Abstract

Performance characteristics are obtained for various optimum and suboptimum photocounting array receiver structures obtained in the preceding paper (Part 1). Probability of error curves are presented for the case of lognormal fading and for a variety of receiver configurations including the BPCM single-detector and array-detector processors and the BPOLM and BPIM single-detector processor. In addition to these approximate optimum processors, the suboptimum aperture integration and MAP receivers are evaluated in order to examine degradation of performance with increasing suboptimality. An optimum amount of diversity is shown to exist for the aperture integration receiver, with a fixed signal energy constraint. In a related paper (Part 3), we examine a bound to the error probability for M-ary signaling.

© 1973 Optical Society of America

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References

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  1. M. C. Teich, S. Rosenberg (Part 1), Appl. Opt. 12, 2616 (1973) [preceding paper].
    [Crossref] [PubMed]
  2. S. Rosenberg, M. C. Teich (Part 3), IEEE Trans. Inform. Theory IT-19, 807 (1973).
    [Crossref]
  3. W. K. Pratt, Laser Communication Systems (Wiley, New York, 1969).
  4. D. Middleton, C. J. Gundersdorf, “On Optimum Detection in Quantum Optics, Part 2: Error Probabilities and Expected Performance,” Technical Report AFAL-TR-68-158, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, (1970) pp. 8–13.
  5. D. L. Fried, R. A. Schmeltzer, Appl. Opt. 6, 1729 (1967).
    [Crossref] [PubMed]
  6. S. Rosenberg, M. C. Teich, J. Appl. Phys. 43, 1256 (1972).
    [Crossref]
  7. M. C. Teich, S. Rosenberg, J. Opto-electron. 3, 63 (1971). Note that Eq. (28) of this article should read B = Q(2) − Λ−1 and |B|1/2 should be replaced by |−B|1/2 throughout. All figures, results, conclusions, and other equations remain unchanged.
    [Crossref]
  8. E. V. Hoversten, R. O. Harger, S. J. Halme, Proc. IEEE 58, 1626 (1970).
    [Crossref]
  9. J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), pp. 527–550.
  10. R. S. Kennedy, E. V. Hoversten, IEEE Trans. Inform. Theory IT-14, 716 (1968).
    [Crossref]
  11. R. L. Mitchell, J. Opt. Soc. Am. 58, 1267 (1968).
    [Crossref]
  12. V. Blumel, L. M. Narducci, R. A. Tuft, J. Opt. Soc. Am. 62, 1309 (1972).
    [Crossref]
  13. A. A. M. Saleh, IEEE J. Quantum Electron. QE-3, 540 (1967).
    [Crossref]
  14. T. Kinsel, Proc. IEEE 58, 1666 (1970).
    [Crossref]
  15. W. N. Peters, R. J. Arguello, IEEE J. Quantum Electron. QE-3, 532 (1967).
    [Crossref]
  16. M. Ross, IEEE Trans. Aero. Electron. Syst. (Supplement) AES-3, 324 (1967).
  17. R. M. Gagliardi, S. Karp, IEEE Trans. Comm. Tech. COM-17, 208 (1969).
    [Crossref]
  18. For abstract of dissertation, see S. Rosenberg, IEEE Trans. Inform. Theory IT-18, 544 (1972).
    [Crossref]
  19. A talk based on portions of this material was presented at the 1972 Annual Meeting of the Optical Society of America; for abstract, see S. Rosenberg, J. Opt. Soc. Am. 62, 353A (1972).
    [Crossref]

1973 (2)

M. C. Teich, S. Rosenberg (Part 1), Appl. Opt. 12, 2616 (1973) [preceding paper].
[Crossref] [PubMed]

S. Rosenberg, M. C. Teich (Part 3), IEEE Trans. Inform. Theory IT-19, 807 (1973).
[Crossref]

1972 (4)

V. Blumel, L. M. Narducci, R. A. Tuft, J. Opt. Soc. Am. 62, 1309 (1972).
[Crossref]

S. Rosenberg, M. C. Teich, J. Appl. Phys. 43, 1256 (1972).
[Crossref]

For abstract of dissertation, see S. Rosenberg, IEEE Trans. Inform. Theory IT-18, 544 (1972).
[Crossref]

A talk based on portions of this material was presented at the 1972 Annual Meeting of the Optical Society of America; for abstract, see S. Rosenberg, J. Opt. Soc. Am. 62, 353A (1972).
[Crossref]

1971 (1)

M. C. Teich, S. Rosenberg, J. Opto-electron. 3, 63 (1971). Note that Eq. (28) of this article should read B = Q(2) − Λ−1 and |B|1/2 should be replaced by |−B|1/2 throughout. All figures, results, conclusions, and other equations remain unchanged.
[Crossref]

1970 (2)

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

T. Kinsel, Proc. IEEE 58, 1666 (1970).
[Crossref]

1969 (1)

R. M. Gagliardi, S. Karp, IEEE Trans. Comm. Tech. COM-17, 208 (1969).
[Crossref]

1968 (2)

R. S. Kennedy, E. V. Hoversten, IEEE Trans. Inform. Theory IT-14, 716 (1968).
[Crossref]

R. L. Mitchell, J. Opt. Soc. Am. 58, 1267 (1968).
[Crossref]

1967 (4)

A. A. M. Saleh, IEEE J. Quantum Electron. QE-3, 540 (1967).
[Crossref]

D. L. Fried, R. A. Schmeltzer, Appl. Opt. 6, 1729 (1967).
[Crossref] [PubMed]

W. N. Peters, R. J. Arguello, IEEE J. Quantum Electron. QE-3, 532 (1967).
[Crossref]

M. Ross, IEEE Trans. Aero. Electron. Syst. (Supplement) AES-3, 324 (1967).

Arguello, R. J.

W. N. Peters, R. J. Arguello, IEEE J. Quantum Electron. QE-3, 532 (1967).
[Crossref]

Blumel, V.

Fried, D. L.

Gagliardi, R. M.

R. M. Gagliardi, S. Karp, IEEE Trans. Comm. Tech. COM-17, 208 (1969).
[Crossref]

Gundersdorf, C. J.

D. Middleton, C. J. Gundersdorf, “On Optimum Detection in Quantum Optics, Part 2: Error Probabilities and Expected Performance,” Technical Report AFAL-TR-68-158, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, (1970) pp. 8–13.

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]

R. S. Kennedy, E. V. Hoversten, IEEE Trans. Inform. Theory IT-14, 716 (1968).
[Crossref]

Jacobs, I. M.

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), pp. 527–550.

Karp, S.

R. M. Gagliardi, S. Karp, IEEE Trans. Comm. Tech. COM-17, 208 (1969).
[Crossref]

Kennedy, R. S.

R. S. Kennedy, E. V. Hoversten, IEEE Trans. Inform. Theory IT-14, 716 (1968).
[Crossref]

Kinsel, T.

T. Kinsel, Proc. IEEE 58, 1666 (1970).
[Crossref]

Middleton, D.

D. Middleton, C. J. Gundersdorf, “On Optimum Detection in Quantum Optics, Part 2: Error Probabilities and Expected Performance,” Technical Report AFAL-TR-68-158, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, (1970) pp. 8–13.

Mitchell, R. L.

Narducci, L. M.

Peters, W. N.

W. N. Peters, R. J. Arguello, IEEE J. Quantum Electron. QE-3, 532 (1967).
[Crossref]

Pratt, W. K.

W. K. Pratt, Laser Communication Systems (Wiley, New York, 1969).

Rosenberg, S.

M. C. Teich, S. Rosenberg (Part 1), Appl. Opt. 12, 2616 (1973) [preceding paper].
[Crossref] [PubMed]

S. Rosenberg, M. C. Teich (Part 3), IEEE Trans. Inform. Theory IT-19, 807 (1973).
[Crossref]

S. Rosenberg, M. C. Teich, J. Appl. Phys. 43, 1256 (1972).
[Crossref]

For abstract of dissertation, see S. Rosenberg, IEEE Trans. Inform. Theory IT-18, 544 (1972).
[Crossref]

A talk based on portions of this material was presented at the 1972 Annual Meeting of the Optical Society of America; for abstract, see S. Rosenberg, J. Opt. Soc. Am. 62, 353A (1972).
[Crossref]

M. C. Teich, S. Rosenberg, J. Opto-electron. 3, 63 (1971). Note that Eq. (28) of this article should read B = Q(2) − Λ−1 and |B|1/2 should be replaced by |−B|1/2 throughout. All figures, results, conclusions, and other equations remain unchanged.
[Crossref]

Ross, M.

M. Ross, IEEE Trans. Aero. Electron. Syst. (Supplement) AES-3, 324 (1967).

Saleh, A. A. M.

A. A. M. Saleh, IEEE J. Quantum Electron. QE-3, 540 (1967).
[Crossref]

Schmeltzer, R. A.

Teich, M. C.

S. Rosenberg, M. C. Teich (Part 3), IEEE Trans. Inform. Theory IT-19, 807 (1973).
[Crossref]

M. C. Teich, S. Rosenberg (Part 1), Appl. Opt. 12, 2616 (1973) [preceding paper].
[Crossref] [PubMed]

S. Rosenberg, M. C. Teich, J. Appl. Phys. 43, 1256 (1972).
[Crossref]

M. C. Teich, S. Rosenberg, J. Opto-electron. 3, 63 (1971). Note that Eq. (28) of this article should read B = Q(2) − Λ−1 and |B|1/2 should be replaced by |−B|1/2 throughout. All figures, results, conclusions, and other equations remain unchanged.
[Crossref]

Tuft, R. A.

Wozencraft, J. M.

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), pp. 527–550.

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

A. A. M. Saleh, IEEE J. Quantum Electron. QE-3, 540 (1967).
[Crossref]

W. N. Peters, R. J. Arguello, IEEE J. Quantum Electron. QE-3, 532 (1967).
[Crossref]

IEEE Trans. Aero. Electron. Syst. (Supplement) (1)

M. Ross, IEEE Trans. Aero. Electron. Syst. (Supplement) AES-3, 324 (1967).

IEEE Trans. Comm. Tech. (1)

R. M. Gagliardi, S. Karp, IEEE Trans. Comm. Tech. COM-17, 208 (1969).
[Crossref]

IEEE Trans. Inform. Theory (3)

For abstract of dissertation, see S. Rosenberg, IEEE Trans. Inform. Theory IT-18, 544 (1972).
[Crossref]

R. S. Kennedy, E. V. Hoversten, IEEE Trans. Inform. Theory IT-14, 716 (1968).
[Crossref]

S. Rosenberg, M. C. Teich (Part 3), IEEE Trans. Inform. Theory IT-19, 807 (1973).
[Crossref]

J. Appl. Phys. (1)

S. Rosenberg, M. C. Teich, J. Appl. Phys. 43, 1256 (1972).
[Crossref]

J. Opt. Soc. Am. (3)

R. L. Mitchell, J. Opt. Soc. Am. 58, 1267 (1968).
[Crossref]

V. Blumel, L. M. Narducci, R. A. Tuft, J. Opt. Soc. Am. 62, 1309 (1972).
[Crossref]

A talk based on portions of this material was presented at the 1972 Annual Meeting of the Optical Society of America; for abstract, see S. Rosenberg, J. Opt. Soc. Am. 62, 353A (1972).
[Crossref]

J. Opto-electron. (1)

M. C. Teich, S. Rosenberg, J. Opto-electron. 3, 63 (1971). Note that Eq. (28) of this article should read B = Q(2) − Λ−1 and |B|1/2 should be replaced by |−B|1/2 throughout. All figures, results, conclusions, and other equations remain unchanged.
[Crossref]

Proc. IEEE (2)

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

T. Kinsel, Proc. IEEE 58, 1666 (1970).
[Crossref]

Other (3)

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), pp. 527–550.

W. K. Pratt, Laser Communication Systems (Wiley, New York, 1969).

D. Middleton, C. J. Gundersdorf, “On Optimum Detection in Quantum Optics, Part 2: Error Probabilities and Expected Performance,” Technical Report AFAL-TR-68-158, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, (1970) pp. 8–13.

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Figures (13)

Fig. 1
Fig. 1

Threshold count kT vs SNR γ for optimum and MAP single-detector photocounting receivers. The parameter σ represents the log-irradiance standard deviation.

Fig. 2
Fig. 2

Probability of error P() vs SNR γ for single-detector photocounting receiver. Odd-numbered curves correspond to the optimum receiver while even-numbered curves correspond to the suboptimum MAP receiver. (a) NB = 1, number of coherent areas = 1; (b) NB = 4, number of coherent areas = 1.

Fig. 3
Fig. 3

Probability of error P() vs dB power loss due to turbulence for a single detector using BPOLM and BPCM formats. Open circles and squares indicate hand-calculated theoretical points obtained from curves such as those in Fig. 2.

Fig. 4
Fig. 4

Optimum threshold boundary kT(n1,n2) for the two-detector array receiver with parametric variation of NB, γ, and R. (a) σ = 0.5; (b) σ = 1.5.

Fig. 5
Fig. 5

Probability of error P() vs SNR γ for two-detector PCM photocounting array receiver. The lower curve of each pair represents the optimum receiver while the upper curve represents the suboptimum MAP receiver. In all cases NB = 2. (a) R = 0; (b) R = 0.5; (c) R 1.0.

Fig. 6
Fig. 6

Probability of error P() vs dB power gain due to diversity (D = 2) for two-detector photocounting array receiver. Circles, squares, and triangles represent hand-calculated theoretical points. The dependence on background and turbulence is presented parametrically.

Fig. 7
Fig. 7

Probability of error P() vs SNR for the aperture integration receiver. The quantity DNB = 8 is kept constant. The upper curve of each pair corresponds to D = 1, NB = 8 while the lower curve corresponds to D = 8, NB = 1.

Fig. 8
Fig. 8

Probability of error P() as a function of total mean signal count NS and diversity D for the aperture integration receiver. NB = 1. (a) σ = 0.5; (b) σ = 1.0.

Fig. 9
Fig. 9

Probability of error P() vs number of diversity paths D for the aperture integration receiver. The dependence on NS is shown parametrically. NB = 1. (a) σ = 0.5; (b) σ = 1.0.

Fig. 10
Fig. 10

Probability of error P() vs number of diversity paths D for the aperture integration receiver. The dependence on NB is shown parametrically. NB = 4. (a) σ = 0.5; (b) σ = 1.0; (c) σ = 1.5.

Fig. 11
Fig. 11

Block diagram of optimum single-detector twin-channel receiver for BPOLM with lognormal fading.

Fig. 12
Fig. 12

Probability of error P() vs SNR γ for single-detector twin-channel receiver using BPIM or BPOLM. The upper of each pair of curves corresponds to decision rule 2 while the lower corresponds to decision rule 3. The effect of fading is shown parametrically. (a) NB = 1; (b) NB = 4.

Fig. 13
Fig. 13

Performance gain of BPOLM over BPCM (in dB) as a function of error rate, turbulence, and background for a single-detector counting receiver. Open circles and squares indicate hand-calculated theoretical points.

Tables (1)

Tables Icon

Table I Gain of Uncorrelated Two-Detector Array Receiver (in dB) over Aperture Integration Detector with D = 2

Equations (13)

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P ( ) = π 0 p ( H 1 H 0 ) + π 1 p ( H 0 H 1 ) ,
P ( ) = 1 2 p ( L > 0 H 0 ) + 1 2 p ( L < 0 H 1 ) ,
P ( ) = 1 2 0 p ( L H 0 ) d L + 1 2 - 0 p ( L H 1 ) d L
P ( C S 1 ) = p ( L 1 L 2 , L 3 , , L M ) = - p ( L 1 ) d L 1 - L 1 - L 1 p ( L 2 , L 3 , , L M ) d L 2 d L M .
P ( ) = 1 - P ( C ) = 1 - - p ( L 1 ) d L 1 - L 1 - L 1 p ( L 2 , L 3 , , L M ) d L 2 d L M .
n ln ( N S N B + 1 ) - N S H 0 H 1 0 ,
n H 0 H 1 k T ,
P ( ) = 1 2 p ( n k T H 0 ) + 1 2 p ( n < k T H 1 ) = 1 2 { n = k T N B n exp ( - N B ) n ! + n = 0 k T - 1 ( N S + N B ) n exp [ - ( N S + N B ) ] n ! } = 1 2 { 1 - n = 0 k T - 1 N B n exp ( - N B ) n ! + n = 0 k T - 1 ( N S + N B ) n exp [ - ( N S + N B ) ] n ! } .
P ( ) = 1 2 p 1 ( n < k T ) + 1 2 p 0 ( n k T ) ,
p 1 ( n 1 , n 2 ) = p 0 ( n 1 , n 2 ) ,
P ( ) = 1 - k = 1 p 1 ( k ) [ n = 0 k - 1 p 0 ( n ) ] ,
P ( ) = { 1 - k = 0 p 1 ( k ) [ n = 0 k p 0 ( n ) ] } + 1 2 n = 0 p 1 ( n ) · p 0 ( n ) .
P ( ) rule 3 = P ( ) rule 2 - 1 2 p ( n 1 = n 0 ) .

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