Conventional and Inverse QRD-RLS Algorithms

  • José A. ApolinárioJrEmail author
  • Maria D. Miranda


This chapter deals with the basic concepts used in the recursive least-squares (RLS) algorithms employing conventional and inverse QR decomposition. The methods of triangularizing the input data matrix and the meaning of the internal variables of these algorithms are emphasized in order to provide details of their most important relations. The notation and variables used herein will be exactly the same used in the previous introductory chapter. For clarity, all derivations will be carried out using real variables and the final presentation of the algorithms (tables and pseudo-codes) will correspond to their complex-valued versions.


Systolic Array Cholesky Factor Plane Rotation Error Energy Optimum Estimation Error 
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.


  1. 1.
    S. Haykin, Adaptive Filter Theory. 4th edition Prentice-Hall, Englewood Cliffs, NJ, USA (2002)Google Scholar
  2. 2.
    P. S. R. Diniz, Adaptive Filtering: Algorithms and Practical Implementation. 3rd edition Springer, New York, NY, USA (2008)Google Scholar
  3. 3.
    G. H. Golub and C. F. Van Loan, Matrix Computations. 2nd edition John Hopkins University Press, Baltimore, MD, USA (1989)Google Scholar
  4. 4.
    R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge University Press, New York, NY, USA (1986)Google Scholar
  5. 5.
    W. Givens, Computation of plane unitary rotations transforming a general matrix to triangular form. Journal of the Society for Industrial and Applied Mathematics, vol. 6, no. 1, pp. 26–50 (March 1958)Google Scholar
  6. 6.
    W. H. Gentleman and H. T. Kung, Matrix triangularization by systolic arrays. SPIE Real-Time Signal Processing IV, vol. 298, pp. 19–26 (January 1981)Google Scholar
  7. 7.
    J. G. McWhirter, Recursive least-squares minimization using a systolic array. SPIE Real-Time Signal Processing VI, vol. 431, pp. 105–112 (January 1983)Google Scholar
  8. 8.
    P. A. Regalia and M. G. Bellanger, On the duality between fast QR methods and lattice methods in least squares adaptive filtering. IEEE Transactions on Signal Processing, vol. 39, no. 4, pp. 879–891 (April 1991)Google Scholar
  9. 9.
    N. E. Hubing and S. T. Alexander, Statistical analysis of initialization methods for RLS adaptive filters. IEEE Transactions on Signal Processing, vol. 39, no. 8, pp. 1793–1804 (August 1991)Google Scholar
  10. 10.
    P. S. R. Diniz and M. G. Siqueira, Fixed-point error analysis of the QR-recursive least square algorithm. IEEE Transactions on Circuits and Systems–II: Analog and Digital Signal Processing, vol. 42, no. 5, pp. 334–348 (May 1995)Google Scholar
  11. 11.
    S. T. Alexander and A. L. Ghirnikar, A method for recursive least squares filtering based upon an inverse QR decomposition. IEEE Transactions on Signal Processing, vol. 41, no. 1, pp. 20–30 (January 1993)Google Scholar
  12. 12.
    J. E. Hudson and T. J. Shepherd, Parallel weight extraction by a systolic least squares algorithm. SPIE Advanced Algorithms and Architectures for Signal Processing IV, vol. 1152, pp. 68–77 (August 1989)Google Scholar
  13. 13.
    S. Ljung and L. Ljung, Error propagation properties of recursive least-squares adaptation algorithms. Automatica, vol. 21, no. 2, pp. 157–167 (March 1985)Google Scholar
  14. 14.
    M. D. Miranda and M. Gerken, Hybrid least squares QR-lattice algorithm using a priori errors. IEEE Transactions on Signal Processing, vol. 45, no. 12, pp. 2900–2911 (December 1997)Google Scholar
  15. 15.
    G. Strang, Computational Science and Engineering. Wellesly-Cambridge Press, Wellesley, MA, USA (2007)Google Scholar
  16. 16.
    B. Haller, J. Götze and J. R. Cavallaro, Efficient implementation of rotation operations for high performance QRD-RLS filtering. IEEE International Conference on Application-Specific Systems, Architectures and Processors, ASAP’97, Zurich, Switzerland, pp. 162–174 (July 1997)Google Scholar
  17. 17.
    J. E. Volder, The CORDIC Trigonometric Computing Technique. IRE Transactions on Electronic Computers, vol. EC-8, no. 3, pp. 330–334 (September 1959)Google Scholar
  18. 18.
    W. M. Gentleman, Least squares computations by Givens transformations without square roots. IMA Journal of Applied Mathematics, vol. 12, no. 3, pp. 329–336 (December 1973)Google Scholar
  19. 19.
    S. Hammarling, A note on modifications to the Givens plane rotation. IMA Journal of Applied Mathematics vol. 13, no. 2, pp. 215–218 (April 1974)Google Scholar
  20. 20.
    J. L. Barlow and I. C. F. Ipsen, Scaled Givens rotations for the solution of linear least squares problems on systolic arrays. SIAM Journal on Scientific and Statistical Computing, vol. 8, no. 5, pp. 716–733 (September 1987)Google Scholar
  21. 21.
    J. Götze and U. Schwiegelshohn, A square root and division free Givens rotation for solving least squares problems on systolic arrays. SIAM Journal on Scientific and Statistical Computing, vol. 12, no. 4, pp. 800–807 (July 1991)Google Scholar
  22. 22.
    E. N. Frantzeskakis and K. J. R. Liu, A class of square root and division free algorithms and architectures for QRD-based adaptive signal processing. IEEE Transactions on Signal Processing, vol. 42, no. 9, pp. 2455–2469 (September 1994)Google Scholar
  23. 23.
    S. F. Hsieh, K. J. R. Liu, and K. Yao, A unified square-root-free approach for QRD-based recursive-least-squares estimation. IEEE Transactions on Signal Processing, vol. 41, no. 3, pp. 1405–1409 (March 1993)Google Scholar

Copyright information

© Springer-Verlag US 2009

Authors and Affiliations

  1. 1.Military Institute of Engineering (IME)Rio de JaneiroBrazil
  2. 2.University of São Paulo (USP)São PauloBrazil

Personalised recommendations