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Systolic Adaptive Beamforming

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Radar Array Processing

Part of the book series: Springer Series in Information Sciences ((SSINF,volume 25))

Abstract

In this chapter we address the important subject of adaptive beamforming or “null-steering” as applied to an adaptive antenna array for the purposes of noise cancellation. The chapter is not intended as a general review or tutorial discussion of the subject. It is concerned solely with the application of systolic arrays to adaptive beamforming networks based on least-squares minimization. The results described here centre upon a particular systolic array first proposed by Gentleman and Kung [5.1] for performing the QR decomposition of a matrix in an efficient row-recursive manner. It will become clear in our subsequent development of the subject that this array provides the basic architectural component for the solution of a wide variety of signal processing problems. Most of the work described in this chapter was carried out during the last five or six years as part of a joint research project between the Royal Signals and Radar Establishment (RSRE) and STC Technology Ltd (STL). The key results have been described in previous publications but no complete overview of the subject has been presented and many important details have not been reported to date. Since the techniques are now being developed for practical application in a number of laboratories worldwide, it seems appropriate to present a more complete discussion in this book.

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References

  1. W.M. Gentleman, H.T. Kung: Matrix triangularization by systolic arrays, in Proc. SPIE, Vol.298, Real Time Signal Processing IV (International Society for Optical Engineering, Bellingham, WA 1989) pp. 19–26

    Google Scholar 

  2. F. Ling, D. Manolakis, J.G. Proakis: A recursive modified Gram-Schmidt algorithm for least-squares estimation. IEEE Trans. ASSP-34, 829–836 (1986)

    Google Scholar 

  3. S. Kalson, K. Yao: Geometrical approach to generalized least-squares estimation with systolic array processing, in Proc. of the 22nd Annual Allerton Conf. on Communications, Control and Computing (1984) pp. 333–342

    Google Scholar 

  4. R. Schreiber, W-P. Tang: On systolic arrays for updating the Cholesky factorization. BIT 26, 451–466 (1986)

    Article  MATH  Google Scholar 

  5. K.C. Sharman, T.S. Durrani: Spatial lattice filter for high-resolution spectral analysis of array data. IEE Proc. F (London) 130, 279–287 (1983)

    Google Scholar 

  6. S. Haykin: Adaptive Filter Theory (Prentice-Hall, Englewood Cliffs, NJ 1986)

    Google Scholar 

  7. S.P. Applebaum, DJ. Chapman: Adaptive arrays with main beam constraints. IEEE Trans. AP-24, 650–662 (1976)

    Google Scholar 

  8. B. Widrow, J.R. Glover, J.M. McCool, J. Kaunitz, CS. Williams, R.H. Hearn, J.R. Zeidler, E. Dong, R.C. Goodlin: Adaptive noise cancelling: Principles and applications. Proc. IEEE 63, 1692–1716 (1975)

    Article  Google Scholar 

  9. H.T. Kung, C.E. Leiserson: Algorithms for VLSI processor arrays, in Introduction to VLSI Systems, ed. by C.A. Mead, L. Conway (Addison-Wesley, Reading, MA 1980) pp. 271–292

    Google Scholar 

  10. S.Y. Kung, K.S. Arun, R.J. Gal-Ezer, D.V. Bhaskar-Rao: Wavefront array processor: Language, architecture, and applications. IEEE Trans. C-31, 1054–1066 (1982)

    Google Scholar 

  11. J.E. Hudson: Adaptive Array Principles (Peregrinus, UK 1981)

    Book  Google Scholar 

  12. O.L. Frost: An algorithm for linearly constrained adaptive array processing. Proc. IEEE 60, 661–675 (1971)

    Google Scholar 

  13. C.L. Lawson, R.J. Hanson: Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ 1974)

    MATH  Google Scholar 

  14. LS. Reed, J.D. Mallett, L.E. Brennan: Rapid convergence rate in adaptive arrays. IEEE Trans. AES-10, 853–863 (1974)

    Google Scholar 

  15. A.S. Householder: Unitary triangularization of a nonsymmetric matrix. J. ACM 5, 339–342 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  16. W. Givens: Computation of plane unitary rotations transforming a general matrix to triangular form. J. Soc. Ind. Appl. Math. 6, 26–50 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  17. R. Schreiber, P.J. Kuekes: Systolic linear algebra machines in digital signal processing, in VLSI and Modern Signal Processing, ed. by S.Y. Kung, H.J. Whitehouse, T. Kailath (Prentice-Hall, Englewood Cliffs, NJ 1985) pp. 389–405

    Google Scholar 

  18. W.M. Gentleman: Least-squares computations by Givens transformations without square roots. J. Inst. Math. Its Appl. 12, 329–336 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  19. CR. Ward, P.J. Hargrave, J.G. McWhirter: A novel algorithm and architecture for adaptive digital beamforming. IEEE Trans. AP-34, 338–346 (1986)

    Google Scholar 

  20. D.T.L. Lee, B. Friedlander, M. Morf: Recursive ladder estimation algorithms. IEEE Trans. ASSP-29, 627–641 (1981)

    MathSciNet  Google Scholar 

  21. D.D. Falconer, L. Ljung: Application of fast Kalman estimation to adaptive equalization. IEEE Trans. COM-26, 1439–1446 (1978)

    Article  Google Scholar 

  22. J.G. McWhirter: Recursive least-squares minimization using a systolic array, in Proc. SPIE, Vol. 431, Real Time Signal Processing IV (Int Soc. for Optical Engineering, Bellingham, WA 1983) pp. 105–112

    Google Scholar 

  23. J.G. McWhirter, T.J. Shepherd: Least-squares lattice algorithm for adaptive channel equalisation-A simplified derivation. IEE (London) Proc, Part F 130, 532–542 (1983)

    Google Scholar 

  24. S.Y. Kung: VLSI Array Processors (Prentice-Hall, Englewood Cliffs, NJ 1988)

    Google Scholar 

  25. Å. Björk: Solving linear least-squares problems by Gram-Schmidt orthogonalization. BIT 7, 1–21 (1967)

    Article  Google Scholar 

  26. R.A. Monzingo, T.W. Miller: Introduction to Adaptive Arrays (Wiley, New York 1980)

    Google Scholar 

  27. T.J. Shepherd, J.G. McWhirter: A pipelined array for linearly constrained least-squares optimisation, in Proc. 1985 IMA Conf. on Mathematics in Signal Processing, ed. by T.S. Durrani, J.B. Abbiss, J.E. Hudson, R.N. Madan, J.G. McWhirter, T.A. Moore (Clarendon, Oxford 1987) pp. 457–483

    Google Scholar 

  28. T.J. Shepherd, J.G. McWhirter: A systolic array for linearly constrained least-squares optimisation, in Proc. 1986 Int. Workshop on Systolic Arrays, ed. by W. Moore, A.M. McCabe, R. Urquhart (Adam Hilger, Bristol 1987) pp. 151–159

    Google Scholar 

  29. C.W. Jim: A comparison of two LMS constrained optimal array structures. Proc. IEEE 65, 1730–1731 (1977)

    Article  Google Scholar 

  30. L.J. Griffiths, C.W. Jim: An alternative approach to linearly constrained adaptive beam-forming. IEEE Trans. AP-30, 27–34 (1982)

    Google Scholar 

  31. S. Kalson, K. Yao: A systolic array for linearly constrained least-squares fitting, in Proc. 1985 IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, Tampa, FL (1985) pp. 977–980

    Google Scholar 

  32. C.Y. Tseng, L.J. Griffiths: A unification and comparison of several adaptive linearly-constrained beamformer structures, in Proc. SPIE, Vol 1152, Advanced Algorithms for Signal Processing IV, ed. by FT. Luk (Int. Soc. for Optical Engineering, Bellingham, WA 1989) pp. 158–256

    Google Scholar 

  33. N.L. Owsley: High-resolution spectrum analysis by dominant-mode enhancement, in VLSI and Modern Signal Processing, ed. by S.Y. Kung, H.J. Whitehouse, T. Kailath (Prentice-Hall, Englewood Cliffs, NJ 1985) pp. 61–82

    Google Scholar 

  34. R. Schreiber: Implementation of adaptive array algorithms. IEEE Trans. ASSP-34,1038–1045 (1986)

    Google Scholar 

  35. A.W. Bojanczyk, F.T. Luk: A novel MVDR beamforming algorithm, in Proc. SPIE, Vol. 826, Advanced Algorithms and Architectures for Signal Processing II, ed. by F.T. Luk (Int. Soc. for Optical Engineering, Bellingham, WA 1987) pp. 12–16

    Google Scholar 

  36. J.G. McWhirter, T.J. Shepherd: Systolic array for MVDR beamforming. IEE (London) Proc. Part F 136, 75–80 (1989)

    Google Scholar 

  37. B. Yang, J.F. Böhme: SystoHc implementation of a general adaptive array processing algorithm, in Proc. 1988 IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, New York (1988) pp. 2785–2789

    Google Scholar 

  38. J.V. McCanny, J.G. McWhirter: Some systolic array developments in the United Kingdom. IEEE Trans. C-20, 51–63 (1987)

    Google Scholar 

  39. D.S. Broomhead, J.G. Harp, J.G. McWhirter, K.J. Palmer, J.G.B. Roberts: A practical comparison of the systolic and wavefront array processing architectures, in Proc. 1985 IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, (Tampa, FL) pp. 296–299

    Google Scholar 

  40. CR. Ward, S.C. Hazon, D.R. Massey, A.J. Urquhart: Practical realizations of parallel adaptive beamforming systems, in Systolic Array Processing, Proc. 1989 Int. Conf. on Systolic Arrays (Prentice-Hall, Hemel Hempstead, UK 1989) pp. 3–12

    Google Scholar 

  41. R.J. Lackey, H.F. Baurle, J. Barile: Application-specific super computer, in Proc. SPIE, Vol. 977, Real Time Signal Processing XI (Int. Soc. for Optical Engineering, Bellingham, WA 1989) pp. 187–195

    Google Scholar 

  42. T.J. Shepherd, J.G. McWhirter, J.E. Hudson: Parallel weight extraction from a systoHc adaptive beamformer, Proc. Second IMA Conf. on Mathematics in Signal Processing, University of Warwick, December 1988 (Oxford University Press, Oxford 1990)

    Google Scholar 

  43. J.E. Hudson, TJ. Shepherd: Parallel weight extraction by a systolic least-squares algoritm, in Proc. SPIE, Vol. 1152, Advanced Algorithms and Architectures for Signal Processing IV, (Int Soc. for Optical Engineering, Bellingham, WA 1989) pp. 68–77

    Google Scholar 

  44. C.R. Ward, P.J. Hargrave, J.G. McWhirter, T.J. Shepherd: A novel accelerated convergence technique for adaptive antenna appUcations, Proc. 6th IEE Int. Conf. on Antennas and Propagation, University of Warwick, 1989, IEE (London) Conf. Publication No. 301 (1989) pp. 331–335

    Google Scholar 

  45. S.C. Pohlig: Hybrid adaptive feedback nulling in the presence of channel mismatch, in Proc. 1988 IEEE Conf. on Acoustics, Speech, and Signal Processing, New York (1988) pp. 1588–1591

    Google Scholar 

  46. L.J. Griffiths: A simple adaptive algorithm for real-time processing in antenna arrays. Proc. IEEE 57, 1696–1704 (1969)

    Article  Google Scholar 

  47. F. Ling, J.G. Proakis: A generalized multichannel least-squares lattice algorithm based on sequential processing stages. IEEE Trans. ASSP-32, 381–389 (1984)

    Google Scholar 

  48. P.S. Lewis: QR algorithm and array architecture for multichannel adaptive least-squares lattice filters, in Proc. 1988 IEEE Conf. on. Acoustics, Speech, and Signal Processing, New York (1988) pp. 2041–2044

    Google Scholar 

  49. F. Ling: Systolic arrays for implementation of order-recursive least-squares adaptive filtering algorithms, in Proc. Int. Conf. on Systolic Arrays, ed. by K. Bromley, S.Y. Kung, E. Swartzlander (Computer Society Press, Washington, DC 1988) pp. 135–144

    Chapter  Google Scholar 

  50. H. Lev-Ari: Modular architectures for adaptive multichannel lattice algorithms. IEEE Trans. ASSP-35, 543–552 (1987)

    Google Scholar 

  51. D. Mansour: A highly parallel architecture for adaptive multichannel algorithms, in Proc. 1986 IEEE Int. Conf on Acoustics, Speech, and Signal Processing, Tokyo (1986) pp. 2931–2934

    Google Scholar 

  52. K.C Sharman, T.S. Durrani: A triangular adaptive lattice filter for spatial signal processing, in Proc. 1983 IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, Boston, MA (1983) pp. 348–351

    Google Scholar 

  53. D.E. Rumelhart, G.E. Hinton, R.J. Williams: Learning internal representations by error propagation, in Parallel Distributed Processing: Vol. 1, ed. by D.E. Rumelhart, J.L. McClelland, (MIT Press, Cambridge, MA 1987) pp. 318–362

    Google Scholar 

  54. D.S. Broomhead, D. Lowe: Multi-variable functional interpolation and adaptive networks. Complex Syst. 2, 321–355 (1988)

    MATH  MathSciNet  Google Scholar 

  55. M.J.D. Powell: Radial basis functions for multi-variable interpolation: A review, in Proc. IMA Conf. on Algorithms for the Approximation of Functions and Data (Oxford University Press, Oxford 1987) pp. 143#x2013;167

    Google Scholar 

  56. C.A. Michelli: Interpolation of scattered data: Distance matrices and conditionally positive definite functions. Constructive Approx. 2, 11–22 (1986)

    Article  Google Scholar 

  57. S. Renais: Radial basis function network for speech pattern classification. Electron. Lett. 25, 437–439 (1989)

    Article  Google Scholar 

  58. T.V. Ho, J. Litva: Systolic array for 2-D adaptive beamforming, in Proc. Int. Conf. on Systolic Arrays, ed. by K. Bromley, S.Y. Kung, E. Swartzlander (Computer Society Press, Washington, DC 1988) pp. 1–10

    Google Scholar 

  59. B. Yang, J.G. Böhme: Linear systolic arrays for constrained least-squares problems, in Second IMA Conf. on Mathematics in Signal Processing, University of Warwick, December 1988 (Oxford University Press, Oxford 1990)

    Google Scholar 

  60. J.E. Voider: The CORDIC trigonometric computing technique. IRE Trans. Electron. Comput. EC-8, 330–334 (1959)

    Article  Google Scholar 

  61. M-J. Chen, K. Yao: Linear systolic array for least-squares estimation, in Proc. Int. Conf. on Systolic Arrays, ed. by K. Bromley, S.Y. Kung, E. Swartzlander (Computer Society Press, Washington, DC 1988) pp. 83–92

    Chapter  Google Scholar 

  62. C.M. Rader: Wafer-scale systolic array for adaptive antenna processing, in Proc. 1988 IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, New York (1988) pp. 2069–2071

    Google Scholar 

  63. D. Heller: Partitioning big matrices for small systolic arrays, in VLSI and Modern Signal Processing, ed. by S.Y. Kung, H J. Whitehouse, T. Kailath, (Prentice-Hall, Englewood Cliffs NJ 1985) pp. 185–199

    Google Scholar 

  64. N. Torralba, J.J. Navarro: A one-dimensional systolic array for solving arbitrarily large least mean square problems, in Proc. Int. Conf. on Systolic Arrays, ed. by K. Bromley, S.Y. Kung, E. Swartzlander (Computer Society Press, Washington, DC 1988) pp. 103–112

    Chapter  Google Scholar 

  65. S.Y. Kung, R.J. Gal-Ezer: Eigenvalue, singular value and least squares solvers via the wavefront array processor, in Proc. Purdue Workshop on Algorithmically Specialized Parallel Computers, ed. by L. Snyder et al. (Academic, New York 1985) pp. 201–212

    Google Scholar 

  66. D.E. Heller, I.C.F. Ipsen: Systolic networks for orthogonal decompositions. SIAM J. Sci. Stat. Comput. 4, 261–269 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  67. S.M. Yuen, K. Abend, R.S. Berkowitz: A recursive least-squares algorithm with multiple inputs and outputs, and a cylindrical systolic implementation, IEEE Trans. ASSP-36, 1917–1923 (1988)

    Google Scholar 

  68. S. Hammarling: A note on modifications to the Givens plane rotation, J. Inst. Math. Its Appl. 13, 215–218 (1974)

    MATH  MathSciNet  Google Scholar 

  69. W.M. Gentleman: Error analysis of QR decompositions by Givens transformations. Linear Algebra Its Appl. 10, 189–197 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  70. F.T. Luk, S. Qiao: Analysis of a recursive least-squares signal processing algorithm. SIAM J. Sci. Stat. Comput. 10, 407–418 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  71. F.T. Luk, E.K. Torng, C.J. Anfinson: A novel fault-tolerant technique for least-squares minimization, VLSI Signal Proc. 1, 181–188 (1989)

    Article  MATH  Google Scholar 

  72. K.J.R. Liu, K. Yao: Gracefully degradable real-time algorithm-based fault-tolerant method for QR recursive least-squares systolic array, in Systolic Array Processors, Proc. 1989 Int. Conf. on Systolic Arrays, ed. by J.V. McCanny, J.G. McWhirter, E. Swartzlander (Prentice-Hall, Hemel Hempstead, UK 1989) pp. 401–410

    Google Scholar 

  73. CR. Ward, P.J. Hargrave, J.G. McWhirter: Adaptive beamforming using real arithmetic, in Proc. SPIE, Vol. 826, Advanced Algorithms and Architectures for Signal Processing II, ed. by F. Luk (Int. Soc. for Optical Engineering, Bellingham, WA 1987) pp. 17–24

    Google Scholar 

  74. R. Kumaresan, D.W. Tufts: Estimating the angles of arrival of multiple plane waves. IEEE Trans. AES-19, 134–139 (1983)

    Google Scholar 

  75. R.O. Schmidt: A signal subspace approach to multiple emitter location and spectral estimation. Ph.D. Thesis, Stanford University (1981)

    Google Scholar 

  76. F.T. Luk: A triangular processor array for computing singular values. Linear Algebra Its Appl. 77, 259–273 (1986)

    Article  MATH  Google Scholar 

  77. GD. de Villiers: A Gentleman-Kung architecture for finding the singular values of a matrix, in Systolic Array Processors, Proc. 1989 Int. Conf. on Systolic Arrays. ed. by J.V. McCanny, J.G. McWhirter, E. Swartzlander (Prentice-Hall, Hemel Hempstead, UK 1989) pp. 545–554

    Google Scholar 

  78. M. Moonen, P. Van Dooren, J. Vandewalle: Updating singular value decompositions. A parallel implementation, in Proc. SPIE, Vol. 1152, Advanced Algorithms for Signal Processing IV, ed. by F.T. Luk (Int. Soc. for Optical Engineering, Bellingham, WA 1989) pp. 80–91

    Google Scholar 

  79. R.E. Kalman: A new approach to linear filtering and prediction problems. Trans. ASME (J.Basic Eng.) 82D, 34–45 (1960)

    Google Scholar 

  80. P.G. Kaminski, A.E. Bryson, S.F. Schmidt: Discrete square root filtering: a survey of current techniques. IEEE Trans. AC-16, 727–736 (1971)

    Google Scholar 

  81. G.J. Bierman: Factorization Methods for Discrete Sequential Estimation (Academic, New York 1977)

    MATH  Google Scholar 

  82. C.C. Paige, M.A. Saunders: Least-squares estimation of discrete linear dynamic systems using orthogonal transformations. SIAM J. Numer. Anal. 14, 181–193 (1977)

    Article  MathSciNet  Google Scholar 

  83. D.B. Duncan, S.D. Horn: Linear dynamicrecursive estimation from the viewpoint of regression analysis. J. Am. Statist. Assoc. 67, 815–821 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  84. A. Andrew: Parallel processing of the Kalman filter, in Proc. 1981 Int. Conf. on Parallel Processing, Columbus, OH (1981) pp. 216–220

    Google Scholar 

  85. J.M. Jover, T. Kailath: A parallel architecture for the Kalman filter measurement update, in Proc. IFAC 9th Triennial World Congress on Adaptive Control, Budapest, Hungary (1984) pp. 1005–1010

    Google Scholar 

  86. M.J. Chen, K. Yao: On realizations of least-squares estimation and Kalman filtering by systolic arrays, in Systolic Arrays, Proc. 1986 Int. Wokshop on Systolic Arrays, ed. by W. Moore, A.M. McCabe, R. Urquhart (Adam Hilger, Bristol 1987) pp. 161–170

    Google Scholar 

  87. M.J. Chen, K. Yao: Systolic Kalman filtering based on QR decomposition, in Proc. SPIE, Vol. 826, Advanced Algorithms and Architectures for Signal Processing II, ed. by F.T. Luk (Int. Soc. for Optical Engineering, Bellingham, WA 1987) pp. 25–32

    Google Scholar 

  88. T.Y. Sung, Y.H. Hu: VLSI implementation of real-time Kalman filter in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, Tokyo (1986) pp. 2223–2226

    Google Scholar 

  89. S.Y. Kung, J.N. Hwang: An efficient tri-array systolic design for real-time Kalman filtering, in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, New York (1988) pp. 2045–2048

    Google Scholar 

  90. F.M.F. Gaston, G.W. Irwin: A systolic square-root information Kalman filter, in Proc. Int. Conf on Systolic Arrays, ed. by K. Bromley, S.Y. Kung, E. Swartzlander (Computer Society Press, Washington, DC 1988) pp. 643–652

    Chapter  Google Scholar 

  91. P. Gosling, J.E. Hudson, J.G. McWhirter, T.J. Shepherd: Direct extraction of the state vector from systolic implementations of the square-root Kalman filter, in Systolic Array Processors, Proc. 1989 Int. Conf. on Systolic Arrays, ed. by J.V. McCanny, J.G. McWhirter, E. Swartzlander (Prentice-Hall, Hemel Hempstead, UK 1989) pp. 42–51

    Google Scholar 

  92. G.W. Irwin, F.M.F. Gaston: A systolic architecture for square-root covariance Kalman filtering, in Systolic Array Processors, Proc. 1989 Int. Conf. on Systolic Arrays, ed. by J.V. McCanny, J.G. McWhirter, E. Swartzlander (Prentice-Hall, Hemel Hempstead UK 1989) pp. 255–263

    Google Scholar 

  93. F.M.F. Gaston, G.W. Irwin: A systoHc square-root covariance Kalman filter, to be published in Proc. Second IMA Conf. on Mathematics in Signal Processing, University of Warwick, December 1988 (Oxford University Press, Oxford 1990)

    Google Scholar 

  94. A.O. Steinhardt: Householder transforms in signal processing. IEEE ASSP Mag. (July 1988)

    Google Scholar 

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Shepherd, T.J., McWhirter, J.G. (1993). Systolic Adaptive Beamforming. In: Haykin, S., Litva, J., Shepherd, T.J. (eds) Radar Array Processing. Springer Series in Information Sciences, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-77347-1_5

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