Linear Systems and Signals : a Primer.

Intro -- 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 &lt -- 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.

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