Matrix: The Cayley-Hamilton Theorem
11.1-3 Computation of an Exponential and a Power of a Matrix
11.2 Introduction to State Space
11.3 A Systematic Procedure to Determine State Equations
11.3-1 Electrical Circuits
11.3-2 State Equations from a Transfer Function
11.4 Solution of State Equations
11.4-1 Laplace Transform Solution of State Equations
11.4-2 Time-Domain Solution of State Equations
11.5 Linear Transformation of a State Vector
11.5-1 Diagonalisation of Matrix A
11.6 Controllability and Observability
11.6-1 Inadequacy of the Transfer Function Description of a System
11.7 State-Space Analysis of Discrete-Time Systems
11.7-1 Solution in State Space
11.7-2 The z-Transform Solution
11.8 MATLAB: Toolboxes and State-Space Analysis
11.8-1 z-Transform Solutions to Discrete-Time, State-Space Systems
11.8-2 Transfer Functions from State-Space Representations
11.8-3 Controllability and Observability of Discrete-Time Systems
11.8-4 Matrix Exponentiation and the Matrix Exponential
11.9 Summary
References
Problems

Index