Table of Contents
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
