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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 Fi