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