Abstract

The spectral efficiency (channel capacity) of the optical direct-detection channel is studied. The modeling of the optical direct-detection channel as a discrete-time Poisson channel is reviewed. Closed-form integral representations for the entropy of random variables with Poisson and negative binomial distributions are derived. The spectral efficiency achievable with an arbitrary input gamma density is expressed in closed integral form. Simple, nonasymptotic upper and lower bounds to the channel capacity are computed. Numerical results are presented and compared with previous bounds and approximations.

© 2007 Optical Society of America

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References

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  1. R. G. Gallager, Information Theory and Reliable Communication (Wiley, 1968).
  2. C. M. Caves and P. D. Drummond, "Quantum limits on bosonic communication rates," Rev. Mod. Phys. 66, 481-537 (1994).
    [CrossRef]
  3. J. M. Kahn and K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 259-272 (2004).
    [CrossRef]
  4. J. P. Gordon, "Quantum effects in communication systems," Proc. IRE 50, 1898-1908 (1962).
    [CrossRef]
  5. S. Shamai (Shitz), "Capacity of a pulse amplitude modulated direct detection photon channel," IEE Proc., Part I: Solid-State Electron Devices 137, 424-430 (1990).
  6. D. Brady and S. Verdú, "The asymptotic capacity of the direct detection photon channel with a bandwidth constraint," in Proceedings of the 28th Annual Allerton Conference on Communication, Control, and Computing (Allerton, 1990), pp. 691-700.
  7. A. Lapidoth and S. M. Moser, "Bounds on the capacity of the discrete-time Poisson channel," in Proceedings of the 41st Allerton Conference on Communication, Control, and Computing, (Allerton, 2003), pp. 201-210.
  8. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  9. A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, 1995).
  10. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2000).
  11. M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, 1974).
  12. G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge U. Press, 1999).
  13. W. Feller, An Introduction to Probability Theory and Its Applications, 2nd ed. (Wiley, 1971), Vol. 2.
  14. I. Csiszár and J. Körner, Information Theory: Coding Theorems for Discrete Memoryless Systems (Academic, 1981).
  15. R. E. Blahut, Principles and Practice of Information Theory (Addison-Wesley, 1987).
  16. I. C. Abou-Faycal, M. D. Trott, and S. Shamai (Shitz), "The capacity of discrete-time memoryless Rayleigh-fading channels," IEEE Trans. Inf. Theory 47, 1290-1301 (2001).
    [CrossRef]
  17. I.S.Gradshteyn, I.M.Ryzhik, and A.Jeffrey (eds.), Tables of Integrals, Series, and Products (Academic, 1994).

2004

J. M. Kahn and K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 259-272 (2004).
[CrossRef]

2001

I. C. Abou-Faycal, M. D. Trott, and S. Shamai (Shitz), "The capacity of discrete-time memoryless Rayleigh-fading channels," IEEE Trans. Inf. Theory 47, 1290-1301 (2001).
[CrossRef]

1994

C. M. Caves and P. D. Drummond, "Quantum limits on bosonic communication rates," Rev. Mod. Phys. 66, 481-537 (1994).
[CrossRef]

1990

S. Shamai (Shitz), "Capacity of a pulse amplitude modulated direct detection photon channel," IEE Proc., Part I: Solid-State Electron Devices 137, 424-430 (1990).

1962

J. P. Gordon, "Quantum effects in communication systems," Proc. IRE 50, 1898-1908 (1962).
[CrossRef]

Abou-Faycal, I. C.

I. C. Abou-Faycal, M. D. Trott, and S. Shamai (Shitz), "The capacity of discrete-time memoryless Rayleigh-fading channels," IEEE Trans. Inf. Theory 47, 1290-1301 (2001).
[CrossRef]

Andrews, G. E.

G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge U. Press, 1999).

Askey, R.

G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge U. Press, 1999).

Blahut, R. E.

R. E. Blahut, Principles and Practice of Information Theory (Addison-Wesley, 1987).

Brady, D.

D. Brady and S. Verdú, "The asymptotic capacity of the direct detection photon channel with a bandwidth constraint," in Proceedings of the 28th Annual Allerton Conference on Communication, Control, and Computing (Allerton, 1990), pp. 691-700.

Caves, C. M.

C. M. Caves and P. D. Drummond, "Quantum limits on bosonic communication rates," Rev. Mod. Phys. 66, 481-537 (1994).
[CrossRef]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2000).

Csiszár, I.

I. Csiszár and J. Körner, Information Theory: Coding Theorems for Discrete Memoryless Systems (Academic, 1981).

Drummond, P. D.

C. M. Caves and P. D. Drummond, "Quantum limits on bosonic communication rates," Rev. Mod. Phys. 66, 481-537 (1994).
[CrossRef]

Feller, W.

W. Feller, An Introduction to Probability Theory and Its Applications, 2nd ed. (Wiley, 1971), Vol. 2.

Gallager, R. G.

R. G. Gallager, Information Theory and Reliable Communication (Wiley, 1968).

Gordon, J. P.

J. P. Gordon, "Quantum effects in communication systems," Proc. IRE 50, 1898-1908 (1962).
[CrossRef]

Ho, K.-P.

J. M. Kahn and K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 259-272 (2004).
[CrossRef]

Kahn, J. M.

J. M. Kahn and K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 259-272 (2004).
[CrossRef]

Körner, J.

I. Csiszár and J. Körner, Information Theory: Coding Theorems for Discrete Memoryless Systems (Academic, 1981).

Lamb, W. E.

M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, 1974).

Lapidoth, A.

A. Lapidoth and S. M. Moser, "Bounds on the capacity of the discrete-time Poisson channel," in Proceedings of the 41st Allerton Conference on Communication, Control, and Computing, (Allerton, 2003), pp. 201-210.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Moser, S. M.

A. Lapidoth and S. M. Moser, "Bounds on the capacity of the discrete-time Poisson channel," in Proceedings of the 41st Allerton Conference on Communication, Control, and Computing, (Allerton, 2003), pp. 201-210.

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2000).

Peres, A.

A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, 1995).

Roy, R.

G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge U. Press, 1999).

Sargent, M.

M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, 1974).

Scully, M. O.

M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, 1974).

Shamai (Shitz), S.

I. C. Abou-Faycal, M. D. Trott, and S. Shamai (Shitz), "The capacity of discrete-time memoryless Rayleigh-fading channels," IEEE Trans. Inf. Theory 47, 1290-1301 (2001).
[CrossRef]

S. Shamai (Shitz), "Capacity of a pulse amplitude modulated direct detection photon channel," IEE Proc., Part I: Solid-State Electron Devices 137, 424-430 (1990).

Trott, M. D.

I. C. Abou-Faycal, M. D. Trott, and S. Shamai (Shitz), "The capacity of discrete-time memoryless Rayleigh-fading channels," IEEE Trans. Inf. Theory 47, 1290-1301 (2001).
[CrossRef]

Verdú, S.

D. Brady and S. Verdú, "The asymptotic capacity of the direct detection photon channel with a bandwidth constraint," in Proceedings of the 28th Annual Allerton Conference on Communication, Control, and Computing (Allerton, 1990), pp. 691-700.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

IEE Proc., Part I: Solid-State Electron Devices

S. Shamai (Shitz), "Capacity of a pulse amplitude modulated direct detection photon channel," IEE Proc., Part I: Solid-State Electron Devices 137, 424-430 (1990).

IEEE J. Sel. Top. Quantum Electron.

J. M. Kahn and K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10, 259-272 (2004).
[CrossRef]

IEEE Trans. Inf. Theory

I. C. Abou-Faycal, M. D. Trott, and S. Shamai (Shitz), "The capacity of discrete-time memoryless Rayleigh-fading channels," IEEE Trans. Inf. Theory 47, 1290-1301 (2001).
[CrossRef]

Proc. IRE

J. P. Gordon, "Quantum effects in communication systems," Proc. IRE 50, 1898-1908 (1962).
[CrossRef]

Rev. Mod. Phys.

C. M. Caves and P. D. Drummond, "Quantum limits on bosonic communication rates," Rev. Mod. Phys. 66, 481-537 (1994).
[CrossRef]

Other

R. G. Gallager, Information Theory and Reliable Communication (Wiley, 1968).

I.S.Gradshteyn, I.M.Ryzhik, and A.Jeffrey (eds.), Tables of Integrals, Series, and Products (Academic, 1994).

D. Brady and S. Verdú, "The asymptotic capacity of the direct detection photon channel with a bandwidth constraint," in Proceedings of the 28th Annual Allerton Conference on Communication, Control, and Computing (Allerton, 1990), pp. 691-700.

A. Lapidoth and S. M. Moser, "Bounds on the capacity of the discrete-time Poisson channel," in Proceedings of the 41st Allerton Conference on Communication, Control, and Computing, (Allerton, 2003), pp. 201-210.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, 1995).

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2000).

M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, 1974).

G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge U. Press, 1999).

W. Feller, An Introduction to Probability Theory and Its Applications, 2nd ed. (Wiley, 1971), Vol. 2.

I. Csiszár and J. Körner, Information Theory: Coding Theorems for Discrete Memoryless Systems (Academic, 1981).

R. E. Blahut, Principles and Practice of Information Theory (Addison-Wesley, 1987).

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Figures (3)

Fig. 1
Fig. 1

Computation of κ 0 ( s ) for several values of ν.

Fig. 2
Fig. 2

Upper and lower bounds to the capacity per degree of freedom.

Fig. 3
Fig. 3

Comparison of spectral efficiency with Gaussian approximation. wrt, with respect to. Differences for (a) low photon counts and (b) large photon counts.

Tables (1)

Tables Icon

Table 1 Evaluation of κ 0 ( s , γ ) for Large Values of s

Equations (100)

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η = log 2 ( 1 + E s N 0 ) ,
η 1 2 { H Geom ( ϵ s ) log 2 [ 2 π exp ( γ e ) ] } ,
H Geom ( ϵ s ) = ( 1 + ϵ s ) log 2 ( 1 + ϵ s ) ϵ s log 2 ( ϵ s ) .
η 1 2 log 2 ϵ s 1 .
η 1 2 log 2 ϵ s + ( 1 + ϵ s ) log 2 ( 1 + 1 ϵ s ) ( 1 + π 24 ϵ s ) log 2 e ,
η 1 2 log 2 ϵ s + o ϵ s ( 1 ) ,
η = ( ϵ s + 1 ) log ( ϵ s + 1 ) + 0 1 ϵ s 2 ( 1 u ) 1 + ϵ u ( 1 u ) d u log u ϵ s γ e .
η 1 2 log 2 ϵ s 0.188 ,
η 1 2 log 2 ( 1 + ϵ s ) .
η ( ϵ s + 1 2 ) log ( 1 2 + ϵ s ) ϵ s log ϵ s 1 2 + log ( 1 + 2 e 1 1 + 2 ϵ s ) ,
z ( t ) = k = 1 W T v k z k ( t ) + z rem ( t ) ,
ρ = ρ 1 ρ 2 ρ W T .
ρ k = α k α k .
α k = n = 0 c n k n k , where c n k = exp ( 1 2 α k 2 ) α k n n ! .
ρ k , l m = 0 2 π 1 2 π c l k c m k * d θ k
= 0 2 π exp ( α k 2 ) α k l + m exp [ j ( l m ) θ k ] 2 π l ! m ! d θ k = { 0 l m α k 2 m m ! exp ( α k 2 ) l = m } .
Pr Z k ( m ) = s k m m ! exp ( s k ) ,
H ( Z ) = m = 0 Pr Z ( m ) log Pr Z ( m ) .
log Γ ( x ) = 0 1 [ 1 u x 1 1 u ( x 1 ) ] d u log u .
Γ ( x ) = a x 0 u x 1 exp ( a u ) d u .
log m ! = log Γ ( m + 1 ) = 0 1 ( 1 u m 1 u m ) d u log u .
Pr Z S ( Z = m S = s ) = s m m ! exp ( s ) .
H Pois ( s ) = s s log s + 0 1 { 1 exp [ s ( 1 u ) ] 1 u s } d u log u .
lim u 1 { 1 exp [ s ( 1 u ) ] 1 u s } 1 log u = s 2 2 .
p S ( s ) = ν ν Γ ( ν ) ϵ s ν s ν 1 exp ( ν s ϵ s ) .
mgf Gamma ( ν , ϵ s ) ( t ) = 0 p S ( s ) exp ( t s ) d s = 1 ( 1 ϵ s t ν ) ν
= ν ν ( ν ϵ s t ) ν .
Pr Z ( m ) = 0 Pr Z S ( m s ) p S ( s ) d s
= 0 exp ( s ) s m m ! ν ν Γ ( ν ) ϵ s ν s ν 1 exp ( ν s ϵ s ) d s
= Γ ( m + ν ) m ! Γ ( ν ) ν ν ϵ s m ( ϵ s + ν ) m + ν .
Pgf ( u ) = m = 0 Pr ( z = m ) u m .
pgf Neg Bin ( ν , ϵ s ) ( u ) = 0 exp ( s ) m = 0 ( u s ) m m ! p S ( s ) d s
= 0 exp ( s ) exp ( u s ) p S ( s ) d s
= mgf Gamma ( ν , ϵ s ) ( u 1 ) = ν ν [ ν + ϵ s ( 1 u ) ] ν .
H NegBin ( ν , ϵ s ) = ( ϵ s + ν ) log ( ϵ s + ν ) ν log ν ϵ s log ϵ s + 0 1 [ 1 pgf ( u ) ] 1 u ν 1 1 u d u log u ,
pgf ( u ) = pgf NegBin ( ν , ϵ s ) ( u ) = ν ν [ ν + ϵ s ( 1 u ) ] ν .
H Geom ( ϵ s ) = H NegBin ( 1 , ϵ s ) = ( ϵ s + 1 ) log ( ϵ s + 1 ) ϵ s log ϵ s .
lim u 1 [ 1 pgf ( u ) ] ( 1 u ν 1 ) ( 1 u ) log u = ϵ s ( 1 ν ) .
S ( ρ ) = Tr ( ρ log ρ ) = λ i λ i log λ i ,
ϵ s , k = 0 s p S k ( s ) d s
= E s h ν 0 = P h ν 0 W .
η = sup p S ( ) I ( S ; Z )
= sup p S ( ) H ( Z ) H ( Z S ) .
I ( S ; Z ) = 0 p S ( s ) [ m = 0 Pr Z S ( m s ) log Pr Z S ( m s ) Pr Z ( m ) ] d s ,
H ( Z S ) = 0 p S ( s ) [ m = 0 Pr Z S ( m s ) log Pr Z S ( m s ) ] d s ,
η = inf Pr z ( ) min γ 0 max s [ I KT ( s ; Z ) γ ( s ϵ s ) ] ,
I KT ( s ; Z ) = m = 0 Pr Z S ( m s ) log Pr Z S ( m s ) Pr Z ( m )
= H Pois ( s ) m = 0 Pr Z S ( m s ) log Pr Z ( m ) .
η ¯ = max s [ I KT ( s ; Z ) γ * ( s ϵ s ) ] ,
I Gamma ( ν , ϵ s ) = ( ϵ s + ν ) log ϵ s + ν ν + ϵ s [ ψ ( ν + 1 ) 1 ] 0 1 { [ 1 pgf ( u ) ] u ν 1 1 u ϵ s } d u log u ,
pgf ( u ) = pgf NegBin ( ν , ϵ s ) ( u ) = ν ν [ ν + ϵ s ( 1 u ) ] ν .
I exp ( ϵ s ) = ( ϵ s + 1 ) log ( ϵ s + 1 ) + 0 1 ϵ s 2 ( 1 u ) 1 + ϵ s ( 1 u ) d u log u ϵ s γ e .
η 1 2 log 2 ϵ s 0.188 ,
p S * ( s ) = α 0 δ ( s ) + ( 1 α 0 ) p Gamma ( ν , ϵ s ) ( s ) ,
Pr Z * ( m ) = { α 0 + ( 1 α 0 ) ν ν ( ϵ s + ν ) ν m = 0 ( 1 α 0 ) Γ ( m + ν ) m ! Γ ( ν ) ν ν ϵ s m ( ϵ s + ν ) m + ν m > 0 } .
I KT ( s ; Z ) γ ( s ϵ s ) = κ 0 ( s , γ ) + κ 1 ( s , γ ) + κ 2 ( s , γ ) ,
κ 0 ( s , γ ) = s + s log s + 0 1 { s u ν 1 1 exp [ s ( 1 u ) ] 1 u } d u log u exp ( s ) log [ 1 + α 0 ( 1 α 0 ) ( 1 + ϵ s ν ) ν ] ,
κ 1 ( s , γ ) = s ( γ log ϵ s + ν ϵ s ) ,
κ 2 ( s , γ ) = ν log ϵ s + ν ν + γ ϵ s log ( 1 α 0 ) .
γ * = log ϵ s + ν ϵ s
max s κ 0 ( s , γ ) = + ,
max s κ 0 ( s , γ ) = κ 0 ( s = 0 , γ ) = 0 .
η ¯ = κ 2 ( s , γ ) = ν log ϵ s + ν ν + γ ϵ s
= ( ϵ s + ν ) log ( ν + ϵ s ) ϵ s log ϵ s ν log ν .
lim s κ 0 ( s , γ ) = 1 2 log ( 2 e ) 0.84657359027997265471 .
log [ 1 + α 0 * 1 α 0 * ( 1 + 2 ϵ s ) 1 2 ] = 1 2 ( 1 + log 2 ) ,
α 0 * = 2 e 1 2 e 1 + ( 1 + 2 ϵ s ) 1 2 ;
η ¯ = 1 2 log ( 2 e ) + 1 2 log ϵ s + ( 1 2 ) 1 2 + ϵ s log ϵ s + ( 1 2 ) ϵ s log ( 1 α 0 * )
= 1 2 + ( ϵ s + 1 2 ) log ( 1 2 + ϵ s ) ϵ s log ϵ s log [ ( 1 + 2 ϵ s ) 1 2 2 e 1 + ( 1 + 2 ϵ s ) 1 2 ]
= ( ϵ s + 1 2 ) log ( 1 2 + ϵ s ) ϵ s log ϵ s 1 2 + log ( 1 + 2 e 1 1 + 2 ϵ s ) .
η ¯ = 1 2 log ϵ s + 2 e 1 1 + 2 ϵ s + O ( ϵ s 1 ) ,
H Pois ( s ) = m = 0 Pr ( m s ) log Pr ( m s )
= m = 0 Pr ( m s ) ( s + m log s log m ! )
= s m = 0 Pr ( m s ) log s m = 0 m Pr ( m s ) + m = 0 Pr ( m s ) log m !
= s s log s + m = 0 Pr ( m s ) log m ! .
m = 0 Pr ( m s ) log m ! = m = 0 exp ( s ) s m m ! 0 1 ( 1 u m 1 u m ) d u log u
= 0 1 m = 0 exp ( s ) s m m ! ( 1 u m 1 u m ) d u log u
= 0 1 ( 1 exp ( s ) exp ( s u ) 1 u s ) d u log u .
H NegBin ( ν , ϵ s ) = m = 0 Pr ( z = m ) log Pr ( z = m )
= m = 0 Pr ( m ) [ log Γ ( m + ν ) m ! Γ ( ν ) + log ν ν ( ϵ s + ν ) ν + m log ϵ s ϵ s + ν ]
= log ν ν ( ϵ s + ν ) ν ϵ s log ϵ s ϵ s + ν m = 0 Pr ( m ) [ log Γ ( m + ν ) m ! Γ ( ν ) ]
= ( ϵ s + ν ) log ( ϵ s + ν ) ν log ν ϵ s log ϵ s m = 0 Pr ( m ) [ log Γ ( m + ν ) m ! Γ ( ν ) ] ,
log Γ ( m + ν ) m ! Γ ( ν ) = 0 1 ( 1 u m + ν 1 1 + u m 1 + u ν 1 1 u ) d u log u
= 0 1 [ ( 1 u ν 1 ) ( 1 u m ) 1 u ] d u log u .
m = 0 Pr ( m ) [ log Γ ( m + ν ) m ! Γ ( ν ) ] = m = 0 Pr ( m ) { 0 1 [ ( 1 u ν 1 ) ( 1 u m ) 1 u ] d u log u }
= 0 1 { ( 1 u ν 1 ) [ 1 pgf ( u ) ] 1 u } d u log u .
I Gamma ( ν , ϵ s ) = I ( S ; Z ) = H ( Z ) H ( Z S ) .
H ( Z S ) = 0 p S ( s ) H ( Z s ) d s = 0 p S ( s ) H Pois ( s ) d s
= 0 p S ( s ) ( s s log s + 0 1 { 1 exp [ s ( 1 u ) ] 1 u s } d u log u ) d s
= ϵ s + 0 1 [ 1 mgf ( u 1 ) 1 u ϵ s ] d u log u 0 p S ( s ) s log s d s .
0 p S ( s ) s log s d s = ϵ s ψ ( ν + 1 ) ϵ s log ν ϵ s .
I ( S ; Z ) = ( ϵ s + ν ) log ϵ s + ν ν + ϵ s [ ψ ( ν + 1 ) 1 ] 0 1 { [ 1 pgf ( u ) ] u ν 1 1 u ϵ s } d u log u .
I KT ( s ; Z ) γ ( s ϵ s ) = H pois ( s ) m = 0 Pr Z S ( m s ) log Pr Z * ( m ) γ ( s ϵ s ) .
Pr Z * ( m ) = { α 0 + ( 1 α 0 ) ν ν ( ϵ s + ν ) ν m = 0 ( 1 α 0 ) Γ ( m + ν ) m ! Γ ( ν ) ν ν ϵ s m ( ϵ s + ν ) m + ν m > 0 } .
m = 0 Pr Z S ( m s ) log Pr Z * ( m ) = Pr Z S ( 0 s ) log Pr Z * ( 0 ) + m = 1 Pr Z S ( m s ) log Pr Z * ( m ) ,
m = 0 Pr Z S ( m s ) log Pr Z * ( m ) = Pr Z S ( 0 s ) log Pr Z * ( 0 ) ( 1 α 0 ) Pr NegBin ( ν , ϵ s ) ( 0 ) + m = 0 Pr Z S ( m s ) log [ ( 1 α 0 ) Pr NegBin ( ν , ϵ s ) ( m ) ] ,
= exp ( s ) log [ 1 + α 0 ( 1 α 0 ) ( 1 + ϵ s ν ) ν ] + log ( 1 α 0 ) + m = 0 Pr Z S ( m s ) log [ Pr NegBin ( ν , ϵ s ) ( m ) ] ,
m = 0 Pr ( m s ) log Pr NegBin ( ν , ϵ s ) ( m ) = ν log ν ϵ s + ν + m = 0 Pr ( m s ) m log ϵ s ϵ s + ν m = 0 Pr ( m s ) 0 1 [ ( 1 u ν 1 ) ( 1 u m ) 1 u ] d u log u
= ν log ν + s log ϵ s ( s + ν ) log ( ϵ s + ν ) 0 1 ( ( 1 u ν 1 ) { 1 exp [ s ( 1 u ) ] } 1 u ) d u log u .
I KT ( s ; Z ) γ ( s ϵ s ) = s + s log s + 0 1 { s u ν 1 1 exp [ s ( 1 u ) ] 1 u } d u log u exp ( s ) log [ 1 + α 0 ( 1 α 0 ) ( 1 + ϵ s ν ) ν ] s ( γ log ϵ s + ν ϵ s ) + ν log ϵ s + ν ν + γ ϵ s log ( 1 α 0 ) .

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