Abstract

The transmittance transfer function of single mode optical fibers operating in both linear and nonlinear regions is presented. For the linear domain, Fresnel sine and cosine integrals are obtained via the Fourier transform. In the nonlinear region dominated by self-phase-modulation effects, the Volterra series is essential to obtain the nonlinear transfer function. A convergence criterion for the Volterra series transfer function (VSTF) approach is described for solving the nonlinear Schrödinger wave propagation equation. Soliton transmission over single fibers is demonstrated as a case study of the application of the VSTF and a modified VSTF with a number of segmented steps whose distance is within the limit of the convergence of the VSTF.

© 2009 Optical Society of America

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References

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  1. M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, 1989).
  2. K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15, 2046-2055 (1997).
    [CrossRef]
  3. K. V. Peddanarappagari and M. Brandt-Pearce, “Study of fiber nonlinearities in communication systems using a Volterra series transfer function approach,” in Proceedings of the 31st Annual Conference on Information Sciences and Systems (1997), pp. 752-757.
  4. K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra series approach for optimizing fiber-optic communications system designs,” J. Lightwave Technol. 16, 2046-2055 (1998).
    [CrossRef]
  5. M. B. Brilliant, “Theory of the analysis of nonlinear systems,” MIT Research Laboratory of Electronics, Tech. Rep. 345, March 3, 1958.
  6. A. Samelis, D. R. Pehlke, and D. Pavlidis, “Volterra series based nonlinear simulation of HBTs using analytically extracted models,” Electron. Lett. 30, 1098-1100 (1994).
    [CrossRef]
  7. J. Tang, “The channel capacity of a multispan DWDM system employing dispersive nonlinear optical fibers and an ideal coherent optical receiver,” J. Lightwave Technol. 20, 1095-1101 (2002).
    [CrossRef]
  8. J. Tang, “A comparison study of the Shannon channel capacity of various nonlinear optical fibers,” J. Lightwave Technol. 24, 2070-2075 (2006).
    [CrossRef]
  9. J. Tang, “The Shannon channel capacity of dispersion-free nonlinear optical fiber transmission,” J. Lightwave Technol. 19, 1104-1109 (2001).
    [CrossRef]
  10. B. Xu and M. Brandt-Pearce, “Comparison of FWM- and XPM-induced crosstalk using the Volterra series transfer function method,” J. Lightwave Technol. 21, 40-54 (2003).
    [CrossRef]
  11. G. P. Agrawal, Optical Communication Systems (Academic, 1997).
  12. A. F. Elrefaie, R. E. Wagner, D. A. Atlas, and D. G. Daut, “Chromatic dispersion limitations in coherent lightwave transmission systems,” J. Lightwave Technol. 6, 704-709 (1998).
    [CrossRef]
  13. R. C. Srinivasan and J. C. Cartledge, “On using fiber transfer functions to characterize laser chirp and fiber dispersion,” IEEE Photon. Technol. Lett. 7, 1327-1329 (1995).
    [CrossRef]
  14. A. V. T. Cartaxo, B. Wedding, and W. Idler, “Influence of fiber nonlinearity on the fiber transfer function: theoretical and experimental analysis,” J. Lightwave Technol. 17, 1806-1813 (1999).
    [CrossRef]
  15. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1968) p. 14.
  16. J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, 2001), pp. 185-213.
  17. L. N. Binh and N. Nguyen, “Generation of high-order multi-bound-solitons and propagation in optical fibers,” Opt. Commun. 282, 2394-2406 (2009).
    [CrossRef]
  18. W. J. Lai, P. Shum, L. N. Binh, and M. Karlsson, “Phase-plane analysis of rational harmonic mode-locking in an erbium-doped fiber ring laser,” IEEE J. Quantum Electron. 41, 426-433 (2005).
    [CrossRef]

2009

L. N. Binh and N. Nguyen, “Generation of high-order multi-bound-solitons and propagation in optical fibers,” Opt. Commun. 282, 2394-2406 (2009).
[CrossRef]

2006

2005

W. J. Lai, P. Shum, L. N. Binh, and M. Karlsson, “Phase-plane analysis of rational harmonic mode-locking in an erbium-doped fiber ring laser,” IEEE J. Quantum Electron. 41, 426-433 (2005).
[CrossRef]

2003

2002

2001

1999

1998

A. F. Elrefaie, R. E. Wagner, D. A. Atlas, and D. G. Daut, “Chromatic dispersion limitations in coherent lightwave transmission systems,” J. Lightwave Technol. 6, 704-709 (1998).
[CrossRef]

K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra series approach for optimizing fiber-optic communications system designs,” J. Lightwave Technol. 16, 2046-2055 (1998).
[CrossRef]

1997

K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15, 2046-2055 (1997).
[CrossRef]

1995

R. C. Srinivasan and J. C. Cartledge, “On using fiber transfer functions to characterize laser chirp and fiber dispersion,” IEEE Photon. Technol. Lett. 7, 1327-1329 (1995).
[CrossRef]

1994

A. Samelis, D. R. Pehlke, and D. Pavlidis, “Volterra series based nonlinear simulation of HBTs using analytically extracted models,” Electron. Lett. 30, 1098-1100 (1994).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Optical Communication Systems (Academic, 1997).

Atlas, D. A.

A. F. Elrefaie, R. E. Wagner, D. A. Atlas, and D. G. Daut, “Chromatic dispersion limitations in coherent lightwave transmission systems,” J. Lightwave Technol. 6, 704-709 (1998).
[CrossRef]

Binh, L. N.

L. N. Binh and N. Nguyen, “Generation of high-order multi-bound-solitons and propagation in optical fibers,” Opt. Commun. 282, 2394-2406 (2009).
[CrossRef]

W. J. Lai, P. Shum, L. N. Binh, and M. Karlsson, “Phase-plane analysis of rational harmonic mode-locking in an erbium-doped fiber ring laser,” IEEE J. Quantum Electron. 41, 426-433 (2005).
[CrossRef]

Brandt-Pearce, M.

B. Xu and M. Brandt-Pearce, “Comparison of FWM- and XPM-induced crosstalk using the Volterra series transfer function method,” J. Lightwave Technol. 21, 40-54 (2003).
[CrossRef]

K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra series approach for optimizing fiber-optic communications system designs,” J. Lightwave Technol. 16, 2046-2055 (1998).
[CrossRef]

K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15, 2046-2055 (1997).
[CrossRef]

K. V. Peddanarappagari and M. Brandt-Pearce, “Study of fiber nonlinearities in communication systems using a Volterra series transfer function approach,” in Proceedings of the 31st Annual Conference on Information Sciences and Systems (1997), pp. 752-757.

Brilliant, M. B.

M. B. Brilliant, “Theory of the analysis of nonlinear systems,” MIT Research Laboratory of Electronics, Tech. Rep. 345, March 3, 1958.

Cartaxo, A. V. T.

Cartledge, J. C.

R. C. Srinivasan and J. C. Cartledge, “On using fiber transfer functions to characterize laser chirp and fiber dispersion,” IEEE Photon. Technol. Lett. 7, 1327-1329 (1995).
[CrossRef]

Daut, D. G.

A. F. Elrefaie, R. E. Wagner, D. A. Atlas, and D. G. Daut, “Chromatic dispersion limitations in coherent lightwave transmission systems,” J. Lightwave Technol. 6, 704-709 (1998).
[CrossRef]

Elrefaie, A. F.

A. F. Elrefaie, R. E. Wagner, D. A. Atlas, and D. G. Daut, “Chromatic dispersion limitations in coherent lightwave transmission systems,” J. Lightwave Technol. 6, 704-709 (1998).
[CrossRef]

Idler, W.

Karlsson, M.

W. J. Lai, P. Shum, L. N. Binh, and M. Karlsson, “Phase-plane analysis of rational harmonic mode-locking in an erbium-doped fiber ring laser,” IEEE J. Quantum Electron. 41, 426-433 (2005).
[CrossRef]

Lai, W. J.

W. J. Lai, P. Shum, L. N. Binh, and M. Karlsson, “Phase-plane analysis of rational harmonic mode-locking in an erbium-doped fiber ring laser,” IEEE J. Quantum Electron. 41, 426-433 (2005).
[CrossRef]

Nguyen, N.

L. N. Binh and N. Nguyen, “Generation of high-order multi-bound-solitons and propagation in optical fibers,” Opt. Commun. 282, 2394-2406 (2009).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1968) p. 14.

Pavlidis, D.

A. Samelis, D. R. Pehlke, and D. Pavlidis, “Volterra series based nonlinear simulation of HBTs using analytically extracted models,” Electron. Lett. 30, 1098-1100 (1994).
[CrossRef]

Peddanarappagari, K. V.

K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra series approach for optimizing fiber-optic communications system designs,” J. Lightwave Technol. 16, 2046-2055 (1998).
[CrossRef]

K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15, 2046-2055 (1997).
[CrossRef]

K. V. Peddanarappagari and M. Brandt-Pearce, “Study of fiber nonlinearities in communication systems using a Volterra series transfer function approach,” in Proceedings of the 31st Annual Conference on Information Sciences and Systems (1997), pp. 752-757.

Pehlke, D. R.

A. Samelis, D. R. Pehlke, and D. Pavlidis, “Volterra series based nonlinear simulation of HBTs using analytically extracted models,” Electron. Lett. 30, 1098-1100 (1994).
[CrossRef]

Proakis, J. G.

J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, 2001), pp. 185-213.

Samelis, A.

A. Samelis, D. R. Pehlke, and D. Pavlidis, “Volterra series based nonlinear simulation of HBTs using analytically extracted models,” Electron. Lett. 30, 1098-1100 (1994).
[CrossRef]

Schetzen, M.

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, 1989).

Shum, P.

W. J. Lai, P. Shum, L. N. Binh, and M. Karlsson, “Phase-plane analysis of rational harmonic mode-locking in an erbium-doped fiber ring laser,” IEEE J. Quantum Electron. 41, 426-433 (2005).
[CrossRef]

Srinivasan, R. C.

R. C. Srinivasan and J. C. Cartledge, “On using fiber transfer functions to characterize laser chirp and fiber dispersion,” IEEE Photon. Technol. Lett. 7, 1327-1329 (1995).
[CrossRef]

Tang, J.

Wagner, R. E.

A. F. Elrefaie, R. E. Wagner, D. A. Atlas, and D. G. Daut, “Chromatic dispersion limitations in coherent lightwave transmission systems,” J. Lightwave Technol. 6, 704-709 (1998).
[CrossRef]

Wedding, B.

Xu, B.

Electron. Lett.

A. Samelis, D. R. Pehlke, and D. Pavlidis, “Volterra series based nonlinear simulation of HBTs using analytically extracted models,” Electron. Lett. 30, 1098-1100 (1994).
[CrossRef]

IEEE J. Quantum Electron.

W. J. Lai, P. Shum, L. N. Binh, and M. Karlsson, “Phase-plane analysis of rational harmonic mode-locking in an erbium-doped fiber ring laser,” IEEE J. Quantum Electron. 41, 426-433 (2005).
[CrossRef]

IEEE Photon. Technol. Lett.

R. C. Srinivasan and J. C. Cartledge, “On using fiber transfer functions to characterize laser chirp and fiber dispersion,” IEEE Photon. Technol. Lett. 7, 1327-1329 (1995).
[CrossRef]

J. Lightwave Technol.

Opt. Commun.

L. N. Binh and N. Nguyen, “Generation of high-order multi-bound-solitons and propagation in optical fibers,” Opt. Commun. 282, 2394-2406 (2009).
[CrossRef]

Other

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, 1989).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1968) p. 14.

J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, 2001), pp. 185-213.

M. B. Brilliant, “Theory of the analysis of nonlinear systems,” MIT Research Laboratory of Electronics, Tech. Rep. 345, March 3, 1958.

K. V. Peddanarappagari and M. Brandt-Pearce, “Study of fiber nonlinearities in communication systems using a Volterra series transfer function approach,” in Proceedings of the 31st Annual Conference on Information Sciences and Systems (1997), pp. 752-757.

G. P. Agrawal, Optical Communication Systems (Academic, 1997).

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Figures (14)

Fig. 1
Fig. 1

Rectangular pulse transmission through a single mode fiber: (a) pulse response, (b) frequency spectrum, (c) step response of the quadratic-phase transmittance function.

Fig. 2
Fig. 2

Input pulse peak power as a function of fiber length of the third-order transfer function.

Fig. 3
Fig. 3

Input pulse peak power as a function of the fiber length of the fifth-order transfer function.

Fig. 4
Fig. 4

Input pulse peak power as a function of fiber length of the seventh-order transfer function.

Fig. 5
Fig. 5

Deviation factor as a function of input optical power for a fiber length of 50 km .

Fig. 6
Fig. 6

Deviation factor as a function of input power for a fiber length of 100 km .

Fig. 7
Fig. 7

Solitonic pulse propagation via the simulation by SSF method.

Fig. 8
Fig. 8

Solitonic pulse propagation via the simulation by VSTF method.

Fig. 9
Fig. 9

Deviation factor between VSTF and SSF methods.

Fig. 10
Fig. 10

Representing a fiber section with eight VSTF segments.

Fig. 11
Fig. 11

Pulse evolution simulation of a single soliton using (a) SSF method, (b) SS-VSTF method.

Fig. 12
Fig. 12

NSD for soliton simulation with 500-segment SS-VSTF.

Fig. 13
Fig. 13

Pulse evolution for 1000-segment fiber using (a) SSF, (b) SS-VSTF method.

Fig. 14
Fig. 14

NSD between the SSF method and the 1000-segment SS-VSTF model.

Equations (28)

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A z + α 0 2 A + β 1 A t + j β 2 2 2 A t 2 + β 3 6 3 A t 3 = j γ A 2 A a 1 ( A 2 A ) t + a 2 A t A 2 + a 3 A 2 t A + j Q R A s r ( t t 1 ) A ( t 1 , z ) 2 d t 1
h ( t ) = 1 j 4 π β 2 exp ( j t 2 4 β 2 ) H ( ω ) = exp ( j β 2 ω 2 ) ,
s ( t ) = 0 t 1 j 4 π β 2 exp ( j t 2 4 β 2 ) d t = 1 j 4 π β 2 [ C ( 1 4 β 2 t ) + j S ( 1 4 β 2 t ) ]
C ( t ) = 0 t cos ( π 2 τ 2 ) d τ ,
S ( t ) = 0 t sin ( π 2 τ 2 ) d τ ,
Y ( ω ) = n = 1 H n ( ω 1 , , ω n 1 , ω ω 1 ω n 1 ) × X ( ω 1 ) X ( ω n 1 ) X ( ω ω 1 ω n 1 ) d ω 1 d ω n 1 ,
A z = α 0 2 A β 1 A t j β 2 2 2 A t 2 β 3 6 3 A t 3 + j γ A 2 A ,
A ( ω , z ) = H 1 ( ω , z ) A ( ω ) + H 3 ( ω 1 , ω 2 , ω ω 1 + ω 2 , z ) A ( ω 1 ) A * ( ω 2 ) A ( ω ω 1 + ω 2 ) d ω 1 d ω 2 + H 5 ( ω 1 , ω 2 , ω 3 , ω 4 , ω ω 1 + ω 2 ω 3 + ω 4 , z ) A ( ω 1 ) A * ( ω 2 ) A ( ω 3 ) A * ( ω 4 ) A ( ω ω 1 + ω 2 ω 3 + ω 4 ) d ω 1 d ω 2 d ω 3 d ω 4 ,
A ( ω , z ) z = G 1 ( ω ) A ( ω , z ) + G 3 ( ω 1 , ω 2 , ω ω 1 + ω 2 ) A ( ω 1 , z ) A * ( ω 2 , z ) A ( ω ω 1 + ω 2 , z ) d ω 1 d ω 2 ,
G 1 ( ω ) = α 0 2 + j β 1 ω + j β 2 2 ω 2 j β 3 6 ω 3 , G 3 ( ω 1 , ω 2 , ω 3 ) = j γ .
z [ H 1 ( ω , z ) A ( ω ) + i H 3 ( ω 1 , ω 2 , ω ω 1 + ω 2 , z ) A ( ω 1 ) A * ( ω 2 ) A ( ω ω 1 + ω 2 ) d ω 1 d ω 2 + H 5 ( ω 1 , ω 2 , ω 3 , ω 4 , ω ω 1 + ω 2 ω 3 + ω 4 , z ) A ( ω 1 ) A * ( ω 2 ) A ( ω 3 ) A * ( ω 4 ) A ( ω ω 1 + ω 2 ω 3 + ω 4 ) d ω 1 d ω 2 d ω 3 d ω 4 ] = G 1 ( ω ) [ H 1 ( ω , z ) A ( ω ) + i H 3 ( ω 1 , ω 2 , ω ω 1 + ω 2 , z ) A ( ω 1 ) A * ( ω 2 ) A ( ω ω 1 + ω 2 ) d ω 1 d ω 2 + H 5 ( ω 1 , ω 2 , ω 3 , ω 4 , ω ω 1 + ω 2 ω 3 + ω 4 , z ) A ( ω 1 ) A * ( ω 2 ) A ( ω 3 ) A * ( ω 4 ) A ( ω ω 1 + ω 2 ω 3 + ω 4 ) d ω 1 d ω 2 d ω 3 d ω 4 ] + G 3 ( ω 1 , ω 2 , ω ω 1 + ω 2 ) [ H 1 ( ω 1 , z ) A ( ω 1 ) + i H 3 ( ω 11 , ω 12 , ω 1 ω 11 + ω 12 , z ) A ( ω 11 ) A * ( ω 12 ) A ( ω 1 ω 11 + ω 12 ) d ω 11 d ω 12 + H 5 ( ω 11 , ω 12 , ω 13 , ω 14 , ω 1 ω 11 + ω 12 ω 13 + ω 14 , z ) A ( ω 11 ) A * ( ω 12 ) A ( ω 13 ) A * ( ω 14 ) A ( ω 1 ω 11 + ω 12 ω 13 + ω 14 ) d ω 11 d ω 12 d ω 13 d ω 14 ] [ H 1 ( ω 2 , z ) A ( ω 2 ) + i H 3 ( ω 21 , ω 22 , ω 2 ω 21 + ω 22 , z ) A ( ω 21 ) A * ( ω 22 ) A ( ω 2 ω 21 + ω 22 ) d ω 21 d ω 22 + H 5 ( ω 21 , ω 22 , ω 23 , ω 24 , ω 2 ω 21 + ω 22 ω 23 + ω 24 , z ) A ( ω 21 ) A * ( ω 22 ) A ( ω 23 ) A * ( ω 24 ) A ( ω 2 ω 21 + ω 22 ω 23 + ω 24 ) d ω 21 d ω 22 d ω 23 d ω 24 ] * [ H 1 ( ω ω 1 + ω 2 , z ) A ( ω ω 1 + ω 2 ) + i H 3 ( ω 31 , ω 32 , ω ω 1 + ω 2 ω 31 + ω 32 , z ) A ( ω 31 ) A * ( ω 32 ) A ( ω ω 1 + ω 2 ω 31 + ω 32 ) d ω 31 d ω 32 + H 5 ( ω 31 , ω 32 , ω 33 , ω 34 , ω ω 1 + ω 2 ω 31 + ω 32 ω 33 + ω 34 , z ) A ( ω 31 ) A * ( ω 32 ) A ( ω 33 ) A * ( ω 34 ) A ( ω ω 1 + ω 2 ω 31 + ω 32 ω 33 + ω 34 ) d ω 31 d ω 32 d ω 33 d ω 34 ] .
z H 1 ( ω , z ) = G 1 ( ω ) H 1 ( ω , z ) .
H 1 ( ω , z ) = e G 1 ( ω ) z = e ( α 0 2 + j β 1 ω + j β 2 2 ω 2 j β 3 6 ω 3 ) z .
z H 3 ( ω 1 , ω 2 , ω ω 1 + ω 2 , z ) A ( ω 1 ) A * ( ω 2 ) A ( ω ω 1 + ω 2 ) d ω 1 d ω 2 = G 3 ( ω 1 , ω 2 , ω ω 1 + ω 2 ) H 1 ( ω 1 , z ) A ( ω 1 ) H 2 * ( ω 2 , z ) A ( ω 2 ) H 1 ( ω ω 1 + ω 2 ) A ( ω ω 1 + ω 2 ) d ω 1 d ω 2 .
H 3 ( ω 1 , ω 2 , ω 3 , z ) z = G 1 ( ω 1 ω 2 + ω 3 ) H 3 ( ω 1 , ω 2 , ω 3 , z ) + G 3 ( ω 1 , ω 2 , ω 3 ) H 1 ( ω 1 , z ) H 1 * ( ω 2 , z ) H 1 ( ω 3 , z ) .
H 3 ( ω 1 , ω 2 , ω 3 , z ) = G 3 ( ω 1 , ω 2 , ω 3 ) e ( G 1 ( ω 1 ) + G 1 * ( ω 2 ) + G 1 ( ω 3 ) ) z e G 1 ( ω 1 ω 2 + ω 3 ) z G 1 ( ω 1 ) + G 1 * ( ω 2 ) + G 1 ( ω 3 ) G 1 ( ω 1 ω 2 + ω 3 ) .
H 5 ( ω 1 , ω 2 , ω 3 , ω 4 , ω 5 , z ) = H 1 ( ω 1 , z ) H 1 * ( ω 2 , z ) H 1 ( ω 3 , z ) H 1 * ( ω 4 , z ) H 1 ( ω 5 , z ) H 1 ( ω 1 ω 2 + ω 3 ω 4 + ω 5 , z ) G 1 ( ω 1 ) + G 1 * ( ω 2 ) + G 1 ( ω 3 ) + G 1 * ( ω 4 ) + G 1 ( ω 5 ) G 1 ( ω 1 ω 2 + ω 3 ω 4 + ω 5 ) × [ G 3 ( ω 1 , ω 2 , ω 3 ω 4 + ω 5 ) G 3 ( ω 3 , ω 4 , ω 5 ) G 1 ( ω 3 ) + G 1 * ( ω 4 ) + G 1 ( ω 5 ) G 1 ( ω 3 ω 4 + ω 5 ) + G 3 ( ω 1 , ω 2 ω 3 + ω 4 , ω 5 ) G 3 * ( ω 2 , ω 3 , ω 4 ) G 1 * ( ω 2 ) + G 1 ( ω 3 ) + G 1 * ( ω 4 ) G 1 * ( ω 2 ω 3 + ω 4 ) + G 3 ( ω 1 ω 2 + ω 3 , ω 4 , ω 5 ) G 3 ( ω 1 , ω 2 , ω 3 ) G 1 ( ω 1 ) + G 1 * ( ω 2 ) + G 1 ( ω 3 ) G 1 ( ω 1 ω 2 + ω 3 ) ] G 3 ( ω 1 , ω 2 , ω 3 ω 4 + ω 5 ) G 3 ( ω 3 , ω 4 , ω 5 ) G 1 ( ω 3 ) + G 1 * ( ω 4 ) + G 1 ( ω 5 ) G 1 ( ω 3 ω 4 + ω 5 ) × H 1 ( ω 1 , z ) H 1 * ( ω 2 , z ) H 1 ( ω 1 ω 2 + ω 3 , z ) H 1 ( ω 1 ω 2 + ω 3 ω 4 + ω 5 , z ) G 1 ( ω 1 ) + G 1 * ( ω 2 ) + G 1 ( ω 3 ω 4 + ω 5 ) G 1 ( ω 1 ω 2 + ω 3 ω 4 + ω 5 ) G 3 ( ω 1 , ω 2 ω 3 + ω 4 , ω 5 ) G 3 * ( ω 2 , ω 3 , ω 4 ) G 1 * ( ω 2 ) + G 1 ( ω 3 ) + G 1 * ( ω 4 ) G 1 * ( ω 2 ω 3 + ω 4 ) × H 1 ( ω 1 , z ) H 1 * ( ω 2 ω 3 + ω 4 , z ) H 1 ( ω 5 , z ) H 1 ( ω 1 ω 2 + ω 3 ω 4 + ω 5 , z ) G 1 ( ω 1 ) + G 1 * ( ω 2 ω 3 + ω 4 ) + G 1 ( ω 5 ) G 1 ( ω 1 ω 2 + ω 3 ω 4 + ω 5 ) G 3 ( ω 1 ω 2 + ω 3 , ω 4 , ω 5 ) G 3 ( ω 1 , ω 2 , ω 3 ) G 1 ( ω 1 ) + G 1 * ( ω 2 ) + G 1 ( ω 3 ) G 1 ( ω 1 ω 2 + ω 3 ) × H 1 ( ω 1 ω 2 + ω 3 , z ) H 1 * ( ω 4 , z ) H 1 ( ω 5 , z ) H 1 ( ω 1 ω 2 + ω 3 ω 4 + ω 5 , z ) G 1 ( ω 1 ω 2 + ω 3 ) + G 1 * ( ω 4 ) + G 1 ( ω 5 ) G 1 ( ω 1 ω 2 + ω 3 ω 4 + ω 5 ) .
n = 1 Y n ( ω 1 , , ω n ) = n = 1 H n ( ω 1 , , ω n ) U ( ω 1 ) U ( ω n ) n = 1 H n ( ω 1 , , ω n ) U max n ,
Y ( ω ) = Y 1 ( ω ) + Y 2 ( ω 1 , ω ω 1 ) d ω 1 + Y 3 ( ω 1 , ω 2 , ω ω 1 ω 2 ) d ω 1 d ω 2 + = H 1 ( ω ) U ( ω ) + H 2 ( ω 1 , ω ω 1 ) U ( ω 1 ) U ( ω ω 1 ) d ω 1 + H 3 ( ω 1 , ω 2 , ω ω 1 ω 2 ) × U ( ω 1 ) U ( ω 2 ) U ( ω ω 1 ω 2 ) d ω 1 d ω 2 + .
Y ( ω ) H 1 ( ω , z ) U max + H 2 ( ω 1 , ω ω 1 ) d ω 1 U max 2 + H 3 ( ω 1 , ω 2 , ω ω 1 ω 2 ) d ω 1 d ω 2 U max 3 + .
H 3 ( ω 1 , ω 2 , ω ω 1 ω 2 ) d ω 1 d ω 2 H 1 ( ω ) U max 2 < 1 ,
H 1 ( ω ) > H 3 ( ω 1 , ω 2 , ω ω 1 ω 2 ) d ω 1 d ω 2 U max 2 .
H 5 ( ω 1 , ω 2 , ω 3 , ω 4 , ω ω 1 ω 2 ω 3 ω 4 ) d ω 1 d ω 2 d ω 3 d ω 4 H 3 ( ω 1 , ω 2 , ω ω 1 ω 2 ) d ω 1 d ω 2 U max 2 < 1
H 5 ( ω 1 , ω 2 , ω 3 , ω 4 , ω ω 1 ω 2 ω 3 ω 4 ) d ω 1 d ω 2 d ω 3 d ω 4 U max 4 H 1 ( ω ) < 1 .
δ n = sup { H n ( ω 1 , , ω ω 1 ω n 1 ) d ω 1 d ω n 1 H 1 ( ω ) } .
U max = ( 1 δ n ) 1 n 1 = inf { ( H 1 ( ω ) H n ( ω 1 , , ω ω 1 ω n 1 ) d ω 1 d ω n 1 ) 1 n 1 } .
P peak = U max 2 = ( 1 δ n ) 2 n 1 = inf { ( H 1 ( ω 1 ) H n ( ω 1 , , ω ω 1 ω n 1 ) d ω 1 d ω n 1 ) 2 n 1 } .
Deviation in % = U 2 ( ω ) U 1 ( ω ) 2 d ω U 2 ( ω ) 2 d ω ,

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