Abstract

We present an optoelectronic implementation of an adaptive-array processor that is capable of performing beam forming and jammer nulling in signals of wide fractional bandwidth that are detected by an array of arbitrary topology. The optical system makes use of a two-dimensional scrolling spatial light modulator to represent an array of input signals in 256 tapped delay lines, two acousto-optic modulators for modulating the feedback error signal, and a photorefractive crystal for representing the adaptive weights as holographic gratings. Gradient-descent learning is used to dynamically adapt the holographic weights to optimally form multiple beams and to null out multiple interference sources, either in the near field or in the far field. Space-integration followed by differential heterodyne detection is used for generating the system’s output. The processor is analyzed to show the effects of exponential weight decay on the optimum solution and on the convergence conditions. Several experimental results are presented that validate the system’s capacity for broadband beam forming and jammer nulling for linear and circular arrays.

© 2004 Optical Society of America

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  1. B. Widrow, S. D. Stearns, Adaptive Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1985).
  2. D. A. B. Miller, “Quantum-well self-electro-optic effect devices,” Opt. Quantum Electron. 22, S61–S98 (1990).
  3. T. L. Worchesky, K. J. Ritter, R. Martin, B. Lane, “Large arrays of spatial light modulators hybridized to silicon integrated circuits,” Appl. Opt. 35, 1180–1186 (1996).
  4. E. A. Wan, “Temporal backpropagation for FIR neural networks,” in Proceedings of the International Joint Conference on Neural Networks (Omnipress, San Diego, Calif., 1990), Vol. 1, pp. 575–580.
  5. P. E. X. Silveira, G. S. Pati, K. H. Wagner, “Optical finite impulse response neural networks,” Appl. Opt. 41, 4162–4180 (2002).
  6. P. E. X. Silveira, K. H. Wagner, “Optical finite impulse response neural networks,” in Algorithms, Devices, and Systems for Optical Information Processing III, B. Javidi, D. Psaltis, eds., Proc. SPIE3804, 72–81 (1999).
  7. P. E. X. Silveira, G. S. Pati, K. H. Wagner, “Optical implementation of a single-layer finite impulse response neural network,” in Proceedings of the International Conference on Optics in Computing 2000, R. A. Lessard, T. V. Galstian, eds., Proc. SPIE4089, 656–667 (2000).
  8. L. J. G. B. Widrow, P. E. Mantey, B. B. Goode, “Adaptive antenna systems,” Proc. IEEE 55, 2143–2161 (1967).
  9. D. Dolfi, F. Michelgabriel, S. Bann, J. P. Huignard, “Two-dimensional optical architecture for time-delay beam forming in a phased-array antenna,” Opt. Lett. 16, 255–257 (1991).
  10. N. A. Riza, “Transmit/receive time-delay beam-forming optical architecture for phased-array antennas,” Appl. Opt. 30, 4594–4595 (1991).
  11. X. S. Yao, L. Maleki, “A novel 2-D programmable photonic time-delay device for millimeter-wave signal-processing applications,” IEEE Photon. Technol. Lett. 6, 1463–1465 (1994).
  12. A. P. Goutzoulis, D. K. Davies, “Hardware-compressive 2-D fiber optic delay line architecture for time steering of phased-array antennas,” Appl. Opt. 29, 5353–5359 (1990).
  13. E. Toughlian, H. Zmuda, “Variable time-delay system for broadband phased array and other transversal filtering applications,” Opt. Eng. 32, 613–617 (1993).
  14. M. Y. Frankel, R. D. Esman, “Dynamic null steering in an ultrawideband time-steered array antenna,” Appl. Opt. 37, 5488–5494 (1998).
  15. J. H. Hong, T. Y. Chang, “Adaptive RF notch filtering using photorefractive two-beam coupling,” IEEE J. Quantum Electron. 30, 313–317 (1994).
  16. J. Rhodes, “Adaptive filter with a time-domain implementation using correlation cancellation loops,” Appl. Opt. 22, 282–287 (1983).
  17. J. H. Hong, “Broadband phased array beamforming,” in Optical Technology for Microwave Applications IV, S.-K. Yao, ed., Proc. SPIE1102, 134–141 (1989).
  18. K. H. Wagner, S. Kraut, L. Griffiths, S. Weaver, R. Weverka, A. Sarto, “Efficient true-time-delay adaptive array processing,” in Radar Processing, Technology, and Applications, W. J. Miceli, ed., Proc. SPIE2845, 287–300 (1996).
  19. G. Kriehn, A. Kiruluta, P. E. X. Silveira, S. Weaver, S. Kraut, K. Wagner, R. T. Weverka, L. Griffiths, “Optical BEAMTAP beam-forming and jammer-nulling system for broadband phased-array antennas,” Appl. Opt. 39, 212–230 (2000).
  20. E. A. Wan, “Time series prediction by using a connectionist network with internal delay lines,” in Time Series Prediction: Forecasting the Future and Understanding the Past, Vol. XVII of the Santa Fe Institute (SFI) Studies in the Science of Complexity, A. S. Weigend, N. A. Gershenfeld, eds. (Addison-Wesley, Reading, Mass., 1993), pp. 195–217.
  21. J. Feinberg, “Assymetric self-defocusing of an optical beam from the photorefractive effect,” J. Opt. Soc. Am. 72, 46–51 (1982).
  22. D. Armitage, “Liquid-crystal display device fundamentals,” in Electro-optical Displays, M. A. Karim, ed. (Marcel Dekker, New York, 1992), Chap. 2, pp. 19–67.
  23. A. Brignon, K. H. Wagner, “Polarization state evolution and eigenmode switching in photorefractive BSO,” Opt. Commun. 101, 239–246 (1993).
  24. R. T. Weverka, K. Wagner, A. Sarto, “Photorefractive processing for large adaptive phased arrays,” Appl. Opt. 35, 1344–1366 (1996).
  25. A. W. Sarto, K. H. Wagner, R. T. Weverka, S. Weaver, E. K. Walge, “Wide angular aperture holograms in photorefractive crystals by the use of orthogonally polarized write and read beams,” Appl. Opt. 35, 5765–5775 (1996).
  26. G. L. Abbas, V. W. S. Chan, T. K. Yee, “Local-oscillator excess noise suppression for homodyne and heterodyne detection,” Opt. Lett. 8, 412–422 (1983).
  27. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  28. P. S. R. Diniz, “LMS-based algorithms,” in Adaptive Filtering: Algorithms and Practical Implementation (Kluwer Academic, Dordrecht, The Netherlands, 1997), Chap. 4, pp. 150–153.
  29. P. E. X. Silveira, “Optoelectronic signal processing using finite impulse response neural networks,” Ph.D., dissertation (University of Colorado at Boulder, Boulder, Colo., 2001).
  30. G. C. Petrisor, A. A. Goldstein, B. K. Jenkins, E. J. Herbulock, A. R. Tanguay, “Convergence of backward-error-propagation learning in photorefractive crystals,” Appl. Opt. 35, 1328–1343 (1996).
  31. K. Y. Hsu, S. H. Lin, P. Yeh, “Conditional convergence of photorefractive perceptron learning,” Opt. Lett. 18, 2135–2137 (1993).
  32. C. Bishop, “Learning and generalization,” in Neural Networks for Pattern Recognition (Clarendon Press, Oxford, UK, 1995), Chap. 9, pp. 338–340.
  33. J. R. T. Compton, Adaptive Antennas (Prentice-Hall, Englewood Cliffs, N.J., 1988).
  34. P. S. R. Diniz, “The least-mean-square (LMS) algorithm,” in Adaptive Filtering: Algorithms and Practical Implementation (Kluwer Academic, Dordrecht, The Netherlands, 1997), Chap. 3, pp. 75–78.

2002

2000

1998

1996

1994

X. S. Yao, L. Maleki, “A novel 2-D programmable photonic time-delay device for millimeter-wave signal-processing applications,” IEEE Photon. Technol. Lett. 6, 1463–1465 (1994).

J. H. Hong, T. Y. Chang, “Adaptive RF notch filtering using photorefractive two-beam coupling,” IEEE J. Quantum Electron. 30, 313–317 (1994).

1993

K. Y. Hsu, S. H. Lin, P. Yeh, “Conditional convergence of photorefractive perceptron learning,” Opt. Lett. 18, 2135–2137 (1993).

E. Toughlian, H. Zmuda, “Variable time-delay system for broadband phased array and other transversal filtering applications,” Opt. Eng. 32, 613–617 (1993).

A. Brignon, K. H. Wagner, “Polarization state evolution and eigenmode switching in photorefractive BSO,” Opt. Commun. 101, 239–246 (1993).

1991

1990

A. P. Goutzoulis, D. K. Davies, “Hardware-compressive 2-D fiber optic delay line architecture for time steering of phased-array antennas,” Appl. Opt. 29, 5353–5359 (1990).

D. A. B. Miller, “Quantum-well self-electro-optic effect devices,” Opt. Quantum Electron. 22, S61–S98 (1990).

1983

1982

1969

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1967

L. J. G. B. Widrow, P. E. Mantey, B. B. Goode, “Adaptive antenna systems,” Proc. IEEE 55, 2143–2161 (1967).

Abbas, G. L.

Armitage, D.

D. Armitage, “Liquid-crystal display device fundamentals,” in Electro-optical Displays, M. A. Karim, ed. (Marcel Dekker, New York, 1992), Chap. 2, pp. 19–67.

Bann, S.

Bishop, C.

C. Bishop, “Learning and generalization,” in Neural Networks for Pattern Recognition (Clarendon Press, Oxford, UK, 1995), Chap. 9, pp. 338–340.

Brignon, A.

A. Brignon, K. H. Wagner, “Polarization state evolution and eigenmode switching in photorefractive BSO,” Opt. Commun. 101, 239–246 (1993).

Chan, V. W. S.

Chang, T. Y.

J. H. Hong, T. Y. Chang, “Adaptive RF notch filtering using photorefractive two-beam coupling,” IEEE J. Quantum Electron. 30, 313–317 (1994).

Compton, J. R. T.

J. R. T. Compton, Adaptive Antennas (Prentice-Hall, Englewood Cliffs, N.J., 1988).

Davies, D. K.

Diniz, P. S. R.

P. S. R. Diniz, “LMS-based algorithms,” in Adaptive Filtering: Algorithms and Practical Implementation (Kluwer Academic, Dordrecht, The Netherlands, 1997), Chap. 4, pp. 150–153.

P. S. R. Diniz, “The least-mean-square (LMS) algorithm,” in Adaptive Filtering: Algorithms and Practical Implementation (Kluwer Academic, Dordrecht, The Netherlands, 1997), Chap. 3, pp. 75–78.

Dolfi, D.

Esman, R. D.

Feinberg, J.

Frankel, M. Y.

Goldstein, A. A.

Goode, B. B.

L. J. G. B. Widrow, P. E. Mantey, B. B. Goode, “Adaptive antenna systems,” Proc. IEEE 55, 2143–2161 (1967).

Goutzoulis, A. P.

Griffiths, L.

G. Kriehn, A. Kiruluta, P. E. X. Silveira, S. Weaver, S. Kraut, K. Wagner, R. T. Weverka, L. Griffiths, “Optical BEAMTAP beam-forming and jammer-nulling system for broadband phased-array antennas,” Appl. Opt. 39, 212–230 (2000).

K. H. Wagner, S. Kraut, L. Griffiths, S. Weaver, R. Weverka, A. Sarto, “Efficient true-time-delay adaptive array processing,” in Radar Processing, Technology, and Applications, W. J. Miceli, ed., Proc. SPIE2845, 287–300 (1996).

Herbulock, E. J.

Hong, J. H.

J. H. Hong, T. Y. Chang, “Adaptive RF notch filtering using photorefractive two-beam coupling,” IEEE J. Quantum Electron. 30, 313–317 (1994).

J. H. Hong, “Broadband phased array beamforming,” in Optical Technology for Microwave Applications IV, S.-K. Yao, ed., Proc. SPIE1102, 134–141 (1989).

Hsu, K. Y.

Huignard, J. P.

Jenkins, B. K.

Kiruluta, A.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kraut, S.

G. Kriehn, A. Kiruluta, P. E. X. Silveira, S. Weaver, S. Kraut, K. Wagner, R. T. Weverka, L. Griffiths, “Optical BEAMTAP beam-forming and jammer-nulling system for broadband phased-array antennas,” Appl. Opt. 39, 212–230 (2000).

K. H. Wagner, S. Kraut, L. Griffiths, S. Weaver, R. Weverka, A. Sarto, “Efficient true-time-delay adaptive array processing,” in Radar Processing, Technology, and Applications, W. J. Miceli, ed., Proc. SPIE2845, 287–300 (1996).

Kriehn, G.

Lane, B.

Lin, S. H.

Maleki, L.

X. S. Yao, L. Maleki, “A novel 2-D programmable photonic time-delay device for millimeter-wave signal-processing applications,” IEEE Photon. Technol. Lett. 6, 1463–1465 (1994).

Mantey, P. E.

L. J. G. B. Widrow, P. E. Mantey, B. B. Goode, “Adaptive antenna systems,” Proc. IEEE 55, 2143–2161 (1967).

Martin, R.

Michelgabriel, F.

Miller, D. A. B.

D. A. B. Miller, “Quantum-well self-electro-optic effect devices,” Opt. Quantum Electron. 22, S61–S98 (1990).

Pati, G. S.

P. E. X. Silveira, G. S. Pati, K. H. Wagner, “Optical finite impulse response neural networks,” Appl. Opt. 41, 4162–4180 (2002).

P. E. X. Silveira, G. S. Pati, K. H. Wagner, “Optical implementation of a single-layer finite impulse response neural network,” in Proceedings of the International Conference on Optics in Computing 2000, R. A. Lessard, T. V. Galstian, eds., Proc. SPIE4089, 656–667 (2000).

Petrisor, G. C.

Rhodes, J.

Ritter, K. J.

Riza, N. A.

Sarto, A.

R. T. Weverka, K. Wagner, A. Sarto, “Photorefractive processing for large adaptive phased arrays,” Appl. Opt. 35, 1344–1366 (1996).

K. H. Wagner, S. Kraut, L. Griffiths, S. Weaver, R. Weverka, A. Sarto, “Efficient true-time-delay adaptive array processing,” in Radar Processing, Technology, and Applications, W. J. Miceli, ed., Proc. SPIE2845, 287–300 (1996).

Sarto, A. W.

Silveira, P. E. X.

P. E. X. Silveira, G. S. Pati, K. H. Wagner, “Optical finite impulse response neural networks,” Appl. Opt. 41, 4162–4180 (2002).

G. Kriehn, A. Kiruluta, P. E. X. Silveira, S. Weaver, S. Kraut, K. Wagner, R. T. Weverka, L. Griffiths, “Optical BEAMTAP beam-forming and jammer-nulling system for broadband phased-array antennas,” Appl. Opt. 39, 212–230 (2000).

P. E. X. Silveira, “Optoelectronic signal processing using finite impulse response neural networks,” Ph.D., dissertation (University of Colorado at Boulder, Boulder, Colo., 2001).

P. E. X. Silveira, G. S. Pati, K. H. Wagner, “Optical implementation of a single-layer finite impulse response neural network,” in Proceedings of the International Conference on Optics in Computing 2000, R. A. Lessard, T. V. Galstian, eds., Proc. SPIE4089, 656–667 (2000).

P. E. X. Silveira, K. H. Wagner, “Optical finite impulse response neural networks,” in Algorithms, Devices, and Systems for Optical Information Processing III, B. Javidi, D. Psaltis, eds., Proc. SPIE3804, 72–81 (1999).

Stearns, S. D.

B. Widrow, S. D. Stearns, Adaptive Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1985).

Tanguay, A. R.

Toughlian, E.

E. Toughlian, H. Zmuda, “Variable time-delay system for broadband phased array and other transversal filtering applications,” Opt. Eng. 32, 613–617 (1993).

Wagner, K.

Wagner, K. H.

P. E. X. Silveira, G. S. Pati, K. H. Wagner, “Optical finite impulse response neural networks,” Appl. Opt. 41, 4162–4180 (2002).

A. W. Sarto, K. H. Wagner, R. T. Weverka, S. Weaver, E. K. Walge, “Wide angular aperture holograms in photorefractive crystals by the use of orthogonally polarized write and read beams,” Appl. Opt. 35, 5765–5775 (1996).

A. Brignon, K. H. Wagner, “Polarization state evolution and eigenmode switching in photorefractive BSO,” Opt. Commun. 101, 239–246 (1993).

K. H. Wagner, S. Kraut, L. Griffiths, S. Weaver, R. Weverka, A. Sarto, “Efficient true-time-delay adaptive array processing,” in Radar Processing, Technology, and Applications, W. J. Miceli, ed., Proc. SPIE2845, 287–300 (1996).

P. E. X. Silveira, K. H. Wagner, “Optical finite impulse response neural networks,” in Algorithms, Devices, and Systems for Optical Information Processing III, B. Javidi, D. Psaltis, eds., Proc. SPIE3804, 72–81 (1999).

P. E. X. Silveira, G. S. Pati, K. H. Wagner, “Optical implementation of a single-layer finite impulse response neural network,” in Proceedings of the International Conference on Optics in Computing 2000, R. A. Lessard, T. V. Galstian, eds., Proc. SPIE4089, 656–667 (2000).

Walge, E. K.

Wan, E. A.

E. A. Wan, “Time series prediction by using a connectionist network with internal delay lines,” in Time Series Prediction: Forecasting the Future and Understanding the Past, Vol. XVII of the Santa Fe Institute (SFI) Studies in the Science of Complexity, A. S. Weigend, N. A. Gershenfeld, eds. (Addison-Wesley, Reading, Mass., 1993), pp. 195–217.

E. A. Wan, “Temporal backpropagation for FIR neural networks,” in Proceedings of the International Joint Conference on Neural Networks (Omnipress, San Diego, Calif., 1990), Vol. 1, pp. 575–580.

Weaver, S.

Weverka, R.

K. H. Wagner, S. Kraut, L. Griffiths, S. Weaver, R. Weverka, A. Sarto, “Efficient true-time-delay adaptive array processing,” in Radar Processing, Technology, and Applications, W. J. Miceli, ed., Proc. SPIE2845, 287–300 (1996).

Weverka, R. T.

Widrow, B.

B. Widrow, S. D. Stearns, Adaptive Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1985).

Widrow, L. J. G. B.

L. J. G. B. Widrow, P. E. Mantey, B. B. Goode, “Adaptive antenna systems,” Proc. IEEE 55, 2143–2161 (1967).

Worchesky, T. L.

Yao, X. S.

X. S. Yao, L. Maleki, “A novel 2-D programmable photonic time-delay device for millimeter-wave signal-processing applications,” IEEE Photon. Technol. Lett. 6, 1463–1465 (1994).

Yee, T. K.

Yeh, P.

Zmuda, H.

E. Toughlian, H. Zmuda, “Variable time-delay system for broadband phased array and other transversal filtering applications,” Opt. Eng. 32, 613–617 (1993).

Appl. Opt.

T. L. Worchesky, K. J. Ritter, R. Martin, B. Lane, “Large arrays of spatial light modulators hybridized to silicon integrated circuits,” Appl. Opt. 35, 1180–1186 (1996).

P. E. X. Silveira, G. S. Pati, K. H. Wagner, “Optical finite impulse response neural networks,” Appl. Opt. 41, 4162–4180 (2002).

N. A. Riza, “Transmit/receive time-delay beam-forming optical architecture for phased-array antennas,” Appl. Opt. 30, 4594–4595 (1991).

A. P. Goutzoulis, D. K. Davies, “Hardware-compressive 2-D fiber optic delay line architecture for time steering of phased-array antennas,” Appl. Opt. 29, 5353–5359 (1990).

M. Y. Frankel, R. D. Esman, “Dynamic null steering in an ultrawideband time-steered array antenna,” Appl. Opt. 37, 5488–5494 (1998).

J. Rhodes, “Adaptive filter with a time-domain implementation using correlation cancellation loops,” Appl. Opt. 22, 282–287 (1983).

G. Kriehn, A. Kiruluta, P. E. X. Silveira, S. Weaver, S. Kraut, K. Wagner, R. T. Weverka, L. Griffiths, “Optical BEAMTAP beam-forming and jammer-nulling system for broadband phased-array antennas,” Appl. Opt. 39, 212–230 (2000).

R. T. Weverka, K. Wagner, A. Sarto, “Photorefractive processing for large adaptive phased arrays,” Appl. Opt. 35, 1344–1366 (1996).

A. W. Sarto, K. H. Wagner, R. T. Weverka, S. Weaver, E. K. Walge, “Wide angular aperture holograms in photorefractive crystals by the use of orthogonally polarized write and read beams,” Appl. Opt. 35, 5765–5775 (1996).

G. C. Petrisor, A. A. Goldstein, B. K. Jenkins, E. J. Herbulock, A. R. Tanguay, “Convergence of backward-error-propagation learning in photorefractive crystals,” Appl. Opt. 35, 1328–1343 (1996).

Bell Syst. Tech. J.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

IEEE J. Quantum Electron.

J. H. Hong, T. Y. Chang, “Adaptive RF notch filtering using photorefractive two-beam coupling,” IEEE J. Quantum Electron. 30, 313–317 (1994).

IEEE Photon. Technol. Lett.

X. S. Yao, L. Maleki, “A novel 2-D programmable photonic time-delay device for millimeter-wave signal-processing applications,” IEEE Photon. Technol. Lett. 6, 1463–1465 (1994).

J. Opt. Soc. Am.

Opt. Commun.

A. Brignon, K. H. Wagner, “Polarization state evolution and eigenmode switching in photorefractive BSO,” Opt. Commun. 101, 239–246 (1993).

Opt. Eng.

E. Toughlian, H. Zmuda, “Variable time-delay system for broadband phased array and other transversal filtering applications,” Opt. Eng. 32, 613–617 (1993).

Opt. Lett.

Opt. Quantum Electron.

D. A. B. Miller, “Quantum-well self-electro-optic effect devices,” Opt. Quantum Electron. 22, S61–S98 (1990).

Proc. IEEE

L. J. G. B. Widrow, P. E. Mantey, B. B. Goode, “Adaptive antenna systems,” Proc. IEEE 55, 2143–2161 (1967).

Other

P. E. X. Silveira, K. H. Wagner, “Optical finite impulse response neural networks,” in Algorithms, Devices, and Systems for Optical Information Processing III, B. Javidi, D. Psaltis, eds., Proc. SPIE3804, 72–81 (1999).

P. E. X. Silveira, G. S. Pati, K. H. Wagner, “Optical implementation of a single-layer finite impulse response neural network,” in Proceedings of the International Conference on Optics in Computing 2000, R. A. Lessard, T. V. Galstian, eds., Proc. SPIE4089, 656–667 (2000).

B. Widrow, S. D. Stearns, Adaptive Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1985).

E. A. Wan, “Temporal backpropagation for FIR neural networks,” in Proceedings of the International Joint Conference on Neural Networks (Omnipress, San Diego, Calif., 1990), Vol. 1, pp. 575–580.

J. H. Hong, “Broadband phased array beamforming,” in Optical Technology for Microwave Applications IV, S.-K. Yao, ed., Proc. SPIE1102, 134–141 (1989).

K. H. Wagner, S. Kraut, L. Griffiths, S. Weaver, R. Weverka, A. Sarto, “Efficient true-time-delay adaptive array processing,” in Radar Processing, Technology, and Applications, W. J. Miceli, ed., Proc. SPIE2845, 287–300 (1996).

P. S. R. Diniz, “LMS-based algorithms,” in Adaptive Filtering: Algorithms and Practical Implementation (Kluwer Academic, Dordrecht, The Netherlands, 1997), Chap. 4, pp. 150–153.

P. E. X. Silveira, “Optoelectronic signal processing using finite impulse response neural networks,” Ph.D., dissertation (University of Colorado at Boulder, Boulder, Colo., 2001).

D. Armitage, “Liquid-crystal display device fundamentals,” in Electro-optical Displays, M. A. Karim, ed. (Marcel Dekker, New York, 1992), Chap. 2, pp. 19–67.

E. A. Wan, “Time series prediction by using a connectionist network with internal delay lines,” in Time Series Prediction: Forecasting the Future and Understanding the Past, Vol. XVII of the Santa Fe Institute (SFI) Studies in the Science of Complexity, A. S. Weigend, N. A. Gershenfeld, eds. (Addison-Wesley, Reading, Mass., 1993), pp. 195–217.

C. Bishop, “Learning and generalization,” in Neural Networks for Pattern Recognition (Clarendon Press, Oxford, UK, 1995), Chap. 9, pp. 338–340.

J. R. T. Compton, Adaptive Antennas (Prentice-Hall, Englewood Cliffs, N.J., 1988).

P. S. R. Diniz, “The least-mean-square (LMS) algorithm,” in Adaptive Filtering: Algorithms and Practical Implementation (Kluwer Academic, Dordrecht, The Netherlands, 1997), Chap. 3, pp. 75–78.

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Figures (24)

Fig. 1
Fig. 1

(a) Adaptive-array processor. Each input undergoes FIR filtering during which the summation of their outputs produces the final output. The weights are controlled by an adaptive algorithm. (b) Schematic representation of an adaptive-array processor implementing the LMS learning rule. The input signals s i (κ) are time delayed, multiplied by the adaptive weights w i, and summed, producing the output o(κ). The output signal is subtracted from the desired signal d(κ), providing a feedback error signal e(κ), which is multiplied by the delayed inputs and is time integrated at the corresponding adaptive weight.

Fig. 2
Fig. 2

Optoelectronic architecture for a sonar adaptive-array processor with a scrolling SLM for delayed input representation. Dynamic gratings recorded in a volume hologram are used for implementation of the adaptive interconnections, followed by spatial integration onto the output detectors. Differential heterodyne detection is used for bipolar signal representation. The output signal is electronically amplified and subtracted from a desired signal, producing a feedback error term that modulates a laser beam by use of an AOM. A second AOM compensates for the Doppler shift of signal frequencies in the first modulator and brings the signal back to baseband while simultaneously introducing a second tone, ω2, used for phase-locked-loop stabilization.

Fig. 3
Fig. 3

Detailed experimental diagram: M, mirror; BS, beam splitter; CBS, cubic beam splitter; PBS, polarizing beam splitter; SF, spatial filter; NDF, neutral-density filter; D, detector; PZ, piezoelectric; CCD, charge coupled device; SLM, spatial light modulator; FLC, ferroelectric liquid crystal; AOM, acousto-optic modulator; I, iris; ES, electronic shutter. Solid lines represent optic signals, and dashed lines represent electronic signals.

Fig. 4
Fig. 4

Two AOMs are used to modulate the error signal and to produce a SSB tone. An input beam is Bragg diffracted by the first AOM, producing an optic beam at its first order of diffraction that is modulated by the error signal e(κ) at rf carrier frequency ω1 and a rf tone at ω2. The second AOM diffracts the same beam at its -1 order, producing a final signal modulated by e(κ) at rf baseband and a tone at the difference frequency ω1 - ω2.

Fig. 5
Fig. 5

Differential heterodyne detection and long-term interferometric stabilization. The signals detected by detectors (Det.) 1 and 2 are subtracted by the differential amplifier, removing common-mode noise and increasing the dynamic range of the detected signal. The differential output signal is fed to the lock-in amplifier, which determines the phase of the reference tone by comparing it with an electronic lock-in reference. Phase drifts are corrected by moving a mirror attached to a piezoelectric actuator in a phase-locked loop.

Fig. 6
Fig. 6

(a) One of the 128 images captured from the SLM, showing a broadband binary chirp in the far field when the SLM operates as an amplitude modulator. (b) Optical Fourier plane of the SLM, showing a thin, straight line that is characteristic of a broadband signal in the far field. The dc spot and multiple diffraction orders are also visible.

Fig. 7
Fig. 7

(a) Binary chirp used as the desired signal. (b) Fourier spectrum of the desired signal. The dashed lines depict the nominal bandwidth of the analog chirp from which the binary chirp was derived. Ampl., amplitude.

Fig. 8
Fig. 8

(a) Processor output before thresholding, showing that the broadband chirp is fully recovered. (b) Processor output after thresholding. (c) Output spectrum, showing the detection of a broadband signal in the frequency range of the desired chirp. The dashed lines depict the initial and final frequencies of the original desired chirp. The dashed-dotted line represents half the sampling frequency. Ampl., amplitude.

Fig. 9
Fig. 9

(a) SLM image diffracted from the PRC, showing the correlation pattern of the far-field desired chirp. The angle α is used in the estimation of the AOA of the far-field signal. (b) Power spectrum of the SLM image diffracted from the PRC, showing that the system responds to a broadband signal in the far field. The angle α1 is used in the estimation of the AOA of the signal.

Fig. 10
Fig. 10

(a) One of the 128 images captured from the SLM, showing a broadband chirp in the near field and a cw jammer in the far field. (b) Image of the SLM in the Fourier plane, showing a dark spot (the cw jammer) and a circular section, both characteristic of a broadband signal in the near field.

Fig. 11
Fig. 11

(a) Open-loop processor output before thresholding, showing that only the cw jammer is detected. (b) Open-loop processor output in the frequency domain, showing a strong peak at the jammer frequency (281 Hz). Ampl., amplitude.

Fig. 12
Fig. 12

(a) Closed-loop processor output before thresholding, showing that the desired chirp is fully recovered. (b) Closed-loop processor output in the frequency domain, showing that a broadband signal is detected while the jammer at 281 Hz has been reduced, although it is still visible as the dominant spectral peak. The dashed lines depict the initial and final frequencies of the original desired chirp. Ampl., amplitude.

Fig. 13
Fig. 13

(a) SLM image diffracted from the PRC, showing the correlation pattern of the near-field desired chirp. (b) Correlation pattern in the SLM Fourier plane, showing a spatiotemporal frequency response to a broadband signal in the near field, and cw jammer nulling of the sidelobe of the jammer within the spectrum of the near-field chirp.

Fig. 14
Fig. 14

Fourier plane of the input scrolling SLM (left-hand side) and the spectrum of the output of the processor after adaptation (right-hand side) for input SINRs of (a) 0 dB, (b) -6 dB, and (c) -12 dB.

Fig. 15
Fig. 15

(a) Fractional output power due to the jammer (solid curve) and the desired signal (dashed curve) as a function of the input SINR. The output SINR is given by the difference between the two curves. (b) Processing gain as a function of the input SINR, showing a nearly linear increase in jammer nulling as the jammer becomes increasingly stronger.

Fig. 16
Fig. 16

(a) One of the 128 images captured from the SLM, showing two broadband chirps as detected by a circular array. (b) Image of the SLM in the Fourier plane, showing a uniform signal distribution over all possible spatial frequencies, which is characteristic of a circular array.

Fig. 17
Fig. 17

(a) Processor output before thresholding, showing that the desired chirp is fully recovered. (b) Closed-loop processor output in the frequency domain, showing that a broadband signal is detected in the same frequency range as the desired signal. The dashed lines depict the initial and final frequencies of the original desired chirp. Ampl., amplitude.

Fig. 18
Fig. 18

(a) Image of the SLM diffracted from the PRC, showing the correlation pattern of the two chirps detected by a circular array. The overlapping scale (in radians) is used for source-position estimation. (b) Correlation pattern in the Fourier plane of the SLM, showing a response toward broadband signals over all transverse spatial frequencies.

Fig. 19
Fig. 19

(a) One of the 128 images captured from the SLM, containing a chirp in the near field and a cw jammer and a broadband jammer in the far field. (b) Image of the SLM in the Fourier plane, showing a circular section (near-field chirp), a dark spot (cw jammer), and a thin, long line (broadband jammer).

Fig. 20
Fig. 20

(a) Processor output before thresholding, showing that the broadband chirp is mostly recovered. (b) Output spectrum, showing a strong response in the frequency range of the desired chirp (dashed lines). Linear filtering of a coherent broadband jammer also contributes to the signal output. Ampl., amplitude.

Fig. 21
Fig. 21

(a) SLM image diffracted from the PRC, showing the correlation pattern of the near-field chirp. (b) Correlation pattern in the SLM Fourier plane, showing the spatiotemporal frequency response to the near-field chirp and the null gratings associated with the broadband and cw jammers.

Fig. 22
Fig. 22

(a) One of the 128 images captured from the SLM, showing a chirp, a cw jammer, and two broadband jammers in the far field (the first from 168.8 to 518.8 Hz and the second from 281.3 to 750 Hz). (b) Image of the SLM in the Fourier plane, showing three long, thin lines (chirp and broadband jammers) and a dark spot (cw jammer).

Fig. 23
Fig. 23

(a) Processor output before thresholding, showing that the broadband chirp is fully recovered. (b) Output spectrum, showing the detection of a broadband signal in the frequency range of the desired chirp (dashed lines), except for a deep null at 350 Hz (the cw jammer frequency). Ampl., amplitude.

Fig. 24
Fig. 24

(a) Diffracted image from the PRC in the SLM image plane, showing the correlation pattern and weights of the near-field chirp, the broadband, and the cw jammers. (b) Correlation pattern in the SLM Fourier plane, showing the system’s response to the near-field chirp, the broadband, and the cw jammers. Note the deep null in the desired chirp spatiotemporal spectrum that is due to the nulling of the overlap of the jammer sidelobe with the chirp spectrum.

Equations (45)

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oκ= i=1Nτ=0T wiτ siκ-τ,
s¯κ=s1κ s1κ-1  s1κ-T s2κ  s2κ-T  sNκ-TT.
w¯=w10 w11   w1T w20   w2T   wNTT.
ξκ=E e2κ=E d2κ-2d κoκ+o2κ.
ξκ=E d2κ-2w¯TE dκs¯κ+w¯TE s¯κs¯Tκw¯=d0-2w¯Tp¯+w¯TR¯¯w¯,
w¯ξ=ξw10ξw11  ξwNTT=-2p¯+2R¯¯w¯.
w¯o=R¯¯-1p¯.
w¯κ+1=w¯κ-ηˆw¯ξ=w¯κ+2η p¯ˆκ-R¯¯ˆ κw¯κ,
w¯κ+1=w¯κ+ηeκs¯κ,
wiτκ=η κ=0κ-1 eκsiκ-τ.
oκ=η i=1Nτ=0T siκ-τκ=0κ-1 siκ-τeκ.
Ei,τAκ; x, y, z, t=Cs expjωt-kz×rectx-iΔxΔxrecty-τΔyΔy×rectt-κ-τΔtΔtsiκ-τ+c.c.,
Ei,τAκ=Cssiκ-τ.
EAx, y, z, t= i=1Nτ=0T Ei,τAx, y, z, t.
Eaxo, tEat- xova=a2Gη1 gxo-aw0E0 expjωt-kz×s˜1t-xova+c.c.,
EBxo, yo, t=Cagxo, yoexpjωt-kzA1et+A2 exp-jω2-ω1t+c.c.,
EBκ=Caeκ,
Vout=V1-V2  2mt|Eav|2+2EavEn*t+c.c.,
EsDt=Es/2 cosωstexpiωt+ϕs+c.c.,
Iht=EsDtErt*+c.c.= EsEr coswstcosϕs-ϕr;
EsSt=Es/2 expiωstexpiωt+ϕs+c.c.,
Iht= EsEr/2 coswst+ϕs-ϕr.
Δnx, y, z; t=β exp-t/τpr0t EAx, y, z; t×EB*x, y, z; texpt/τprdt+c.c.,
Δni,τκ=Cγκp=0P/2κ=0T e2p-1T+1+κ×siκ-τγ-1,
Δni,τκ=Cκγκ-1p=0P/2κ=0T e2p-1T+1+κ×siκ-τ,
oκ=ηdZ0gei=1Nτ=0T siκ-τΔni,τκ,
oκ=Cκγκ-1i=1Nτ=0T siκ-τp=0P/2κ=0T e2p-1× T+1+κsiκ-τ,
Δti=aisin θ/v,
τi= t0Δt+a sin θvΔti,
θ=arcsin- tanαvΔta.
θ=arccos- tanα1vΔta.
SINRo= PdPj= ηdPoηjPo= ηdηj,
ηd=E|rodκ|2E|oκ|2E|dκ|2- E|rdjκ|2E|dκ|2E|jκ|2,
ηj=E|rojκ|2E|oκ|2E|jκ|2- E|rjdκ|2E|dκ|2E|jκ|2,
Pg= SINRoSINRi,
ti=Rv1-cosβi-θ,
τi=t0Δt+ RvΔt- RvΔtcosβi-θ,
Δw¯=η R¯¯w¯-p¯+ γ-1w¯,
w¯d= I¯¯+R¯¯-11-γη-1w¯o,
v¯d= I¯¯+Λ¯¯-11-γη-1v¯o,
vld=vlo11+ 1-γηλl,
wiτκ+1 =γwiτκ+ηe κsiκ-τ.
w¯κ=l=1NT qlγ-ηλlκ+R¯¯-1p¯,
-1<γ-ηλl<1
γ-1λmax<η< γ+1λmax,

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