Abstract

The Wigner distribution function (WDF) offers comprehensive insight into a signal, for it employs both space (or time) and frequency simultaneously. Whenever optical signals are involved, the importance of the WDF is significantly higher because of the diffraction (or dispersion) behavior of optical signals. Novel optical implementations of the WDF and of the inverse Wigner transform are proposed. Both implementations are based on bulk optics elements incorporating joint transform correlator architecture. A similar implementation is derived for the ambiguity function, which is related to the WDF through Fourier transformation.

© 1998 Optical Society of America

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References

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  1. J. C. Wood, D. T. Barry, “Tomographic time–frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2103 (1994).
    [CrossRef]
  2. F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time–frequency signal representations,” IEEE Signal Process. Mag. 9(2), 21–67 (1992).
    [CrossRef]
  3. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  4. K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
    [CrossRef]
  5. H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992).
    [CrossRef]
  6. D. Dragoman, M. Dragoman, “Wigner-transform implementation in the time-frequency domain,” Appl. Opt. 35, 7025–7030 (1996).
    [CrossRef] [PubMed]
  7. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1250 (1966).
    [CrossRef] [PubMed]

1996

1994

J. C. Wood, D. T. Barry, “Tomographic time–frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2103 (1994).
[CrossRef]

1992

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time–frequency signal representations,” IEEE Signal Process. Mag. 9(2), 21–67 (1992).
[CrossRef]

H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

1982

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

1978

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

1966

Barry, D. T.

J. C. Wood, D. T. Barry, “Tomographic time–frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2103 (1994).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Boudreaux-Bartels, G. F.

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time–frequency signal representations,” IEEE Signal Process. Mag. 9(2), 21–67 (1992).
[CrossRef]

Brenner, K. H.

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

Dragoman, D.

Dragoman, M.

Goodman, J. W.

Hlawatsch, F.

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time–frequency signal representations,” IEEE Signal Process. Mag. 9(2), 21–67 (1992).
[CrossRef]

Lohmann, A. W.

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

Weaver, C. S.

Weber, H.

H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

Wood, J. C.

J. C. Wood, D. T. Barry, “Tomographic time–frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2103 (1994).
[CrossRef]

Appl. Opt.

IEEE Signal Process. Mag.

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time–frequency signal representations,” IEEE Signal Process. Mag. 9(2), 21–67 (1992).
[CrossRef]

IEEE Trans. Signal Process.

J. C. Wood, D. T. Barry, “Tomographic time–frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals,” IEEE Trans. Signal Process. 42, 2094–2103 (1994).
[CrossRef]

J. Mod. Opt.

H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

Opt. Commun.

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

K. H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1-D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Direct optical implementation of the WDF.

Fig. 2
Fig. 2

Optical implementation of the AF.

Fig. 3
Fig. 3

Optical implementation of the inverse Wigner transform.

Equations (15)

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W x ,   ν =   u x + x 2 u * x - x 2 exp - i 2 π ν x d x =   ũ ν + ν 2 ũ * ν - ν 2 exp i 2 π x ν d ν ,
A μ ,   y =   u x + y 2 u * x - y 2 exp - i 2 π μ x d x ,
A μ ,   y =   W x ,   ν exp i 2 π y ν - μ x d x d ν .
F x ,   x = u x + x 2 u * x - x 2
FT x u x + x 2 =   u x + x 2 exp - i 2 π ν x d x = 2   exp i 4 π ν x ũ 2 ν ,
FT x u * x - x 2 =   u * x - x 2 exp - i 2 π ν x d x = 2   exp - i 4 π ν x ũ * 2 ν ,
W x ,   ν 2 = 4 exp i 2 π ν x ũ ν ν * exp - i 2 π ν x ũ - ν ,
t x ,   y = 2 δ y cos 2 π xs 0 .
T t ,   s = ũ s - s 0 + ũ s + s 0 .
L + t ,   s = exp i 2 π st ,     L - t ,   s = exp - i 2 π st .
T 1 t ,   s = ũ s - s 0 exp [ i 2 π ( s s 0 ) t ] L + ( t ,   s s 0 ) L + t ,   s - s 0 + ũ s + s 0 exp [ i 2 π ( s s 0 ) t ] L + ( t ,   s s 0 ) .
T 1 u ,   v = ũ v - s 0 exp i 2 π v - s 0 u + ũ - v - s 0 exp - i 2 π v + s 0 u .
L t ,   s = exp - i   π 2 λ f t - s 2 exp i   π 2 λ f t + s 2 = exp i   2 π λ f   ts .
A μ ,   y = exp i π μ y ũ y μ * exp - i π μ y ũ y ,
u 2 x       W x ,   ν exp i 4 π x ν d ν .

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