LMS Adaptive Algorithm: Tác động phổ trên tốc độ hội tụ - Luận án tiến sĩ
Luận án tiến sĩ phân tích ảnh hưởng phổ trên tốc độ hội tụ của thuật toán LMS thích ứng. Nghiên cứu hiệu suất MSE và MSD, so sánh với LMS/Newton.
stanford university
Electrical Engineering
Luan An
luận án
Năm xuất bản
Số trang
199
Thời gian đọc
30 phút
Lượt xem
1
Lượt tải
0
Phí lưu trữ
50 Point
Mục lục chi tiết
Tóm tắt nội dung
I. LMS Adaptive Algorithm Core Principles
The Least Mean Squares algorithm represents a cornerstone in statistical signal processing. This adaptive filtering technique solves linear estimation problems by iteratively updating weight vectors. The algorithm minimizes mean square error between desired and actual outputs. LMS operates on streaming data, adjusting parameters in real-time without requiring complete dataset knowledge. The method's simplicity and robustness enable deployment across diverse applications. Digital communications, echo cancellation, and GPS systems rely heavily on LMS implementation. System identification and noise canceling benefit from its computational efficiency. The algorithm's popularity stems from low complexity and stable performance characteristics.
1.1. Wiener Solution Approximation
LMS iteratively approaches the optimal Wiener solution through gradient descent methods. Each iteration brings weight vectors closer to theoretical optimum. The algorithm updates weights based on instantaneous error gradients. Convergence depends on step size parameter selection and input signal characteristics. Smaller step sizes ensure stability but slow convergence. Larger values accelerate learning but risk instability. The Wiener solution minimizes expected square error for linear estimation problems.
1.2. Adaptive Filtering Applications
Adaptive filtering technology powers numerous commercial systems. Antenna arrays use LMS for beamforming and interference suppression. Active vibration suppression systems employ the algorithm for real-time control. Adaptive control mechanisms in industrial processes depend on LMS stability. Echo cancellers in telecommunications eliminate feedback loops efficiently. The algorithm's versatility enables deployment in resource-constrained environments. Performance remains consistent across varying operational conditions.
1.3. Algorithm Computational Efficiency
LMS requires minimal computational resources per iteration. Operations involve simple multiplications and additions without matrix inversions. Memory requirements scale linearly with filter length. Real-time implementation becomes feasible on modest hardware platforms. The algorithm processes samples sequentially without batch requirements. Computational complexity remains constant regardless of data history. This efficiency enables high-speed signal processing applications.
II. Spectral Properties Impact on Convergence Rate
Eigenvalue spread significantly affects LMS convergence behavior. The autocorrelation matrix eigenvalues determine adaptation speed across different modes. Large eigenvalue spread causes slow convergence in minimum eigenvalue directions. Maximum eigenvalue modes converge rapidly while minimum modes lag substantially. This spectral disparity creates performance bottlenecks in practical applications. Input signal correlation structure directly influences eigenvalue distribution. White noise inputs exhibit unity eigenvalue spread, ensuring uniform convergence. Colored signals with strong correlation produce high eigenvalue spreads. Performance degradation becomes severe when spread exceeds several orders of magnitude. Understanding spectral effects enables better algorithm parameter selection.
2.1. Eigenvalue Spread Definition
Eigenvalue spread quantifies autocorrelation matrix conditioning. The ratio between maximum and minimum eigenvalues defines spread magnitude. Unity spread indicates perfectly conditioned problems with equal eigenvalues. High spreads signal ill-conditioned problems requiring careful parameter tuning. Spread values above 100 typically cause noticeable performance degradation. The metric predicts convergence time differences across signal modes. Spectral analysis reveals fundamental limitations of standard LMS implementation.
2.2. Mode Dependent Convergence
Different eigenmodes converge at vastly different rates. Time constants for each mode inversely relate to corresponding eigenvalues. Dominant eigenvalue modes reach steady state quickly. Weak eigenvalue modes require extended adaptation periods. Overall convergence time depends on slowest mode behavior. Step size selection must accommodate the minimum eigenvalue constraint. Balancing fast and slow mode convergence presents fundamental tradeoffs.
2.3. Correlation Structure Effects
Input signal correlation determines autocorrelation matrix structure. Highly correlated signals produce concentrated eigenvalue spectra. Narrow eigenvalue distributions accelerate overall convergence. Broadband signals with weak correlation exhibit better conditioning. Preprocessing techniques can improve spectral properties. Whitening filters reduce eigenvalue spread but add computational cost. Understanding correlation effects guides preprocessing strategy selection.
III. LMS Newton Algorithm Spectral Normalization
The LMS/Newton algorithm eliminates eigenvalue spread effects through input transformation. Linear pre-transformation equalizes autocorrelation matrix eigenvalues. This normalization makes convergence rate independent of input spectral properties. The algorithm multiplies input vectors by inverse square root of autocorrelation matrix. Transformed inputs exhibit unity eigenvalue spread regardless of original statistics. LMS/Newton serves as theoretical performance benchmark for adaptive algorithms. All eigenmodes converge at identical rates with proper step size selection. However, practical implementation requires complete input statistics knowledge. The autocorrelation matrix must be known or accurately estimated. This requirement limits real-world deployment possibilities. LMS/Newton remains valuable for theoretical analysis and performance bounds.
3.1. Input Pre Transformation Method
Pre-transformation whitens input signals before weight adaptation. The transformation matrix equals inverse square root of autocorrelation. Matrix computation requires eigenvalue decomposition or Cholesky factorization. Transformed signals exhibit uncorrelated components with equal variances. This preprocessing eliminates spectral-dependent convergence issues. Implementation complexity far exceeds standard LMS requirements. The approach demonstrates theoretical limits of convergence acceleration.
3.2. Theoretical Benchmark Role
LMS/Newton establishes performance upper bounds for gradient methods. Comparing standard LMS against LMS/Newton reveals spectral impact. Performance ratios quantify degradation from non-ideal spectral properties. The benchmark enables fair algorithm comparisons across different inputs. Researchers use LMS/Newton to validate new adaptive algorithms. Theoretical analysis becomes simpler with equalized eigenvalue assumptions. The algorithm guides development of practical spectral normalization techniques.
3.3. Practical Implementation Challenges
Autocorrelation matrix knowledge rarely exists in real applications. Estimation requires extensive data collection and computational resources. Matrix inversion adds significant complexity to each adaptation cycle. Estimation errors propagate through transformation, degrading performance. Recursive estimation methods introduce additional convergence dynamics. The benefits often fail to justify implementation costs. Alternative approaches like normalized LMS offer partial solutions with lower complexity.
IV. Mean Square Error and Deviation Metrics
Performance assessment requires precise convergence criteria. Mean square error measures output prediction accuracy over time. MSE quantifies squared difference between desired and actual outputs. The metric captures both bias and variance components of estimation error. Mean square deviation evaluates weight vector accuracy directly. MSD measures squared distance from current weights to Wiener solution. This criterion assesses parameter estimation quality independent of output. Both metrics provide complementary performance perspectives. MSE reflects application-level performance in signal estimation tasks. MSD reveals underlying parameter convergence behavior. Transient analysis examines evolution from initial conditions to steady state. Learning curves plot MSE or MSD versus iteration number. Steady-state analysis characterizes long-term performance after convergence.
4.1. MSE Learning Curve Analysis
Learning curves visualize convergence trajectories over iterations. Initial MSE depends on starting weight vector quality. Curves typically exhibit exponential decay toward steady-state values. Multiple time constants appear with eigenvalue spread presence. Dominant modes decay quickly while weak modes persist longer. Area under learning curve quantifies total convergence cost. This integral metric summarizes overall convergence speed effectively.
4.2. MSD Performance Characteristics
MSD directly measures weight vector estimation accuracy. The metric remains meaningful even without desired output observations. Theoretical analysis often simplifies with MSD formulations. Weight deviation components align with autocorrelation matrix eigenvectors. Each component evolves according to corresponding eigenvalue dynamics. MSD steady-state values indicate misadjustment levels. Lower misadjustment requires smaller step sizes, slowing convergence.
4.3. Transient Versus Steady State Behavior
Transient phase begins from initial weight configuration. Weights move toward Wiener solution following gradient descent paths. Convergence speed depends on step size and eigenvalue spectrum. Steady state emerges when weights hover randomly around optimal values. Fluctuations result from gradient noise in stochastic approximation. Steady-state MSE exceeds minimum achievable error by misadjustment amount. The misadjustment-convergence tradeoff guides step size selection.
V. Stationary Signal Statistics Transient Analysis
Stationary environments maintain constant input and output statistics. Autocorrelation matrices and cross-correlation vectors remain fixed. The Wiener solution stays constant throughout adaptation process. Transient analysis focuses on convergence from initial conditions. Uniform random initial conditions provide worst-case convergence scenarios. Weight vectors start from random positions in parameter space. Analysis reveals how quickly LMS approaches acceptable performance levels. Simple expressions relate LMS performance to LMS/Newton benchmarks. Spectral properties of input determine relative performance ratios. Initial condition statistics influence transient behavior significantly. Deterministic initializations represent special cases of general framework. Area under learning curves serves as primary convergence metric.
5.1. Uniform Random Initialization
Random initial weights model complete parameter uncertainty. Uniform distributions span feasible parameter ranges. This initialization represents maximum initial uncertainty scenarios. Convergence analysis becomes more complex but realistic. Performance averaging over random starts yields robust predictions. Worst-case convergence times emerge from unfavorable initializations. The approach provides conservative performance estimates for system design.
5.2. Convergence Speed Criteria
Area under learning curve quantifies total convergence cost. Smaller areas indicate faster overall convergence to target performance. The metric integrates transient excess error over all iterations. Comparison with LMS/Newton areas reveals spectral impact magnitude. Ratios between areas provide dimensionless performance measures. These ratios depend primarily on eigenvalue spread characteristics. Simple formulas enable rapid performance prediction without simulation.
5.3. Initial Condition Impact
Starting weight quality dramatically affects convergence trajectories. Good initializations reduce transient duration substantially. Poor starts require extended adaptation before acceptable performance. Initial condition statistics enter performance expressions explicitly. Deterministic starts from zero weights represent common practical cases. Informed initializations from prior knowledge accelerate convergence. The analysis accommodates arbitrary initial distribution assumptions.
VI. Nonstationary Environments Tracking Performance
Nonstationary scenarios involve time-varying optimal solutions. The Wiener solution changes according to random walk models. Weight vectors track moving targets rather than converging. Random walk models capture gradual system changes realistically. Step-by-step Wiener solution variations follow Gaussian distributions. Variance of changes determines nonstationarity severity. Steady-state analysis replaces transient convergence examination. Tracking error measures distance between adaptive and optimal weights. MSE and MSD steady-state values quantify tracking accuracy. Performance expressions relate LMS tracking to LMS/Newton benchmarks. Spectral properties influence tracking capability similarly to convergence. Striking similarities emerge between stationary and nonstationary formulas. These connections reveal fundamental relationships between convergence and tracking.
6.1. Random Walk Model
Wiener solution evolution follows discrete-time random walk. Each time step adds independent Gaussian perturbation. Perturbation variance controls nonstationarity rate. Larger variances create faster solution changes. The model captures gradual system drift realistically. Abrupt changes require different modeling approaches. Random walk assumptions enable tractable analytical solutions.
6.2. Steady State Tracking Error
Tracking performance reaches equilibrium between adaptation and drift. Steady-state error balances algorithm learning against solution changes. Faster adaptation reduces lag error but increases gradient noise. Optimal step size minimizes total steady-state error. This optimum differs from stationary environment settings. Eigenvalue spread affects optimal parameter selection significantly. Performance degradation from spectral effects persists in tracking scenarios.
6.3. Convergence Tracking Connections
Mathematical expressions for convergence and tracking show remarkable similarity. Both depend on eigenvalue spread in analogous ways. Input statistics influence both scenarios through identical mechanisms. Initial condition effects parallel nonstationarity impact. These connections suggest unified theoretical framework. Understanding one scenario provides insight into the other. The relationships simplify algorithm analysis and design procedures.
Tải xuống file đầy đủ để xem toàn bộ nội dung
Tải đầy đủ (199 trang)Trích đoạn nội dung luận án
Tải xuống để đọc toàn bộSPECTRAL EFFECTS ON THE RATE OF CONVERGENCE OF THE LMS ADAPTIVE ALGORITHM A DISSERTATION SUBMITTED TO THE DEPARTMENT OF ELECTRICAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Aaron E. Flores December 2005 UMI Number: 3197432 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted.
Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3197432 Copyright 2006 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.
ProQuest Information and Learning Company 300 North Zeeb Road P. Box 1346 Ann Arbor, MI 48106-1346 © Copyright by Aarón E. Flores 2006 All Rights Reserved ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Bernard Widrow Principal Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.
Thomas Cover I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. bully Approved for the University Committee on Graduate Studies. 111 Abstract In the field of statistical signal processing, a fundamental problem is that of linearly esti- mating a random process given observations of a related random process. The usual ob- jective is to find the weights of the linear estimation that minimizes on average the square of the error.
Adaptive algorithms are used to iteratively update a weight vector, an approx- imation of the optimal solution (also called Wiener solution), as input data is presented in a streaming way. The Least Mean Square (LMS) algorithm is one of the most popular adaptive algorithms because it is simple and robust, as it is used in a wide variety of ap- plications, including digital communications, echo cancellers, system identification, GPS systems, noise canceling, antenna arrays, adaptive control, active vibration suppression systems, and many other commercial applications of significance. However, the perfor- mance of LMS may vary greatly when the autocorrelation matrix of its input has a high eigenvalue spread, so there is a need to be able to predict its performance in practice. The LMS/Newton algorithm is a variation of the LMS algorithm, it linearly pre-transforms the input so that the eigenvalues of the new input autocorrelation matrix are equal to each other.
This makes the LMS/Newton algorithm immune to the eigenvalue spread problem of LMS. However, since it requires knowledge of the input statistics often not available in practice, the LMS/Newton algorithm is mainly used as a theoretical benchmark for adap- tive algorithms. In this thesis, we study the performance of LMS relative to LMS/Newton. The performance of LMS is assessed in terms of the mean square error (MSE), and the mean square deviation (MSD) of its weight vector from the Wiener solution.
The analysis is done for stationary and nonstationary signal statistics. In the stationary case, transient behavior results when the adaptive weight vector starts from initial conditions and proceeds iv toward the Wiener solution, reaching a steady-state where the adaptive weight vector hov- ers randomly about the Wiener solution. The transient phase is thoroughly analyzed under uniform random initial conditions to evaluate how fast LMS converges to a good approx- imation of the Wiener solution. For general statistics of the initial conditions, including deterministic initial conditions as a special case, the areas under the MSE and MSD learn- ing curves are used as the convergence speed criteria.
Simple expressions are obtained for the transient performance of LMS relative to LMS/Newton’s in terms of the statistics of the input and initial conditions. In the nonstationary case, the Wiener solution varies randomly according to a random walk model, and the adapting weight vector is tracking a moving target. In this case, a steady state MSE and MSD criteria is used, obtaining simple expres- sions for the tracking performance of LMS relative to that of LMS/Newton in terms of the statistics of the input and changes of the Wiener solution. The expressions found for the stationary and nonstationary cases have a striking resemblance, showing some connections between transient and tracking behavior of the LMS and LMS/Newton algorithms.
When a transversal adaptive filter is considered, the input autocorrelation matrix is Toeplitz; this allows the expressions found for the performance of LMS to be translated into the frequency domain. Regarding the Wiener solution as an impulse response, we refer to the magnitude square of its Fourier transform as the Wiener solution spectrum. For the Transient analysis, it is found that when adapting from zero initial conditions, the ratio between the areas under the learning curves of LMS and LMS/Newton is given by the inner product of the normalized input power spectrum and Wiener solution spectrum. With our model of nonstationarity, the Wiener impulse response changes from sample time to sample time.
Take the Fourier transform of the average changes and take the magnitude square of this Fourier transform; we call this the spectrum of the Wiener solution changes. It is found that the steady state performance of LMS relative to LMS/Newton is given by the inner product of the normalized input power spectrum and spectrum of the Wiener solution changes. Our results imply that when zero initial conditions are used the transient performance of LMS is better than that of LMS/Newton, in spite of a high a eigenvalue spread, given that the input power spectrum is similar to the Wiener solution spectrum (e. a low-pass input and a low-pass Wiener filter.
In the nonstationary case, if the input spectrum is similar to the spectrum of the changes in the Wiener solution, LMS tracks better than LMS/Newton in steady-state. On the other hand, if the above spectra are dissimilar, then LMS per- forms worse than LMS/Newton, explaining the slow transient performance of LMS often observed in equalization tasks. Our results allow prediction of LMS performance in appli- cations where approximate prior knowledge of the spectra is available. This is illustrated with examples in system identification and channel equalization.
vi Acknowledgements Throughout my years at Stanford, there are a number of people who were important for the development of this thesis. First, I want to express my deepest gratitude to Prof. Bernard Widrow, who, much more than an adviser, became a friend to me. His technical expertise and strong engineering intuition, together with his patience, warmth and kind personality, have been a model for me to follow.
I am grateful to him for the invaluable experience I gained during my years by his side as a teaching assistant. He strongly supported my passion for teaching; and his drive to do research motivated by real problems, always reminded me of the ultimate goal of academic work. It was an honor and pleasure being his Ph. ) I want thank Prof.
Tom Cover for agreeing to be my associate advisor, his comments and feedback after my oral presentation were enthusiastic and motivating. I would also like to specially thank Prof. Julius Smith, for accepting to be my third reader, and for all his suggestions and enlightening discussions we had, ultimately shaping several of the chapters of this thesis. I would like to use this opportunity to express my admiration for the outstanding faculty at Stanford, in particular Profs.
Stephen Boyd, Thomas Kailath, and Robert M. Gray, who motivated and inspired my academic spirit since my first years at Stanford. I want to express my deepest gratitude to the administrative staff, specially Joice de Bolt, Marianne Marx, Diane Shankle, and Denise Murphy. They were always there for me when I needed them, and in several occasions were responsible for important and fortunate mile- stones of my path at Stanford, including my teaching experience and the many rewards that came with it.
Naturally, I thank all the Zoomates I have interacted with over the years, — past and Vil present — for the time we had together. I want to especially thank Max for all his help throughout my years in the Zoo, his tireless dedication as a sysadmin undoubtedly made the life of all his fellow Zoomates a lot easier; but more than that, I want to thank him for the many enjoyable discussions we had about technical matters, and interesting “life” topics of relevance, such as blackjack and poker. I would also like to thank Gabriel for all his help, specially for his earlier research work in Sweet Hall, which was the seed for the main ideas in this thesis. During my years at Stanford I was fortunate to have incredible friends, from both inside and outside the university.
My life as a student would have been completely different without them. They are too many to be mentioned by name, but I deeply thank them for their unconditional support in difficult times, and for all the great fun we had together over the years. I am specially grateful to my parents Ramona and Aurelio, my sister Judith, and my girlfriend Silvia, with whom my life has been blessed; thanks to their love, my experience ‘at Stanford has been a complete one. Finally, I acknowledge the financial aid from CONACYT, which helped supporting me during my first years at Stanford.
Vili Contents Abstract Acknowledgements viii 1 Introduction 1. ees +> 2 Adaptive Algorithms woOnm 2.1 Introduction to Adaptive Algorithms. Q Q HQ Q LH Ko 2.1 The steepest descentmethod.2 Derivationof the LMS algorithim.3 Convergence of the mean weight vectorforLMS. The LMS/Newton Algorthm.2 Derivation of the LMS/Newton algorithm .3 Convergence of the mean weight vector for LMS/Newton 18 2.
eee eee 21 3 Transient Analysis with Stationary Statistics 3.1 Mean Square Error (MSE) and Mean Square Deviation(MSD) .2 MSEand MSD analysisforLMS.1 Dynamics of the Weight-Error Power Vector. MSE and MSD analysis for LMS/Newton .1 Dynamics of the Weight Error Power Vector.4 Eigenvalue spread problenofLMS.5 Analysis with Random iniialcondiions. LMS Transient Efficiency 57 4. HQ HH HQ Q Q Q k kia 58 4.1 Transient Energy with random Initialcondilons.2 LMS Transient Efficiency.
LMS MSE Transient Efficiency .2 LMS MSD Transient Efficiency. 64 LMS Transient Efficiency in terms of Spectra 68 5.1 Adaptive Transversal Filter. ee ee ee 69 5. Q Q Q HQ HQ HQ ng kia 70 5.1 Continuous Fourier Spectra.
LMS Transient Efficiency in termsofSpecra .1 The R-norm and R-!-norm in the Fourierdomain .2 LMS MSE Transient Efficiency in terms of Fourier Spectra. LMS MSD Transient Efficiency in terms of Fourier Spectra 79 5. HH n Q Q Q V vn va 83 55 Applicaions. 97 Nonstationary Analysis 100 61 Nonstationarntymodel.
Ặ Q0 Q HQ eee 62 MSEandMSD AnalysisfoerLMS.1 Dynamics of the weight error power vector .2 MSE Steady state performance under Nonstationary conditions. MSD Steady state performance under Nonstationary conditions. MSE and MSD AnalysisforLMS/Newton.1 Dynamics of the weight error power vector .2 MSE Steady state performance under Nonstationary conditions. MSD Steady state performance under Nonstationary conditions .4 LMS Nonstationary Efficiency.
ee eee eee 113 6.1 LMS MSE Nonstationary Efficlency.2 LMS MSD Nonstationary Effclency. cu Q Q Q HQ HQ kg V na 120 6.5 LMS Nonstationary Efficiency in terms of Spectra.1 Fourier Spectrum of the Wiener solution steps.2 LMS MSE Nonstationary Efficiency in terms of spectra .3 LMS MSD Nonstationary Efficiency in terms of spectra. ee kia 129 7 Conclusions and Future Work 137 7. HH HH HQ kh va 137 7.
Nội dung được bảo vệ bản quyền — Tải xuống đầy đủ
Từ khóa và chủ đề nghiên cứu
Câu hỏi thường gặp
Luận án "LMS adaptive algorithm: Spectral effects trên tốc độ hội tụ" nghiên cứu về vấn đề gì?
Luận án tiến sĩ phân tích ảnh hưởng phổ trên tốc độ hội tụ của thuật toán LMS thích ứng. Nghiên cứu hiệu suất MSE và MSD, so sánh với LMS/Newton.
Luận án "LMS adaptive algorithm: Spectral effects trên tốc độ hội tụ" được bảo vệ tại trường nào?
Luận án này được bảo vệ tại stanford university. Năm bảo vệ: 2006.
Luận án "LMS adaptive algorithm: Spectral effects trên tốc độ hội tụ" thuộc chuyên ngành gì?
Luận án "LMS adaptive algorithm: Spectral effects trên tốc độ hội tụ" thuộc chuyên ngành Electrical Engineering. Danh mục: Kỹ Thuật Điện Tử.
Luận án "LMS adaptive algorithm: Spectral effects trên tốc độ hội tụ" có bao nhiêu trang?
Luận án "LMS adaptive algorithm: Spectral effects trên tốc độ hội tụ" có 199 trang. Bạn có thể xem trước một phần tài liệu ngay trên trang web trước khi tải về.
Cách tải luận án "LMS adaptive algorithm: Spectral effects trên tốc độ hội tụ" về máy như thế nào?
Để tải luận án về máy, bạn nhấn nút "Tải xuống ngay" trên trang này, sau đó hoàn tất thanh toán phí lưu trữ. File sẽ được tải xuống ngay sau khi thanh toán thành công. Hỗ trợ qua Zalo: 0559 297 239.