Abstract

A single-threshold processor is derived for a wide class of classical binary decision problems involving the likelihood-ratio detection of a signal embedded in noise. The class of problems we consider encompasses the case of multiple independent (but not necessarily identically distributed) observations of a nonnegative (nonpositive) signal, embedded in additive, independent, and noninterfering noise, where the range of the signal and noise is discrete. We show that a comparison of the sum of the observations with a unique threshold comprises optimum processing, if a weak condition on the noise is satisfied, independent of the signal. Examples of noise densities that satisfy and violate our condition are presented. The results are applied to a generalized photocounting optical communication system, and it is shown that most components of the system can be incorporated into our model. The continuous case is treated elsewhere [ IEEE Trans. Inf. Theory IT-25, (March , 1979)].

© 1978 Optical Society of America

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References

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  1. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part 1 (Wiley, New York, 1968).
  2. B. Reiffen, H. Sherman, Proc. IEEE 51, 1316 (1963).
    [CrossRef]
  3. C. W. Helstrom, IEEE Trans. Inf. Theory IT-10, 275 (1964).
    [CrossRef]
  4. W. K. Pratt, Laser Communication Systems (Wiley, New York, 1969).
  5. Proc. IEEE (Special Issue on Optical Communications), 58 (1970).
  6. E. Hoversten, “Optical Communication Theory,” in Laser Handbook, F. T. Arrechi, E. O. Schulz-DuBois, Eds. (North-Holland, Amsterdam, 1972), Vol. 2.
  7. M. C. Teich, R. Y. Yen, IEEE Trans. Aerosp. Electron. Syst. AES-8, 13 (1972).
    [CrossRef]
  8. R. Y. Yen, P. Diament, M. C. Teich, IEEE Trans. Inf. Theory IT-18, 302 (1972).
    [CrossRef]
  9. M. C. Teich, S. Rosenberg, Appl. Opt. 12, 2616 (1973).
    [CrossRef] [PubMed]
  10. S. Rosenberg, M. C. Teich, Appl. Opt. 12, 2625 (1973).
    [CrossRef] [PubMed]
  11. E. V. Hoversten, D. L. Snyder, R. O. Harger, K. Kurimoto, IEEE Trans. Commun. COM-22, 17 (1974).
    [CrossRef]
  12. D. L. Snyder, Random Point Processes (Wiley, New York, 1975).
  13. R. M. Gagliardi, S. Karp, Optical Communications (Wiley, New York, 1976).
  14. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976).
  15. S. D. Personick, Bell Syst. Tech. J. 50, 3075 (1971).
  16. S. D. Personick, Proc. IEEE 65, 1670 (1977).
    [CrossRef]
  17. S. D. Personick, P. Balaban, J. H. Bobsin, P. R. Kumar, IEEE Trans. Commun COM-25, 541 (1977).
    [CrossRef]
  18. M. C. Teich, P. R. Prucnal, G. Vannucci, Opt. Lett. 1, 208 (1977).
    [CrossRef] [PubMed]
  19. M. C. Teich, B. I. Cantor, IEEE J. Quantum Electron. QE-14 (December, 1978).
  20. G. W. Flint, IEEE Trans. Mil. Electron. MIL-8, 22 (1964).
    [CrossRef]
  21. J. W. Goodman, Proc. IEEE 53, 1688 (1965).
    [CrossRef]
  22. W. W. Peterson, T. G. Birdsall, W. C. Fox, Trans. IRE Prof. Group Inf. Theory PGIT-4, 171 (1954).
    [CrossRef]
  23. W. P. Tanner, J. A. Swets, Psychol. Rev. 61, 401 (1954).
    [CrossRef] [PubMed]
  24. H. B. Barlow, J. Opt. Soc. Am. 46, 634 (1956).
    [CrossRef] [PubMed]
  25. M. C. Teich, P. R. Prucnal, J. Opt. Soc. Am. 67, 1426 (1977).
  26. W. J. McGill, J. Math. Psychol. 4, 351 (1967).
    [CrossRef]
  27. M. C. Teich, W. J. McGill, Phys. Rev. Lett. 36, 754 (1976).
    [CrossRef]
  28. P. R. Prucnal, M. C. Teich, IEEE Trans. Inf. Theory IT-25, (March, 1979).
  29. The finite difference is Δ[f(n)]k ≡ f(n + k) − f(n). If no subscript is used, k is assumed to be unity. We employ the second difference Δ2[f(n)]k,j ≡ [f(n + j) − f(n)] − [f(n + j − k) − f(n − k)] with j = 1 to obtain Δ2[f(n)]k.
  30. D. C. Murdoch, Linear Algebra (Wiley, New York, 1970), p. 165.
  31. The product rule can be seen fromΔ[f(n)g(n)h(n)]=f(n+1)g(n+1)h(n+1)−f(n+1)g(n+1)h(n)+f(n+1)g(n+1)h(n)−f(n+1)g(n)h(n)+f(n+1)g(n)h(n)−f(n)g(n)h(n)=f(n+1)g(n+1)Δ[h(n)]+f(n+1)Δ[g(n)]h(n)+Δ[f(n)]g(n)h(n),which extends easily to the case of an N-fold product.
  32. W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1957).
  33. M. C. Teich, P. R. Prucnal, unpublished.
  34. M. Woodroofe, Probability with Applications (McGraw-Hill, New York, 1975).
  35. R. J. Glauber, “Photon Counting and Field Correlations,” in Physics of Quantum Electronics, P. L. Kelley, B. Lax, P. E. Tannenwald, Eds. (McGraw-Hill, New York, 1966), pp. 788–811.
  36. P. Diament, M. C. Teich, J. Opt. Soc. Am. 60, 682 (1970).
    [CrossRef]
  37. P. R. Prucnal, M. C. Teich, J. Opt. Soc. Am., to be published.
  38. P. Diament, M. C. Teich, J. Opt. Soc. Am. 60, 1489 (1970).
    [CrossRef]
  39. P. Diament, M. C. Teich, Appl. Opt. 10, 1664 (1971).
    [CrossRef] [PubMed]
  40. B. I. Cantor, M. C. Teich, J. Opt. Soc. Am. 65, 786 (1975).
    [CrossRef]
  41. P. R. Prucnal, Proc. IEEE, to be published.

1979 (1)

P. R. Prucnal, M. C. Teich, IEEE Trans. Inf. Theory IT-25, (March, 1979).

1978 (1)

M. C. Teich, B. I. Cantor, IEEE J. Quantum Electron. QE-14 (December, 1978).

1977 (4)

S. D. Personick, Proc. IEEE 65, 1670 (1977).
[CrossRef]

S. D. Personick, P. Balaban, J. H. Bobsin, P. R. Kumar, IEEE Trans. Commun COM-25, 541 (1977).
[CrossRef]

M. C. Teich, P. R. Prucnal, G. Vannucci, Opt. Lett. 1, 208 (1977).
[CrossRef] [PubMed]

M. C. Teich, P. R. Prucnal, J. Opt. Soc. Am. 67, 1426 (1977).

1976 (1)

M. C. Teich, W. J. McGill, Phys. Rev. Lett. 36, 754 (1976).
[CrossRef]

1975 (1)

1974 (1)

E. V. Hoversten, D. L. Snyder, R. O. Harger, K. Kurimoto, IEEE Trans. Commun. COM-22, 17 (1974).
[CrossRef]

1973 (2)

1972 (2)

M. C. Teich, R. Y. Yen, IEEE Trans. Aerosp. Electron. Syst. AES-8, 13 (1972).
[CrossRef]

R. Y. Yen, P. Diament, M. C. Teich, IEEE Trans. Inf. Theory IT-18, 302 (1972).
[CrossRef]

1971 (2)

S. D. Personick, Bell Syst. Tech. J. 50, 3075 (1971).

P. Diament, M. C. Teich, Appl. Opt. 10, 1664 (1971).
[CrossRef] [PubMed]

1970 (3)

1967 (1)

W. J. McGill, J. Math. Psychol. 4, 351 (1967).
[CrossRef]

1965 (1)

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

1964 (2)

G. W. Flint, IEEE Trans. Mil. Electron. MIL-8, 22 (1964).
[CrossRef]

C. W. Helstrom, IEEE Trans. Inf. Theory IT-10, 275 (1964).
[CrossRef]

1963 (1)

B. Reiffen, H. Sherman, Proc. IEEE 51, 1316 (1963).
[CrossRef]

1956 (1)

1954 (2)

W. W. Peterson, T. G. Birdsall, W. C. Fox, Trans. IRE Prof. Group Inf. Theory PGIT-4, 171 (1954).
[CrossRef]

W. P. Tanner, J. A. Swets, Psychol. Rev. 61, 401 (1954).
[CrossRef] [PubMed]

Balaban, P.

S. D. Personick, P. Balaban, J. H. Bobsin, P. R. Kumar, IEEE Trans. Commun COM-25, 541 (1977).
[CrossRef]

Barlow, H. B.

Birdsall, T. G.

W. W. Peterson, T. G. Birdsall, W. C. Fox, Trans. IRE Prof. Group Inf. Theory PGIT-4, 171 (1954).
[CrossRef]

Bobsin, J. H.

S. D. Personick, P. Balaban, J. H. Bobsin, P. R. Kumar, IEEE Trans. Commun COM-25, 541 (1977).
[CrossRef]

Cantor, B. I.

M. C. Teich, B. I. Cantor, IEEE J. Quantum Electron. QE-14 (December, 1978).

B. I. Cantor, M. C. Teich, J. Opt. Soc. Am. 65, 786 (1975).
[CrossRef]

Diament, P.

Feller, W.

W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1957).

Flint, G. W.

G. W. Flint, IEEE Trans. Mil. Electron. MIL-8, 22 (1964).
[CrossRef]

Fox, W. C.

W. W. Peterson, T. G. Birdsall, W. C. Fox, Trans. IRE Prof. Group Inf. Theory PGIT-4, 171 (1954).
[CrossRef]

Gagliardi, R. M.

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

Glauber, R. J.

R. J. Glauber, “Photon Counting and Field Correlations,” in Physics of Quantum Electronics, P. L. Kelley, B. Lax, P. E. Tannenwald, Eds. (McGraw-Hill, New York, 1966), pp. 788–811.

Goodman, J. W.

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

Harger, R. O.

E. V. Hoversten, D. L. Snyder, R. O. Harger, K. Kurimoto, IEEE Trans. Commun. COM-22, 17 (1974).
[CrossRef]

Helstrom, C. W.

C. W. Helstrom, IEEE Trans. Inf. Theory IT-10, 275 (1964).
[CrossRef]

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976).

Hoversten, E.

E. Hoversten, “Optical Communication Theory,” in Laser Handbook, F. T. Arrechi, E. O. Schulz-DuBois, Eds. (North-Holland, Amsterdam, 1972), Vol. 2.

Hoversten, E. V.

E. V. Hoversten, D. L. Snyder, R. O. Harger, K. Kurimoto, IEEE Trans. Commun. COM-22, 17 (1974).
[CrossRef]

Karp, S.

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

Kumar, P. R.

S. D. Personick, P. Balaban, J. H. Bobsin, P. R. Kumar, IEEE Trans. Commun COM-25, 541 (1977).
[CrossRef]

Kurimoto, K.

E. V. Hoversten, D. L. Snyder, R. O. Harger, K. Kurimoto, IEEE Trans. Commun. COM-22, 17 (1974).
[CrossRef]

McGill, W. J.

M. C. Teich, W. J. McGill, Phys. Rev. Lett. 36, 754 (1976).
[CrossRef]

W. J. McGill, J. Math. Psychol. 4, 351 (1967).
[CrossRef]

Murdoch, D. C.

D. C. Murdoch, Linear Algebra (Wiley, New York, 1970), p. 165.

Personick, S. D.

S. D. Personick, Proc. IEEE 65, 1670 (1977).
[CrossRef]

S. D. Personick, P. Balaban, J. H. Bobsin, P. R. Kumar, IEEE Trans. Commun COM-25, 541 (1977).
[CrossRef]

S. D. Personick, Bell Syst. Tech. J. 50, 3075 (1971).

Peterson, W. W.

W. W. Peterson, T. G. Birdsall, W. C. Fox, Trans. IRE Prof. Group Inf. Theory PGIT-4, 171 (1954).
[CrossRef]

Pratt, W. K.

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

Prucnal, P. R.

P. R. Prucnal, M. C. Teich, IEEE Trans. Inf. Theory IT-25, (March, 1979).

M. C. Teich, P. R. Prucnal, J. Opt. Soc. Am. 67, 1426 (1977).

M. C. Teich, P. R. Prucnal, G. Vannucci, Opt. Lett. 1, 208 (1977).
[CrossRef] [PubMed]

P. R. Prucnal, Proc. IEEE, to be published.

P. R. Prucnal, M. C. Teich, J. Opt. Soc. Am., to be published.

M. C. Teich, P. R. Prucnal, unpublished.

Reiffen, B.

B. Reiffen, H. Sherman, Proc. IEEE 51, 1316 (1963).
[CrossRef]

Rosenberg, S.

Sherman, H.

B. Reiffen, H. Sherman, Proc. IEEE 51, 1316 (1963).
[CrossRef]

Snyder, D. L.

E. V. Hoversten, D. L. Snyder, R. O. Harger, K. Kurimoto, IEEE Trans. Commun. COM-22, 17 (1974).
[CrossRef]

D. L. Snyder, Random Point Processes (Wiley, New York, 1975).

Swets, J. A.

W. P. Tanner, J. A. Swets, Psychol. Rev. 61, 401 (1954).
[CrossRef] [PubMed]

Tanner, W. P.

W. P. Tanner, J. A. Swets, Psychol. Rev. 61, 401 (1954).
[CrossRef] [PubMed]

Teich, M. C.

P. R. Prucnal, M. C. Teich, IEEE Trans. Inf. Theory IT-25, (March, 1979).

M. C. Teich, B. I. Cantor, IEEE J. Quantum Electron. QE-14 (December, 1978).

M. C. Teich, P. R. Prucnal, J. Opt. Soc. Am. 67, 1426 (1977).

M. C. Teich, P. R. Prucnal, G. Vannucci, Opt. Lett. 1, 208 (1977).
[CrossRef] [PubMed]

M. C. Teich, W. J. McGill, Phys. Rev. Lett. 36, 754 (1976).
[CrossRef]

B. I. Cantor, M. C. Teich, J. Opt. Soc. Am. 65, 786 (1975).
[CrossRef]

S. Rosenberg, M. C. Teich, Appl. Opt. 12, 2625 (1973).
[CrossRef] [PubMed]

M. C. Teich, S. Rosenberg, Appl. Opt. 12, 2616 (1973).
[CrossRef] [PubMed]

R. Y. Yen, P. Diament, M. C. Teich, IEEE Trans. Inf. Theory IT-18, 302 (1972).
[CrossRef]

M. C. Teich, R. Y. Yen, IEEE Trans. Aerosp. Electron. Syst. AES-8, 13 (1972).
[CrossRef]

P. Diament, M. C. Teich, Appl. Opt. 10, 1664 (1971).
[CrossRef] [PubMed]

P. Diament, M. C. Teich, J. Opt. Soc. Am. 60, 682 (1970).
[CrossRef]

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

P. R. Prucnal, M. C. Teich, J. Opt. Soc. Am., to be published.

M. C. Teich, P. R. Prucnal, unpublished.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part 1 (Wiley, New York, 1968).

Vannucci, G.

Woodroofe, M.

M. Woodroofe, Probability with Applications (McGraw-Hill, New York, 1975).

Yen, R. Y.

M. C. Teich, R. Y. Yen, IEEE Trans. Aerosp. Electron. Syst. AES-8, 13 (1972).
[CrossRef]

R. Y. Yen, P. Diament, M. C. Teich, IEEE Trans. Inf. Theory IT-18, 302 (1972).
[CrossRef]

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

S. D. Personick, Bell Syst. Tech. J. 50, 3075 (1971).

IEEE J. Quantum Electron. (1)

M. C. Teich, B. I. Cantor, IEEE J. Quantum Electron. QE-14 (December, 1978).

IEEE Trans. Aerosp. Electron. Syst. (1)

M. C. Teich, R. Y. Yen, IEEE Trans. Aerosp. Electron. Syst. AES-8, 13 (1972).
[CrossRef]

IEEE Trans. Commun (1)

S. D. Personick, P. Balaban, J. H. Bobsin, P. R. Kumar, IEEE Trans. Commun COM-25, 541 (1977).
[CrossRef]

IEEE Trans. Commun. (1)

E. V. Hoversten, D. L. Snyder, R. O. Harger, K. Kurimoto, IEEE Trans. Commun. COM-22, 17 (1974).
[CrossRef]

IEEE Trans. Inf. Theory (3)

R. Y. Yen, P. Diament, M. C. Teich, IEEE Trans. Inf. Theory IT-18, 302 (1972).
[CrossRef]

C. W. Helstrom, IEEE Trans. Inf. Theory IT-10, 275 (1964).
[CrossRef]

P. R. Prucnal, M. C. Teich, IEEE Trans. Inf. Theory IT-25, (March, 1979).

IEEE Trans. Mil. Electron. (1)

G. W. Flint, IEEE Trans. Mil. Electron. MIL-8, 22 (1964).
[CrossRef]

J. Math. Psychol. (1)

W. J. McGill, J. Math. Psychol. 4, 351 (1967).
[CrossRef]

J. Opt. Soc. Am. (5)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

M. C. Teich, W. J. McGill, Phys. Rev. Lett. 36, 754 (1976).
[CrossRef]

Proc. IEEE (4)

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

S. D. Personick, Proc. IEEE 65, 1670 (1977).
[CrossRef]

B. Reiffen, H. Sherman, Proc. IEEE 51, 1316 (1963).
[CrossRef]

Proc. IEEE (Special Issue on Optical Communications), 58 (1970).

Psychol. Rev. (1)

W. P. Tanner, J. A. Swets, Psychol. Rev. 61, 401 (1954).
[CrossRef] [PubMed]

Trans. IRE Prof. Group Inf. Theory (1)

W. W. Peterson, T. G. Birdsall, W. C. Fox, Trans. IRE Prof. Group Inf. Theory PGIT-4, 171 (1954).
[CrossRef]

Other (15)

The finite difference is Δ[f(n)]k ≡ f(n + k) − f(n). If no subscript is used, k is assumed to be unity. We employ the second difference Δ2[f(n)]k,j ≡ [f(n + j) − f(n)] − [f(n + j − k) − f(n − k)] with j = 1 to obtain Δ2[f(n)]k.

D. C. Murdoch, Linear Algebra (Wiley, New York, 1970), p. 165.

The product rule can be seen fromΔ[f(n)g(n)h(n)]=f(n+1)g(n+1)h(n+1)−f(n+1)g(n+1)h(n)+f(n+1)g(n+1)h(n)−f(n+1)g(n)h(n)+f(n+1)g(n)h(n)−f(n)g(n)h(n)=f(n+1)g(n+1)Δ[h(n)]+f(n+1)Δ[g(n)]h(n)+Δ[f(n)]g(n)h(n),which extends easily to the case of an N-fold product.

W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1957).

M. C. Teich, P. R. Prucnal, unpublished.

M. Woodroofe, Probability with Applications (McGraw-Hill, New York, 1975).

R. J. Glauber, “Photon Counting and Field Correlations,” in Physics of Quantum Electronics, P. L. Kelley, B. Lax, P. E. Tannenwald, Eds. (McGraw-Hill, New York, 1966), pp. 788–811.

E. Hoversten, “Optical Communication Theory,” in Laser Handbook, F. T. Arrechi, E. O. Schulz-DuBois, Eds. (North-Holland, Amsterdam, 1972), Vol. 2.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part 1 (Wiley, New York, 1968).

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

D. L. Snyder, Random Point Processes (Wiley, New York, 1975).

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

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976).

P. R. Prucnal, M. C. Teich, J. Opt. Soc. Am., to be published.

P. R. Prucnal, Proc. IEEE, to be published.

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Figures (3)

Fig. 1
Fig. 1

The likelihood ratio Λ(n) vs the observations ni for the case N = 2. The solution Λ(n) = λ is represented by the multiple curved intersections of Λ(n) with the dotted plane. The decision regions R0 are cross-hatched and represent the coordinates (n1, n2) for which Λ(n) < λ. The decision regions R1 are unshaded and represent the coordinates (n1, n2) for which Λ(n) ≥ λ. This case exhibits multiple curved decision boundaries.

Fig. 2
Fig. 2

The transformed likelihood ratio Λ(m) for the case N = 2 where Λ(m) is monotonic with respect to m1. The solution Λ(m) = λ is represented by a single curved intersection of Λ(m) with the dotted plane. The region R0 is cross-hatched and represents the coordinates (m1, m2) for which Λ(m) < λ. The decision region R1 is unshaded and represents the coordinates (m1, m2) for which Λ(m) ≥ λ. This case exhibits a decision boundary λ″(m2) which is single valued, and therefore single-threshold processing.

Fig. 3
Fig. 3

A generalized photocounting optical communication system. The solid boxes represent components that can be included in our model, whereas the dashed boxes represent components that cannot be included in our model.

Equations (60)

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Λ ( n ) = p ( n | H 1 ) p ( n | H 0 ) H 1 < H 0 λ ,
n 1 H 1 < H 0 λ ,
m 1 = i = 1 N n i ,
m 1 H 1 < H 0 λ ( m 2 , , m N ) = λ ( n ) ,
i = 1 N n i H 1 < H 0 λ ,
p i ( n i | H 1 ) = k = u 0 u 1 p D i ( n i k ) p S i ( k )
p i ( n i | H 0 ) = p D i ( n i ) ,
Λ ( n ) = i = 1 N Λ i ( n i ) H 1 < H 0 λ ,
Λ i ( n i ) = p i ( n i | H 1 ) / p i ( n i | H 0 ) .
Λ i ( n i ) = [ k = u 0 u 1 p D i ( n i k ) p S i ( k ) ] / p D i ( n i ) .
Δ 2 [ log p D i ( d i ) ] k 0 k 0 , d i , i ,
Δ 2 [ log p D i ( d i ) ] k 0 k 0 , d i , i ,
i = 1 N n i H 1 < H 0 λ ( n ) ,
[ log p D i ( n i + 1 k ) log p D i ( n i + 1 ) ] [ log p D i ( n i k ) log p D i ( n i ) ] 0 k 0 , n i , i ,
p D i ( n i + 1 k ) p D i ( n i + 1 ) p D i ( n i k ) p D i ( n i ) 0 k 0 , n i , i .
Δ [ Λ i ( n i ) ] = k = u 0 u 1 p D i ( n i + 1 k ) p S i ( k ) p D i ( n i + 1 ) k = u 0 u 1 p D i ( n i k ) p S i ( k ) p D i ( n i ) ,
Δ [ Λ i ( n i ) ] = ( u 0 u 0 ) p D i ( n i u 0 ) p S i ( u 0 ) p D i ( n i ) + ( u 1 u 1 ) p D i ( n i + 1 u 1 ) p S i ( u 1 ) p D i ( n i + 1 ) + k = u 0 u 1 p S i ( k ) [ p D i ( n i + 1 k ) p D i ( n i + 1 ) p D i ( n i k ) p D i ( n i ) ] .
Δ [ Λ i ( n i ) ] 0 n i , i
Δ [ Λ i ( n i ) ] 0 n i , i
m = A n ,
A = [ N 1 / 2 N 1 / 2 N 1 / 2 e 21 e 22 e 2 N e N 1 e N 2 e N N ] .
m 1 = N 1 / 2 i = 1 N n i .
n ( m ) = A T m ,
n i ( m ) = k = 1 N e k i m k .
Δ n i [ f ( n ) ] f ( n 1 , , n i + 1 , , n N ) f ( n 1 , , n i , , n N ) .
Δ m j { Λ [ n ( m ) ] } = Δ m j { i = 1 N Λ i [ n i ( m ) ] } .
Δ m j { Λ [ n ( m ) ] } = { k = 1 N 1 Λ k [ n k ( m ) ] } Δ m j { Λ N [ n N ( m ) ] } + i = 2 N 1 ( { k = 1 i 1 Λ k [ n k ( m ) ] } Δ m j { Λ i [ n i ( m ) ] } { k = i + 1 N Λ k [ n k ( m ) + 1 ] } ) + Δ m j { Λ 1 [ n 1 ( m ) ] } { k = 2 N Λ k [ n k ( m ) + 1 ] } .
Δ m j [ n i ( m ) ] = Δ m j ( k = 1 N e k i m k ) ,
Δ m j [ n i ( m ) ] = e j i ( m j + 1 ) + k = 1 k j N e k i m k k = 1 N e k i m k .
Δ m j [ n i ( m ) ] = e j i .
Δ m j { Λ i [ n i ( m ) ] } = Δ n i [ Λ i ( n i ) ] Δ m j [ n i ( m ) ] .
Δ m j { Λ i [ n i ( m ) ] } = e j i Δ n i [ Λ i ( n i ) ] .
Δ m 1 { Λ i [ n i ( m ) ] } = N 1 / 2 Δ n i [ Λ i ( n i ) ] .
i = 1 N n i
Δ n i [ m 1 ( n i ) ] = Δ n i ( N 1 / 2 n i ) ,
Δ n i [ m 1 ( n i ) ] = N 1 / 2 .
Δ n i { Λ i [ n i ( m ) ] } = Δ m 1 { Λ i [ n i ( m ) ] } Δ n i [ m 1 ( n i ) ]
Δ N 1 / 2 m 1 [ Λ i ( n i ) ] = N 1 Δ [ Λ i ( n i ) ] .
Δ N 1 / 2 m 1 [ Λ i ( n i ) ] 0 n i , i .
Δ n i [ Λ ( n ) ] 0 n i .
Δ N 1 / 2 m 1 [ Λ i ( n i ) ] 0 n i , i .
Δ n i [ Λ ( n ) ] 0 n i .
i = 1 N n i H 1 < H 0 λ ( n )
p D i ( d i ) = u ( d i ) ( a 1 d i ) ( a a 1 r d i ) ( a r ) ,
Δ 2 [ ln p D i ( d i ) ] k = ln [ ( d i + 1 k ) ( a 1 d i ) ( r d i ) ( a a 1 r + d i + 1 k ) ( d i + 1 ) ( a 1 d i + k ) ( r d i + k ) ( a a 1 r + d i + 1 ) ] < 0 d i 0 ,
p D i ( d i ) = [ u ( d i ) u ( d i r ) ] ( p / q d i ) ( ( 1 p ) / q r d i ) ( 1 / q r ) ,
p D ( d i ) = u ( d i ) ( M + d i 1 d i ) ( M M + d i ) M ( d i M + d i ) d i ,
p D i ( d i ) = u ( d i ) ( 2 π δ ) 1 / 2 d i 3 / 2 d i × exp [ ( 2 δ d i ) 1 ( d i d i ) 2 ] δ 1 ,
Δ 2 [ ln p D i ( d i ) ] k = ln d i + 1 k d i + 1 + ln ( p / q ) + d i ( p / q ) + d i k + ln r d i r d i + k + ln [ ( 1 p ) / q ] + r d i 1 + k [ ( 1 p ) / q ] + r d i 1 .
1 q p q ,
Δ 2 [ ln p D i ( n i ) ] k 0 d i 0 d i r .
p D i ( d i ) = u ( d i ) ( 2 d i ) ! d i d i / [ 2 d i ( d i ! ) 2 ( 1 + 2 d i ) d i + 1 / 2 ]
Δ 2 [ ln p D i ( d i ) ] k = ln ( 2 d i + 1 + 2 k ) ( d i + 1 ) ( 2 d i + 1 ) ( d i + 1 + k ) > 0 d i .
P F = n i = λ p D i ( n i ) α ,
P F = m 1 = λ ( n ) p D i ( m ) α ,
n i ( m 1 , , m j 1 , a , m j + 1 , , m N ) n i ( a ) and n i ( m 1 , , m j 1 , a + 1 , m j + 1 , , m N ) n i ( a + 1 ) is u = n i ( a ) + ( m j a ) Δ m j [ n i ( a ) ] ,
υ = Λ [ n i ( a ) ] + [ n i n i ( a ) ] Δ n i { Λ [ n i ( a ) ] } ,
υ = Λ [ n i ( a ) ] + ( m j a ) Δ m j [ n i ( a ) ] Δ n i { Λ [ n i ( a ) ] } .
Δ m j { Λ [ n i ( m ) ] } = Δ n i { Λ [ n i ( m ) ] } Δ m j [ n i ( m ) ] .
Δ [ f ( n ) g ( n ) h ( n ) ] = f ( n + 1 ) g ( n + 1 ) h ( n + 1 ) f ( n + 1 ) g ( n + 1 ) h ( n ) + f ( n + 1 ) g ( n + 1 ) h ( n ) f ( n + 1 ) g ( n ) h ( n ) + f ( n + 1 ) g ( n ) h ( n ) f ( n ) g ( n ) h ( n ) = f ( n + 1 ) g ( n + 1 ) Δ [ h ( n ) ] + f ( n + 1 ) Δ [ g ( n ) ] h ( n ) + Δ [ f ( n ) ] g ( n ) h ( n ) ,

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