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Analysis of Adaptive Filter Algorithms with an Application to Harmonic Noise Cancellation in Distribution Power Line Communications by Jin-Der Wang Center for Communications and Signal Processing Department of Electrical and Computer Engineering North Carolina State University August 1985 CCSP-TR-85/15 ABSTRACT WANG, JIN-DER. Analysis of Adaptive filter Algorithms with an Application to Harmonic Noise Cancellation in Distribution Power Line Communications. (Under the direction of Dr. Henry Joel Trussell.) Some well-known adaptive algorithms such as the least mean squares (LMS) and the recursive least squares (RLS) algorithms are reviewed, and the properties of some newly developed fast RLS algorithms such as the fast Kalman (FK), the fast a priori estimation sequential technique (FAEST), and the fast transversal filter (FTF) algorithms are investigated. These adaptive algorithms are applied to the problem of harmonic noise cancellation in distribution power line communica­ tions. The fast RLS algorithms were derived independently, from different approaches, and no connection had previously been made. They are rederived with a unified approach, and their mathematical equivalence is shown by examining certain algorithmic quantities and initial conditions. Based on this work, an improved rescue variable is then proposed which can detect the tendency of algo­ rithm divergence earlier than previously proposed rescue variables. Adaptive filters are applied to harmonic noise cancellation m distribution power line communications. It is shown that most filter taps can be constrained to zero to reduce computational complexity. The constrained LMS and RLS algo­ rithms are derived for a general constraint problem, and the constrained FK algorithm is derived for harmonic noise cancellation. Simulations of applying adaptive filters to noise cancellation were done in the presence and absence of hard limiters using noise recorded in a power distribution substation. It is shown that both constrained and unconstrained adaptive noise cancellation filters can be used effectively in tandem with an optimal hard limiter. ii BIOGRAPHY Jin-Der Wang was born in Chia-Yi, Taiwan on November, 13, 1955. He received his BSEE and MSEE degrees with high honor from the National Chiao­ Tung University (NCTU), Taiwan, in 1978 and 1980, respectively. He worked as a research assistant and a teaching assistant from 1978 to 1980 in the Electronics Engineering and Computer Science Departments, NCTU. He worked as a lecturer from 1980 to 1982 (during military service) in the Physics and Electrical Engineering Department of the Naval Academy, Taiwan. There, he taught courses on microcomputer applications, electronics, communication sys­ tems, and data structures. Since 1982, he has been conducting research in the Electrical Engineering Department of North Carolina State University, Raleigh, while working towards his PhD. iii ACKNOWLEDGEMENT I wish to express my deep appreciation to Dr. H. J. Trussell, my advisor, for his supervision and guidance throughout the course of this research at all stages. This work would not be finished without his candid help and encouragement. I am thankful to the Carolina Power and Light Company and Dr. J. B. O'Neal for sup­ porting the research. I also want to thank Mr. J. L. Faber for his helpful discussion during the preparation of this dissertation. Special thanks go to my wife, my son, and my daughter for their understand­ ing and support.. iv TABLE OF CONTENTS LIST OF TABLES . VI LIST OF FIGURES . vii 1. INTRODUCTION . 1.1 An Overview of Adaptive Filters . 1.2 Introduction to Adaptive Noise Cancelling ~ .. 2 1.3 An Overview of Power Line Communications . 4 1.3.1 Filtering of Power Line Noise .. 6 1.4 Outline of Dissertation . t 2. INTRODUCTION TO ADAPTIVE ALGORITHMS . 8 2.1 The Least Mean Squar-es Algorithm . 8 2.2 The Least Squares Algorithm . L1 2.2.1 The Recursive Least Squares Algorithm . 13 2.2.2 The ~'K, F AEST, and FTF Algorithms .. 13 3. UNIFIED DERIVATION OF FK, FAEST, AND FTF ALGORITHMS .. 15 3.1 Introduction . 15 3.2 Review of the Fast Kalman Algorithm .. t6 3.3 Unification of the FK, F AEST, and FTF Algorithms .. 26 3.4 An Improved Rescue Variable . 37 3.5 Summary of Unified Derivation of Fast RLS Algorithms .. 40 4. CONSTRAINED ADAPTIVE ALGORITHMS .. 40 4.1 Introduction . 40 4.2 The Constrained LMS Algorithm .. 41 4.3 The Constrained Recursive Least Squares Algorithm .. 48 4.4 The Constrained Fast Kalman Algorithm .. 55 4.5 Summary of the Constrained Fast Kalman Algorithm .. 71 5. ADAPTIVE HARMONIC NOISE CANCELLATION IN DISTRIBUTION POWER LINE COMMUNICATIONS . 72 5.1 Intr-oductlon to Distribution Power Line Communications .. 72 5.2 The OLC Noise Model . 74 5.3 Simulation Conditions . 76 5.3.1 Cornplex Demodulation . 84 5.4 The Optimal Har-monic Cancelling Filter .. 85 5.4.1 The Demodulated DLC Noise .. 87 5.4.2 FIR-ALE Ilarmonic Noise Cancelling Filters . 88 5.4.3 (IR-ALE fla~monic Noise Cancelling Filters .. 98 5.5 DLC Harmonic Noise Cancellers . LOO 5.5.1 The LMS Noise Canceller . LOO 5.5.2 Implementation of DLC Noise Cancellers . ros 5.5.3 The Constrained LMS ~'ilter .. 106 5.5.4 Detection Error Rates of the LMS Noise Canceller . l08 5.5.5 Summary of the Application of the LMS Noise Canceller . t12 v 5.5.6 Comparison of the LMS and LS Noise Carieef ler-s 11:~ 5.6 Including a liard Lirniter l l f 5.6.l Detection Err-or Rl1tes of Includ ing a Hard Limiter l20 5.6.2 Sumnlary of" the Addition of a Hard Limiter l:.!3 6. SUMMARY AND FUR"rHER ItESEARCH l~5 6.1 Summary of Contributions l25 6.2 Further Research 128 7. LIST OF REJ:t'ERENCES 130 8. APPENDICES 137 8.1 The Derivation of the Constrained LMS Algorithm Using the Lagrange-Multipler Technique l37 8.2 The Constrained F AEST and FTF algorithms 141 Vi LIST OF TABLES Table 5.1 Error Count for- Unconstrained LMS Noise Canceller .. LtD Table 5.2 Error Count ror Constrained LMS Noise Canceller .. 112 Table 5.3 Error Count for Noise Canceller with/without Clipper . 1:!2 Vll LIST OF FlCURES Fig. 1.1 Schematic Diagram or the Adaptive Noise Cancelling Concept, 3 Fig. 4.1 The Constrained ~'\i1ter Configuration 57 Fig. 5.1a A Standard DLe Time- Domain Noise Recorded in a Substation 77 Fig. 5.tb A Standard DLC Noise Spectrum Recorded in a Substation 77 Fig. 5.2a An Expanded OLe Noise Spectrum Recorded in a Substation .. 78 Fig. 5.2b An Expanded DLe Noise Spectrum Recorded in a Substation .. 78 Fig. 5.2e An Expanded DLC Noise Spectrum Recorded in a Substation . 1:9 Fig. 5.2d An Expanded DLC Noise Spectrum Recorded in a Substation .. 79 Fig. 5.3a A Block Diagram of the OLe Receiver without Clipper . H2 Fig. 5.3b A Block Diagram of the OLe Receiver with Clipper . 82 Fig. 5.4 A Simplified DLC Signal Spectrum before Demodulation .. 83 Fig. 5.5 Signal Spectrum after Real Demodulation . 83 Fig. 5.6a Signal Spectrum after Complex Demodulation . 85 Fig. 5.6b Expanded Demodulated Signal Spectrum .. 85 Fig. 5.6c Demodulated Signal Spectrum after Right-Shifting .. 86 Fig. 5.6d Real Reconstructed Demodulated Signal Spectrum .. 86 Fig. 5.7 The Schematic Diagram of the Adaptive Line Enchancer . 90 Fig. 5.8 Demodulated OLC Noise Spectrum . 96 Fig. 5.9 Adaptive Filter Response . n6 Fig. 5.10 Adaptive Filter Spectrum ~ . 97 Fig. 5.11 Demodulated Time Domain Noise . 101 Fig. 5.12 Demodulated Noise Spectrum . 101 Fig. 5.13 Time-Domain Resldual Noise . 102 Fig. 5.14 Power Spectrum of the Residual Error .. 102 Fig. 5.15 LMS Filter Response at the 1500th Iteration . l04 Fig. 5.16 LMS Filter Spectrum at the L500th Iteration . 104 Fig. 5.11 Adaptive Noise Canceller Requiring Signal Interruption (Method 1) . l05 Fig. 5.18 Adaptive Noise Canceller Using Out-Of-Band Noise (Method 2) . 106 Fig. 5.19 Adaptive Noise Canceller Using Out-Or-Band Noise (Method 3) .. 107 Fig. 5.20 Constrained LMS Filter Response .. 109 Fig. 5.21 Power Spectrum of Constrained LMS Filter . 109 Fig. 5.22 FK Filter Spectrum at the L500th Iteration, 8 =0. L .. 116 Fig. 5.23 FK Filter Spectrum at the 1500th Iteration, 8 =0.01 .. 1L6 Fig. 5.24 A DLC Noise Cancelling System with Clipper (Method 1) .. 117 Fig. 5.25 A DLC Noise Cancelling System with Clipper (Method 3) .. l17 Fig. 5.26 Noise Spectrum After Clipping at 0.002 . l19 Fig. 5.21 DLC Noise Spectrum after Clipping and Noise Cancellation .. l19 e)f) Fig. 5.28 Error Rates vs, Clipping Values at Different Signal Powers . t Wl'" Fig. 5.29 Error Rates vs, Clipping Values at Different Signal Powers . 123 CHAPTER 1 INTRODUCTION 1.1. An Overview of Adaptive Filters Adaptive digital filtering techniques have been shown to be effective in a wide range of practical applications where the signal is nonstationary or there is insuffi­ cient a priori knowledge ~f the expected signal and noise parameters to design optimal fixed filters. The increased computational speed and performance capabil­ ities of modern digital hardware has made it possible to use adaptive filtering tech­ niques in a wide range of applications in recent years. Representative applications include noise cancellation [1]-[3], echo cancellation [4]-[8]' data equalization [9]-[11]' predictive deconvolution [12], instantaneous frequency tracking [13]-[15]' adaptive predictive coding [45]-[47] speech processing [16]-[19]' intrusion detection [20]-[21], spectral estimation [22]-[23], beamforming [24]-[28], radar clutter rejection [29], data communication [30]-[31]' system identification [32]-[:33], narrow-band interfer­ ence rejection [34], narrow-band signal enchancement [35]-[42]' and narrow-band signal detection [43]-[44]. As discussed in the above references, a variety of adap­ tive estimation techniques have been used to provide time-varying estimates of the Wiener filter parameters.

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