## Table of contents for Fundamentals of signals and systems / M.J. Roberts.

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```Fundamentals of Signals and Systems
Chapter 1 - Introduction
1.1	Signals and Systems Defined	1-1
1.2	Types of Signals	1-2
Continuous-Time, Continuous-Value, Discrete-Time, Discrete-
Value, Random
1.3	Examples of Systems	1-8
A Mechanical System
Differential Equations
A Fluid System
A Discrete-Time System
Difference Equations
Feedback Systems
Stability
1.4	A Familiar Signal and System Example	1-16
Power Spectrum, Signal-to-Noise Ratio
1.5	Use of MATLAB	1-21
Chapter 2 - Mathematical Description of Continuous-Time Signals
2.1	Introduction and Goals	2-1
2.2	Functions and Functional Notation	2-2
Domain, Range
2.3	Signal Functions	2-2
Continuous-Time Functions
Complex Exponentials and Sinusoids
Cyclic and Radian Frequency
Functions with Discontinuities
Singularity Functions and Related Functions
The Unit Step Function
The Signum Function
The Unit Ramp Function
The Unit Impulse
Generalized Derivatives, Equivalence, Sampling and
Scaling
Properties
The Periodic Impulse or Impulse Train
A Coordinated Notation for Singularity Functions
The Unit Rectangle Function
The Unit Triangle Function
The Unit Sinc Function
The Dirichlet Function
MATLAB m-Files for Some Singularity Functions and
Related Functions
2.4	Functions and Combinations of Functions	2-24
Functional Notation - Conventions and Conversions
Combinations of Functions
2.5	Scaling and Shifting	2-26
Amplitude Scaling
Time Shifting
Time Scaling
Simultaneous Scaling and Shifting
Significance of Sequence
2.6	Differentiation and Integration	2-39
Antiderivative, Indefinite Integral, Definite Integral,
Cumulative Integral
2.7	Even and Odd Functions	2-42
Sums, Products, Differences, Quotients, Derivatives, Integrals
2.8	Periodic Functions		2-46
Least Common Multiple, Greatest Common Divisor
2.9	Signal Energy and Power	2-50
Energy Signals, Power Signals
2.10	Summary of Important Points	2-55
Exercises				2-56
Chapter 3 - Mathematical Description of Discrete-Time Signals
3.1	Introduction and Goals	3-1
3.2	Signal Functions	3-2
Sampling and Discrete Time
Uniform Sampling, Sequential-State Machines
Exponentials and Sinusoids
Periods, Normalized Frequency
Singularity Functions
The Unit Impulse or Kronecker Delta Function
The Unit Sequence
The Signum Function
The Rectangle Function
The Unit Ramp Function
The Periodic Impulse or Impulse Train
3.3	Scaling and Shifting	2-68
Amplitude Scaling
Time Shifting
Time Scaling
Compression, Expansion, Decimation, Interpolation
3.4	Differencing and Accumulation	3-22
Approximation of Derivatives, Forward, Backward and Central
Differences
3.5	Even and Odd Functions	3-26
Sums, Products, Differences and Quotients
Accumulation
3.6	Periodic Functions	3-30
3.7	Signal Energy and Power	3-31
3.8	Summary of Important Points	3-34
Exercises	3-36
Chapter 4 - Properties of Continuous-Time Systems
4.1	Introduction and Goals	4-1
4.2	Block Diagrams and System Terminology 	4-2
Components, Amplifiers, Summing Junctions, Integrators
4.3	System Modeling	4-7
Differential Equations, Model Complexity, Feedback and
Stability
4.4	System Properties	4-14
Introductory Example
Zero-Input Response, Zero-State Response
Homogeneity
Time Invariance
Linearity and Superposition
Linear, Time-Invariant Systems, Non-Linearity and
Linearization
Stability
Bounded-Input-Bounded-Output
Incremental Linearity
Causality
Memory
Static and Dynamic Systems
Static Non-Linearity
Invertibility
Inverse System
4.5	Eigenfunctions of LTI Systems	4-38
Natural Radian Frequency, Damping Factor, Critical Radian
Frequency,
Damping Ratio, Complex Exponential Excitation
4.6	Summary of Important Points	4-41
Exercises	4-42
Chapter 5 - Properties of Discrete-Time Systems
5.1	Introduction and Goals	5-1
5.2	Block Diagrams and System Terminology 	5-1
Delay
5.3	System Modeling	5-2
Approximation of Differential Equations with Difference
Equations, Feedback,
Oscillation, Stability
5.4	System Properties	5-10
Stability Criterion
5.5	Eigenfunctions of LTI Systems	5-14
Complex Exponential Excitation
5.6	Summary of Important Points	5-14
Exercises	5-15
Chapter 6 - Time-Domain Analysis of Continuous-Time Systems
6.1	Introduction and Goals	6-1
6.2	The Convolution Integral	6-1
Impulse Response
Calculation from Differential Equations
Convolution
Derivation from Rectangular-Pulse Approximation
Convolution Properties
Convolution with an Impulse, Graphical Convolution Methods, Time
Shifting, Commutativity, Associativity, Distributivity, Differentiation,
Area, Scaling, Convergence of Convolution Integral
System Interconnections
Stability and Impulse Response
Responses of Systems to Standard Signals
Unit-Step Response, Complex Exponential Response,
Transfer
Function, Frequency Response
6.3	Block-Diagram Realization of Differential Equations		6-31
Preference for Integrators over Differentiators, Direct Forms
1 and 2
6.4	Summary of Important Points	6-37
Exercises	6-39
Chapter 7 - Time-Domain Analysis of Discrete-Time Systems
7.1	Introduction and Goals	7-1
7.2	The Convolution Sum		7-1
Impulse Response
Calculation from Difference Equations
Convolution
Development, Graphical Method, Array Method
Convolution Properties
Convolution with an Impulse, Time Shifting, Commutativity,
Associativity, Distributivity, Differencing, Sum, Convergence of
Convolution Sum
Numerical Convolution
Approximating the Convolution Integral
System Interconnections
Stability and Impulse Response
Responses to Standard Signals
Unit-Sequence Response, Complex Exponential Response,
Transfer
Function, Frequency Response
7.3	Block-Diagram Realization of Difference Equations		7-31
Direct Forms 1 and 2
7.4	Summary of Important Points	7-35
Exercises	7-36
Chapter 8 - The Continuous-Time Fourier Series
8.1	Introduction and Goals	8-1
8.2	Periodic Excitation and Response of LTI Systems	8-2
8.3	Basic Concepts and Development of the Fourier Series	8-5
Linearity and Complex-Exponential Excitation
Illustrative Example, Trigonometric and Complex Forms
Derivation of the Fourier Series
Inner Products and Orthogonality,
Limitations on Functions Representable by a Fourier Series
The Dirichlet Conditions
The Trigonometric Fourier Series
Relation to Complex Form
Cyclic-Frequency and Radian-Frequency Forms
Periodicity of Fourier Series Representations
8.4	Calculation of the Fourier Series		8-19
Sinusoidal Signals
Non-sinusoidal Signals
The Fourier Series of Periodic Signals over an Integer Number
of
Fundamental Periods
Minimum Mean-Squared Error of Fourier-Series Partial Sums
The Fourier Transform of Even and Odd Periodic Signals
8.5	Numerical Computation of the Fourier Series		8-32
The Discrete Fourier Transform (DFT)
8.6	Convergence of the Fourier Series	8-38
Continuous Signals
Signals with Discontinuities and The Gibbs Phenomenon
8.7	 Properties of the Fourier Series	8-42
Linearity
Time Shifting
Frequency Shifting
Time Reversal
Time Scaling
Change of Period
Time Differentiation
Time Integration
Effect of Average Value
Multiplication-Convolution Duality
Periodic Convolution Integral
Conjugation
Parseval?s Theorem
8.8	Use of Tables and Properties		8-60
8.9	Bandlimited Signals		8-63
8.10	Responses of LTI Systems with Periodic Excitation		8-68
Harmonic Response
8.11	Summary of Important Points	8-70
Exercises	8-71
Chapter 9 - The Discrete-Time Fourier Series
9.1	Introduction and Goals	9-1
9.2	Periodic Excitation and Response of LTI Systems	9-1
9.3	Basic Concepts and Development of the Fourier Series	9-3
Linearity and Complex-Exponential Excitation
Illustrative Example, Trigonometric and Complex Forms
Derivation of the Fourier Series
Completeness of a Finite Sum, Vector and Matrix Methods,
Orthogonality, Projections, Summation Forms, Relation to the Discrete
Fourier Transform (DFT), Complex and Trigonometric Forms
Representing Periodic Functions with the DTFS
Conservation of Information
9.4	Properties of the Fourier Series	9-30
Linearity, Time Shifting, Frequency Shifting, Conjugation and
Time
Reversal
Time Scaling
Restrictions Stemming from Discrete Time
Change of Period
Multiplication-Convolution Duality
Periodic Convolution Sum
First Backward Difference
Accumulation
Even and Odd Signals
Parseval?s Theorem
9.5	Convergence of the Fourier Series	9-49
9.6	Responses of LTI Systems with Periodic Excitation		9-52
Harmonic Response
9.7	Summary of Important Points		9-57
Exercises	9-58
Chapter 10 - The Continuous-Time Fourier Transform
10.1	Introduction and Goals	10-1
10.2	Aperiodic Excitation and Response of LTI Systems		10-1
Inadequacy of the Fourier Series
10.3	Basic Concepts and Development of the Fourier Transform	10-3
The Transition from the Fourier Series to the Fourier
Transform
Normalization and Notation Changes
Definition of the Fourier Transform
Cyclic and Radian Frequency Forms, Time and Frequency
Domains,
Physical Interpretation
10.4	Convergence and the Generalized Fourier Transform	10-14
Convergence Factor, Negative Frequency,
10.5	Numerical Computation of the Fourier Transform		10-21
10.6	Properties of the Fourier Transform	10-25
Linearity
Time Shifting and Frequency Shifting
Time Scaling and Frequency Scaling
The Uncertainty Principle
Transform of a Conjugate
Even Magnitude and Odd Phase
Multiplication-Convolution Duality
System Interconnection, Frequency Response
Time Differentiation
Transforms of Periodic Signals
Parseval?s Theorem
Integral Definition of an Impulse
Duality
Total-Area Integral Using Fourier Transforms
Time Integration
10.7	Summary of Important Points	10-48
Exercises 10-50
Chapter 11 - The Discrete-Time Fourier Transform
11.1	Introduction and Goals	11-1
11.2	Basic Concepts and Development of the Fourier Transform	11-1
Graphical Illustration
Analytical Derivation
Definition of the Fourier Transform
Cyclic and Radian Frequency Forms, Four Fourier Methods
Comparison Summary
11.3	Convergence of the Fourier Transform	11-6
11.4	Numerical Computation of the Fourier Transform	11-6
11.5	Properties of the Fourier Transform	11-7
Linearity
Time Shifting and Frequency Shifting
Time Scaling and Frequency Scaling
Limitations Due to Discrete Time
Transform of a Conjugate
Differencing and Accumulation
Time Reversal
Multiplication-Convolution Duality
System Interconnection, Frequency Response
Accumulation Definition of a Periodic Impulse Function
Parseval?s Theorem
11.6	Relations Among Fourier Methods	11-24
Discrete-Periodic Duality
CTFT and CTFS
Information Equivalence, Periodic Extension
CTFT and DTFT
Sampling and Impulse Sampling
DTFT and DTFS
Information Equivalence, Periodic Extension
Method Comparison Examples
11.7	Summary of Important Points		11-45
Exercises	11-46
Chapter 12 - Continuous-Time Fourier Transform Analysis of Signals
and Systems
12.1	Introduction and Goals	12-1
12.2	Frequency Response		12-1
Cascade and Parallel Connections, Power Spectrum, Audio
Amplifiers and
Filters, Graphic Equalizer
12.3	Ideal Filters	12-9
Distortion
Distortionless System
Filter Classifications
Lowpass, Highpass, Bandpass, Bandstop
Ideal Filter Frequency Responses
Distortionless in Passband
Bandwidth
Impulse Responses and Causality
Impulse and Frequency Responses of Causal Filters
The Power Spectrum
Noise Removal
12.4	Practical Passive Filters	12-18
The RC Lowpass Filter
Impedance and Voltage Division
The RLC Bandpass Filter
Resonance
Examples of Filters
Automobile Suspension
12.5	Log-Magnitude Frequency-Response Graphs and Bode Diagrams	12-27
Component Diagrams
Poles and Zeros
One-Real-Pole System
One-Real-Zero-System
Integrators and Differentiators
Frequency-Independent Gain
Complex Pole and Zero Pairs
Natural Radian Frequency, Damping Ratio
12.6	Practical Active Filters	12-44
Operational Amplifiers
Inverting and Non-Inverting Amplifiers, Integrator,
Lowpass Filter
12.7	Communication Systems	12-56
Modulation
Problems of Communicating at a Distance, Frequency
Multiplexing
Double-Sideband Suppressed-Carrier Modulation
Sidebands, Demodulation, Synchronous Demodulation, Local
Oscillator
Double-Sideband Transmitted-Carrier Modulation
Envelope Detector, Asynchronous Demodulation,
Overmodulation
Single-Sideband Modulation and Demodulation
Pulse Amplitude Modulation
12.8	Impulse Sampling	12-70
Aliases, The Sampling Theorem, Intrerpolation
12.9	Summary of Important Points	12-76
Exercises	12-78
Chapter 13 - Discrete-Time Fourier Transform Analysis of Signals and
Systems
13.1	Introduction and Goals	13-1
13.2	Ideal Filters	13-1
Distortion
Distortionless System
Filter Classifications
Lowpass, Highpass, Bandpass, Bandstop
Ideal Filter Frequency Responses
Distortionless in Passband
Impulse Responses and Causality
Impulse and Frequency Responses of Causal Filters
Filtering Images
Pixels, Discrete Space, Causality,
13.3	Practical Filters	13-12
Lowpass, Highpass, Bandpass, Bandstop, Responses to Standard
Signals,
Moving Average Filters, Advantages over Continuous-Time Filters
13.4	Summary of Important Points		13-25
Exercises	13-26
Chapter 14 - Sampling and the Discrete Fourier Transform
14.1	Introduction and Goals	14-1
14.2	Representing a Continuous-Time Signal by Samples		14-2
Qualitative Concepts
The Sampling Theorem
Aliases, Nyquist Rate, Oversampling, Undersampling
Aliasing
Anti-Aliasing Filter
Timelimited and Bandlimited Signals
Interpolation
Sinc Function, Zero-Order Hold, First-Order Hold
Sampling a Sinusoid
14.3	Bandlimited Periodic Signals		14-22
Described by a Finite Set of Numbers
14.4	The Discrete Fourier Transform and its Relation to Other Fourier
Methods			14-27
Sampling, Windowing, Frequency-Domain Sampling, Periodic Repetition
Approximating the CTFT with the DFT
Computing the CTFS with the DFT
Computing the DTFS with the DFT
Approximating the DTFT with the DFT
Approximating Continuous-Time Convolution with the DFT
Approximating Discrete-Time Convolution with the DFT
Approximating Continuous-Time Periodic Convolution with the
DFT
Computing Discrete-Time Periodic Convolution with the DFT
14.5	The Fast Fourier Transform		14-36
Number of Complex Multiplications and Additions, Importance of
an Integer
Power of 2 as Number of Points
14.6	Summary of Important Points	14-38
Exercises	14-40
Chapter 15 - The Laplace Transform
15.1	Introduction and Goals	15-1
15.2	Development of the Laplace Transform	15-2
Derivation and Definition
Generalization of Fourier Transform, Natural Result of
Complex-
Exponential Excitation
Region of Convergence
Relation to Causality
The Unilateral Laplace Transform
15.3	Properties of the Laplace Transform		15-16
Linearity
Time Shifting
Complex-Frequency Shifting
Time Scaling
Frequency Scaling
First Time Derivative
Nth Time Derivative
Complex-Frequency Differentiation
Multiplication-Convolution Duality
Integration
Initial Value Theorem
Final Value Theorem
Conditions for Validity
15.4	The Inverse Laplace Transform Using Partial-Fraction Expansion
15-25
Proper Rational Functions of s, Handling Repeated Poles
15.5	Laplace Transform - Fourier Transform Equivalence	15-38
15.6	Solution of Differential Equations with Initial Conditions	15-39
15.7	Transfer Functions from Circuit and System Diagrams	15-41
15.8	System Stability	15-47
Pole Locations, Marginal Stability
15.9	Parallel, Cascade and Feedback Connections		15-50
Loop Transfer Function, Beneficial Effects
15.10	System Responses to Standard Signals	15-56
Unit Step Response
Stable and Unstable First and Second Order Systems,
Damping,
Resonance
Response to a Sinusoid Applied at a Finite Time
Steady-State or Forced Response
15.11	Pole-Zero Diagrams and Graphical Calculation of Frequency Response

15-66
Vector Interpretation, Influences of Poles and Zeros, Lowpass,
Bandpass,
Highpass Examples
15.12	Standard Realizations of Systems	15-77
Canonical Direct-Form 2, Cascade, Parallel
15.13	Summary of Important Points		15-82
Exercises	15-84
Chapter 16 - The z Transform
16.1	Introduction and Goals	16-1
16.2	Development of the z Transform	16-1
Definition and Derivation
Generalization of Discrete-Time Fourier Transform,
Natural Result of
Complex-Exponential Excitation
Region of Convergence
The Unilateral z Transform
16.3	Properties of the z Transform		16-8
Linearity
Time Shifting
Delay and Advance, Relation to Block Diagrams
Change of Scale
Relation to Frequency Shifting
Initial Value Theorem
z-Domain Differentiation
Convolution in Discrete Time
Differencing
Accumulation
Final Value Theorem
Conditions for Validity
16.4	The Inverse z Transform	16-17
Synthetic Division, Partial-Fraction Expansion
16.5	Solution of Difference Equations with Initial Conditions	16-23
16.6	The Relationships Among the z Transform, the Discrete-Time Fourier
Transform and the Laplace Transform	16-25
s-Plane-to-z-Plane Mapping, Relation to Aliasing
16.7	Transfer Functions	16-29
16.8	System Stability	16-31
Pole Locations, Marginal Stability
16.9	Parallel, Cascade and Feedback Connections		16-32
Loop Transfer Function
16.10	System Responses to Standard Signals	16-33
Unit Sequence Response
First and Second Order Systems
Response to a Sinusoid Applied at a Finite Time
Steady-State or Forced Response
16.11	Pole-Zero Diagrams and the Graphical Calculation of Frequency
Response
16-41
Vector Interpretation, Influences of Poles and Zeros
16.12	Simulating Continuous-Time Systems with Discrete-Time Systems
16-47
Discretization, Impulse-Invariant Design
16.13	Sampled-Data Systems	16-49
Analog-to-Digital and Digital-to-Analog Converters, Zero-Order
Hold,
Sampling-Rate Effects
16.14	Standard Realizations of Systems	16-57
Canonical Direct-Form 2, Cascade, Parallel
16.15	Summary of Important Points		16-60
Exercises	16-62
Appendix A - Useful Mathematical Relations
Appendix B - The Continuous-Time Fourier Series Pairs
Appendix C - Discrete-Time Fourier Series Pairs
Appendix D - Continuous-Time Fourier Transform Pairs
Appendix E - Discrete-Time Fourier Transform Pairs
Appendix F - Laplace Transform Pairs
Appendix G - z Transform Pairs
Web Appendix A - Introduction to MATLAB
Web Appendix B - Method for Finding Least Common Multiples
Web Appendix C - Example of Linearizing and Small Signal Analysis
Web Appendix D - Derivations of Convolution Properties
Web Appendix E - An Exploration of Impulse Properties using
Convolution
Web Appendix F - Derivations of the Properties of the Continuous-
Time Fourier Series
Web Appendix G - Derivations of the Properties of the Discrete-
Time Fourier Series
Web Appendix H - Derivations of the Properties of the Continuous-
Time Fourier Transform
Web Appendix I - Derivations of the Properties of the Discrete-
Time Fourier Transform
Web Appendix J - Frequency Response of an Ideal Lowpass Filter
Web Appendix K - Sampling Methods
Web Appendix L - The DFT in Relation to the Other Fourier Methods
- With Examples
Web Appendix M - The Fast Fourier Transform
Web Appendix N - Derivations of the Properties of the Laplace
Transform
Web Appendix O - Derivations of the Properties of the z Transform
Web Appendix P - Complex Numbers and Complex Functions
Web Appendix Q - Differential and Difference Equations
Web Appendix R - Vectors and Matrices
```

Library of Congress Subject Headings for this publication:

Signal processing -- Textbooks.
System analysis -- Textbooks.