Adaptive Filtering pp 309-359 | Cite as

# QR-Decomposition-Based RLS Filters

Chapter

## Abstract

The application of QR decomposition [1] to triangularize the input data matrix results in an alternative method for the implementation of the recursive least-squares (RLS) method previously discussed. The main advantages brought about by the recursive least-squares algorithm based on QR decomposition are its possible implementation in systolic arrays [2]–[4] and its improved numerical behavior when quantization effects are taken into account [5].

## Keywords

Information Matrix Adaptive Filter Systolic Array Real Time Signal Processing Input Data Matrix
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.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.G. H. Golub and C. F. Van Loan,
*Matrix Computations*, John Hopkins University Press, Baltimore, MD, 2nd edition, 1989.MATHGoogle Scholar - 2.W. H. Gentleman and H. T. Kung, “Matrix triangularization by systolic arrays,”
*Proc. of SPIE, Real Time Signal Processing*IV, vol. 298, pp. 19–26, 1981.CrossRefGoogle Scholar - 3.J. G. McWhirter, “Recursive least-squares minimization using a systolic array,”
*Proc. of SPIE, Real Time Signal Processing*VI, vol. 431, pp. 105–112, 1983.CrossRefGoogle Scholar - 4.J. M. Cioffi, “The fast adaptive ROTOR’s RLS algorithm,”
*IEEE Trans, on Acoust., Speech, and Signal Processing*, vol. 38, pp. 631–653, April 1990.CrossRefGoogle Scholar - 5.I. K. Proudler, J. G. McWhirter, and Y. J. Shepherd, “Fast QRD-based algorithms for least squares linear prediction,”
*Proc. IMA Conference on Mathematics in Signal Processing*, Warwick, England, pp. 465–488, Dec. 1988.Google Scholar - 6.M. G. Bellanger, “The FLS-QR algorithm for adaptive filtering,”
*Signal Processing*, vol. 17, pp. 291–304, Aug. 1984.MathSciNetCrossRefGoogle Scholar - 7.M. G. Bellanger and P. A. Regalia, “The FLS-QR algorithm for adaptive filtering: The case of multichannel signals,”
*Signal Processing*, vol. 22, pp. 115–126, March 1991.MATHCrossRefGoogle Scholar - 8.P. A. Regalia and M. G. Bellanger, “On the duality between fast QR methods and lattice methods in least squares adaptive filtering,”
*IEEE Trans, on Signal Processing*, vol. 39, pp. 879–891, April 1991.CrossRefGoogle Scholar - 9.J. A. Apolinario Jr., and P. S. R. Diniz, “A new fast QR algorithm based on
*a priori*errors,”*IEEE Signal Processing Letters*, vol. 4, pp. 307–309, Nov. 1997.CrossRefGoogle Scholar - 10.M. D. Miranda and M. Gerken, “A hybrid QR-lattice least squares algorithm using
*a priori*errors,”*IEEE Trans, on Signal Processing*, vol. 45, pp. 2900–2911, Dec. 1997.CrossRefGoogle Scholar - 11.A. A. Rontogiannis and S. Theodoridis, “New fast QR decomposition least squares adaptive algorithms,”
*IEEE Trans, on Signal Processing*, vol. 46, pp. 2113–2121, Aug. 1998.CrossRefGoogle Scholar - 12.J. A. Apolinario Jr., M. G. Siqueira, and P. S. R. Diniz, “On fast QR algorithm based on backward prediction errors: new result and comparisons,”
*Proc. first IEEE Balkan Conf. on Signal Processing, Communications, Circuits, and Systems*, Istanbul, Turkey, pp. 1–4, CD-ROM, June 2000.Google Scholar - 13.C. R. Ward, P. J. Hargrave, and J. G. McWhirter, “A novel algorithm and architecture for adaptive digital beamforming,”
*IEEE Trans, on Antennas and Propagation*, vol. 34, pp. 338–346, March 1986.CrossRefGoogle Scholar - 14.Z. Chi, J. Ma, and K. Parhi, “Hybrid annihilation transformation (HAT) for pipelining QRD-based least-square adaptive filters,”
*IEEE Trans, on Circuits and Systems-II: Analog and Digital Signal Processing*, vol. 48, pp. 661–674, July 2001.CrossRefGoogle Scholar - 15.C. A. Mead and L. A. Conway,
*Introduction to VLSI Systems*, Addison-Wesley, Reading, MA, 1980.Google Scholar - 16.W. H. Gentleman, “Least squares computations by Givens transformations without square roots,”
*Inst. Maths. Applies.*, vol. 12, pp. 329–336, 1973.MathSciNetMATHCrossRefGoogle Scholar - 17.W. H. Gentleman, “Error analysis of QR decompositions by Givens transformations,”
*Linear Algebra and its Applications*, vol. 10, pp. 189–197, 1975.MathSciNetMATHCrossRefGoogle Scholar - 18.H. Leung and S. Haykin, “Stability of recursive QRD-LS algorithms using finite-precision systolic array implementation,”
*IEEE Trans, on Acoust., Speech, and Signal Processing*, vol. 37, pp. 760–763, May 1989.CrossRefGoogle Scholar - 19.K. J. R. Liu, S.-F. Hsieh, K. Yao, and C.-T. Chiu, “Dynamic range, stability, and fault-tolerant capability of finite-precision RLS systolic array based on Givens rotations,”
*IEEE Trans, on Circuits and Systems*, vol. 38, pp. 625–636, June 1991.CrossRefGoogle Scholar - 20.P. S. R. Diniz and M. G. Siqueira, “Fixed-point error analysis of the QR-recursive least squares algorithm,”
*IEEE Trans, on Circuits and Systems II: Analog and Digital Signal Processing*, vol. 43, pp. 334–348, May 1995.CrossRefGoogle Scholar - 21.P. A. Regalia, “Numerical stability properties of a QR-based fast least squares algorithm,”
*IEEE Trans, on Signal Processing*, vol. 41, pp. 2096–2109, June 1993.MATHCrossRefGoogle Scholar - 22.M. G. Siqueira, P. S. R. Diniz, and A. Alwan, “Infinite precision analysis of the fast QR-recursive least squares algorithm,”
*Proc. IEEE Intern. Symposium on Circuits and Systems*, London, England, pp. 2.293–2.296 , May 1994.Google Scholar - 23.P. S. Lewis, “QR-based algorithms for multichannel adaptive least squares lattice filters,”
*IEEE Trans, on Acoust., Speech, and Signal Processing*, vol. 38, pp. 421–432, May 1990.MATHCrossRefGoogle Scholar - 24.I. K. Proudler, J. G. McWhirter, and T. J. Shepherd, “Computationally efficient QR decomposition approach to least squares adaptive filtering,”
*IEE Proceedings-Part F*, vol. 148, pp. 341–353, Aug. 1991.Google Scholar - 25.B. Yang, and J. F. Bohme, “Rotation-based RLS algorithms: Unified derivations, numerical properties, and parallel implementations,”
*IEEE Trans, on Signal Processing*, vol. 40, pp. 1151–1166, May 1992.MATHCrossRefGoogle Scholar - 26.F. Ling, “Givens rotation based least squares lattice and related algorithms,”
*IEEE Trans, on Signal Processing*, vol. 39, pp. 1541–1551, July 1991.CrossRefGoogle Scholar - 27.J. M. Cioffi, “The fast Householder filters RLS adaptive filter,”
*Proc. IEEE Intern. Conf, on Acoust., Speech, Signal Processing*, Albuquerque, NM, pp. 1619–1622, 1990.Google Scholar - 28.K. J. R. Liu, S.-F. Hsieh, and K. Yao, “Systolic block Householder transformation for RLS algorithm with two-level pipelined implementation,”
*IEEE Trans, on Signal Processing*, vol. 40, pp. 946–957, April 1992.CrossRefGoogle Scholar - 29.Z.-S. Liu and J. Li, “A QR-based least mean squares algorithm for adaptive parameter estimation,”
*IEEE Trans, on Circuits and Systems-II: Analog and Digital Signal Processing*, vol. 45, pp. 321–329, March 1998.Google Scholar - 30.A. Ghirnikar and S. T. Alexander, “Stable recursive least squares filtering using an inverse QR decomposition,”
*Proc. IEEE Intern. Conf. on Acoust., Speech, Signal Processing*, Albuquerque, NM, pp. 1623–1626, 1990.Google Scholar - 31.S. T. Alexander and A. Ghirnikar, “A method for recursive least squares filtering based upon an inverse QR decomposition,”
*IEEE Trans, on Signal Processing*, vol. 41, pp. 20–30, Jan. 1993.MATHCrossRefGoogle Scholar - 32.M. Syed and V. J. Mathews, “QR-Decomposition based algorithms for adaptive Volterra filtering,”
*IEEE Trans, on Circuits and Systems I: Fundamental Theory and Applications*, vol. 40, pp. 372–382, June 1993.MATHCrossRefGoogle Scholar

## Copyright information

© Springer Science+Business Media New York 2002