Signal Processing First

PREFACE ......Page 19
1 Introduction ......Page 25
1-1 Mathematical Representation of Signals ......Page 26
1-2 Mathematical Representation of Systems ......Page 28
1-3 Thinking About Systems ......Page 29
1-4 The Next Step ......Page 30
2 Sinusoids ......Page 31
2-1 Tuning Fork Experiment ......Page 32
2-2 Review of Sine and Cosine Functions ......Page 33
2-3 Sinusoidal Signals ......Page 35
2-3.1 Relation of Frequency to Period ......Page 36
2-3.2 Phase Shift and Time Shift ......Page 37
2-4 * Sampling and Plotting Sinusoids ......Page 39
2-5.1 Review of Complex Numbers ......Page 41
2-5.2 Complex Exponential Signals ......Page 42
2-5.3 The Rotating Phasor Interpretation ......Page 43
2-5.4 Inverse Euler Formulas ......Page 45
2-6 Phasor Addition ......Page 46
2-6.2 Phasor Addition Rule ......Page 47
2-6.3 Phasor Addition Rule: Example ......Page 48
2-6.4 MATLAB Demo of Phasors ......Page 49
2-6.5 Summary of the Phasor Addition Rule ......Page 50
2-7.1 Equations from Laws of Physics ......Page 51
2-8 Time Signals: More Than Formulas ......Page 53
2-9 Summary and Links ......Page 54
2-10 Problems ......Page 55
3-1 The Spectrum of a Sum of Sinusoids ......Page 60
3-1.2 Graphical Plot of the Spectrum ......Page 62
3-2.1 Multiplication of Sinusoids ......Page 63
3-2.2 Beat Note Waveform ......Page 64
3-2.3 Amplitude Modulation ......Page 65
3-3 Periodic Waveforms ......Page 67
3-3.1 Synthetic Vowel ......Page 68
3-3.2 Example of a Nonperiodic Signal ......Page 69
3-4 Fourier Series ......Page 71
3-4.2 Fourier Series Derivation ......Page 72
3-5 Spectrum of the Fourier Series ......Page 74
3-6 Fourier Analysis of Periodic Signals ......Page 75
3-6.1 The Square Wave ......Page 76
3-6.2 Spectrum for a Square Wave ......Page 77
3-6.3 Synthesis of a Square Wave ......Page 78
3-6.4 Triangle Wave ......Page 79
3-6.5 Synthesis of a Triangle Wave ......Page 80
3-7 Time-Frequency Spectrum ......Page 81
3-7.2 Spectrogram Analysis ......Page 83
3-8.1 Chirp or Linearly Swept Frequency ......Page 84
3-8.2 A Closer Look at Instantaneous Frequency ......Page 86
3-9 Summary and Links ......Page 87
3-10 Problems ......Page 88
4-1 Sampling ......Page 95
4-1.1 Sampling Sinusoidal Signals ......Page 97
4-1.2 The Concept of Aliasing ......Page 99
4-1.3 Spectrum of a Discrete-Time Signal ......Page 100
4-1.4 The Sampling Theorem ......Page 101
4-1.5 Ideal Reconstruction ......Page 102
4-2.2 Over-Sampling ......Page 103
4-2.3 Aliasing Due to Under-Sampling ......Page 105
4-2.4 Folding Due to Under-Sampling ......Page 106
4-2.5 Maximum Reconstructed Frequency ......Page 107
4-3 Strobe Demonstration ......Page 108
4-3.1 Spectrum Interpretation ......Page 111
4-4.1 Interpolation with Pulses ......Page 112
4-4.2 Zero-Order Hold Interpolation ......Page 113
4-4.4 Cubic Spline Interpolation ......Page 114
4-4.5 Over-Sampling Aids Interpolation ......Page 115
4-4.6 Ideal Bandlimited Interpolation ......Page 116
4-5 The Sampling Theorem ......Page 117
4-6 Summary and Links ......Page 118
4-7 Problems ......Page 120
5 FIR Filters ......Page 125
5-2 The Running-Average Filter ......Page 126
5-3 The General FIR Filter ......Page 129
5-3.1 An Illustration of FIR Filtering ......Page 130
5-3.2.1 Unit Impulse Sequence ......Page 131
5-3.2.2 Unit Impulse Response Sequence ......Page 132
5-3.2.3 The Unit-Delay System ......Page 133
5-3.3.1 Computing the Output of a Convolution ......Page 134
5-4.1 Building Blocks ......Page 135
5-4.1.3 Unit Delay ......Page 136
5-4.2.1 Other Block Diagrams ......Page 137
5-5 Linear Time-Invariant (LTI) Systems ......Page 139
5-5.1 Time Invariance ......Page 140
5-5.3 TheFIRCase ......Page 141
5-6.1 Derivation of the Convolution Sum ......Page 142
5-6.2 Some Properties of LTI Systems ......Page 144
5-6.2.3 Associative Property of Convolution ......Page 145
5-7 Cascaded LTI Systems ......Page 146
5-8 Example of FIR Filtering ......Page 148
5-10 Problems ......Page 150
6-1 Sinusoidal Response of FIR Systems ......Page 154
6-2 Superposition and the Frequency Response ......Page 156
6-3 Steady-State and Transient Response ......Page 159
6-4.1 Relation to Impulse Response and Difference Equation ......Page 161
6-4.3 Conjugate Symmetry ......Page 162
6-5.1 Delay System ......Page 163
6-5.2 First-Difference System ......Page 164
6-5.3 A Simple Lowpass Filter ......Page 166
6-6 Cascaded LTI Systems ......Page 167
6-7 Running-Average Filtering ......Page 169
6-7.1 Plotting the Frequency Response ......Page 170
6-7.2 Cascade of Magnitude and Phase ......Page 172
6-7.3 Experiment: Smoothing an Image ......Page 173
6-8 Filtering Sampled Continuous-Time Signals ......Page 175
6-8.1 Example: Lowpass Averager ......Page 176
6-8.2 Interpretation of Delay ......Page 178
6-9 Summary and Links ......Page 179
6-10 Problems ......Page 181
7 z-Transforms ......Page 187
7-1 Definition of the z-Transform ......Page 188
7-2 The z-Transform and Linear Systems ......Page 189
7-2.1 The z-Transform of an FIR Filter ......Page 190
7-3 Properties of the z-Transform ......Page 191
7-3.2 The Time-Delay Property of the z-Transform ......Page 192
7-4.1 Unit-Delay Operator ......Page 193
7-4.3 Operator Notation in Block Diagrams ......Page 194
7-5 Convolution and the z-Transform ......Page 195
7-5.1 Cascading Systems ......Page 197
7-5.2 Factoring z-Polynomials ......Page 198
7-6 Relationship Between the z-Domain and the c2)-Domain ......Page 199
7-6.1 The z-Plane and the Unit Circle ......Page 200
7-6.2 The Zeros and Poles of H(z) ......Page 201
7-6.3 Significance of the Zeros of H(z) ......Page 202
7-6.4 Nulling Filters ......Page 203
7-6.5 Graphical Relation Between z and a) ......Page 204
7-7.1 The L-Point Running-Sum Filter ......Page 205
7-7.2 A Complex Bandpass Filter ......Page 207
7-7.3 A Bandpass Filter with Real Coefficients ......Page 209
7-8 Practical Bandpass Filter Design ......Page 210
7-9.2 Locations of the Zeros of FIR Linear-Phase Systems ......Page 213
7-10 Summary and Links ......Page 214
7-11 Problems ......Page 215
8 IIR Filters ......Page 220
8-1 The General IIR Difference Equation ......Page 221
8-2 Time-Domain Response ......Page 222
8-2.1 Linearity and Time Invariance of IIR Filters ......Page 223
8-2.2 Impulse Response of a First-Order IIR System ......Page 224
8-2.3 Response to Finite-Length Inputs ......Page 225
8-2.4 Step Response of a First-Order Recursive System ......Page 226
8-3 System Function of an IIR Filter ......Page 228
8-3.1 The General First-Order Case ......Page 229
8-3.2.1 Direct Form I Structure ......Page 230
8-3.2.2 Direct Form II Structure ......Page 231
8-3.2.3 The Transposed Form Structure ......Page 232
8-3.4 Summary of the Method ......Page 233
8-4 Poles and Zeros ......Page 234
8-4.2 Pole Locations and Stability ......Page 235
8-5 Frequency Response of an IIR Filter ......Page 236
8-5.1 Frequency Response using MATLAB ......Page 237
8-5.2 Three-Dimensional Plot of a System Function ......Page 238
8-7 The Inverse ^-Transform and Some Applications ......Page 240
8-7.1 Revisiting the Step Response of a First-Order System ......Page 241
8-7.2 A General Procedure for Inverse z-Transformation ......Page 242
8-8 Steady-State Response and Stability ......Page 244
8-9.1 z-Transform of Second-Order Filters ......Page 247
8-9.2 Structures for Second-Order IIR Systems ......Page 248
8-9.3 Poles and Zeros ......Page 249
8-9.4 Impulse Response of a Second-Order IIR System ......Page 250
8-9.4.1 Real Poles ......Page 251
8-9.5 Complex Poles ......Page 252
8-10 Frequency Response of Second-Order IIR Filter ......Page 255
8-10.2 3-dB Bandwidth ......Page 256
8-10.3 Three-Dimensional Plot of System Functions ......Page 257
8-11 Example of an IIR Lowpass Filter ......Page 260
8-12 Summary and Links ......Page 261
8-13 Problems ......Page 262
9 Continuous-Time Signals and LTI Systems ......Page 269
9-1.1 Two-Sided Infinite-Length Signals ......Page 270
9-1.2 One-Sided Signals ......Page 271
9-2 The Unit Impulse ......Page 272
9-2.1 Sampling Property of the Impulse ......Page 274
9-2.4 Derivative of the Unit Step ......Page 276
9-3.1 Some Basic Continuous-Time Systems ......Page 278
9-4 Linear Time-Invariant Systems ......Page 279
9-4.2 Linearity ......Page 280
9-4.3 The Convolution Integral ......Page 281
9-4.4 Properties of Convolution ......Page 283
9-5.1 Integrator ......Page 284
9-6 Convolution of Impulses ......Page 285
9-7.1 Delayed Unit-Step Input ......Page 287
9-7.2 Evaluation of Discrete Convolution ......Page 291
9-7.3 Square-Pulse Input ......Page 292
9-7.4 Very Narrow Square Pulse Input ......Page 293
9-8.1 Cascade and Parallel Combinations ......Page 294
9-8.2 Differentiation and Integration of Convolution ......Page 296
9-8.3 Stability and Causality ......Page 297
9-9 Using Convolution to Remove Multipath Distortion ......Page 300
9-10 Summary ......Page 302
9-11 Problems ......Page 303
10-1 The Frequency Response Function for LTI Systems ......Page 309
10-1.1 Plotting the Frequency Response ......Page 311
10-1.2 Magnitude and Phase Changes ......Page 312
10-2 Response to Real Sinusoidal Signals ......Page 313
10-2.2 Symmetry of H(ja>) ......Page 314
10-2.3 Response to a General Sum of Sinusoids ......Page 317
10-2.4 Periodic Input Signals ......Page 318
10-3.1 Ideal Delay System ......Page 319
10-3.2 Ideal Lowpass Filter ......Page 320
10-3.4 Ideal Bandpass Filter ......Page 321
10-4 Application of Ideal Filters ......Page 322
10-5 Time-Domain or Frequency-Domain? ......Page 324
10-6 Summary/Future ......Page 325
10-7 Problems ......Page 326
11 Continuous-Time Fourier Transform ......Page 331
11-1 Definition of the Fourier Transform ......Page 332
11-2.1 Limit of the Fourier Series ......Page 334
11-3 Existence and Convergence of the Fourier Transform ......Page 336
11-4.1 Right-Sided Real Exponential Signals ......Page 337
11-4.2 Rectangular Pulse Signals ......Page 338
11-4.3 Bandlimited Signals ......Page 340
11-4.4 Impulse in Time or Frequency ......Page 341
11-4.5 Sinusoids ......Page 342
11-4.6 Periodic Signals ......Page 343
11-5.1 The Scaling Property ......Page 346
11-5.2 Symmetry Properties of Fourier Transform Pairs ......Page 348
11-6.1 Frequency Response ......Page 350
11-6.2 Fourier Transform of a Convolution ......Page 351
11-6.3.1 Convolution of Two Bandlimited Functions ......Page 352
11-6.3.2 Product of Two Sine Functions ......Page 353
11-6.3.3 Partial Fraction Expansions ......Page 354
11-7.1 Time Delay ......Page 356
11-7.2 Differentiation ......Page 357
11-7.3 Systems Described by Differential Equations ......Page 358
11-8.1 The General Signal Multiplication Property ......Page 359
11-8.2 The Frequency Shifting Property ......Page 360
11-10 Using the Fourier Transform for Multipath Analysis ......Page 361
11-11 Summary ......Page 365
11-12 Problems ......Page 366
12-1 Linear Time-Invariant Systems ......Page 370
12-1.1 Cascade and Parallel Configurations ......Page 371
12-1.2 Ideal Delay ......Page 372
12-1.3.1 Ideal Lowpass Filter ......Page 375
12-1.3.2 Other Ideal Frequency Selective Filters ......Page 376
12-1.4 Example of Filtering in the Frequency-Domain ......Page 377
12-1.5 Compensation for the Effect of an LTI Filter ......Page 379
12-2.1 Double-Sideband Amplitude Modulation ......Page 382
12-2.2 DSBAM with Transmitted Carrier (DSBAM-TC) ......Page 386
12-2.3 Frequency Division Multiplexing ......Page 390
12-3.1 The Sampling Theorem and Aliasing ......Page 392
12-3.2 Bandlimited Signal Reconstruction ......Page 394
12-3.3 Bandlimited Interpolation ......Page 396
12-3.4 Ideal C-to-D and D-to-C Converters ......Page 397
12-3.5 The Discrete-Time Fourier Transform ......Page 399
12-3.6 The Inverse DTFT ......Page 400
12-3.7 Discrete-Time Filtering of Continuous-Time Signals ......Page 401
12-4 Summary ......Page 404
12-5 Problems ......Page 405
13 Computing the Spectrum ......Page 413
13-1 Finite Fourier Sum ......Page 414
13-2 Too Many Fourier Transforms? ......Page 415
13-2.1 Relation of the DTFT to the CTFT ......Page 416
13-3 Time-Windowing ......Page 417
13-4 Analysis of a Sum of Sinusoids ......Page 419
13-4.1 DTFT of a Windowed Sinusoid ......Page 422
13-5 Discrete Fourier Transform ......Page 423
13-5.1 The Inverse DFT ......Page 424
13-5.2 Summary of the DFT Representation ......Page 425
13-5.4 Negative Frequencies and the DFT ......Page 426
13-5.5 DFT Example ......Page 427
13-6 Spectrum Analysis of Finite-Length Signals ......Page 429
13-7 Spectrum Analysis of Periodic Signals ......Page 431
13-8 The Spectrogram ......Page 432
13-8.1 Spectrogram Display ......Page 433
13-8.3 Spectrogram of a Sampled Periodic Signal ......Page 434
13-8.4 Resolution of the Spectrogram ......Page 435
13-8.4.1 Resolution Experiment ......Page 436
13-8.5 Spectrogram of a Musical Scale ......Page 437
13-8.6 Spectrogram of a Speech Signal ......Page 439
13-8.7 Filtered Speech ......Page 442
13-9.1 Derivation of the FFT ......Page 444
13-9.1.1 FFT Operation Count ......Page 445
13-10 Summary and Links ......Page 447
13-11 Problems ......Page 448
A Complex Numbers ......Page 451
A-2.1 Rectangular Form ......Page 452
A-2.2 Polar Form ......Page 453
A-2.3 Conversion: Rectangular and Polar ......Page 454
A-3 Euler's Formula ......Page 455
A-4 Algebraic Rules for Complex Numbers ......Page 456
A-5 Geometric Views of Complex Operations ......Page 458
A-5.1 Geometric View of Addition ......Page 459
A-5.2 Geometric View of Subtraction ......Page 460
A-5.5 Geometric View of the Inverse, z~l ......Page 461
A-6 Powers and Roots ......Page 462
A-6.1 Roots of Unity ......Page 463
A-6.1.1 Procedure for Finding Multiple Roots ......Page 464
A-8 Problems ......Page 465
B Programming in MATLAB ......Page 467
B-2 Matrix Operations and Variables ......Page 468
B-2.2.1 A Review of Matrix Multiplication ......Page 469
B-3 Plots and Graphics ......Page 470
B-4 Programming Constructs ......Page 471
B-6 Writing a MATLAB Function ......Page 472
B-6.1 Creating A Clip Function ......Page 473
B-7 Programming Tips ......Page 475
B-7.3 Vectorizing Logical Operations ......Page 476
B-7.5 The Find Function ......Page 477
B-7.7 Programming Style ......Page 478
C Laboratory Projects ......Page 479
C-1.1.2 Movies: MATLAB Tutorials ......Page 481
C-1.2 Warm-up ......Page 482
C-1.2.2 MATLAB Script Files ......Page 483
C-1.3 Laboratory: Manipulating Sinusoids with MATLAB ......Page 484
C-1.4 Lab Review Questions ......Page 485
C-2.1.2 Background: Telephone Touch-Tone Dialing ......Page 487
C-2.2.1 Signal Concatenation ......Page 488
C-2.3 Warm-up: DTMF Synthesis ......Page 489
C-2.3.1 DTMF Dial Function ......Page 490
C-2.3.2 Simple Bandpass Filter Design ......Page 491
C-2.4.1 Filter Bank Design: dtmfdesign.m ......Page 492
C-2.4.2 A Scoring Function: dtmf score .m ......Page 493
C-2.4.3 DTMF Decode Function: dtmfrun.m ......Page 494
C-2.4.5 Telephone Numbers ......Page 495
C-2.4.6 Demo ......Page 496
C-3.2.1 Discrete-Time Convolution Demo ......Page 497
C-3.2.2 Continuous-Time Convolution Demo ......Page 498
C-3.4.1 Continuous-Time Convolution ......Page 499
C-3.4.3 Discrete-Time Convolution ......Page 500
D CD-ROM Demos ......Page 502
Index ......Page 506