Preface xiAbout the Author xvAcknowledgments xviiAbout the Companion Website xix1 Review of Linear Algebra 11.1 Introduction 11.2 Vectors, Scalars, and Bases 2Worked Exercise: Linear Combinations on the Left-hand Side of the Scalar Product 31.3 Vector Representation in Different Bases 71.4 Linear Operators 121.5 Representation of Linear Operators 141.6 Eigenvectors and Eigenvalues 181.7 General Method of Solution of a Matrix Equation 211.8 The Closure Relation 231.9 Representation of Linear Operators in Terms of Eigenvectors and Eigenvalues 241.10 The Dirac Notation 25Worked Exercise: The Bra of the Action of an Operator on a Ket 281.11 Exercises 30Interlude: Signals and Systems: What is it About? 352 Representation of Signals 372.1 Introduction 372.2 The Convolution 38Worked Exercise: First Example of Convolution 42Worked Exercise: Second Example of Convolution 442.3 The Impulse Function, or Dirac Delta 462.4 Convolutions with Impulse Functions 50Worked Exercise: The Convolution with delta(t . a) 522.5 Impulse Functions as a Basis: The Time Domain Representation of Signals 532.6 The Scalar Product 602.7 Orthonormality of the Basis of Impulse Functions 62Worked Exercise: Proof of Orthonormality of the Basis of Impulse Functions 642.8 Exponentials as a Basis: The Frequency Domain Representation of Signals 652.9 The Fourier Transform 72Worked Exercise: The Fourier Transform of the Rectangular Function 742.10 The Algebraic Meaning of Fourier Transforms 75Worked Exercise: Projection on the Basis of Exponentials 782.11 The Physical Meaning of Fourier Transforms 802.12 Properties of Fourier Transforms 852.12.1 Fourier Transform and the DC level 852.12.2 Property of Reality 862.12.3 Symmetry Between Time and Frequency 882.12.4 Time Shifting 882.12.5 Spectral Shifting 90Worked Exercise: The Property of Spectral Shifting and AM Modulation 912.12.6 Differentiation 922.12.7 Integration 932.12.8 Convolution in the Time Domain 962.12.9 Product in the Time Domain 97Worked Exercise: The Fourier Transform of a Physical Sinusoidal Wave 982.12.10 The Energy of a Signal and Parseval's Theorem 1012.13 The Fourier Series 102Worked Exercise: The Fourier Series of a Square Wave 1082.14 Exercises 1093 Representation of Systems 1133.1 Introduction and Properties 1133.1.1 Linearity 1143.1.2 Time Invariance 114Worked Exercise: Example of a Time Invariant System 116Worked Exercise: An Example of a Time Variant System 1173.1.3 Causality 1173.2 Operators Representing Linear and Time Invariant Systems 1183.3 Linear Systems as Matrices 1193.4 Operators in Dirac Notation 1213.5 Statement of the Problem 1233.6 Eigenvectors and Eigenvalues of LTI Operators 1233.7 General Method of Solution 1243.7.1 Step 1: Defining the Problem 1243.7.2 Step 2: Finding the Eigenvalues 1253.7.3 Step 3: The Representation in the Basis of Eigenvectors 1263.7.4 Step 4: Solving the Equation and Returning to the Original Basis 129Worked Exercise: Input is an Eigenvector 130Worked Exercise: Input is an Explicit Linear Combination of Eigenvectors 131Worked Exercise: An Arbitrary Input 1323.8 The Physical Meaning of Eigenvalues: The Impulse and Frequency Responses 133Worked Exercise: Impulse and Frequency Responses of a Harmonic Oscillator 136Worked Exercise: How can the Frequency Response be Measured? 139Worked Exercise: The Transient of a Harmonic Oscillator 142Worked Exercise: Charge and Discharge in an RC Circuit 1453.9 Frequency Conservation in LTI Systems 1473.10 Frequency Conservation in Other Fields 1483.10.1 Snell's Law 1493.10.2 Wavefunctions and Heisenberg's Uncertainty Principle 1503.11 Exercises 1524 Electric Circuits as LTI Systems 1574.1 Electric Circuits as LTI Systems 1574.2 Phasors, Impedances, and the Frequency Response 158Worked Exercise: An RLC Circuit as a Harmonic Oscillator 1634.3 Exercises 1645 Filters 1655.1 Ideal Filters 1655.2 Example of a Low-pass Filter 1675.3 Example of a High-pass Filter 1705.4 Example of a Band-pass Filter 1715.5 Exercises 1726 Introduction to the Laplace Transform 1756.1 Motivation: Stability of LTI Systems 1756.2 The Laplace Transform as a Generalization of the Fourier Transform 1796.3 Properties of Laplace Transforms 1816.4 Region of Convergence 1826.5 Inverse Laplace Transform by Inspection 185Worked Exercise: Example of Inverse Laplace Transform by Inspection 185Worked Exercise: Impulse Response of a Harmonic Oscillator 1876.6 Zeros and Poles 188Worked Exercise: Finding the Zeros and Poles 189Worked Exercise: Poles of a Harmonic Oscillator 1906.7 The Unilateral Laplace Transform 1916.7.1 The Differentiation Property of the Unilateral Fourier Transform 193Worked Exercise: Differentiation Property of the Unilateral Fourier Transform Involving Higher Order Derivatives 195Worked Exercise: Example of Differentiation Using the Unilateral Fourier Transform 196Worked Exercise: Discharge of an RC Circuit 1976.7.2 Generalization to the Unilateral Laplace Transform 1986.8 Exercises 199Interlude: Discrete Signals and Systems: Why do we Need Them? 2037 The Sampling Theorem and the Discrete Time Fourier Transform (DTFT) 2057.1 Discrete Signals 2057.2 Fourier Transforms of Discrete Signals and the Sampling Theorem 2077.3 The Discrete Time Fourier Transform (DTFT) 216Worked Exercise: Example of a Matlab Routine to Calculate the Dtft 218Worked Exercise: Fourier Transform from the DTFT 2217.4 The Inverse DTFT 2237.5 Properties of the DTFT 2247.5.1 'Time' shifting 2257.5.2 Difference 2267.5.3 Sum 2287.5.4 Convolution in the 'Time' Domain 2297.5.5 Product in the Time Domain 2307.5.6 The Theorem that Should not be: Energy of Discrete Signals 2317.6 Concluding Remarks 2357.7 Exercises 2358 The Discrete Fourier Transform (DFT) 2398.1 Discretizing the Frequency Domain 2398.2 The DFT and the Fast Fourier Transform (fft) 246Worked Exercise: Getting the Centralized DFT Using the Command fft 250Worked Exercise: Getting the Fourier Transform with the fft 254Worked Exercise: Obtaining the Inverse Fourier Transform Using the ifft 2568.3 The Circular Time Shift 2588.4 The Circular Convolution 2598.5 Relationship Between Circular and Linear Convolutions 2648.6 Parseval's Theorem for the DFT 2698.7 Exercises 2709 Discrete Systems 2759.1 Introduction and Properties 2759.1.1 Linearity 2769.1.2 'Time' invariance 2769.1.3 Causality 2769.1.4 Stability 2769.2 Linear and Time Invariant Discrete Systems 277Worked Exercise: Further Advantages of Frequency Domain 2799.3 Digital Filters 2839.4 Exercises 28510 Introduction to the z-transform 28710.1 Motivation: Stability of LTI Systems 28710.2 The z-transform as a Generalization of the DTFT 289Worked Exercise: Example of z-transform 29010.3 Relationship Between the z-transform and the Laplace Transform 29210.4 Properties of the z-transform 29310.4.1 'Time' shifting 29410.4.2 Difference 29410.4.3 Sum 29410.4.4 Convolution in the Time Domain 29410.5 The Transfer Function of Discrete LTI Systems 29510.6 The Unilateral z-transform 29510.7 Exercises 297References with Comments 299Appendix A: Laplace Transform Property of Product in the Time Domain 301Appendix B: List of Properties of Laplace Transforms 303Index 305
Emiliano R. Martins majored in electrical engineering at the University of São Paulo (Brazil), then obtained a master's degree in electrical engineering from the same university, another master's degree in photonics from the Erasmus Mundus Master in Photonics (European consortium), and a PhD in physics from the University of St. Andrews (UK). He has been teaching signals and systems in the Department of Electrical and Computer Engineering of the University of São Paulo (Brazil) since 2016. He is also the author of Essentials of Semiconductor Device Physics.
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