Abstract
In Chap. 2, we discussed the recursive laws of the Normal Equations, and in Chap. 4, we saw how these properties can be used to obtain fast processing schemes for solving the Normal Equations in the recursive case based on the Givens reduction. This chapter is devoted to the recursive least-squares (RLS) algorithms based on a transversal predictor structure. Contrary to the order in this book, recursive solutions of the Normal Equations were first investigated for the case of a transversal prediction error filter. The first commonly recognized algorithm of this type was derived by Godard [5.1] in 1974. But the RLS algorithm was apparently found independently by several authors. The earliest reference seems to be Plackett [5.2]. The RLS algorithm exploits the fact that an actual solution of the Normal Equations can be computed recursively from the previous (one time-step delayed) solution plus some update information, which depends on the actual (incoming) process sample. Therefore, the RLS algorithm exhibits the structure of a Kalman filter [5.3,4]. See also Chap. 10 for a discussion of the relationships between parameter estimation and Kalman filter theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Chapter 5
D. Godard: Channel equalization using a Kalman filter for fast data transmission. IBM J. Res. Dev. 18, 267–273 (1974)
R.L. Plackett: Some theorems in least squares. Biometrika 37, 149 (1950)
R.E. Kalman: A new approach to linear filtering and prediction problems. J. Basic Eng. 82, 35–45 (1960)
M.D. Srinath, P.K. Rajasekaran: An Introduction to Statistical Signal Processing with Applications (Wiley, New York 1979)
L. Ljung, M. Morf, D.D. Falconer: Fast calculation of gain matrices for recursive estimation schemes. Int. J. Control 27, 1–19 (1978)
T.C. Hsia: System Identification (Lexington Books, Lexington, MA 1977)
L. Ljung, T. Söderström: Theory and Practice of Recursive Identification (MIT Press, Cambridge, MA 1983)
J. Sherman, W.J. Morrison: Adjustment of an inverse matrix corresponding to the changes in the elements of a given column or a given row of the original matrix. Ann. Math. Stat. 20, 621 (1949)
J.M. Cioffi, T. Kailath: Fast, recursive least-squares transversal filters for adaptive filtering. IEEE Trans. ASSP 32, 304–337 (1984)
G. Carayannis, D. Manolakis, N. Kalouptsidis: A fast sequential algorithm for least-squares filtering and prediction. IEEE Trans. ASSP 31, 1394–1402 (1983)
N. Kalouptsidis, G. Carayannis, D. Manolakis: A fast covariance type algorithm for sequential LS filtering and prediction. IEEE Trans. Autom. Control 29, 752–755 (1984)
B. Widrow, M.E. Hoff, Jr.: Adaptive switching circuits. Weston Conf. Rec, IRE, Part 4, 96–104 (1960)
B. Widrow, J.M. McCool, M.G. Larimore, C.R. Johnson: Stationary and nonstationary learning characteristics of the LMS adaptive filter. Proc. IEEE 64, 1151–1161 (1976)
B. Widrow, S.D. Stearns: Adaptive Signal Processing (Prentice-Hall, Englewood Cliffs, NJ 1985)
J.E. Potter: New Statistical Formulas. Memo 40, Instrumentation Laboratory, Massachusetts Institute of Technology (1963)
G. Kubin: Stabilization of the RLS algorithm in the absence of persistent excitation. Proc. Int. Conf. on ASSP, New York (1988) pp. 1369–1372
G.J. Bierman: Factorization Methods for Discrete Sequential Estimation (Academic, New York 1977)
M. Morf: Fast Algorithms for Multivariable Systems. Ph.D. Dissertation, Stanford University, Stanford, CA (1974)
M. Morf, T. Kailath, L. Ljung: Fast algorithms for recursive identification. Proc. IEEE Conf. Dec. Control, 916–921 (1976)
E. Eleftheriou, D.D. Falconer: Tracking properties and steady-state performance of RLS adaptive filter algorithms. IEEE Trans. ASSP 34, 1097–1110 (1986)
J.M. Cioffi: Fast Transversal Filters for Communications Applications. Ph.D. Dissertation, Stanford University, Stanford, CA (1984)
J.M. Cioffi, T. Kailath: Windowed fast transversal filters adaptive algorithms with normalization. IEEE Trans. ASSP 33, 607–625 (1985)
C. Samson: A unified treatment of fast Kalman algorithms for identification. Int. J. Control 35, 909–934 (1982)
M.L. Honig, D.G. Messerschmitt: Adaptive Filters (Cluwer, Boston, MA 1984)
S.T. Alexander: Adaptive Signal Processing: Theory and Applications (Springer, New York 1986)
D.T.L. Lee: Canonical Ladder Form Realizations and Fast Estimation Algorithms. Ph.D. Dissertation, Stanford University, Stanford, CA (1980)
N.J. Bershad: On the optimum gain parameter in LMS adaptation. IEEE Trans. ASSP 35, 1065–1067 (1987)
N.J. Bershad: Behavior of ε-normalized LMS algorithm with Gaussian input. IEEE Trans. ASSP 35, 636–644 (1987)
N.J. Bershad, P.L. Feintuch: A normalized frequency domain LMS adaptive algorithm. IEEE Trans. ASSP 34, 452–461 (1986)
C.F. Cowan, P.M. Grant: Adaptive Filters (Prentice-Hall, Englewood Cliffs, NJ 1985)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Strobach, P. (1990). Recursive Least-Squares Transversal Algorithms. In: Linear Prediction Theory. Springer Series in Information Sciences, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-75206-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-75206-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-75208-7
Online ISBN: 978-3-642-75206-3
eBook Packages: Springer Book Archive