Principles of Adaptive Filters and Self-learning Systems

wc_buaa · 发表于 2024-5-1 14:46:43

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wc_buaa · 发表于 2024-5-1 15:07:30

1 Adaptive Filtering.............................................................................................3
1.1 Linear Adaptive Filters.............................................................................5
1.1.1 Linear Adaptive Filter Algorithms ..............................................7
1.2 Nonlinear Adaptive Filters........................................................................9
1.2.1 Adaptive Volterra Filters.............................................................9
1.3 Nonclassical Adaptive Systems ..............................................................10
1.3.1 Artificial Neural Networks........................................................10
1.3.2 Fuzzy Logic...............................................................................11
1.3.3 Genetic Algorithms ...................................................................11
1.4 A Brief History and Overview of Classical Theories..............................12
1.4.1 Linear Estimation Theory..........................................................12
1.4.2 Linear Adaptive Filters..............................................................13
1.4.3 Adaptive Signal Processing Applications..................................14
1.4.4 Adaptive Control.......................................................................16
1.5 A Brief History and Overview of Nonclassical Theories........................17
1.5.1 Artificial Neural Networks........................................................17
1.5.2 Fuzzy Logic...............................................................................18
1.5.3 Genetic Algorithms ...................................................................18
1.6 Fundamentals of Adaptive Networks......................................................19
1.7 Choice of Adaptive Filter Algorithm......................................................23
2 Linear Systems and Stochastic Processes......................................................25
2.1 Basic Concepts of Linear Systems..........................................................27
2.2 Discrete-time Signals and Systems .........................................................29
2.3 The Discrete Fourier Transform (DFT)..................................................31
2.3.1 Discrete Linear Convolution using the DFT..............................32
2.3.2 Digital Sampling Theory...........................................................33
2.3.2.1 Analogue Interpretation Formula...............................37
2.4 The Fast Fourier Transform....................................................................37
2.5 The z-Transform.....................................................................................40
2.5.1 Relationship between Laplace Transform and z-Transform......40
2.5.1.1 Bilateral z-Transform ................................................41
2.5.1.2 Unilateral z-Transform..............................................42
2.5.1.3 Region of Convergence (ROC) for the z-Transform .42
2.5.1.4 Region of Convergence (ROC) for General Signals..43
2.5.2 General Properties of the DFT and z-Transform.......................44
2.6 Summary of Discrete-Time LSI Systems................................................46
2.7 Special Classes of Filters........................................................................48
2.7.1 Phase Response from Frequency Magnitude Response.............50
2.8 Linear Algebra Summary........................................................................51
2.8.1 Vectors......................................................................................51
2.8.2 Linear Independence, Vector Spaces, and Basic Vectors..........52
2.8.3 Matrices.....................................................................................53
2.8.4 Linear Equations .......................................................................55
2.8.5 Special Matrices........................................................................56
2.8.6 Quadratic and Hermitian Forms................................................59
2.8.7 Eigenvalues and Eigenvectors...................................................59
2.9 Introduction to Stochastic Processes.......................................................61
2.10 Random Signals......................................................................................63
2.11 Basic Descriptive Models of Random Signals........................................64
2.11.1 The Mean Square Value and Variance......................................64
2.11.2 The Probability Density Function..............................................65
2.11.3 Jointly Distributed Random Variables.......................................68
2.11.4 The Expectation Operator .........................................................68
2.11.5 The Autocorrelation and Related Functions..............................69
2.11.6 Power Spectral Density Functions.............................................72
2.11.7 Coherence Function...................................................................73
2.11.8 Discrete Ergodic Random Signal Statistics...............................74
2.11.9 Autocovariance and Autocorrelation Matrices..........................75
2.11.10 Spectrum of a Random Process.................................................76
2.11.11 Filtering of Random Processes..................................................78
2.11.12 Important Examples of Random Processes ...............................80
2.11.12.1Gaussian Process.......................................................80
2.11.12.2White Noise...............................................................80
2.11.12.3White Sequences .......................................................81
2.11.12.4Gauss-Markov Processes...........................................81
2.11.12.5The Random Telegraph Wave...................................81
2.12 Exercises.................................................................................................82
2.12.1 Problems....................................................................................82
3 Optimisation and Least Square Estimation..................................................89
3.1 Optimisation Theory...............................................................................89
3.2 Optimisation Methods in Digital Filter Design.......................................91
3.3 Least Squares Estimation........................................................................95
3.4 Least Squares Maximum Likelihood Estimator......................................97
3.5 Linear Regression – Fitting Data to a Line.............................................98
3.6 General Linear Least Squares.................................................................99
3.7 A Ship Positioning Example of LSE.....................................................100
3.8 Acoustic Positioning System Example..................................................104
3.9 Measure of LSE Precision ....................................................................108
3.10 Measure of LSE Reliability...................................................................109
3.11 Limitations of LSE................................................................................110
3.12 Advantages of LSE...............................................................................110
3.13 The Singular Value Decomposition......................................................111
3.13.1 The Pseudoinverse...................................................................112
3.13.2 Computation of the SVD.........................................................112
3.13.2.1 The Jacobi Algorithm..............................................112
3.13.2.2 The QR Algorithm...................................................115
3.14 Exercises...............................................................................................116
3.14.1 Problems..................................................................................116
4 Parametric Signal and System Modelling ..................................................119
4.1 The Estimation Problem.......................................................................120
4.2 Deterministic Signal and System Modelling.........................................121
4.2.1 The Least Squares Method......................................................122
4.2.2 The Padé Approximation Method...........................................124
4.2.3 Prony’s Method.......................................................................127
4.2.3.1 All-Pole Modelling using Prony’s Method..............130
4.2.3.2 Linear Prediction.....................................................131
4.2.3.3 Digital Wiener Filter................................................132
4.2.4 Autocorrelation and Covariance Methods...............................133
4.3 Stochastic Signal Modelling ................................................................137
4.3.1 Autoregressive Moving Average Models................................137
4.3.2 Autoregressive Models............................................................139
4.3.3 Moving Average Models.........................................................140
4.4 The Levinson-Durbin Recursion and Lattice Filters.............................141
4.4.1 The Levinson-Durbin Recursion Development.......................142
4.4.1.1 Example of the Levinson-Durbin Recursion............145
4.4.2 The Lattice Filter.....................................................................146
4.4.3 The Cholesky Decomposition .................................................149
4.4.4 The Levinson Recursion..........................................................151
4.5 Exercises...............................................................................................154
4.5.1 Problems..................................................................................154
5 Optimum Wiener Filter................................................................................159
5.1 Derivation of the Ideal Continuous-time Wiener Filter ........................160
5.2 The Ideal Discrete-time FIR Wiener Filter...........................................162
5.2.1 General Noise FIR Wiener Filtering .......................................164
5.2.2 FIR Wiener Linear Prediction.................................................165
5.3 Discrete-time Causal IIR Wiener Filter ................................................167
5.3.1 Causal IIR Wiener Filtering....................................................169
5.3.2 Wiener Deconvolution ............................................................170
5.4 Exercises...............................................................................................171
5.4.1 Problems..................................................................................171
6 Optimal Kalman Filter.................................................................................173
6.1 Background to The Kalman Filter ........................................................173
6.2 The Kalman Filter.................................................................................174
6.2.1 Kalman Filter Examples..........................................................181
6.3 Kalman Filter for Ship Motion..............................................................185
6.3.1 Kalman Tracking Filter Proper................................................186
6.3.2 Simple Example of a Dynamic Ship Models...........................189
6.3.3 Stochastic Models ...................................................................192
6.3.4 Alternate Solution Models.......................................................192
6.3.5 Advantages of Kalman Filtering..............................................193
6.3.6 Disadvantages of Kalman Filtering.........................................193
6.4 Extended Kalman Filter........................................................................194
6.5 Exercises...............................................................................................194
6.5.1 Problems..................................................................................194
7 Power Spectral Density Analysis.................................................................197
7.1 Power Spectral Density Estimation Techniques ...................................198
7.2 Nonparametric Spectral Density Estimation.........................................199
7.2.1 Periodogram Power Spectral Density Estimation....................199
7.2.2 Modified Periodogram – Data Windowing.............................203
7.2.3 Bartlett’s Method – Periodogram Averaging ..........................205
7.2.4 Welch’s Method......................................................................206
7.2.5 Blackman-Tukey Method........................................................208
7.2.6 Performance Comparisons of Nonparametric Models.............209
7.2.7 Minimum Variance Method ....................................................209
7.2.8 Maximum Entropy (All Poles) Method...................................212
7.3 Parametric Spectral Density Estimation................................................215
7.3.1 Autoregressive Methods..........................................................215
7.3.1.1 Yule-Walker Approach............................................216
7.3.1.2 Covariance, Least Squares and Burg Methods........217
7.3.1.3 Model Order Selection for the
Autoregressive Methods..........................................218
7.3.2 Moving Average Method ........................................................218
7.3.3 Autoregressive Moving Average Method................................219
7.3.4 Harmonic Methods..................................................................219
7.3.4.1 Eigendecomposition of the Autocorrelation
Matrix......................................................................219
7.3.4.1.1 Pisarenko’s Method.................................221
7.3.4.1.2 MUSIC.....................................................222
8 Adaptive Finite Impulse Response Filters...................................................227
8.1 Adaptive Interference Cancelling .........................................................228
8.2 Least Mean Squares Adaptation ...........................................................230
8.2.1 Optimum Wiener Solution.......................................................231
8.2.2 The Method of Steepest Gradient Descent Solution................233
8.2.3 The LMS Algorithm Solution..................................................235
8.2.4 Stability of the LMS Algorithm...............................................237
8.2.5 The Normalised LMS Algorithm.............................................239
8.3 Recursive Least Squares Estimation.....................................................239
8.3.1 The Exponentially Weighted Recursive Least
Squares Algorithm...................................................................240
8.3.2 Recursive Least Squares Algorithm Convergence...................243
8.3.2.1 Convergence of the Filter Coefficients in
the Mean..................................................................243
8.3.2.2 Convergence of the Filter Coefficients in
the Mean Square......................................................244
8.3.2.3 Convergence of the RLS Algorithm in
the Mean Square......................................................244
8.3.3 The RLS Algorithm as a Kalman Filter...................................244
8.4 Exercises...............................................................................................245
8.4.1 Problems..................................................................................245
9 Frequency Domain Adaptive Filters ...........................................................247
9.1 Frequency Domain Processing..............................................................247
9.1.1 Time Domain Block Adaptive Filtering..................................248
9.1.2 Frequency Domain Adaptive Filtering....................................249
9.1.2.1 The Overlap-Save Method.......................................251
9.1.2.2 The Overlap-Add Method .......................................254
9.1.2.3 The Circular Convolution Method...........................255
9.1.2.4 Computational Complexity......................................256
9.2 Exercises...............................................................................................256
9.2.1 Problems..................................................................................256
10 Adaptive Volterra Filters.............................................................................257
10.1 Nonlinear Filters...................................................................................257
10.2 The Volterra Series Expansion.............................................................259
10.3 A LMS Adaptive Second-order Volterra Filter ....................................259
10.4 A LMS Adaptive Quadratic Filter ........................................................261
10.5 A RLS Adaptive Quadratic Filter .........................................................262
10.6 Exercises...............................................................................................264
10.6.1 Problems..................................................................................264
11 Adaptive Control Systems............................................................................267
11.1 Main Theoretical Issues........................................................................268
11.2 Introduction to Model-reference Adaptive Systems..............................270
11.2.1 The Gradient Approach...........................................................271
11.2.2 Least Squares Estimation ........................................................273
11.2.3 A General Single-input-single-output MRAS..........................274
11.2.4 Lyapunov’s Stability Theory...................................................277
11.3 Introduction to Self-tuning Regulators..................................................280
11.3.1 Indirect Self-tuning Regulators................................................282
11.3.2 Direct Self-tuning Regulators..................................................283
11.4 Relations between MRAS and STR......................................................284
11.5 Applications..........................................................................................285
12 Introduction to Neural Networks................................................................289
12.1 Artificial Neural Networks....................................................................289
12.1.1 Definitions...............................................................................290
12.1.2 Three Main Types...................................................................290
12.1.3 Specific Artificial Neural Network Paradigms........................292
12.1.4 Artificial Neural Networks as Black Boxed............................293
12.1.5 Implementation of Artificial Neural Networks........................294
12.1.6 When to Use an Artificial Neural Network.............................295
12.1.7 How to Use an Artificial Neural Network...............................295
12.1.8 Artificial Neural Network General Applications.....................296
12.1.9 Simple Application Examples.................................................297
12.1.9.1 Sheep Eating Phase Identification from Jaw
Sounds.....................................................................298
12.1.9.2 Hydrate Particle Isolation in SEM Images...............298
12.1.9.3 Oxalate Needle Detection in Microscope Images....299
12.1.9.4 Water Level Determination from Resonant
Sound Analysis........................................................299
12.1.9.5 Nonlinear Signal Filtering .......................................299
12.1.9.6 A Motor Control Example.......................................300
12.2 A Three-layer Multi-layer Perceptron Model.......................................300
12.2.1 MLP Backpropagation-of-error Learning................................302
12.2.2 Derivation of Backpropagation-of-error Learning ..................303
12.2.2.1 Change in Error due to Output Layer Weights........303
12.2.2.2 Change in Error due to Hidden Layer Weights........304
12.2.2.3 The Weight Adjustments.........................................305
12.2.2.4 Additional Momentum Factor .................................307
12.2.3 Notes on Classification and Function Mapping.......................308
12.2.4 MLP Application and Training Issues.....................................308
13 Introduction to Fuzzy Logic Systems..........................................................313
13.1 Basic Fuzzy Logic ................................................................................313
13.1.1 Fuzzy Logic Membership Functions .......................................314
13.1.2 Fuzzy Logic Operations ..........................................................315
13.1.3 Fuzzy Logic Rules...................................................................316
13.1.4 Fuzzy Logic Defuzzification...................................................317
13.2 Fuzzy Logic Control Design.................................................................318
13.2.1 Fuzzy Logic Controllers..........................................................319
13.2.1.1 Control Rule Construction.......................................319
13.2.1.2 Parameter Tuning ....................................................321
13.2.1.3 Control Rule Revision.............................................322
13.3 Fuzzy Artificial Neural Networks.........................................................322
13.4 Fuzzy Applications...............................................................................323
14 Introduction to Genetic Algorithms............................................................325
14.1 A General Genetic Algorithm...............................................................326
14.2 The Common Hypothesis Representation.............................................327
14.3 Genetic Algorithm Operators................................................................329
14.4 Fitness Functions ..................................................................................330
14.5 Hypothesis Searching............................................................................330
14.6 Genetic Programming...........................................................................331
14.7 Applications of Genetic Programming..................................................332
14.7.1 Filter Circuit Design Applications of GAs and GP .................333
14.7.2 Tic-tac-to Game Playing Application of GAs .........................334
15 Applications of Adaptive Signal Processing ...............................................339
15.1 Adaptive Prediction..............................................................................340
15.2 Adaptive Modelling..............................................................................342
15.3 Adaptive Telephone Echo Cancelling...................................................343
15.4 Adaptive Equalisation of Communication Channels.............................344
15.5 Adaptive Self-tuning Filters..................................................................346
15.6 Adaptive Noise Cancelling...................................................................346
15.7 Focused Time Delay Estimation for Ranging.......................................348
15.7.1 Adaptive Array Processing......................................................349
15.8 Other Adaptive Filter Applications.......................................................350
15.8.1 Adaptive 3-D Sound Systems..................................................350
15.8.2 Microphone Arrays..................................................................351
15.8.3 Network and Acoustic Echo Cancellation...............................352
15.8.4 Real-world Adaptive Filtering Applications............................353
16 Generic Adaptive Filter Structures.............................................................355
16.1 Sub-band Adaptive Filters ....................................................................355
16.2 Sub-space Adaptive Filters...................................................................358
16.2.1 MPNN Model..........................................................................360
16.2.2 Approximately Piecewise Linear Regression Model...............362
16.2.3 The Sub-space Adaptive Filter Model.....................................364
16.2.4 Example Applications of the SSAF Model..............................366
16.2.4.1 Loudspeaker 3-D Frequency Response Model........367
16.2.4.2 Velocity of Sound in Water 3-D Model...................369
16.3 Discussion and Overview of the SSAF.................................................370