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1 Introduction 1

1-1 Mathematical Representation of Signals 2

1-2 Mathematical Representation of Systems 4

1-3 Thinking About Systems 5

1-4 The Next Step 6

2 Sinusoids 7

2-1 Tuning Fork Experiment 8

2-2 Review of Sine and Cosine Functions 9

2-3 Sinusoidal Signals 11

2-3.1 Relation of Frequency to Period 12

2-3.2 Phase Shift and Time Shift 13

2-4 Sampling and Plotting Sinusoids 15

2-5.1 Review of Complex Numbers 17

2-5 Complex Exponentials and Phasors 17

2-5.2 Complex Exponential Signals 18

2-5.3 The Rotating Phasor Interpretation 19

2-5.4 Inverse Euler Formulas 21

2-6 Phasor Addition 22

2-6.1 Addition of Complex Numbers 23

2-6.2 Phasor Addition Rule 23

2-6.3 Phasor Addition Rule:Example 24

2-6.4 MATLAB Demo of Phasors 25

2-6.5 Summary of the Phasor Addition Rule 26

2-7 Physics of the Tuning Fork 27

2-7.1 Equations from Laws of Physics 27

2-7.2 General Solution to the Differential Equation 29

2-7.3 Listening to Tones 29

2-8 Time Signals:More Than Formulas 29

2-9 Summary and Links 30

2-10 Problems 31

3 Spectrum Representation 36

3-1 The Spectrum of a Sum of Sinusoids 36

3-1.1 Notation Change 38

3-1.2 Graphical Plot of the Spectrum 38

3-2 Beat Notes 39

3-2.1 Multiplication of Sinusoids 39

3-2.2 Beat Note Waveform 40

3-2.3 Amplitude Modulation 41

3-3 Periodic Waveforms 43

3-3.1 Synthetic Vowel 44

3-3.2 Example of a Nonperiodic Signal 45

3-4 Fourier Series 47

3-4.2 Fourier Series Derivation 48

3-4.1 Fourier Series:Analysis 48

3-5 Spectrum of the Fourier Series 50

3-6 Fourier Analysis of Periodic Signals 51

3-6.1 The Square Wave 52

3-6.1.1 DC Value of a Square Wave 53

3-6.2 Spectrum for a Square Wave 53

3-6.3 Synthesis of a Square Wave 54

3-6.4 Triangle Wave 55

3-6.5 Synthesis of a Triangle Wave 56

3-6.6 Convergence of Fourier Synthesis 57

3-7 Time-Frequency Spectrum 57

3-7.1 Stepped Frequency 59

3-7.2 Spectrogram Analysis 59

3-8.1 Chirp or Linearly Swept Frequency 60

3-8 Frequency Modulation:Chirp Signals 60

3-8.2 A Closer Look at Instantaneous Frequency 62

3-9 Summary and Links 63

3-10 Problems 64

4 Sampling and Aliasing 71

4-1 Sampling 71

4-1.1 Sampling Sinusoidal Signals 73

4-1.2 The Concept of Aliasing 75

4-1.3 Spectrum of a Discrete-Time Signal 76

4-1.4 The Sampling Theorem 77

4-1.5 Ideal Reconstruction 78

4-2 Spectrum View of Sampling and Reconstruction 79

4-2.1 Spectrum of a Discrete-Time Signal Obtained by Sampling 79

4-2.2 Over-Sampling 79

4-2.3 Aliasing Due to Under-Sampling 81

4-2.4 Folding Due to Under-Sampling 82

4-2.5 Maximum Reconstructed Frequency 83

4-3 Strobe Demonstration 84

4-3.1 Spectrum Interpretation 87

4-4 Discrete-to-Continuous Conversion 88

4-4.1 Interpolation with Pulses 88

4-4.2 Zero-Order Hold Interpolation 89

4-4.3 Linear Interpolation 90

4-4.4 Cubic Spline Interpolation 90

4-4.5 Over-Sampling Aids Interpolation 91

4-4.6 Ideal Bandlimited Interpolation 92

4-5 The Sampling Theorem 93

4-6 Summary and Links 94

4-7 Problems 96

5 FIR Filters 101

5-1 Discrete-Time Systems 102

5-2 The Running-Average Filter 102

5-3 The General FIR Filter 105

5-3.1 An Illustration of FIR Filtering 106

5-3.2 The Unit Impulse Response 107

5-3.2.1 Unit Impulse Sequence 107

5-3.2.2 Unit Impulse Response Sequence 108

5-3.2.3 The Unit-Delay System 109

5-3.3 Convolution and FIR Filters 110

5-3.3.1 Computing the Output of a Convolution 110

5-4 Implementation of FIR Filters 111

5-4.1 Building Blocks 111

5-3.3.2 Convolution in MATLAB 111

5-4.1.1 Multiplier 112

5-4.1.2 Adder 112

5-4.1.3 Unit Delay 112

5-4.2 Block Diagrams 113

5-4.2.1 Other Block Diagrams 113

5-4.2.2 Internal Hardware Details 115

5-5 Linear Time-Invariant(LTI)Systems 115

5-5.1 Time Invariance 116

5-5.2 Linearity 117

5-5.3 The FIR Case 117

5-6 Convolution and LTI Systems 118

5-6.1 Derivation of the Convolution Sum 118

5-6.2 Some Properties of LTI Systems 120

5-6.2.3 Associative Property of Convolution 121

5-6.2.1 Convolution as an Operator 121

5-6.2.2 Commutative Property of Convolution 121

5-7 Cascaded LTI Systems 122

5-8 Example of FIR Filtering 124

5-9 Summary and Links 126

5-10 Problems 126

6 Frequency Response of FIR Filters 130

6-1 Sinusoidal Response of FIR Systems 130

6-2 Superposition and the Frequency Response 132

6-3 Steady-State and Transient Response 135

6-4 Properties of the Frequency Response 137

6-4.1 Relation to Impulse Response and Difference Equation 137

6-4.2 Periodicity of H(ej?) 138

6-4.3 Conjugate Symmetry 138

6-5.1 Delay System 139

6-5 Graphical Representation of the Frequency Response 139

6-5.2 First-Difference System 140

6-5.3 A Simple Lowpass Filter 142

6-6 Cascaded LTI Systems 143

6-7 Running-Average Filtering 145

6-7.1 Plotting the Frequency Response 146

6-7.2 Cascade of Magnitude and Phase 148

6-7.3 Experiment:Smoothing an Image 149

6-8 Filtering Sampled Continuous-Time Signals 151

6-8.1 Example:Lowpass Averager 152

6-8.2 Interpretation of Delay 154

6-9 Summary and Links 155

6-10 Problems 157

7 z-Transforms 163

7-1 Definition of the z-Transform 164

7-2 The z-Transform and Linear Systems 165

7-2.1 The z-Transform of an FIR Filter 166

7-3 Properties of the z-Transform 167

7-3.1 The Superposition Property of the z-Transform 168

7-3.2 The Time-Delay Property of the z-Transform 168

7-3.3 A General z-Transform Formula 169

7-4 The z-Transform as an Operator 169

7-4.1 Unit-Delay Operator 169

7-4.2 Operator Notation 170

7-4.3 Operator Notation in Block Diagrams 170

7-5 Convolution and the z-Transform 171

7-5.1 Cascading Systems 173

7-5.2 Factoring z-Polynomials 174

7-5.3 Deconvolution 175

7-6 Relationship Between the z-Domain and the ?-Domain 175

7-6.1 The z-Plane and the Unit Circle 176

7-6.2 The Zeros and Poles of H(z) 177

7-6.3 Significance of the Zeros of H(z) 178

7-6.4 Nulling Filters 179

7-6.5 Graphical Relation Between z and ? 180

7-7 Useful Filters 181

7-7.1 The L-Point Running-Sum Filter 181

7-7.2 A Complex Bandpass Filter 183

7-7.3 A Bandpass Filter with Real Coefficients 185

7-8 Practical Bandpass Filter Design 186

7-9.2 Locations of the Zeros of FIR Linear-Phase Systems 189

7-9.1 The Linear-Phase Condition 189

7-9 Properties of Linear-Phase Filters 189

7-10 Summary and Links 190

7-11 Problems 191

8 IIR Filters 196

8-1 The General IIR Difference Equation 197

8-2 Time-Domain Response 198

8-2.1 Linearity and Time Invariance of IIR Filters 199

8-2.2 Impulse Response of a First-Order IIR System 200

8-2.3 Response to Finite-Length Inputs 201

8-2.4 Step Response of a First-Order Recursive System 202

8-3 System Function of an IIR Filter 204

8-3.1 The General First-Order Case 205

8-3.2.1 Direct Form Ⅰ Structure 206

8-3.2 The System Function and Block-Diagram Structures 206

8-3.2.2 Direct Form Ⅱ Structure 207

8-3.2.3 The Transposed Form Structure 208

8-3.3 Relation to the Impulse Response 209

8-3.4 Summary of the Method 209

8-4 Poles and Zeros 210

8-4.1 Poles or Zeros at the Origin or Infinity 211

8-4.2 Pole Locations and Stability 211

8-5 Frequency Response of an IIR Filter 212

8-5.1 Frequency Response using MATLAB 213

8-5.2 Three-Dimensional Plot of a System Function 214

8-6 Three Domains 216

8-7 The Inverse z-Transform and Some Applications 216

8-7.1 Revisiting the Step Response of a First-Order System 217

8-7.2 A General Procedure for Inverse z-Transformation 218

8-8 Steady-State Response and Stability 220

8-9 Second-Order Filters 223

8-9.1 z-Transform of Second-Order Filters 223

8-9.2 Structures for Second-Order IIR Systems 224

8-9.3 Poles and Zeros 225

8-9.4 Impulse Response of a Second-Order IIR System 226

8-9.4.1 Real Poles 227

8-9.5 Complex Poles 228

8-10 Frequency Response of Second-Order IIR Filter 231

8-10.1 Frequency Response via MATLAB 232

8-10.2 3-dB Bandwidth 232

8-10.3 Three-Dimensional Plot of System Functions 233

8-11 Example of an IIR Lowpass Filter 236

8-12 Summary and Links 237

8-13 Problems 238

9 Continuous-Time Signals and LTI Systems 245

9-1 Continuous-Time Signals 246

9-1.1 Two-Sided Infinite-Length Signals 246

9-1.2 One-Sided Signals 247

9-1.3 Finite-Length Signals 248

9-2 The Unit Impulse 248

9-2.1 Sampling Property of the Impulse 250

9-2.2 Mathematical Rigor 252

9-2.3 Engineering Reality 252

9-2.4 Derivative of the Unit Step 252

9-3 Continuous-Time Systems 254

9-3.1 Some Basic Continuous-Time Systems 254

9-4 Linear Time-Invariant Systems 255

9-3.3 Analogous Discrete-Time Systems 255

9-3.2 Continuous-Time Outputs 255

9-4.1 Time-Invariance 256

9-4.2 Linearity 256

9-4.3 The Convolution Integral 257

9-4.4 Properties of Convolution 259

9-5 Impulse Responses of Basic LTI Systems 260

9-5.1 Integrator 260

9-5.2 Differentiator 261

9-5.3 Ideal Delay 261

9-6 Convolution of Impulses 261

9-7 Evaluating Convolution Integrals 263

9-7.1 Delayed Unit-Step Input 263

9-7.2 Evaluation of Discrete Convolution 267

9-7.3 Square-Pulse Input 268

9-7.4 Very Narrow Square Pulse Input 269

9-7.5 Discussion of Convolution Examples 270

9-8 Properties of LTI Systems 270

9-8.1 Cascade and Parallel Combinations 270

9-8.2 Differentiation and Integration of Convolution 272

9-8.3 Stability and Causality 273

9-9 Using Convolution to Remove Multipath Distortion 276

9-10 Summary 278

9-11 Problems 279

10 Frequency Response 285

10-1 The Frequency Response Function for LTI Systems 285

10-1.1 Plotting the Frequency Response 287

10-1.2 Magnitude and Phase Changes 288

10-1.1.1 Logarithmic Plot 288

10-2 Response to Real Sinusoidal Signals 289

10-2.1 Cosine Inputs 290

10-2.2 Symmetry of H(jω) 290

10-2.3 Response to a General Sum of Sinusoids 293

10-2.4 Periodic Input Signals 294

10-3 Ideal Filters 295

10-3.1 Ideal Delay System 295

10-3.2 Ideal Lowpass Filter 296

10-3.3 Ideal Highpass Filter 297

10-3.4 Ideal Bandpass Filter 297

10-4 Application of Ideal Filters 298

10-5 Time-Domain or Frequency-Domain? 300

10-6 Summary/Future 301

10-7 Problems 302

11 Continuous-Time Fourier Transform 307

11-1 Definition of the Fourier Transform 308

11-2 Fourier Transform and the Spectrum 310

11-2.1 Limit of the Fourier Series 310

11-3 Existence and Convergence of the Fourier Transform 312

11-4 Examples of Fourier Transform Pairs 313

11-4.1 Right-Sided Real Exponential Signals 313

11-4.1.1 Bandwidth and Decay Rate 314

11-4.2 Rectangular Pulse Signals 314

11-4.3 Bandlimited Signals 316

11-4.4 Impulse in Time or Frequency 317

11-4.5 Sinusoids 318

11-4.6 Periodic Signals 319

11-5.1 The Scaling Property 322

11-5 Properties of Fourier Transform Pairs 322

11-5.2 Symmetry Properties of Fourier Transform Pairs 324

11-6 The Convolution Property 326

11-6.1 Frequency Response 326

11-6.2 Fourier Transform of a Convolution 327

11-6.3 Examples of the Use of the Convolution Property 328

11-6.3.1 Convolution of Two Bandlimited Functions 328

11-6.3.2 Product of Two Sinc Functions 329

11-6.3.3 Partial Fraction Expansions 330

11-7 Basic LTI Systems 332

11-7.1 Time Delay 332

11-7.2 Differentiation 333

11-7.3 Systems Described by Differential Equations 334

11-8.1 The General Signal Multiplication Property 335

11-8 The Multiplication Property 335

11-8.2 The Frequency Shifting Property 336

11-9 Table of Fourier Transform Properties and Pairs 337

11-10 Using the Fourier Transform for Multipath Analysis 337

11-11 Summary 341

11-12 Problems 342

12 Filtering,Modulation,and Sampling 346

12-1 Linear Time-Invariant Systems 346

12-1.1 Cascade and Parallel Configurations 347

12-1.2 Ideal Delay 348

12-1.3 Frequency Selective Filters 351

12-1.3.1 Ideal Lowpass Filter 351

12-1.3.2 Other Ideal Frequency Selective Filters 352

12-1.4 Example of Filtering in the Frequency-Domain 353

12-1.5 Compensation for the Effect of an LTI Filter 355

12-2 Sinewave Amplitude Modulation 358

12-2.1 Double-Sideband Amplitude Modulation 358

12-2.2 DSBAM with Transmitted Carrier(DSBAM-TC) 362

12-2.3 Frequency Division Multiplexing 366

12-3 Sampling and Reconstruction 368

12-3.1 The Sampling Theorem and Aliasing 368

12-3.2 Bandlimited Signal Reconstruction 370

12-3.3 Bandlimited Interpolation 372

12-3.4 Ideal C-to-D and D-to-C Converters 373

12-3.5 The Discrete-Time Fourier Transform 375

12-3.6 The Inverse DTFT 376

12-3.7 Discrete-Time Filtering of Continuous-Time Signals 377

12-4 Summary 380

12-5 Problems 381

13 Computing the Spectrum 389

13-1 Finite Fourier Sum 390

13-2 Too Many Fourier Transforms? 391

13-2.1 Relation of the DTFT to the CTFT 392

13-2.2 Relation of the DFT to the DTFT 393

13-2.3 Relation of the DFT to the CTFT 393

13-3 Time-Windowing 393

13-4 Analysis of a Sum of Sinusoids 395

13-4.1 DTFT of a Windowed Sinusoid 398

13-5 Discrete Fourier Transform 399

13-5.1 The Inverse DFT 400

13-5.2 Summary of the DFT Representation 401

13-5.3 The Fast Fourier Transform(FFT) 402

13-5.4 Negative Frequencies and the DFT 402

13-5.5 DFT Example 403

13-6 Spectrum Analysis of Finite-Length Signals 405

13-7 Spectrum Analysis of Periodic Signals 407

13-8 The Spectrogram 408

13-8.1 Spectrogram Display 409

13-8.2 Spectrograms in MATLAB 410

13-8.3 Spectrogram of a Sampled Periodic Signal 410

13-8.4 Resolution of the Spectrogram 411

13-8.4.1 Resolution Experiment 412

13-8.5 Spectrogram of a Musical Scale 413

13-8.6 Spectrogram of a Speech Signal 415

13-8.7 Filtered Speech 418

13-9 The Fast Fourier Transform(FFT) 420

13-9.1 Derivation of the FFT 420

13-9.1.1 FFT Operation Count 421

13-10 Summary and Links 423

13-11 Problems 424

A Complex Numbers 427

A-1 Introduction 428

A-2 Notation for Complex Numbers 428

A-2.1 Rectangular Form 428

A-2.2 Polar Form 429

A-2.3 Conversion:Rectangular and Polar 430

A-2.4 Difficulty in Second or Third Quadrant 431

A-3 Euler s Formula 431

A-3.1 Inverse Euler Formulas 432

A-4 Algebraic Rules for Complex Numbers 432

A-4.1 Complex Number Exercises 434

A-5 Geometric Views of Complex Operations 434

A-5.1 Geometric View of Addition 435

A-5.2 Geometric View of Subtraction 436

A-5.3 Geometric View of Multiplication 437

A-5.4 Geometric View of Division 437

A-5.5 Geometric View of the Inverse,z-1 437

A-5.6 Geometric View of the Conjugate,z＊ 438

A-6 Powers and Roots 438

A-6.1 Roots of Unity 439

A-6.1.1 Procedure for Finding Multiple Roots 440

A-7 Summary and Links 441

A-8 Problems 441

B Programming in MATLAB 443

B-1 MATLAB Help 444

B-2 Matrix Operations and Variables 444

B-2.2.1 A Review of Matrix Multiplication 445

B-2.1 The Colon Operator 445

B-2.2 Matrix and Array Operations 445

B-2.2.2 Pointwise Array Operations 446

B-3 Plots and Graphics 446

B-3.1 Figure Windows 447

B-3.2 Multiple Plots 447

B-3.3 Printing and Saving Graphics 447

B-4 Programming Constructs 447

B-4.1 MATLAB Built-in Functions 448

B-4.2 Program Flow 448

B-5 MATLAB Scripts 448

B-6 Writing a MATLAB Function 448

B-6.1 Creating A Clip Function 449

B-7 Programming Tips 451

B-6.2 Debugging a MATLAB M-file 451

B-7.2 Repeating Rows or Columns 452

B-7.3 Vectorizing Logical Operations 452

B-7.1 Avoiding Loops 452

B-7.4 Creating an Impulse 453

B-7.5 The Find Function 453

B-7.6 Seek to Vectorize 454

B-7.7 Programming Style 454

C Laboratory Projects 455

C-1 Introduction to MATLAB 457

C-1.1 Pre-Lab 457

C-1.1.1 Overview 457

C-1.1.2 Movies:MATLAB Tutorials 457

C-1.2 Warm-up 458

C-1.1.3 Getting Started 458

C-1.2.1 MATLAB Array Indexing 459

C-1.2.2 MATLAB Script Files 459

C-1.2.3 MATLAB Sound(optional) 460

C-1.3 Laboratory:Manipulating Sinusoids with MATLAB 460

C-1.3.1 Theoretical Calculations 461

C-1.3.2 Complex Amplitude 461

C-1.4 Lab Review Questions 461

C-2 Encoding and Decoding Touch-Tone Signals 463

C-2.1 Introduction 463

C-2.1.1 Review 463

C-2.1.2 Background:Telephone Touch-Tone Dialing 463

C-2.2 Pre-Lab 464

C-2.2.1 Signal Concatenation 464

C-2.1.3 DTMF Decoding 464

C-2.2.2 Comment on Efficiency 465

C-2.2.3 Encoding from a Table 465

C-2.2.4 Overlay Plotting 465

C-2.3 Warm-up:DTMF Synthesis 465

C-2.3.1 DTMF Dial Function 466

C-2.3.2 Simple Bandpass Filter Design 467

C-2.4 Lab:DTMF Decoding 468

C-2.4.1 Filter Bank Design:dtmfdesign.m 468

C-2.4.2 A Scoring Function:dtmfscore.m 469

C-2.4.3 DTMF Decode Function:dtmfrun.m 470

C-2.4.4 Testing 471

C-2.4.5 Telephone Numbers 471

C-2.4.6 Demo 472

C-3 Two Convolution GUIs 473

C-3.1 Introduction 473

C-3.2 Pre-Lab:Run the GUIs 473

C-3.2.1 Discrete-Time Convolution Demo 473

C-3.2.2 Continuous-Time Convolution Demo 474

C-3.3 Warm-up:Run the GUIs 475

C-3.3.1 Continuous-Time Convolution GUI 475

C-3.3.2 Discrete Convolution GUI 475

C-3.4 Lab Exercises 475

C-3.4.1 Continuous-Time Convolution 475

C-3.4.2 Continuous-Time Convolution Again 476

C-3.4.3 Discrete-Time Convolution 476

D CD-ROM Demos 478

Index 482