Signals and Systems For Dummies

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