About the Authors xiii -- Preface xiv -- Acknowledgments xvii -- About the Companion Website xix -- 1 Introduction to Systems and Signals 1 -- 1.1 Example Applications 2 -- 1.1.1 Three-Dimensional World Models by LIDAR Signals 3 -- 1.1.2 Modeling the Brain Networks from the Brain Signals 3 -- 1.1.3 Detecting the Buildings from the Remote-Sensed Satellite Images 4 -- 1.1.4 Noise Reduction in Old Records 4 -- 1.2 Relationship Between Signals and Systems 4 -- 1.3 Mathematical Representation of Signals and Systems 5 -- 1.3.1 Signals Represented by Functions 6 -- 1.3.2 Types of Signals 6 -- 1.3.3 Energy of a Signal 9 -- 1.3.4 Power of a Signal 10 -- 1.4 Operations on the Time Variable of Signals 10 -- 1.4.1 Time Shift 11 -- 1.4.2 Time Reverse 12 -- 1.4.3 Time Scale 13 -- 1.4.4 Time Scale and Shift 15 -- 1.5 Signals with Symmetry Properties 19 -- 1.5.1 Periodic Signals 21 -- 1.5.1.1 Continuous Time Periodic Signals 22 -- 1.5.1.2 Discrete Time Periodic Signals 23 -- 1.5.2 Even and Odd Signals 24 -- 1.6 Complex Signals Represented by Complex Functions 28 -- 1.6.1 Complex Numbers Represented in Cartesian Coordinate System 28 -- 1.6.2 Complex Numbers Represented in Polar Coordinate System and Euler's Number 30 -- 1.6.3 Complex Functions 33 -- 1.7 Chapter Summary 35 -- Problems 36 -- 2 Basic Building Blocks of Signals 43 -- 2.1 LEGO Functions of Signals 43 -- 2.2 King of the Functions: Exponential Function 44 -- 2.2.1 Real Exponential Function 44 -- 2.2.1.1 Continuous Time Real Exponential Function 44 -- 2.2.1.2 Discrete Time Real Exponential Function 46 -- 2.2.2 Complex Exponential Function 47 -- 2.2.2.1 Continuous Time Complex Exponential Functions 48 -- 2.2.2.2 Harmonically Related Complex Exponential 49 -- 2.2.2.3 Complex Exponential Function for Discrete Time Signals 53 -- 2.3 Unit Impulse Function 55 -- 2.3.1 Discrete Time Unit Impulse Function or Dirac-Delta Function 55 -- 2.3.2 Continuous Time Unit Impulse Function 56 -- 2.3.3 Comparison of Discrete Time and Continuous Time Unit Impulse Functions 57 -- 2.4 Unit Step Function 58 -- 2.4.1 Discrete Time Unit Step Function 58 -- 2.4.2 Relationship Between the Discrete Time Unit Step and Unit Impulse Functions 58 -- 2.4.3 Continuous Time Unit Step Function 60 -- 2.4.4 Comparison of Discrete Time and Continuous Time Unit Step functions 61 -- 2.4.4.1 Relationship Between the Continuous Time Unit Step and Unit Impulse Functions 61 -- 2.5 Chapter Summary 65 -- Problems 65 -- 3 Basic Building Blocks and Properties of Systems 69 -- 3.1 Representation of Systems by Equations 69 -- 3.2 Interconnection of Basic Systems: Series, Parallel, Hybrid, and Feedback Control Systems 70 -- 3.2.1 Series Systems 70 -- 3.2.2 Parallel Systems 71 -- 3.2.3 Hybrid Systems 71 -- 3.2.3.1 Feedback Control Systems 72 -- 3.2.3.2 An Example of System Modeling: Neurons as a Subsystem of Human Brain 73 -- 3.3 Properties of Systems 74 -- 3.3.1 Memory 75 -- 3.3.2 Causality 76 -- 3.3.3 Invertibility 77 -- 3.3.4 Stability 79 -- 3.3.5 Time Invariance 80 -- 3.3.6 Linearity and Superposition Property 81 -- 3.4 Basic Building Blocks of Systems and Their Properties 85 -- 3.4.1 Scalar Multiplier 85 -- 3.4.2 Adder 85 -- 3.4.3 Multiplier 85 -- 3.4.4 Integrator 86 -- 3.4.5 Differentiator 86 -- 3.4.6 Unit Delay Operator 87 -- 3.4.7 Unit Advance Operator 87 -- 3.5 Chapter Summary 89 -- Problems 89 -- 4 Representation of Linear Time-Invariant Systems by Impulse Response and Convolution Operation 95 -- 4.1 Representation of LTI Systems by Impulse Response 96 -- 4.1.1 Representation of Discrete Time Linear Time-Invariant Systems by Impulse Response 97 -- 4.1.2 Representation of Continuous Time Linear Time-Invariant System 98 -- 4.1.3 Convolution Operation in Continuous Time 101 -- 4.1.4 Convolution Operation in Discrete Time Systems 105 -- 4.1.5 Cross-correlation and Autocorrelation Operations 107 -- 4.2 Properties of Impulse Response for LTI Systems 110 -- 4.2.1 Impulse Response of Memoryless LTI Systems 110 -- 4.2.2 Impulse Response of Causal LTI Systems 110 -- 4.2.3 Inverse of Impulse Response for LTI Systems 111 -- 4.2.4 Impulse Response of Stable LTI Systems 114 -- 4.2.5 Unit Step Response 115 -- 4.3 An Application of Convolution in Machine Learning 116 -- 4.4 Chapter Summary 118 -- Problems 118 -- 5 Representation of LTI Systems by Differential and Difference Equations 123 -- 5.1 Linear Constant-Coefficient Differential Equations 123 -- 5.2 Representation of a Continuous Time LTI System by Differential Equations 124 -- 5.3 Solving the Linear Constant Coefficient Differential Equations That Represent LTI Systems 126 -- 5.3.1 Finding the Particular Solution 127 -- 5.3.2 Finding the Homogeneous Solution 128 -- 5.3.3 Finding the General Solution 129 -- 5.3.4 Transfer Function of a Continuous Time LTI System 134 -- 5.4 Linear Constant Coefficient Difference Equations 136 -- 5.4.1 Representation of a Discrete Time LTI Systems by Difference Equations 136 -- 5.4.2 Solution to Linear Constant Coefficient Difference Equations 137 -- 5.4.3 Transfer Function of a Discrete Time LTI System 139 -- 5.5 Relationship Between the Impulse Response and Difference or Differential Equations 140 -- 5.6 Block Diagram Representation of Differential Equations for LTI Systems 144 -- 5.7 Chapter Summary 147 -- Problems 148 -- 6 Fourier Series Representation of Continuous Time Periodic Signals 155 -- 6.1 History 156 -- 6.2 Mathematical Representation of Waves and Harmony 157 -- 6.3 Dirichlet Conditions 160 -- 6.4 Fourier Theorem 162 -- 6.4.1 Proof Sketch for the Fourier Theorem 162 -- 6.4.2 Terminology 163 -- 6.5 Frequency Domain and Hilbert Spaces 164 -- 6.6 Response of a Linear Time-Invariant System to the Continuous Time Complex Exponential Input Signal 170 -- 6.6.1 Eigenfunctions and Eigenvalues of a Continuous Time LTI Systems 171 -- 6.7 Convergence of the Fourier Series and Gibbs Phenomenon 173 -- 6.8 Properties of Fourier Series for Continuous Time Functions 174 -- 6.8.1 Linearity Property 174 -- 6.8.2 Time Shifting Property 174 -- 6.8.3 Time Scale Property 175 -- 6.8.4 Time Reversal Property 175 -- 6.8.5 Convolution Property 175 -- 6.8.6 Multiplication Property 176 -- 6.8.7 Conjugate Symmetry 176 -- 6.8.8 Parseval's Equality 177 -- 6.8.9 Differentiation Property 177 -- 6.9 Trigonometric Fourier Series for Continuous Time Functions 180 -- 6.10 Trigonometric Fourier Series for Continuous Time Even and Odd Functions 182 -- 6.11 Chapter Summary 185 -- Problems 186 -- 7 Fourier Series Representation of Discrete Time Periodic Signals 191 -- 7.1 Fourier Series Theorem for Discrete Time Functions 191 -- 7.2 Discrete Time Fourier Series Representation in Hilbert Space 193 -- 7.3 Properties of Discrete Time Fourier Series 199 -- 7.3.1 Difference Property 203 -- 7.3.2 Convolution Property 205 -- 7.3.3 Multiplication Property 208 -- 7.4 Discrete Time LTI Systems with Periodic Input and Output Pairs 211 -- 7.4.1 Eigenfunctions, Eigenvalues, and Transfer Functions of a Discrete Time LTI Systems 212 -- 7.4.2 Relationship Between the Fourier Series of Periodic Input and Output Pairs of Discrete Time LTI Systems 213 -- 7.5 Chapter Summary 215 -- Problems 215 -- 8 Continuous Time Fourier Transform and Its Extension to Laplace Transform 221 -- 8.1 Fourier Series Extension to Aperiodic Functions 222 -- 8.2 Existence and Convergence of the Fourier Transforms: Dirichlet Conditions 224 -- 8.3 Fourier Transforms 225 -- 8.4 Comparison of Fourier Series and Fourier Transform 226 -- 8.5 Frequency Content of Fourier Transform 227 -- 8.6 Representation of LTI Systems in Frequency Domain by Frequency Response 232 -- 8.7 Relationship Between the Fourier Series and Fourier Transform of Periodic Functions 235 -- 8.8 Properties of Fourier Transform: For Continuous Time Signals and Systems 238 -- 8.8.1 Basic Properties of Continuous Time Fourier Transform 239 -- 8.8.2 Continuous Time Linear Time-Invariant Systems in Frequency Domain 256 -- 8.9 Laplace Transforms as an Extension of Continuous Time Fourier Transforms 259 -- 8.9.1 One-Sided Laplace Transform 260 -- 8.9.2 Region of Convergence in Laplace Transforms 261 -- 8.10 Inverse of Laplace Transform 265 -- 8.11 Continuous Time Linear Time-Invariant Systems in Laplace Domain 268 -- 8.11.1 Eigenvalues and Transfer Functions in s-Domain 269 -- 8.12 Chapter Summary 272 -- Problems 273 -- 9 Discrete Time Fourier Transform and Its Extension to z-Transforms 281 -- 9.1 Fourier Series Extension to Discrete Time Aperiodic Functions 281 -- 9.1.1

Discrete Time Fourier Transform 282 -- 9.2 Dirichlet Conditions Are Relaxed for the Existence of Discrete Time Fourier Transform 284 -- 9.3 Fourier Transform of Discrete Time Periodic Functions 293 -- 9.4 Properties of Fourier Transforms for Discrete Time Signals and Systems 297 -- 9.4.1 Basic Properties of Discrete Time Fourier Transform 297 -- 9.5 Discrete Time Linear Time-Invariant Systems in Frequency Domain 307 -- 9.6 Representation of Discrete Time LTI Systems 310 -- 9.6.1 Impulse Response 311 -- 9. ...