Abstract

In this paper we study slotted ALOHA random access optical code division multiple access packet-switching networks with a chip-level receiver structure. We use generalized optical orthogonal codes (GOOCs) and the overlapping pulse position modulation (OPPM) signaling technique to improve system performance. The impact of physical layer parameters such as the GOOC cross-correlation value and the OPPM overlapping index on key performance benchmarks such as network throughput and delay are analyzed. We also study the stability of the network based on the number of backlogged users, using the expected state drift. It has been shown that using GOOC instead of strict optical orthogonal codes improves the network average delay and throughput. Moreover, by comparing M-ary OPPM and conventional OOK signaling with a fair criterion we have demonstrated that OPPM signaling can substantially improve both steady state and transient network characteristics. This improvement can be obtained without introducing additional complexity in the receiver and transmitter structure.

© 2011 OSA

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  1. Z. A. El-Sahn, Y. M. Abdelmalek, H. M. H. Shalaby, and S. A. El-Badawy, "Performance of the R3T random-access OCDMA protocol in noisy environment," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1396‒1402 (2007).
    [CrossRef]
  2. H. M. H. Shalaby, "Optical CDMA random access protocols with and without pretransmission coordination," J. Lightwave Technol. 21, (11), 2455‒2462 (2003).
    [CrossRef]
  3. M. Ibrahimi, F. Farnoud, and J. A. Salehi, "A packet-based photonic label switching router for a multirate all-optical CDMA-based GMPLS switch," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1522‒1530 (2007).
    [CrossRef]
  4. B. M. Ghaffari, M. D. Matinfar, and J. A. Salehi, "Wireless optical CDMA LAN: digital implementation analysis," IEEE J. Sel. Areas Commun. 27, (9), 1676‒1686 (2009).
    [CrossRef]
  5. A. E. Willner, P. Saghari, and V. R. Arbab, "Advanced techniques to increase the number of users and bit rate in OCDMA networks," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1403‒1414 (2007).
    [CrossRef]
  6. C. S. Hsu and V. O. K. Li, "Performance analysis of slotted fiber-optic code division multiple access (CDMA) packet networks," IEEE Trans. Commun. 45, (7), 819‒828 (1997).
    [CrossRef]
  7. C. S. Hsu and V. O. K. Li, "Performance analysis of unslotted fiber optic code division multiple access (CDMA) packet networks," IEEE Trans. Commun. 45, (8), 978‒987 (1997).
    [CrossRef]
  8. M. A. A. Mohamed, H. M. H. Shalaby, and E. A. El-Badawy, "Performance analysis of an optical CDMA MAC protocol with variable-size sliding window," J. Lightwave Technol. 24, (10), 3590‒3597 (2006).
    [CrossRef]
  9. H. M. H. Shalaby, "A proposal of an optical CDMA random access protocol," Proc. IEEE Communications Conf. (ICC), June 2004, Paris, France, pp. 1709‒1713.
  10. S. Mashhadi and J. A. Salehi, "Optimum code structure for positive optical CDMA using normalized divergence maximization criterion," IEEE Trans. Commun. 56, 1414‒1421 (2008).
    [CrossRef]
  11. J. A. Salehi and C. A. Brackett, "Code division multiple-access techniques in optical fiber networks—part II: system performance analysis," IEEE Trans. Commun. 37, 834‒842 (1989).
    [CrossRef]
  12. S. Mashhadi and J. A. Salehi, "Code division multiple-access techniques in optical fiber networks—part III: optical AND gate receiver structure with generalized optical orthogonal codes," IEEE Trans. Commun. 45, 1457‒1468 (2006).
    [CrossRef]
  13. B. Ghaffari and J. A. Salehi, "Multiclass, multistage, and multilevel fiber-optic CDMA signaling techniques based on advanced binary optical logic gate elements," IEEE Trans. Commun. 57, 1424‒1432 (2009).
    [CrossRef]
  14. H. M. H. Shalaby, "Direct detection optical overlapping PPM-CDMA communication systems with double optical hardlimiters," J. Lightwave Technol. 17, 1158‒1165 (1999).
    [CrossRef]
  15. B. M. Ghaffari, Energy/throughput efficient signalings for optical CDMA systems [Ph.D. thesis], Sharif Univ. of Technology, 2010.
  16. D. Raychaudhuri, "Performance analysis of random access packet switched code division multiple access systems," IEEE Trans. Commun. 29, (6), 895‒901 (1981).
    [CrossRef]
  17. S. S. Lam and L. Kleinrock, "Packet switching in a multiclass broadcast channel: dynamic control procedures," IEEE Trans. Commun. 23, (9), 891‒904 (1975).
    [CrossRef]
  18. L. Kleinrock and S. S. Lam, "Packet switching in a multiclass broadcast channel: performance evaluation," IEEE Trans. Commun. 23, (4), 410‒423 (1975).
    [CrossRef]
  19. A. B. Carleial and M. E. Hellman, "Bistable behavior of ALOHA-type systems," IEEE Trans. Commun. 23, (4), 401‒410 (1975).
    [CrossRef]

2009 (2)

B. M. Ghaffari, M. D. Matinfar, and J. A. Salehi, "Wireless optical CDMA LAN: digital implementation analysis," IEEE J. Sel. Areas Commun. 27, (9), 1676‒1686 (2009).
[CrossRef]

B. Ghaffari and J. A. Salehi, "Multiclass, multistage, and multilevel fiber-optic CDMA signaling techniques based on advanced binary optical logic gate elements," IEEE Trans. Commun. 57, 1424‒1432 (2009).
[CrossRef]

2008 (1)

S. Mashhadi and J. A. Salehi, "Optimum code structure for positive optical CDMA using normalized divergence maximization criterion," IEEE Trans. Commun. 56, 1414‒1421 (2008).
[CrossRef]

2007 (3)

A. E. Willner, P. Saghari, and V. R. Arbab, "Advanced techniques to increase the number of users and bit rate in OCDMA networks," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1403‒1414 (2007).
[CrossRef]

Z. A. El-Sahn, Y. M. Abdelmalek, H. M. H. Shalaby, and S. A. El-Badawy, "Performance of the R3T random-access OCDMA protocol in noisy environment," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1396‒1402 (2007).
[CrossRef]

M. Ibrahimi, F. Farnoud, and J. A. Salehi, "A packet-based photonic label switching router for a multirate all-optical CDMA-based GMPLS switch," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1522‒1530 (2007).
[CrossRef]

2006 (2)

M. A. A. Mohamed, H. M. H. Shalaby, and E. A. El-Badawy, "Performance analysis of an optical CDMA MAC protocol with variable-size sliding window," J. Lightwave Technol. 24, (10), 3590‒3597 (2006).
[CrossRef]

S. Mashhadi and J. A. Salehi, "Code division multiple-access techniques in optical fiber networks—part III: optical AND gate receiver structure with generalized optical orthogonal codes," IEEE Trans. Commun. 45, 1457‒1468 (2006).
[CrossRef]

2003 (1)

1999 (1)

1997 (2)

C. S. Hsu and V. O. K. Li, "Performance analysis of slotted fiber-optic code division multiple access (CDMA) packet networks," IEEE Trans. Commun. 45, (7), 819‒828 (1997).
[CrossRef]

C. S. Hsu and V. O. K. Li, "Performance analysis of unslotted fiber optic code division multiple access (CDMA) packet networks," IEEE Trans. Commun. 45, (8), 978‒987 (1997).
[CrossRef]

1989 (1)

J. A. Salehi and C. A. Brackett, "Code division multiple-access techniques in optical fiber networks—part II: system performance analysis," IEEE Trans. Commun. 37, 834‒842 (1989).
[CrossRef]

1981 (1)

D. Raychaudhuri, "Performance analysis of random access packet switched code division multiple access systems," IEEE Trans. Commun. 29, (6), 895‒901 (1981).
[CrossRef]

1975 (3)

S. S. Lam and L. Kleinrock, "Packet switching in a multiclass broadcast channel: dynamic control procedures," IEEE Trans. Commun. 23, (9), 891‒904 (1975).
[CrossRef]

L. Kleinrock and S. S. Lam, "Packet switching in a multiclass broadcast channel: performance evaluation," IEEE Trans. Commun. 23, (4), 410‒423 (1975).
[CrossRef]

A. B. Carleial and M. E. Hellman, "Bistable behavior of ALOHA-type systems," IEEE Trans. Commun. 23, (4), 401‒410 (1975).
[CrossRef]

Abdelmalek, Y. M.

Z. A. El-Sahn, Y. M. Abdelmalek, H. M. H. Shalaby, and S. A. El-Badawy, "Performance of the R3T random-access OCDMA protocol in noisy environment," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1396‒1402 (2007).
[CrossRef]

Arbab, V. R.

A. E. Willner, P. Saghari, and V. R. Arbab, "Advanced techniques to increase the number of users and bit rate in OCDMA networks," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1403‒1414 (2007).
[CrossRef]

Brackett, C. A.

J. A. Salehi and C. A. Brackett, "Code division multiple-access techniques in optical fiber networks—part II: system performance analysis," IEEE Trans. Commun. 37, 834‒842 (1989).
[CrossRef]

Carleial, A. B.

A. B. Carleial and M. E. Hellman, "Bistable behavior of ALOHA-type systems," IEEE Trans. Commun. 23, (4), 401‒410 (1975).
[CrossRef]

El-Badawy, S. A.

Z. A. El-Sahn, Y. M. Abdelmalek, H. M. H. Shalaby, and S. A. El-Badawy, "Performance of the R3T random-access OCDMA protocol in noisy environment," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1396‒1402 (2007).
[CrossRef]

El-Badawy, E. A.

El-Sahn, Z. A.

Z. A. El-Sahn, Y. M. Abdelmalek, H. M. H. Shalaby, and S. A. El-Badawy, "Performance of the R3T random-access OCDMA protocol in noisy environment," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1396‒1402 (2007).
[CrossRef]

Farnoud, F.

M. Ibrahimi, F. Farnoud, and J. A. Salehi, "A packet-based photonic label switching router for a multirate all-optical CDMA-based GMPLS switch," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1522‒1530 (2007).
[CrossRef]

Ghaffari, B.

B. Ghaffari and J. A. Salehi, "Multiclass, multistage, and multilevel fiber-optic CDMA signaling techniques based on advanced binary optical logic gate elements," IEEE Trans. Commun. 57, 1424‒1432 (2009).
[CrossRef]

Ghaffari, B. M.

B. M. Ghaffari, M. D. Matinfar, and J. A. Salehi, "Wireless optical CDMA LAN: digital implementation analysis," IEEE J. Sel. Areas Commun. 27, (9), 1676‒1686 (2009).
[CrossRef]

B. M. Ghaffari, Energy/throughput efficient signalings for optical CDMA systems [Ph.D. thesis], Sharif Univ. of Technology, 2010.

Hellman, M. E.

A. B. Carleial and M. E. Hellman, "Bistable behavior of ALOHA-type systems," IEEE Trans. Commun. 23, (4), 401‒410 (1975).
[CrossRef]

Hsu, C. S.

C. S. Hsu and V. O. K. Li, "Performance analysis of slotted fiber-optic code division multiple access (CDMA) packet networks," IEEE Trans. Commun. 45, (7), 819‒828 (1997).
[CrossRef]

C. S. Hsu and V. O. K. Li, "Performance analysis of unslotted fiber optic code division multiple access (CDMA) packet networks," IEEE Trans. Commun. 45, (8), 978‒987 (1997).
[CrossRef]

Ibrahimi, M.

M. Ibrahimi, F. Farnoud, and J. A. Salehi, "A packet-based photonic label switching router for a multirate all-optical CDMA-based GMPLS switch," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1522‒1530 (2007).
[CrossRef]

Kleinrock, L.

S. S. Lam and L. Kleinrock, "Packet switching in a multiclass broadcast channel: dynamic control procedures," IEEE Trans. Commun. 23, (9), 891‒904 (1975).
[CrossRef]

L. Kleinrock and S. S. Lam, "Packet switching in a multiclass broadcast channel: performance evaluation," IEEE Trans. Commun. 23, (4), 410‒423 (1975).
[CrossRef]

Lam, S. S.

L. Kleinrock and S. S. Lam, "Packet switching in a multiclass broadcast channel: performance evaluation," IEEE Trans. Commun. 23, (4), 410‒423 (1975).
[CrossRef]

S. S. Lam and L. Kleinrock, "Packet switching in a multiclass broadcast channel: dynamic control procedures," IEEE Trans. Commun. 23, (9), 891‒904 (1975).
[CrossRef]

Li, V. O. K.

C. S. Hsu and V. O. K. Li, "Performance analysis of slotted fiber-optic code division multiple access (CDMA) packet networks," IEEE Trans. Commun. 45, (7), 819‒828 (1997).
[CrossRef]

C. S. Hsu and V. O. K. Li, "Performance analysis of unslotted fiber optic code division multiple access (CDMA) packet networks," IEEE Trans. Commun. 45, (8), 978‒987 (1997).
[CrossRef]

Mashhadi, S.

S. Mashhadi and J. A. Salehi, "Optimum code structure for positive optical CDMA using normalized divergence maximization criterion," IEEE Trans. Commun. 56, 1414‒1421 (2008).
[CrossRef]

S. Mashhadi and J. A. Salehi, "Code division multiple-access techniques in optical fiber networks—part III: optical AND gate receiver structure with generalized optical orthogonal codes," IEEE Trans. Commun. 45, 1457‒1468 (2006).
[CrossRef]

Matinfar, M. D.

B. M. Ghaffari, M. D. Matinfar, and J. A. Salehi, "Wireless optical CDMA LAN: digital implementation analysis," IEEE J. Sel. Areas Commun. 27, (9), 1676‒1686 (2009).
[CrossRef]

Mohamed, M. A. A.

Raychaudhuri, D.

D. Raychaudhuri, "Performance analysis of random access packet switched code division multiple access systems," IEEE Trans. Commun. 29, (6), 895‒901 (1981).
[CrossRef]

Saghari, P.

A. E. Willner, P. Saghari, and V. R. Arbab, "Advanced techniques to increase the number of users and bit rate in OCDMA networks," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1403‒1414 (2007).
[CrossRef]

Salehi, J. A.

B. Ghaffari and J. A. Salehi, "Multiclass, multistage, and multilevel fiber-optic CDMA signaling techniques based on advanced binary optical logic gate elements," IEEE Trans. Commun. 57, 1424‒1432 (2009).
[CrossRef]

B. M. Ghaffari, M. D. Matinfar, and J. A. Salehi, "Wireless optical CDMA LAN: digital implementation analysis," IEEE J. Sel. Areas Commun. 27, (9), 1676‒1686 (2009).
[CrossRef]

S. Mashhadi and J. A. Salehi, "Optimum code structure for positive optical CDMA using normalized divergence maximization criterion," IEEE Trans. Commun. 56, 1414‒1421 (2008).
[CrossRef]

M. Ibrahimi, F. Farnoud, and J. A. Salehi, "A packet-based photonic label switching router for a multirate all-optical CDMA-based GMPLS switch," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1522‒1530 (2007).
[CrossRef]

S. Mashhadi and J. A. Salehi, "Code division multiple-access techniques in optical fiber networks—part III: optical AND gate receiver structure with generalized optical orthogonal codes," IEEE Trans. Commun. 45, 1457‒1468 (2006).
[CrossRef]

J. A. Salehi and C. A. Brackett, "Code division multiple-access techniques in optical fiber networks—part II: system performance analysis," IEEE Trans. Commun. 37, 834‒842 (1989).
[CrossRef]

Shalaby, H. M. H.

Willner, A. E.

A. E. Willner, P. Saghari, and V. R. Arbab, "Advanced techniques to increase the number of users and bit rate in OCDMA networks," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1403‒1414 (2007).
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

B. M. Ghaffari, M. D. Matinfar, and J. A. Salehi, "Wireless optical CDMA LAN: digital implementation analysis," IEEE J. Sel. Areas Commun. 27, (9), 1676‒1686 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (3)

A. E. Willner, P. Saghari, and V. R. Arbab, "Advanced techniques to increase the number of users and bit rate in OCDMA networks," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1403‒1414 (2007).
[CrossRef]

Z. A. El-Sahn, Y. M. Abdelmalek, H. M. H. Shalaby, and S. A. El-Badawy, "Performance of the R3T random-access OCDMA protocol in noisy environment," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1396‒1402 (2007).
[CrossRef]

M. Ibrahimi, F. Farnoud, and J. A. Salehi, "A packet-based photonic label switching router for a multirate all-optical CDMA-based GMPLS switch," IEEE J. Sel. Top. Quantum Electron. 13, (5), 1522‒1530 (2007).
[CrossRef]

IEEE Trans. Commun. (10)

D. Raychaudhuri, "Performance analysis of random access packet switched code division multiple access systems," IEEE Trans. Commun. 29, (6), 895‒901 (1981).
[CrossRef]

S. S. Lam and L. Kleinrock, "Packet switching in a multiclass broadcast channel: dynamic control procedures," IEEE Trans. Commun. 23, (9), 891‒904 (1975).
[CrossRef]

L. Kleinrock and S. S. Lam, "Packet switching in a multiclass broadcast channel: performance evaluation," IEEE Trans. Commun. 23, (4), 410‒423 (1975).
[CrossRef]

A. B. Carleial and M. E. Hellman, "Bistable behavior of ALOHA-type systems," IEEE Trans. Commun. 23, (4), 401‒410 (1975).
[CrossRef]

C. S. Hsu and V. O. K. Li, "Performance analysis of slotted fiber-optic code division multiple access (CDMA) packet networks," IEEE Trans. Commun. 45, (7), 819‒828 (1997).
[CrossRef]

C. S. Hsu and V. O. K. Li, "Performance analysis of unslotted fiber optic code division multiple access (CDMA) packet networks," IEEE Trans. Commun. 45, (8), 978‒987 (1997).
[CrossRef]

S. Mashhadi and J. A. Salehi, "Optimum code structure for positive optical CDMA using normalized divergence maximization criterion," IEEE Trans. Commun. 56, 1414‒1421 (2008).
[CrossRef]

J. A. Salehi and C. A. Brackett, "Code division multiple-access techniques in optical fiber networks—part II: system performance analysis," IEEE Trans. Commun. 37, 834‒842 (1989).
[CrossRef]

S. Mashhadi and J. A. Salehi, "Code division multiple-access techniques in optical fiber networks—part III: optical AND gate receiver structure with generalized optical orthogonal codes," IEEE Trans. Commun. 45, 1457‒1468 (2006).
[CrossRef]

B. Ghaffari and J. A. Salehi, "Multiclass, multistage, and multilevel fiber-optic CDMA signaling techniques based on advanced binary optical logic gate elements," IEEE Trans. Commun. 57, 1424‒1432 (2009).
[CrossRef]

J. Lightwave Technol. (3)

Other (2)

B. M. Ghaffari, Energy/throughput efficient signalings for optical CDMA systems [Ph.D. thesis], Sharif Univ. of Technology, 2010.

H. M. H. Shalaby, "A proposal of an optical CDMA random access protocol," Proc. IEEE Communications Conf. (ICC), June 2004, Paris, France, pp. 1709‒1713.

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Figures (12)

Fig. 1
Fig. 1

Optical CDMA network with N users connected to a star coupler.

Fig. 2
Fig. 2

Packet flow in a random access CDMA system.

Fig. 3
Fig. 3

M-ary overlapping PPM signaling for a 100010001100 signature code with parameters M = 4 , γ = 4 , L = 12 , and w = 4 .

Fig. 4
Fig. 4

Expected state drift d ( n ) and occupancy probability μ ( n ) for stable and bistable demo schemes.

Fig. 5
Fig. 5

Network throughput versus offered traffic with the binomial packet arrival model.

Fig. 6
Fig. 6

Effect of retransmission probability on throughput for varying transmission probability and signaling. λ = 3 .

Fig. 7
Fig. 7

Throughput–delay characteristics. p r = 0 . 3 .

Fig. 8
Fig. 8

Expected drift in state for OOK and OPPM signaling with λ = 3 .

Fig. 9
Fig. 9

Undesirable region of OOK and OPPM signaling with λ = 3 .

Fig. 10
Fig. 10

f β s for the entire probability plane. In the case of bistable conditions, the first stable point is considered.

Fig. 11
Fig. 11

Average delay versus average throughput. Each curve is plotted in fixed p 0 and varying p r in the 0.05 to 0.95 interval.

Fig. 12
Fig. 12

Optimum p r value versus p 0 . The solid curve is fitted to discrete points.

Equations (30)

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

P S = s | Z = z = z s P c s ( z ) P E z s ( z ) .
β = E s = E E S | Z = E s = 0 Z s Z s P c s Z P E Z s Z = z = 1 N z P c ( z ) f Z ( z ) .
f Z ( z ) = N z p z (1 − p) N z .
β = z = 1 N z N z y N z 1 y N N z P c ( z ) ,
β = y 1 y N N 1 1 + ( N 1 ) y / N ( 1 y / N ) P c ( 2 ) + ( N 1 ) ( N 2 ) 2 ! ( y / N ) 2 ( 1 y / N ) 2 P c ( 3 ) + + ( y / N ) N 1 ( 1 y / N ) N 1 P c ( N ) .
Pr { S-RTX = k , U-NTX = l | x t = n } = ξ 0 = l N n ξ r = k n b [ l , ξ 0 , P E ( ξ 0 + ξ r ) ] . b [ k , ξ r , P c ( ξ 0 + ξ r ) ] b ( ξ 0 , N n , p 0 ) . b ( ξ r , n , p r ) .
p n m = j = 0 min ( n , N m ) Pr { S-RTX = j , U-NTX = m n + j | x t = n } for  m n j = 0 min ( m , N n ) Pr { S-RTX = n m + j , U-NTX = j | x t = n } for  m n .
μ T = μ T P ,
f Z ( z | x = n ) = j = max ( z n , 0 ) min ( z , N n ) Pr { NTX = j , RTX = z j | x = n } = j = max ( z n , 0 ) min ( z , N n ) b ( j , N n , p 0 ) b ( z j , n , p r ) ,
β = z = 1 N z P c ( z ) n = 0 N f Z ( z | x = n ) μ ( n ) .
D = n ̄ β .
T = ( M 1 ) τ γ + τ = ( M 1 + γ ) L γ T c .
P e = 1 2 Pr ( error | 0 ) + 1 2 Pr ( error | 1 ) .
Pr ( e r r o r | 0 ) = Pr ( α 1 > 0 , α 2 > 0 , , α w > 0 ) ,
Pr ( α 1 > 0 , α 2 > 0 , , α w > 0 ) = 1 Pr ( α 1 = 0  or  α 2 = 0  or   or  α w = 0 ) = 1 + k = 1 w ( 1 ) k w k Pr ( α 1 = α 2 = = α k = 0 ) .
Pr ( α 1 = α 2 = = α k = 0 ) = 1 + l = 1 λ ( 1 ) l k l ρ l N 1 ,
ρ l = Pr ( α 1 = = α l = 1 | one interfering user ) = p l + w 1 1 p l + 1 + + w l λ l p λ = i = l λ w l i l p i .
P e = 1 2 1 + k = 1 w ( 1 ) k w k 1 + l = 1 λ i = l λ ( 1 ) l k l w l i l N p i .
Pr { β i = 1 } = Pr { α i 1 > 0 , α i 2 > 0 , , α i w > 0 } .
P s = Pr { β i = 1 ,  some  i 0 } ;
P s ( M 1 ) Pr { β 1 = 1 } .
P e = ( M 1 ) 1 + k = 1 w ( 1 ) k w k 1 + l = 1 λ i = l λ ( 1 ) l k l w l i l N p i ,
j = 1 λ w j p j = w 2 2 L .
j = 1 λ w j p j = P OPPM .
p j = P OPPM w 2 / 2 L .
d ( n ) = m = 0 N ( m n ) p n m .
N ( L , w , λ ) ( L 1 ) L 2 L λ w w 1 w λ .
β ( n 0 ) = z = 1 N z P c ( z ) f Z ( z | x = n 0 ) .
f β s = μ OPPM s β OPPM s μ OOK s β OOK s .
p r = 0 . 9 p 0 0 . 16 10 . 04 ( p 0 ) 3 14 . 92 ( p 0 ) 2 + 5 . 194 ( p 0 ) 0 . 3501 0 . 16 < p 0 < 0 . 8 0 . 06 p 0 0 . 8 .