Advertisement

On Pipelined Implementations of QRD-RLS Adaptive Filters

  • Jun MaEmail author
  • Keshab K. Parhi
Chapter

Abstract

This chapter discusses the pipelined systolic implementations pipelined implementation of QR-decomposition-based recursive least-squares (QRD-RLS) adaptive filters. Theannihilation-reording look-ahead technique is presented as an attractive technique for pipelining of Givens rotation (or CO-ordinate Rotation DIgital Computer (CORDIC)) based adaptive filters. It is an exact look-ahead and is based on CORDIC arithmetic, CORDIC!arithmetic which is known to be numerically stable. The conventional look-ahead is based on multiply–add arithmetic. The annihilation-reording look-ahead technique transforms an orthogonal sequential adaptive filtering algorithm into an equivalent orthogonal concurrent one by creating additional concurrency in the algorithm. Parallelism in the transformed algorithm is explored, and different implementation styles including pipelining, block processing, and incremental block processing are presented. Their complexity are also studied and compared. The annihilation-reording look-ahead is employed to develop fine-grain pipelined QRD-RLS adaptive filters. fine-grain pipelining Both implicit and explicit weight extraction algorithms are considered. The proposed pipelined architectures can be operated at arbitrarily high sample rate without degrading the filter convergence behavior. Stability under finite-precision arithmetic are studied and proved for the proposed architectures. The complexity of the pipelined architectures are analyzed and compared.

Keywords

Systolic Array Recursive Little Square Very Large Scale Integra Block Processing Little Mean Square Algorithm 
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.
    S. Haykin, Adaptive Filter Theory. 2nd edition Prentice-Hall, Englewood Cliffs, NJ, USA (1991)zbMATHGoogle Scholar
  2. 2.
    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
  3. 3.
    Y. H. Hu, Cordic-based VLSI architectures for digital signal processing. IEEE Signal Processing Magazine, no. 3, vol. 9, pp. 16–35 (July 1992)Google Scholar
  4. 4.
    G.J. Hekstra and E. F. Deprettere, Floating point CORDIC. 11th Symposium on Computer Arithmetic, Windsor, Canada, pp. 130–137 (June 1993)Google Scholar
  5. 5.
    E. Rijpkema, G. Hekstra, E. Deprettere, and J. Ma, A strategy for determining a Jacobi specific dataflow processor. IEEE International Conference on Application-Specific Systems, Architectures and Processors, Zurich, Switzerland, pp. 53–64 (July 1997)Google Scholar
  6. 6.
    J. G. McWhirter, Recursive least-squares minimization using a systolic array. Proc. SPIE: Real Time Signal Processing VI, vol. 431, pp. 105–112 (January 1983)Google Scholar
  7. 7.
    W. M. Gentleman and H. T. Kung, Matrix triangularization by systolic arrays. Proceedings SPIE: Real-Time Signal Processing IV, vol. 298, pp. 298–303 (January 1981)Google Scholar
  8. 8.
    T. J. Shepherd, J. G. McWhirter, and J. E.Hudson, Parallel weight extraction from a systolic adaptive beamformer. Mathematics in Signal Processing II (J.G. McWhirter ed.), pp. 775–790, Clarendon Press, Oxford (1990)Google Scholar
  9. 9.
    J. G. McWhirter and T. J. Shepherd, Systolic array processor for MVDR beamforming. IEE Proceedings, vol. 136, pp. 75–80 (April 1989)Google Scholar
  10. 10.
    K. K. Parhi, Algorithm transformation techniques for concurrent processors. Proceedings of the IEEE, vol. 77, pp. 1879–1895 (December 1989)Google Scholar
  11. 11.
    K. K. Parhi and D. G. Messerschmitt, Pipeline interleaving and parallelism in recursive digital filters–Part I: Pipelining using scattered look-ahead and decomposition. IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, pp. 1118–1134 (July 1989)Google Scholar
  12. 12.
    K. K. Parhi and D. G. Messerschmitt, Pipeline interleaving and parallelism in recursive digital filters–Part II: Pipelined incremental block filtering. IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, pp. 1099–1117 (July 1989)Google Scholar
  13. 13.
    A. P. Chandrakasan, S. Sheng, and R. W. Broderson, Low-power CMOS digital design. IEEE Journal on Solid-State Circuits, vol. 27, pp. 473–484 (April 1992)Google Scholar
  14. 14.
    K. K. Parhi, VLSI Digital Signal Processing Systems, Design and Implementation. John Wiley & Sons, New York, NY, USA (1999)Google Scholar
  15. 15.
    K. J. Raghunath and K. K. Parhi, Pipelined RLS adaptive filtering using Scaled Tangent Rotations (STAR). IEEE Transactions on Signal Processing, vol. 40, pp. 2591–2604 (October 1996)Google Scholar
  16. 16.
    T. H. Y. Meng, E. A. Lee, and D. G. Messerschmitt, Least-squares computation at arbitrarily high speeds, International Conference on Acoustics, Speech, and Signal Processing, ICASSP’1987, Dallas, USA, pp. 1398–1401 (April 1987)Google Scholar
  17. 17.
    G. H. Golub and C. F. V. Loan, Matrix Computation. Johns Hopkins University Press, Baltimore, MD, USA (1989)Google Scholar
  18. 18.
    J. M. Cioffi, The fast adaptive ROTOR RLS algorithm. IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 38, pp. 631–653 (April 1990)Google Scholar
  19. 19.
    S. F. Hsieh, K. J. R. Liu, and K. Yao, A unified square-root-free Givens rotation approach for QRD-based recursive least squares estimation. IEEE Transactions on Signal Processing, vol. 41, pp. 1405–1409 (March 1993)Google Scholar
  20. 20.
    S. Hammarling, A note on modifications to Givens plane rotation. IMA Journal of Applied Mathematics, vol. 13, no. 2, pp. 215–218 (1974)zbMATHMathSciNetGoogle Scholar
  21. 21.
    J. L. Barlow and I. C. F. Ipsen, Scaled Givens rotations for solution of linear least-squares problems on systolic arrays. SIAM Journal on Scientific and Statistical Computing, vol. 13, no. 5, pp. 716–733 (September 1987)Google Scholar
  22. 22.
    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
  23. 23.
    E. Franrzeskakis 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, pp. 2455–2469 (September 1994)Google Scholar
  24. 24.
    J. Ma, K. K. Parhi, and E. F. Deprettere, Annihilation reordering lookahead pipelined CORDIC based RLS adaptive filters and their application to adaptive beamforming. IEEE Transactions on Signal Processing, vol. 48, pp. 2414–2431 (August 2000)Google Scholar
  25. 25.
    C. E. Leiserson, F. Rose, and J. Saxe, Optimizing synchronous circuitry by retiming. 3rd Caltech Conference on VLSI, Pasadena, USA, pp. 87–116 (March 1983)Google Scholar
  26. 26.
    K. K. Parhi, High-level algorithm and architecture transformations for DSP synthesis. Journal of VLSI Signal Processing, vol. 9, pp. 121–143 January 1995Google Scholar
  27. 27.
    E. F. Deprettere, P. Held, and P. Wielage, Model and methods for regular array design. International Journal of High Speed Electronics; Special issue on Massively Parallel Computing–Part II, vol. 4(2), pp. 133–201 (1993)Google Scholar
  28. 28.
    H. 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
  29. 29.
    M. Moonen and E. F. Deprettere, A fully pipelined RLS-based array for channel equalization. Journal of VLSI Signal Processing, vol. 14, pp. 67–74 (October 1996)Google Scholar
  30. 30.
    R. Gooch and J. Lundell, The CM array: an adaptive beamformer for constant modulus signals. International Conference on Acoustics, Speech, and Signal Processing, ICASSP’1986, Tokyo, Japan, pp. 2523–2526 (April 1986)Google Scholar
  31. 31.
    C.-T. Pan and R. J. Plemmons, Least squares modifications with inverse factorizations: parallel implications. Journal of Computational and Applied Mathematics, vol. 27, pp. 109–127 (September 1989)Google Scholar
  32. 32.
    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. SP-41, no. 1, pp. 20–30 (January 1993)Google Scholar
  33. 33.
    O. L. Frost III, An algorithm for linearly constrained adaptive array processing. Proceedings of the IEEE, vol. 60, no. 8, pp. 926–935 (August 1972)Google Scholar
  34. 34.
    R. L. Hanson and C. L. Lawson, Extensions and applications of the Householder algorithm for solving linear least squares problems. AMS Mathematics of Computation, vol. 23, no. 108, pp. 787-812 (October 1969)Google Scholar
  35. 35.
    S. P. Applebaum and D. J. Chapman, Adaptive arrays with main beam constraints. IEEE Transactions on Antennas and Propagation, vol. AP-24, no. 5, pp. 650–662 (September 1976)Google Scholar

Copyright information

© Springer-Verlag US 2009

Authors and Affiliations

  1. 1.Shanghai Jiaotong UniversityShanghaiChina
  2. 2.University of MinnesotaMinneapolisUSA

Personalised recommendations