Signals and Systems For Dummies

The concepts covered in a typical signals and systems course are often considered by engineering students to be some of the most difficult to master. Thankfully, Signals & Systems For Dummies is your intuitive guide to this tricky course, walking you step-by-step through some of the more complex theories and mathematical formulas in a way that is easy to understand.

From Laplace Transforms to Fourier Analyses, Signals & Systems For Dummies explains in plain English the difficult concepts that can trip you up. Perfect as a study aid or to complement your classroom texts, this friendly, hands-on guide makes it easy to figure out the fundamentals of signal and system analysis.

• Serves as a useful tool for electrical and computer engineering students looking to grasp signal and system analysis
• Provides helpful explanations of complex concepts and techniques related to signals and systems
• Includes worked-through examples of real-world applications using Python, an open-source software tool, as well as a custom function module written for the book
• Brings you up-to-speed on the concepts and formulas you need to know

Signals & Systems For Dummies is your ticket to scoring high in your introductory signals and systems course.

Introduction 1

About This Book 1

Conventions Used in This Book 1

What You’re Not to Read 2

Foolish Assumptions 2

How This Book Is Organized 2

Part I: Getting Started with Signals and Systems 3

Part II: Exploring the Time Domain 3

Part III: Picking Up the Frequency Domain 3

Part IV: Entering the s- and z-Domains 3

Part V: The Part of Tens 4

Icons Used in This Book 4

Where to Go from Here 4

Part I: Getting Started with Signals and Systems 7

Chapter 1: Introducing Signals and Systems 9

Applying Mathematics 10

Getting Mixed Signals and Systems 11

Going on and on and on 11

Working in spurts: Discrete-time signals and systems 13

Classifying Signals 14

Periodic 14

Aperiodic 15

Random 15

Signals and Systems in Other Domains 16

Viewing signals in the frequency domain 16

Traveling to the s- or z-domain and back 18

Testing Product Concepts with Behavioral Level Modeling 18

Staying abstract to generate ideas 19

Working from the top down 19

Relying on mathematics 20

Exploring Familiar Signals and Systems 20

MP3 music player 21

Smartphone 22

Automobile cruise control 22

Using Computer Tools for Modeling and Simulation 23

Getting the software 24

Exploring the interfaces 25

Seeing the Big Picture 26

Chapter 2: Brushing Up on Math 29

Revealing Unknowns with Algebra 29

Solving for two variables 30

Checking solutions with computer tools 30

Exploring partial fraction expansion 31

Making Nice Signal Models with Trig Functions 35

Manipulating Numbers: Essential Complex Arithmetic 36

Believing in imaginary numbers 37

Operating with the basics 39

Applying Euler’s identities 41

Applying the phasor addition formula 42

Catching Up with Calculus 44

Differentiation 44

Integration 45

System performance 47

Geometric series 48

Finding Polynomial Roots 50

Chapter 3: Continuous-Time Signals and Systems 51

Considering Signal Types 52

Exponential and sinusoidal signals 52

Singularity and other special signal types 55

Getting Hip to Signal Classifications 60

Deterministic and random 60

Periodic and aperiodic 62

Considering power and energy 63

Even and odd signals 68

Transforming Simple Signals 69

Time shifting 69

Flipping the time axis 70

Putting it together: Shift and flip 70

Superimposing signals 71

Checking Out System Properties 72

Linear and nonlinear 73

Time-invariant and time varying 73

Causal and non-causal 74

Memory and memoryless 74

Bounded-input bounded-output 75

Choosing Linear and Time-Invariant Systems 75

Chapter 4: Discrete-Time Signals and Systems 77

Exploring Signal Types 77

Exponential and sinusoidal signals 78

Special signals 80

Surveying Signal Classifications in the Discrete-Time World 83

Deterministic and random signals 84

Periodic and aperiodic 85

Recognizing energy and power signals 88

Computer Processing: Capturing Real Signals in Discrete-Time 89

Capturing and reading a wav file 90

Finding the signal energy 91

Classifying Systems in Discrete-Time 92

Checking linearity 92

Investigating time invariance 93

Looking into causality 93

Figuring out memory 94

Testing for BIBO stability 95

Part II: Exploring the Time Domain 97

Chapter 5: Continuous-Time LTI Systems and the Convolution Integral 99

Establishing a General Input/Output Relationship 100

LTI systems and the impulse response 100

Developing the convolution integral 101

Looking at useful convolution integral properties 103

Working with the Convolution Integral 105

Seeing the general solution first 105

Solving problems with finite extent signals 107

Dealing with semi-infinite limits 111

Stepping Out and More 116

Step response from impulse response 116

BIBO stability implications 117

Causality and the impulse response 117

Chapter 6: Discrete-Time LTI Systems and the Convolution Sum 119

Specializing the Input/Output Relationship 120

Using LTI systems and the impulse response (sequence) 120

Getting to the convolution sum 121

Simplifying with Convolution Sum Properties and Techniques 124

Applying commutative, associative, and distributive properties 124

Convolving with the impulse function 126

Transforming a sequence 126

Solving convolution of finite duration sequences 128

Working with the Convolution Sum 133

Using spreadsheets and a tabular approach 133

Attacking the sum directly with geometric series 136

Connecting the step response and impulse response 144

Checking the BIBO stability 145

Checking for system causality 146

Chapter 7: LTI System Differential and Difference Equations in the Time Domain 149

Getting Differential 150

Introducing the general Nth-order system 150

Considering sinusoidal outputs in steady state 151

Finding the frequency response in general Nth-order LCC differential equations 153

Checking out the Difference Equations 156

Modeling a system using a general Nth-order LCC difference equation 156

Using recursion to find the impulse response of a first-order system 158

Considering sinusoidal outputs in steady state 159

Solving for the general Nth-order LCC difference equation frequency response 161

Part III: Picking Up the Frequency Domain 163

Chapter 8: Line Spectra and Fourier Series of Periodic Continuous-Time Signals 165

Sinusoids in the Frequency Domain 166

Viewing signals from the amplitude, phase, and frequency parameters 167

Forming magnitude and phase line spectra plots 168

Working with symmetry properties for real signals 171

Exploring spectral occupancy and shared resources 171

Establishing a sum of sinusoids: Periodic and aperiodic 172

General Periodic Signals: The Fourier Series Representation 175

Analysis: Finding the coefficients 176

Synthesis: Returning to a general periodic signal, almost 178

Checking out waveform examples 179

Working problems with coefficient formulas and properties 186

Chapter 9: The Fourier Transform for Continuous-Time Signals and Systems 191

Tapping into the Frequency Domain for Aperiodic Energy Signals 192

Working with the Fourier series 192

Using the Fourier transform and its inverse 194

Getting amplitude and phase spectra 197

Seeing the symmetry properties for real signals 197

Finding energy spectral density with Parseval’s theorem 201

Applying Fourier transform theorems 203

Checking out transform pairs 208

Getting Around the Rules with Fourier Transforms in the Limit 210

Handling singularity functions 210

Unifying the spectral view with periodic signals 211

LTI Systems in the Frequency Domain 213

Checking out the frequency response 214

Evaluating properties of the frequency response 214

Getting connected with cascade and parallel systems 216

Ideal filters 216

Realizable filters 218

Chapter 10: Sampling Theory 219

Seeing the Need for Sampling Theory 220

Periodic Sampling of a Signal: The ADC 221

Analyzing the Impact of Quantization Errors in the ADC 226

Analyzing Signals in the Frequency Domain 228

Impulse train to impulse train Fourier transform theorem 229

Finding the spectrum of a sampled bandlimited signal 230

Aliasing and the folded spectrum 233

Applying the Low-Pass Sampling Theorem 233

Reconstructing a Bandlimited Signal from Its Samples: The DAC 234

Interpolating with an ideal low-pass filter 236

Using a realizable low-pass filter for interpolation 239

Chapter 11: The Discrete-Time Fourier Transform for Discrete-Time Signals 241

Getting to Know DTFT 242

Checking out DTFT properties 243

Relating the continuous-time spectrum to the discrete-time spectrum 244

Getting even (or odd) symmetry properties for real signals 245

Studying transform theorems and pairs 249

Working with Special Signals 252

Getting mean-square convergence 252

Finding Fourier transforms in the limit 255

LTI Systems in the Frequency Domain 258

Taking Advantage of the Convolution Theorem 260

Chapter 12: The Discrete Fourier Transform and Fast Fourier Transform Algorithms 263

Establishing the Discrete Fourier Transform 264

The DFT/IDFT Pair 265

DFT Theorems and Properties 270

Carrying on from the DTFT 271

Circular sequence shift 272

Circular convolution 274

Computing the DFT with the Fast Fourier Transform 277

Decimation-in-time FFT algorithm 277

Computing the inverse FFT 280

Application Example: Transform Domain Filtering 280

Making circular convolution perform linear convolution 281

Using overlap and add to continuously filter sequences 281

Part IV: Entering the s- and z-Domains 283

Chapter 13: The Laplace Transform for Continuous-Time 285

Seeing Double: The Two-Sided Laplace Transform 286

Finding direction with the ROC 286

Locating poles and zeros 288

Checking stability for LTI systems with the ROC 289

Checking stability of causal systems through pole positions 290

Digging into the One-Sided Laplace Transform 290

Checking Out LT Properties 292

Transform theorems 292

Transform pairs 296

Getting Back to the Time Domain 298

Dealing with distinct poles 299

Working double time with twin poles 299

Completing inversion 299

Using tables to complete the inverse Laplace transform 300

Working with the System Function 302

Managing nonzero initial conditions 303

Checking the frequency response with pole-zero location 304

Chapter 14: The z-Transform for Discrete-Time Signals 307

The Two-Sided z-Transform 308

The Region of Convergence 309

The significance of the ROC 309

Plotting poles and zeros 311

The ROC and stability for LTI systems 311

Finite length sequences 313

Returning to the Time Domain 315

Working with distinct poles 316

Managing twin poles 316

Performing inversion 317

Using the table-lookup approach 317

Surveying z-Transform Properties 320

Transform theorems 321

Transform pairs 322

Leveraging the System Function 323

Applying the convolution theorem 324

Finding the frequency response with pole-zero geometry 325

Chapter 15: Putting It All Together: Analysis and Modeling Across Domains 327

Relating Domains 328

Using PyLab for LCC Differential and Difference Equations 329

Continuous time 330

Discrete time 332

Mashing Domains in Real-World Cases 334

Problem 1: Analog filter design with a twist 334

Problem 2: Solving the DAC ZOH droop problem in the z-domain 340

Part V: The Part of Tens 343

Chapter 16: More Than Ten Common Mistakes to Avoid When Solving Problems 345

Miscalculating the Folding Frequency 345

Getting Confused about Causality 346

Plotting Errors in Sinusoid Amplitude Spectra 346

Missing Your Arctan Angle 347

Being Unfamiliar with Calculator Functions 347

Foregoing the Return to LCCDE 348

Ignoring the Convolution Output Interval 348

Forgetting to Reduce the Numerator Order before Partial Fractions 348

Forgetting about Poles and Zeros from H(z) 349

Missing Time Delay Theorems 349

Disregarding the Action of the Unit Step in Convolution 349

Chapter 17: Ten Properties You Never Want to Forget 351

LTI System Stability 351

Convolving Rectangles 351

The Convolution Theorem 352

Frequency Response Magnitude 352

Convolution with Impulse Functions 352

Spectrum at DC 353

Frequency Samples of N-point DFT 353

Integrator and Accumulator Unstable 353

The Spectrum of a Rectangular Pulse 354

Odd Half-Wave Symmetry and Fourier Series Harmonics 354

Index 355

Mark Wickert, PhD, is a Professor of Electrical and Computer Engineering at the University of Colorado, Colorado Springs. He is a member of the IEEE and is doing real signals and systems problem solving as a consultant with local industry.