Abstract

We present the design of efficient all-optical code-division multiplexing (AOCDM) systems that can transmit multiple-bit-rate (MBR) data signals over a common optical fiber. This is achieved when the proposed strict optical orthogonal code (OOC) of autocorrelation and cross-correlation constraints of 1 are used but without performance degradation compared with the use of conventional OOC. We describe the design of various strict OOC’s by employing the useful concept of slot distances, and methods of code construction are also presented. Moreover, we give the principle of MBR data transmissions in an AOCDM system. It is shown that AOCDM systems using the proposed OOC can effectively transmit multiuser MBR and equal-bit-rate (EBR) data with no increase of system complexity. In principle, optimal strict OOC’s need the same or a slightly larger system bandwidth compared with optimal conventional OOC’s for EBR operation, whereas the former can require a smaller system bandwidth and have a better system performance than the latter for MBR transmissions. A new, to our knowledge, family of strict OOC’s is also introduced, whose code words can have nonconstant weights to support multiuser communications with different transmission quality. Furthermore, we design low-cost AOCDM transmitters that are based on an efficient gain-switching scheme that does not require an electro-optic intensity modulator to on–off modulate an optical clock pulse stream at each transmitter. The basic operation principle is also experimentally demonstrated.

© 2000 Optical Society of America

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References

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  1. C. DeCusatis, “Optical data communication: fundamentals and future directions,” Opt. Eng. 37, 3082–3099 (1998).
    [CrossRef]
  2. P. R. Prucnal, M. A. Santoro, T. R. Fan, “Spread spectrum fiber-optic local area network using optical processing,” J. Lightwave Technol. LT-4, 547–554 (1986).
    [CrossRef]
  3. P. R. Prucnal, M. A. Santoro, S. K. Sehgal, “Ultrafast all-optical synchronous multiple access fiber networks,” IEEE J. Sel. Areas Commun. SAC-4, 1484–1493 (1986).
    [CrossRef]
  4. F. R. K. Chung, J. A. Salehi, V. K. Wei, “Optical orthogonal codes: design, analysis, and applications,” IEEE Trans. Inf. Theory 35, 595–604 (1989).
    [CrossRef]
  5. J. A. Salehi, “Code division multiple-access techniques in optical fiber networks—part I: fundamental principles,” IEEE Trans. Commun. 37, 824–833 (1989).
    [CrossRef]
  6. H. Chung, P. V. Kumar, “Optical orthogonal codes—new bounds and an optimal construction,” IEEE Trans. Inf. Theory 36, 866–873 (1990).
    [CrossRef]
  7. S. V. Maric, M. D. Hahm, E. L. Titlebaum, “Construction and performance analysis of a new family of optical orthogonal codes for CDMA fiber-optic networks,” IEEE Trans. Commun. 43, 485–489 (1995).
    [CrossRef]
  8. J. A. Salehi, C. A. Brackett, “Code division multiple-access techniques in optical fiber networks—part II: systems performance analysis,” IEEE Trans. Commun. 37, 834–842 (1989).
    [CrossRef]
  9. M. Azizoglu, J. A. Salehi, Y. Li, “Optical CDMA via temporal codes,” IEEE Trans. Commun. 40, 1162–1170 (1992).
    [CrossRef]
  10. R. M. Gagliardi, A. J. Mendez, M. R. Dale, E. Park, “Fiber-optic digital video multiplexing using optical CDMA,” J. Lightwave Technol. 11, 20–26 (1993).
    [CrossRef]
  11. R. Petrovic, S. Holmes, “Orthogonal codes for CDMA optical fibre LANs with variable bit interval,” Electron. Lett. 26, 662–664 (1990).
    [CrossRef]
  12. J.-G. Zhang, “Novel techniques and architecture for all-optical code-division and time-division multiple access networks,” Ph.D. dissertation (University of Parma, Parma, Italy, 1994) (in Italian).
  13. J.-G. Zhang, “Variable-bit-rate video transmission systems using optical fiber code-division multiplexing scheme,” IEEE Trans. Consumer Electron. 42, 874–884 (1996).
    [CrossRef]
  14. J.-G. Zhang, “Strict optical orthogonal codes for purely asynchronous code-division multiple-access applications,” Appl. Opt. 35, 6996–6999 (1996).
    [CrossRef] [PubMed]
  15. J.-G. Zhang, “Novel optical fiber code-division multiple access networks supporting real-time multichannel variable-bit-rate (VBR) video distributions,” IEEE Trans. Broadcast. 43, 339–349 (1997).
    [CrossRef]
  16. Y. Wu, B. Caron, “Digital television terrestrial broadcasting,” IEEE Commun. Mag. 32, 46–52 (1994).
    [CrossRef]
  17. T. Klove, “Bounds on the size of optimal difference triangle sets,” IEEE Trans. Inf. Theory 34, 355–361 (1988).
    [CrossRef]
  18. C. A. Siller, M. Shafi, SONET/SDH—A Sourcebook of Synchronous Networking (Institute of Electrical and Electronics Engineers, New York, 1996).
  19. Y.-G. Lu, P. Zhou, J. Cheng, “A directly modulated pulse-compressed and time-multiplexed optical source for high-speed multiple-access networks,” IEEE Photonics Technol. Lett. 5, 905–907 (1993).
    [CrossRef]
  20. L. P. Barry, J. Debeau, R. Boittin, “40-nm tunable source of picosecond pulses at 10 GHz by external injection into a gain-switched FP laser,” in Proceedings of the 20th European Conference on Optical Communication (Istituto Internazionale delle Comunicazioni, Genova, Italy, 1994), pp. 369–372.

1998

C. DeCusatis, “Optical data communication: fundamentals and future directions,” Opt. Eng. 37, 3082–3099 (1998).
[CrossRef]

1997

J.-G. Zhang, “Novel optical fiber code-division multiple access networks supporting real-time multichannel variable-bit-rate (VBR) video distributions,” IEEE Trans. Broadcast. 43, 339–349 (1997).
[CrossRef]

1996

J.-G. Zhang, “Variable-bit-rate video transmission systems using optical fiber code-division multiplexing scheme,” IEEE Trans. Consumer Electron. 42, 874–884 (1996).
[CrossRef]

J.-G. Zhang, “Strict optical orthogonal codes for purely asynchronous code-division multiple-access applications,” Appl. Opt. 35, 6996–6999 (1996).
[CrossRef] [PubMed]

1995

S. V. Maric, M. D. Hahm, E. L. Titlebaum, “Construction and performance analysis of a new family of optical orthogonal codes for CDMA fiber-optic networks,” IEEE Trans. Commun. 43, 485–489 (1995).
[CrossRef]

1994

Y. Wu, B. Caron, “Digital television terrestrial broadcasting,” IEEE Commun. Mag. 32, 46–52 (1994).
[CrossRef]

1993

Y.-G. Lu, P. Zhou, J. Cheng, “A directly modulated pulse-compressed and time-multiplexed optical source for high-speed multiple-access networks,” IEEE Photonics Technol. Lett. 5, 905–907 (1993).
[CrossRef]

R. M. Gagliardi, A. J. Mendez, M. R. Dale, E. Park, “Fiber-optic digital video multiplexing using optical CDMA,” J. Lightwave Technol. 11, 20–26 (1993).
[CrossRef]

1992

M. Azizoglu, J. A. Salehi, Y. Li, “Optical CDMA via temporal codes,” IEEE Trans. Commun. 40, 1162–1170 (1992).
[CrossRef]

1990

H. Chung, P. V. Kumar, “Optical orthogonal codes—new bounds and an optimal construction,” IEEE Trans. Inf. Theory 36, 866–873 (1990).
[CrossRef]

R. Petrovic, S. Holmes, “Orthogonal codes for CDMA optical fibre LANs with variable bit interval,” Electron. Lett. 26, 662–664 (1990).
[CrossRef]

1989

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

F. R. K. Chung, J. A. Salehi, V. K. Wei, “Optical orthogonal codes: design, analysis, and applications,” IEEE Trans. Inf. Theory 35, 595–604 (1989).
[CrossRef]

J. A. Salehi, “Code division multiple-access techniques in optical fiber networks—part I: fundamental principles,” IEEE Trans. Commun. 37, 824–833 (1989).
[CrossRef]

1988

T. Klove, “Bounds on the size of optimal difference triangle sets,” IEEE Trans. Inf. Theory 34, 355–361 (1988).
[CrossRef]

1986

P. R. Prucnal, M. A. Santoro, T. R. Fan, “Spread spectrum fiber-optic local area network using optical processing,” J. Lightwave Technol. LT-4, 547–554 (1986).
[CrossRef]

P. R. Prucnal, M. A. Santoro, S. K. Sehgal, “Ultrafast all-optical synchronous multiple access fiber networks,” IEEE J. Sel. Areas Commun. SAC-4, 1484–1493 (1986).
[CrossRef]

Azizoglu, M.

M. Azizoglu, J. A. Salehi, Y. Li, “Optical CDMA via temporal codes,” IEEE Trans. Commun. 40, 1162–1170 (1992).
[CrossRef]

Barry, L. P.

L. P. Barry, J. Debeau, R. Boittin, “40-nm tunable source of picosecond pulses at 10 GHz by external injection into a gain-switched FP laser,” in Proceedings of the 20th European Conference on Optical Communication (Istituto Internazionale delle Comunicazioni, Genova, Italy, 1994), pp. 369–372.

Boittin, R.

L. P. Barry, J. Debeau, R. Boittin, “40-nm tunable source of picosecond pulses at 10 GHz by external injection into a gain-switched FP laser,” in Proceedings of the 20th European Conference on Optical Communication (Istituto Internazionale delle Comunicazioni, Genova, Italy, 1994), pp. 369–372.

Brackett, C. A.

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

Caron, B.

Y. Wu, B. Caron, “Digital television terrestrial broadcasting,” IEEE Commun. Mag. 32, 46–52 (1994).
[CrossRef]

Cheng, J.

Y.-G. Lu, P. Zhou, J. Cheng, “A directly modulated pulse-compressed and time-multiplexed optical source for high-speed multiple-access networks,” IEEE Photonics Technol. Lett. 5, 905–907 (1993).
[CrossRef]

Chung, F. R. K.

F. R. K. Chung, J. A. Salehi, V. K. Wei, “Optical orthogonal codes: design, analysis, and applications,” IEEE Trans. Inf. Theory 35, 595–604 (1989).
[CrossRef]

Chung, H.

H. Chung, P. V. Kumar, “Optical orthogonal codes—new bounds and an optimal construction,” IEEE Trans. Inf. Theory 36, 866–873 (1990).
[CrossRef]

Dale, M. R.

R. M. Gagliardi, A. J. Mendez, M. R. Dale, E. Park, “Fiber-optic digital video multiplexing using optical CDMA,” J. Lightwave Technol. 11, 20–26 (1993).
[CrossRef]

Debeau, J.

L. P. Barry, J. Debeau, R. Boittin, “40-nm tunable source of picosecond pulses at 10 GHz by external injection into a gain-switched FP laser,” in Proceedings of the 20th European Conference on Optical Communication (Istituto Internazionale delle Comunicazioni, Genova, Italy, 1994), pp. 369–372.

DeCusatis, C.

C. DeCusatis, “Optical data communication: fundamentals and future directions,” Opt. Eng. 37, 3082–3099 (1998).
[CrossRef]

Fan, T. R.

P. R. Prucnal, M. A. Santoro, T. R. Fan, “Spread spectrum fiber-optic local area network using optical processing,” J. Lightwave Technol. LT-4, 547–554 (1986).
[CrossRef]

Gagliardi, R. M.

R. M. Gagliardi, A. J. Mendez, M. R. Dale, E. Park, “Fiber-optic digital video multiplexing using optical CDMA,” J. Lightwave Technol. 11, 20–26 (1993).
[CrossRef]

Hahm, M. D.

S. V. Maric, M. D. Hahm, E. L. Titlebaum, “Construction and performance analysis of a new family of optical orthogonal codes for CDMA fiber-optic networks,” IEEE Trans. Commun. 43, 485–489 (1995).
[CrossRef]

Holmes, S.

R. Petrovic, S. Holmes, “Orthogonal codes for CDMA optical fibre LANs with variable bit interval,” Electron. Lett. 26, 662–664 (1990).
[CrossRef]

Klove, T.

T. Klove, “Bounds on the size of optimal difference triangle sets,” IEEE Trans. Inf. Theory 34, 355–361 (1988).
[CrossRef]

Kumar, P. V.

H. Chung, P. V. Kumar, “Optical orthogonal codes—new bounds and an optimal construction,” IEEE Trans. Inf. Theory 36, 866–873 (1990).
[CrossRef]

Li, Y.

M. Azizoglu, J. A. Salehi, Y. Li, “Optical CDMA via temporal codes,” IEEE Trans. Commun. 40, 1162–1170 (1992).
[CrossRef]

Lu, Y.-G.

Y.-G. Lu, P. Zhou, J. Cheng, “A directly modulated pulse-compressed and time-multiplexed optical source for high-speed multiple-access networks,” IEEE Photonics Technol. Lett. 5, 905–907 (1993).
[CrossRef]

Maric, S. V.

S. V. Maric, M. D. Hahm, E. L. Titlebaum, “Construction and performance analysis of a new family of optical orthogonal codes for CDMA fiber-optic networks,” IEEE Trans. Commun. 43, 485–489 (1995).
[CrossRef]

Mendez, A. J.

R. M. Gagliardi, A. J. Mendez, M. R. Dale, E. Park, “Fiber-optic digital video multiplexing using optical CDMA,” J. Lightwave Technol. 11, 20–26 (1993).
[CrossRef]

Park, E.

R. M. Gagliardi, A. J. Mendez, M. R. Dale, E. Park, “Fiber-optic digital video multiplexing using optical CDMA,” J. Lightwave Technol. 11, 20–26 (1993).
[CrossRef]

Petrovic, R.

R. Petrovic, S. Holmes, “Orthogonal codes for CDMA optical fibre LANs with variable bit interval,” Electron. Lett. 26, 662–664 (1990).
[CrossRef]

Prucnal, P. R.

P. R. Prucnal, M. A. Santoro, S. K. Sehgal, “Ultrafast all-optical synchronous multiple access fiber networks,” IEEE J. Sel. Areas Commun. SAC-4, 1484–1493 (1986).
[CrossRef]

P. R. Prucnal, M. A. Santoro, T. R. Fan, “Spread spectrum fiber-optic local area network using optical processing,” J. Lightwave Technol. LT-4, 547–554 (1986).
[CrossRef]

Salehi, J. A.

M. Azizoglu, J. A. Salehi, Y. Li, “Optical CDMA via temporal codes,” IEEE Trans. Commun. 40, 1162–1170 (1992).
[CrossRef]

J. A. Salehi, “Code division multiple-access techniques in optical fiber networks—part I: fundamental principles,” IEEE Trans. Commun. 37, 824–833 (1989).
[CrossRef]

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

F. R. K. Chung, J. A. Salehi, V. K. Wei, “Optical orthogonal codes: design, analysis, and applications,” IEEE Trans. Inf. Theory 35, 595–604 (1989).
[CrossRef]

Santoro, M. A.

P. R. Prucnal, M. A. Santoro, T. R. Fan, “Spread spectrum fiber-optic local area network using optical processing,” J. Lightwave Technol. LT-4, 547–554 (1986).
[CrossRef]

P. R. Prucnal, M. A. Santoro, S. K. Sehgal, “Ultrafast all-optical synchronous multiple access fiber networks,” IEEE J. Sel. Areas Commun. SAC-4, 1484–1493 (1986).
[CrossRef]

Sehgal, S. K.

P. R. Prucnal, M. A. Santoro, S. K. Sehgal, “Ultrafast all-optical synchronous multiple access fiber networks,” IEEE J. Sel. Areas Commun. SAC-4, 1484–1493 (1986).
[CrossRef]

Shafi, M.

C. A. Siller, M. Shafi, SONET/SDH—A Sourcebook of Synchronous Networking (Institute of Electrical and Electronics Engineers, New York, 1996).

Siller, C. A.

C. A. Siller, M. Shafi, SONET/SDH—A Sourcebook of Synchronous Networking (Institute of Electrical and Electronics Engineers, New York, 1996).

Titlebaum, E. L.

S. V. Maric, M. D. Hahm, E. L. Titlebaum, “Construction and performance analysis of a new family of optical orthogonal codes for CDMA fiber-optic networks,” IEEE Trans. Commun. 43, 485–489 (1995).
[CrossRef]

Wei, V. K.

F. R. K. Chung, J. A. Salehi, V. K. Wei, “Optical orthogonal codes: design, analysis, and applications,” IEEE Trans. Inf. Theory 35, 595–604 (1989).
[CrossRef]

Wu, Y.

Y. Wu, B. Caron, “Digital television terrestrial broadcasting,” IEEE Commun. Mag. 32, 46–52 (1994).
[CrossRef]

Zhang, J.-G.

J.-G. Zhang, “Novel optical fiber code-division multiple access networks supporting real-time multichannel variable-bit-rate (VBR) video distributions,” IEEE Trans. Broadcast. 43, 339–349 (1997).
[CrossRef]

J.-G. Zhang, “Variable-bit-rate video transmission systems using optical fiber code-division multiplexing scheme,” IEEE Trans. Consumer Electron. 42, 874–884 (1996).
[CrossRef]

J.-G. Zhang, “Strict optical orthogonal codes for purely asynchronous code-division multiple-access applications,” Appl. Opt. 35, 6996–6999 (1996).
[CrossRef] [PubMed]

J.-G. Zhang, “Novel techniques and architecture for all-optical code-division and time-division multiple access networks,” Ph.D. dissertation (University of Parma, Parma, Italy, 1994) (in Italian).

Zhou, P.

Y.-G. Lu, P. Zhou, J. Cheng, “A directly modulated pulse-compressed and time-multiplexed optical source for high-speed multiple-access networks,” IEEE Photonics Technol. Lett. 5, 905–907 (1993).
[CrossRef]

Appl. Opt.

Electron. Lett.

R. Petrovic, S. Holmes, “Orthogonal codes for CDMA optical fibre LANs with variable bit interval,” Electron. Lett. 26, 662–664 (1990).
[CrossRef]

IEEE Commun. Mag.

Y. Wu, B. Caron, “Digital television terrestrial broadcasting,” IEEE Commun. Mag. 32, 46–52 (1994).
[CrossRef]

IEEE J. Sel. Areas Commun.

P. R. Prucnal, M. A. Santoro, S. K. Sehgal, “Ultrafast all-optical synchronous multiple access fiber networks,” IEEE J. Sel. Areas Commun. SAC-4, 1484–1493 (1986).
[CrossRef]

IEEE Photonics Technol. Lett.

Y.-G. Lu, P. Zhou, J. Cheng, “A directly modulated pulse-compressed and time-multiplexed optical source for high-speed multiple-access networks,” IEEE Photonics Technol. Lett. 5, 905–907 (1993).
[CrossRef]

IEEE Trans. Broadcast.

J.-G. Zhang, “Novel optical fiber code-division multiple access networks supporting real-time multichannel variable-bit-rate (VBR) video distributions,” IEEE Trans. Broadcast. 43, 339–349 (1997).
[CrossRef]

IEEE Trans. Commun.

J. A. Salehi, “Code division multiple-access techniques in optical fiber networks—part I: fundamental principles,” IEEE Trans. Commun. 37, 824–833 (1989).
[CrossRef]

S. V. Maric, M. D. Hahm, E. L. Titlebaum, “Construction and performance analysis of a new family of optical orthogonal codes for CDMA fiber-optic networks,” IEEE Trans. Commun. 43, 485–489 (1995).
[CrossRef]

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

M. Azizoglu, J. A. Salehi, Y. Li, “Optical CDMA via temporal codes,” IEEE Trans. Commun. 40, 1162–1170 (1992).
[CrossRef]

IEEE Trans. Consumer Electron.

J.-G. Zhang, “Variable-bit-rate video transmission systems using optical fiber code-division multiplexing scheme,” IEEE Trans. Consumer Electron. 42, 874–884 (1996).
[CrossRef]

IEEE Trans. Inf. Theory

F. R. K. Chung, J. A. Salehi, V. K. Wei, “Optical orthogonal codes: design, analysis, and applications,” IEEE Trans. Inf. Theory 35, 595–604 (1989).
[CrossRef]

H. Chung, P. V. Kumar, “Optical orthogonal codes—new bounds and an optimal construction,” IEEE Trans. Inf. Theory 36, 866–873 (1990).
[CrossRef]

T. Klove, “Bounds on the size of optimal difference triangle sets,” IEEE Trans. Inf. Theory 34, 355–361 (1988).
[CrossRef]

J. Lightwave Technol.

P. R. Prucnal, M. A. Santoro, T. R. Fan, “Spread spectrum fiber-optic local area network using optical processing,” J. Lightwave Technol. LT-4, 547–554 (1986).
[CrossRef]

R. M. Gagliardi, A. J. Mendez, M. R. Dale, E. Park, “Fiber-optic digital video multiplexing using optical CDMA,” J. Lightwave Technol. 11, 20–26 (1993).
[CrossRef]

Opt. Eng.

C. DeCusatis, “Optical data communication: fundamentals and future directions,” Opt. Eng. 37, 3082–3099 (1998).
[CrossRef]

Other

C. A. Siller, M. Shafi, SONET/SDH—A Sourcebook of Synchronous Networking (Institute of Electrical and Electronics Engineers, New York, 1996).

J.-G. Zhang, “Novel techniques and architecture for all-optical code-division and time-division multiple access networks,” Ph.D. dissertation (University of Parma, Parma, Italy, 1994) (in Italian).

L. P. Barry, J. Debeau, R. Boittin, “40-nm tunable source of picosecond pulses at 10 GHz by external injection into a gain-switched FP laser,” in Proceedings of the 20th European Conference on Optical Communication (Istituto Internazionale delle Comunicazioni, Genova, Italy, 1994), pp. 369–372.

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Figures (9)

Fig. 1
Fig. 1

Configuration for on–off keying AOCDM transmission system. PD, photodetector.

Fig. 2
Fig. 2

Block diagram of AOCDM decoder.

Fig. 3
Fig. 3

Examples of violations of λ c = 1 or/and λ a = 1 for MBR AOCDM transmission system with (32, 4, 1) OOC of two code words C 1 = {1, 5, 12, 31} and C 2 = {1, 10, 13, 28}.5

Fig. 4
Fig. 4

BER versus number of active AOCDM transmitters for various w and L f .

Fig. 5
Fig. 5

Block diagram of proposed cost-effective AOCDM transmitters. OF, optical fiber.

Fig. 6
Fig. 6

Illustration of signal waveforms at proposed AOCDM transmitter.

Fig. 7
Fig. 7

Experimental setup of optical pulse generation and modulation. HP, Hewlett-Packard.

Fig. 8
Fig. 8

(a) Measured non-return-to-zero electronic data bits 10001 (with emitter-coupled logic) and (b) resulting optical bit pattern 10001 from gain-switched laser diode. Time base, 5.2 ns/div.

Fig. 9
Fig. 9

Measured results. Output optical spectra of a gain-switched FP laser diode (a) without optical injection locking and (b) with optical injection locking. (c) Without optical injection locking, the optical pulses at the output of a 2.2-km dispersion-shifted single-mode optical fiber. With optical injection locking, the optical pulses at the output of a dispersion-shifted optical fiber of lengths (d) 2.2 km, (e) 6.7 km, and (f) 8.8 km.

Tables (4)

Tables Icon

Table 1 Strict OOC’s Obtained from Construction Method B

Tables Icon

Table 2 Comparison of Strict OOC’s Generated by Construction Method B with Those Obtained from Nonstrict OOC’s

Tables Icon

Table 3 Comparison of Newly Obtained Strict OOC’s with Truncated Costas OOC’s (Nonstrict) of a Prime Number of 61a

Tables Icon

Table 4 Use of an (ni, 4, 1) UL-S-OOC for EBR and MBR Data Transmissions

Equations (40)

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

Ci=b1i, b2i,, bji,, bwin.
djki=bki-bji-1
1b1i<b2i<<bwin,
0dj,j+1i<djki<djli<d1win-2,j+1<k<l<w;  j1. 
dlj,mkt,t+n+Δτ  n+ymt+n+Δτk+Δτ-yltj-1,  Δτ0,
Di=bwi-2,
D=maxDi|i=1, 2,, M.
t=0n-1 xtxt+τ=wfor τ=01otherwise
t=0n-1 xtyt+τ1
dlj,mkt,t+n+Δτ>D
dklidpqi,  i=1, 2,, M
dklidpqj,  i, j=1, 2,, M
 3.n2D+3.
An×M=C1||Ci|OD+1×M|CM|=BD+2×M|OD+1×M,
NM, wMww-1+1.
NM, w22w-x-22x+2Mx+12w-x-22+1+Mx+1x+24-12+1,
x=w-2for M=1w-2 or w-1for M>0 or w>5,
NM, w2162x2-2x+3M+2x-5+2w-x-14x+1×x2w-x-1M+2+G3xx+1+1,
G=0 for x=1 orM=16M-6 for x=2,M218M-24 for x=3,M29M-12x-1 for x4,2M810M-20x-1 for x4,M9.
   Φn, w, 1n-1ww-1,
Φn, w, 12x+2n-22w-x-2x+12w-x-22+x+22
Mww-1+12Dc+3ns2nc-1. 
w¯=1w2wMw,
iw=d12id13id14id1wid23id24id2wid34id3widw-1,wi.
djki=l=jk-2dl,l+1i+1+dk-1,ki,j=1, 2,, k-2;k=3, 4,, w,
hi=i+3M2+1 for i=2, 4, 6,, Mi+5M+32 for i=1, 3, 5,, M-1.
hi=i+3M2+1 for i=1, 3, 5,, Mi+5M+32 for i=2, 4, 6,, M-1.
Let C=Ci|i=1, 2,, M be a D+2, w, 1 APS-OOC, where Ci=1, b2i,, bwiD+2.
niDi+D+3,  i=1, 2,, M,
Ci=1, b2i,, bwini,  i=1, 2,, M.
RiniDi+D+3,  i=1, 2,, M,
fi1Di+D+3τ,  i=1, 2,, M,
τ=1Dmin+D+3fmax,
τ=1Ds1+D+3f1,
Riniminni|i=1,, M=2D+3,
fi12D+3τ,  i=1,, M,
pI1I1=i=0M-1M-1iw22ni1-w22nM-1-iδI1-i,
BER=12i=ThM-1M-1iw22ni1-w22nM-1-i,
djkidjki,  i,i=1,2,  , M
n2D+3,

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