Abstract

An electrooptical realization of an optimal phase estimator for phase modulated communication signals is described in this work. The realization uses an electrooptical processor to perform a 1-D convolution in a 2-D space and a computer to complete the calculations. The processor is a realization of the time domain recursive nonlinear filter. This paper describes the nonlinear filter theory, the electrooptical realization, and the performance of the processor.

© 1983 Optical Society of America

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  1. H. Stark, Opt. Eng. 13, 243 (1974).
    [CrossRef]
  2. E. L. Hall, R. P. Kruger, A. F. Turner, Opt. Eng. 13, 250 (1974).
    [CrossRef]
  3. R. E. Brooks, R. F. Kemp, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 119 (1980).
  4. J. R. Benton, F. Corbett, R. Tuft, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 126 (1980).
  5. D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 228 (1980).
  6. J. R. Leger, J. Cederquist, S. H. Lee, Opt. Eng. 21, 557 (1982).
    [CrossRef]
  7. R. P. Akins, R. A. Athale, S. H. Lee, Opt. Eng. 19, 347 (1980).
    [CrossRef]
  8. J. W. Goodman, A. R. Dias, L. M. Woody. Opt. Lett. 2, 1 (1978).
    [CrossRef] [PubMed]
  9. J. W. Goodman, M. S. Song, Appl. Opt. 21, 502 (1982).
    [CrossRef] [PubMed]
  10. D. Psaltis, D. Casasent, M. Carlotto, Opt. Lett. 4, 348 (1979).
    [CrossRef] [PubMed]
  11. H. J. Caulfield, D. Dvore, J. W. Goodman, W. Rhodes, Appl. Opt. 20, 2263 (1981).
    [CrossRef] [PubMed]
  12. B. V. K. Vijaya Kumar, D. Casasent, Appl. Opt. 20, 3707 (1981).
    [CrossRef]
  13. D. Casasent, Appl. Opt. 21, 1859 (1982).
    [CrossRef] [PubMed]
  14. D. Casasent et al., Proc. Soc. Photo-Opt. Instrum. Eng. 295, 179 (1981).
  15. C. Neuman et al., in Proceedings, Electro-Optical System Design Conference (Industrial & Scientific Conference Management, Chicago, 1981).
  16. T. Sato, K. Sasaki, R. Yamamoto, Appl. Opt. 17, 717 (1978).
    [CrossRef] [PubMed]
  17. R. S. Bucy, J. Astronaut. Sci. 17, 80 (1969).
  18. R. S. Bucy, Proc. IEEE 58, 854 (1970).
    [CrossRef]
  19. R. S. Bucy, in Proceedings, Second Symposium on Nonlinear Estimating Theory and Its Applications, San Diego, 51–58H. W. Sorenson, Ed: (Western Periodicals Co., N. Hollywood, Calif., 1971).
  20. R. S. Bucy, P. D. Joseph, Filtering for Stochastic Processes with Applications to Guidance, (Interscience, New York, 1968).
  21. R. S. Bucy, M. J. Merritt, D. S. Miller, in Proceedings, Second Symposium on Nonlinear Estimating Theory and Its Applications, San Diego, 59–87H. W. Sorenson, Ed: (Western Periodicals Co., N. Hollywood, Calif., 1971).
  22. R. S. Bucy, K. D. Senne, C. Hecht, “New Methods for Nonlinear Filtering,” (Francais d’Automatique Informatique, Recherche Operational, Dunod, Paris, 1973), pp. 3–54.
  23. W. H. Steier, W. E. Stephens, R. Morris, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 116 (1980).
  24. W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978), pp. 226–229.
  25. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 113.
  26. W. E. Stephens, “An Experimental Hybrid Optical–Digital Realization of the Coherent Phase Demodulator,” Ph.D. Dissertation, U. Southern California, Aug.1981.
  27. A. J. Viterbi, Principles of Coherent Communication (McGraw-Hill, New York, 1966).

1982 (3)

1981 (3)

1980 (5)

W. H. Steier, W. E. Stephens, R. Morris, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 116 (1980).

R. P. Akins, R. A. Athale, S. H. Lee, Opt. Eng. 19, 347 (1980).
[CrossRef]

R. E. Brooks, R. F. Kemp, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 119 (1980).

J. R. Benton, F. Corbett, R. Tuft, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 126 (1980).

D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 228 (1980).

1979 (1)

1978 (2)

1974 (2)

H. Stark, Opt. Eng. 13, 243 (1974).
[CrossRef]

E. L. Hall, R. P. Kruger, A. F. Turner, Opt. Eng. 13, 250 (1974).
[CrossRef]

1970 (1)

R. S. Bucy, Proc. IEEE 58, 854 (1970).
[CrossRef]

1969 (1)

R. S. Bucy, J. Astronaut. Sci. 17, 80 (1969).

Akins, R. P.

R. P. Akins, R. A. Athale, S. H. Lee, Opt. Eng. 19, 347 (1980).
[CrossRef]

Athale, R. A.

R. P. Akins, R. A. Athale, S. H. Lee, Opt. Eng. 19, 347 (1980).
[CrossRef]

Benton, J. R.

J. R. Benton, F. Corbett, R. Tuft, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 126 (1980).

Brooks, R. E.

R. E. Brooks, R. F. Kemp, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 119 (1980).

Bucy, R. S.

R. S. Bucy, Proc. IEEE 58, 854 (1970).
[CrossRef]

R. S. Bucy, J. Astronaut. Sci. 17, 80 (1969).

R. S. Bucy, M. J. Merritt, D. S. Miller, in Proceedings, Second Symposium on Nonlinear Estimating Theory and Its Applications, San Diego, 59–87H. W. Sorenson, Ed: (Western Periodicals Co., N. Hollywood, Calif., 1971).

R. S. Bucy, K. D. Senne, C. Hecht, “New Methods for Nonlinear Filtering,” (Francais d’Automatique Informatique, Recherche Operational, Dunod, Paris, 1973), pp. 3–54.

R. S. Bucy, in Proceedings, Second Symposium on Nonlinear Estimating Theory and Its Applications, San Diego, 51–58H. W. Sorenson, Ed: (Western Periodicals Co., N. Hollywood, Calif., 1971).

R. S. Bucy, P. D. Joseph, Filtering for Stochastic Processes with Applications to Guidance, (Interscience, New York, 1968).

Carlotto, M.

Casasent, D.

Caulfield, H. J.

Cederquist, J.

J. R. Leger, J. Cederquist, S. H. Lee, Opt. Eng. 21, 557 (1982).
[CrossRef]

Corbett, F.

J. R. Benton, F. Corbett, R. Tuft, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 126 (1980).

Dias, A. R.

Dvore, D.

Goodman, J. W.

Hall, E. L.

E. L. Hall, R. P. Kruger, A. F. Turner, Opt. Eng. 13, 250 (1974).
[CrossRef]

Hecht, C.

R. S. Bucy, K. D. Senne, C. Hecht, “New Methods for Nonlinear Filtering,” (Francais d’Automatique Informatique, Recherche Operational, Dunod, Paris, 1973), pp. 3–54.

Joseph, P. D.

R. S. Bucy, P. D. Joseph, Filtering for Stochastic Processes with Applications to Guidance, (Interscience, New York, 1968).

Kemp, R. F.

R. E. Brooks, R. F. Kemp, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 119 (1980).

Kruger, R. P.

E. L. Hall, R. P. Kruger, A. F. Turner, Opt. Eng. 13, 250 (1974).
[CrossRef]

Lee, S. H.

J. R. Leger, J. Cederquist, S. H. Lee, Opt. Eng. 21, 557 (1982).
[CrossRef]

R. P. Akins, R. A. Athale, S. H. Lee, Opt. Eng. 19, 347 (1980).
[CrossRef]

Leger, J. R.

J. R. Leger, J. Cederquist, S. H. Lee, Opt. Eng. 21, 557 (1982).
[CrossRef]

Merritt, M. J.

R. S. Bucy, M. J. Merritt, D. S. Miller, in Proceedings, Second Symposium on Nonlinear Estimating Theory and Its Applications, San Diego, 59–87H. W. Sorenson, Ed: (Western Periodicals Co., N. Hollywood, Calif., 1971).

Miller, D. S.

R. S. Bucy, M. J. Merritt, D. S. Miller, in Proceedings, Second Symposium on Nonlinear Estimating Theory and Its Applications, San Diego, 59–87H. W. Sorenson, Ed: (Western Periodicals Co., N. Hollywood, Calif., 1971).

Morris, R.

W. H. Steier, W. E. Stephens, R. Morris, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 116 (1980).

Neuman, C.

C. Neuman et al., in Proceedings, Electro-Optical System Design Conference (Industrial & Scientific Conference Management, Chicago, 1981).

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 113.

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978), pp. 226–229.

Psaltis, D.

Rhodes, W.

Sasaki, K.

Sato, T.

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 113.

Senne, K. D.

R. S. Bucy, K. D. Senne, C. Hecht, “New Methods for Nonlinear Filtering,” (Francais d’Automatique Informatique, Recherche Operational, Dunod, Paris, 1973), pp. 3–54.

Song, M. S.

Sorenson, H. W.

R. S. Bucy, M. J. Merritt, D. S. Miller, in Proceedings, Second Symposium on Nonlinear Estimating Theory and Its Applications, San Diego, 59–87H. W. Sorenson, Ed: (Western Periodicals Co., N. Hollywood, Calif., 1971).

R. S. Bucy, in Proceedings, Second Symposium on Nonlinear Estimating Theory and Its Applications, San Diego, 51–58H. W. Sorenson, Ed: (Western Periodicals Co., N. Hollywood, Calif., 1971).

Stark, H.

H. Stark, Opt. Eng. 13, 243 (1974).
[CrossRef]

Steier, W. H.

W. H. Steier, W. E. Stephens, R. Morris, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 116 (1980).

Stephens, W. E.

W. H. Steier, W. E. Stephens, R. Morris, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 116 (1980).

W. E. Stephens, “An Experimental Hybrid Optical–Digital Realization of the Coherent Phase Demodulator,” Ph.D. Dissertation, U. Southern California, Aug.1981.

Tuft, R.

J. R. Benton, F. Corbett, R. Tuft, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 126 (1980).

Turner, A. F.

E. L. Hall, R. P. Kruger, A. F. Turner, Opt. Eng. 13, 250 (1974).
[CrossRef]

Vijaya Kumar, B. V. K.

Viterbi, A. J.

A. J. Viterbi, Principles of Coherent Communication (McGraw-Hill, New York, 1966).

Woody, L. M.

Yamamoto, R.

Appl. Opt. (5)

J. Astronaut. Sci. (1)

R. S. Bucy, J. Astronaut. Sci. 17, 80 (1969).

Opt. Eng. (4)

H. Stark, Opt. Eng. 13, 243 (1974).
[CrossRef]

E. L. Hall, R. P. Kruger, A. F. Turner, Opt. Eng. 13, 250 (1974).
[CrossRef]

J. R. Leger, J. Cederquist, S. H. Lee, Opt. Eng. 21, 557 (1982).
[CrossRef]

R. P. Akins, R. A. Athale, S. H. Lee, Opt. Eng. 19, 347 (1980).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (1)

R. S. Bucy, Proc. IEEE 58, 854 (1970).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (5)

R. E. Brooks, R. F. Kemp, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 119 (1980).

J. R. Benton, F. Corbett, R. Tuft, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 126 (1980).

D. Casasent, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 228 (1980).

D. Casasent et al., Proc. Soc. Photo-Opt. Instrum. Eng. 295, 179 (1981).

W. H. Steier, W. E. Stephens, R. Morris, Proc. Soc. Photo-Opt. Instrum. Eng. 218, 116 (1980).

Other (9)

W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978), pp. 226–229.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 113.

W. E. Stephens, “An Experimental Hybrid Optical–Digital Realization of the Coherent Phase Demodulator,” Ph.D. Dissertation, U. Southern California, Aug.1981.

A. J. Viterbi, Principles of Coherent Communication (McGraw-Hill, New York, 1966).

C. Neuman et al., in Proceedings, Electro-Optical System Design Conference (Industrial & Scientific Conference Management, Chicago, 1981).

R. S. Bucy, in Proceedings, Second Symposium on Nonlinear Estimating Theory and Its Applications, San Diego, 51–58H. W. Sorenson, Ed: (Western Periodicals Co., N. Hollywood, Calif., 1971).

R. S. Bucy, P. D. Joseph, Filtering for Stochastic Processes with Applications to Guidance, (Interscience, New York, 1968).

R. S. Bucy, M. J. Merritt, D. S. Miller, in Proceedings, Second Symposium on Nonlinear Estimating Theory and Its Applications, San Diego, 59–87H. W. Sorenson, Ed: (Western Periodicals Co., N. Hollywood, Calif., 1971).

R. S. Bucy, K. D. Senne, C. Hecht, “New Methods for Nonlinear Filtering,” (Francais d’Automatique Informatique, Recherche Operational, Dunod, Paris, 1973), pp. 3–54.

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Figures (10)

Fig. 1
Fig. 1

Block diagram of homodyne method to extract a relative phase signal from a modulated carrier in the presence of additive narrowband Gaussian noise.

Fig. 2
Fig. 2

Block diagram of the computation loop of the time domain recursive nonlinear filter. Circulating in the computation loop is the 2-D probability of the state equations of the phase.

Fig. 3
Fig. 3

Schematic of the electrooptical processor (EOP) that convolves the 2-D probability density of the state equations (pn|n) with a fixed Gaussian kernel function. This operation produces a 1-D convolution of the 2-D space in the output plane.

Fig. 4
Fig. 4

Block diagram of the computation steps in converting the ordinary convolution in 1-D over the 2-D space to a circular convolution in 1-D over the 2-D space via the overlap and add method. The ordinary convolution is in the long (phase rate) direction. The tails of the convolution are contained in subarrays A and D. They are removed and added to the subarrays C and B thus making a 72 × 96 array.

Fig. 5
Fig. 5

Three-dimensional computer illustrations of the CCD sensor view of the optical convolution of an input 2-D circular Gaussian pattern. The crosshatch lines represent the edges of the cells when the 96 × 96 array is reduced to 16 × 16. (a) shows that there is no convolution in the phase direction as the intensities fall between the cell edge lines. (b) shows that in the phase rate direction, there is a convolution between the optical function and its input. The resultant output is an elliptical Gaussian pattern.

Fig. 6
Fig. 6

Accuracy measurement of the convolution in the phase rate direction of forty-five input normalized circular Gaussian patterns with a variance of 1.0 and different means. The error bars represent the scatter of data while the dashed curve is the theoretical output of the convolution.

Fig. 7
Fig. 7

System block diagram of the experimental hybrid optical–digital TDRNF. The optical processor is used to compute 1-D convolutions in phase rate while the computer derives the phase estimate and updates the state probability density from incoming measurements.

Fig. 8
Fig. 8

System block diagram of the noise contaminated measurement simulator.

Fig. 9
Fig. 9

(a)–(f) is a sequence of photos showing the evolution of the shifted probability density matrix for the nonlinear filter. These photos show the LED matrix input into the EOP starting with iteration (4) and stopping with iteration (9). The phase rate axis is the horizontal axis with +π/Δ in the left corner and −π/Δ in the right corner. The phase axis is the vertical axis with +π at the bottom and −π on the top. The 1-D convolution is along the horizontal axis.

Fig. 10
Fig. 10

(a)–(f) is a sequence of computer-generated probability density plots after processing by the EOP and enhancement by the incoming measurements. These densities start at iteration (4) and stop at iteration (9), thus they directly correspond to the photos of Fig. 9. From each density the optimal estimate is calculated for that time increment.

Tables (1)

Tables Icon

Table I Summary of Monte Carlo Simulation to Determine the Effects of Optical Errors on the Running of the Hybrid Digital–Optical Phase Estimator

Equations (12)

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z ( t ) = A sin [ ω 0 t + θ ( t ) ] + ν ( t ) ,
ν ( t ) = ν 2 ( t ) cos ω 0 ( t ) + ν 1 ( t ) sin ω 0 ( t ) ,
[ z 1 ( t ) z 2 ( t ) ] = [ cos θ ( t ) sin θ ( t ) ] + [ ν 1 ( t ) ν 2 ( t ) ] ,
[ z 1 ( n ) z 2 ( n ) ] = [ cos θ ( n ) sin θ ( n ) ] + [ ν 1 ( n ) ν 2 ( n ) ] .
[ x 1 ( n + 1 ) x 2 ( n + 1 ) ] = [ x 1 ( n ) x 2 ( n ) ] + [ Δ x 2 ( n ) Δ u 2 ( n ) ] .
p n + 1 | n + 1 ( y 1 , y 2 ) = S n + 1 ( y 1 ) π / Δ π / Δ i = × exp [ ( y 2 x 2 + 2 π i / Δ ) 2 q Δ ] · k = j = p n | n ( y 1 x 2 Δ + 2 π ( k j ) x 2 + 2 π j / Δ ) d x 2 ,
S n + 1 ( y 1 ) = C 0 exp { Δ r [ z 1 cos ( y 1 ) + z 2 sin ( y 1 ) ] } ,
p n + 1 | n + 1 ( y 1 y 2 ) = S n + 1 ( y 1 ) [ η ( y 2 ) * p ˜ n | n ( y 1 y 2 ) ] ,
p ˜ n | n ( y 1 y 2 ) = p n | n ( y 1 y 2 Δ y 2 ) .
x ˆ 1 = tan 1 { E [ sin ( x 1 ) | z n ] / E [ cos ( x 1 ) | z n ] } ,
E [ sin ( x 1 ) | z n ] = π / Δ π / Δ π π sin ( x 1 ) p n | n ( x 1 , x 2 ) d x 1 d x 2 , E [ cos ( x 1 ) | z n ] = π / Δ π / Δ π π cos ( x 1 ) p n | n ( x 1 , x 2 ) d x 1 d x 2 , p n | n ( x 1 , x 2 ) = j = k = p n | n ( x 1 + 2 π j x 2 + 2 π k / Δ ) .
I i ( x , y ) = K | h [ a ( x b u ) , a ( y b υ ) ] | 2 I o ( u , υ ) d u d υ ,

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