Abstract

A signal may have compact support, be band-limited (i.e., its Fourier transform has compact support), or neither (“unbounded”). We determine conditions for the linear canonical transform of a signal having these properties. We examine the significance of these conditions for special cases of the linear canonical transform and consider the physical significance of our results.

© 2008 Optical Society of America

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References

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  1. H. Baher, Analog & Digital Signal Processing (Wiley, 1990), p. 121.
  2. A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, 1962), p. 215.
  3. R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain (American Mathematical Society, 1934), p. 12.
  4. D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, Opt. Eng. 45, 088201 (2006).
    [CrossRef]
  5. B. M. Hennelly and J. T. Sheridan, J. Opt. Soc. Am. A 22, 917 (2005).
    [CrossRef]
  6. M. J. Bastiaans and K. B. Wolf, in Seventh International Symposium on Signal Processing and Its Applications (IEEE, 2003), Vol. 1, pp. 589-592.
    [CrossRef]
  7. B. Barshan, M. Alper Kutay, and H. M. Ozaktas, Opt. Commun. 135, 32 (1997).
    [CrossRef]
  8. S.-C. Pei and J.-J. Ding, J. Opt. Soc. Am. A 17, 2355 (2000).
    [CrossRef]
  9. K. K. Sharma and S. D. Joshi, Opt. Commun. 265, 454 (2006).
    [CrossRef]
  10. A. Stern, Signal Processing 86, 1421 (2006).
    [CrossRef]

2006

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, Opt. Eng. 45, 088201 (2006).
[CrossRef]

K. K. Sharma and S. D. Joshi, Opt. Commun. 265, 454 (2006).
[CrossRef]

A. Stern, Signal Processing 86, 1421 (2006).
[CrossRef]

2005

2000

1997

B. Barshan, M. Alper Kutay, and H. M. Ozaktas, Opt. Commun. 135, 32 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

B. Barshan, M. Alper Kutay, and H. M. Ozaktas, Opt. Commun. 135, 32 (1997).
[CrossRef]

K. K. Sharma and S. D. Joshi, Opt. Commun. 265, 454 (2006).
[CrossRef]

Opt. Eng.

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, Opt. Eng. 45, 088201 (2006).
[CrossRef]

Signal Processing

A. Stern, Signal Processing 86, 1421 (2006).
[CrossRef]

Other

M. J. Bastiaans and K. B. Wolf, in Seventh International Symposium on Signal Processing and Its Applications (IEEE, 2003), Vol. 1, pp. 589-592.
[CrossRef]

H. Baher, Analog & Digital Signal Processing (Wiley, 1990), p. 121.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, 1962), p. 215.

R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain (American Mathematical Society, 1934), p. 12.

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Tables (1)

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Table 1 Conditions on the Input Signal (Columns) and the Parameters (Rows) a

Equations (11)

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L T { f ( x ) } ( x ) = { 1 b e j π 4 f ( x ) e j π b ( a x 2 2 x x + d x 2 ) d x b 0 d e j 2 c d x 2 f ( d x ) b = 0 } .
( x k ) = ( a b c d ) ( x k ) ,
L T { f ( x ) } ( x ) = e j π ( d x 2 ) b f ( x ) e j 2 π x x b d x .
L T { f ( x ) } ( x ) = e j π ( d x 2 ) b F ( 2 π x b ) ,
L T { f ( x ) } ( x ) = d e j c d x 2 2 f ( d x ) .
( 0 1 1 0 ) ( a b 0 a 1 ) ( 0 1 1 0 ) = ( a 1 0 b a ) .
L T { f ( x ) } ( x ) = Γ Γ f ( x ) e j π b ( a x 2 2 x x ) d x .
I [ L T { f ( x ) } ( x ) ] ( k ) = Γ Γ f ( x ) e j π b ( a x 2 ) e j π b ( 2 x x ) e j k x d x d x .
I [ L T { f ( x ) } ( x ) ] ( k ) = Γ Γ f ( x ) e j π b ( a x 2 ) δ ( k 2 π x b ) d x = f ( b k 2 π ) e j a b k 2 4 π .
( a b c d ) = ( 1 0 d b 1 ) ( 0 b 1 b 0 ) ( 1 0 a b 1 ) .
( a b c d ) ( 0 1 1 0 ) = ( b a d c ) ,

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