# Essentials of Signals and Systems

1. Edition March 2023

336 Pages, Softcover*Textbook*

**978-1-119-90921-7**

Novel approach to the theory of signals and systems in an introductory, accessible textbook

Signals and Systems have the reputation of being a difficult subject. Essentials of Signals and Systems is a standalone textbook aiming to change this reputation with a novel approach to this subject, teaching the essential concepts of signals and systems in a clear, friendly, intuitive, and accessible way.

The overall vision of the book is that traditional approaches to signals and systems are unnecessarily convoluted, and that students' learning experiences are much improved by making a clear connection between the theory of representation of signal and systems, and the theory of representation of vectors and matrices in linear algebra. The author begins by reviewing the theory of representation in linear algebra, emphasizing that vectors are represented by different coordinates when the basis is changed, and that the basis of eigenvectors is special because it diagonalizes the operator. Thus, in each step of the theory of representation of signals and systems, the author shows the analogous step in linear algebra. With such an approach, students can easily understand that signals are analogous to vectors, that systems are analogous to matrices, and that Fourier transforms are a change to the basis that diagonalizes LTI operators.

The text emphasizes the key concepts in the analysis of linear and time invariant systems, demonstrating both the algebraic and physical meaning of Fourier transforms. The text carefully connects the most important transforms (Fourier series, Discrete Time Fourier Transform, Discrete Fourier Transforms, Laplace and z-transforms), emphasizing their relationships and motivations. The continuous and discrete time domains are neatly connected, and the students are shown step-by-step how to use the fft function, using simple examples.

Incorporating learning objectives and problems, and supported with simple Matlab codes to illustrate concepts, the text presents to students the foundations to allow the reader to pursue more advanced topics in later courses.

Developed from lecture notes already tested with more than 600 students over six years, Essentials of Signals and Systems covers sample topics such as:

* Basic concepts of linear algebra that are pertinent to signals and systems.

* Theory of representation of signals, with an emphasis on the notion of Fourier transforms as a change of basis, and on their physical meaning.

* Theory of representation of linear and time invariant systems, emphasizing the role of Fourier transforms as a change to the basis of eigenvectors, and the physical meaning of the impulse and frequency responses.

* What signals and systems have to do with phasors and impedances, and the basics of filter design.

* The Laplace transform as an extension of Fourier transforms.

* Discrete signals and systems, the sampling theorem, the Discrete Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT), and how to use the fast fourier transform (fft).

* The z-transform as an extension of the Discrete Time Fourier Transform.

Essentials of Signals and Systems is an immensely helpful textbook on the subject for undergraduate students of electrical and computer engineering. The information contained within is also pertinent to those in physics and related fields involved in the understanding of signals and system processing, including those working on related practical applications.

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 delta(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