CONTINUOUS SIGNALS AND SYSTEMS

WITH MATLAB SECOND EDITION

Taan S. EIAIi Benedict College, South Carolina

Mohammad A. Karim Old Dominion University, Virginia

C\ CRC Press W / Taylor & Francis Group

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

Contents

Preface xiii

Authors xv

Chapter 1 Signal Representation 1 1.1 Examples of Continuous Signals 1 1.2 The Continuous Signal 2 1.3 Periodic and Nonperiodic Signals 3 1.4 General Form of Sinusoidal Signals 4 1.5 Energy and Power Signals 6 1.6 The Shifting Operation 7 1.7 The Reflection Operation 8 1.8 Even and Odd Functions 10 1.9 Time Scaling 13 1.10 The Unit Step Signal 15 1.11 The Signum Signal 17 1.12 The Ramp Signal 17 1.13 The Sampling Signal 18 1.14 The Impulse Signal 19 1.15 Some Insights: Signals in the Real World 20

1.15.1 The Step Signal 21 1.15.2 The Impulse Signal 21 1.15.3 The Sinusoidal Signal 22 1.15.4 The Ramp Signal 23 1.15.5 Other Signals 23

1.16 End-of-Chapter Examples 24 1.17 End-of-Chapter Problems 42 References 50

Chapter 2 Continuous Systems 51 2.1 Definition of a System 51 2.2 Input and Output 51 2.3 Linear Continuous System 52 2.4 Time-Invariant System 54 2.5 Systems without Memory 56 2.6 Causal Systems 57 2.7 The Inverse of a System 58 2.8 Stahle Systems 60 2.9 Convolution 61 2.10 Simple Block Diagrams 62

2.11 Graphical Convolution 66 2.12 Differential Equations and Physical Systems 71 2.13 Homogeneous Differential Equations

and Their Solutions 71 2.13.1 Case When the Roots Are All Distinct 72 2.13.2 Case When Two Roots Are Real and Equal 72 2.13.3 Case When Two Roots Are Complex 72

2.14 Nonhomogeneous Differential Equations and Their Solutions 73 2.14.1 How Do We Find the Particular Solution? 74

2.15 The Stability of Linear Continuous Systems: The Characteristic Equation 77

2.16 Block Diagram Representation of Linear Systems 82 2.16.1 Integrator 82 2.16.2 Adder 82 2.16.3 Subtractor 82 2.16.4 Multiplier 83

2.17 From Block Diagrams to Differential Equations 83 2.18 From Differential Equations to Block Diagrams 85 2.19 The Impulse Response 87 2.20 Some Insights: Calculating y(t) 89

2.20.1 How Can We Find These Eigenvalues? 90 2.20.2 Stability and Eigenvalues 90

2.21 End-of-Chapter Examples 91 2.22 End-of-Chapter Problems 117 References 125

Chapter 3 Fourier Series 127 3.1 Review of Complex Numbers 127

3.1.1 Definition 127 3.1.2 Addition 127 3.1.3 Subtraction 127 3.1.4 Multiplication 128 3.1.5 Division 129 3.1.6 From Rectangular to Polar 129 3.1.7 From Polar to Rectangular 130

3.2 Orthogonal Functions 131 3.3 Periodic Signals 132 3.4 Conditions for Writing a Signal as a Fourier Series Sum 132 3.5 Basis Functions 133 3.6 The Magnitude and the Phase Spectra 134 3.7 Fourier Series and the Sin-Cos Notation 134 3.8 Fourier Series Approximation and the Resulting Error 138 3.9 The Theorem of Parseval 139 3.10 Systems with Periodic Inputs 140

3.11 A Formula for Finding y(t) When x(t) Is Periodic: The Steady-State Response 142

3.12 Some Insight: Why the Fourier Series? 144 3.12.1 No Exact Sinusoidal Representation for x(t) 144 3.12.2 The Frequency Components 144

3.13 End-of-Chapter Examples 145 3.14 End-of-Chapter Problems 158 References 160

Chapter 4 The Fourier Transform and Linear Systems 161 4.1 Definition 161 4.2 Introduction 161 4.3 The Fourier Transform Pairs 163 4.4 Energy of Non-Periodic Signals 175 4.5 The Energy Spectral Density of a Linear System 177 4.6 Some Insights: Notes and a Useful Formula 178 4.7 End-of-Chapter Examples 179 4.8 End-of-Chapter Problems 191 References 197

Chapter 5 The Laplace Transform and Linear Systems 199 5.1 Definition 199 5.2 The Bilateral Laplace Transform 199 5.3 The Unilateral Laplace Transform 200 5.4 The Inverse Laplace Transform 201 5.5 Block Diagrams Using the Laplace Transform 205

5.5.1 Parallel Systems 206 5.5.2 Series Systems 207

5.6 Representation of Transfer Functions as Block Diagrams 207 5.7 Procedure for Drawing the Block Diagram

from the Transfer Function 209 5.8 Solving LTI Systems Using the Laplace Transform 211 5.9 Solving Differential Equations Using

the Laplace Transform 213 5.10 The Final Value Theorem 215 5.11 The Initial Value Theorem 216 5.12 Some Insights: Poles and Zeros 216

5.12.1 The Poles of the System 216 5.12.2 The Zeros of the System 217 5.12.3 The Stability of the System 217

5.13 End-of-Chapter Examples 217 5.14 End-of-Chapter Problems 242 References 252

Chapter 6 State-Space and Linear Systems 253 6.1 Introduction 253 6.2 A Review of Matrix Algebra 254

6.2.1 Definition, General Terms, and Notations 254 6.2.2 The Identity Matrix 254 6.2.3 Adding Two Matrices 254 6.2.4 Subtracting Two Matrices 255 6.2.5 Multiplying a Matrix by a Constant 255 6.2.6 Determinant of a 2 x 2 Matrix 256 6.2.7 Transpose of a Matrix 256 6.2.8 Inverse of a Matrix 256 6.2.9 Matrix Multiplication 257 6.2.10 Diagonal Form of a Matrix 257 6.2.11 Exponent of a Matrix 258 6.2.12 A Special Matrix 258 6.2.13 Observation 259 6.2.14 Eigenvalues of a Matrix 260 6.2.15 Eigenvectors of a Matrix 260

6.3 General Representation of Systems in State-Space 278 6.4 General Solution of State-Space Equations Using

the Laplace Transform 279 6.5 General Solution of the State-Space Equations

in Real Time 280 6.6 Ways of Evaluating eAt 281

6.6.1 First Method: A is a Diagonal Matrix 281 ~a b~ 281

of Any Form 281

6.6.2 Second Method: A is of the Form L0 aj

6.6.3 Third Method: Numerical Evaluation, A 6.6.4 Fourth Method: The Cayley-Hamilton Approach 281 6.6.5 Fifth Method: The Inverse-Laplace Method 283 6.6.6 Sixth Method: Using the General Form

of O(f) = eA'and Its Properties 285 6.7 Some Insights: Poles and Stability 289 6.8 End-of-Chapter Examples 290 6.9 End-of-Chapter Problems 340 References 349

Chapter 7 Block Diagrams 351 7.1 Introduction 351 7.2 Basic Block Diagram Components 352

7.2.1 The Ideal Integrator 352 7.2.2 The Adder 352 7.2.3 The Subtractor 353 7.2.4 The Multiplier 353

7.3 Block Diagrams as Interconnected Subsystems 353 7.3.1 The General Transfer Function Representation 353 7.3.2 The Parallel Representation 354 7.3.3 The Series Representation 354 7.3.4 The Basic Feedback Representation 355

7.4 The Controllable Canonical Form (CCF) Block Diagrams with Basic Blocks 356

7.5 The Observable Canonical Form (OCF) Block Diagrams with Basic Blocks 358

7.6 The Diagonal Form (DF) Block Diagrams with Basic Blocks 360 7.6.1 Distinct Roots Case 360 7.6.2 Repeated Roots Case 361

7.7 The Parallel Block Diagrams with Subsystems 362 7.7.1 Distinct Roots Case 362 7.7.2 Repeated Roots Case 363

7.8 The Series Block Diagrams with Subsystems 364 7.8.1 Distinct Real Roots Case 365 7.8.2 Mixed Complex and Real Roots Case 365

7.9 Block Diagram Reduction Rules 365 7.9.1 Using the Reduction Rules 365 7.9.2 Using Mason's Rule 366

7.10 End of Chapter Examples 367 7.10.1 From Block Diagrams with Basic Block Components

to Transfer Functions 367 7.10.2 From Block Diagrams with Interconnected Subsystems

to Transfer Functions 370 7.11 End of Chapter Problems 375 References 377

Chapter 8 Analog Filter Design 379

By Dr. Khaled Younis

8.1 Introduction 379 8.2 Analog Filter Specifications 380 8.3 Butterworth Filter Approximation 383 8.4 Chebyshev Filters 386

8.4.1 Type I Chebyshev Approximation 386 8.4.2 Inverse Chebyshev Filter (Type II Chebyshev Filters) 389

8.5 Elliptic Filter Approximation 391 8.6 Bessel Filters 392 8.7 Analog Frequency Transformation 393 8.8 Analog Filter Design Using Matlab 396

8.8.1 Order Estimation Functions 397 8.8.2 Analog Prototype Design Functions 398 8.8.3 Complete Classical HR Filter Design 398 8.8.4 Analog Frequency Transformation 400

8.9 How Do We Find the Cutoff Frequency Analytically? 401 8.10 Limitations 405 8.11 Comparison between Analog Filter Types 405 8.12 Implementation of Analog Filters 406

8.13 Some Insights: Filters with High Gain versus Filters with Low Gain and the Relation between the Time Constant and the Cutoff Frequency for First-Order Circuits and the Series RLC Circuit 408

8.14 End of Chapter Examples 409 8.15 End of Chapter Problems 442 References 449

Chapter 9 Introduction to Nonlinear Systems 451 9.1 Introduction 451 9.2 Linear and Nonlinear Differential Equations 452 9.3 The Process of Linearization 453

9.3.1 Linearization of a Nonlinear System Given by a Differential Equation 454

9.3.2 Linearization When/(z) Is a Function of the State Vector Only 456

9.3.3 Linearization When/(z) Is a Function of the State Vector and the Input x(t) 460

9.4 Some Insights: The Meaning of Linear and Nonlinear 464 9.5 End-of-Chapter Examples 466 9.6 End-of-Chapter Problems 491 References 497

Index 499