Abstract

The nonlinear Schrödinger equation can be solved by split-step methods, where in each step, linear dispersion and nonlinear effects are treated separately. This paper considers the optimal design of an FIR filter as the time-domain implementation for the linear part. The objective is to minimize the integral of the squared error between the FIR frequency response and the desired dispersion characteristics over the band of interest. This least square (LS) problem is solved in two approaches: the normal equation approach gives the explicit solution, whereas the singular value decomposition approach, which is based on the theory of discrete prolate spheroidal sequences, provides geometrical insights and reveals that the normal equation could be ill-conditioned. In addition, the frequency response might exhibit singular behaviors such as overshoot. We propose two filters that both can mitigate these shortcomings: the regularized LS filter achieves this by adding a regularization term to the objective function; the quadratically constrained quadratic programming-based filter addresses overshooting more efficiently by imposing a maximum magnitude constraint on the frequency response. Numerical results show that these filters can suppress the overshoots, control the squared error, reduce the filter length and lower the computational complexity. For both single channel and wavelength-division multiplexing channels, the proposed methods generate similar outputs as the standard split-step Fourier method.

© 2012 IEEE

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  2. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).
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  4. Q. Chang, E. Jia, W. Sun, "Difference schemes for solving the generalized nonlinear Schrödinger equation," J. Comput. Phys. 148, 397-415 (1999).
  5. O. Sinkin, Z. Holzlöhner, R. , C. J. Menyuk, "Optimization of the split-step Fourier method in modeling optical-fiber communications systems," J. Lightw. Technol. 21, 61-68 (2003).
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  7. A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Sel. Areas Commun. 15, 751-765 (1997).
  8. T. Kremp, W. Freude, "Fast split-step wavelet collocation method for WDM system parameter optimization," J. Lightw. Technol. 23, 1491-1502 (2005).
  9. L. R. Watkins, Y. R. Zhou, "Modeling propagation in optical fibers using wavelets," J. Lightw. Technol. 12, 1536-1542 (1994).
  10. M. Delfour, M. Fortin, G. Payr, "Finite-difference solutions of a non-linear Schrödinger equation," J. Comput. Phys. 44, 277-288 (1981).
  11. K. Peddanarappagari, M. Brandt-Pearce, "Volterra series approach for optimizing fiber-optic communications system designs," J. Lightw. Technol. 16, 2046-2055 (1998).
  12. X. Li, X. Chen, M. Qasmi, "A broad-band digital filtering approach for time-domain simulation of pulse propagation in optical fiber," J. Lightw. Technol. 23, 864-875 (2005).
  13. K. He, X. Li, "An efficient approach for time-domain simulation of pulse propagation in optical fiber," J. Lightw. Technol. 28, 2912-2918 (2010).
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  17. A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics (Springer-Verlag, 2007).
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  27. G. Golub, C. Van Loan, Matrix Computations (Johns Hopkins Univ. Press, 1996).
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  36. R. Farhoudi, K. Mehrany, "Time-domain split-step method with variable step-sizes in vectorial pulse propagation by using digital filters," Opt. Commun. 283, 2518-2524 (2010).
  37. S. Savory, "Digital filters for coherent optical receivers," Opt. Exp. 16, 804-817 (2008).

2010 (3)

K. He, X. Li, "An efficient approach for time-domain simulation of pulse propagation in optical fiber," J. Lightw. Technol. 28, 2912-2918 (2010).

L. Trefethen, "Householder triangularization of a quasimatrix," IMA J. Numer. Anal. 30, 887-897 (2010).

R. Farhoudi, K. Mehrany, "Time-domain split-step method with variable step-sizes in vectorial pulse propagation by using digital filters," Opt. Commun. 283, 2518-2524 (2010).

2009 (1)

L. Zhu, X. Li, E. Mateo, G. Li, "Complementary FIR filter pair for distributed impairment compensation of WDM fiber transmission," IEEE Photon. Technol. Lett. 21, 292-294 (2009).

2008 (2)

Q. Zhang, M. Hayee, "Symmetrized split-step Fourier scheme to control global simulation accuracy in fiber-optic communication systems," J. Lightw. Technol. 26, 302-316 (2008).

S. Savory, "Digital filters for coherent optical receivers," Opt. Exp. 16, 804-817 (2008).

2005 (2)

T. Kremp, W. Freude, "Fast split-step wavelet collocation method for WDM system parameter optimization," J. Lightw. Technol. 23, 1491-1502 (2005).

X. Li, X. Chen, M. Qasmi, "A broad-band digital filtering approach for time-domain simulation of pulse propagation in optical fiber," J. Lightw. Technol. 23, 864-875 (2005).

2004 (1)

Z. Battles, L. Trefethen, "An extension of MATLAB to continuous functions and operators," SIAM J. Sci. Comput. 25, 1743-1770 (2004).

2003 (1)

O. Sinkin, Z. Holzlöhner, R. , C. J. Menyuk, "Optimization of the split-step Fourier method in modeling optical-fiber communications systems," J. Lightw. Technol. 21, 61-68 (2003).

1999 (1)

Q. Chang, E. Jia, W. Sun, "Difference schemes for solving the generalized nonlinear Schrödinger equation," J. Comput. Phys. 148, 397-415 (1999).

1998 (1)

K. Peddanarappagari, M. Brandt-Pearce, "Volterra series approach for optimizing fiber-optic communications system designs," J. Lightw. Technol. 16, 2046-2055 (1998).

1997 (1)

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Sel. Areas Commun. 15, 751-765 (1997).

1994 (1)

L. R. Watkins, Y. R. Zhou, "Modeling propagation in optical fibers using wavelets," J. Lightw. Technol. 12, 1536-1542 (1994).

1984 (1)

T. Taha, M. Ablowitz, "Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation," J. Comput. Phys. 55, 203-230 (1984).

1983 (1)

B. Hermansson, D. Yevick, "Numerical investigation of soliton interaction," Electronics Lett. 19, 570-571 (1983).

1982 (1)

E. Phan-Huy Hao, "Quadratically constrained quadratic programming: Some applications and a method for solution," Math. Meth. Oper. Res. 26, 105-119 (1982).

1981 (1)

M. Delfour, M. Fortin, G. Payr, "Finite-difference solutions of a non-linear Schrödinger equation," J. Comput. Phys. 44, 277-288 (1981).

1978 (1)

D. Slepian, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: The discrete case," Bell Syst. Tech. J. 57, 1371-1430 (1978).

1973 (1)

R. Hardin, F. Tappert, "Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations," SIAM Rev. 15, 423 (1973).

1947 (1)

N. Levinson, "The Wiener RMS error criterion in filter design and prediction," J. Math. Phys. 25, 261-278 (1947).

Bell Syst. Tech. J. (1)

D. Slepian, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: The discrete case," Bell Syst. Tech. J. 57, 1371-1430 (1978).

Electronics Lett. (1)

B. Hermansson, D. Yevick, "Numerical investigation of soliton interaction," Electronics Lett. 19, 570-571 (1983).

IEEE Photon. Technol. Lett. (1)

L. Zhu, X. Li, E. Mateo, G. Li, "Complementary FIR filter pair for distributed impairment compensation of WDM fiber transmission," IEEE Photon. Technol. Lett. 21, 292-294 (2009).

IEEE J. Sel. Areas Commun. (1)

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Sel. Areas Commun. 15, 751-765 (1997).

IMA J. Numer. Anal. (1)

L. Trefethen, "Householder triangularization of a quasimatrix," IMA J. Numer. Anal. 30, 887-897 (2010).

J. Lightw. Technol. (4)

K. Peddanarappagari, M. Brandt-Pearce, "Volterra series approach for optimizing fiber-optic communications system designs," J. Lightw. Technol. 16, 2046-2055 (1998).

X. Li, X. Chen, M. Qasmi, "A broad-band digital filtering approach for time-domain simulation of pulse propagation in optical fiber," J. Lightw. Technol. 23, 864-875 (2005).

T. Kremp, W. Freude, "Fast split-step wavelet collocation method for WDM system parameter optimization," J. Lightw. Technol. 23, 1491-1502 (2005).

O. Sinkin, Z. Holzlöhner, R. , C. J. Menyuk, "Optimization of the split-step Fourier method in modeling optical-fiber communications systems," J. Lightw. Technol. 21, 61-68 (2003).

J. Comput. Phys. (3)

Q. Chang, E. Jia, W. Sun, "Difference schemes for solving the generalized nonlinear Schrödinger equation," J. Comput. Phys. 148, 397-415 (1999).

T. Taha, M. Ablowitz, "Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation," J. Comput. Phys. 55, 203-230 (1984).

M. Delfour, M. Fortin, G. Payr, "Finite-difference solutions of a non-linear Schrödinger equation," J. Comput. Phys. 44, 277-288 (1981).

J. Lightw. Technol. (1)

K. He, X. Li, "An efficient approach for time-domain simulation of pulse propagation in optical fiber," J. Lightw. Technol. 28, 2912-2918 (2010).

J. Lightw. Technol. (2)

Q. Zhang, M. Hayee, "Symmetrized split-step Fourier scheme to control global simulation accuracy in fiber-optic communication systems," J. Lightw. Technol. 26, 302-316 (2008).

L. R. Watkins, Y. R. Zhou, "Modeling propagation in optical fibers using wavelets," J. Lightw. Technol. 12, 1536-1542 (1994).

J. Math. Phys. (1)

N. Levinson, "The Wiener RMS error criterion in filter design and prediction," J. Math. Phys. 25, 261-278 (1947).

Math. Meth. Oper. Res. (1)

E. Phan-Huy Hao, "Quadratically constrained quadratic programming: Some applications and a method for solution," Math. Meth. Oper. Res. 26, 105-119 (1982).

Opt. Commun. (1)

R. Farhoudi, K. Mehrany, "Time-domain split-step method with variable step-sizes in vectorial pulse propagation by using digital filters," Opt. Commun. 283, 2518-2524 (2010).

Opt. Exp. (1)

S. Savory, "Digital filters for coherent optical receivers," Opt. Exp. 16, 804-817 (2008).

SIAM J. Sci. Comput. (1)

Z. Battles, L. Trefethen, "An extension of MATLAB to continuous functions and operators," SIAM J. Sci. Comput. 25, 1743-1770 (2004).

SIAM Rev. (1)

R. Hardin, F. Tappert, "Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations," SIAM Rev. 15, 423 (1973).

Other (16)

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics (Springer-Verlag, 2007).

Y. Ye, Interior Point Algorithms: Theory and Analysis (Wiley, 1997).

S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge Univ. Press, 2004).

J. Hunter, B. Nachtergaele, Applied Analysis (World Scientific, 2001).

W. Rudin, Real and Complex Analysis (McGraw-Hill, 1987).

J. Nocedal, S. Wright, Numerical Optimization (Springer-Verlag, 2006).

G. Golub, C. Van Loan, Matrix Computations (Johns Hopkins Univ. Press, 1996).

Å. Björck, Numerical Methods for Least Squares Problems (SIAM, 1996).

Y. Zhu, Optimal design of dispersion filter for time-domain implementation of split-step method in optical fiber communication Master's thesis McGill Univ.MontrealCanada (2011).

G. W. Stewart, Afternotes Goes to Graduate School (SIAM, 1998).

"YALMIP wiki," http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Main.HomePage.

M. Grant, S. Boyd, "CVX: Matlab software for disciplined convex programming," http://sedumi.ie.lehigh.edu/.

"SeDuMi: A Matlab toolbox for optimization over symmetric cones," [Online]. Available: http://sedumi.ie.lehigh.edu/.

A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete-Time Signal Processing (Prentice Hall, 1999).

H. Fu, "Revisiting large-scale convolution and FFT on parallel computation platforms," (2010) http://www.stanford.edu/~haohuan/pdf/ncar_talk.pdf.

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