Abstract

We present a novel optical code division multiple access (OCDMA) scheme based on spatial optical heterodyning for free-space optical communication systems. In this technique, in particular, the decoding process is established by means of a spatial optical heterodyne receiver. The spatial heterodyning OCDMA introduced can be considered to be in the class of spread-space techniques, which implies that there is no structural limitation on the shape of time domain signals, e.g., digital or analog modulation. For the sake of simplicity and practicality we considered on–off keying modulation. However, there is no limitation on using more advanced modulation schemes such as pulse position modulation. In this scheme a set of independent orthogonal spatial channels at both transmitting and receiving apertures is needed. Thus this class of OCDMA is applicable only to near-field free-space systems. The structures of the encoder and decoder are presented, and the wave propagation is completely analyzed by means of the Fresnel approximation for the free-space kernel. Only the effect of multiuser interference is considered. The effects of background noise and atmospheric turbulence on the performance of the system are avoided in order to highlight the main capabilities and features of spatial optical heterodyning OCDMA. A thorough statistical analysis is carried out, and the characteristic function of the sampled output is computed. The bit error rate is obtained by means of Gaussian and saddle-point approximations. Results show that the signal-to-interference ratio is approximately a function of processing gain and the number of users, as expected. The optical spatial heterodyning OCDMA technique can prove to be of importance in optical switching and free-space optical communication systems.

© 2009 Optical Society of America

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References

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  1. V. W. S. Chan “Free-space optical communications,” J. Lightwave Technol., vol. 24, no. 12, pp. 4750–4762, Dec. 2006.
    [CrossRef]
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    [CrossRef]
  4. J. A. Salehi, C. A. Brackett, “Code division multiple access techniques in optical fiber networks—part II: system performance analysis,” IEEE Trans. Commun., vol. 37, no. 8, pp. 834–842, Aug. 1989.
    [CrossRef]
  5. J. A. Salehi, A. M. Weiner, J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol., vol. 8, no. 3, pp. 478–491, March 1990.
    [CrossRef]
  6. J. A. Salehi, A. G. Paek, “Holographic CDMA,” IEEE Trans. Commun., vol. 43, pp. 2434–2438, Sept. 1995.
    [CrossRef]
  7. J. A. Salehi, M. Abtahi, “All-optical holographic code division multiple access switch,” U.S. patent 6,844,947, Jan. 18, 2005.
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    [CrossRef]
  9. D. A. B. Miller, “Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths,” Appl. Opt., vol. 39, pp. 1681–1699, 2000.
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. J. W. Goodman, Statistical Optics. New York, NY: Wiely, 1985.
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    [CrossRef]
  15. H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—II,” Bell Syst. Tech. J., vol. 40, pp. 65–84, Jan. 1961.
    [CrossRef]
  16. H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III: the dimension of the space of essentially time- and band-limited signals,” Bell Syst. Tech. J., vol. 41, pp. 1295–1336, July 1962.
    [CrossRef]
  17. J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere. Berlin, Germany: Springer-Verlag, 1978, pp. 171–222.
    [CrossRef]
  18. J. H. Shapiro, “Normal-mode approach to wave propagation in the turbulent atmosphere,” Appl. Opt., vol. 13, pp. 2614–2619, 1974.
    [CrossRef] [PubMed]
  19. C. Flammer, Spheroidal Wave Functions. Stanford, CA: Stanford Univ. Press, 1957.

2006 (1)

2002 (1)

M. Abtahi, J. A. Salehi, “Spread-space holographic CDMA technique: basic analysis and applications,” IEEE Trans. Wireless Commun., vol. 1, no. 2, pp. 311–321, April 2002.
[CrossRef]

2000 (2)

1995 (1)

J. A. Salehi, A. G. Paek, “Holographic CDMA,” IEEE Trans. Commun., vol. 43, pp. 2434–2438, Sept. 1995.
[CrossRef]

1992 (1)

A. M. Weiner, D. E. Learid, J. S. Patel, J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron., vol. 28, pp. 908–920, April 1992.
[CrossRef]

1990 (1)

J. A. Salehi, A. M. Weiner, J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol., vol. 8, no. 3, pp. 478–491, March 1990.
[CrossRef]

1989 (2)

J. A. Salehi, “Code division multiple access techniques in optical fiber networks—part I: fundamental principles,” IEEE Trans. Commun., vol. 37, no. 8, pp. 824–833, Aug. 1989.
[CrossRef]

J. A. Salehi, C. A. Brackett, “Code division multiple access techniques in optical fiber networks—part II: system performance analysis,” IEEE Trans. Commun., vol. 37, no. 8, pp. 834–842, Aug. 1989.
[CrossRef]

1974 (1)

1962 (1)

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III: the dimension of the space of essentially time- and band-limited signals,” Bell Syst. Tech. J., vol. 41, pp. 1295–1336, July 1962.
[CrossRef]

1961 (2)

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J., vol. 40, pp. 43–63, Jan. 1961.
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—II,” Bell Syst. Tech. J., vol. 40, pp. 65–84, Jan. 1961.
[CrossRef]

Abtahi, M.

M. Abtahi, J. A. Salehi, “Spread-space holographic CDMA technique: basic analysis and applications,” IEEE Trans. Wireless Commun., vol. 1, no. 2, pp. 311–321, April 2002.
[CrossRef]

J. A. Salehi, M. Abtahi, “All-optical holographic code division multiple access switch,” U.S. patent 6,844,947, Jan. 18, 2005.

Brackett, C. A.

J. A. Salehi, C. A. Brackett, “Code division multiple access techniques in optical fiber networks—part II: system performance analysis,” IEEE Trans. Commun., vol. 37, no. 8, pp. 834–842, Aug. 1989.
[CrossRef]

Chan, V. W. S.

Einarsson, G.

G. Einarsson, Principles of Lightwave Communications. New York, NY: Wiley, 1996.

Flammer, C.

C. Flammer, Spheroidal Wave Functions. Stanford, CA: Stanford Univ. Press, 1957.

Gagliardi, R. M.

R. M. Gagliardi, S. Karp, Optical Communications. New York, NY: Wiley, 1976.

Goodman, J. W.

J. W. Goodman, Statistical Optics. New York, NY: Wiely, 1985.

Heritage, J. P.

J. A. Salehi, A. M. Weiner, J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol., vol. 8, no. 3, pp. 478–491, March 1990.
[CrossRef]

Kannari, F.

Karp, S.

R. M. Gagliardi, S. Karp, Optical Communications. New York, NY: Wiley, 1976.

Landau, H. J.

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III: the dimension of the space of essentially time- and band-limited signals,” Bell Syst. Tech. J., vol. 41, pp. 1295–1336, July 1962.
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—II,” Bell Syst. Tech. J., vol. 40, pp. 65–84, Jan. 1961.
[CrossRef]

Learid, D. E.

A. M. Weiner, D. E. Learid, J. S. Patel, J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron., vol. 28, pp. 908–920, April 1992.
[CrossRef]

Miller, D. A. B.

Paek, A. G.

J. A. Salehi, A. G. Paek, “Holographic CDMA,” IEEE Trans. Commun., vol. 43, pp. 2434–2438, Sept. 1995.
[CrossRef]

Patel, J. S.

A. M. Weiner, D. E. Learid, J. S. Patel, J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron., vol. 28, pp. 908–920, April 1992.
[CrossRef]

Pollack, H. O.

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III: the dimension of the space of essentially time- and band-limited signals,” Bell Syst. Tech. J., vol. 41, pp. 1295–1336, July 1962.
[CrossRef]

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J., vol. 40, pp. 43–63, Jan. 1961.
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—II,” Bell Syst. Tech. J., vol. 40, pp. 65–84, Jan. 1961.
[CrossRef]

Salehi, J. A.

M. Abtahi, J. A. Salehi, “Spread-space holographic CDMA technique: basic analysis and applications,” IEEE Trans. Wireless Commun., vol. 1, no. 2, pp. 311–321, April 2002.
[CrossRef]

J. A. Salehi, A. G. Paek, “Holographic CDMA,” IEEE Trans. Commun., vol. 43, pp. 2434–2438, Sept. 1995.
[CrossRef]

J. A. Salehi, A. M. Weiner, J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol., vol. 8, no. 3, pp. 478–491, March 1990.
[CrossRef]

J. A. Salehi, “Code division multiple access techniques in optical fiber networks—part I: fundamental principles,” IEEE Trans. Commun., vol. 37, no. 8, pp. 824–833, Aug. 1989.
[CrossRef]

J. A. Salehi, C. A. Brackett, “Code division multiple access techniques in optical fiber networks—part II: system performance analysis,” IEEE Trans. Commun., vol. 37, no. 8, pp. 834–842, Aug. 1989.
[CrossRef]

J. A. Salehi, M. Abtahi, “All-optical holographic code division multiple access switch,” U.S. patent 6,844,947, Jan. 18, 2005.

Shapiro, J. H.

J. H. Shapiro, “Normal-mode approach to wave propagation in the turbulent atmosphere,” Appl. Opt., vol. 13, pp. 2614–2619, 1974.
[CrossRef] [PubMed]

J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere. Berlin, Germany: Springer-Verlag, 1978, pp. 171–222.
[CrossRef]

Shirakawa, A.

Slepian, D.

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J., vol. 40, pp. 43–63, Jan. 1961.
[CrossRef]

Takasago, K.

Takekawa, M.

Weiner, A. M.

A. M. Weiner, D. E. Learid, J. S. Patel, J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron., vol. 28, pp. 908–920, April 1992.
[CrossRef]

J. A. Salehi, A. M. Weiner, J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol., vol. 8, no. 3, pp. 478–491, March 1990.
[CrossRef]

Wullert, J. R.

A. M. Weiner, D. E. Learid, J. S. Patel, J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron., vol. 28, pp. 908–920, April 1992.
[CrossRef]

Appl. Opt. (3)

Bell Syst. Tech. J. (3)

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J., vol. 40, pp. 43–63, Jan. 1961.
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—II,” Bell Syst. Tech. J., vol. 40, pp. 65–84, Jan. 1961.
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III: the dimension of the space of essentially time- and band-limited signals,” Bell Syst. Tech. J., vol. 41, pp. 1295–1336, July 1962.
[CrossRef]

IEEE J. Quantum Electron. (1)

A. M. Weiner, D. E. Learid, J. S. Patel, J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron., vol. 28, pp. 908–920, April 1992.
[CrossRef]

IEEE Trans. Commun. (3)

J. A. Salehi, “Code division multiple access techniques in optical fiber networks—part I: fundamental principles,” IEEE Trans. Commun., vol. 37, no. 8, pp. 824–833, Aug. 1989.
[CrossRef]

J. A. Salehi, C. A. Brackett, “Code division multiple access techniques in optical fiber networks—part II: system performance analysis,” IEEE Trans. Commun., vol. 37, no. 8, pp. 834–842, Aug. 1989.
[CrossRef]

J. A. Salehi, A. G. Paek, “Holographic CDMA,” IEEE Trans. Commun., vol. 43, pp. 2434–2438, Sept. 1995.
[CrossRef]

IEEE Trans. Wireless Commun. (1)

M. Abtahi, J. A. Salehi, “Spread-space holographic CDMA technique: basic analysis and applications,” IEEE Trans. Wireless Commun., vol. 1, no. 2, pp. 311–321, April 2002.
[CrossRef]

J. Lightwave Technol. (2)

J. A. Salehi, A. M. Weiner, J. P. Heritage, “Coherent ultrashort light pulse code-division multiple access communication systems,” J. Lightwave Technol., vol. 8, no. 3, pp. 478–491, March 1990.
[CrossRef]

V. W. S. Chan “Free-space optical communications,” J. Lightwave Technol., vol. 24, no. 12, pp. 4750–4762, Dec. 2006.
[CrossRef]

Other (6)

J. H. Shapiro, “Imaging and optical communication through atmospheric turbulence,” in J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere. Berlin, Germany: Springer-Verlag, 1978, pp. 171–222.
[CrossRef]

C. Flammer, Spheroidal Wave Functions. Stanford, CA: Stanford Univ. Press, 1957.

J. A. Salehi, M. Abtahi, “All-optical holographic code division multiple access switch,” U.S. patent 6,844,947, Jan. 18, 2005.

R. M. Gagliardi, S. Karp, Optical Communications. New York, NY: Wiley, 1976.

J. W. Goodman, Statistical Optics. New York, NY: Wiely, 1985.

G. Einarsson, Principles of Lightwave Communications. New York, NY: Wiley, 1996.

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Figures (15)

Fig. 1
Fig. 1

Principles of spatial heterodyning OCDMA.

Fig. 2
Fig. 2

Arbitrary shape transmitting and receiving apertures and their waveforms.

Fig. 3
Fig. 3

Single transmitter and receiver diagram.

Fig. 4
Fig. 4

Typical structure of spatial heterodyning OCDMA network.

Fig. 5
Fig. 5

Block diagram of the receiver in the noise-free case.

Fig. 6
Fig. 6

Specific example of transmitting and receiving apertures.

Fig. 7
Fig. 7

Example of a transmitter’s mask with 15 × 15 spatial modes using PSWFs as spatial communication modes.

Fig. 8
Fig. 8

Magnitude of a transmitter’s mask used for the encoding process.

Fig. 9
Fig. 9

Receiver’s mask used for encoding of the LO electric field.

Fig. 10
Fig. 10

Received field from the channel after the phase canceller mask. Independent spatial modes (sinc functions) are distinct. 100 spatial modes are clearly separated. The effect of interference in comparison with Fig. 9 is obvious.

Fig. 11
Fig. 11

BER of a spatial heterodyne OCDMA receiver in the noise-free case using the Gaussian approximation method.

Fig. 12
Fig. 12

BER of a spatial homodyne OCDMA receiver in the noise-free case using the Gaussian approximation method. In case 1 P L = 10 5 P T , case 2 P L = 10 4 P T , and case 3 P L = 10 3 P T . Note that the optimum threshold is used.

Fig. 13
Fig. 13

BER versus normalized threshold for 2500 modes in the spatial homodyne OCDMA noise-free case

Fig. 14
Fig. 14

Comparison of spatial heterodyne and homodyne OCDMA receivers in the noise-free case, using the Gaussian approximation method. In case 1 P L = 10 5 P T , and in case 2 P L = 10 4 P T . Note that the optimum threshold is used for the homodyne receiver.

Fig. 15
Fig. 15

Comparison of saddle-point and Gaussian approximations for a noise-free spatial heterodyne OCDMA receiver.

Tables (1)

Tables Icon

Table 1 Definitions and Typical Values of System Parameters Used for Numerical Analysis of Theoretical Results and Space-Domain Simulation

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

S T ( x , y ) = m , n b m n a T m n ( x , y ) ,
S T ( i ) ( x , y ) = P S N m = 1 n x crit n = 1 n y crit b m n ( i ) a T m n ( x , y ) .
S R ( i ) ( p , q ) = P S N e j ( 2 π λ L ) j λ L m = 1 n x crit n = 1 n y crit b m n ( i ) × A T F ( x , y ) a T m n ( x , y ) e j ( 2 π λ L ) ( x p + y q ) d x d y ,
F ( x , y ) = exp ( j π λ L ( x 2 + y 2 ) ) .
S R ( i ) ( p , q ) = P S N e j ( 2 π λ L ) j λ L m = 1 n x crit n = 1 n y crit b m n ( i ) I { a T x m ( x ) } I { a T y n ( y ) } = P S N e j ( 2 π λ L ) j λ L m = 1 n x crit n = 1 n y crit b m n ( i ) a R p m ( p ) a R q n ( q ) = P S N e j ( 2 π λ L ) j λ L m = 1 n x crit n = 1 n y crit b m n ( i ) a R m n ( p , q ) .
S L ( i ) ( p , q ) = P L N e j ( 2 π λ L ) j λ L m = 1 n x crit n = 1 n y crit b m n ( i ) F * ( p , q ) a R m n ( p , q ) .
P i = A R | S R ( i ) ( p , q ) + F ( p , q ) S L ( i ) ( p , q ) | 2 d p d q = ( P S N + P L N ) 2 m = 1 n x crit n = 1 n y crit k = 1 n x crit j = 1 n y crit b m n ( i ) b j k ( i ) × A R ( 1 λ L ) 2 a R m n ( p , q ) a R j k * ( p , q ) d p d q .
P i = ( P S N + P L N ) 2 k = 1 n x crit j = 1 n y crit ( b m n ( i ) ) 2 = ( P S + P L ) 2 .
S T ( x , y ) = i = 0 M d i S T ( i ) ( x , y ) exp ( j ω 0 t + j ϕ i ) ,
S R ( p , q ) = P S N e j ( 2 π λ L ) j λ L i = 0 M d i exp ( j ω 0 t + j ϕ i ) { m = 1 n x crit n = 1 n y crit b m n ( i ) a R p m ( p ) a R q n ( q ) } = i = 0 M d i exp ( j ω 0 t + j ϕ i ) S R ( i ) ( p , q ) .
S L ( 0 ) ( p , q ) = P L N e j ( 2 π λ L ) j λ L exp ( j ω L t + j ϕ L ) F * ( p , q ) S R ( 0 ) ( p , q ) .
Q i j m = 1 n x crit n = 1 n y crit b m n ( i ) b m n ( j ) .
P R = A R I ( p , q ) d p d q = A R | S R ( p , q ) + F ( p , q ) S L ( 0 ) ( p , q ) | 2 d p d q = A R { | S R ( p , q ) | 2 + | S L ( 0 ) ( p , q ) | 2 + 2 Re { S R ( p , q ) F * ( p , q ) S L ( 0 ) ( p , q ) * } } d p d q = P S N { i = 0 M d i 2 Q i i 2 + 2 i = 0 M j = i + 1 M d i d j Q i j cos ( ϕ i ϕ j ) } + P L N Q 00 + 2 P S P L N i = 0 M d i Q 0 i cos ( ω d t + ϕ i ϕ L ) ,
h r ( t ) = { 1 T cos ( ω d t ) 0 < t < T 0 else } ,
V = P S P L N i = 0 M d i Q 0 i cos ( ϕ i ϕ L ) = P S P L N i = 0 M d i { m = 1 n x crit n = 1 n y crit b m n ( i ) b m n ( 0 ) } cos ( ϕ i ϕ L ) .
V = d 0 + 1 N i = 1 M d i Q 0 i cos ( ϕ i ϕ L ) .
V = ( P L + 2 P S P L d 0 + P S d 0 2 + P S i = 1 M d i 2 ) + 2 P S N i = 1 M j = i + 1 M d i d j Q i j cos ( ϕ i ϕ j ) + 2 ( P S P L N + P S N d 0 ) i = 1 M d i Q 0 i cos ( ϕ i ϕ L ) .
λ x m S 0 m ( p , C x ) = 1 1 sin ( C x ( x p ) ) π ( x p ) S 0 m ( x , C x ) d x ,
a T m n ( x , y ) = 1 λ x m λ y n x 0 y 0 S 0 m ( x x 0 , C x ) S 0 n ( y y 0 , C y ) ,
N = n x crit n y crit = 4 x 0 p 0 λ L 4 y 0 q 0 λ L = A T A R ( λ L ) 2 .
S T ( i ) ( x , y ) = m = 1 n x crit n = 1 n y crit P S N λ x m λ y n x 0 y 0 b m n ( i ) S 0 m ( x x 0 , C x ) S 0 n ( y y 0 , C y ) .
S R ( i ) ( p , q ) e j ( 2 π λ L ) j λ L m = 1 n x crit n = 1 n y crit P S N λ x m λ y n x 0 y 0 b m n ( i ) S 0 m ( p p 0 , C x ) S 0 n ( q q 0 , C y ) ,
S L ( 0 ) ( p , q ) = e j ( 2 π λ L ) j λ L exp ( j ω L t + j ϕ L ) F ( p , q ) × P L N λ x 0 λ y 0 x 0 y 0 m = 1 n x crit n = 1 n y crit S 0 m ( p p 0 , C x ) S 0 n ( q q 0 , C y ) .
S T ( i ) ( x , y ) = P S N m = 1 n x crit n = 1 n y crit b m n ( i ) exp ( j 2 π ( m x x 0 + n y y 0 ) ) rect ( x 2 x 0 ) rect ( y 2 y 0 ) ,
S R ( i ) ( p , q ) P S N e j ( 2 π λ L ) j λ L m = 1 n x crit n = 1 n y crit b m n ( i ) sinc ( 2 x 0 ( p λ L m x 0 ) ) sinc ( 2 y 0 ( q λ L m y 0 ) ) ,
V = d 0 + 1 N i = 1 M d i Q 0 i cos ( ϕ i ϕ L ) .
P { Q 0 i = 2 m } = ( 1 2 ) N ( N N 2 + m ) , m = N 2 , , N 2 .
E { Q 0 i } = { 0 i 0 N i = 0 } ,
Var { Q 0 i } = { N i 0 0 i = 0 } .
V N ( d 0 , k 2 N ) .
Pe = k = 1 M ( 1 2 ) M + 1 ( M k ) erfc ( N 4 k ) .
ψ z ( s ) = E x { E y { exp ( s x y ) } } = E x { ψ y ( s x ) } .
f Y ( y ) = 1 π 1 y 2 W ( y ) ,
W ( y ) = { 1 | y | < 1 0 else } ,
ψ y ( s ) = I 0 ( s ) ,
f Z ( z ) = k = N 2 k 0 N 2 ( 1 2 ) N ( N N 2 + k ) 1 | 2 k | f Y ( z 2 k ) + ( 1 2 ) N ( N N 2 ) δ ( z ) ,
ψ z ( s ) = m = N 2 N 2 ( 1 2 ) N ( N N 2 + m ) I 0 ( 2 m s ) .
ψ V ( s ) = exp ( s d 0 ) ( m = N 2 N 2 ( 1 2 ) N ( N N 2 + m ) I 0 ( 2 m s N ) ) k .
V N ( P S ( k + d 0 ) + P L + 2 P S P L d 0 , 2 P S P L k N + P S 2 ( k + d 0 ) ( k + d 0 1 ) N + 2 P S 3 P L k N d 0 ) .
ψ V ( s ) = exp ( s ( P L + 2 P S P L d 0 + P S d 0 + P S k ) ) { ( m = N 2 N 2 ( 1 2 ) N ( N N 2 + m ) I 0 ( 4 s P S N m ) ) k [ ( k 1 ) 2 ] + ( m = N 2 N 2 ( 1 2 ) N ( N N 2 + m ) I 0 ( 4 s ( P S P L N + P S N d 0 ) m ) ) k } .