Abstract

An averaged threshold receiver is developed for an optical communication system consisting of a symmetric binary, pulse-code modulated transmitter, a lognormal channel, and an array of independent photocounting detectors. When compared to previously described receivers, it is shown to be a much simpler structure to implement and to provide generally lower bit error rates. Probability of error curves demonstrating this improved performance are presented for various combinations of turbulence strength, background radiation level, SNR, number of diversity channels, and, in the newly developed processor, number of bits used for threshold averaging.

© 1977 Optical Society of America

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References

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  1. M. C. Teich, S. Rosenberg, Appl. Opt. 12, 2616 (1973).
    [Crossref] [PubMed]
  2. E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
    [Crossref]
  3. W. K. Pratt, Laser Communications Systems (Wiley, New York, 1969), Chap. 9.
  4. R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), Chap. 5.
  5. S. Rosenberg, M. C. Teich, Appl. Opt. 12, 2625 (1973).
    [Crossref] [PubMed]
  6. P. Diament, M. C. Teich, J. Opt. Soc. Am. 60, 1489 (1970).
    [Crossref]
  7. S. Karp, J. R. Clark, IEEE Trans. Inf. Theory IT-16, 672 (1970).
    [Crossref]
  8. R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
    [Crossref]
  9. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).
  10. D. L. Fried, G. E. Mevers, M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967).
    [Crossref]
  11. R. L. Mitchell, J. Opt. Soc. Am. 58, 1267 (1968).
    [Crossref]
  12. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 16.
  13. H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Chap. 2.
  14. H. E. Slazer, R. Zucker, R. Capuano, Handbook of Mathematical Functions, M. Abramowitz, I. Stegun, Eds. (Dover, New York, 1965), p. 924.

1973 (2)

1970 (4)

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[Crossref]

P. Diament, M. C. Teich, J. Opt. Soc. Am. 60, 1489 (1970).
[Crossref]

S. Karp, J. R. Clark, IEEE Trans. Inf. Theory IT-16, 672 (1970).
[Crossref]

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[Crossref]

1968 (1)

1967 (1)

Capuano, R.

H. E. Slazer, R. Zucker, R. Capuano, Handbook of Mathematical Functions, M. Abramowitz, I. Stegun, Eds. (Dover, New York, 1965), p. 924.

Clark, J. R.

S. Karp, J. R. Clark, IEEE Trans. Inf. Theory IT-16, 672 (1970).
[Crossref]

Diament, P.

Fried, D. L.

Gagliardi, R. M.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), Chap. 5.

Halme, S. J.

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[Crossref]

Harger, R. O.

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[Crossref]

Hoversten, E. V.

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[Crossref]

Karp, S.

S. Karp, J. R. Clark, IEEE Trans. Inf. Theory IT-16, 672 (1970).
[Crossref]

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), Chap. 5.

Keister, M. P.

Lawrence, R. S.

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[Crossref]

Mevers, G. E.

Mitchell, R. L.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 16.

Pratt, W. K.

W. K. Pratt, Laser Communications Systems (Wiley, New York, 1969), Chap. 9.

Rosenberg, S.

Slazer, H. E.

H. E. Slazer, R. Zucker, R. Capuano, Handbook of Mathematical Functions, M. Abramowitz, I. Stegun, Eds. (Dover, New York, 1965), p. 924.

Strohbehn, J. W.

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[Crossref]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).

Teich, M. C.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Chap. 2.

Zucker, R.

H. E. Slazer, R. Zucker, R. Capuano, Handbook of Mathematical Functions, M. Abramowitz, I. Stegun, Eds. (Dover, New York, 1965), p. 924.

Appl. Opt. (2)

IEEE Trans. Inf. Theory (1)

S. Karp, J. R. Clark, IEEE Trans. Inf. Theory IT-16, 672 (1970).
[Crossref]

J. Opt. Soc. Am. (3)

Proc. IEEE (2)

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[Crossref]

E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
[Crossref]

Other (6)

W. K. Pratt, Laser Communications Systems (Wiley, New York, 1969), Chap. 9.

R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976), Chap. 5.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 16.

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Chap. 2.

H. E. Slazer, R. Zucker, R. Capuano, Handbook of Mathematical Functions, M. Abramowitz, I. Stegun, Eds. (Dover, New York, 1965), p. 924.

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Figures (16)

Fig. 1
Fig. 1

Block diagram of general optical communication system. Specific receiver structures will be examined for channels through the clear-air turbulent atmosphere.

Fig. 2
Fig. 2

Block diagram of approximate optimum receiver.

Fig. 3
Fig. 3

Block diagram of MAP receiver.

Fig. 4
Fig. 4

Block diagram of averaged threshold receiver.

Figs. 5
Figs. 5

Total probability of error P(E) vs SNR γ for various receiver structures. For each combination of background radiation level NB, number of diversity channels D, and log-intensity standard deviation σ, P(E) curves are denoted by 0 for σ = 0, 1 for MAP receiver, 2 for approximate optimum receiver, 3 for infinite averaged threshold receiver, 4 for ten bit averaged threshold receiver, 5 for twenty-five bit averaged threshold receiver. NB = 4, D = 1, σ = 1.5.

Fig. 6
Fig. 6

NB = 4, D = 1, σ = 1.0 (see Fig. 5 caption).

Fig. 7
Fig. 7

NB = 4, D = 1, σ = 0.5 (see Fig. 5 caption).

Fig. 8
Fig. 8

NB = 2, D = 2, σ = 1.5 (see Fig. 5 caption).

Fig. 9
Fig. 9

NB = 2, D = 2, σ = 0.5 (see Fig. 5 caption).

Fig. 10
Fig. 10

NB = 1, D = 4, σ = 1.5 (see Fig. 5 caption).

Fig. 11
Fig. 11

NB = 1, D = 4, σ = 1.0 (see Fig. 5 caption).

Fig. 12
Fig. 12

NB = 1, D = 4, σ = 0.5 (see Fig. 5 caption).

Fig. 13
Fig. 13

NB = 40, D = 1, σ = 1.5 (see Fig. 5 caption).

Fig. 14
Fig. 14

NB = 40, D = 1, σ = 1.0 (see Fig. 5 caption).

Fig. 15
Fig. 15

NB = 10, D = 4, σ = 1.5 (see Fig. 5 caption).

Fig. 16
Fig. 16

Upper bound on number of averaging bits Nmax vs ratio of data rate to perpendicular wind velocity, R/v with dependence on log-intensity standard deviation, σ and Fresnel zone size (λL)1/2 shown parametrically. Lower curve in each doublet corresponds to. σ = 1.0 and the upper one to σ = 0.5 or 1.5, where the doublets are labeled by (1) for (λL)1/2 = 1.0, (2) for (λL)1/2 = 0.1, and (3) for (λL)1/2 = 0.01.

Equations (28)

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p ( Z ) = 1 ( 2 π ) 1 / 2 σ Z exp { - [ ln ( Z ) + 1 2 σ 2 ] 2 2 σ 2 } ,
I = Z I S + I B if 1 is sent ,
I = I B if 0 is sent .
p ( n W ) = [ 1 / ( n ! ) ] W n exp ( - W ) .
W = η h ν T A I ( r , t ) d r d t ,
W = Z N S + N B if 1 is sent ,
W = N B if 0 is sent .
p ( n H 0 ) = i = 1 D p ( n i H 0 ) = i = 1 D 1 n i ! ( N B ) n i exp ( - N B ) ,
p ( n H 1 ) = i = 1 D p ( n i H 1 ) = i = 1 D 0 1 n i ! ( Z i N S + N B ) n i exp ( - Z i N S - N B ) p ( Z i ) d Z i .
p ( n i H 1 ) = 1 n i ! ( Z i o N S + N B ) n i exp ( - Z i o N S - N B ) exp { - [ ln ( Z i o ) + ( 1 / 2 ) σ 2 ] 2 2 σ 2 } { 1 + σ 2 Z i o N s [ 1 - n i N B ( Z i o N S + N B ) 2 ] } 1 / 2 ,
n i Z i o N S Z i o N S + N B - Z i o N S - ln ( Z i o ) + 1 2 σ 2 σ 2 = 0 for i = 1 , 2 , D .
L ( n ) = ln p ( n H 1 ) p ( n H 0 ) ,
P ( E ) = 1 2 [ P ( L < 0 H 1 ) + P ( L > 0 H 0 ) ] .
L = i = 1 D 0 ( Z N S N B + 1 ) n i exp ( - Z N S ) p ( Z ) d Z .
L = i = 1 D ( n i ln ( Z i o N S N B + 1 ) - Z i o N S - ( ln Z i o + 1 2 σ 2 ) 2 2 σ 2 - 1 2 ln { 1 + σ 2 Z i o N S [ 1 - n i N B ( Z i o N S + N B ) 2 ] } )
L = ln p ( n Z , H 1 ) p ( n Z , H 0 ) = i = 1 D n i ln ( Z ^ i N S N B + 1 ) - Z ^ i N S ,
Z i [ p ( Z i n i ) ] Z i = Z ^ i = 0 i = 1 , 2 , D ,
Z i [ p ( Z i n ¯ i ) ] Z i = Z ^ i = 0             i = 1 , 2 , D ,
n ¯ i = 1 N j = 1 N n i j = i 1 , 2 , D .
N n ¯ i Z ^ i N S Z ^ i N S + 2 N B - 1 2 N Z ^ i N S - [ ln ( Z ^ i ) + 1 2 σ 2 ] σ 2 = 0.
n ¯ i - 1 2 Z ^ i N S - N B = 0.
L = n i ln ( 2 n ¯ i N B - 1 ) - 2 n ¯ i + 2 N B .
N 0.546 N B 1 / 3 ( λ L ) 5 / 18 l 0 1 / 9 ( R N S σ v ) 2 / 3 × ( 1 2 N S + N B + e σ 2 N S 2 )
( Δ Z ) 2 ( Δ Z ^ ) 2 .
( Δ Z ) 2 = ( d Z d t ) 2 N 2 R 2 = 24.6 v 2 σ 2 N 2 ( λ L ) 5 / 6 l 0 1 / 9 R 2 .
( Δ Z ^ ) 2 = 4 N N S 2 ( 1 2 N S + N B + e σ 2 N S 2 ) .
N 0.546 ( λ L ) 5 / 18 l 0 1 / 9 ( R v σ N S ) 2 / 3 ( 1 2 N S + N B + e σ 2 N S 2 ) .
N max 0.274 ( λ L ) 5 / 8 ( R v ) 2 / 3 ( e σ 2 σ 2 ) 1 / 3 .

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