- ISBN: 9780133943283 | 0133943283
- Cover: Hardcover
- Copyright: 4/1/2015
For upper-level undergraduate courses in deterministic and stochastic signals and system engineering
An Integrative Approach to Signals, Systems and Inference
Signals, Systems and Inference is a comprehensive text that builds on introductory courses in time- and frequency-domain analysis of signals and systems, and in probability. Directed primarily to upper-level undergraduates and beginning graduate students in engineering and applied science branches, this new textbook pioneers a novel course of study. Instead of the usual leap from broad introductory subjects to highly specialized advanced subjects, this engaging and inclusive text creates a study track for a transitional course. Properties and representations of deterministic signals and systems are reviewed and elaborated on, including group delay and the structure and behavior of state-space models.
The text also introduces and interprets correlation functions and power spectral densities for describing and processing random signals. Application contexts include pulse amplitude modulation, observer-based feedback control, optimum linear filters for minimum mean-square-error estimation, and matched filtering for signal detection. Model-based approaches to inference are emphasized, in particular for state estimation, signal estimation, and signal detection. The text explores ideas, methods and tools common to numerous fields involving signals, systems and inference: signal processing, control, communication, time-series analysis, financial engineering, biomedicine, and many others. Signals, Systems, and Inference is a long-awaited and flexible text that can be used for a rigorous course in a broad range of engineering and applied science curricula.
1 Introduction 11
2 Signals and Systems 13
2.1 Signals, Systems, Models, Properties . . . . . . . . . . . . . . . . . . 13
2.1.1 System Properties . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Linear, Time-Invariant Systems . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Impulse-Response Representation of LTI Systems . . . . . . . 17
2.2.2 Eigenfunction and Transform Representation of LTI Systems 19
2.2.3 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Deterministic Signals and their Fourier Transforms . . . . . . . . . . 23
2.3.1 Signal Classes and their Fourier Transforms . . . . . . . . . . 23
2.3.2 Parseval’s Identity, Energy Spectral Density, Deterministic Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 The Bilateral Laplace and Z-Transforms . . . . . . . . . . . . . . . . 29
2.4.1 The Bilateral Z-Transform . . . . . . . . . . . . . . . . . . . 29
2.4.2 The Inverse Z-Transform . . . . . . . . . . . . . . . . . . . . 32
2.4.3 The Bilateral Laplace Transform . . . . . . . . . . . . . . . . 33
2.5 Discrete-Time Processing of Continuous-Time Signals . . . . . . . . 34
2.5.1 Basic Structure for DT Processing of CT Signals . . . . . . . 34
2.5.2 DT Filtering, and Overall CT Response . . . . . . . . . . . . 37
2.5.3 Non-Ideal D/C converters . . . . . . . . . . . . . . . . . . . . 39
2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6.2 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . 57
2.6.3 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . 70
3 Frequency Domain Amplitude, Phase and Group Delay 81
3.1 Fourier Transform Magnitude and Phase . . . . . . . . . . . . . . . . 81
3.2 Group Delay and The Effect of Nonlinear Phase . . . . . . . . . . . 85
3.2.1 Narrowband Input Signals . . . . . . . . . . . . . . . . . . . . 85
3.2.2 Broadband Input Signals . . . . . . . . . . . . . . . . . . . . 87
3.3 All-Pass and Minimum-Phase Systems . . . . . . . . . . . . . . . . . 91
3.3.1 All-Pass Systems . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.3.2 Minimum-Phase Systems . . . . . . . . . . . . . . . . . . . . 94
3.3.3 The Group Delay of Minimum-Phase Systems . . . . . . . . . 94
3.4 Spectral Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.5.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.5.2 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . 109
3.5.3 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . 122
4 Pulse Amplitude Modulation 125
4.1 Baseband Pulse Amplitude Modulation . . . . . . . . . . . . . . . . 125
4.1.1 The Transmitted Signal . . . . . . . . . . . . . . . . . . . . . 126
4.1.2 The Received Signal . . . . . . . . . . . . . . . . . . . . . . . 127
4.1.3 Frequency-Domain Characterizations . . . . . . . . . . . . . . 128
4.1.4 Inter-Symbol Interference at the Receiver . . . . . . . . . . . 131
4.2 Nyquist Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.3 Passband Pulse Amplitude Modulation . . . . . . . . . . . . . . . . . 136
4.3.1 Frequency-Shift Keying (FSK) . . . . . . . . . . . . . . . . . 137
4.3.2 Phase-Shift Keying (PSK) . . . . . . . . . . . . . . . . . . . . 137
4.3.3 Quadrature Amplitude Modulation . . . . . . . . . . . . . . . 139
4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.4.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.4.2 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . 149
4.4.3 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . 153
5 State-Space Models 163
5.1 System Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.3 State-Space Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.3.1 DT State-Space Models . . . . . . . . . . . . . . . . . . . . . 177
5.3.2 CT State-Space Models . . . . . . . . . . . . . . . . . . . . . 180
5.3.3 Defining Properties of State-Space Models . . . . . . . . . . . 183
5.4 State-Space Models from LTI Input—Output Models . . . . . . . . . 185
5.5 Equilibria and Linearization of Nonlinear State-Space Models . . . . 191
5.5.1 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.5.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.6.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.6.2 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . 204
5.6.3 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . 207
6 LTI State-Space Models 211
6.1 Continuous-Time and Discrete-Time LTI Models . . . . . . . . . . . 211
6.2 Zero-Input Response and Modal Representation . . . . . . . . . . . . 214
6.2.1 Undriven CT Systems . . . . . . . . . . . . . . . . . . . . . . 214
6.2.2 Undriven DT Systems . . . . . . . . . . . . . . . . . . . . . . 221
6.2.3 Asymptotic Stability of LTI Systems . . . . . . . . . . . . . . 224
6.3 General Response in Modal Coordinates . . . . . . . . . . . . . . . . 229
6.3.1 Driven CT Systems . . . . . . . . . . . . . . . . . . . . . . . 229
6.3.2 Driven DT Systems . . . . . . . . . . . . . . . . . . . . . . . 231
6.3.3 Similarity Transformations and Diagonalization . . . . . . . . 234
6.4 Transfer Functions, Hidden Modes, Reachability, Observability . . . 240
6.4.1 Input-State-Output Structure of CT Systems . . . . . . . . . 241
6.4.2 Input-State-Output Structure of DT Systems . . . . . . . . . 249
6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
6.5.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 260
6.5.2 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . 270
6.5.3 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . 276
7 State Observers and State Feedback 281
7.1 Plant and Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
7.2 State Estimation and Observers . . . . . . . . . . . . . . . . . . . . . 283
7.2.1 Real-Time Simulation . . . . . . . . . . . . . . . . . . . . . . 284
7.2.2 The State Observer . . . . . . . . . . . . . . . . . . . . . . . 286
7.2.3 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . 287
7.3 State Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . 298
7.3.1 Open-Loop Control . . . . . . . . . . . . . . . . . . . . . . . . 299
7.3.2 Closed-Loop Control via LTI State Feedback . . . . . . . . . 299
7.3.3 LTI State Feedback Design . . . . . . . . . . . . . . . . . . . 301
7.4 Observer-Based Feedback Control . . . . . . . . . . . . . . . . . . . . 308
7.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
7.5.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 314
7.5.2 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . 323
7.5.3 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . 326
8 Probabilistic Models 329
8.1 The Basic Probability Model . . . . . . . . . . . . . . . . . . . . . . 329
8.2 Conditional Probability, Bayes’ Rule, and Independence . . . . . . . 330
8.3 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
8.4 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 333
8.5 Jointly Distributed Random Variables . . . . . . . . . . . . . . . . . 335
8.6 Expectations, Moments and Variance . . . . . . . . . . . . . . . . . . 337
8.7 Correlation and Covariance for Bivariate Random Variables . . . . . 340
8.8 A Vector-Space Interpretation of Correlation Properties . . . . . . . 345
8.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
8.9.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 347
8.9.2 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . 349
8.9.3 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . 354
9 Estimation 359
9.1 Estimation of a Continuous Random Variable . . . . . . . . . . . . . 359
9.2 From Estimates to the Estimator . . . . . . . . . . . . . . . . . . . . 365
9.2.1 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
9.3 Linear Minimum Mean Square Error Estimation . . . . . . . . . . . 371
9.3.1 Linear Estimation of One Random Variable From a Single
Measurement of Another . . . . . . . . . . . . . . . . . . . . 371
9.3.2 Multiple Measurements . . . . . . . . . . . . . . . . . . . . . 376
9.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
9.4.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 381
9.4.2 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . 387
9.4.3 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . 395
10 Hypothesis Testing 401
10.1 Binary Pulse Amplitude Modulation in Noise . . . . . . . . . . . . . 401
10.2 Hypothesis Testing with Minimum Error Probability . . . . . . . . . 403
10.2.1 Deciding with Minimum Conditional Probability of Error . . 404
10.2.2 MAP Decision Rule for Minimum Overall Probability of Error 405
10.2.3 Hypothesis Testing in Coded Digital Communication . . . . . 408
10.3 Binary Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . 411
10.3.1 False Alarm, Miss, and Detection . . . . . . . . . . . . . . . . 413
10.3.2 The Likelihood Ratio Test . . . . . . . . . . . . . . . . . . . . 415
10.3.3 Neyman-PearsonDecision Rule and Receiver Operating Characteristic
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
10.4 Minimum Risk Decisions . . . . . . . . . . . . . . . . . . . . . . . . . 421
10.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
10.5.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 423
10.5.2 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . 429
10.5.3 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . 436
10.5.4 Problems Commented Out in 2010 . . . . . . . . . . . . . . . 444
11 Random Processes 447
11.1 Definition and Examples of a Random Process . . . . . . . . . . . . 447
11.2 First and Second Moment Characterization of Random Processes . . 452
11.3 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
11.3.1 Strict-Sense Stationarity . . . . . . . . . . . . . . . . . . . . . 453
11.3.2 Wide-Sense Stationarity . . . . . . . . . . . . . . . . . . . . . 454
11.3.3 Some Properties of WSS Correlation and Covariance Functions
11.4 Further Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
11.6 Linear Estimation of Random Processes . . . . . . . . . . . . . . . . 460
11.6.1 Linear Prediction . . . . . . . . . . . . . . . . . . . . . . . . . 460
11.6.2 Linear FIR Filtering . . . . . . . . . . . . . . . . . . . . . . . 462
11.7 LTI Filtering of WSS Processes . . . . . . . . . . . . . . . . . . . . . 463
11.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
11.8.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 470
11.8.2 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . 477
11.8.3 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . 485
11.8.4 Unused Problems . . . . . . . . . . . . . . . . . . . . . . . . . 495
11.8.5 Possible Examples . . . . . . . . . . . . . . . . . . . . . . . . 496
12 Power Spectral Density 497
12.1 Spectral Distribution of Expected Instantaneous Power . . . . . . . . 498
12.1.1 Power Spectral Density (PSD) . . . . . . . . . . . . . . . . . 498
12.1.2 Fluctuation Spectral Density . . . . . . . . . . . . . . . . . . 502
12.1.3 Cross-Spectral Density . . . . . . . . . . . . . . . . . . . . . . 507
12.2 Expected Time-Averaged Power Spectrum and the Einstein-Wiener-Khinchin Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
12.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
12.3.1 Revealing Cyclic Components . . . . . . . . . . . . . . . . . . 513
12.3.2 Modeling Filters . . . . . . . . . . . . . . . . . . . . . . . . . 514
12.3.3 Whitening Filters . . . . . . . . . . . . . . . . . . . . . . . . . 517
12.3.4 Sampling Bandlimited Random Processes . . . . . . . . . . . 518
12.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
12.4.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 523
12.4.2 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . 530
12.4.3 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . 535
13 Signal Estimation 547
13.1 LMMSE Estimation for Random Variables . . . . . . . . . . . . . . . 547
13.2 FIR Wiener Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
13.3 The Unconstrained DT Wiener Filter . . . . . . . . . . . . . . . . . 555
13.4 Causal DT Wiener Filtering . . . . . . . . . . . . . . . . . . . . . . . 564
13.5 Optimal Observers and Kalman Filtering . . . . . . . . . . . . . . . 571
13.5.1 Causal Estimation of a Signal Corrupted by Additive Noise . 571
13.5.2 Observer Implementation of the Wiener Filter . . . . . . . . 573
13.5.3 Optimal State Estimates and Kalman Filtering . . . . . . . . 575
13.6 Estimation of CT Signals . . . . . . . . . . . . . . . . . . . . . . . . 576
13.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
13.7.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 578
13.7.2 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . 598
13.7.3 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . 601
14 Signal Detection 603
14.1 Hypothesis Testing with Multiple Measurements . . . . . . . . . . . 604
14.2 Detecting a Known Signal in IID Gaussian Noise . . . . . . . . . . . 606
14.2.1 The Optimal Solution . . . . . . . . . . . . . . . . . . . . . . 607
14.2.2 Characterizing Performance . . . . . . . . . . . . . . . . . . . 609
14.2.3 Matched Filtering . . . . . . . . . . . . . . . . . . . . . . . . 612
14.3 Extensions of Matched-Filter Detection . . . . . . . . . . . . . . . . 614
14.3.1 Infinite-Duration, Finite-Energy Signals . . . . . . . . . . . . 614
14.3.2 Maximizing SNR for Signal Detection in White Noise . . . . 614
14.3.3 Detection in Colored Noise . . . . . . . . . . . . . . . . . . . 617
14.3.4 Continuous-Time Matched Filters . . . . . . . . . . . . . . . 620
14.3.5 Matched FIltering and Nyquist Pulse Design . . . . . . . . . 621
14.3.6 Unknown Arrival Time, and Pulse Compression . . . . . . . . 622
14.4 Signal Discrimination in IID Gaussian Noise . . . . . . . . . . . . . . 624
14.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
14.5.1 Basic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 631
14.5.2 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . 640
14.5.3 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . 650
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