Preface ........................................................ xi
The Cover ...................................................... xv
Acknowledgments .............................................. xvii
Prologue .................................................... xxiii
1 Signals and Systems ........................................ 1
1.1 Signals, Systems, Models, and Properties ................... 1
1.1.1 System Properties .................................. 3
1.2 Linear, Time-Invariant Systems ............................. 5
1.2.1 Impulse-Response Representation of LTI Systems ...... 5
1.2.2 Eigenfunction and Transform Representation of
LTI Systems ......................................... 6
1.2.3 Fourier Transforms ................................. 10
1.3 Deterministic Signals and Their Fourier Transforms ........ 11
1.3.1 Signal Classes and Their Fourier Transforms ........ 11
1.3.2 Parseval's Identity, Energy Spectral Density,
and Deterministic Autocorrelation .................. 14
1.4 Bilateral Laplace and Z-Transforms ........................ 16
1.4.1 The Bilateral z-Transform .......................... 16
1.4.2 The Bilateral Laplace Transform .................... 20
1.5 Discrete-Time Processing of Continuous-Time Signals ....... 21
1.5.1 Basic Structure for DT Processing of CT Signals .... 22
1.5.2 DT Filtering and Overall CT Response ............... 24
1.5.3 Nonideal D/C Converters ............................ 26
1.6 Further Reading ........................................... 28
Problems .................................................. 29
Basic Problems ............................................ 29
Advanced Problems ......................................... 40
Extension Problems ........................................ 51
2 Amplitude, Phase, and Group Delay ......................... 62
2.1 Fourier Transform Magnitude and Phase ..................... 62
2.2 Group Delay and the Effect of Nonlinear Phase ............. 66
2.2.1 Narrowband Input Signals ........................... 66
2.2.2 Broadband Input Signals ............................ 68
2.3 All-Pass and Minimum-Phase Systems ........................ 73
2.3.1 All-Pass Systems ................................... 73
2.3.2 Minimum-Phase Systems .............................. 75
2.4 Spectral Factorization .................................... 78
2.5 Further Reading ........................................... 80
Problems .................................................. 80
Basic Problems ............................................ 80
Advanced Problems ......................................... 88
Extension Problems ....................................... 100
3 Pulse-Amplitude Modulation ............................... 102
3.1 Baseband Pulse-Amplitude Modulation ...................... 103
3.1.1 The Transmitted Signal ............................ 103
3.1.2 The Received Signal ............................... 105
3.1.3 Frequency-Domain Characterizations ................ 105
3.1.4 Intersymbol Interference at the Receiver .......... 108
3.2 Nyquist Pulses ........................................... 110
3.3 Passband Pulse-Amplitude Modulation ...................... 113
3.3.1 Frequency-Shift Keying (FSK) ...................... 114
3.3.2 Phase-Shift Keying (PSK) .......................... 114
3.3.3 Quadrature-Amplitude Modulation (QAM) ............. 116
3.4 Further Reading .......................................... 118
Problems ................................................. 119
Basic Problems ........................................... 119
Advanced Problems ........................................ 123
Extension Problems ....................................... 126
4 State-Space Models ....................................... 133
4.1 System Memory ............................................ 133
4.2 Illustrative Examples .................................... 134
4.3 State-Space Models ....................................... 146
4.3.1 DT State-Space Models ............................. 146
4.3.2 CT State-Space Models ............................. 149
4.3.3 Defining Properties of State-Space Models ......... 151
4.4 State-Space Models from LTI Input-Output
Models ................................................... 153
4.5 Equilibria and Linearization of Nonlinear State-Space
Models ................................................... 158
4.5.1 Equilibrium ....................................... 158
4.5.2 Linearization ..................................... 161
4.6 Further Reading .......................................... 164
Problems ................................................. 165
Basic Problems ........................................... 165
Advanced Problems ........................................ 169
Extension Problems ....................................... 172
5 LTI State-Space Models ................................... 174
5.1 Continuous-Time and Discrete-Time LTI Models ............. 174
5.2 Zero-Input Response and Modal Representation ............. 177
5.2.1 Undriven CT Systems ............................... 177
5.2.2 Undriven DT Systems ............................... 185
5.2.3 Asymptotic Stability of LTI Systems ............... 187
5.3 General Response in Modal Coordinates .................... 191
5.3.1 Driven CT Systems ................................. 191
5.3.2 Driven DT Systems ................................. 194
5.3.3 Similarity Transformations and Diagonalization .... 196
5.4 Transfer Functions, Hidden Modes, Reachability,
and Observability ........................................ 202
5.4.1 Input-State-Output Structure of CT Systems ........ 202
5.4.2 Input-State-Output Structure of DT Systems ........ 210
5.5 Further Reading .......................................... 219
Problems ................................................. 220
Basic Problems ........................................... 220
Advanced Problems ........................................ 228
Extension Problems ....................................... 233
6 State Observers and State Feedback ....................... 236
6.1 Plant and Model .......................................... 237
6.2 State Estimation and Observers ........................... 239
6.2.1 Real-Time Simulation .............................. 239
6.2.2 The State Observer ................................ 241
6.2.3 Observer Design ................................... 243
6.3 State Feedback Control ................................... 252
6.3.1 Open-Loop Control ................................. 252
6.3.2 Closed-Loop Control via LTI State Feedback ........ 253
6.3.3 LTI State Feedback Design ......................... 254
6.4 Observer-Based Feedback Control .......................... 262
6.5 Further Reading .......................................... 267
Problems ................................................. 267
Basic Problems ........................................... 267
Advanced Problems ........................................ 274
Extension Problems ....................................... 277
7 Probabilistic Models ..................................... 279
7.1 The Basic Probability Model .............................. 279
7.2 Conditional Probability, Bayes' Rule, and Independence ... 280
7.3 Random Variables ......................................... 283
7.4 Probability Distributions ................................ 283
7.5 Jointly Distributed Random Variables ..................... 285
7.6 Expectations, Moments, and Variance ...................... 287
7.7 Correlation and Covariance for Bivariate Random
Variables ................................................ 290
7.8 A Vector-Space Interpretation of Correlation Properties .. 294
7.9 Further Reading .......................................... 296
Problems ...................................................... 297
Basic Problems ................................................ 297
Advanced Problems ............................................. 298
Extension Problems ............................................ 302
8 Estimation ............................................... 306
8.1 Estimation of a Continuous Random Variable ............... 307
8.2 From Estimates to the Estimator .......................... 312
8.2.1 Orthogonality ..................................... 317
8.3 Linear Minimum Mean Square Error Estimation .............. 318
8.3.1 Linear Estimation of One Random Variable from
a Single Measurement of Another ................... 318
8.3.2 Multiple Measurements ............................. 323
8.4 Further Reading .......................................... 327
Problems ................................................. 328
Basic Problems ........................................... 328
Advanced Problems ........................................ 332
Extension Problems ....................................... 338
9 Hypothesis Testing ....................................... 343
9.1 Binary Pulse-Amplitude Modulation in Noise ............... 343
9.2 Hypothesis Testing with Minimum Error Probability ........ 345
9.2.1 Deciding with Minimum Conditional Probability
of Error .......................................... 346
9.2.2 MAP Decision Rule for Minimum Overall Probability
of Error .......................................... 347
9.2.3 Hypothesis Testing in Coded Digital
Communication ..................................... 350
9.3 Binary Hypothesis Testing ................................ 353
9.3.1 False Alarm, Miss, and Detection .................. 354
9.3.2 The Likelihood Ratio Test ......................... 356
9.3.3 Neyman-Pearson Decision Rule and Receiver
Operating Characteristic .......................... 357
9.4 Minimum Risk Decisions ................................... 361
9.5 Further Reading .......................................... 363
Problems ................................................. 363
Basic Problems ........................................... 363
Advanced Problems ........................................ 368
Extension Problems ....................................... 373
10 Random Processes ......................................... 380
10.1 Definition and Examples of a Random Process .............. 380
10.2 First- and Second-Moment Characterization of Random
Processes ................................................ 385
10.3 Stationarity ............................................. 386
10.3.1 Strict-Sense Stationarity ......................... 386
10.3.2 Wide-Sense Stationarity ........................... 386
10.3.3 Some Properties of WSS Correlation and Covariance
Functions ......................................... 388
10.4 Ergodicity ............................................... 391
10.5 Linear Estimation of Random Processes .................... 392
10.5.1 Linear Prediction ................................. 392
10.5.2 Linear FIR Filtering .............................. 394
10.6 LTI Filtering of WSS Processes ........................... 395
10.7 Further Reading .......................................... 401
Problems ................................................. 401
Basic Problems ........................................... 401
Advanced Problems ........................................ 406
Extension Problems ....................................... 412
11 Power Spectral Density ................................... 421
11.1 Spectral Distribution of Expected Instantaneous
Power .................................................... 422
11.1.1 Power Spectral Density ............................ 422
11.1.2 Fluctuation Spectral Density ...................... 426
11.1.3 Cross-Spectral Density ............................ 431
11.2 Expected Time-Averaged Power Spectrum and the
Einstein-Wiener-Khinchin Theorem ......................... 432
11.3 Applications ............................................. 437
11.3.1 Revealing Cyclic Components ....................... 437
11.3.2 Modeling Filters .................................. 439
11.3.3 Whitening Filters ................................. 443
11.3.4 Sampling Bandlimited Random Processes ............. 444
11.4 Further Reading .......................................... 444
Problems ................................................. 445
Basic Problems ........................................... 445
Advanced Problems ........................................ 451
Extension Problems ....................................... 455
12 Signal Estimation ........................................ 464
12.1 LMMSE Estimation for Random Variables .................... 465
12.2 FIR Wiener Filters ....................................... 467
12.3 The Unconstrained DT Wiener Filter ....................... 472
12.4 Causal DT Wiener Filtering ............................... 480
12.5 Optimal Observers and Kalman Filtering ................... 487
12.5.1 Causal Wiener Filtering of a Signal Corrupted
by Additive Noise ................................. 487
12.5.2 Observer Implementation of the Wiener Filter ...... 489
12.5.3 Optimal State Estimates and Kalman Filtering ...... 491
12.6 Estimation of CT Signals ................................. 492
12.7 Further Reading .......................................... 493
Problems ................................................. 493
Basic Problems ........................................... 493
Advanced Problems ........................................ 504
Extension Problems ....................................... 510
13 Signal Detection ......................................... 511
13.1 Hypothesis Testing with Multiple Measurements ............ 512
13.2 Detecting a Known Signal in I.I.D. Gaussian Noise ........ 514
13.2.1 The Optimal Solution .............................. 515
13.2.2 Characterizing Performance ........................ 517
13.2.3 Matched Filtering ................................. 519
13.3 Extensions of Matched-Filter Detection ................... 522
13.3.1 Infinite-Duration, Finite-Energy Signals .......... 522
13.3.2 Maximizing SNR for Signal Detection in White
Noise ............................................. 522
13.3.3 Detection in Colored Noise ........................ 525
13.3.4 Continuous-Time Matched Filters ................... 528
13.3.5 Matched Filtering and Nyquist Pulse Design ........ 529
13.3.6 Unknown Arrival Time and Pulse Compression ........ 530
13.4 Signal Discrimination in I.I.D. Gaussian Noise ........... 532
13.5 Further Reading .......................................... 538
Problems ................................................. 538
Basic Problems ................................................ 538
Advanced Problems ............................................. 543
Extension Problems ............................................ 552
Bibliography .................................................. 555
Index ......................................................... 561
|
