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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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
F. Ling, D. Manolakis, J.G. Proakis: A recursive modified Gram-Schmidt algorithm for least-squares estimation. IEEE Trans. ASSP-34, 829–836 (1986)
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
R. Schreiber, W-P. Tang: On systolic arrays for updating the Cholesky factorization. BIT 26, 451–466 (1986)
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)
S. Haykin: Adaptive Filter Theory (Prentice-Hall, Englewood Cliffs, NJ 1986)
S.P. Applebaum, DJ. Chapman: Adaptive arrays with main beam constraints. IEEE Trans. AP-24, 650–662 (1976)
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)
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
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)
J.E. Hudson: Adaptive Array Principles (Peregrinus, UK 1981)
O.L. Frost: An algorithm for linearly constrained adaptive array processing. Proc. IEEE 60, 661–675 (1971)
C.L. Lawson, R.J. Hanson: Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ 1974)
LS. Reed, J.D. Mallett, L.E. Brennan: Rapid convergence rate in adaptive arrays. IEEE Trans. AES-10, 853–863 (1974)
A.S. Householder: Unitary triangularization of a nonsymmetric matrix. J. ACM 5, 339–342 (1958)
W. Givens: Computation of plane unitary rotations transforming a general matrix to triangular form. J. Soc. Ind. Appl. Math. 6, 26–50 (1958)
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
W.M. Gentleman: Least-squares computations by Givens transformations without square roots. J. Inst. Math. Its Appl. 12, 329–336 (1973)
CR. Ward, P.J. Hargrave, J.G. McWhirter: A novel algorithm and architecture for adaptive digital beamforming. IEEE Trans. AP-34, 338–346 (1986)
D.T.L. Lee, B. Friedlander, M. Morf: Recursive ladder estimation algorithms. IEEE Trans. ASSP-29, 627–641 (1981)
D.D. Falconer, L. Ljung: Application of fast Kalman estimation to adaptive equalization. IEEE Trans. COM-26, 1439–1446 (1978)
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
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)
S.Y. Kung: VLSI Array Processors (Prentice-Hall, Englewood Cliffs, NJ 1988)
Å. Björk: Solving linear least-squares problems by Gram-Schmidt orthogonalization. BIT 7, 1–21 (1967)
R.A. Monzingo, T.W. Miller: Introduction to Adaptive Arrays (Wiley, New York 1980)
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
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
C.W. Jim: A comparison of two LMS constrained optimal array structures. Proc. IEEE 65, 1730–1731 (1977)
L.J. Griffiths, C.W. Jim: An alternative approach to linearly constrained adaptive beam-forming. IEEE Trans. AP-30, 27–34 (1982)
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
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
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
R. Schreiber: Implementation of adaptive array algorithms. IEEE Trans. ASSP-34,1038–1045 (1986)
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
J.G. McWhirter, T.J. Shepherd: Systolic array for MVDR beamforming. IEE (London) Proc. Part F 136, 75–80 (1989)
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
J.V. McCanny, J.G. McWhirter: Some systolic array developments in the United Kingdom. IEEE Trans. C-20, 51–63 (1987)
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
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
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
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)
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
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
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
L.J. Griffiths: A simple adaptive algorithm for real-time processing in antenna arrays. Proc. IEEE 57, 1696–1704 (1969)
F. Ling, J.G. Proakis: A generalized multichannel least-squares lattice algorithm based on sequential processing stages. IEEE Trans. ASSP-32, 381–389 (1984)
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
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
H. Lev-Ari: Modular architectures for adaptive multichannel lattice algorithms. IEEE Trans. ASSP-35, 543–552 (1987)
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
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
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
D.S. Broomhead, D. Lowe: Multi-variable functional interpolation and adaptive networks. Complex Syst. 2, 321–355 (1988)
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
C.A. Michelli: Interpolation of scattered data: Distance matrices and conditionally positive definite functions. Constructive Approx. 2, 11–22 (1986)
S. Renais: Radial basis function network for speech pattern classification. Electron. Lett. 25, 437–439 (1989)
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
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)
J.E. Voider: The CORDIC trigonometric computing technique. IRE Trans. Electron. Comput. EC-8, 330–334 (1959)
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
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
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
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
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
D.E. Heller, I.C.F. Ipsen: Systolic networks for orthogonal decompositions. SIAM J. Sci. Stat. Comput. 4, 261–269 (1983)
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)
S. Hammarling: A note on modifications to the Givens plane rotation, J. Inst. Math. Its Appl. 13, 215–218 (1974)
W.M. Gentleman: Error analysis of QR decompositions by Givens transformations. Linear Algebra Its Appl. 10, 189–197 (1975)
F.T. Luk, S. Qiao: Analysis of a recursive least-squares signal processing algorithm. SIAM J. Sci. Stat. Comput. 10, 407–418 (1989)
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)
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
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
R. Kumaresan, D.W. Tufts: Estimating the angles of arrival of multiple plane waves. IEEE Trans. AES-19, 134–139 (1983)
R.O. Schmidt: A signal subspace approach to multiple emitter location and spectral estimation. Ph.D. Thesis, Stanford University (1981)
F.T. Luk: A triangular processor array for computing singular values. Linear Algebra Its Appl. 77, 259–273 (1986)
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
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
R.E. Kalman: A new approach to linear filtering and prediction problems. Trans. ASME (J.Basic Eng.) 82D, 34–45 (1960)
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)
G.J. Bierman: Factorization Methods for Discrete Sequential Estimation (Academic, New York 1977)
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)
D.B. Duncan, S.D. Horn: Linear dynamicrecursive estimation from the viewpoint of regression analysis. J. Am. Statist. Assoc. 67, 815–821 (1972)
A. Andrew: Parallel processing of the Kalman filter, in Proc. 1981 Int. Conf. on Parallel Processing, Columbus, OH (1981) pp. 216–220
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
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
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
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
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
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
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
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
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)
A.O. Steinhardt: Householder transforms in signal processing. IEEE ASSP Mag. (July 1988)
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-642-77347-1_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-77349-5
Online ISBN: 978-3-642-77347-1
eBook Packages: Springer Book Archive