Abstract

In a fiber-optic link, the continuous semiclassical and discrete photon statistics models are widely used to characterize the output probability distribution of an optical amplifier. The semiclassical model describes the output as a chi-square distribution while the discrete photon statistics model describes it through the noncentral negative binomial (NNB) distribution. In this paper, we present a way of understanding the discrete photon statistical distribution at the output of the amplifier and the convergence of the same to its semiclassical counter part. We highlight this convergence through the relation between the NNB and chi-square distribution demonstrated through deviation analysis and simulations. The penalty incurred in bit error rate analysis due to semiclassical approximation over NNB is also discussed.

© 2011 Optical Society of America

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References

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  1. E. Desurvire, Erbium Doped Fiber Amplifiers: Principles and Applications (Wiley Interscience, 2002).
  2. E. Desurvire, Erbium Doped Fiber Amplifiers: Device and System Developments (Wiley Interscience, 2002).
  3. P. A. Humblet and M. Azizoglu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. 9, 1576–1582 (1991).
    [CrossRef]
  4. A. J. Weiss, “On the performance of electrical equalization in optical fiber transmission systems,” IEEE Photon. Technol. Lett. 15, 1225–1227 (2003).
    [CrossRef]
  5. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective, 2nd ed. (Elsevier, 2002).
  6. T. Li and M. C. Teich, “Performance of a lightwave system incorporating a cascade of erbium-doped fiber amplifiers,” Opt. Commun. 91, 41–45 (1992).
    [CrossRef]
  7. W. S. Wong, H. A. Haus, L. A. Jiang, P. B. Hansen, and M. Margalit, “Photon statistics of amplified spontaneous emission noise in 10 Gbit/s optically preamplified direct-detection receiver,” Opt. Lett. 23, 1832–1834 (1998).
    [CrossRef]
  8. G. Grimmett and D. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford University, 2001).
  9. T. Li and M. C. Teich, “Photon point process for travelling wave laser amplifiers,” IEEE J. Quantum Electron. 29, 2568–2578(1993).
    [CrossRef]
  10. T. Li and M. Teich, “Bit error rate for a lightwave communication system incorporating an erbium doped fiber amplifier,” Electron. Lett. 27, 598–599 (1991).
    [CrossRef]
  11. R. Noe, “Optical amplifier performance in digital optical communication systems,” Electr. Eng. 83, 15–20 (2001).
    [CrossRef]
  12. A. Mecozzi, “Quantum and semiclassical theory of noise in optical transmission lines employing in-line erbium amplifiers,” J. Opt. Soc. Am. B 17, 607–617 (2000).
    [CrossRef]
  13. M. C. Teich and P. Diament, “Multiply stochastic representations for K distributions and their Poisson transforms,” J. Opt. Soc. Am. 6, 80–91 (1989).
    [CrossRef]
  14. S. H. Ong, “Some notes on the non-central negative binomial distribution,” Metrika 34, 225–236 (1987).
    [CrossRef]
  15. S. H. Ong, “On a class of discrete distributions arising from the birth-death-with-immigration process,” Metrika 43, 221–235(1996).
    [CrossRef]
  16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).
  17. A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 5th ed. (Tata McGraw-Hill, 2002).

2003

A. J. Weiss, “On the performance of electrical equalization in optical fiber transmission systems,” IEEE Photon. Technol. Lett. 15, 1225–1227 (2003).
[CrossRef]

2001

R. Noe, “Optical amplifier performance in digital optical communication systems,” Electr. Eng. 83, 15–20 (2001).
[CrossRef]

2000

1998

1996

S. H. Ong, “On a class of discrete distributions arising from the birth-death-with-immigration process,” Metrika 43, 221–235(1996).
[CrossRef]

1993

T. Li and M. C. Teich, “Photon point process for travelling wave laser amplifiers,” IEEE J. Quantum Electron. 29, 2568–2578(1993).
[CrossRef]

1992

T. Li and M. C. Teich, “Performance of a lightwave system incorporating a cascade of erbium-doped fiber amplifiers,” Opt. Commun. 91, 41–45 (1992).
[CrossRef]

1991

T. Li and M. Teich, “Bit error rate for a lightwave communication system incorporating an erbium doped fiber amplifier,” Electron. Lett. 27, 598–599 (1991).
[CrossRef]

P. A. Humblet and M. Azizoglu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. 9, 1576–1582 (1991).
[CrossRef]

1989

M. C. Teich and P. Diament, “Multiply stochastic representations for K distributions and their Poisson transforms,” J. Opt. Soc. Am. 6, 80–91 (1989).
[CrossRef]

1987

S. H. Ong, “Some notes on the non-central negative binomial distribution,” Metrika 34, 225–236 (1987).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

Azizoglu, M.

P. A. Humblet and M. Azizoglu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. 9, 1576–1582 (1991).
[CrossRef]

Desurvire, E.

E. Desurvire, Erbium Doped Fiber Amplifiers: Principles and Applications (Wiley Interscience, 2002).

E. Desurvire, Erbium Doped Fiber Amplifiers: Device and System Developments (Wiley Interscience, 2002).

Diament, P.

M. C. Teich and P. Diament, “Multiply stochastic representations for K distributions and their Poisson transforms,” J. Opt. Soc. Am. 6, 80–91 (1989).
[CrossRef]

Grimmett, G.

G. Grimmett and D. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford University, 2001).

Hansen, P. B.

Haus, H. A.

Humblet, P. A.

P. A. Humblet and M. Azizoglu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. 9, 1576–1582 (1991).
[CrossRef]

Jiang, L. A.

Li, T.

T. Li and M. C. Teich, “Photon point process for travelling wave laser amplifiers,” IEEE J. Quantum Electron. 29, 2568–2578(1993).
[CrossRef]

T. Li and M. C. Teich, “Performance of a lightwave system incorporating a cascade of erbium-doped fiber amplifiers,” Opt. Commun. 91, 41–45 (1992).
[CrossRef]

T. Li and M. Teich, “Bit error rate for a lightwave communication system incorporating an erbium doped fiber amplifier,” Electron. Lett. 27, 598–599 (1991).
[CrossRef]

Margalit, M.

Mecozzi, A.

Noe, R.

R. Noe, “Optical amplifier performance in digital optical communication systems,” Electr. Eng. 83, 15–20 (2001).
[CrossRef]

Ong, S. H.

S. H. Ong, “On a class of discrete distributions arising from the birth-death-with-immigration process,” Metrika 43, 221–235(1996).
[CrossRef]

S. H. Ong, “Some notes on the non-central negative binomial distribution,” Metrika 34, 225–236 (1987).
[CrossRef]

Papoulis, A.

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 5th ed. (Tata McGraw-Hill, 2002).

Pillai, S. U.

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 5th ed. (Tata McGraw-Hill, 2002).

Ramaswami, R.

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective, 2nd ed. (Elsevier, 2002).

Sivarajan, K. N.

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective, 2nd ed. (Elsevier, 2002).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

Stirzaker, D.

G. Grimmett and D. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford University, 2001).

Teich, M.

T. Li and M. Teich, “Bit error rate for a lightwave communication system incorporating an erbium doped fiber amplifier,” Electron. Lett. 27, 598–599 (1991).
[CrossRef]

Teich, M. C.

T. Li and M. C. Teich, “Photon point process for travelling wave laser amplifiers,” IEEE J. Quantum Electron. 29, 2568–2578(1993).
[CrossRef]

T. Li and M. C. Teich, “Performance of a lightwave system incorporating a cascade of erbium-doped fiber amplifiers,” Opt. Commun. 91, 41–45 (1992).
[CrossRef]

M. C. Teich and P. Diament, “Multiply stochastic representations for K distributions and their Poisson transforms,” J. Opt. Soc. Am. 6, 80–91 (1989).
[CrossRef]

Weiss, A. J.

A. J. Weiss, “On the performance of electrical equalization in optical fiber transmission systems,” IEEE Photon. Technol. Lett. 15, 1225–1227 (2003).
[CrossRef]

Wong, W. S.

Electr. Eng.

R. Noe, “Optical amplifier performance in digital optical communication systems,” Electr. Eng. 83, 15–20 (2001).
[CrossRef]

Electron. Lett.

T. Li and M. Teich, “Bit error rate for a lightwave communication system incorporating an erbium doped fiber amplifier,” Electron. Lett. 27, 598–599 (1991).
[CrossRef]

IEEE J. Quantum Electron.

T. Li and M. C. Teich, “Photon point process for travelling wave laser amplifiers,” IEEE J. Quantum Electron. 29, 2568–2578(1993).
[CrossRef]

IEEE Photon. Technol. Lett.

A. J. Weiss, “On the performance of electrical equalization in optical fiber transmission systems,” IEEE Photon. Technol. Lett. 15, 1225–1227 (2003).
[CrossRef]

J. Lightwave Technol.

P. A. Humblet and M. Azizoglu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. 9, 1576–1582 (1991).
[CrossRef]

J. Opt. Soc. Am.

M. C. Teich and P. Diament, “Multiply stochastic representations for K distributions and their Poisson transforms,” J. Opt. Soc. Am. 6, 80–91 (1989).
[CrossRef]

J. Opt. Soc. Am. B

Metrika

S. H. Ong, “Some notes on the non-central negative binomial distribution,” Metrika 34, 225–236 (1987).
[CrossRef]

S. H. Ong, “On a class of discrete distributions arising from the birth-death-with-immigration process,” Metrika 43, 221–235(1996).
[CrossRef]

Opt. Commun.

T. Li and M. C. Teich, “Performance of a lightwave system incorporating a cascade of erbium-doped fiber amplifiers,” Opt. Commun. 91, 41–45 (1992).
[CrossRef]

Opt. Lett.

Other

G. Grimmett and D. Stirzaker, Probability and Random Processes, 3rd ed. (Oxford University, 2001).

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective, 2nd ed. (Elsevier, 2002).

E. Desurvire, Erbium Doped Fiber Amplifiers: Principles and Applications (Wiley Interscience, 2002).

E. Desurvire, Erbium Doped Fiber Amplifiers: Device and System Developments (Wiley Interscience, 2002).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972).

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 5th ed. (Tata McGraw-Hill, 2002).

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Figures (7)

Fig. 1
Fig. 1

Noncentral negative binomial distribution for G = 101 , λ o = 80 , and Poisson distribution with mean ( G λ o = 8080 ) at the output of the amplifier.

Fig. 2
Fig. 2

Noncentral negative binomial distribution at the output of the amplifier of gain G = 101 , M = 8 for a coherent input ( λ o = 80 ) with its components amplified signal (S) distribution and noise (N) distribution.

Fig. 3
Fig. 3

Noncentral chi-square distribution at the output of the amplifier with gain ( G = 101 , M = 8 ) for a coherent input ( λ o = 80 ) and gamma distribution as a windowing function for i = 1000 , 4000.

Fig. 4
Fig. 4

Probability distribution of “0” transmission (noise) and “1” transmission ( signal + noise ) for an amplifier with gain G = 11 , M = 2 , 16 and coherent input with λ o = 80 .

Fig. 5
Fig. 5

Probability distribution of “0” transmission (noise) and “1” transmission ( signal + noise ) for an amplifier with gain G = 101 , M = 2 , 8 and coherent input with λ o = 80 .

Fig. 6
Fig. 6

Normalized deviation ( | Δ f ( i / i 0 ) | max ( | Δ f ( i ) | ) ) between chi-squared and NNB distribution for different values of gain ( G = 101 , 11, 4) when a “0” is transmitted.

Fig. 7
Fig. 7

Normalized deviation ( Δ f ( i / i 0 ) max ( Δ f ( i ) ) ) between chi-squared and NNB distribution for different values of gain ( G = 101 , 11, 4) when a “0” is transmitted.

Tables (6)

Tables Icon

Table 1 Analysis of Deviation between the NNB and Chi-Square Distributions for the Case of “0” Transmission ( λ o = 0 , M = 2 ) Using Closed-Form Theoretical Expressions for the Deviation at the Peak Location i 0 and Through Simulation

Tables Icon

Table 2 Analysis of Deviation between the NNB and Chi-Square Distributions for the Case of “0” Transmission ( λ o = 0 , M = 8 ) Using Closed-Form Theoretical Expressions for the Deviation at the Peak Location i 0 and through Simulation

Tables Icon

Table 3 Analysis of Deviation between the NNB and Chi-Square Distributions for the case of “1” Transmission ( λ o = 80 , M = 2 ) Using Closed-Form Theoretical Expressions for the Deviation at the Peak Location i 0 and Numerically

Tables Icon

Table 4 Error Analysis for the Case of “1” transmission ( λ o = 80 , M = 8 ) Using Closed-Form Theoretical Expressions for the Deviation at the Peak Location i 0 and Through Simulation

Tables Icon

Table 5 Crossover Location ( i th ) of the Semiclassical Distribution for Different Values of Gain with M = 8 and the Deviation ( i i th ) of the New Threshold Location from the NNB

Tables Icon

Table 6 Number of Excess Photons Required in the Quantum Model to Achieve the Same BER as the Semiclassical Model for Different Amplifier Gains

Equations (66)

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G Noise ( z ) = [ 1 1 n ASE ( z 1 ) ] M .
P 0 ( n ) = ( n + M 1 n ) ( n ASE ) n ( 1 + n ASE ) n + M .
G ASc ( z ) = [ 1 1 n ASE ( z 1 ) ] M exp ( n out ( z 1 ) 1 n ASE ( z 1 ) ) .
P 1 ( n ) = ( n ASE ) n ( 1 + n ASE ) n + M exp ( n out 1 + n ASE ) L n M 1 ( n out n ASE ( 1 + n ASE ) ) ,
g V ( ω ) = E [ e i ω V ] ,
= ( 1 1 i n ASE ω ) M exp ( i n out ω 1 i n ASE ω ) .
f V , 1 ( v ) = 1 n ASE exp ( n out n ASE v n ASE ) ( v n ASE n ASE n out ) ( M 1 ) 2 I M 1 ( 2 v n out n ASE ) .
f V , 0 ( v ) = 1 n ASE ( M 1 ) ! ( v n ASE ) M 1 exp ( v n ASE ) .
Pr ( Y = i ) = 0 v i e v i ! f V ( v ) d v = 0 w i ( v ) f ( v ) d v .
var ( S + N ) var ( w ( λ o + M ) G ( v ) ) ( 2 λ o + M ) G ( G 1 ) ( λ o + M ) G 2 ( G 1 ) 1.
| Δ f ( i ) | = | f Y ( i ) f V ( i ) | .
f Y ( i ) = 0 w i ( v ) f V ( v ) d v ,
f V ( v ) = f V ( i ) + f V ( i ) ( v i ) + f V ( i ) 2 ! ( v i ) 2 + .
| Δ f ( i ) | = | 0 w i ( v ) [ f V ( i ) + f V ( i ) ( v i ) + f V ( i ) 2 ! ( v i ) 2 + ] d v f V ( i ) | .
0 w i ( v ) f V ( i ) d v = f V ( i ) 0 w i ( v ) d v = f V ( i ) ,
f V ( i ) 0 w i ( v ) ( v i ) d v = f V ( i ) [ ( i + 1 ) 0 v i + 1 e v d v ( i + 1 ) ! i 0 v i e v d v ( i ) ! ] ,
= f V ( i ) [ ( i + 1 ) i ] ,
= f V ( i ) .
| Δ f ( i ) | = | f V ( i ) + f V ( i ) 0 w i ( v ) ( v i ) 2 2 d v | ,
= | f V ( i ) + f V ( i ) ( i 2 ) | .
| Δ f ( i ) | = | [ U + ( U + U 2 ) ( i 2 ) ] f V , 0 ( i ) | ,
i 0 ( theory ) = ( M 1 ) n ASE .
| Δ f ( i 0 ) | = | ( 1 2 n ASE ) f V , 0 ( ( M 1 ) n ASE ) | ,
% Δ f = Δ f ( i 0 ) f V , 0 ( i 0 ) × 100.
| Δ f ( i ) | = | f V , 1 ( i ) + f V , 1 ( i ) ( i 2 ) | ,
| Δ f ( i ) | = | [ U + ( U + U 2 ) ( i 2 ) ] f V , 1 ( i ) | ,
i 0 ( theory ) = ceil [ n out + 4 ( M 1 2 1 4 ) n ASE + 2 ( M 1 2 1 4 ) n out n ASE 4 ] .
| Δ f ( i 0 ) | = | [ U + ( U + U 2 ) ( i 0 2 ) ] f V , 0 ( i ) | ,
% Δ f = | Δ f ( i 0 ) | f V , 1 ( i 0 ) × 100.
Δ f Theory , 0 / 1 ( i ) = f Y , 0 / 1 ( i ) f V , 0 / 1 ( i ) .
Δ f Theory , 0 ( i ) = f Y , 0 ( i ) f V , 0 ( i ) ,
= g 0 ( i ) f V , 0 ( i ) ,
Δ f Theory , 1 ( i ) = g 1 ( i ) f V , 1 ( i ) ,
f Y , 0 ( i ) = ( g 0 ( i ) + 1 ) f V , 0 ( i ) ,
f Y , 1 ( i ) = ( g 1 ( i ) + 1 ) f V , 1 ( i ) .
f Y , 0 ( i ) = ( g 0 ( i th ) + 1 ) f V , 0 ( i th ) + ( i i th ) [ ( g 0 ( i th ) + 1 ) f V , 0 ( i th ) ] + ,
f Y , 1 ( i ) = ( g 1 ( i th ) + 1 ) f V , 1 ( i th ) + ( i i th ) [ ( g 1 ( i th ) + 1 ) f V , 1 ( i th ) ] + ,
0 = f V , 0 ( i th ) ( ( g 0 ( i th ) + 1 ) ( g 1 ( i th ) + 1 ) ) + ( i i th ) { [ ( g 0 ( i th ) + 1 ) f V , 0 ( i th ) ] [ ( g 1 ( i th ) + 1 ) f V , 1 ( i th ) ] } .
( i i th ) = f V , 0 ( i th ) ( g 0 ( i th ) g 1 ( i th ) ) { [ ( g 1 ( i th ) + 1 ) f V , 1 ( i th ) ] [ ( g 0 ( i th ) + 1 ) f V , 0 ( i th ) ] } ,
( g 0 ( i th ) g 1 ( i th ) ) { [ ( g 1 ( i th ) + 1 ) ( ( M 1 ) 2 1 i th 1 n ASE + n out n ASE 1 i th 1 4 i th ) ] [ ( g 0 ( i th ) + 1 ) ( ( M 1 ) i th 1 n ASE ) ] } .
BER ( semiclassical ) = i th f V , 0 ( v ) d v + 0 i th f V , 1 ( v ) d v ,
BER ( semiclassical ) = 1 2 [ [ 1 γ ( M , i th n ASE ) ] + k = 0 exp ( n out n ASE ) ( n out n ASE ) k k ! γ ( k + M , i th n ASE ) ] ,
BER ( quantum ) = 0 i th P 1 ( n ) + i th P 0 ( n ) .
Excess Penalty = 10 log 10 ( Additional no. of photons required in NNB model λ o ) .
var ( S + N ) var ( w ( λ o + M ) G ( v ) ) ( 2 λ o + M ) G ( G 1 ) ( λ o + M ) G = ( G 1 ) ( 2 M λ 0 + M ) .
f U k ( u ) = u k e u Γ ( k ) .
f V ( v ) = a ( M + 1 ) / 2 ( v / λ ) ( M 1 ) / 2 e ( λ + a v ) I M 1 ( 2 λ a v ) ,
μ 1 Γ ( k ) u k 1 e u d u = x = 0 k 1 μ x e μ / x ! k = 1 , 2 , .
Pr ( U k > v ) = i = 0 k 1 v i e v i ! , Pr ( U k > V ) = 0 Pr ( U k > V | V = v ) f V ( v ) d v = 0 Pr ( U k > v ) f V ( v ) d v = 0 ( i = 0 k 1 v i e v i ! ) f V ( v ) d v = i = 0 k 1 ( 0 v i e v i ! f V ( v ) d v ) = i = 0 k 1 Pr ( Y = i ) , Pr ( U k > V ) = Pr ( Y < k ) .
Pr ( Y = i ) = ( 0 v i e v i ! f V ( v ) d v ) .
f V ( v ) = 1 n ASE exp ( n out n ASE v n ASE ) ( v n ASE n ASE n out ) ( M 1 ) 2 I M 1 ( 2 v n out n ASE ) .
P ( Y = i ) = ( n ASE ) i ( 1 + n ASE ) i + M exp ( n out 1 + n ASE ) L i M 1 [ n out n ASE ( 1 + n ASE ) ] .
f X ( x ) = 1 n ASE exp ( n out n ASE x n ASE ) ( x n ASE n ASE n out ) ( M 1 ) 2 I M 1 ( 2 x n out n ASE ) ,
I N ( z ) = exp ( z ) 2 π z ( 1 + ( 1 2 N ) ( 1 + 2 N ) 8 z + ) exp ( z ) 2 π z .
f X ( x ) = 1 n ASE exp ( n out n ASE ) ( 1 n out ) ( M 1 ) 2 2 π 2 n out n ASE exp ( x n ASE + 2 x n out n ASE ) x ( M 1 2 1 4 ) .
x n out x ( M 1 2 1 4 ) n ASE = 0.
x = n out + 4 ( M 1 2 1 4 ) n ASE + 2 ( M 1 2 1 4 ) n out n ASE 4 ,
i o ( theory ) = ceil [ n out + 4 ( M 1 2 1 4 ) n ASE + 2 ( M 1 2 1 4 ) n out n ASE 4 ] .
BER = 1 2 [ i th f V , 0 ( v ) d v + 0 i th f V , 1 ( v ) d v ] .
i th f V , 0 ( v ) d v = [ 1 0 i th 1 n ASE ( M 1 ) ! ( v n ASE ) M 1 exp ( v n ASE ) d v ] = [ 1 γ ( M , i th n ASE ) ] ,
f V , 1 ( v ) = 1 n ASE exp ( n out n ASE v n ASE ) ( v n ASE n ASE n out ) ( M 1 ) 2 × [ ( v n out n ASE ) M 1 k = 0 ( v n ASE n out n ASE ) k k ! Γ ( M 1 + k + 1 ) ] .
f V , 1 ( v ) = k = 0 exp ( n out n ASE ) ( n out n ASE ) k k ! f V , 0 ( k + M 1 , v ) ,
f V , 0 ( k + M 1 , v ) = [ 1 n ASE ( k + M 1 ) ! ( v n ASE ) ( k + M 1 ) exp ( v n ASE ) ] .
0 i th f V , 1 ( v ) = k = 0 exp ( n out n ASE ) ( n out n ASE ) k k ! 0 i th f V , 0 ( k + M 1 , v ) ,
= k = 0 exp ( n out n ASE ) ( n out n ASE ) k k ! γ ( k + M , i th n ASE ) .
BER = 1 2 [ [ 1 γ ( M , i th n ASE ) ] + k = 0 exp ( n out n ASE ) ( n out n ASE ) k k ! γ ( k + M , i th n ASE ) ] .

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