Introduction to Applied Statistical Signal Analysis (eBook)

Front Cover 1
Title Page 4
Copyright Page 5
Table of Contents 6
Preface 14
Dedication 16
Acknowledgments 18
List of symbols 20
Chapter 1 Introduction and terminology 24
1.1 Introduction 24
1.2 Signal terminology 26
1.2.1 Domain Types 26
1.2.2 Amplitude Types 28
1.2.3 Basic Signal Forms 29
1.2.4 The Transformed Domain—The Frequency Domain 31
1.2.5 General Amplitude Properties 32
1.3 Analog to digital conversion 33
1.4 Measures of signal properties 34
1.4.1 Time Domain 34
1.4.2 Frequency Domain 35
References 36
Chapter 2 Empirical modeling and approximation 38
2.1 Introduction 38
2.2 Model development 39
2.3 Generalized least squares 44
2.4 Generalities 46
2.5 Models from linearization 47
2.6 Orthogonal polynomials 51
2.7 Interpolation and extrapolation 56
2.7.1 Lagrange Polynomials 57
2.7.2 Spline Interpolation 61
2.8 Overview 66
References 66
Exercises 67
Chapter 3 Fourier analysis 74
3.1 Introduction 74
3.2 Review of fourier series 76
3.2.1 Definition 76
3.2.2 Convergence 83
3.3 Overview of fourier transform relationships 84
3.3.1 Continuous versus Discrete Time 84
3.3.2 Discrete Time and Frequency 86
3.4 Discrete fourier transform 87
3.4.1 Definition Continued 87
3.4.2 Partial Summary of DFT Properties and Theorems 88
3.5 Fourier analysis 91
3.5.1 Frequency Range and Scaling 92
3.5.2 The Effect of Discretizing Frequency 93
3.5.3 The Effect of Truncation 96
3.5.4 Windowing 100
3.5.5 Resolution 102
3.5.6 Detrending 105
3.6 Procedural summary 105
3.7 Selected applications 105
References 109
Exercises 110
Appendices 115
Appendix 3.1 DFT of ionosphere data 115
Appendix 3.2 Review of properties of orthogonal functions 116
Appendix 3.3 The fourier transform 117
Appendix 3.4 Data and spectral windows 121
Chapter 4 Probability concepts and signal characteristics 124
4.1 Introduction 124
4.2 Introduction to random variables 125
4.2.1 Probability Descriptors 125
4.2.2 Moments of Random Variables 131
4.2.3 Gaussian Random Variable 133
4.3 Joint probability 135
4.3.1 Bivariate Distributions 135
4.3.2 Moments of Bivariate Distributions 136
4.4 Concept of sampling and estimation 138
4.4.1 Sample Moments 138
4.4.2 Significance of the Estimate 142
4.5 Density function estimation 145
4.5.1 General Principle for .2 Approach 145
4.5.2 Detailed Procedure for .2 Approach 147
4.5.3 Quantile-Quantile Approach 150
4.6 Correlation and regression 153
4.6.1 Estimate of Correlation 153
4.6.2 Simple Regression Model 155
4.7 General properties of estimators 159
4.7.1 Convergence 159
4.7.2 Recursion 160
4.7.3 Maximum Likelihood Estimation 161
4.8 Random numbers and signal characteristics 162
4.8.1 Random Number Generation 163
4.8.2 Change of Mean and Variance 164
4.8.3 Density Shaping 165
References 168
Exercises 169
Appendices 177
Appendix 4.1 Plots and formulas for five probability density functions 177
Chapter 5 Introduction to random processes and signal properties 178
5.1 Introduction 178
5.2 Definition of stationarity 179
5.3 Definition of moment functions 182
5.3.1 General Definitions 182
5.3.2 Moments of Stationary Processes 183
5.4 Time averages and ergodicity 185
5.5 Estimating correlation functions 189
5.5.1 Estimator Definition 189
5.5.2 Estimator Bias 191
5.5.3 Consistency and Ergodicity 191
5.5.4 Sampling Properties 193
5.5.5 Asymptotic Distributions 194
5.6 Correlation and signal structure 199
5.6.1 General Moving Average 199
5.6.2 First-Order MA 200
5.6.3 Second-Order MA 204
5.6.4 Overview 204
5.7 Assessing stationarity of signals 205
5.7.1 Multiple Segments—Parametric 207
5.7.2 Multiple Segments—Nonparametric 212
References 216
Exercises 217
Appendices 220
Appendix 5.1 Variance of autocovariance estimate 220
Appendix 5.2 Stationarity tests 221
Chapter 6 Random signals, linear systems, and power spectra 224
6.1 Introduction 224
6.2 Power spectra 224
6.2.1 Empirical Approach 224
6.2.2 Theoretical Approach 226
6.3 System definition review 228
6.3.1 Basic Definitions 228
6.3.2 Relationships between Input and Output 231
6.4 Systems and signal structure 233
6.4.1 Moving Average Process 233
6.4.2 Structure with Autoregressive Systems 234
6.4.3 Higher-Order AR Systems 238
6.5 Time series models for spectral density 242
References 248
Exercises 249
Chapter 7 Spectral analysis for random signals: Nonparametric methods 252
7.1 Spectral estimation concepts 252
7.1.1 Developing Procedures 256
7.1.2 Sampling Moments of Estimators 257
7.2 Sampling distribution for spectral estimators 262
7.2.1 Spectral Estimate for White Noise 262
7.2.2 Sampling Properties for General Random Processes 265
7.3 Consistent estimators—Direct methods 267
7.3.1 Periodogram Averaging 267
7.3.2 Confidence Limits 271
7.3.3 Summary of Procedure for Spectral Averaging 281
7.3.4 Welch Method 282
7.3.5 Spectral Smoothing 282
7.3.6 Additional Applications 286
7.4 Consistent estimators—Indirect methods 287
7.4.1 Spectral and Lag Windows 287
7.4.2 Important Details for Using FFT Algorithms 289
7.4.3 Statistical Characteristics of BT Approach 290
7.5 Autocorrelation estimation 298
References 300
Exercises 301
Appendices 304
Appendix 7.1 Variance of periodogram 304
Appendix 7.2 Proof of variance of BT spectral smoothing 306
Appendix 7.3 Window characteristics 307
Appendix 7.4 Lag window functions 308
Appendix 7.5 Spectral estimates from smoothing 309
Chapter 8 Random signal modeling and parametric spectral estimation 310
8.1 Introduction 310
8.2 Model development 311
8.3 Random data modeling approach 316
8.3.1 Basic Concepts 316
8.3.2 Solution of General Model 320
8.3.3 Model Order 323
8.3.4 Levinson-Durbin Algorithm 328
8.3.5 Burg Method 332
8.3.6 Summary of Signal Modeling 336
8.4 Power spectral density estimation 337
8.4.1 Definition and Properties 337
8.4.2 Statistical Properties 341
8.4.3 Other Spectral Estimation Methods 343
8.4.4 Comparison of Nonparametric and Parametric Methods 345
References 346
Exercises 347
Appendices 350
Appendix 8.1 Matrix form of Levinson-Durbin recursion 350
Chapter 9 Theory and application of cross correlation and coherence 354
9.1 Introduction 354
9.2 Properties of cross correlation functions 356
9.2.1 Theoretical Function 356
9.2.2 Estimators 357
9.3 Detection of time-limited signals 362
9.3.1 Basic Concepts 363
9.3.2 Application of Pulse Detection 365
9.3.3 Random Signals 366
9.3.4 Time Difference of Arrival 368
9.3.5 Marine Seismic Signal Analysis 370
9.3.6 Procedure for Estimation 370
9.4 Cross spectral density functions 372
9.4.1 Definition and Properties 372
9.4.2 Properties of Cross Spectral Estimators 374
9.5 Applications 377
9.6 Tests for correlation between time series 378
9.6.1 Coherence Estimators 378
9.6.2 Statistical Properties of Estimators 381
9.6.3 Confidence Limits 382
9.6.4 Procedure for Estimation 385
9.6.5 Application 385
References 387
Exercises 388
Chapter 10 Envelopes and kernel functions 390
10.1 The Hilbert transform and analytic functions 390
10.1.1 Introduction 390
10.1.2 Hilbert Transform 391
10.1.3 Analytic Signal 393
10.1.4 Discrete Hilbert Transform 396
10.2 Point processes and continuous signals via kernel functions 398
10.2.1 Concept 398
10.2.2 Nerve Activity and the Spike Density Function 401
References 405
Exercises 406
Appendices 408
Table A Values of the Standardized Normal cdf F(z) 408
Table B Student’s t Distribution 410
Table C Chi-Square Distribution 411
Table D Critical Points for the Q-Q Plot Correlation Coefficient Test for Normality 412
Table E F Distribution Significance Limit for 97.5th Percentile 413
Table F Percentage Points of Run Distribution 415
Index 416