Fast Recursive Least-Squares Ladder Algorithms

  • Peter Strobach
Part of the Springer Series in Information Sciences book series (SSINF, volume 21)


Chapters 7 and 8 have illuminated the classes of ladder algorithms that are based on a pure order recursive construction of the ladder form. In this type of algorithm, the central problem appeared to be the order recursive updating of the covariance Cm(t) according to
$$ {{{\text{C}}}_{{\text{m}}}}{\text{(t)}}{\mkern 1mu} {\mkern 1mu} {\text{ = }}{\mkern 1mu} {{{\text{C}}}_{{{\text{m - 1}}}}}{\text{(t)}}{\mkern 1mu} {\text{ + }}{\mkern 1mu} \ldots {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\text{.}} $$
Two solutions to the problem (9.1) have been presented. The first approach, discussed in Chap. 7, was based on the incorporation of transversal predictor parameters as intermediate recursion variables in Levinson-type algorithms. Alternatively, Chap. 8 introduced the algorithms of the PORLA type where the objective was to replace the Levinson-type recursion by inner product recursions, where inner products, also termed “generalized residual energies”, were used as intermediate recursion variables, hence avoiding the explicit computation of transversal predictor parameters.


Reflection Coefficient Projection Operator Residual Energy Residual Vector Differential Covariance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


Chapter 9

  1. 9.1
    D.T.L. Lee: Canonical Ladder Form Realizations and Fast Estimation Algorithms. Ph.D. Dissertation, Stanford University, Stanford, CA (1980)Google Scholar
  2. 9.2
    G.W. Stewart: Error and perturbation bounds for subspaces associated with certain eigenvalue problems. SIAM Rev. 15, 727–764 (1973)CrossRefMATHMathSciNetGoogle Scholar
  3. 9.3
    A. Björck, G.H. Golub: Numerical methods for computing angles between linear subspaces. Math. Comp. 27, 579–594 (1973)CrossRefMATHMathSciNetGoogle Scholar
  4. 9.4
    P.A. Wedin: Perturbation theory for pseudoinverses. B.I.T. 13, 217–232 (1973)MATHMathSciNetGoogle Scholar
  5. 9.5
    S. Afriat: Orthogonal and oblique projections and the characteristics of pairs of vector spaces. Proc. Cambridge Philos. Soc. 53, 800–816 (1957)CrossRefADSMathSciNetGoogle Scholar
  6. 9.6
    H. Zassenhaus: Angles of inclination in correlation theory. Am. Math. Mon. 71, 218–219 (1964)CrossRefMathSciNetGoogle Scholar
  7. 9.7
    T.N.E. Greville: Solutions of the matrix equation XAX = X, and relations between oblique and orthogonal projectors. SIAM J. Appl. Math. 26, 828–832 (1974)CrossRefMATHMathSciNetGoogle Scholar
  8. 9.8
    H.J. Landau, H.O. Pollak: Prolate spheroidal wave functions, Fourier analysis and uncertainty — II Bell Syst. Tech. J. XI, 65–84 (1961)MathSciNetGoogle Scholar
  9. 9.9
    J.E. Mazo: On the angle between two Fourier subspaces. Bell Syst. Tech. J. 56, 411–426 (1977)MATHMathSciNetGoogle Scholar
  10. 9.10
    I.C. Gohberg, M.G. Krein: Introduction to the Theory of linear Nonselfadjoint Operators (American Math. Soc, Providence, RI 1969)Google Scholar
  11. 9.11
    H. Bart, I.C. Gohberg, M.A. Kaashoek: Minimal Factorization of Matrix and Operator Functions (Birkhäuser, Basel 1979)MATHGoogle Scholar
  12. 9.12
    P. Van Dooren, P. Dewilde: Minimal factorization of rational matrices. Proc. 17th IEEE Conf. Dec. Control, San Diego (1979) pp. 170–171Google Scholar
  13. 9.13
    J.M. Cioffi, T. Kailath: Fast, recursive least-squares transversal filters for adaptive filtering. IEEE Trans. ASSP 32, 304–337 (1984)CrossRefMATHGoogle Scholar
  14. 9.14
    M.L. Honig, D.G. Messerschmitt: Adaptive Filters (Cluwer, Boston, MA 1984)MATHGoogle Scholar
  15. 9.15
    S.T. Alexander: Adaptive Signal Processing: Theory and Applications (Springer, New York 1986)MATHGoogle Scholar
  16. 9.16
    D.T.L. Lee, M. Morf, B. Friedlander: Recursive least-squares ladder estimation algorithms. IEEE Trans. ASSP 29, 627–641 (1981)CrossRefMATHMathSciNetGoogle Scholar
  17. 9.17
    F. Ling, D. Manolakis, J.G. Proakis: Numerically robust least-squares lattice-ladder algorithms with direct updating of the reflection coefficients. IEEE Trans. ASSP 34, 837–845 (1986)CrossRefGoogle Scholar
  18. 9.18
    L.J. Griffiths: A continuously adaptive filter implemented as a lattice structure. Proc. IEEE Int. Conf. ASSP, Hartford (1977) pp. 683–686Google Scholar
  19. 9.19
    L.J. Griffiths: An adaptive lattice structure for noise cancelling applications. Proc. IEEE Int. Conf. ASSP, Tulsa (1978) pp. 87–90Google Scholar
  20. 9.20
    J. Makhoul, R. Viswanathan: Adaptive lattice methods for linear prediction. Proc. IEEE Int. Conf. ASSP, Tulsa (1978) pp. 83–86Google Scholar
  21. 9.21
    B. Friedlander: Lattice filters for adaptive processing. Proc. IEEE 70, 829–867 (1982)CrossRefGoogle Scholar
  22. 9.22
    C. Samson, V.U. Reddy: Fixed-point error analysis of the normalized ladder algorithm. IEEE Trans. ASSP 31, 1177–1191 (1983)CrossRefMATHGoogle Scholar
  23. 9.23
    B. Porat, B. Friedlander, M. Morf: Square-root covariance ladder algorithms. IEEE Trans. Autom. Control 27, 813–829 (1982)CrossRefMATHMathSciNetGoogle Scholar
  24. 9.24
    B. Porat, T. Kailath: Normalized lattice algorithms for least-squares FIR system identification. IEEE Trans. ASSP 31, 122–128 (1983)CrossRefMATHMathSciNetGoogle Scholar
  25. 9.25
    D.T.L. Lee, B. Friedlander, M. Morf: Recursive ladder algorithms for ARMA modeling. IEEE Trans. Autom. Control 27, 753–764 (1981)CrossRefMathSciNetGoogle Scholar
  26. 9.26
    N. Ahmed, M. Morf, D.T.L. Lee, P.H. Ang: A VLSI speech analysis chip set based on square-root normalized ladder forms. Proc. IEEE Int. Conf. ASSP, Atlanta (1981) pp. 648–653Google Scholar
  27. 9.27
    D.T.L. Lee, M. Morf: Generalized CORDIC for digital signal processing. Proc. IEEE Int. Conf. ASSP, Paris (1982) pp. 1748–1751Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Peter Strobach
    • 1
  1. 1.ZFE IS — Forschung für Informatik und SoftwareSIEMENS AG, Zentralabteilung Forschung und EntwicklungMünchen 83Fed. Rep. of Germany

Personalised recommendations