Description
Book SynopsisTable of ContentsPreface xi
About the Author xv
Acknowledgments xvii
About the Companion Website xix
1 Review of Linear Algebra 1
1.1 Introduction 1
1.2 Vectors, Scalars, and Bases 2
Worked Exercise: Linear Combinations on the Left-hand Side of the Scalar Product 3
1.3 Vector Representation in Different Bases 7
1.4 Linear Operators 12
1.5 Representation of Linear Operators 14
1.6 Eigenvectors and Eigenvalues 18
1.7 General Method of Solution of a Matrix Equation 21
1.8 The Closure Relation 23
1.9 Representation of Linear Operators in Terms of Eigenvectors and Eigenvalues 24
1.10 The Dirac Notation 25
Worked Exercise: The Bra of the Action of an Operator on a Ket 28
1.11 Exercises 30
Interlude: Signals and Systems: What is it About? 35
2 Representation of Signals 37
2.1 Introduction 37
2.2 The Convolution 38
Worked Exercise: First Example of Convolution 42
Worked Exercise: Second Example of Convolution 44
2.3 The Impulse Function, or Dirac Delta 46
2.4 Convolutions with Impulse Functions 50
Worked Exercise: The Convolution with δ(t − a) 52
2.5 Impulse Functions as a Basis: The Time Domain Representation of Signals 53
2.6 The Scalar Product 60
2.7 Orthonormality of the Basis of Impulse Functions 62
Worked Exercise: Proof of Orthonormality of the Basis of Impulse Functions 64
2.8 Exponentials as a Basis: The Frequency Domain Representation of Signals 65
2.9 The Fourier Transform 72
Worked Exercise: The Fourier Transform of the Rectangular Function 74
2.10 The Algebraic Meaning of Fourier Transforms 75
Worked Exercise: Projection on the Basis of Exponentials 78
2.11 The Physical Meaning of Fourier Transforms 80
2.12 Properties of Fourier Transforms 85
2.12.1 Fourier Transform and the DC level 85
2.12.2 Property of Reality 86
2.12.3 Symmetry Between Time and Frequency 88
2.12.4 Time Shifting 88
2.12.5 Spectral Shifting 90
Worked Exercise: The Property of Spectral Shifting and AM Modulation 91
2.12.6 Differentiation 92
2.12.7 Integration 93
2.12.8 Convolution in the Time Domain 96
2.12.9 Product in the Time Domain 97
Worked Exercise: The Fourier Transform of a Physical Sinusoidal Wave 98
2.12.10 The Energy of a Signal and Parseval’s Theorem 101
2.13 The Fourier Series 102
Worked Exercise: The Fourier Series of a Square Wave 108
2.14 Exercises 109
3 Representation of Systems 113
3.1 Introduction and Properties 113
3.1.1 Linearity 114
3.1.2 Time Invariance 114
Worked Exercise: Example of a Time Invariant System 116
Worked Exercise: An Example of a Time Variant System 117
3.1.3 Causality 117
3.2 Operators Representing Linear and Time Invariant Systems 118
3.3 Linear Systems as Matrices 119
3.4 Operators in Dirac Notation 121
3.5 Statement of the Problem 123
3.6 Eigenvectors and Eigenvalues of LTI Operators 123
3.7 General Method of Solution 124
3.7.1 Step 1: Defining the Problem 124
3.7.2 Step 2: Finding the Eigenvalues 125
3.7.3 Step 3: The Representation in the Basis of Eigenvectors 126
3.7.4 Step 4: Solving the Equation and Returning to the Original Basis 129
Worked Exercise: Input is an Eigenvector 130
Worked Exercise: Input is an Explicit Linear Combination of Eigenvectors 131
Worked Exercise: An Arbitrary Input 132
3.8 The Physical Meaning of Eigenvalues: The Impulse and Frequency Responses 133
Worked Exercise: Impulse and Frequency Responses of a Harmonic Oscillator 136
Worked Exercise: How can the Frequency Response be Measured? 139
Worked Exercise: The Transient of a Harmonic Oscillator 142
Worked Exercise: Charge and Discharge in an RC Circuit 145
3.9 Frequency Conservation in LTI Systems 147
3.10 Frequency Conservation in Other Fields 148
3.10.1 Snell’s Law 149
3.10.2 Wavefunctions and Heisenberg’s Uncertainty Principle 150
3.11 Exercises 152
4 Electric Circuits as LTI Systems 157
4.1 Electric Circuits as LTI Systems 157
4.2 Phasors, Impedances, and the Frequency Response 158
Worked Exercise: An RLC Circuit as a Harmonic Oscillator 163
4.3 Exercises 164
5 Filters 165
5.1 Ideal Filters 165
5.2 Example of a Low-pass Filter 167
5.3 Example of a High-pass Filter 170
5.4 Example of a Band-pass Filter 171
5.5 Exercises 172
6 Introduction to the Laplace Transform 175
6.1 Motivation: Stability of LTI Systems 175
6.2 The Laplace Transform as a Generalization of the Fourier Transform 179
6.3 Properties of Laplace Transforms 181
6.4 Region of Convergence 182
6.5 Inverse Laplace Transform by Inspection 185
Worked Exercise: Example of Inverse Laplace Transform by Inspection 185
Worked Exercise: Impulse Response of a Harmonic Oscillator 187
6.6 Zeros and Poles 188
Worked Exercise: Finding the Zeros and Poles 189
Worked Exercise: Poles of a Harmonic Oscillator 190
6.7 The Unilateral Laplace Transform 191
6.7.1 The Differentiation Property of the Unilateral Fourier Transform 193
Worked Exercise: Differentiation Property of the Unilateral Fourier Transform Involving Higher Order Derivatives 195
Worked Exercise: Example of Differentiation Using the Unilateral Fourier Transform 196
Worked Exercise: Discharge of an RC Circuit 197
6.7.2 Generalization to the Unilateral Laplace Transform 198
6.8 Exercises 199
Interlude: Discrete Signals and Systems: Why do we Need Them? 203
7 The Sampling Theorem and the Discrete Time Fourier Transform (DTFT) 205
7.1 Discrete Signals 205
7.2 Fourier Transforms of Discrete Signals and the Sampling Theorem 207
7.3 The Discrete Time Fourier Transform (DTFT) 216
Worked Exercise: Example of a Matlab Routine to Calculate the Dtft 218
Worked Exercise: Fourier Transform from the DTFT 221
7.4 The Inverse DTFT 223
7.5 Properties of the DTFT 224
7.5.1 ‘Time’ shifting 225
7.5.2 Difference 226
7.5.3 Sum 228
7.5.4 Convolution in the ‘Time’ Domain 229
7.5.5 Product in the Time Domain 230
7.5.6 The Theorem that Should not be: Energy of Discrete Signals 231
7.6 Concluding Remarks 235
7.7 Exercises 235
8 The Discrete Fourier Transform (DFT) 239
8.1 Discretizing the Frequency Domain 239
8.2 The DFT and the Fast Fourier Transform (fft) 246
Worked Exercise: Getting the Centralized DFT Using the Command fft 250
Worked Exercise: Getting the Fourier Transform with the fft 254
Worked Exercise: Obtaining the Inverse Fourier Transform Using the ifft 256
8.3 The Circular Time Shift 258
8.4 The Circular Convolution 259
8.5 Relationship Between Circular and Linear Convolutions 264
8.6 Parseval’s Theorem for the DFT 269
8.7 Exercises 270
9 Discrete Systems 275
9.1 Introduction and Properties 275
9.1.1 Linearity 276
9.1.2 ‘Time’ invariance 276
9.1.3 Causality 276
9.1.4 Stability 276
9.2 Linear and Time Invariant Discrete Systems 277
Worked Exercise: Further Advantages of Frequency Domain 279
9.3 Digital Filters 283
9.4 Exercises 285
10 Introduction to the z-transform 287
10.1 Motivation: Stability of LTI Systems 287
10.2 The z-transform as a Generalization of the DTFT 289
Worked Exercise: Example of z-transform 290
10.3 Relationship Between the z-transform and the Laplace Transform 292
10.4 Properties of the z-transform 293
10.4.1 ‘Time’ shifting 294
10.4.2 Difference 294
10.4.3 Sum 294
10.4.4 Convolution in the Time Domain 294
10.5 The Transfer Function of Discrete LTI Systems 295
10.6 The Unilateral z-transform 295
10.7 Exercises 297
References with Comments 299
Appendix A: Laplace Transform Property of Product in the Time Domain 301
Appendix B: List of Properties of Laplace Transforms 303
Index 305
