Advertisement

Finite and Infinite-Precision Properties of QRD-RLS Algorithms

  • Paulo S.R. DinizEmail author
  • Marcio G. Siqueira
Chapter

Abstract

This chapter analyzes the finite and infinite-precision properties of QR-decomposition recursive least-squares (QRD-RLS) algorithms with emphasis on the conventional QRD-RLS and fast QRD-lattice (FQRD-lattice) formulations. The analysis encompasses deriving mean squared values of internal variables in steady-state and also the mean squared error of the deviations of the same variables assuming fixed-point arithmetic. In particular, analytical expressions for the excess of mean squared error and for the variance of the deviation in the tap coefficients of the QRD-RLS algorithm are derived, and the analysis is extended to the error signal of the FQRD-lattice algorithm. All the analytical results are confirmed to be accurate through computer simulations. Conclusions follow.

Keywords

Internal Variable Quantization Error Precision Analysis Systolic Array Array Implementation 
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.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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 Transactions on Circuits and Systems, vol. 38, no. 6, pp. 625–636 (June 1991)Google Scholar
  2. 2.
    S. Leung and S. Haykin, Stability of recursive QRD-LS algorithms using finite-precision systolic array implementation. IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, no. 5, pp. 760–763 (May 1989)Google Scholar
  3. 3.
    S. Haykin, Adaptive Filter Theory. Prentice-Hall, Englewood Cliffs, NJ, USA (1991)Google Scholar
  4. 4.
    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
  5. 5.
    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
  6. 6.
    P. S. R. Diniz and M. G. Siqueira, Finite precision analysis of the QR-recursive least squares algorithm. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 42, pp. 334–348 (May 1995)Google Scholar
  7. 7.
    A. Papoulis, Probability, Random Variables, and Stochastic Processes. McGraw-Hill Book Company, New York, NY, USA (1965)Google Scholar
  8. 8.
    C. Caraiscos and B. Liu, A roundoff error analysis of the LMS adaptive algorithm. IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-32, no, 1, pp. 34–41 (February 1984)Google Scholar
  9. 9.
    N. Kalouptsidis and S. Theodoridis, Adaptive System Identification and Signal Processing Algorithms, Prentice-Hall, Upper Saddle River, NJ, USA (1993)Google Scholar
  10. 10.
    M. G. Siqueira and P. S. R. Diniz, Infinite precision analysis of the QR-recursive least squares algorithm. IEEE International Symposium on Circuit and Systems, ISCAS’93, Chicago, USA, pp. 878–881 (May 1993)Google Scholar
  11. 11.
    C. G. Samson and V. U. Reddy, Fixed point error analysis of the normalized ladder algorithm. IEEE Transactions on Audio, Speech, and Signal Processing, vol. ASSP-31, no. 5, pp. 1177–1191 (October 1983)Google Scholar
  12. 12.
    M. G. Siqueira, P. S. R. Diniz, and A. Alwan, Infinite precision analysis of the fast QR decomposition RLS algorithm. IEEE International Symposium on Circuits and Systems, ISCAS’94, London, UK, vol. 2, pp. 293–296 (May–June 1994)Google Scholar

Copyright information

© Springer-Verlag US 2009

Authors and Affiliations

  1. 1.Federal University of Rio de Janeiro (UFRJ)Rio de JaneiroBrazil
  2. 2.Cisco Systems IncSan JoseUSA

Personalised recommendations