Abstract

Linear canonical transform (LCT) is very useful and powerful in signal processing and optics. In this paper, discrete LCT (DLCT) is proposed to approximate LCT by utilizing the discrete dilated Hermite functions. The Wigner distribution function is also used to investigate DLCT performances in the time–frequency domain. Compared with the existing digital computation of LCT, our proposed DLCT possess additivity and reversibility properties with no oversampling involved. In addition, the length of input/output signals will not be changed before and after the DLCT transformations, which is consistent with the time–frequency area-preserving nature of LCT; meanwhile, the proposed DLCT has very good approximation of continuous LCT.

© 2011 Optical Society of America

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2010 (3)

2009 (3)

P. D’Alberto and A. Nicolau, “Adaptive Winograd’s matrix multiplications,” ACM Trans. Math. Softw. 36, 1–23 (2009).
[CrossRef]

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641–648 (2009).
[CrossRef]

K. K. Sharma, “Fractional Laplace transform,” Signal Image Video Process. 4, 377–379 (2009).
[CrossRef]

2008 (3)

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]

U. Gopinathan, G. Situ, T. J. Naughton, and J. T. Sheridan, “Noninterferometric phase retrieval using a fractional Fourier system,” J. Opt. Soc. Am. 25, 108–115 (2008).
[CrossRef]

S. C. Pei, J. J. Ding, W. L. Hsue, and K. W. Chang, “Generalized commuting matrices and their eigenvectors for DFTs, offset DFTs, and other periodic operations,” IEEE Trans. Signal Process. 56, 3891–3904 (2008).
[CrossRef]

2007 (2)

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

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983–990 (2007).
[CrossRef]

2006 (3)

S. C. Pei, W. L. Hsue, and J. J. Ding, “Discrete fractional Fourier transform based on new nearly tridiagonal commuting matrices,” IEEE Trans. Signal Process. 54, 3815–3828 (2006).
[CrossRef]

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421–1425 (2006).
[CrossRef]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).
[CrossRef] [PubMed]

2005 (1)

2003 (1)

M. J. Bastiaans and K. B. Wolf, “Phase reconstruction from intensity measurements in linear systems,” J. Opt. Soc. Am. 20, 1046–1049 (2003).
[CrossRef]

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)

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

S. C. Pei and J. J. Ding, “Simplified fractional Fourier transforms,” J. Opt. Soc. Am. 17, 2355–2367 (2000).
[CrossRef]

S. C. Pei and J. J. Ding, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338–1353 (2000).
[CrossRef]

1998 (1)

1997 (3)

1996 (3)

D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its applications 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]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

1994 (3)

S. Abe and J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef] [PubMed]

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

A. Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. 11, 3–13 (1994).
[CrossRef]

1989 (1)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

1982 (3)

B. W. Dickinson and K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 25–31 (1982).
[CrossRef]

F. A. Grünbaum, “The eigenvectors of the discrete Fourier transform: a version of the Hermite functions,” J. Math. Anal. Appl. 88, 355–363 (1982).
[CrossRef]

M. Nazarathy and J. Shamir, “First-order optics—a canonical operator representation: Lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
[CrossRef]

1981 (1)

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

1980 (2)

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

T. A. C. M. Claasen and W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis, part I: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

1979 (1)

1970 (2)

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
[CrossRef]

L. I. Bluestein, “A linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. AU-18, 451–455 (1970).
[CrossRef]

1969 (1)

V. Strassen, “Gaussian elimination is not optimal,” Numer. Math. 13, 354–356 (1969).
[CrossRef]

1932 (1)

E. P. Wigner, “On the quantum correlation for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Abe, S.

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1970).

Agarwal, G. S.

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

Allen, R. L.

R. L. Allen and D. W. Mills, Signal Analysis: Time, Frequency, Scale, and Structure (Wiley-Interscience, 2004).

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Arikan, O.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Bagini, V.

Barshan, B.

B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans and K. B. Wolf, “Phase reconstruction from intensity measurements in linear systems,” J. Opt. Soc. Am. 20, 1046–1049 (2003).
[CrossRef]

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

Bernardo, L. M.

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

Bluestein, L. I.

L. I. Bluestein, “A linear filtering approach to the computation of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust. AU-18, 451–455 (1970).
[CrossRef]

Bozdagi, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

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]

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

Chang, K. W.

S. C. Pei, J. J. Ding, W. L. Hsue, and K. W. Chang, “Generalized commuting matrices and their eigenvectors for DFTs, offset DFTs, and other periodic operations,” IEEE Trans. Signal Process. 56, 3891–3904 (2008).
[CrossRef]

Chen, D.

S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications (Prentice-Hall, 1996).

Claasen, T. A. C. M.

T. A. C. M. Claasen and W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis, part I: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

Clary, S.

D. H. Mugler, S. Clary, and Y. Wu, “Discrete Hermite expansion of digital signals: applications to ECG signals,” in Proceedings of 2002 IEEE 10th Digital Signal Processing Workshop and 2nd Signal Processing Education Workshop (IEEE, 2002), pp. 262–267.
[CrossRef]

Collins, S. A.

D’Alberto, P.

P. D’Alberto and A. Nicolau, “Adaptive Winograd’s matrix multiplications,” ACM Trans. Math. Softw. 36, 1–23 (2009).
[CrossRef]

Dickinson, B. W.

B. W. Dickinson and K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 30, 25–31 (1982).
[CrossRef]

Ding, J. J.

S. C. Pei, J. J. Ding, W. L. Hsue, and K. W. Chang, “Generalized commuting matrices and their eigenvectors for DFTs, offset DFTs, and other periodic operations,” IEEE Trans. Signal Process. 56, 3891–3904 (2008).
[CrossRef]

S. C. Pei, W. L. Hsue, and J. J. Ding, “Discrete fractional Fourier transform based on new nearly tridiagonal commuting matrices,” IEEE Trans. Signal Process. 54, 3815–3828 (2006).
[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, “Closed-form discrete fractional and affine Fourier transforms,” IEEE Trans. Signal Process. 48, 1338–1353 (2000).
[CrossRef]

S. C. Pei and J. J. Ding, “Simplified fractional Fourier transforms,” J. Opt. Soc. Am. 17, 2355–2367 (2000).
[CrossRef]

Gopinathan, U.

U. Gopinathan, G. Situ, T. J. Naughton, and J. T. Sheridan, “Noninterferometric phase retrieval using a fractional Fourier system,” J. Opt. Soc. Am. 25, 108–115 (2008).
[CrossRef]

Gori, F.

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

Grünbaum, F. A.

F. A. Grünbaum, “The eigenvectors of the discrete Fourier transform: a version of the Hermite functions,” J. Math. Anal. Appl. 88, 355–363 (1982).
[CrossRef]

Healy, J. J.

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641–648 (2009).
[CrossRef]

Hennelly, B. M.

Hesselink, L.

Hsue, W. L.

S. C. Pei, J. J. Ding, W. L. Hsue, and K. W. Chang, “Generalized commuting matrices and their eigenvectors for DFTs, offset DFTs, and other periodic operations,” IEEE Trans. Signal Process. 56, 3891–3904 (2008).
[CrossRef]

S. C. Pei, W. L. Hsue, and J. J. Ding, “Discrete fractional Fourier transform based on new nearly tridiagonal commuting matrices,” IEEE Trans. Signal Process. 54, 3815–3828 (2006).
[CrossRef]

Hua, J.

James, D. F. V.

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

Koç, A.

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]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).
[CrossRef] [PubMed]

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

B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

Lai, Y. C.

S. C. Pei and Y. C. Lai, “Signal scaling by centered discrete dilated Hermite functions,” IEEE Trans. Signal Process. (submitted for publication).

Li, B.-Z.

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983–990 (2007).
[CrossRef]

Li, G.

Liu, L.

Marsden, J. E.

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed. (Springer-Verlag, 1999).

Mecklenbrauker, W. F. G.

T. A. C. M. Claasen and W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis, part I: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

Mills, D. W.

R. L. Allen and D. W. Mills, Signal Analysis: Time, Frequency, Scale, and Structure (Wiley-Interscience, 2004).

Mugler, D. H.

D. H. Mugler, S. Clary, and Y. Wu, “Discrete Hermite expansion of digital signals: applications to ECG signals,” in Proceedings of 2002 IEEE 10th Digital Signal Processing Workshop and 2nd Signal Processing Education Workshop (IEEE, 2002), pp. 262–267.
[CrossRef]

Mukunda, N.

Namias, V.

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

Naughton, T. J.

U. Gopinathan, G. Situ, T. J. Naughton, and J. T. Sheridan, “Noninterferometric phase retrieval using a fractional Fourier system,” J. Opt. Soc. Am. 25, 108–115 (2008).
[CrossRef]

Nazarathy, M.

Nicolau, A.

P. D’Alberto and A. Nicolau, “Adaptive Winograd’s matrix multiplications,” ACM Trans. Math. Softw. 36, 1–23 (2009).
[CrossRef]

Ozaktas, H. M.

A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals,” J. Opt. Soc. Am. A 27, 1288–1302 (2010).
[CrossRef]

A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate algorithm for the computation of complex linear canonical transforms,” J. Opt. Soc. Am. A 27, 1896–1908 (2010).
[CrossRef]

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]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).
[CrossRef] [PubMed]

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

B. Barshan, M. A. Kutay, and H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

Palma, C.

Papoulis, A.

A. Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. 11, 3–13 (1994).
[CrossRef]

Pei, S. C.

S. C. Pei, J. J. Ding, W. L. Hsue, and K. W. Chang, “Generalized commuting matrices and their eigenvectors for DFTs, offset DFTs, and other periodic operations,” IEEE Trans. Signal Process. 56, 3891–3904 (2008).
[CrossRef]

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J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed. (Springer-Verlag, 1999).

K. B. Wolf, “Canonical transforms,” in Integral Transforms in Science and Engineering, K.B.Wolf, ed. (Plenum, 1979), pp. 381–416.

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[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Effects of special cases of LCT on the WDF [24]. The parallelograms indicate the support region of (a) the original signal and the signal after (b) FT, (c) FrFT, (d) scaling, (e) Fresnel transform (chirp convolution), and (f) chirp multiplication.

Fig. 2
Fig. 2

First six orders of CDDHFs for the scaling parameter σ = 1.0 (solid curve) and 2.8 (dashed curve).

Fig. 3
Fig. 3

Illustrations of tested functions, F 1 6 , on time and time–frequency domains.

Fig. 4
Fig. 4

Real (“Re”) and imaginary (“Im”) part simulation results of (a)  F 1 / M 1 and (b)  F 2 / M 1 using the proposed DLCT method (solid curve), CM-CC-CM method (dashed curve), and Method II in [24] (dotted curve).

Fig. 5
Fig. 5

Simulation results of (a)  F 3 / M 1 , (b)  F 4 / M 2 , (c)  F 5 / M 3 , and (d)  F 6 / M 4 using the proposed DLCT method.

Fig. 6
Fig. 6

Real (“Re”) and imaginary (“Im”) part comparisons between the proposed method (curves with plus sign marker) and sampled version of Eq. (36) (curves with circle marker) of (a)  G 1 / M 1 , (b)  G 2 / M 3 , and (c)  G 3 / M 4 .

Fig. 7
Fig. 7

Real (“Re”) and imaginary (“Im”) part simulation results of composited DLCT signals of (a)  F 1 / M 2 M 1 , (b)  F 3 / M 3 M 1 , and (c)  F 5 / M 4 M 3 from LHS (solid curve) and RHS (dashed curve) of Eq. (37).

Fig. 8
Fig. 8

Simulation results of IDLCT signals of (a)  F 2 / M 1 , (b)  F 3 / M 2 , (c)  F 4 / M 3 , and (d)  F 5 / M 4 by Eq. (39).

Tables (5)

Tables Icon

Table 1 Descriptions of Tested Functions F 1 6

Tables Icon

Table 2 Experimented LCT Matrices M 1 4 in Terms of ( a , b , c , d ) and ( ξ , σ , α )

Tables Icon

Table 3 Comparisons for Method II [24] and Proposed Method [Eq. (34)]

Tables Icon

Table 4 MSEs between LHS and RHS of Eq. (37)

Tables Icon

Table 5 MSEs of IDLCT Signals by Eq. (39)

Equations (41)

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

O LCT ( a , b , c , d ) ( x ( t ) ) = 1 i b · exp [ i π ( d b u 2 2 b u t + a b t 2 ) ] x ( t ) d t when     b 0 , O LCT ( a , 0 , c , d ) ( x ( t ) ) = d · exp ( i π d c t 2 ) x ( d · t ) when     b = 0 ,
SL 2 ( R ) = { M [ a b c d ] : a , b , c , d R and a d b c = 1 } .
O LCT ( a 3 , b 3 , c 3 , d 3 ) ( x ( t ) ) = O LCT ( a 2 , b 2 , c 2 , d 2 ) { O LCT ( a 1 , b 1 , c 1 , d 1 ) ( x ( t ) ) } , where     [ a 3 b 3 c 3 d 3 ] = [ a 2 b 2 c 2 d 2 ] [ a 1 b 1 c 1 d 1 ] .
x ( t ) = O ILCT ( a , b , c , d ) { O LCT ( a , b , c , d ) ( x ( t ) ) } where     O ILCT ( a , b , c , d ) = O LCT ( d , b , c , a ) .
W x ( t ) ( t , f ) = x ( t + τ / 2 ) · x * ( t + τ / 2 ) exp ( i 2 π f τ ) d τ ,
W O LCT ( a , b , c , d ) ( x ( t ) ) ( t , f ) = W x ( t ) ( t , f ) , where     [ t f ] = [ a b c d ] [ t f ] = [ a t + b f c t + d f ] .
FT ( x ( t ) ) = i · O LCT ( 0 , 1 , 1 , 0 ) ( x ( t ) ) = exp ( i 2 π u t ) x ( t ) d t
W F T ( x ( t ) ) ( f , t ) = W x ( t ) ( t , f ) ,
FrFT α ( x ( t ) ) = 1 i cot α · exp [ i π ( cot α · u 2 2 csc α · u t + cot α · t 2 ) ] x ( t ) d t .
FrFT α ( x ( t ) ) = exp ( i α / 2 ) · O LCT ( cos α , sin α , sin α , cos α ) ( x ( t ) ) .
W FrFT α ( x ( t ) ) ( t cos α + f sin α , t sin α + f cos α ) = W x ( t ) ( t , f ) .
FrFT β { FrFT α ( x ( t ) ) } = FrFT α { FrFT β ( x ( t ) ) } = FrFT α + β ( x ( t ) ) .
Scal σ ( x ( t ) ) = σ 1 · x ( σ 1 t ) = O LCT ( σ , 0 , 0 , σ 1 ) ( x ( t ) ) ,
W Scal σ ( x ( t ) ) ( σ t , σ 1 f ) = W x ( t ) ( t , f ) .
Fres λ , z ( x ( t ) ) = exp ( i π z / λ ) i λ z · exp [ i π 1 λ z ( u t ) 2 ] x ( t ) d t ,
Fres λ , z ( x ( t ) ) = exp ( i π z / λ ) i λ z · chirp ( t λ z ) * x ( t ) where     chirp ( t ) = exp ( i π t 2 )
Fres λ , z ( x ( t ) ) = exp ( i π z / λ ) · O LCT ( 1 , λ z , 0 , 1 ) ( x ( t ) ) or chirp ( t λ z ) * x ( t ) = i λ z · O LCT ( 1 , λ z , 0 , 1 ) ( x ( t ) ) .
W CC ( t + η f , f ) = W x ( t ) ( t , f ) .
exp ( i π ξ t 2 ) · x ( t ) = O LCT ( 1 , 0 , ξ , 1 ) ( x ( t ) ) .
W CM ( t , ξ t + f ) = W x ( t ) ( t , f ) .
Φ p ( t ) = 1 2 p p ! π exp ( t 2 / 2 ) H p ( t ) , p = 0 , 1 , 2 , ... ,
H p ( t ) = ( 1 ) p exp ( t 2 ) d p d t p exp ( t 2 ) ,
FrFT α ( Φ p ( t ) ) = exp ( i p α ) · Φ p ( t ) .
A σ = P + σ 4 FPF 1 ,
P [ m , n ] = { ( m N 1 2 ) 2 if     m = n 0 otherwise , F [ m , n ] = exp [ i 2 π N ( m N 1 2 ) ( n N 1 2 ) ] 0 m , n N 1.
[ a b c d ] = [ 1 0 ( d 1 ) / b 1 ] [ 1 b 0 1 ] [ 1 0 ( a 1 ) / b 1 ] ,
( 1 )     x 1 ( t ) = e i π a 1 b t 2 · x ( t ) , ( 2 )     x 2 ( u ) = 1 i b [ e i π 1 b t 2 * x 1 ( t ) ] , ( 3 )     x 3 ( u ) = e i π d 1 b u 2 · x 2 ( u ) .
[ a b c d ] = [ 1 0 ξ 1 ] [ σ 0 0 σ 1 ] [ cos α sin α sin α cos α ] ,
ξ = a c + b d a 2 + b 2 , σ = a 2 + b 2 , α = cos 1 ( a σ ) = sin 1 ( b σ ) .
O DLCT ( cos α , sin α , sin α , cos α ) ( Φ p ; 1 CDD HFs [ n ] ) = exp ( i α / 2 ) exp ( i p α ) · Φ p ; 1 CDD HFs [ n ] .
O DLCT ( cos α , sin α , sin α , cos α ) ( x [ n ] ) = E D α E T x [ n ] , where     D α [ m , n ] = { exp [ i ( m + 1 2 ) α ] , 0 , if     m = n     and     0 m , n N 1 oth erw ise ,
x [ n ] = p = 0 N 1 c p , 1 Φ p ; 1 CDD HFs [ n ] with inner product coefficients     c p , 1 = x [ n ] , Φ p ; 1 CDD HFs [ n ] and x σ [ n ] = p = 0 N 1 c p , 1 Φ p ; σ CDD HFs [ n ] ,
O DLCT ( σ , 0 , 0 , σ 1 ) ( x [ n ] ) = E σ E T x [ n ] .
O DLCT ( 1 , 0 , ξ , 1 ) ( x [ n ] ) = D ξ x [ n ] where     D ξ [ m , n ] = { exp { i π ξ [ n N 1 2 ] 2 / N } , if     m = n and 0 m , n N 1 0 , otherwise .
O DLCT ( a , b , c , d ) ( x [ n ] ) = { ( D ξ ) ( E σ E T ) ( E D α E T ) } x [ n ] = { D ξ E σ D α E T } x [ n ] .
W O DLCT ( a , b , c , d ) ( x ( t ) ) ( t , f ) = W x ( t ) ( t , f ) where     [ t f ] = [ a b c d ] [ t f ] = [ 1 0 ξ 1 ] [ σ 0 0 σ 1 ] [ cos α sin α sin α cos α ] [ t f ] = [ t σ cos α + f σ sin α t ( ξ σ cos α σ 1 sin α ) + f ( ξ σ sin α + σ 1 cos α ) ] .
O LCT ( a , b , c , d ) ( exp ( 2 π s t 2 ) ) = 1 a + i 2 b s · exp [ a d 1 + i 2 s b d 2 π s b 2 i π a b π 2 u 2 ] .
D ξ 3 E σ 3 D α 3 E T ( x [ n ] ) = D ξ 2 E σ 2 D α 2 E T { D ξ 1 E σ 1 D α 1 E T ( x [ n ] ) } , where     [ a 3 b 3 c 3 d 3 ] = [ a 2 b 2 c 2 d 2 ] [ a 1 b 1 c 1 d 1 ] .
O IDLCT ( a , b , c , d ) { O DLCT ( a , b , c , d ) ( x [ n ] ) } = O DLCT ( d , b , c , a ) { O DLCT ( a , b , c , d ) ( x [ n ] ) } = x [ n ] i . e . ,     [ a b c d ] 1 [ a b c d ] = [ 1 0 0 1 ] ,
O IDLCT ( a , b , c , d ) = inv ( O DLCT ( a , b , c , d ) ) = ( O DLCT ( a , b , c , d ) ) H = E D α E σ T D ξ ,
O IDLCT ( a , b , c , d ) = O DLCT ( d , b , c , a ) = D ξ ^ E σ ^ D α ^ E T .

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