Abstract

The linear canonical transform (LCT) with a, b, c, d parameter plays an important role in quantum mechanics, optics, and signal processing. The eigenfunctions of the LCT are also important because they describe the self-imaging phenomenon in optical systems. However, the existing solutions for the eigenfunctions of the LCT are divided into many cases and they lack a systematic way to solve these eigenfunctions. In this paper, we find a linear, second-order, self-adjoint differential commuting operator that commutes with the LCT operator. Hence, the commuting operator and the LCT share the same eigenfunctions with different eigenvalues. The commuting operator is very general and simple when it is compared to the existing multiple-parameter differential equations. Then, the eigenfunctions can be derived systematically. The eigenvalues of the commuting operator have closed-form relationships with the eigenvalues of the LCT. We also simplify the eigenfunctions for |a+d|>2 and a+d=±2, b0 into the more compact closed form instead of the integral form. For |a+d|>2, the eigenfunctions are related to the parabolic cylinder functions.

© 2013 Optical Society of America

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2012 (1)

2011 (2)

S.-C. Pei and Y.-C. Lai, “Discrete linear canonical transforms based on dilated Hermite functions,” J. Opt. Soc. Am. A 28, 1695–1708 (2011).
[CrossRef]

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

2010 (1)

S.-C. Pei and J.-J. Ding, “Fractional Fourier transform, Wigner distribution, and filter design for stationary and nonstationary random processes,” IEEE Trans. Signal Process. 58, 4079–4092 (2010).
[CrossRef]

2009 (1)

C. A. Muñoz, J. Rueda-Paz, and K. B. Wolf, “Discrete repulsive oscillator wavefunctions,” J. Phys. A 42, 485210 (2009).

2008 (1)

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

2007 (1)

C. Candan, “On higher order approximations for Hermite-Gaussian functions and discrete fractional Fourier transforms,” IEEE Signal Process. Lett. 14, 699–702 (2007).
[CrossRef]

2006 (1)

C. Yuce, A. Kilic, and A. Coruh, “Inverted oscillator,” Phys. Scr. 74, 114–116 (2006).
[CrossRef]

2005 (1)

2002 (1)

S.-C. Pei and J.-J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11–26 (2002).
[CrossRef]

2000 (3)

M. Izzetoğlu, B. Onaral, P. Chitrapu, and N. Bilgutay, “Discrete time processing of linear scale invariant signals and systems,” Proc. SPIE 4116, 110–118 (2000).
[CrossRef]

L. Barker, C. Candan, T. Hakioglu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

C. Candan, M. A. Kutay, and H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

1996 (3)

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. Series B 38, 209–219 (1996).
[CrossRef]

D. F. James and G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996).
[CrossRef]

1995 (2)

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

R. G. Baraniuk and D. Jones, “Unitary equivalence: a new twist on signal processing,” IEEE Trans. Signal Process. 43, 2269–2282 (1995).
[CrossRef]

1994 (1)

1993 (2)

R. G. Baraniuk, “Signal transform covariant to scale changes,” Electron. Lett. 29, 1675–1676 (1993).
[CrossRef]

L. Cohen, “The scale representation,” IEEE Trans. Signal Process. 41, 3275–3292 (1993).
[CrossRef]

1988 (1)

S. Tarzi, “The inverted harmonic oscillator: some statistical properties,” J. Phys. A 21, 3105–3111 (1988).
[CrossRef]

1987 (1)

1986 (1)

G. Barton, “Quantum mechanics of the inverted oscillator potential,” Ann. Phys. 166, 322–363 (1986).
[CrossRef]

1980 (2)

P. G. L. Leach, “Sl(3, R) and the repulsive oscillator,” J. Phys. A 13, 1991–2000 (1980).
[CrossRef]

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” IMA J. Appl. Math. 25, 241–265 (1980).

1974 (1)

E. G. Kalnins and W. Miller, “Lie theory and separation of variables. 5. The equations iUt + Uxx = 0 and iUt + Uxxc/x2U = 0,” J. Math. Phys. 15, 1728–1737 (1974).

1966 (1)

Abe, S.

Agarwal, G. S.

D. F. James and G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

Aldrmaz, S.

A. Serbes, S. Aldrmaz, and L. Durak-Ata, “Eigenfunctions of the linear canonical transform,” in Proceedings of 20th Signal Processing and Communications Applications Conference (2012), pp. 1–4.

Alieva, T.

T. Alieva and M. J. Bastiaans, “Properties of eigenfunctions of the canonical integral transform,” in Proceedings of IEEE EURASIP Workshop on Nonlinear Signal and Image Processing (1999), Vol. 2, pp. 585–587.

Baraniuk, R. G.

R. G. Baraniuk and D. Jones, “Unitary equivalence: a new twist on signal processing,” IEEE Trans. Signal Process. 43, 2269–2282 (1995).
[CrossRef]

R. G. Baraniuk, “Signal transform covariant to scale changes,” Electron. Lett. 29, 1675–1676 (1993).
[CrossRef]

Barker, L.

L. Barker, C. Candan, T. Hakioglu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

Barton, G.

G. Barton, “Quantum mechanics of the inverted oscillator potential,” Ann. Phys. 166, 322–363 (1986).
[CrossRef]

Bastiaans, M. J.

T. Alieva and M. J. Bastiaans, “Properties of eigenfunctions of the canonical integral transform,” in Proceedings of IEEE EURASIP Workshop on Nonlinear Signal and Image Processing (1999), Vol. 2, pp. 585–587.

Bernardo, L. M.

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996).
[CrossRef]

Bilgutay, N.

M. Izzetoğlu, B. Onaral, P. Chitrapu, and N. Bilgutay, “Discrete time processing of linear scale invariant signals and systems,” Proc. SPIE 4116, 110–118 (2000).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 2002).

Candan, C.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

C. Candan, “On higher order approximations for Hermite-Gaussian functions and discrete fractional Fourier transforms,” IEEE Signal Process. Lett. 14, 699–702 (2007).
[CrossRef]

L. Barker, C. Candan, T. Hakioglu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

C. Candan, M. A. Kutay, and H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

Chitrapu, P.

M. Izzetoğlu, B. Onaral, P. Chitrapu, and N. Bilgutay, “Discrete time processing of linear scale invariant signals and systems,” Proc. SPIE 4116, 110–118 (2000).
[CrossRef]

Cohen, L.

L. Cohen, “The scale representation,” IEEE Trans. Signal Process. 41, 3275–3292 (1993).
[CrossRef]

Coruh, A.

C. Yuce, A. Kilic, and A. Coruh, “Inverted oscillator,” Phys. Scr. 74, 114–116 (2006).
[CrossRef]

Ding, J. J.

J. J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. thesis (National Taiwan University, 2001).

Ding, J.-J.

S.-C. Pei and J.-J. Ding, “Fractional Fourier transform, Wigner distribution, and filter design for stationary and nonstationary random processes,” IEEE Trans. Signal Process. 58, 4079–4092 (2010).
[CrossRef]

S.-C. Pei and J.-J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11–26 (2002).
[CrossRef]

Durak-Ata, L.

A. Serbes, S. Aldrmaz, and L. Durak-Ata, “Eigenfunctions of the linear canonical transform,” in Proceedings of 20th Signal Processing and Communications Applications Conference (2012), pp. 1–4.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).

Griffiths, D. J.

D. J. Griffiths, Introduction to Quantum Mechanics (Pearson Prentice-Hall, 2005).

Hakioglu, T.

L. Barker, C. Candan, T. Hakioglu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

Hanson, S. G.

Healy, J. J.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

Hennelly, B. M.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Fast numerical algorithm for the linear canonical transform,” J. Opt. Soc. Am. A 22, 928–937 (2005).
[CrossRef]

Izzetoglu, M.

M. Izzetoğlu, B. Onaral, P. Chitrapu, and N. Bilgutay, “Discrete time processing of linear scale invariant signals and systems,” Proc. SPIE 4116, 110–118 (2000).
[CrossRef]

James, D. F.

D. F. James and G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

Jones, D.

R. G. Baraniuk and D. Jones, “Unitary equivalence: a new twist on signal processing,” IEEE Trans. Signal Process. 43, 2269–2282 (1995).
[CrossRef]

Kalnins, E. G.

E. G. Kalnins and W. Miller, “Lie theory and separation of variables. 5. The equations iUt + Uxx = 0 and iUt + Uxxc/x2U = 0,” J. Math. Phys. 15, 1728–1737 (1974).

Kelly, D. P.

D. Li, D. P. Kelly, R. Kirner, and J. T. Sheridan, “Speckle orientation in paraxial optical systems,” Appl. Opt. 51, A1–A10 (2012).
[CrossRef]

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

Kilic, A.

C. Yuce, A. Kilic, and A. Coruh, “Inverted oscillator,” Phys. Scr. 74, 114–116 (2006).
[CrossRef]

Kirner, R.

Koç, A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Kogelnik, H.

Kutay, M. A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

C. Candan, M. A. Kutay, and H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

L. Barker, C. Candan, T. Hakioglu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Lai, Y.-C.

Leach, P. G. L.

P. G. L. Leach, “Sl(3, R) and the repulsive oscillator,” J. Phys. A 13, 1991–2000 (1980).
[CrossRef]

Li, D.

Li, T.

Mendlovic, D.

Miller, W.

E. G. Kalnins and W. Miller, “Lie theory and separation of variables. 5. The equations iUt + Uxx = 0 and iUt + Uxxc/x2U = 0,” J. Math. Phys. 15, 1728–1737 (1974).

Muñoz, C. A.

C. A. Muñoz, J. Rueda-Paz, and K. B. Wolf, “Discrete repulsive oscillator wavefunctions,” J. Phys. A 42, 485210 (2009).

Mustard, D.

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. Series B 38, 209–219 (1996).
[CrossRef]

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” IMA J. Appl. Math. 25, 241–265 (1980).

Oberhettinger, F.

F. Oberhettinger, Tables of Mellin Transforms (Springer-Verlag, 1974).

Onaral, B.

M. Izzetoğlu, B. Onaral, P. Chitrapu, and N. Bilgutay, “Discrete time processing of linear scale invariant signals and systems,” Proc. SPIE 4116, 110–118 (2000).
[CrossRef]

Ozaktas, H. M.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

C. Candan, M. A. Kutay, and H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

L. Barker, C. Candan, T. Hakioglu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Pei, S.-C.

S.-C. Pei and Y.-C. Lai, “Discrete linear canonical transforms based on dilated Hermite functions,” J. Opt. Soc. Am. A 28, 1695–1708 (2011).
[CrossRef]

S.-C. Pei and J.-J. Ding, “Fractional Fourier transform, Wigner distribution, and filter design for stationary and nonstationary random processes,” IEEE Trans. Signal Process. 58, 4079–4092 (2010).
[CrossRef]

S.-C. Pei and J.-J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11–26 (2002).
[CrossRef]

Rueda-Paz, J.

C. A. Muñoz, J. Rueda-Paz, and K. B. Wolf, “Discrete repulsive oscillator wavefunctions,” J. Phys. A 42, 485210 (2009).

Serbes, A.

A. Serbes, S. Aldrmaz, and L. Durak-Ata, “Eigenfunctions of the linear canonical transform,” in Proceedings of 20th Signal Processing and Communications Applications Conference (2012), pp. 1–4.

Sheridan, J. T.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Tarzi, S.

S. Tarzi, “The inverted harmonic oscillator: some statistical properties,” J. Phys. A 21, 3105–3111 (1988).
[CrossRef]

Wolf, K. B.

C. A. Muñoz, J. Rueda-Paz, and K. B. Wolf, “Discrete repulsive oscillator wavefunctions,” J. Phys. A 42, 485210 (2009).

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).

Wornell, G. W.

G. W. Wornell, Signal Processing with Fractals: A Wavelet-Based Approach (Prentice-Hall, 1996).

Yuce, C.

C. Yuce, A. Kilic, and A. Coruh, “Inverted oscillator,” Phys. Scr. 74, 114–116 (2006).
[CrossRef]

Yura, H. T.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Ann. Phys. (1)

G. Barton, “Quantum mechanics of the inverted oscillator potential,” Ann. Phys. 166, 322–363 (1986).
[CrossRef]

Appl. Opt. (2)

Electron. Lett. (1)

R. G. Baraniuk, “Signal transform covariant to scale changes,” Electron. Lett. 29, 1675–1676 (1993).
[CrossRef]

IEEE Signal Process. Lett. (1)

C. Candan, “On higher order approximations for Hermite-Gaussian functions and discrete fractional Fourier transforms,” IEEE Signal Process. Lett. 14, 699–702 (2007).
[CrossRef]

IEEE Trans. Signal Process. (6)

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

S.-C. Pei and J.-J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11–26 (2002).
[CrossRef]

S.-C. Pei and J.-J. Ding, “Fractional Fourier transform, Wigner distribution, and filter design for stationary and nonstationary random processes,” IEEE Trans. Signal Process. 58, 4079–4092 (2010).
[CrossRef]

L. Cohen, “The scale representation,” IEEE Trans. Signal Process. 41, 3275–3292 (1993).
[CrossRef]

R. G. Baraniuk and D. Jones, “Unitary equivalence: a new twist on signal processing,” IEEE Trans. Signal Process. 43, 2269–2282 (1995).
[CrossRef]

C. Candan, M. A. Kutay, and H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

IMA J. Appl. Math. (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” IMA J. Appl. Math. 25, 241–265 (1980).

J. Aust. Math. Soc. Series B (1)

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. Series B 38, 209–219 (1996).
[CrossRef]

J. Eur. Opt. Soc. Rapid Pub. (1)

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

J. Math. Phys. (1)

E. G. Kalnins and W. Miller, “Lie theory and separation of variables. 5. The equations iUt + Uxx = 0 and iUt + Uxxc/x2U = 0,” J. Math. Phys. 15, 1728–1737 (1974).

J. Opt. Soc. Am. A (4)

J. Phys. A (4)

P. G. L. Leach, “Sl(3, R) and the repulsive oscillator,” J. Phys. A 13, 1991–2000 (1980).
[CrossRef]

S. Tarzi, “The inverted harmonic oscillator: some statistical properties,” J. Phys. A 21, 3105–3111 (1988).
[CrossRef]

C. A. Muñoz, J. Rueda-Paz, and K. B. Wolf, “Discrete repulsive oscillator wavefunctions,” J. Phys. A 42, 485210 (2009).

L. Barker, C. Candan, T. Hakioglu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209–2222 (2000).
[CrossRef]

Opt. Commun. (1)

D. F. James and G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

Opt. Eng. (1)

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. 35, 732–740 (1996).
[CrossRef]

Opt. Lett. (1)

Phys. Scr. (1)

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Figures (2)

Fig. 1.
Fig. 1.

Different cases for the eigenfunctions of the LCT, modified from [14].

Fig. 2.
Fig. 2.

Illustration of the eigenexpansion of the LCT. (b) The input signal x(t) and (a) its Wigner distribution function (WDF) WDFx(t,f). For M=[1,0.4;4.75,0.9], the first eigenfunction is listed in (c) while (d) shows the second eigenfunction. The resultant LCT XM(t) is plotted in (f). (e) stands for the WDF of XM(t). (g) and (h) indicate the eigenfunctions multiplied by the eigenvalues λM=|λM|ejλM=ejλM, respectively. In (g) and (h), it is obvious that the magnitudes remain unchanged, the additional phase terms are appended, and the real as well as imaginary parts differ from those in (c) and (d).

Tables (3)

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Table 1. Summary on the Eigenfunctions and the Eigenvalues [14]

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Table 2. Commuting Operators for Some Special Cases of the LCT

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Table 3. Summary on LCT Eigenfunctions and Eigenvalues

Equations (142)

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XM(u)=(LMx)(u)=1j2πbx(t)eja2bt2ej1btuejd2bu2dt,b0,t,uR.
XM(t)=dejcd2t2x(td),b=0.
LM1M2=LM1LM2,LM1=LM1.
LM{tx(t)}=(jbddt+td)XM(t),
LM{ddtx(t)}=(addtjct)XM(t),
LMt=(jbDt+td)LM,
LMDt=(aDtjct)LM.
Ψn(σ,τ)(u)=1σ2nn!jπHn(uσejπ4)ej1τ2σ2u2.
CM=bDt2+jad2(tDt+Dtt)+ct2,
LMCM=bLMDt2+jad2LM(tDt+Dtt)+cLMt2.
LMDt2=(aDtjct)2LM,
LM(tDt)=(jbDt+td)(aDtjct)LM,
LM(Dtt)=(aDtjct)(jbDt+td)LM,
LMt2=(jbDt+td)2LM.
[b(aDtjct)2+jad2(jbDt+td)(aDtjct)+jad2(aDtjct)(jbDt+td)+c(jbDt+td)2]LM=[(ba2+jad2(j2ab)cb2)Dt2+(jabc+jad2(ad+bc)+jbcd)(tDt+Dtt)+(bc2+jad2(j2cd)+cd2)t2]LM.
LMCM=[b(adbc)Dt2+j(ad)(adbc)2(tDt+Dtt)+c(adbc)t2]LM=CMLM.
(A+B)=A+B,(AB)=BA.
Dt=Dt,(Dt2)=Dt2,t=t,a=a*,
CM=(bDt2)+[jad2(tDt+Dtt)]+(ct2)
=(Dt2)b+(tDt+Dtt)(jad2)+(t2)c.
CM=bDt2+(tDtDtt)(jad2)+ct2,
CMψμM(t)=μMψμM(t),LMψμM(t)=λMψμM(t),
LMψμM(t)2=|λM|ψμM(t)2.
(12σ2Dt2jτtDt+1+τ22σ2t2jτ2)ψn(t)=(n+1/2)ψn(t).
(bDt2+j(ad)tDt+ct2jad2)ψn(t)=(2b/σ2)(n+1/2)ψn(t).
CMψn(t)=(bDt2+jad2(tDt+Dtt)+ct2)ψn(t)=sgn(b)4(a+d)2(n+1/2)ψn(t).
μM=sgn(b)4(a+d)2(n+1/2).
λM=ej(n+1/2)α=(cosα+jsinα)(n+1/2).
cosα=(a+d)/2,
sinα=12sgn(b)4(a+d)2,
λM=[a+d2+jsgn(b)4(a+d)22]μMsgn(b)4(a+d)2.
|λM|=|cosα+jsinα|(n+1/2)=1.
ct2ψμM(t)=μMψμM(t).
t2ψμM(t)=μMcψμM(t).
ψμM(1,B)(t)=δ(t+μM/c),
ψμM(2,B)(t)=δ(tμM/c),
LMψμM(1,B)(t)=ej12μMψμM(1,B)(t),
LMψμM(2,B)(t)=ej12μMψμM(2,B)(t).
ψμM(1,B)(t),ψμM(1,B)(t)=δ(μMμM),
ψμM(1,B)(t),ψμM(2,B)(t)={δ(μMμM)μM=μM=0,0otherwise,
ψμM(2,B)(t),ψμM(2,B)(t)=δ(μMμM).
[The eigen value ofψμM+4nπ(1,B)(t)]=ej12(μM+4nπ)=ej12μM,
[The eigen value ofψμM+4mπ(2,B)(t)]=ej12(μM+4mπ)=ej12μM,
[The eigen value ofnAnψμM+4nπ(1,B)(t)+BnψμM+4nπ(2,B)(t)]=ej12μM.
LMψμM(1,B)(t)=jej12μMψμM(2,B)(t),
LMψμM(2,B)(t)=jej12μMψμM(1,B)(t).
LMψμM(1,C)(t)=jej12μMψμM(1,C)(t),
LMψμM(2,C)(t)=jej12μMψμM(2,C)(t),
ψμM(1,C)(t)=12(ψμM(1,B)(t)+ψμM(2,B)(t)),
ψμM(2,C)(t)=12(ψμM(1,B)(t)ψμM(2,B)(t)).
[abcd]=[a1b1c1d1][1η01][a1b1c1d1]1=M1M0M11,
η=ba12,c1=da2ba1,b1=2b(d1a11)da,a10,
Dt2ψμM0(t)=μM0ηψμM0(t).
ψμM0(1,D)(t)=12πejμM0ηt,
ψμM0(2,D)(t)=12πejμM0ηt.
LM1Dt2ψμM0(t)=μM0ηLM1ψμM0(t).
η(a1Dtjc1t)2LM1ψμM0(t)=μM0LM1ψμM0(t).
[a12ηDt2ja1c1η(tDt+Dtt)c12ηt2]ψμM(t)=μM0ψμM(t).
a12η=b,ja1c1η=jad2,c12η=c.
CMψμM(t)=μMψμM(t),
eαξ2+βξdξ=παeβ24α,
ψμM(1,D)(t)=12πejda4bt2ejtμM/b,
ψμM(2,D)(t)=12πejda4bt2ejtμM/b.
ψμM(1,D)(t),ψμM(1,D)(t)=ψμM0(1,D)(t),ψμM0(1,D)(t)=δ(μM0μM0),
[abcd]=[a1b1c1d1][1η01][a1b1c1d1]1.
ψμM0(1,E)(t)=12(ψμM0(1,D)(t)+ψμM0(2,D)(t)),
=1πcos(μM0ηt),
ψμM0(2,E)(t)=12(ψμM0(1,D)(t)ψμM0(2,D)(t)),
=jπsin(μM0ηt).
ψμM(1,E)(t)=1πejda4bt2cos(μMbt),
ψμM(2,E)(t)=jπejda4bt2sin(μMbt),
[abcd]=[a1b1c1d1][σ100σ][a1b1c1d1]1,
σ=a+d±(a+d)242>0,
s=sgn(σσ1),
b1=sba1(a+d)24,
c1=2a1scs(da)+(a+d)24,
d1=12a1(s(da)(a+d)24+1),
ψμM(t)=LM1h(t),
σh(σt)=λh(t).
hω(t)={12πt12+jωt(0,),0t(,0],
σhω(σt)=σ(σt)12+jω2π=σσ12+jωt12+jω2π=σjωt12+jω2π=ejωlogσhω(t).
12j(tDt+Dtt)hω(t)=ωhω(t).
ψμM0(1,F1)(t)=hω(t)={12πt12+jωt(0,),0t(,0],
ψμM0(2,F1)(t)=hω(t)={12π(t)12+jωt(,0),0t[0,).
12jLM1[(tDt+Dtt)ψμM0(t)]=ωLM1ψμM0(t).
12jLM1[(2tDt+I)ψμM0(t)]=ωψμM(t).
12j[2(jb1Dt+td1)(a1Dtjc1t)+1]ψμM(t)=ωψμM(t).
[a1b1Dt2+2a1d112j(tDt+Dtt)c1d1t2]ψμM(t)=ωψμM(t).
CMψμM(t)=sgn(σσ1)ω(a+d)24ψμM(t).
μM=sgn(σσ1)ω(a+d)24.
λM=σψμM0(σt)ψμM0(t)=σ(σt)12+jωt12+jω=σjω=ejωlogσ=ejμMsgn(σσ1)(a+d)24logσ.
ψμM(1,F1)(t)=LM1ψμM0(1,F1)(t)=1j2πb10(eja12b1ξ2ejtξb1ejd12b1t2)ξ12+jω2πdξ
0eατ2βττζ1dτ=(2α)ζ2Γ(ζ)eβ28αDζ(β2α),
Γ(z)=0ettz1dt.
(Dz2+ν+1214z2)Dν(z)=0.
U(a,z)=Da12(z).
α=ja12b1,β=jtb1,ζ=12+jω,
ψμM(1,F1)(t)=NωD12jω(jejπ4ta1b1)ej2a1d114a1b1t2,
Nω=1j4π2b1(ja1b1)14jω2Γ(12+jω).
ψμM(1,F1)(t)=NωD12jω(ej3π4tη)ejda4bt2,
|Nω|=eπω42πη14|Γ(12+jω)|,
ψμM(2,F1)(t)=|Nω|D12jω(ej3π4tη)ejda4bt2.
ψμM(1,F1)(t),ψμM(1,F1)(t)=ψμM0(1,F1)(t),ψμM0(1,F1)(t)=δ(μM0μM0).
|D12jω(z)|=|U(jω,z)||e14z2z12|
ψμM(1,F2)(t)=d1ejc1d12t2ψμM0(1,F1)(td1),
=d1jω2πejc1d12t2t12+jω
[a1b1c1d1]=[a10a1cad1/a1].
ψμM(1,F2)(t)=12πejc2(ad)t2t12+jμMda,
μM=(da)ω
ψμM(2,F2)(t)=12πejc2(ad)t2(t)12+jμMda
[abcd]=[a1b1c1d1][σ100σ][a1b1c1d1]1,
σ=ad±(a+d)242,s=sgn(σ1σ)
ψμM0(1,G)(t)=12(ψμM0(1,F)(t)+ψμM0(2,F)(t)),
ψμM0(2,G)(t)=12(ψμM0(1,F)(t)ψμM0(2,F)(t)),
μM=sgn(σ1σ)ω(a+d)24.
λM(1,G)=σψμM0(1,G)(σt)ψμM0(1,G)(t)=jσψμM0(1,G)(σt)ψμM0(1,G)(t)=jejωlogσ=jejμMsgn(σ1σ)(a+d)24logσ.
λM(2,G)=jejωlogσ=jejμMsgn(σ1σ)(a+d)24logσ.
ψμM(1,G1)(t)=21/2|Nω|ejda4bt2[D12jω(ej3π4tη)+D12jω(ej3π4tη)],
ψμM(2,G1)(t)=21/2|Nω|ejda4bt2[D12jω(ej3π4tη)D12jω(ej3π4tη)].
ψμM(1,G2)(t)={12πejc2(ad)t2|t|12+jμMda,t0,0t=0,
ψμM(2,G2)(t)=12πejc2(ad)t2|t|12+jμMdasgn(t),
Dn(ξ)=2n2Hn(ξ/2)e14ξ2,n=0,1,2,.
Mα=[cosαsinαsinαcosα],Mβ=[coshβsinhβsinhβcoshβ],
CMα=cosα(Dt2t2),CMβ=coshβ(Dt2+t2).
Dn(tejπ/4σ)ejτ2σ2t2.
|Dn(tejπ/4σ)ejτ2σ2t2|=|Dn(tejπ/4σ)|=|2n/2Hn(t2ejπ/4σ)ejt24σ2|=|2n/2Hn(t2ejπ/4σ)|ast.
λn={eα(n1/2)a+d>2,e(αjπ)(n1/2)a+d<2,
TTet=etT=(eT)et,
TTejt=ej(tT)=(ejT)ejt.
x(t)=12rect(t2)={1/2|t|<1,0otherwise.
x(t)=μMaμMψμM(t).
aμM=ψμM(t),x(t)=ψμM*(t)x(t)dt.
XM(t)=μMλMaμMψμM(t),
δ(tα)=12πe±jω(tα)dω.
δ(tα),δ(tβ)=14π2ejω1(tα)ejω2(tβ)dω1dω2dt.
δ(tα),δ(tβ)=12πejω1αejω2βδ(ω1+ω2)dω1dω2.
δ(tα),δ(tβ)=12πejω2(αβ)dω2=δ(αβ).
12πejω1t,12πejω2t=12πej(ω1ω2)tdt.
hω1(t),hω2(t)=12π0t1j(ω1ω2)dt=12π0ej(ω1ω2)logtt1dt.
hω1(t),hω2(t)=12πej(ω1ω2)τdτ.
OF(1,0,c,1)(ϕCc,h(t))=jejcu2/2[δ(u4nπ|c|+h)+δ(u+4nπ|c|+h)].
OF(1,0,c,1)(ϕCc,h(t))=jejcu2/2[δ(u+4nπ|c|+h)+δ(u4nπ|c|+h)].

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