Adaptive Filters

A. H. Sayed, Adaptive Filters, Wiley, NJ, 2008.


Adaptive filtering is a topic of immense practical and theoretical value, having applications in areas ranging from digital and wireless communications to biomedical systems. Now, preserving the style and main features of the earlier award-winning publication, Fundamentals of Adaptive Filtering (2005 Terman Award), the author offers readers and instructors a concentrated, systematic, and up-to-date treatment of the subject in this valuable new book. Adaptive Filters allows readers to gain a gradual and solid introduction to the subject, its applications to a variety of topical problems, existing limitations, and extensions of current theories. The book consists of eleven parts—each part containing a series of focused lectures and ending with bibliographic comments, problems, and computer projects with MATLAB® solutions available to all readers. Additional features include:

  • Numerous tables, figures, and projects
  • Special focus on geometric constructions, physical intuition, linear-algebraic concepts, and vector notation
  • Background material on random variables, linear algebra, and complex gradients collected in three introductory chapters
  • Complete solutions manual available for instructors
  • MATLAB® solutions available for all computer projects

Adaptive Filters offers a fresh, focused look at the subject in a manner that will entice students, challenge experts, and appeal to practitioners and instructors.

About the Author

Ali H. Sayed is Professor of Electrical Engineering at UCLA, where he established and directs the Adaptive Systems Laboratory. He is a Fellow of the IEEE for his contributions to adaptive filtering and estimation algorithms. His research has attracted several recognitions including the 2003 Kuwait Prize, 2005 Terman Award, and several IEEE Best Paper Awards.


More details can be found in the preface of the book.The preface explains the distinctive features of the treatment, including the special roles played by geometric constructions, energy conservation arguments, system-theoretic arguments, uniformity, and extensive material not covered elsewhere. The book can be used to design different course offerings according to instructor needs in terms of breadth and depth of coverage.

Table of Contents

Preface and Acknowledgments.Notation and Symbols.


A. Random Variables. A.1 Variance of a Random Variable. A.2 Dependent Random Variables. A.3 Complex-Valued Random Variables. A.4 Vector-Valued Random Variables. A.5 Gaussian Random Vectors.

B. Linear Algebra. B.1 Hermitian and Positive-Definite Matrices. B.2 Range Spaces and Nullspaces of Matrices. B.3 Schur Complements. B.4 Cholesky Factorization. B.5 QR Decomposition. B.6 Singular Value Decomposition. B.7 Kronecker Products.

C. Complex Gradients. C.1 Cauchy-Riemann Conditions. C.2 Scalar Arguments. C.3 Vector Arguments.


1. Scalar-Valued Data. 1.1 Estimation Without Observations. 1.2 Estimation Given Dependent Observations. 1.3 Orthogonality Principle. 1.4 Gaussian Random Variables.

2. Vector-Valued Data. 2.1 Optimal Estimator in the Vector Case. 2.2 Spherically Invariant Gaussian Variables. 2.3 Equivalent Optimization Criterion. Summary and Notes. Problems and Computer Projects.


3. Normal Equations. 3.1 Mean-Square Error Criterion. 3.2 Minimization by Differentiation. 3.3 Minimization by Completion-of-Squares. 3.4 Minimization of the Error Covariance Matrix. 3.5 Optimal Linear Estimator.

4. Orthogonality Principle. 4.1 Design Examples. 4.2 Orthogonality Condition. 4.3 Existence of Solutions. 4.4 Nonzero-Mean Variables.

5. Linear Models. 5.1 Estimation using Linear Relations. 5.2 Application: Channel Estimation. 5.3 Application: Block Data Estimation. 5.4 Application: Linear Channel Equalization. 5.5 Application: Multiple-Antenna Receivers.

6. Constrained Estimation. 6.1 Minimum-Variance Unbiased Estimation. 6.2 Example: Mean Estimation. 6.3 Application: Channel and Noise Estimation. 6.4 Application: Decision Feedback Equalization. 6.5 Application: Antenna Beamforming.

7. Kalman Filter. 7.1 Innovations Process. 7.2 State-Space Model. 7.3 Recursion for the State Estimator. 7.4 Computing the Gain Matrix. 7.5 Riccati Recursion. 7.6 Covariance Form. 7.7 Measurement and Time-Update Form. Summary and Notes. Problems and Computer Projects.


8. Steepest-Descent Technique. 8.1 Linear Estimation Problem. 8.2 Steepest-Descent Method. 8.3 More General Cost Functions.

9. Transient Behavior. 9.1 Modes of Convergence. 9.2 Optimal Step-Size. 9.3 Weight-Error Vector Convergence. 9.4 Time Constants. 9.5 Learning Curve. 9.6 Contour Curves of the Error Surface. 9.7 Iteration-Dependent Step-Sizes. 9.8 Newton’s Method.

10. LMS Algorithm. 10.1 Motivation. 10.2 Instantaneous Approximation. 10.3 Computational Cost. 10.4 Least-Perturbation Property. 10.5 Application: Adaptive Channel Estimation. 10.6 Application: Adaptive Channel Equalization. 10.7 Application: Decision-Feedback Equalization. 10.8 Ensemble-Average Learning Curves.

11. Normalized LMS Algorithm. 11.1 Instantaneous Approximation. 11.2 Computational Cost. 11.3 Power Normalization. 11.4 Least-Perturbation Property.

12. Other LMS-Type Algorithms. 12.1 Non-Blind Algorithms. 12.2 Blind Algorithms. 12.3 Some Properties.

13. Affine Projection Algorithm. 13.1 Instantaneous Approximation. 13.2 Computational Cost. 13.3 Least-Perturbation Property. 13.4 Affine Projection Interpretation.

14. RLS Algorithm. 14.1 Instantaneous Approximation. 14.2 Computational Cost. Summary and Notes. Problems and Computer Projects.


15. Energy Conservation. 15.1 Performance Measure. 15.2 Stationary Data Model. 15.3 Energy Conservation Relation. 15.4 Variance Relation. 15.A Interpretations of the Energy Relation.

16. Performance of LMS. 16.1 Variance Relation. 16.2 Small Step-Sizes. 16.3 Separation Principle. 16.4 White Gaussian Input. 16.5 Statement of Results. 16.6 Simulation Results.

17. Performance of NLMS. 17.1 Separation Principle. 17.2 Simulation Results. 17.A Relating NLMS to LMS.

18. Performance of Sign-Error LMS. 18.1 Real-Valued Data. 18.2 Complex-Valued Data. 18.3 Simulation Results.

19. Performance of RLS and Other Filters. 19.1 Performance of RLS. 19.2 Performance of Other Filters. 19.3 Performance Table for Small Step-Sizes.

20. Nonstationary Environments. 20.1 Motivation. 20.2 Nonstationary Data Model. 20.3 Energy Conservation Relation. 20.4 Variance Relation.

21. Tracking Performance. 21.1 Performance of LMS. 21.2 Performance of NLMS. 21.3 Performance of Sign-Error LMS. 21.4 Performance of RLS. 21.5 Comparison of Tracking Performance. 21.6 Comparing RLS and LMS. 21.7 Performance of Other Filters. 21.8 Performance Table for Small Step-Sizes. Summary and Notes. Problems and Computer Projects.


22. Weighted Energy Conservation. 22.1 Data Model. 22.2 Data-Normalized Adaptive Filters. 22.3 Weighted Energy Conservation Relation. 22.4 Weighted Variance Relation.

23. LMS with Gaussian Regressors. 23.1 Mean and Variance Relations. 23.2 Mean Behavior. 23.3 Mean-Square Behavior. 23.4 Mean-Square Stability. 23.5 Steady-State Performance. 23.6 Small Step-Size Approximations. 23.A Convergence Time.

24. LMS with non-Gaussian Regressors. 24.1 Mean and Variance Relations.  24.2 Mean-Square Stability and Performance. 24.3 Small Step-Size Approximations. 24.A Independence and Averaging Analysis.

25. Data-Normalized Filters. 25.1 NLMS Filter.  25.2 Data-Normalized Filters. 25.A Stability Bound. 25.B Stability of NLMS. Summary and Notes. Problems and Computer Projects.


26. Transform Domain Adaptive Filters. 26.1 Transform-Domain Filters. 26.2 DFT-Domain LMS. 26.3 DCT-Domain LMS. 26.A DCT-Transformed Regressors.

27. Efficient Block Convolution. 27.1 Motivation. 27.2 Block Data Formulation. 27.3 Block Convolution.

28. Block and Subband Adaptive Filters. 28.1 DFT Block Adaptive Filters. 28.2 Subband Adaptive Filters. 28.A Another Constrained DFT Block Filter. 28.B Overlap-Add Block Adaptive Filters. Summary and Notes. Problems and Computer Projects.


29. Least-Squares Criterion. 29.1 Least-Squares Problem. 29.2 Geometric Argument. 29.3 Algebraic Arguments. 29.4 Properties of Least-Squares Solution. 29.5 Projection Matrices. 29.6 Weighted Least-Squares. 29.7 Regularized Least-Squares. 29.8 Weighted Regularized Least-Squares.

30. Recursive Least-Squares. 30.1 Motivation. 30.2 RLS Algorithm. 30.3 Regularization. 30.4 Conversion Factor. 30.5 Time-Update of the Minimum Cost. 30.6 Exponentially-Weighted RLS Algorithm.

31. Kalman Filtering and RLS. 31.1 Equivalence in Linear Estimation. 31.2 Kalman Filtering and Recursive Least-Squares. 31.A Extended RLS Algorithms.

32. Order and Time-Update Relations. 32.1 Backward Order-Update Relations. 32.2 Forward Order-Update Relations. 32.3 Time-Update Relation. Summary and Notes. Problems and Computer Projects.


33. Norm and Angle Preservation. 33.1 Some Difficulties. 33.2 Square-Root Factors. 33.3 Norm and Angle Preservation. 33.4 Motivation for Array Methods.

34. Unitary Transformations. 34.1 Givens Rotations. 34.2 Householder Transformations.

35. QR and Inverse QR Algorithms. 35.1 Inverse QR Algorithm. 35.2 QR Algorithm. 35.3 Extended QR Algorithm. 35.A Array Algorithms for Kalman Filtering. Summary and Notes. Problems and Computer Projects.


36. Hyperbolic Rotations. 36.1 Hyperbolic Givens Rotations. 36.2 Hyperbolic Householder Transformations. 36.3 Hyperbolic Basis Rotations.

37. Fast Array Algorithm. 37.1 Time-Update of the Gain Vector. 37.2 Time-Update of the Conversion Factor. 37.3 Initial Conditions. 37.4 Array Algorithm. 37.A Chandrasekhar Filter.

38. Regularized Prediction Problems. 38.1 Regularized Backward Prediction. 38.2 Regularized Forward Prediction. 38.3 Low-Rank Factorization.

39. Fast Fixed-Order Filters. 39.1 Fast Transversal Filter. 39.2 FAEST Filter. 39.3 Fast Kalman Filter. 39.4 Stability Issues. Summary and Notes. Problems and Computer Projects.


40. Three Basic Estimation Problems. 40.1 Motivation for Lattice Filters. 40.2 Joint Process Estimation. 40.3 Backward Estimation Problem. 40.4 Forward Estimation Problem. 40.5 Time and Order-Update Relations.

41. Lattice Filter Algorithms. 41.1 Significance of Data Structure. 41.2 A Posteriori-Based Lattice Filter. 41.3 A Priori-Based Lattice Filter.

42. Error-Feedback Lattice Filters. 42.1 A Priori Error-Feedback Lattice Filter. 42.2 A Posteriori Error-Feedback Lattice Filter. 42.3 Normalized Lattice Filter.

43. Array Lattice Filters. 43.1 Order-Update of Output Estimation Errors.  43.2 Order-Update of Backward Estimation Errors. 43.3 Order-Update of Forward Estimation Errors. 43.4 Significance of Data Structure. Summary and Notes. Problems and Computer Projects.


44. Indefinite Least-Squares. 44.1 Indefinite Least-Squares. 44.2 Recursive Minimization Algorithm. 44.3 Time-Update of the Minimum Cost. 44.4 Singular Weighting Matrices. 44.A Stationary Points. 44.B Inertia Conditions.

45. Robust Adaptive Filters. 45.1 A Posteriori-Based Robust Filters. 45.2 e-NLMS Algorithm. 45.3 A Priori-Based Robust Filters. 45.4 LMS Algorithm. 45.A Hoo Filters.

46. Robustness Properties. 46.1 Robustness of LMS. 46.2 Robustness of e-NLMS. 46.3 Robustness of RLS. Summary and Notes. Problems and Computer Projects.



Author Index.

Subject Index.