Abstract

We derive expressions for the capacity of the pulse-position-modulated (PPM) direct-detection photon-counting channel in the presence of noise photons in addition to the signal-dependent shot noise that is normally considered in studying photon counting at low photon rates. We note that even a small mean number of noise photons per PPM count bin significantly decreases the capacity of the channel. These results are useful for comparisons of performance that are obtained by use of real coding and synchronization algorithms with photon-counting PPM schemes that are currently being considered for deep-space optical communications.

© 2000 Optical Society of America

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References

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  1. H. Hemmati, K. Wilson, M. K. Sue, L. J. Harcke, M. Wilhelm, C.-C. Chen, J. Lesh, Y. Feria, D. Rascoe, F. Lansing, J. W. Layland, “Comparative study of optical and radio-frequency communication systems for a deep space mission,” (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 15February1997).
  2. J. R. Pierce, “Optical channels,” IEEE Trans. Commun. COM-26, 1819–1821 (1978).
    [CrossRef]
  3. J. R. Pierce, E. C. Posner, E. R. Rodemich, “The capacity of the photon counting channel,” IEEE Trans. Inf. Theory IT-27, 61–77 (1981).
    [CrossRef]
  4. J. R. Pierce, E. C. Posner, Introduction to Communication Science and Systems (Plenum, New York, 1980).
    [CrossRef]
  5. R. J. McEliece, “Practical codes for photon communication,” IEEE Trans. Inf. Theory IT-27, 393–398 (1981).
    [CrossRef]
  6. R. M. Gagliardi, Optical Communications, 2nd ed. (Wiley, New York, 1995).
  7. A. D. Wyner, “Capacity and error exponent for the direct detection photon channel. I,” IEEE Trans. Inf. Theory 34, 1449–1461 (1988).
    [CrossRef]
  8. A. D. Wyner, “Capacity and error exponent for the direct detection photon channel. II,” IEEE Trans. Inf. Theory 34, 1462–1471 (1988).
    [CrossRef]
  9. I. Bar-David, G. Kaplan, “Information rates for photon limited overlapping pulse position modulation channels,” IEEE Trans. Inf. Theory IT-30, 455–464 (1984).
    [CrossRef]
  10. S. M. Tseng, M. R. Bell, “Joint synchronization and decoding for photon limited optical channels: code design and complexity reduction,” IEEE Trans. Commun. (to be published).
  11. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948); reprinted in C. E. Shannon, W. W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, Ill., 1949), pp. 29–125.
  12. R. G. Gallager, Information Theory and Reliable Communication (Wiley, New York, 1968).
  13. D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).
    [CrossRef]
  14. B. Saleh, Photoelectron Statistics (Springer-Verlag, New York, 1978).
    [CrossRef]

1988

A. D. Wyner, “Capacity and error exponent for the direct detection photon channel. I,” IEEE Trans. Inf. Theory 34, 1449–1461 (1988).
[CrossRef]

A. D. Wyner, “Capacity and error exponent for the direct detection photon channel. II,” IEEE Trans. Inf. Theory 34, 1462–1471 (1988).
[CrossRef]

1984

I. Bar-David, G. Kaplan, “Information rates for photon limited overlapping pulse position modulation channels,” IEEE Trans. Inf. Theory IT-30, 455–464 (1984).
[CrossRef]

1981

J. R. Pierce, E. C. Posner, E. R. Rodemich, “The capacity of the photon counting channel,” IEEE Trans. Inf. Theory IT-27, 61–77 (1981).
[CrossRef]

R. J. McEliece, “Practical codes for photon communication,” IEEE Trans. Inf. Theory IT-27, 393–398 (1981).
[CrossRef]

1978

J. R. Pierce, “Optical channels,” IEEE Trans. Commun. COM-26, 1819–1821 (1978).
[CrossRef]

1948

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948); reprinted in C. E. Shannon, W. W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, Ill., 1949), pp. 29–125.

Bar-David, I.

I. Bar-David, G. Kaplan, “Information rates for photon limited overlapping pulse position modulation channels,” IEEE Trans. Inf. Theory IT-30, 455–464 (1984).
[CrossRef]

Bell, M. R.

S. M. Tseng, M. R. Bell, “Joint synchronization and decoding for photon limited optical channels: code design and complexity reduction,” IEEE Trans. Commun. (to be published).

Chen, C.-C.

H. Hemmati, K. Wilson, M. K. Sue, L. J. Harcke, M. Wilhelm, C.-C. Chen, J. Lesh, Y. Feria, D. Rascoe, F. Lansing, J. W. Layland, “Comparative study of optical and radio-frequency communication systems for a deep space mission,” (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 15February1997).

Feria, Y.

H. Hemmati, K. Wilson, M. K. Sue, L. J. Harcke, M. Wilhelm, C.-C. Chen, J. Lesh, Y. Feria, D. Rascoe, F. Lansing, J. W. Layland, “Comparative study of optical and radio-frequency communication systems for a deep space mission,” (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 15February1997).

Gagliardi, R. M.

R. M. Gagliardi, Optical Communications, 2nd ed. (Wiley, New York, 1995).

Gallager, R. G.

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

Harcke, L. J.

H. Hemmati, K. Wilson, M. K. Sue, L. J. Harcke, M. Wilhelm, C.-C. Chen, J. Lesh, Y. Feria, D. Rascoe, F. Lansing, J. W. Layland, “Comparative study of optical and radio-frequency communication systems for a deep space mission,” (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 15February1997).

Hemmati, H.

H. Hemmati, K. Wilson, M. K. Sue, L. J. Harcke, M. Wilhelm, C.-C. Chen, J. Lesh, Y. Feria, D. Rascoe, F. Lansing, J. W. Layland, “Comparative study of optical and radio-frequency communication systems for a deep space mission,” (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 15February1997).

Kaplan, G.

I. Bar-David, G. Kaplan, “Information rates for photon limited overlapping pulse position modulation channels,” IEEE Trans. Inf. Theory IT-30, 455–464 (1984).
[CrossRef]

Lansing, F.

H. Hemmati, K. Wilson, M. K. Sue, L. J. Harcke, M. Wilhelm, C.-C. Chen, J. Lesh, Y. Feria, D. Rascoe, F. Lansing, J. W. Layland, “Comparative study of optical and radio-frequency communication systems for a deep space mission,” (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 15February1997).

Layland, J. W.

H. Hemmati, K. Wilson, M. K. Sue, L. J. Harcke, M. Wilhelm, C.-C. Chen, J. Lesh, Y. Feria, D. Rascoe, F. Lansing, J. W. Layland, “Comparative study of optical and radio-frequency communication systems for a deep space mission,” (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 15February1997).

Lesh, J.

H. Hemmati, K. Wilson, M. K. Sue, L. J. Harcke, M. Wilhelm, C.-C. Chen, J. Lesh, Y. Feria, D. Rascoe, F. Lansing, J. W. Layland, “Comparative study of optical and radio-frequency communication systems for a deep space mission,” (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 15February1997).

McEliece, R. J.

R. J. McEliece, “Practical codes for photon communication,” IEEE Trans. Inf. Theory IT-27, 393–398 (1981).
[CrossRef]

Miller, M. I.

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).
[CrossRef]

Pierce, J. R.

J. R. Pierce, E. C. Posner, E. R. Rodemich, “The capacity of the photon counting channel,” IEEE Trans. Inf. Theory IT-27, 61–77 (1981).
[CrossRef]

J. R. Pierce, “Optical channels,” IEEE Trans. Commun. COM-26, 1819–1821 (1978).
[CrossRef]

J. R. Pierce, E. C. Posner, Introduction to Communication Science and Systems (Plenum, New York, 1980).
[CrossRef]

Posner, E. C.

J. R. Pierce, E. C. Posner, E. R. Rodemich, “The capacity of the photon counting channel,” IEEE Trans. Inf. Theory IT-27, 61–77 (1981).
[CrossRef]

J. R. Pierce, E. C. Posner, Introduction to Communication Science and Systems (Plenum, New York, 1980).
[CrossRef]

Rascoe, D.

H. Hemmati, K. Wilson, M. K. Sue, L. J. Harcke, M. Wilhelm, C.-C. Chen, J. Lesh, Y. Feria, D. Rascoe, F. Lansing, J. W. Layland, “Comparative study of optical and radio-frequency communication systems for a deep space mission,” (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 15February1997).

Rodemich, E. R.

J. R. Pierce, E. C. Posner, E. R. Rodemich, “The capacity of the photon counting channel,” IEEE Trans. Inf. Theory IT-27, 61–77 (1981).
[CrossRef]

Saleh, B.

B. Saleh, Photoelectron Statistics (Springer-Verlag, New York, 1978).
[CrossRef]

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948); reprinted in C. E. Shannon, W. W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, Ill., 1949), pp. 29–125.

Snyder, D. L.

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).
[CrossRef]

Sue, M. K.

H. Hemmati, K. Wilson, M. K. Sue, L. J. Harcke, M. Wilhelm, C.-C. Chen, J. Lesh, Y. Feria, D. Rascoe, F. Lansing, J. W. Layland, “Comparative study of optical and radio-frequency communication systems for a deep space mission,” (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 15February1997).

Tseng, S. M.

S. M. Tseng, M. R. Bell, “Joint synchronization and decoding for photon limited optical channels: code design and complexity reduction,” IEEE Trans. Commun. (to be published).

Wilhelm, M.

H. Hemmati, K. Wilson, M. K. Sue, L. J. Harcke, M. Wilhelm, C.-C. Chen, J. Lesh, Y. Feria, D. Rascoe, F. Lansing, J. W. Layland, “Comparative study of optical and radio-frequency communication systems for a deep space mission,” (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 15February1997).

Wilson, K.

H. Hemmati, K. Wilson, M. K. Sue, L. J. Harcke, M. Wilhelm, C.-C. Chen, J. Lesh, Y. Feria, D. Rascoe, F. Lansing, J. W. Layland, “Comparative study of optical and radio-frequency communication systems for a deep space mission,” (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 15February1997).

Wyner, A. D.

A. D. Wyner, “Capacity and error exponent for the direct detection photon channel. II,” IEEE Trans. Inf. Theory 34, 1462–1471 (1988).
[CrossRef]

A. D. Wyner, “Capacity and error exponent for the direct detection photon channel. I,” IEEE Trans. Inf. Theory 34, 1449–1461 (1988).
[CrossRef]

Bell Syst. Tech. J.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948); reprinted in C. E. Shannon, W. W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, Ill., 1949), pp. 29–125.

IEEE Trans. Commun.

J. R. Pierce, “Optical channels,” IEEE Trans. Commun. COM-26, 1819–1821 (1978).
[CrossRef]

IEEE Trans. Inf. Theory

J. R. Pierce, E. C. Posner, E. R. Rodemich, “The capacity of the photon counting channel,” IEEE Trans. Inf. Theory IT-27, 61–77 (1981).
[CrossRef]

R. J. McEliece, “Practical codes for photon communication,” IEEE Trans. Inf. Theory IT-27, 393–398 (1981).
[CrossRef]

A. D. Wyner, “Capacity and error exponent for the direct detection photon channel. I,” IEEE Trans. Inf. Theory 34, 1449–1461 (1988).
[CrossRef]

A. D. Wyner, “Capacity and error exponent for the direct detection photon channel. II,” IEEE Trans. Inf. Theory 34, 1462–1471 (1988).
[CrossRef]

I. Bar-David, G. Kaplan, “Information rates for photon limited overlapping pulse position modulation channels,” IEEE Trans. Inf. Theory IT-30, 455–464 (1984).
[CrossRef]

Other

S. M. Tseng, M. R. Bell, “Joint synchronization and decoding for photon limited optical channels: code design and complexity reduction,” IEEE Trans. Commun. (to be published).

R. M. Gagliardi, Optical Communications, 2nd ed. (Wiley, New York, 1995).

J. R. Pierce, E. C. Posner, Introduction to Communication Science and Systems (Plenum, New York, 1980).
[CrossRef]

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

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).
[CrossRef]

B. Saleh, Photoelectron Statistics (Springer-Verlag, New York, 1978).
[CrossRef]

H. Hemmati, K. Wilson, M. K. Sue, L. J. Harcke, M. Wilhelm, C.-C. Chen, J. Lesh, Y. Feria, D. Rascoe, F. Lansing, J. W. Layland, “Comparative study of optical and radio-frequency communication systems for a deep space mission,” (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Calif., 15February1997).

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Figures (8)

Fig. 1
Fig. 1

Four frames of 8-ary PPM that correspond to the transmitted symbol sequence 2, 6, 7, 2.

Fig. 2
Fig. 2

DMC of the PPM photon-counting channel.

Fig. 3
Fig. 3

Probability of obtaining the correct symbol decision P C plotted as a function of the mean signal photon count μ s for values of M = 2, 4, 8, 16, 32, 64, 128, 256 and a mean noise-photon count of μ n = 1.

Fig. 4
Fig. 4

Probability of obtaining a symbol error, P E = (M - 1)δ, plotted as a function of the mean signal photon count μ s for values of M = 2, 4, 8, 16, 32, 64, 128, 256 and a mean noise-photon count of μ n = 1.

Fig. 5
Fig. 5

Probability of erasure ∊ plotted as a function of the mean signal photon count μ s for values of M = 2, 4, 8, 16, 32, 64, 128, 256 and a mean noise-photon count of μ n = 1.

Fig. 6
Fig. 6

Capacity of the PPM photon-counting channel for μ n = 1 and various values of M.

Fig. 7
Fig. 7

Capacity of the PPM photon-counting channel for M = 8 and various values of the mean noise-photon count μ n .

Fig. 8
Fig. 8

Capacity of the PPM photon-counting channel for M = 16 and various values of the mean noise-photon count μ n .

Equations (34)

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PR=ATARλ2R2PT,
CPC=1ln 2hfkT bits/photon,
PY|X=PcδδδδPcδδδδPcδδδδPc,
Pc=1-M-1δ-.
C=maxpxP IX; Y,
IX; Y=HX-HX|Y
=HY-HY|X,
HX=x pxlog1px,
HY=y pylog1py,
HX|Y=xy px, ylog1px|y,
HY|X=xy px, ylog1py|x.
Ixk; Y=j=1M+1 pyj|xklogpyj|xkpyj,
IX; Y=k=1M pxkIxk; Y.
Ixk; Y=C,for all xk such that pxk>0,Ixk; YC,for all xk such that pxk=0.
pxk, yj=pyj|xkpxk=1-M-1δ-/Mj=k/Mj=M+1δ/Mjk,j=1,, M.
pyj=k=1M pxk, yj=j=M+11-M-1δ-/M+M-1δ/Mj=1,, M=1-/Mj=1,, Mj=M+1.
logpyj|xkpyj=log1-M-1δ-1-/Mj=1,, M,k=jlogδ1-/Mj=1,, M, k=1,, M,kjlog/=0j=M+1.
Ixk; Y=j=1M+1 pyj|xklogpyj|xkpyj=1-M-1δ+log1-M-1δ+1-/M+M-1δ logδ1-/M.
C=1-M-1δ+log1-M-1δ+1-/M+M-1δ logδ1-/M=1-log M-log1-+M-1δ log δ+1-M-1δ+log1-M-1δ+.
C=1-log M-log1-+M-1δ log δ+1-M-1δ+log1-M-1δ+.
μ0=λ0Δt=λnΔt=μn,
μ1=λ1Δt=λs+λnΔt=μs+μn.
PK0=k=exp-λnΔtλnΔtkk!=exp-μnμnkk!,k=0, 1, 2,,
PK1=k=exp-λs+λsΔtλs+λnΔtkk!=exp-μs+μnμs+μnkk!,k=0, 1, 2,.
Pc=PK1>K01K1>K02K1>K0M-1=k1=1 PK1=k1PK01<k1K02<k1×K0M-1<k1=k1=1 PK1=k1PK0<K1M-1because K01,, K0M-1 are independent identically distributed Poisson with a mean μ0=k1=1 PK1=k1k0=0k1-1 PK0=k0M-1=k1=1exp-μs+μnμs+μnkk!×k0=0k1-1exp-μnμnkk!M-1=k1=1exp-μ1μ1kk!k0=0k1-1exp-μ0μ0kk!M-1.
k0=0k1-1exp-μ0μ0kk!M-1=Γk1, μ0Γk1,
Γk1, μ0=μ0 sk1-1 exp-sds
Γk1=0 sk1-1 exp-sds
Pc=k1=1exp-μ1μ1k1k1!Γk1, μ0Γk1M-1.
γμ0, μ1, M=k1=1exp-μ1μ1k1k1!Γk1, μ0Γk1M-1.
δμ0, μ1, M=j0=1 PK0=j0k0=0j0-1 PK0=k0M-2×k1=0j0=1 PK1=k1=j0=1exp-μ0μ0j0j0!Γj0, μ0Γj0M-2Γj0, μ1Γj0.
γ+M-1δ+=1,
μ0, μ1, M=1-M-1δμ0, μ1, M-γμ0, μ1, M.
C=1-log M-log1-+M-1δ log δ+1-M-1δ+log1-M-1δ+,γ=k1=1exp-μ1μ1k1k1!Γk1, μ0Γk1M-1,δ=j0=1exp-μ0μ0j0j0!Γj0, μ0Γj0M-2Γj0, μ1Γj0,=1-M-1δμ0, μ1, M-γμ0, μ1, M.

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