Abstract

Two recently described transforms are shown to be related. The Radon–Wigner transform is the squared modulus of the fractional Fourier transform. This new theorem may serve to translate signal and image processing results between different signal representations. Some consequences regarding moments are presented, including a new fractional-Fourier-transform uncertainty relation. Implications for processing are suggested.

© 1994 Optical Society of America

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References

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  1. J. C. Wood, D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).
  2. J. C. Wood, D. T. Barry, “Radon transformation of the Wigner spectrum,” in Advanced Signal Processing, Algorithms, Architectures, and Implementations III, F. T. Luk, ed., Proc. Soc. Photo-Opt. Eng.1770, 358–375 (1992).
    [Crossref]
  3. W. Li, “Wigner distribution method equivalent to dechirp method for detecting a chirp signal,” IEEE Trans. Acoust. Speech Signal Process. 35, 1210–1211 (1987).
    [Crossref]
  4. S. Kay, G. F. Boudreau-Bartels, “On the optimality of the Wigner distribution for detection,” Proc. Int. Conf. Acoust. Speech Signal Process. 3, 1017–1020 (1985).
  5. P. Flandrin, “On detection-estimation procedures in the time-frequency plane,” Proc. Int. Conf. Acoust. Speech Signal Process. 4, 2331–2334 (1986).
  6. N. G. de Bruijn, “A theory of generalized functions with applications to Wigner distribution and Weyl correspondence,” Nieuwe Arch. Wiskunde 21, 205–280 (1973).
  7. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [Crossref]
  8. A. C. McBride, F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [Crossref]
  9. B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982).
    [Crossref]
  10. L. B. Almeida, “The angular Fourier transform,” submitted to IMA J. Appl. Math and to IEEE Trans. Signal Process.
  11. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [Crossref]
  12. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. Parts I and II,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [Crossref]
  13. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [Crossref]

1993 (3)

1992 (1)

J. C. Wood, D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).

1987 (2)

W. Li, “Wigner distribution method equivalent to dechirp method for detecting a chirp signal,” IEEE Trans. Acoust. Speech Signal Process. 35, 1210–1211 (1987).
[Crossref]

A. C. McBride, F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[Crossref]

1986 (1)

P. Flandrin, “On detection-estimation procedures in the time-frequency plane,” Proc. Int. Conf. Acoust. Speech Signal Process. 4, 2331–2334 (1986).

1985 (1)

S. Kay, G. F. Boudreau-Bartels, “On the optimality of the Wigner distribution for detection,” Proc. Int. Conf. Acoust. Speech Signal Process. 3, 1017–1020 (1985).

1982 (1)

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

1980 (1)

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

1973 (1)

N. G. de Bruijn, “A theory of generalized functions with applications to Wigner distribution and Weyl correspondence,” Nieuwe Arch. Wiskunde 21, 205–280 (1973).

Almeida, L. B.

L. B. Almeida, “The angular Fourier transform,” submitted to IMA J. Appl. Math and to IEEE Trans. Signal Process.

Barry, D. T.

J. C. Wood, D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).

J. C. Wood, D. T. Barry, “Radon transformation of the Wigner spectrum,” in Advanced Signal Processing, Algorithms, Architectures, and Implementations III, F. T. Luk, ed., Proc. Soc. Photo-Opt. Eng.1770, 358–375 (1992).
[Crossref]

Boudreau-Bartels, G. F.

S. Kay, G. F. Boudreau-Bartels, “On the optimality of the Wigner distribution for detection,” Proc. Int. Conf. Acoust. Speech Signal Process. 3, 1017–1020 (1985).

de Bruijn, N. G.

N. G. de Bruijn, “A theory of generalized functions with applications to Wigner distribution and Weyl correspondence,” Nieuwe Arch. Wiskunde 21, 205–280 (1973).

Dickinson, B. W.

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

Flandrin, P.

P. Flandrin, “On detection-estimation procedures in the time-frequency plane,” Proc. Int. Conf. Acoust. Speech Signal Process. 4, 2331–2334 (1986).

Kay, S.

S. Kay, G. F. Boudreau-Bartels, “On the optimality of the Wigner distribution for detection,” Proc. Int. Conf. Acoust. Speech Signal Process. 3, 1017–1020 (1985).

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[Crossref]

Li, W.

W. Li, “Wigner distribution method equivalent to dechirp method for detecting a chirp signal,” IEEE Trans. Acoust. Speech Signal Process. 35, 1210–1211 (1987).
[Crossref]

Lohmann, A. W.

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[Crossref]

Mendlovic, D.

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[Crossref]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. Parts I and II,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[Crossref]

Namias, V.

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

Ozaktas, H. M.

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[Crossref]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. Parts I and II,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[Crossref]

Steiglitz, K.

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

Wood, J. C.

J. C. Wood, D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).

J. C. Wood, D. T. Barry, “Radon transformation of the Wigner spectrum,” in Advanced Signal Processing, Algorithms, Architectures, and Implementations III, F. T. Luk, ed., Proc. Soc. Photo-Opt. Eng.1770, 358–375 (1992).
[Crossref]

IEEE Trans. Acoust. Speech Signal Process. (2)

W. Li, “Wigner distribution method equivalent to dechirp method for detecting a chirp signal,” IEEE Trans. Acoust. Speech Signal Process. 35, 1210–1211 (1987).
[Crossref]

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

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[Crossref]

J. Inst. Math. Appl. (1)

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

J. Opt. Soc. Am. A (2)

Nieuwe Arch. Wiskunde (1)

N. G. de Bruijn, “A theory of generalized functions with applications to Wigner distribution and Weyl correspondence,” Nieuwe Arch. Wiskunde 21, 205–280 (1973).

Opt. Commun. (1)

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[Crossref]

Proc. Int. Conf. Acoust. Speech Signal Process. (3)

S. Kay, G. F. Boudreau-Bartels, “On the optimality of the Wigner distribution for detection,” Proc. Int. Conf. Acoust. Speech Signal Process. 3, 1017–1020 (1985).

P. Flandrin, “On detection-estimation procedures in the time-frequency plane,” Proc. Int. Conf. Acoust. Speech Signal Process. 4, 2331–2334 (1986).

J. C. Wood, D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” Proc. Int. Conf. Acoust. Speech Signal Process. 4, 257–261 (1992).

Other (2)

J. C. Wood, D. T. Barry, “Radon transformation of the Wigner spectrum,” in Advanced Signal Processing, Algorithms, Architectures, and Implementations III, F. T. Luk, ed., Proc. Soc. Photo-Opt. Eng.1770, 358–375 (1992).
[Crossref]

L. B. Almeida, “The angular Fourier transform,” submitted to IMA J. Appl. Math and to IEEE Trans. Signal Process.

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Figures (1)

Fig. 1
Fig. 1

Signals uA and uB, overlapping in x and ν, are represented in Wigner distributions WA and WB and by their fractional Fourier transforms uPA and uPB. Only real quantities are indicated, and the Wigner interference cross terms, which usually disappear in the fractional Fourier transform, are not shown.

Equations (27)

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u ( x + x / 2 ) u * ( x x / 2 ) exp ( 2 π i ν x ) d x = W ( x , ν ) .
u ( x ) exp ( 2 π i ν x ) d x = ũ ( ν ) .
W ( x , ν ) = ũ ( ν + ν / 2 ) ũ * ( ν ν / 2 ) exp ( 2 π i x ν ) d ν .
W ( x , ν ) d ν = | u ( x ) | 2 ,
W ( x , ν ) d x = | ũ ( ν ) | 2 ,
W ( x / 2 , ν ) exp ( 2 π i ν x ) d ν = u ( x ) u ( o ) * , | u ( o ) | 2 = W ( o , ν ) d ν ,
u 0 ( x ) W [ u 0 ] = W 0 ( x , ν ) ,
W 0 ( x , ν ) W 0 [ x cos ( Φ ) ν sin ( Φ ) , ν cos ( Φ ) + x sin ( Φ ) ] = W P ( x , ν ) ,
W P ( x , ν ) W 1 [ W P ] = u P ( x ) ; P = Φ / 90 ° .
υ 0 ( x , y ) υ 0 [ x cos ( Φ ) y sin ( Φ ) , y cos ( Φ ) + x sin ( Φ ) ] = υ ( x , y ) ,
υ ( x , y ) υ ( x , y ) d y = R [ υ 0 ; Φ ] = V ( x ; Φ ) .
u 0 ( x ) W 0 ( x , ν ) ,
W 0 ( x , ν ) R [ W 0 ; Φ ] = V ( x ; Φ ) .
u 0 ( x ) RW [ u 0 ] = V ( x ; Φ ) .
u 0 ( x ) FRT [ u 0 ] = u P ( x ) ; P = Φ / 90 ° .
u P ( x ) | u P ( x ) | 2 = V ( x ; Φ ) .
RW [ u 0 ] = | FRT [ u 0 ] | 2 = | u P | 2 ; P = Φ / 90 ° .
x 2 ¯ = x 2 | u ( x ) | 2 d x = x 2 W ( x , ν ) d x d ν .
ν 2 ¯ = ν 2 W ( x , ν ) d ν d x .
x P 2 ¯ = x 2 W P ( x , ν ) d x d ν = x 2 R W P ( x ) d x ,
ν P 2 ¯ = ν 2 W P ( x , ν ) d ν d x .
ν P = x P + 1 ,
x P + 1 2 ¯ = x 2 W P + 1 ( x , ν ) d ν d x .
x P = x 0 cos ( ϕ ) ν 0 sin ( ϕ ) ; x P + 1 = ν 0 cos ( ϕ ) + x 0 sin ( ϕ ) .
x P 2 ¯ + x P + 1 2 ¯ = x 0 2 ¯ + x 1 2 ¯ .
x 2 | u P ( x ) | 2 d x + x 2 | u P + 1 ( x ) | 2 d x = const .
( Δ x P ) ( Δ x P + 1 ) 1 / ( 4 π ) ,

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