Linear systems and signals: a primer

Linear Systems and Signals: A Primer
Contents
Preface
Part I
Time Domain Analysis
Chapter 1 Introduction to Signals and
Systems
1.1 Signals and Their Classification
1.2 Discrete Time Signals
1.2.1 Discrete Time Simulation of Analog Systems
1.3 Periodic Signals
1.4 Power and Energy in Signals
1.4.1 Energy and Power Signal Examples
References
Chapter 2 Special Functions and a
System Point of View
2.1 The Unit Step or Heaviside Function
2.2 Dirac’s Delta Function d(t)
2.3 The Complex Exponential Function
2.4 Kronecker Delta Function
2.5 A System Point of View
2.5.1 Systems With Memory and Causality
2.5.2 Linear Systems
2.5.3 Time Invariant Systems
2.5.4 Stable Systems
2.6 Summary
References
Chapter 3 The Continuous Time Convolution Theorem
3.1 Introduction
3.2 The System Step Response
3.2.1 A System at Rest
3.2.2 Step Response s(t)
3.3 The System Impulse Response h(t)
3.4 Continuous Time Convolution Theorem
3.5 Summary
References
Chapter 4 Examples and Applications of
the Convolution Theorem
4.1 A First Example
4.2 A Second Example: Convolving with an
Impulse Train
4.3 A Third Example: Cascaded Systems
4.4 Systems and Linear Di˛erential Equations
4.4.1 Example: A Second Order System
4.5 Continuous Time LTI System Not at Rest
4.6 Matched Filter Theorem
4.6.1 Monte Carlo Computer Simulation
4.7 Summary
References
Chapter 5 Discrete Time Convolution
Theorem
5.1 Discrete Time IR
5.2 Discrete Time Convolution Theorem
5.3 Example: Discrete Convolution
5.4 Discrete Convolution Using a Matrix
5.5 Discrete Time Di˛erence Equations
5.5.1 Example: A Discrete Time Model of the RL Circuit
5.5.2 Example: The Step Response of a RL Circuit
5.5.3 Example: The Impulse Response of the RL Circuit
5.5.4 Example: Application of the Convolution Theorem to Compute the Step Response
5.6 Generalizing the Results: Discrete TimeSystem of Order N
5.6.1 Constant-Coe˝cient Di˛erence Equation of Order N
5.6.2 Recursive Formulation of the Response y[n]
5.6.3 Computing the Impulse Response h[n]
5.7 Summary
References
Chapter 6 Examples: Discrete Time
Systems
6.1 Example: Second Order System
6.2 Numerical Analysis of a Discrete System
6.3 Summary
References
Chapter 7 Discrete LTI Systems: State Space Analysis
7.1 Eigenanalysis of a Discrete System
7.2 State Space Representation and Analysis
7.3 Solution of the State Space Equations
7.3.1 Computing An
7.4 Example: State Space Analysis
7.4.1 Computing the Impulse Response h[n]
7.5 Analyzing a Damped Pendulum
7.5.1 Solution
7.5.2 Solving the Di˛erential Equation Numerically
7.5.3 Numerical Solution with Negligible Damping
7.6 Summary
References
Part II System Analysis Based on Transformation Theory
Chapter 8 The Fourier Transform Applied to LTI Systems
8.1 The Integral Transform
8.2 The Fourier Transform
8.3 Properties of the Fourier Transform
8.3.1 Convolution
8.3.2 Time Shifting Theorem
8.3.3 Linearity of the Fourier Transform
8.3.4 Di˛erentiation in the Time Domain
8.3.5 Integration in the Time Domain
8.3.6 Multiplication in the Time Domain
8.3.7 Convergence of the Fourier Transform
8.3.8 The Frequency Response of a Continuous Time LTI System
8.3.9 Further Theorems Based on the Fourier Transform
8.4 Applications and Insights Based on the Fourier Transform
8.4.1 Interpretation of the Fourier Transform
8.4.2 Fourier Transform of a Pulse (t)
8.4.3 Uncertainty Principle
8.4.4 Transfer Function of a Piece of Conducting Wire
8.5 Example: Fourier Transform of e? tu(t)
8.5.1 Fourier Transform of u(t)
8.6 The Transfer Function of the RC Circuit
8.7 Fourier Transform of a Sinusoid and aCosinusoid
8.8 Modulation and a Filter
8.8.1 A Design Example
8.8.2 Frequency Translation and Modulation
8.9 Nyquist-Shannon Sampling Theorem
8.9.1 Examples
8.10 Summary
References
Chapter 9 The Laplace Transform and
LTI Systems
9.1 Introduction
9.2 Definition of the Laplace Transform
9.2.1 Convergence of the Laplace Transform
9.3 Examples of the Laplace Transformation
9.3.1 An Exponential Function
9.3.2 The Dirac Impulse
9.3.3 The Step Function
9.3.4 The Damped Cosinusoid
9.3.5 The Damped Sinusoid
9.3.6 Laplace Transform of e-ajt
9.4 Properties of the Laplace Transform
9.4.1 Convolution
9.4.2 Time Shifting Theorem
9.4.3 Linearity of the Laplace Transform
9.4.4 Di˛erentiation in the Time Domain
9.4.5 Integration in the Time Domain
9.4.6 Final Value Theorem
9.5 The Inverse Laplace Transformation
9.5.1 Proper Rational Function: M < N
9.5.2 Improper Rational Function: M  N
9.5.3 Example: Inverse with a Multiple Pole
9.5.4 Example: Inverse without a Multiple Pole
9.5.5 Example: Inverse with Complex Poles
9.6 Table of Laplace Transforms
9.7 Systems and the Laplace Transform
9.8 Example: System Analysis Based on the
Laplace Transform
9.9 Linear Di˛erential Equations and Laplace
9.9.1 Capacitor
9.9.2 Inductor
9.10 Example: RC Circuit at Rest
9.11 Example: RC Circuit Not at Rest
9.12 Example: Second Order Circuit Not at
Rest
9.13 Forced Response and Transient
9.13.1 An Example with a Harmonic Driving Function
9.14 The Transfer Function H(w)
9.15 Transfer Function with Second Order Real
Poles
9.16 Transfer Function for a Second Order
System with Complex Poles
9.17 Summary
References
Chapter 10 The z-Transform and Discrete
LTI Systems
10.1 The z-Transform
10.1.1 Region of Convergence
10.2 Examples of the z-Transform
10.2.1 The Kronecker Delta d[n]
10.2.2 The Unit Step u[n]
10.2.3 The Sequence anu[n]
10.3 Table of z-Transforms
10.4 Properties of the z-Transform
10.4.1 Convolution
10.4.2 Time Shifting Theorem
10.4.3 Linearity of the z-transform
10.5 The Inverse z-Transform
10.5.1 Example: Repeated Pole
10.5.2 Example: Making use of Shifting Theorem
10.5.3 Example: Using Linearity and the Shifting Theorem
10.6 System Transfer Function for Discrete
Time LTI systems
10.7 System Analysis using the z-Transform
10.7.1 Step Response with a Given Impulse Response
10.8 Example: System Not at Rest
10.9 Example: First Order System
10.9.1 Recursive Formulation
10.9.2 Zero Input Response
10.9.3 The Zero State Response
10.9.4 The System Transfer Function H(z)
10.9.5 Impulse Response h[n]
10.10 Second Order System Not at Rest
10.10.1 Numerical Example
10.11 Discrete Time Simulation
10.12 Summary
References
Chapter 11 Signal Flow Graph
Representation
11.1 Block Diagrams
11.2 Block Diagram Simplification
11.3 The Signal Flow Graph
11.4 Mason’s Rule: The Transfer Function
11.5 A First Example: Third Order Low Pass
Filter
11.5.1 Making Use of a Graph
11.6 A Second Example: Canonical Feedback System
11.7 A Third Example: Transfer Function of a Block Diagram
11.8 Summary
References
Chapter 12 Fourier Analysis of Discrete-Time Systems and Signals
12.1 Introduction
12.2 Fourier Transform of a Discrete Signal
12.3 Properties of the Fourier Transform of Discrete Signals
12.4 LTI Systems and Di˛erence Equations
12.5 Example: Discrete Pulse Sequence
12.6 Example: A Periodic Pulse Train
12.7 The Discrete Fourier Transform
12.8 Inverse Discrete Fourier Transform
12.9 Increasing Frequency Resolution
12.10 Example: Pulse with 1 and N Samples
12.11 Example: Lowpass Filter with the DFT
12.12 The Fast Fourier Transform
12.13 Summary
References
Part III Stochastic Processes and
Linear Systems
Chapter 13 Introduction to Random Processes and Ergodicity
13.1 A Random Process
13.1.1 A Discrete Random Process: A Set of Dice
13.1.2 A Continuous Random Process: A Wind Electricity Farm
13.2 Random Variables and Distributions
13.2.1 First Order Distribution
13.2.2 Second Order Distribution
13.3 Statistical Averages
13.3.1 The Ensemble Mean
13.3.2 The Ensemble Correlation
13.3.3 The Ensemble Cross-Correlation
13.4 Properties of Random Processes
13.4.1 Statistical Independence
13.4.2 Uncorrelated
13.4.3 Orthogonal Processes
13.4.4 A Stationary Random Process
13.5 Time Averages and Ergodicity
13.5.1 Implications for a Stationary Random Process
13.5.2 Ergodic Random Processes
13.6 A First Example
13.6.1 Ensemble or Statistical Averages
13.6.2 Time Averages
13.6.3 Ergodic in the Mean and the Autocorrelation
13.7 A Second Example
13.7.1 Ensemble or Statistical Averages
13.7.2 Time Averages
13.8 A Third Example
13.9 Summary
References
Chapter 14 Spectral Analysis of Random
Processes
14.1 Correlation and Power Spectral Density
14.1.1 Properties of the Autocorrelation for a WSS Process
14.1.2 Power Spectral Density of a WSS Random Process
14.1.3 Cross-Power Spectral Density
14.2 White Noise and a Constant Signal (DC)
14.2.1 White Noise
14.2.2 A Constant Signal
14.3 Linear Systems with a Random Process
as Input
14.3.1 Cross-Correlation Between Input and Response
14.3.2 Relationship Between PSD of Input and Response
14.4 Practical Applications
14.4.1 Multipath Propagation
14.4.2 White Noise Filtering
14.5 Summary
References
Chapter 15 Discrete Time Filter Design in
the Presence of Noise
15.1 Introduction
15.2 The Prefilter
15.3 Linear Mean-Square Estimation
15.4 Prefilter Design During Pilot Frames
15.5 Evaluating Efvvyg and Efs[n]vg
15.6 Design Example
15.7 Summary
References
About the Author
Index
About the Author
Index