Abstract

A discrete orthogonal Gauss–Hermite transform (DOGHT) is introduced for the analysis of optical pulse properties in the time and frequency domains. Gaussian quadrature nodes and weights are used to calculate the expansion coefficients. The discrete orthogonal properties of the DOGHT are similar to the ones satisfied by the discrete Fourier transform so the two transforms have many common characteristics. However, it is demonstrated that the DOGHT produces a more compact representation of pulses in the time and frequency domains and needs less expansion coefficients for a given accuracy. It is shown that it can be used advantageously for propagation analysis of optical signals in the linear and nonlinear regimes.

© 2003 Optical Society of America

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  1. D. Marcuse, “Pulse distortion in single-mode fibers,” Appl. Opt. 19, 1653–1660 (1980).
    [CrossRef] [PubMed]
  2. R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 423 (1973).
  3. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
    [CrossRef]
  4. T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
    [CrossRef]
  5. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  6. H. da Silva and J. O’Reilly, “Optical pulse modeling with Hermite–Gaussian functions,” Opt. Lett. 14, 526–528 (1989).
    [CrossRef] [PubMed]
  7. S. Dijali, A. Dienes, and J. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
    [CrossRef]
  8. T. Tang, “The Hermite spectral method for Gaussian-type functions,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 594–606 (1993).
    [CrossRef]
  9. P. Lazaridis, G. Debarge, and P. Gallion, “Exact solutions for linear propagation of chirped pulses using a chirped Gauss–Hermite orthogonal basis,” Opt. Lett. 22, 685–687 (1997).
    [CrossRef] [PubMed]
  10. A. Bogush and R. Elkins, “Gaussian field expansions for large aperture antennas,” IEEE Trans. Antennas Propag. 34, 228–243 (1986).
    [CrossRef]
  11. J. Murphy and R. Padman, “Phase centers of horn antennas using Gaussian beam mode analysis,” IEEE Trans. Antennas Propag. 38, 1306–1310 (1990).
    [CrossRef]
  12. M. Rao, T. Sarkar, T. Anjali, and R. S. Adve, “Simultaneous extrapolation in time and frequency domains using Hermite expansions,” IEEE Trans. Antennas Propag. 47, 1108–1115 (1999).
    [CrossRef]
  13. T. Oliveira e Silva and H. J. W. Belt, “On the determination of the optimal center and scale factor for truncated Hermite series,” presented at the European Conference on Signal Processing (EUSIPCO-96) Trieste, Italy, September 10–13, 1996, paper PFT.11.
  14. J.-B. Martens, “The Hermite transform theory,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1595–1606 (1990).
    [CrossRef]
  15. A. M. van Dijk and J.-B. Martens, “Image representation and compression with steered Hermite transforms,” Signal Process. 56, 1–16 (1997).
    [CrossRef]
  16. L. Lo Conte, R. Merletti, and G. Sandri, “Hermite expansions of compact support waveforms: applications to myoelectric signals,” IEEE Trans. Biomed. Eng. 41, 1147–1159 (1994).
    [CrossRef] [PubMed]
  17. A. I. Rasiah, R. Togneri, and Y. Attikiouzel, “Modelling 1-D signals using Hermite basis functions,” IEE Proc. Vision Image Signal Process. 144(6), 345–354 (1997).
    [CrossRef]
  18. A. J. Jerri, “The application of general discrete transforms to computing orthogonal series and solving boundary value problems,” Bull. Calcutta Math. Soc. 71, 177–187 (1979).
  19. P. Lazaridis, G. Debarge, and P. Gallion, “Split-step-Gauss-Hermite algorithm for fast and accurate simulation of soliton propagation,” Int. J. Numer. Model. 14, 325–329 (2001).
    [CrossRef]
  20. N. J. Zabusky and M. D. Kruskal, “Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240–243 (1965).
    [CrossRef]
  21. J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
    [CrossRef]

2001

P. Lazaridis, G. Debarge, and P. Gallion, “Split-step-Gauss-Hermite algorithm for fast and accurate simulation of soliton propagation,” Int. J. Numer. Model. 14, 325–329 (2001).
[CrossRef]

1999

M. Rao, T. Sarkar, T. Anjali, and R. S. Adve, “Simultaneous extrapolation in time and frequency domains using Hermite expansions,” IEEE Trans. Antennas Propag. 47, 1108–1115 (1999).
[CrossRef]

1997

A. M. van Dijk and J.-B. Martens, “Image representation and compression with steered Hermite transforms,” Signal Process. 56, 1–16 (1997).
[CrossRef]

A. I. Rasiah, R. Togneri, and Y. Attikiouzel, “Modelling 1-D signals using Hermite basis functions,” IEE Proc. Vision Image Signal Process. 144(6), 345–354 (1997).
[CrossRef]

P. Lazaridis, G. Debarge, and P. Gallion, “Exact solutions for linear propagation of chirped pulses using a chirped Gauss–Hermite orthogonal basis,” Opt. Lett. 22, 685–687 (1997).
[CrossRef] [PubMed]

1994

L. Lo Conte, R. Merletti, and G. Sandri, “Hermite expansions of compact support waveforms: applications to myoelectric signals,” IEEE Trans. Biomed. Eng. 41, 1147–1159 (1994).
[CrossRef] [PubMed]

1993

T. Tang, “The Hermite spectral method for Gaussian-type functions,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 594–606 (1993).
[CrossRef]

1990

S. Dijali, A. Dienes, and J. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

J.-B. Martens, “The Hermite transform theory,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1595–1606 (1990).
[CrossRef]

J. Murphy and R. Padman, “Phase centers of horn antennas using Gaussian beam mode analysis,” IEEE Trans. Antennas Propag. 38, 1306–1310 (1990).
[CrossRef]

1989

1986

A. Bogush and R. Elkins, “Gaussian field expansions for large aperture antennas,” IEEE Trans. Antennas Propag. 34, 228–243 (1986).
[CrossRef]

1984

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[CrossRef]

1980

1979

A. J. Jerri, “The application of general discrete transforms to computing orthogonal series and solving boundary value problems,” Bull. Calcutta Math. Soc. 71, 177–187 (1979).

1974

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

1973

R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 423 (1973).

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

1965

N. J. Zabusky and M. D. Kruskal, “Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240–243 (1965).
[CrossRef]

Ablowitz, M. J.

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[CrossRef]

Adve, R. S.

M. Rao, T. Sarkar, T. Anjali, and R. S. Adve, “Simultaneous extrapolation in time and frequency domains using Hermite expansions,” IEEE Trans. Antennas Propag. 47, 1108–1115 (1999).
[CrossRef]

Anjali, T.

M. Rao, T. Sarkar, T. Anjali, and R. S. Adve, “Simultaneous extrapolation in time and frequency domains using Hermite expansions,” IEEE Trans. Antennas Propag. 47, 1108–1115 (1999).
[CrossRef]

Attikiouzel, Y.

A. I. Rasiah, R. Togneri, and Y. Attikiouzel, “Modelling 1-D signals using Hermite basis functions,” IEE Proc. Vision Image Signal Process. 144(6), 345–354 (1997).
[CrossRef]

Bogush, A.

A. Bogush and R. Elkins, “Gaussian field expansions for large aperture antennas,” IEEE Trans. Antennas Propag. 34, 228–243 (1986).
[CrossRef]

da Silva, H.

Debarge, G.

P. Lazaridis, G. Debarge, and P. Gallion, “Split-step-Gauss-Hermite algorithm for fast and accurate simulation of soliton propagation,” Int. J. Numer. Model. 14, 325–329 (2001).
[CrossRef]

P. Lazaridis, G. Debarge, and P. Gallion, “Exact solutions for linear propagation of chirped pulses using a chirped Gauss–Hermite orthogonal basis,” Opt. Lett. 22, 685–687 (1997).
[CrossRef] [PubMed]

Dienes, A.

S. Dijali, A. Dienes, and J. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

Dijali, S.

S. Dijali, A. Dienes, and J. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

Elkins, R.

A. Bogush and R. Elkins, “Gaussian field expansions for large aperture antennas,” IEEE Trans. Antennas Propag. 34, 228–243 (1986).
[CrossRef]

Gallion, P.

P. Lazaridis, G. Debarge, and P. Gallion, “Split-step-Gauss-Hermite algorithm for fast and accurate simulation of soliton propagation,” Int. J. Numer. Model. 14, 325–329 (2001).
[CrossRef]

P. Lazaridis, G. Debarge, and P. Gallion, “Exact solutions for linear propagation of chirped pulses using a chirped Gauss–Hermite orthogonal basis,” Opt. Lett. 22, 685–687 (1997).
[CrossRef] [PubMed]

Hardin, R. H.

R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 423 (1973).

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Jerri, A. J.

A. J. Jerri, “The application of general discrete transforms to computing orthogonal series and solving boundary value problems,” Bull. Calcutta Math. Soc. 71, 177–187 (1979).

Kruskal, M. D.

N. J. Zabusky and M. D. Kruskal, “Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240–243 (1965).
[CrossRef]

Lazaridis, P.

P. Lazaridis, G. Debarge, and P. Gallion, “Split-step-Gauss-Hermite algorithm for fast and accurate simulation of soliton propagation,” Int. J. Numer. Model. 14, 325–329 (2001).
[CrossRef]

P. Lazaridis, G. Debarge, and P. Gallion, “Exact solutions for linear propagation of chirped pulses using a chirped Gauss–Hermite orthogonal basis,” Opt. Lett. 22, 685–687 (1997).
[CrossRef] [PubMed]

Lo Conte, L.

L. Lo Conte, R. Merletti, and G. Sandri, “Hermite expansions of compact support waveforms: applications to myoelectric signals,” IEEE Trans. Biomed. Eng. 41, 1147–1159 (1994).
[CrossRef] [PubMed]

Marcuse, D.

Martens, J.-B.

A. M. van Dijk and J.-B. Martens, “Image representation and compression with steered Hermite transforms,” Signal Process. 56, 1–16 (1997).
[CrossRef]

J.-B. Martens, “The Hermite transform theory,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1595–1606 (1990).
[CrossRef]

Merletti, R.

L. Lo Conte, R. Merletti, and G. Sandri, “Hermite expansions of compact support waveforms: applications to myoelectric signals,” IEEE Trans. Biomed. Eng. 41, 1147–1159 (1994).
[CrossRef] [PubMed]

Murphy, J.

J. Murphy and R. Padman, “Phase centers of horn antennas using Gaussian beam mode analysis,” IEEE Trans. Antennas Propag. 38, 1306–1310 (1990).
[CrossRef]

O’Reilly, J.

Padman, R.

J. Murphy and R. Padman, “Phase centers of horn antennas using Gaussian beam mode analysis,” IEEE Trans. Antennas Propag. 38, 1306–1310 (1990).
[CrossRef]

Rao, M.

M. Rao, T. Sarkar, T. Anjali, and R. S. Adve, “Simultaneous extrapolation in time and frequency domains using Hermite expansions,” IEEE Trans. Antennas Propag. 47, 1108–1115 (1999).
[CrossRef]

Rasiah, A. I.

A. I. Rasiah, R. Togneri, and Y. Attikiouzel, “Modelling 1-D signals using Hermite basis functions,” IEE Proc. Vision Image Signal Process. 144(6), 345–354 (1997).
[CrossRef]

Sandri, G.

L. Lo Conte, R. Merletti, and G. Sandri, “Hermite expansions of compact support waveforms: applications to myoelectric signals,” IEEE Trans. Biomed. Eng. 41, 1147–1159 (1994).
[CrossRef] [PubMed]

Sarkar, T.

M. Rao, T. Sarkar, T. Anjali, and R. S. Adve, “Simultaneous extrapolation in time and frequency domains using Hermite expansions,” IEEE Trans. Antennas Propag. 47, 1108–1115 (1999).
[CrossRef]

Satsuma, J.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

Smith, J.

S. Dijali, A. Dienes, and J. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

Taha, T. R.

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[CrossRef]

Tang, T.

T. Tang, “The Hermite spectral method for Gaussian-type functions,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 594–606 (1993).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Tappert, F. D.

R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 423 (1973).

Togneri, R.

A. I. Rasiah, R. Togneri, and Y. Attikiouzel, “Modelling 1-D signals using Hermite basis functions,” IEE Proc. Vision Image Signal Process. 144(6), 345–354 (1997).
[CrossRef]

van Dijk, A. M.

A. M. van Dijk and J.-B. Martens, “Image representation and compression with steered Hermite transforms,” Signal Process. 56, 1–16 (1997).
[CrossRef]

Yajima, N.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

Zabusky, N. J.

N. J. Zabusky and M. D. Kruskal, “Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240–243 (1965).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Bull. Calcutta Math. Soc.

A. J. Jerri, “The application of general discrete transforms to computing orthogonal series and solving boundary value problems,” Bull. Calcutta Math. Soc. 71, 177–187 (1979).

IEE Proc. Vision Image Signal Process.

A. I. Rasiah, R. Togneri, and Y. Attikiouzel, “Modelling 1-D signals using Hermite basis functions,” IEE Proc. Vision Image Signal Process. 144(6), 345–354 (1997).
[CrossRef]

IEEE J. Quantum Electron.

S. Dijali, A. Dienes, and J. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26, 1158–1164 (1990).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process.

J.-B. Martens, “The Hermite transform theory,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1595–1606 (1990).
[CrossRef]

IEEE Trans. Antennas Propag.

A. Bogush and R. Elkins, “Gaussian field expansions for large aperture antennas,” IEEE Trans. Antennas Propag. 34, 228–243 (1986).
[CrossRef]

J. Murphy and R. Padman, “Phase centers of horn antennas using Gaussian beam mode analysis,” IEEE Trans. Antennas Propag. 38, 1306–1310 (1990).
[CrossRef]

M. Rao, T. Sarkar, T. Anjali, and R. S. Adve, “Simultaneous extrapolation in time and frequency domains using Hermite expansions,” IEEE Trans. Antennas Propag. 47, 1108–1115 (1999).
[CrossRef]

IEEE Trans. Biomed. Eng.

L. Lo Conte, R. Merletti, and G. Sandri, “Hermite expansions of compact support waveforms: applications to myoelectric signals,” IEEE Trans. Biomed. Eng. 41, 1147–1159 (1994).
[CrossRef] [PubMed]

Int. J. Numer. Model.

P. Lazaridis, G. Debarge, and P. Gallion, “Split-step-Gauss-Hermite algorithm for fast and accurate simulation of soliton propagation,” Int. J. Numer. Model. 14, 325–329 (2001).
[CrossRef]

J. Comput. Phys.

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

N. J. Zabusky and M. D. Kruskal, “Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240–243 (1965).
[CrossRef]

Prog. Theor. Phys. Suppl.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput.

T. Tang, “The Hermite spectral method for Gaussian-type functions,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 594–606 (1993).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) Rev.

R. H. Hardin and F. D. Tappert, “Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 423 (1973).

Signal Process.

A. M. van Dijk and J.-B. Martens, “Image representation and compression with steered Hermite transforms,” Signal Process. 56, 1–16 (1997).
[CrossRef]

Other

T. Oliveira e Silva and H. J. W. Belt, “On the determination of the optimal center and scale factor for truncated Hermite series,” presented at the European Conference on Signal Processing (EUSIPCO-96) Trieste, Italy, September 10–13, 1996, paper PFT.11.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

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Tables (6)

Tables Icon

Table 1 DOGHT of sech(t) for N=8

Tables Icon

Table 2 Comparison of DOGHT and FFT for the Linear Propagation of a Chirped Hyperbolic Secant Pulse

Tables Icon

Table 3 SSF First-Order Soliton Propagation Simulation a

Tables Icon

Table 4 SSGH First-Order Soliton Propagation Simulation a

Tables Icon

Table 5 SSF Second-Order Soliton Propagation Simulation a

Tables Icon

Table 6 SSGH Second-Order Soliton Propagation Simulation a

Equations (42)

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

cn=i=0N-1wiftiThn(ti),
f˜tiT=n=0N-1Cnhn(ti),
hn(t)=1(2nn!π)1/2exp-t22Hn(t).
-+hm(t)hn(t)dt=δmn
wi=2[hN(ti)]2.
h0=1π1/4 exp-t22,h1=2th0,
hn+1=1n (2thn-n-1hn-1).
hn=2nhn-1,
i=0N-1wihm(ti)hn(ti)=δmn,
n=0N-1wjhn(ti)hn(tj)=δij.
f˜tiT=ftiT,
E˜=1T n=0N-1|cn|2=1T i=0N-1wiftiT2,
E=-+|f(t)|2dt.
F˜(ω)=-+f˜(t)exp(-jωt)dt=-+ n=0N-1cnhn(Tt)exp(-jωt)dt=2π|T| n=0N-1(-j)ncnhnωT
T=tN-1tmax,
f˜(ti)=n=0N-1cnhn(tiT).
j φ(z, t)z=122φ(z, t)t2,
Φ(z, ω)=Φ(0, ω)expj ωz22.
Φn(0, ω)=2π(-j)nhn(ω).
φn(z, t)=hnt1+z2(1+z2)1/4×expj(n+1/2)arctan z-j2zt21+z2.
u(z, t)=n=0N-1cnφn(z, t),
cn=i=0N-1wiu(0, ti)hn(ti).
cn=i=0N-1wiu0, tiThn(ti).
u(z, t)=n=0N-1cnφn(zT2, tT).
u(0, t)=sec hπ2 texp-j αt22.
uz=-i2 2ut2+j|u|2u.
uz=j|u|2u,
uNL(z0+Δz, t)=exp[jΔz|u(z0, t)|2]u(z0, t).
uz=-j22ut2
u(z0+Δz, t)=F-1expj Δz2 ωF[uNL(z0+Δz, t)].
cn=i=0N-1wiuNLz0+Δz, tiThn(ti),
u(z0+Δz, t)=n=0N-1Cnφn[(Δz)T2, tiT].
wihn(ti),
φn[(Δz)T2, tiT]
u(z=0, t)=A sec h(t),
uex(z, t)=exp-j z2sec ht
uex(z, t)=4 exp-j z2[cosh 3t+3 exp(-4jz)cosh t]cosh 4t+4 cosh 2t+3 cos 4z,
max|u(z=10, ti)-uex(z=10, ti)|.
max|u(z=1, ti)-uex(z=1, ti)|.
f˜tiT=n=0N-1Cnhn(ti)=n=0N-1j=0N-1wjftjThn(tj)hn(ti).
f˜tiT=j=0N-1ftjTn=0N-1wjhn(ti)hn(tj).
f˜tiT=j=0N-1ftjTδij=ftiT.

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