Abstract

In this paper, we study a new class of two-dimensional codes, here called multilevel prime codes, with expanded code cardinality by relaxing the maximum cross-correlation function to any arbitrary positive integer. Besides having asymptotically optimal cardinality and zero autocorrelation sidelobes, these multilevel prime codes can be partitioned into a tree structure of multiple levels of subsets of code matrices. In each level, the number of subsets, the number of code matrices per subset, and the cross-correlation function of each subset are related to the level number. The performance of the new codes in an optical code-division multiple-access system with hard-limiting detection is analyzed. Our results show that the unique partition property of the multilevel prime codes supports a trade-off between code cardinality and performance for meeting different system requirements, such as user capacity and throughput.

© 2009 Optical Society of America

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  1. L. Tancevski, I. Andonovic, “Hybrid wavelength hopping/time spreading schemes for use in massive optical networks with increased security,” J. Lightwave Technol., vol. 14, no. 12, pp. 2636–2647, Dec. 1996.
    [Crossref]
  2. G.-C. Yang, W. C. Kwong, “Performance comparison of multiwavelength CDMA and WDMA+CDMA for fiber-optic networks,” IEEE Trans. Commun., vol. 45, no. 11, pp. 1426–1434, Nov. 1997.
    [Crossref]
  3. G.-C. Yang, W. C. Kwong, Prime Codes With Applications to CDMA Optical and Wireless Networks. Artech House, Norwood, MA, 2002.
  4. R. M. H. Yim, L. R. Chen, J. Bajcsy, “Design and performance of 2-D codes for wavelength-time optical CDMA,” IEEE Photon. Technol. Lett., vol. 14, no. 5, pp. 714–716, May 2002.
    [Crossref]
  5. W. C. Kwong, G.-C. Yang, V. Baby, C.-S. Brès, P. R. Prucnal, “Multiple-wavelength optical orthogonal codes under prime-sequence permutations for optical CDMA,” IEEE Trans. Commun., vol. 53, no. 1, pp. 117–123, Jan. 2005.
    [Crossref]
  6. C.-Y. Chang, G.-C. Yang, W. C. Kwong, “Wavelength-time codes with maximum cross-correlation function of two for multicode-keying optical CDMA,” J. Lightwave Technol., vol. 24, no. 3, pp. 1093–1100, Mar. 2006.
    [Crossref]
  7. E. L. Titlebaum, L. H. Sibul, “Time-frequency hop signals part II: Coding based upon quadratic congruences,” IEEE Trans. Aerosp. Electron. Syst., vol. 17, no. 4, pp. 494–500, July 1981.
    [Crossref]
  8. S. V. Maric, E. L. Titlebaum, “Frequency hop multiple access codes based upon the theory of cubic congruences,” IEEE Trans. Aerosp. Electron. Syst., vol. 26, no. 6, pp. 1035–1039, Nov. 1990.
    [Crossref]
  9. S. V. Maric, E. L. Titlebaum, “A class of frequency hop codes with nearly ideal characteristics for use in multiple-access spread-spectrum communications and radar and sonar systems,” IEEE Trans. Commun., vol. 40, no. 9, pp. 1442–1447, Sept. 1992.
    [Crossref]
  10. J.-H. Tien, G.-C. Yang, C.-Y. Chang, W. C. Kwong, “Design and analysis of 2-D codes with the maximum cross-correlation value of two for optical CDMA,” J. Lightwave Technol., vol. 26, no. 22, pp. 3632–3639, Nov. 2008.
    [Crossref]
  11. T.-C. Wang, G.-C. Yang, C.-Y. Chang, W. C. Kwong, “A new family of 2-D codes for fiber-optic CDMA systems with and without the chip-synchronous assumption,” J. Lightwave Technol., vol. 27, no. 14, pp. 2612–2620, July 2009.
    [Crossref]
  12. G.-C. Yang, J.-Y. Jaw, “Performance analysis and sequence designs of synchronous code-division multiple-access systems with multimedia services,” IEE Proc.-Commun., vol. 141, no. 6, pp. 371, Dec. 1994.
    [Crossref]
  13. W. C. Kwong, P. A. Perrier, P. R. Prucnal, “Performance comparison of asynchronous and synchronous code-division multiple-access techniques for fiber-optic local area networks,” IEEE Trans. Commun., vol. 39, no. 11, pp. 1625–1634, Nov. 1991.
    [Crossref]
  14. J. Wu, D. J. Costello, “New multilevel codes over GF(q),” IEEE Trans. Inf. Theory, vol. 38, no. 3, pp. 933–939, May 1992.
    [Crossref]
  15. J.-J. Chen, G.-C. Yang, “CDMA fiber-optic systems with optical hard limiters,” J. Lightwave Technol., vol. 19, no. 7, pp. 950–958, July 2001.

2009 (1)

T.-C. Wang, G.-C. Yang, C.-Y. Chang, W. C. Kwong, “A new family of 2-D codes for fiber-optic CDMA systems with and without the chip-synchronous assumption,” J. Lightwave Technol., vol. 27, no. 14, pp. 2612–2620, July 2009.
[Crossref]

2008 (1)

J.-H. Tien, G.-C. Yang, C.-Y. Chang, W. C. Kwong, “Design and analysis of 2-D codes with the maximum cross-correlation value of two for optical CDMA,” J. Lightwave Technol., vol. 26, no. 22, pp. 3632–3639, Nov. 2008.
[Crossref]

2006 (1)

C.-Y. Chang, G.-C. Yang, W. C. Kwong, “Wavelength-time codes with maximum cross-correlation function of two for multicode-keying optical CDMA,” J. Lightwave Technol., vol. 24, no. 3, pp. 1093–1100, Mar. 2006.
[Crossref]

2005 (1)

W. C. Kwong, G.-C. Yang, V. Baby, C.-S. Brès, P. R. Prucnal, “Multiple-wavelength optical orthogonal codes under prime-sequence permutations for optical CDMA,” IEEE Trans. Commun., vol. 53, no. 1, pp. 117–123, Jan. 2005.
[Crossref]

2002 (1)

R. M. H. Yim, L. R. Chen, J. Bajcsy, “Design and performance of 2-D codes for wavelength-time optical CDMA,” IEEE Photon. Technol. Lett., vol. 14, no. 5, pp. 714–716, May 2002.
[Crossref]

2001 (1)

J.-J. Chen, G.-C. Yang, “CDMA fiber-optic systems with optical hard limiters,” J. Lightwave Technol., vol. 19, no. 7, pp. 950–958, July 2001.

1997 (1)

G.-C. Yang, W. C. Kwong, “Performance comparison of multiwavelength CDMA and WDMA+CDMA for fiber-optic networks,” IEEE Trans. Commun., vol. 45, no. 11, pp. 1426–1434, Nov. 1997.
[Crossref]

1996 (1)

L. Tancevski, I. Andonovic, “Hybrid wavelength hopping/time spreading schemes for use in massive optical networks with increased security,” J. Lightwave Technol., vol. 14, no. 12, pp. 2636–2647, Dec. 1996.
[Crossref]

1994 (1)

G.-C. Yang, J.-Y. Jaw, “Performance analysis and sequence designs of synchronous code-division multiple-access systems with multimedia services,” IEE Proc.-Commun., vol. 141, no. 6, pp. 371, Dec. 1994.
[Crossref]

1992 (2)

J. Wu, D. J. Costello, “New multilevel codes over GF(q),” IEEE Trans. Inf. Theory, vol. 38, no. 3, pp. 933–939, May 1992.
[Crossref]

S. V. Maric, E. L. Titlebaum, “A class of frequency hop codes with nearly ideal characteristics for use in multiple-access spread-spectrum communications and radar and sonar systems,” IEEE Trans. Commun., vol. 40, no. 9, pp. 1442–1447, Sept. 1992.
[Crossref]

1991 (1)

W. C. Kwong, P. A. Perrier, P. R. Prucnal, “Performance comparison of asynchronous and synchronous code-division multiple-access techniques for fiber-optic local area networks,” IEEE Trans. Commun., vol. 39, no. 11, pp. 1625–1634, Nov. 1991.
[Crossref]

1990 (1)

S. V. Maric, E. L. Titlebaum, “Frequency hop multiple access codes based upon the theory of cubic congruences,” IEEE Trans. Aerosp. Electron. Syst., vol. 26, no. 6, pp. 1035–1039, Nov. 1990.
[Crossref]

1981 (1)

E. L. Titlebaum, L. H. Sibul, “Time-frequency hop signals part II: Coding based upon quadratic congruences,” IEEE Trans. Aerosp. Electron. Syst., vol. 17, no. 4, pp. 494–500, July 1981.
[Crossref]

Andonovic, I.

L. Tancevski, I. Andonovic, “Hybrid wavelength hopping/time spreading schemes for use in massive optical networks with increased security,” J. Lightwave Technol., vol. 14, no. 12, pp. 2636–2647, Dec. 1996.
[Crossref]

Baby, V.

W. C. Kwong, G.-C. Yang, V. Baby, C.-S. Brès, P. R. Prucnal, “Multiple-wavelength optical orthogonal codes under prime-sequence permutations for optical CDMA,” IEEE Trans. Commun., vol. 53, no. 1, pp. 117–123, Jan. 2005.
[Crossref]

Bajcsy, J.

R. M. H. Yim, L. R. Chen, J. Bajcsy, “Design and performance of 2-D codes for wavelength-time optical CDMA,” IEEE Photon. Technol. Lett., vol. 14, no. 5, pp. 714–716, May 2002.
[Crossref]

Brès, C.-S.

W. C. Kwong, G.-C. Yang, V. Baby, C.-S. Brès, P. R. Prucnal, “Multiple-wavelength optical orthogonal codes under prime-sequence permutations for optical CDMA,” IEEE Trans. Commun., vol. 53, no. 1, pp. 117–123, Jan. 2005.
[Crossref]

Chang, C.-Y.

T.-C. Wang, G.-C. Yang, C.-Y. Chang, W. C. Kwong, “A new family of 2-D codes for fiber-optic CDMA systems with and without the chip-synchronous assumption,” J. Lightwave Technol., vol. 27, no. 14, pp. 2612–2620, July 2009.
[Crossref]

J.-H. Tien, G.-C. Yang, C.-Y. Chang, W. C. Kwong, “Design and analysis of 2-D codes with the maximum cross-correlation value of two for optical CDMA,” J. Lightwave Technol., vol. 26, no. 22, pp. 3632–3639, Nov. 2008.
[Crossref]

C.-Y. Chang, G.-C. Yang, W. C. Kwong, “Wavelength-time codes with maximum cross-correlation function of two for multicode-keying optical CDMA,” J. Lightwave Technol., vol. 24, no. 3, pp. 1093–1100, Mar. 2006.
[Crossref]

Chen, J.-J.

J.-J. Chen, G.-C. Yang, “CDMA fiber-optic systems with optical hard limiters,” J. Lightwave Technol., vol. 19, no. 7, pp. 950–958, July 2001.

Chen, L. R.

R. M. H. Yim, L. R. Chen, J. Bajcsy, “Design and performance of 2-D codes for wavelength-time optical CDMA,” IEEE Photon. Technol. Lett., vol. 14, no. 5, pp. 714–716, May 2002.
[Crossref]

Costello, D. J.

J. Wu, D. J. Costello, “New multilevel codes over GF(q),” IEEE Trans. Inf. Theory, vol. 38, no. 3, pp. 933–939, May 1992.
[Crossref]

Jaw, J.-Y.

G.-C. Yang, J.-Y. Jaw, “Performance analysis and sequence designs of synchronous code-division multiple-access systems with multimedia services,” IEE Proc.-Commun., vol. 141, no. 6, pp. 371, Dec. 1994.
[Crossref]

Kwong, W. C.

T.-C. Wang, G.-C. Yang, C.-Y. Chang, W. C. Kwong, “A new family of 2-D codes for fiber-optic CDMA systems with and without the chip-synchronous assumption,” J. Lightwave Technol., vol. 27, no. 14, pp. 2612–2620, July 2009.
[Crossref]

J.-H. Tien, G.-C. Yang, C.-Y. Chang, W. C. Kwong, “Design and analysis of 2-D codes with the maximum cross-correlation value of two for optical CDMA,” J. Lightwave Technol., vol. 26, no. 22, pp. 3632–3639, Nov. 2008.
[Crossref]

C.-Y. Chang, G.-C. Yang, W. C. Kwong, “Wavelength-time codes with maximum cross-correlation function of two for multicode-keying optical CDMA,” J. Lightwave Technol., vol. 24, no. 3, pp. 1093–1100, Mar. 2006.
[Crossref]

W. C. Kwong, G.-C. Yang, V. Baby, C.-S. Brès, P. R. Prucnal, “Multiple-wavelength optical orthogonal codes under prime-sequence permutations for optical CDMA,” IEEE Trans. Commun., vol. 53, no. 1, pp. 117–123, Jan. 2005.
[Crossref]

G.-C. Yang, W. C. Kwong, “Performance comparison of multiwavelength CDMA and WDMA+CDMA for fiber-optic networks,” IEEE Trans. Commun., vol. 45, no. 11, pp. 1426–1434, Nov. 1997.
[Crossref]

W. C. Kwong, P. A. Perrier, P. R. Prucnal, “Performance comparison of asynchronous and synchronous code-division multiple-access techniques for fiber-optic local area networks,” IEEE Trans. Commun., vol. 39, no. 11, pp. 1625–1634, Nov. 1991.
[Crossref]

G.-C. Yang, W. C. Kwong, Prime Codes With Applications to CDMA Optical and Wireless Networks. Artech House, Norwood, MA, 2002.

Maric, S. V.

S. V. Maric, E. L. Titlebaum, “A class of frequency hop codes with nearly ideal characteristics for use in multiple-access spread-spectrum communications and radar and sonar systems,” IEEE Trans. Commun., vol. 40, no. 9, pp. 1442–1447, Sept. 1992.
[Crossref]

S. V. Maric, E. L. Titlebaum, “Frequency hop multiple access codes based upon the theory of cubic congruences,” IEEE Trans. Aerosp. Electron. Syst., vol. 26, no. 6, pp. 1035–1039, Nov. 1990.
[Crossref]

Perrier, P. A.

W. C. Kwong, P. A. Perrier, P. R. Prucnal, “Performance comparison of asynchronous and synchronous code-division multiple-access techniques for fiber-optic local area networks,” IEEE Trans. Commun., vol. 39, no. 11, pp. 1625–1634, Nov. 1991.
[Crossref]

Prucnal, P. R.

W. C. Kwong, G.-C. Yang, V. Baby, C.-S. Brès, P. R. Prucnal, “Multiple-wavelength optical orthogonal codes under prime-sequence permutations for optical CDMA,” IEEE Trans. Commun., vol. 53, no. 1, pp. 117–123, Jan. 2005.
[Crossref]

W. C. Kwong, P. A. Perrier, P. R. Prucnal, “Performance comparison of asynchronous and synchronous code-division multiple-access techniques for fiber-optic local area networks,” IEEE Trans. Commun., vol. 39, no. 11, pp. 1625–1634, Nov. 1991.
[Crossref]

Sibul, L. H.

E. L. Titlebaum, L. H. Sibul, “Time-frequency hop signals part II: Coding based upon quadratic congruences,” IEEE Trans. Aerosp. Electron. Syst., vol. 17, no. 4, pp. 494–500, July 1981.
[Crossref]

Tancevski, L.

L. Tancevski, I. Andonovic, “Hybrid wavelength hopping/time spreading schemes for use in massive optical networks with increased security,” J. Lightwave Technol., vol. 14, no. 12, pp. 2636–2647, Dec. 1996.
[Crossref]

Tien, J.-H.

J.-H. Tien, G.-C. Yang, C.-Y. Chang, W. C. Kwong, “Design and analysis of 2-D codes with the maximum cross-correlation value of two for optical CDMA,” J. Lightwave Technol., vol. 26, no. 22, pp. 3632–3639, Nov. 2008.
[Crossref]

Titlebaum, E. L.

S. V. Maric, E. L. Titlebaum, “A class of frequency hop codes with nearly ideal characteristics for use in multiple-access spread-spectrum communications and radar and sonar systems,” IEEE Trans. Commun., vol. 40, no. 9, pp. 1442–1447, Sept. 1992.
[Crossref]

S. V. Maric, E. L. Titlebaum, “Frequency hop multiple access codes based upon the theory of cubic congruences,” IEEE Trans. Aerosp. Electron. Syst., vol. 26, no. 6, pp. 1035–1039, Nov. 1990.
[Crossref]

E. L. Titlebaum, L. H. Sibul, “Time-frequency hop signals part II: Coding based upon quadratic congruences,” IEEE Trans. Aerosp. Electron. Syst., vol. 17, no. 4, pp. 494–500, July 1981.
[Crossref]

Wang, T.-C.

T.-C. Wang, G.-C. Yang, C.-Y. Chang, W. C. Kwong, “A new family of 2-D codes for fiber-optic CDMA systems with and without the chip-synchronous assumption,” J. Lightwave Technol., vol. 27, no. 14, pp. 2612–2620, July 2009.
[Crossref]

Wu, J.

J. Wu, D. J. Costello, “New multilevel codes over GF(q),” IEEE Trans. Inf. Theory, vol. 38, no. 3, pp. 933–939, May 1992.
[Crossref]

Yang, G.-C.

T.-C. Wang, G.-C. Yang, C.-Y. Chang, W. C. Kwong, “A new family of 2-D codes for fiber-optic CDMA systems with and without the chip-synchronous assumption,” J. Lightwave Technol., vol. 27, no. 14, pp. 2612–2620, July 2009.
[Crossref]

J.-H. Tien, G.-C. Yang, C.-Y. Chang, W. C. Kwong, “Design and analysis of 2-D codes with the maximum cross-correlation value of two for optical CDMA,” J. Lightwave Technol., vol. 26, no. 22, pp. 3632–3639, Nov. 2008.
[Crossref]

C.-Y. Chang, G.-C. Yang, W. C. Kwong, “Wavelength-time codes with maximum cross-correlation function of two for multicode-keying optical CDMA,” J. Lightwave Technol., vol. 24, no. 3, pp. 1093–1100, Mar. 2006.
[Crossref]

W. C. Kwong, G.-C. Yang, V. Baby, C.-S. Brès, P. R. Prucnal, “Multiple-wavelength optical orthogonal codes under prime-sequence permutations for optical CDMA,” IEEE Trans. Commun., vol. 53, no. 1, pp. 117–123, Jan. 2005.
[Crossref]

J.-J. Chen, G.-C. Yang, “CDMA fiber-optic systems with optical hard limiters,” J. Lightwave Technol., vol. 19, no. 7, pp. 950–958, July 2001.

G.-C. Yang, W. C. Kwong, “Performance comparison of multiwavelength CDMA and WDMA+CDMA for fiber-optic networks,” IEEE Trans. Commun., vol. 45, no. 11, pp. 1426–1434, Nov. 1997.
[Crossref]

G.-C. Yang, J.-Y. Jaw, “Performance analysis and sequence designs of synchronous code-division multiple-access systems with multimedia services,” IEE Proc.-Commun., vol. 141, no. 6, pp. 371, Dec. 1994.
[Crossref]

G.-C. Yang, W. C. Kwong, Prime Codes With Applications to CDMA Optical and Wireless Networks. Artech House, Norwood, MA, 2002.

Yim, R. M. H.

R. M. H. Yim, L. R. Chen, J. Bajcsy, “Design and performance of 2-D codes for wavelength-time optical CDMA,” IEEE Photon. Technol. Lett., vol. 14, no. 5, pp. 714–716, May 2002.
[Crossref]

IEE Proc.-Commun. (1)

G.-C. Yang, J.-Y. Jaw, “Performance analysis and sequence designs of synchronous code-division multiple-access systems with multimedia services,” IEE Proc.-Commun., vol. 141, no. 6, pp. 371, Dec. 1994.
[Crossref]

IEEE Photon. Technol. Lett. (1)

R. M. H. Yim, L. R. Chen, J. Bajcsy, “Design and performance of 2-D codes for wavelength-time optical CDMA,” IEEE Photon. Technol. Lett., vol. 14, no. 5, pp. 714–716, May 2002.
[Crossref]

IEEE Trans. Aerosp. Electron. Syst. (2)

E. L. Titlebaum, L. H. Sibul, “Time-frequency hop signals part II: Coding based upon quadratic congruences,” IEEE Trans. Aerosp. Electron. Syst., vol. 17, no. 4, pp. 494–500, July 1981.
[Crossref]

S. V. Maric, E. L. Titlebaum, “Frequency hop multiple access codes based upon the theory of cubic congruences,” IEEE Trans. Aerosp. Electron. Syst., vol. 26, no. 6, pp. 1035–1039, Nov. 1990.
[Crossref]

IEEE Trans. Commun. (4)

S. V. Maric, E. L. Titlebaum, “A class of frequency hop codes with nearly ideal characteristics for use in multiple-access spread-spectrum communications and radar and sonar systems,” IEEE Trans. Commun., vol. 40, no. 9, pp. 1442–1447, Sept. 1992.
[Crossref]

W. C. Kwong, G.-C. Yang, V. Baby, C.-S. Brès, P. R. Prucnal, “Multiple-wavelength optical orthogonal codes under prime-sequence permutations for optical CDMA,” IEEE Trans. Commun., vol. 53, no. 1, pp. 117–123, Jan. 2005.
[Crossref]

G.-C. Yang, W. C. Kwong, “Performance comparison of multiwavelength CDMA and WDMA+CDMA for fiber-optic networks,” IEEE Trans. Commun., vol. 45, no. 11, pp. 1426–1434, Nov. 1997.
[Crossref]

W. C. Kwong, P. A. Perrier, P. R. Prucnal, “Performance comparison of asynchronous and synchronous code-division multiple-access techniques for fiber-optic local area networks,” IEEE Trans. Commun., vol. 39, no. 11, pp. 1625–1634, Nov. 1991.
[Crossref]

IEEE Trans. Inf. Theory (1)

J. Wu, D. J. Costello, “New multilevel codes over GF(q),” IEEE Trans. Inf. Theory, vol. 38, no. 3, pp. 933–939, May 1992.
[Crossref]

J. Lightwave Technol. (5)

J.-J. Chen, G.-C. Yang, “CDMA fiber-optic systems with optical hard limiters,” J. Lightwave Technol., vol. 19, no. 7, pp. 950–958, July 2001.

L. Tancevski, I. Andonovic, “Hybrid wavelength hopping/time spreading schemes for use in massive optical networks with increased security,” J. Lightwave Technol., vol. 14, no. 12, pp. 2636–2647, Dec. 1996.
[Crossref]

C.-Y. Chang, G.-C. Yang, W. C. Kwong, “Wavelength-time codes with maximum cross-correlation function of two for multicode-keying optical CDMA,” J. Lightwave Technol., vol. 24, no. 3, pp. 1093–1100, Mar. 2006.
[Crossref]

J.-H. Tien, G.-C. Yang, C.-Y. Chang, W. C. Kwong, “Design and analysis of 2-D codes with the maximum cross-correlation value of two for optical CDMA,” J. Lightwave Technol., vol. 26, no. 22, pp. 3632–3639, Nov. 2008.
[Crossref]

T.-C. Wang, G.-C. Yang, C.-Y. Chang, W. C. Kwong, “A new family of 2-D codes for fiber-optic CDMA systems with and without the chip-synchronous assumption,” J. Lightwave Technol., vol. 27, no. 14, pp. 2612–2620, July 2009.
[Crossref]

Other (1)

G.-C. Yang, W. C. Kwong, Prime Codes With Applications to CDMA Optical and Wireless Networks. Artech House, Norwood, MA, 2002.

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Figures (5)

Fig. 1
Fig. 1

Tree structure of the ( w × P , w , 0 , n ) multilevel prime codes.

Fig. 2
Fig. 2

Error probability of the ( 16 × 89 , 16 , 0 , λ c ) multilevel prime codes for λ c = n = { 1 , 2 , 3 } .

Fig. 3
Fig. 3

Error probabilities of the ( 16 × P , 16 , 0 , 3 ) trilevel prime codes for P = { 79 , 89 , 97 } .

Fig. 4
Fig. 4

Error probabilities of the ( L × 89 , w , 0 , 3 ) trilevel prime codes for L = w = { 8 , 16 , 24 } .

Fig. 5
Fig. 5

Error probabilities of the ( 8 × 23 , 8 , 0 , 2 ) bilevel prime codes, optimized as a function of the number of simultaneous users K.

Tables (2)

Tables Icon

Table 1 Matrices of the ( 4 × 5 , 4 , 0 , 2 ) Bilevel Prime Codes

Tables Icon

Table 2 Matrices of the ( 5 × 5 , 5 , 0 , 3 ) Trilevel Prime Codes

Equations (20)

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

s i n , i n 1 , , i 1 , j = ( i n j n ) P ( i n 1 j n 1 ) P P ( i 1 j )
j = 0 n q n , j = 1 ,
j = 0 n j q n , j = w 2 2 L N = w 2 P ,
P e = 1 2 k = 0 w ( 1 ) k ( w k ) [ j = 0 n q n , j ( w k j ) ( w j ) ] K 1 ,
q n , j = 1 2 × h n , j P ( P n 1 ) .
h n , 0 = 2 P ( P n 1 ) h n , n h n , n 1 h n , 1 ,
h n , 1 = w ( P n 1 ) n h n , n 2 h n , 2 ,
h n , n = ( w n ) ( P 1 ) ,
h n , j = ( h n 1 , j 1 + h n 2 , j 1 + + h j 1 , j 1 h j 1 , j 1 ) h j , j
h 1 , 1 = ( w 1 ) ( P 1 ) = w ( P 1 ) .
h 2 , 2 = ( w 2 ) ( P 1 ) = w ( w 1 ) 2 ( P 1 ) .
h 2 , 1 = w ( P 2 1 ) 2 h 2 , 2 = w ( P + 2 w ) ( P 1 ) .
s i n , i n 1 , , i 1 , j = ( i n j n ) P ( i n 1 j n 1 ) P P ( i 1 j ) P τ ,
s i n , i n 1 , , i 1 , j = ( i n j n ) P ( i n 1 j n 1 ) P P ( i 1 j )
( ( i n i n ) j n ) P ( ( i n 1 i n 1 ) j n 1 ) P P ( ( i 1 i 1 ) j ) P τ = 0 .
s i n , i n 1 , , i 1 , j = ( i n j n ) P ( i n 1 j n 1 ) P P ( i l + 1 j l + 1 ) P ( i l j l ) P ( i 1 j ) P τ ,
s i n , i n 1 , , i 1 , j = ( i n j n ) P ( i n 1 j n 1 ) P P ( i l + 1 j l + 1 ) P ( i l j l ) P ( i 1 j )
( ( i n i n ) j n ) P ( ( i n 1 i n 1 ) j n 1 ) P P ( ( i l + 1 i l + 1 ) j l + 1 ) P ( ( i l i l ) j l ) P P ( ( i 1 i 1 ) j ) P τ = ( ( i l i l ) j l ) P P ( ( i 1 i 1 ) j ) P τ = 0 .
h n , j h n 1 , j h n 1 , j 1 = h n 1 , j h n 2 , j h n 2 , j 1 = = h j , j h j 1 , j h j 1 , j 1 ,
h n , j = h n 1 , j + h n 1 , j 1 h j 1 , j 1 × h j , j .