Abstract

We have numerically investigated a chaotic laser diode transmitter–receiver array scheme (CLDTRAS), which is a secure digital communication scheme using a difference between two types of transmitter–receiver array consisting of two self-pulsating laser diodes (LDs), i.e., a receiver LD and a transmitter LD. By analyzing the bit error rate, particularly its dependence on the parameter mismatches of the hardware and channel noise and on the correlation coefficient between a transmitter LD and receiver LD, we examined the problems of sensitivity to parameter mismatches and channel noise and a dependence on chaos synchronization between a transmitter LD and a receiver LD. The former makes communication difficult, and the latter makes it possible for an eavesdropper to estimate the receiver LD using chaos synchronization and to forge the hardware. Then we studied the effects of the bit error rate for various values of the threshold, which determines a binary message, and for various numbers of transmitters–receivers making up a LD transmitter–receiver array. It has been shown that a highly noise-tolerant and hardware-dependent communication scheme can be achieved with the LD transmitter–receiver array, whose transmitter and receiver LDs are asynchronous with respect to each other, by choosing the proper threshold and increasing the number of LD transmitters–receivers. Since it is possible to communicate without chaos synchronization, it becomes difficult to forge hardware and to eavesdrop with the forged hardware even if the key is stolen.

© 2007 Optical Society of America

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  1. L. Pecora and T. Carroll, "Synchronization in chaotic systems," Phys. Rev. Lett. 64, 821-824 (1990).
    [CrossRef] [PubMed]
  2. J. Otsubo, "Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback," IEEE J. Quantum Electron. 38, 1141-1154 (2002).
    [CrossRef]
  3. R. J. Jones, P. Rees, P. S. Spencer, and K. A. Shore, "Chaos and synchronization of self-pulsating laser diodes," J. Opt. Soc. Am. B 18, 166-172 (2001).
    [CrossRef]
  4. U. Parlirz, L. O. Chua, L. Kocarev, K. S. Halle, and A. Shang, "Transmission of digital signals by chaotic synchronization," Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973-977 (1992).
    [CrossRef]
  5. S. Ebisawa and S. Komatsu, "Optical chaotic communication using laser diode transmitter/receiver array," Jpn. J. Appl. Phys. 43, 5910-5917 (2004).
    [CrossRef]
  6. S. Ebisawa and S. Komatsu, "Digital communication based on the complexity of chaotic LD-transmitter/receiver array," in Proceedings of Winter International Symposium on Information and Communication Technologies (2004), pp. 434-439.
  7. Y. Liu, H. F. Chen, J. M. Liu, P. Davis, and T. Aida, "Synchronization of optical-feedback-induced chaos in semiconductor lasers by optical injection," Phys. Rev. A 63, 031802 (2001).
    [CrossRef]
  8. S. Sivaprakasam and K. A. Store, "Message encoding and decoding using chaotic external-cavity diode lasers," IEEE J. Quantum Electron. 36, 35-39 (2000).
    [CrossRef]
  9. F. L. Lin and J.-MLiu, "Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback," IEEE J. Quantum Electron. 39, 562-568 (2003).
    [CrossRef]
  10. C. Juang, T. M. Hwang, J. Juang, and W.-WLin, "A synchronization scheme using self-pulsating laser diodes in optical chaotic communication," IEEE J. Quantum Electron. 36, 300-304 (2000).
    [CrossRef]
  11. S. Sato, M. Sano, and Y. Sawada, "Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high-dimensional chaotic systems," Prog. Theor. Phys. 77, 1-5 (1987).
    [CrossRef]
  12. H. Nagashima and Y. Baba, Kaosunyumon (Introduction of Chaos) (Baifukan, Tokyo, 1992), p. 42 [in Japanese].
  13. R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford U. Press, 1994), Sections 4 and 9.

2004

S. Ebisawa and S. Komatsu, "Optical chaotic communication using laser diode transmitter/receiver array," Jpn. J. Appl. Phys. 43, 5910-5917 (2004).
[CrossRef]

2003

F. L. Lin and J.-MLiu, "Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback," IEEE J. Quantum Electron. 39, 562-568 (2003).
[CrossRef]

2002

J. Otsubo, "Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback," IEEE J. Quantum Electron. 38, 1141-1154 (2002).
[CrossRef]

2001

R. J. Jones, P. Rees, P. S. Spencer, and K. A. Shore, "Chaos and synchronization of self-pulsating laser diodes," J. Opt. Soc. Am. B 18, 166-172 (2001).
[CrossRef]

Y. Liu, H. F. Chen, J. M. Liu, P. Davis, and T. Aida, "Synchronization of optical-feedback-induced chaos in semiconductor lasers by optical injection," Phys. Rev. A 63, 031802 (2001).
[CrossRef]

2000

S. Sivaprakasam and K. A. Store, "Message encoding and decoding using chaotic external-cavity diode lasers," IEEE J. Quantum Electron. 36, 35-39 (2000).
[CrossRef]

C. Juang, T. M. Hwang, J. Juang, and W.-WLin, "A synchronization scheme using self-pulsating laser diodes in optical chaotic communication," IEEE J. Quantum Electron. 36, 300-304 (2000).
[CrossRef]

1992

U. Parlirz, L. O. Chua, L. Kocarev, K. S. Halle, and A. Shang, "Transmission of digital signals by chaotic synchronization," Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973-977 (1992).
[CrossRef]

1990

L. Pecora and T. Carroll, "Synchronization in chaotic systems," Phys. Rev. Lett. 64, 821-824 (1990).
[CrossRef] [PubMed]

1987

S. Sato, M. Sano, and Y. Sawada, "Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high-dimensional chaotic systems," Prog. Theor. Phys. 77, 1-5 (1987).
[CrossRef]

Aida, T.

Y. Liu, H. F. Chen, J. M. Liu, P. Davis, and T. Aida, "Synchronization of optical-feedback-induced chaos in semiconductor lasers by optical injection," Phys. Rev. A 63, 031802 (2001).
[CrossRef]

Baba, Y.

H. Nagashima and Y. Baba, Kaosunyumon (Introduction of Chaos) (Baifukan, Tokyo, 1992), p. 42 [in Japanese].

Carroll, T.

L. Pecora and T. Carroll, "Synchronization in chaotic systems," Phys. Rev. Lett. 64, 821-824 (1990).
[CrossRef] [PubMed]

Chen, H. F.

Y. Liu, H. F. Chen, J. M. Liu, P. Davis, and T. Aida, "Synchronization of optical-feedback-induced chaos in semiconductor lasers by optical injection," Phys. Rev. A 63, 031802 (2001).
[CrossRef]

Chua, L. O.

U. Parlirz, L. O. Chua, L. Kocarev, K. S. Halle, and A. Shang, "Transmission of digital signals by chaotic synchronization," Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973-977 (1992).
[CrossRef]

Davis, P.

Y. Liu, H. F. Chen, J. M. Liu, P. Davis, and T. Aida, "Synchronization of optical-feedback-induced chaos in semiconductor lasers by optical injection," Phys. Rev. A 63, 031802 (2001).
[CrossRef]

Ebisawa, S.

S. Ebisawa and S. Komatsu, "Optical chaotic communication using laser diode transmitter/receiver array," Jpn. J. Appl. Phys. 43, 5910-5917 (2004).
[CrossRef]

S. Ebisawa and S. Komatsu, "Digital communication based on the complexity of chaotic LD-transmitter/receiver array," in Proceedings of Winter International Symposium on Information and Communication Technologies (2004), pp. 434-439.

Halle, K. S.

U. Parlirz, L. O. Chua, L. Kocarev, K. S. Halle, and A. Shang, "Transmission of digital signals by chaotic synchronization," Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973-977 (1992).
[CrossRef]

Hilborn, R. C.

R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford U. Press, 1994), Sections 4 and 9.

Hwang, T. M.

C. Juang, T. M. Hwang, J. Juang, and W.-WLin, "A synchronization scheme using self-pulsating laser diodes in optical chaotic communication," IEEE J. Quantum Electron. 36, 300-304 (2000).
[CrossRef]

Jones, R. J.

Juang, C.

C. Juang, T. M. Hwang, J. Juang, and W.-WLin, "A synchronization scheme using self-pulsating laser diodes in optical chaotic communication," IEEE J. Quantum Electron. 36, 300-304 (2000).
[CrossRef]

Juang, J.

C. Juang, T. M. Hwang, J. Juang, and W.-WLin, "A synchronization scheme using self-pulsating laser diodes in optical chaotic communication," IEEE J. Quantum Electron. 36, 300-304 (2000).
[CrossRef]

Kocarev, L.

U. Parlirz, L. O. Chua, L. Kocarev, K. S. Halle, and A. Shang, "Transmission of digital signals by chaotic synchronization," Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973-977 (1992).
[CrossRef]

Komatsu, S.

S. Ebisawa and S. Komatsu, "Optical chaotic communication using laser diode transmitter/receiver array," Jpn. J. Appl. Phys. 43, 5910-5917 (2004).
[CrossRef]

S. Ebisawa and S. Komatsu, "Digital communication based on the complexity of chaotic LD-transmitter/receiver array," in Proceedings of Winter International Symposium on Information and Communication Technologies (2004), pp. 434-439.

Lin, F. L.

F. L. Lin and J.-MLiu, "Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback," IEEE J. Quantum Electron. 39, 562-568 (2003).
[CrossRef]

Lin, W.-W

C. Juang, T. M. Hwang, J. Juang, and W.-WLin, "A synchronization scheme using self-pulsating laser diodes in optical chaotic communication," IEEE J. Quantum Electron. 36, 300-304 (2000).
[CrossRef]

Liu, J. M.

Y. Liu, H. F. Chen, J. M. Liu, P. Davis, and T. Aida, "Synchronization of optical-feedback-induced chaos in semiconductor lasers by optical injection," Phys. Rev. A 63, 031802 (2001).
[CrossRef]

Liu, J.-M

F. L. Lin and J.-MLiu, "Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback," IEEE J. Quantum Electron. 39, 562-568 (2003).
[CrossRef]

Liu, Y.

Y. Liu, H. F. Chen, J. M. Liu, P. Davis, and T. Aida, "Synchronization of optical-feedback-induced chaos in semiconductor lasers by optical injection," Phys. Rev. A 63, 031802 (2001).
[CrossRef]

Nagashima, H.

H. Nagashima and Y. Baba, Kaosunyumon (Introduction of Chaos) (Baifukan, Tokyo, 1992), p. 42 [in Japanese].

Otsubo, J.

J. Otsubo, "Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback," IEEE J. Quantum Electron. 38, 1141-1154 (2002).
[CrossRef]

Parlirz, U.

U. Parlirz, L. O. Chua, L. Kocarev, K. S. Halle, and A. Shang, "Transmission of digital signals by chaotic synchronization," Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973-977 (1992).
[CrossRef]

Pecora, L.

L. Pecora and T. Carroll, "Synchronization in chaotic systems," Phys. Rev. Lett. 64, 821-824 (1990).
[CrossRef] [PubMed]

Rees, P.

Sano, M.

S. Sato, M. Sano, and Y. Sawada, "Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high-dimensional chaotic systems," Prog. Theor. Phys. 77, 1-5 (1987).
[CrossRef]

Sato, S.

S. Sato, M. Sano, and Y. Sawada, "Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high-dimensional chaotic systems," Prog. Theor. Phys. 77, 1-5 (1987).
[CrossRef]

Sawada, Y.

S. Sato, M. Sano, and Y. Sawada, "Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high-dimensional chaotic systems," Prog. Theor. Phys. 77, 1-5 (1987).
[CrossRef]

Shang, A.

U. Parlirz, L. O. Chua, L. Kocarev, K. S. Halle, and A. Shang, "Transmission of digital signals by chaotic synchronization," Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973-977 (1992).
[CrossRef]

Shore, K. A.

Sivaprakasam, S.

S. Sivaprakasam and K. A. Store, "Message encoding and decoding using chaotic external-cavity diode lasers," IEEE J. Quantum Electron. 36, 35-39 (2000).
[CrossRef]

Spencer, P. S.

Store, K. A.

S. Sivaprakasam and K. A. Store, "Message encoding and decoding using chaotic external-cavity diode lasers," IEEE J. Quantum Electron. 36, 35-39 (2000).
[CrossRef]

IEEE J. Quantum Electron.

S. Sivaprakasam and K. A. Store, "Message encoding and decoding using chaotic external-cavity diode lasers," IEEE J. Quantum Electron. 36, 35-39 (2000).
[CrossRef]

F. L. Lin and J.-MLiu, "Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback," IEEE J. Quantum Electron. 39, 562-568 (2003).
[CrossRef]

C. Juang, T. M. Hwang, J. Juang, and W.-WLin, "A synchronization scheme using self-pulsating laser diodes in optical chaotic communication," IEEE J. Quantum Electron. 36, 300-304 (2000).
[CrossRef]

J. Otsubo, "Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback," IEEE J. Quantum Electron. 38, 1141-1154 (2002).
[CrossRef]

Int. J. Bifurcation Chaos Appl. Sci. Eng.

U. Parlirz, L. O. Chua, L. Kocarev, K. S. Halle, and A. Shang, "Transmission of digital signals by chaotic synchronization," Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 973-977 (1992).
[CrossRef]

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys.

S. Ebisawa and S. Komatsu, "Optical chaotic communication using laser diode transmitter/receiver array," Jpn. J. Appl. Phys. 43, 5910-5917 (2004).
[CrossRef]

Phys. Rev. A

Y. Liu, H. F. Chen, J. M. Liu, P. Davis, and T. Aida, "Synchronization of optical-feedback-induced chaos in semiconductor lasers by optical injection," Phys. Rev. A 63, 031802 (2001).
[CrossRef]

Phys. Rev. Lett.

L. Pecora and T. Carroll, "Synchronization in chaotic systems," Phys. Rev. Lett. 64, 821-824 (1990).
[CrossRef] [PubMed]

Prog. Theor. Phys.

S. Sato, M. Sano, and Y. Sawada, "Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high-dimensional chaotic systems," Prog. Theor. Phys. 77, 1-5 (1987).
[CrossRef]

Other

H. Nagashima and Y. Baba, Kaosunyumon (Introduction of Chaos) (Baifukan, Tokyo, 1992), p. 42 [in Japanese].

R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford U. Press, 1994), Sections 4 and 9.

S. Ebisawa and S. Komatsu, "Digital communication based on the complexity of chaotic LD-transmitter/receiver array," in Proceedings of Winter International Symposium on Information and Communication Technologies (2004), pp. 434-439.

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Figures (11)

Fig. 1
Fig. 1

Scheme of the chaotic LD transmitter–receiver array system. This consists of several pairs of transmitter and receiver LDs, which operate in a self-pulsating mode using a high-frequency drive current, and every pair in the array operates independently of each other. The small output of the transmitter LD is injected into the corresponding receiver LD. ND, neutral density filter; OI, optical isolator; BS, beam splitter; PD, photodetector.

Fig. 2
Fig. 2

Numerical simulation of communication CLDTRAS. (a) Binary message signal used in CLDTRAS. (b) Time series of keys containing information of parameters between transmitter and receiver LDs. (c) Transmitting signal in chaotic LD transmitter–receiver array scheme. Other people without the key cannot decode a chaotic encoded message. (d) Lyapunov exponents calculated from encoded message with key. (e) Message signal decoded using key.

Fig. 3
Fig. 3

Bit error rate plotted against parameter mismatches for (a) CMO and (b) CLDTRAS. The shaded areas appear as black valleys in both figures, where the bit error rate is small and communication can be achieved with a certain bit error. These show that CLDTRAS is a hardware-dependent scheme, similar to CMO. Since the area where communication can be achieved with a small bit error in CLDTRAS is narrower than that in CMO, it makes the construction of the hardware difficult.

Fig. 4
Fig. 4

(a) Bit error rate plotted against SNR in CMO. The bit error rate gradually increases with channel noise. (b) Lyapunov exponents plotted against SNR in conventional CLDTRAS for a type 1 array indicating signal 0 (square) and a type 2 array indicating signal 1 (triangle). The broken line indicates the threshold of the Lyapunov exponent ( λ th ) , which is intermediate of the Lyapunov exponents for the type 1 and type 2 arrays without noise. (b) shows that it is difficult to communicate with only a slight channel noise. These figures show that the conventional CLDTRAS is more sensitive to channel noise than the CMO.

Fig. 5
Fig. 5

Correlation coefficient between transmitter and receiver LDs plotted against parameter mismatches of carrier lifetime and differential gain. By comparison with Figs. 3(a) and 3(b), the CMO and the conventional CLDTRAS are found to depend on the synchronization between the transmitter and receiver LDs.

Fig. 6
Fig. 6

Bifurcation diagram of transmitter LD output photon number S versus mismatch of injection current between transmitter and receiver LDs. When δ I = 0.013 and δ I = 0.056 , their outputs are chaotic.

Fig. 7
Fig. 7

Bit error rate of improved CLDTRAS discretely plotted against mismatch of carrier lifetime and differential gain in cases in which (a) λ th is intermediate of the Lyapunov exponents for the type 1 and type 2 arrays without noise, (b) λ th is 3.80, (c) λ th is 4.30, (d) λ th is 4.80, (e) λ th is 5.30, and (f) λ th is 5.80. (a) shows that for the plots in the graph where the bit error rate is zero and somewhat scattered over a large area it is difficult to communicate securely. However, by using an appropriate threshold of Lyapunov exponents λ th , it can be observed that the bit error rate is small along the line δ a = 1.15 × δ τ + 0.016 ; that is, only a receiver with a particular piece of hardware can decode a message. (e) Communication without bit errors can be achieved along the line δ a = 1.15 × δ τ + 0.016 where the transmitter and receiver LDs are synchronized. This shows that it is difficult to forge hardware by synchronization.

Fig. 8
Fig. 8

Bit error rate plotted against SNR in CLDTRAS with (a) array of asynchronous transmitters–receivers at ( δ τ = 0.042 , δ a = 0.018 ) in Fig. 7(e), and (b) array that consists of two arrays used in Fig. 7(e).

Fig. 9
Fig. 9

Schematic of state space for the LD transmitter–receiver array. The points belonging to LD transmitters–receivers named LDTR1 and LDTR2 in Σ are diffused in Σ 1 and in Σ 2 , respectively.

Fig. 10
Fig. 10

Bit error rate plotted against SNR in CLDTRAS with (a) array of two asynchronous transmitters–receivers at ( δ τ = 0.068 , δ a = 0.026 ) , and (b) array that consists of two arrays used in (a).

Fig. 11
Fig. 11

Bit error rate plotted against number of LD transmitters–receivers, of which the entire array is composed. It is shown that the bit error rate gradually decreases with the number of LD transmitters–receivers.

Tables (2)

Tables Icon

Table 1 Largest Lyapunov Exponents Calculated for Various Arrays

Tables Icon

Table 2 Thresholds of Lyapunov Exponents Calculated for Various Arrays

Equations (163)

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

N 1
N 2
d N 1 d t = a 1 ξ 1 V 1 ( N 1 N g 1 ) S N 1 τ s N 1 N 2 T 12 + I ( 1 + m sin 2 π f t ) ,
d N 2 d t = a 2 ξ 2 V 2 ( N 2 N g 2 ) S N 2 τ s N 2 N 1 T 21 ,
d S d t = [ a 1 ξ 1 ( N 1 N g 1 ) + a 2 ξ 2 ( N 2 N g 2 ) G th ] S + C N 1 V 1 τ s .
S ^
N ^ 1
N ^ 2
d N ^ 1 d t = a ^ 1 ξ 1 V 1 ( N ^ 1 N g 1 ) S ^ N ^ 1 τ ^ s N ^ 1 N ^ 2 T 12 + I ^ ( 1 + m sin 2 π f t ) ,
d N ^ 2 d t = a ^ 2 ξ 2 V 2 ( N ^ 2 N g 2 ) S ^ N ^ 2 τ ^ s N ^ 2 N ^ 1 T 21 ,
d S ^ d t = [ a ^ 1 ξ 1 ( N ^ 1 N g 1 ) + a ^ 2 ξ 2 ( N ^ 2 N g 2 ) G th ] S ^ + C N ^ 1 V 1 τ ^ s + κ τ in S .
τ s
G t h
N g
τ in
m I sin 2 π f t
V 1
κ = ( 1 r 0 2 ) b / r 0
r 0
a 1 = 3.08 × 10 12 m 3 s 1
a 2 = 1.232 × 10 11 m 3 s 1
N g 1 = 1.4 × 10 24 m 3
N g 2 = 1.6 × 10 24 m 3
e = 1.602 × 10 19   C
T 12 = 2.65 × 10 9   s
T 21 = 4.45 × 10 9   s
ξ 1 = 0.2034
ξ 2 = 0.1449
C = 1.573 × 10 5
τ s = 3.0 × 10 9   s
G th = 3.92 × 10 11 s 1
V 1 = 7.2 × 10 17 m 3
V 2 = 1.0296 × 10 16 m 3
τ in = 7.2 × 10 12   s
r 0 = 0.11
I = 19.8   mA
f = 3.4   GHz
N 1 ( 0 ) = N 2 ( 0 ) = S ( 0 ) = 0
a ^ 1 = a 1 ( 1 + δ a ) ,
τ ^ s = τ s ( 1 + δ τ ) ,
I = I ^ ( 1 + δ I ) .
δ a
δ τ
δ I
δ a = 0
δ τ = 0
δ I = 0
λ t h
λ th
a n , i
π n , k = ( a n , 1 , a n , 2 , a n , 3 , , a n , i , , a n , k 1 , a n , k ) .
Π N = ( π 1 , k , π 2 , k , , π n , k , , π N , k ) = ( a 1 , 1 , a 1 , 2 , , a 1 , k , a 2 , 1 , a 2 , 2 , , a N , 1 , a N , 2 , , a N , k ) .
P n , i ( a n , i , a n , i + 1 )
( i = 1 , 2 , …  , k 1 )
( a n , i , a n , i + 1 )
( i = 1 , 2 , …  , k 1 )
( a n , i , a n , i + 1 )
ε n , i , i
( a n , i , a n , i + 1 )
( a n , i , a n , i + 1 )
10 2
( a n , i + 1 a n , i + 1 ) 2 + ( a n , i a n , i ) 2
λ = 1 τ ln | ε n , i + 1 , i + 1 ε n , i , i | ,
i
δ τ
δ a
( δ τ = 0 , δ a = 0 )
( δ τ = 0 , δ a = 0 )
δ a = 1.15 × δ τ
δ τ
δ a
δ τ
δ a
( δ τ = 0 , δ a = 0 )
δ a = 1.15 × δ τ
ρ = max Δ t ( P t ( t ) P t ) ( P r ( t + Δ t ) P r ) ( P t ( t ) P t ) 2 ( P r ( t + Δ t ) P r ) 2 ,
P t
P r
δ τ
δ a
( δ τ = 0 , δ a = 0 )
δ a = 1.15 × δ τ
δ τ
δ a
δ τ
δ a
δ a = 1.15 × δ τ
± 0.2
( δ I = 0 )
δ I
δ I
δ I
δ I
λ 1 = 4.62683
λ 2 = 4.19725
δ τ
δ a
λ th = 4.41239
λ th
λ th
δ τ
δ a
λ th
δ τ
δ a
λ th = 5.30
δ a = 1.15 × δ τ + 0.016
( δ τ = 0.042 , δ a = 0.018 )
( δ I = 0 )
δ I
δ I
δ a
δ τ
( δ τ = 0.042 , δ a = 0.018 )
P n , i ( a n , i , a n , i + 1 )
P 1 , i
P 2 , j
P 1 , i
P 2 , j
P 1 , i + 1
P 2 , j + 1
P 1 , i
P 2 , j
Σ 1
Σ 2
Σ 1
Σ 2
P 1 , i
P 2 , j
Σ 1
Σ 2
δ a = 0.026
δ τ = 0.068
δ I
λ th
λ 1 = 6.77934
λ 2 = 5.65657
δ I
δ I
δ I
δ I
P 1 , i
P 2 , j
P 1 , i
P 2 , i
Σ 1
Σ 2
( λ th )
δ I = 0.013
δ I = 0.056
λ th
λ th
λ th
λ th
λ th
λ th
λ th
δ a = 1.15 × δ τ + 0.016
δ a = 1.15 × δ τ + 0.016
( δ τ = 0.042 , δ a = 0.018 )
Σ 1
Σ 2
( δ τ = 0.068 , δ a = 0.026 )

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