Abstract

A sampling theorem applicable to that class of linear systems characterized by sufficiently slowly varying line-spread functions is developed. For band-limited inputs such systems can be exactly characterized with knowledge of the sampled system line-spread function and the corresponding sampled input. The desired sampling rate is shown to be determined by both the system and the input. The corresponding output is shown to be band limited. A discrete matrix representation of the specific system class is also presented. Applications to digital processing and coherent space-variant system representation are suggested.

© 1976 Optical Society of America

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References

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  1. T. Kailath, “Channel Characterization: Time-Variant Dispersive Channels,” in Lectures on Communications System Theory, edited by E. J. Baghdady (McGraw-Hill, New York, 1960), pp. 95–124.
  2. N. Liskov, “Analytical Techniques for Linear Time-Varying Systems,” Ph.D. dissertation (Electrical Engineering Research Laboratory, Cornell University, Ithaca, N. Y.1964) (unpublished), pp. 31–52.
  3. T. S. Huang, “Digital Computer Analysis of Linear Shift-Variant Systems,” in Proc. NASA/ERA SeminarDecember, 1969 (unpublished), pp. 83–87.
  4. A. W. Lohmann and D. P. Paris, “Space-Variant Image Formation,” J. Opt. Soc. Am. 55, 1007–1013 (1965).
  5. Here, and in the material to follow, “band limited” refers specifically to that case where the spectrum is nonzero only over a single interval centered about zero frequency. It appears, however, that the results can be extended to any spectrum with finite support by application of corresponding sampling theorems. For example, see D. A. Linden, “A Discussion of Sampling Theorems,” Proc. IRE 47, 1219–1226 (1959).
    [Crossref]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  7. L. M. Deen, J. F. Walkup, and M. O. Hagler, “Representations of Space-Variant Optical Systems Using Volume Holograms,” Appl. Opt. 14, 2438–2446 (1975).
    [Crossref] [PubMed]
  8. L. M. Deen, “Holographic Representations of Optical Systems,” M. S. thesis (Department of Electrical Engineering, Texas Tech University, Lubbock, Tex., 1975) (unpublished), pp. 37–60.
  9. R. J. Marks and T. F. Krile, “Holographic Representation of Space-Variant Systems; System Theory,” to appear in Appl. Opt.
  10. R. J. Marks, “Holographic Recording of Optical Space-Variant Systems,” M. S. thesis (Rose-Hulman Institute of Technology, Terre Haute. Ind., 1973) (unpublished), pp. 74–93.
  11. R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holograpy (Academic, New York/London, 1971), pp. 466–467.
  12. D. Slepian, “On Bandwidth,” Proc. IEEE 64, 292 (1976).
    [Crossref]
  13. K. Yao and J. B. Thomas, “On Truncation Error Bounds for Sampling Representations of Band-Limited Signals,” IEEE Trans. Aerospace Electron. Syst. 2, 640–647 (1966).
    [Crossref]
  14. A. A. Sawchuk, “Space-Variant Restoration by Coordinate Transformation,” J. Opt. Soc. Am. 64, 138–144 (1974).
    [Crossref]

1976 (1)

D. Slepian, “On Bandwidth,” Proc. IEEE 64, 292 (1976).
[Crossref]

1975 (1)

1974 (1)

1966 (1)

K. Yao and J. B. Thomas, “On Truncation Error Bounds for Sampling Representations of Band-Limited Signals,” IEEE Trans. Aerospace Electron. Syst. 2, 640–647 (1966).
[Crossref]

1965 (1)

1959 (1)

Here, and in the material to follow, “band limited” refers specifically to that case where the spectrum is nonzero only over a single interval centered about zero frequency. It appears, however, that the results can be extended to any spectrum with finite support by application of corresponding sampling theorems. For example, see D. A. Linden, “A Discussion of Sampling Theorems,” Proc. IRE 47, 1219–1226 (1959).
[Crossref]

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holograpy (Academic, New York/London, 1971), pp. 466–467.

Collier, R. J.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holograpy (Academic, New York/London, 1971), pp. 466–467.

Deen, L. M.

L. M. Deen, J. F. Walkup, and M. O. Hagler, “Representations of Space-Variant Optical Systems Using Volume Holograms,” Appl. Opt. 14, 2438–2446 (1975).
[Crossref] [PubMed]

L. M. Deen, “Holographic Representations of Optical Systems,” M. S. thesis (Department of Electrical Engineering, Texas Tech University, Lubbock, Tex., 1975) (unpublished), pp. 37–60.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hagler, M. O.

Huang, T. S.

T. S. Huang, “Digital Computer Analysis of Linear Shift-Variant Systems,” in Proc. NASA/ERA SeminarDecember, 1969 (unpublished), pp. 83–87.

Kailath, T.

T. Kailath, “Channel Characterization: Time-Variant Dispersive Channels,” in Lectures on Communications System Theory, edited by E. J. Baghdady (McGraw-Hill, New York, 1960), pp. 95–124.

Krile, T. F.

R. J. Marks and T. F. Krile, “Holographic Representation of Space-Variant Systems; System Theory,” to appear in Appl. Opt.

Lin, L. H.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holograpy (Academic, New York/London, 1971), pp. 466–467.

Linden, D. A.

Here, and in the material to follow, “band limited” refers specifically to that case where the spectrum is nonzero only over a single interval centered about zero frequency. It appears, however, that the results can be extended to any spectrum with finite support by application of corresponding sampling theorems. For example, see D. A. Linden, “A Discussion of Sampling Theorems,” Proc. IRE 47, 1219–1226 (1959).
[Crossref]

Liskov, N.

N. Liskov, “Analytical Techniques for Linear Time-Varying Systems,” Ph.D. dissertation (Electrical Engineering Research Laboratory, Cornell University, Ithaca, N. Y.1964) (unpublished), pp. 31–52.

Lohmann, A. W.

Marks, R. J.

R. J. Marks, “Holographic Recording of Optical Space-Variant Systems,” M. S. thesis (Rose-Hulman Institute of Technology, Terre Haute. Ind., 1973) (unpublished), pp. 74–93.

R. J. Marks and T. F. Krile, “Holographic Representation of Space-Variant Systems; System Theory,” to appear in Appl. Opt.

Paris, D. P.

Sawchuk, A. A.

Slepian, D.

D. Slepian, “On Bandwidth,” Proc. IEEE 64, 292 (1976).
[Crossref]

Thomas, J. B.

K. Yao and J. B. Thomas, “On Truncation Error Bounds for Sampling Representations of Band-Limited Signals,” IEEE Trans. Aerospace Electron. Syst. 2, 640–647 (1966).
[Crossref]

Walkup, J. F.

Yao, K.

K. Yao and J. B. Thomas, “On Truncation Error Bounds for Sampling Representations of Band-Limited Signals,” IEEE Trans. Aerospace Electron. Syst. 2, 640–647 (1966).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Aerospace Electron. Syst. (1)

K. Yao and J. B. Thomas, “On Truncation Error Bounds for Sampling Representations of Band-Limited Signals,” IEEE Trans. Aerospace Electron. Syst. 2, 640–647 (1966).
[Crossref]

J. Opt. Soc. Am. (2)

Proc. IEEE (1)

D. Slepian, “On Bandwidth,” Proc. IEEE 64, 292 (1976).
[Crossref]

Proc. IRE (1)

Here, and in the material to follow, “band limited” refers specifically to that case where the spectrum is nonzero only over a single interval centered about zero frequency. It appears, however, that the results can be extended to any spectrum with finite support by application of corresponding sampling theorems. For example, see D. A. Linden, “A Discussion of Sampling Theorems,” Proc. IRE 47, 1219–1226 (1959).
[Crossref]

Other (8)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

T. Kailath, “Channel Characterization: Time-Variant Dispersive Channels,” in Lectures on Communications System Theory, edited by E. J. Baghdady (McGraw-Hill, New York, 1960), pp. 95–124.

N. Liskov, “Analytical Techniques for Linear Time-Varying Systems,” Ph.D. dissertation (Electrical Engineering Research Laboratory, Cornell University, Ithaca, N. Y.1964) (unpublished), pp. 31–52.

T. S. Huang, “Digital Computer Analysis of Linear Shift-Variant Systems,” in Proc. NASA/ERA SeminarDecember, 1969 (unpublished), pp. 83–87.

L. M. Deen, “Holographic Representations of Optical Systems,” M. S. thesis (Department of Electrical Engineering, Texas Tech University, Lubbock, Tex., 1975) (unpublished), pp. 37–60.

R. J. Marks and T. F. Krile, “Holographic Representation of Space-Variant Systems; System Theory,” to appear in Appl. Opt.

R. J. Marks, “Holographic Recording of Optical Space-Variant Systems,” M. S. thesis (Rose-Hulman Institute of Technology, Terre Haute. Ind., 1973) (unpublished), pp. 74–93.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holograpy (Academic, New York/London, 1971), pp. 466–467.

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Figures (1)

FIG. 1
FIG. 1

Generation of a sample line-spread function and corresponding sample-transfer function for an arbitrary coherent space-variant system.

Equations (34)

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g ( x ) = - f ( ξ ) h ( x - ξ ; ξ ) d ξ ,
g ( x ) = - f ( ξ ) h ( x - ξ ) d ξ = f ( x ) * h ( x ) .
G ( f x ) = F x [ g ( x ) ] = - f ( ξ ) F x [ h ( x ; ξ ) ] exp ( - j 2 π f x ξ ) d ξ = F ξ F x [ f ( ξ ) h ( x ; ξ ) ] ν = f x ,
F x [ p ( x ; ξ ) ] - p ( x ; ξ ) exp ( - j 2 π f x x ) d x
F ξ [ p ( x ; ξ ) ] - p ( x ; ξ ) exp ( - j 2 π v ξ ) d ξ .
H x ( f x ; ξ ) F x [ h ( x ; ξ ) ] .
G ( f x ) = F ξ [ f ( ξ ) H x ( f x ; ξ ) ] v = f x .
H ξ ( x ; v ) F ξ [ h ( x ; ξ ) ] .
H ξ ( x ; v ) = 0 for v > W v for all x .
f ( ξ ) h ( x ; ξ ) .
2 W = 2 W f + 2 W v .
f ( ξ ) h ( x ; ξ ) = n = - f ( ξ n ) h ( x ; ξ n ) sinc 2 W ( ξ - ξ n ) ,
rect ( x ) { 1 , x 1 2 , 0 , x > 1 2 .
G ( f x ) = 1 2 W n f ( ξ n ) H x ( f x ; ξ n ) × exp ( - j 2 π f x ξ n ) rect ( f x 2 W ) ,
g ( x ) = n f ( ξ n ) h ( x - ξ n ; ξ n ) * sinc ( 2 W x ) .
f x = x / λ f ,
S = 4 W a
h ( x - ξ ; ξ ) = 2 f 0 sinc 2 f 0 [ x - M ξ ] = 2 f 0 sinc 2 f 0 [ ( x - ξ ) - ( M - 1 ) ξ ] ,
h ( x ; ξ ) = 2 f 0 sinc 2 f 0 [ x - ( M - 1 ) ξ ] .
H x ( f x ; ξ ) = exp [ - j 2 π ( M - 1 ) f x ξ ] rect ( f x / 2 f 0 ) .
G ( f x ) = F ( M f x ) rect ( f x / 2 f 0 ) ,
F ( f x ) = F x [ f ( x ) ] .
H ξ ( x ; v ) = 1 M - 1 exp [ - j 2 π v ( x M - 1 ) ] rect ( v 2 f 0 ( M - 1 ) ) .
2 W v = 2 f 0 M - 1 .
G ( f x ) = 1 2 W n f ( ξ n ) exp ( - j 2 π f x M ξ n ) rect ( f x 2 W ) rect ( f x 2 f 0 ) .
G ( f x ) = 1 2 W n g ( x n ) exp ( - j 2 π f x x n ) rect ( f x 2 W ) ,
x n = n / 2 W .
n g ( x n ) exp ( - j 2 π f x ( n - m ) 2 W ) rect ( f x 2 W ) = n f ( ξ n ) H x ( f x ; ξ n ) exp ( j 2 π f x ( m - n ) 2 W ) rect ( f x 2 W ) .
Ĥ x ( f x ; ξ n ) = H x ( f x ; ξ n ) rect ( f x / 2 W ) ,
- exp ( - j 2 π f x ( n - m ) 2 W ) rect ( f x 2 W ) d f x = 2 W sinc ( n - m ) = 2 W δ n m ,
g ( x m ) = 1 2 W n f ( ξ n ) ĥ ( x m - ξ n ; ξ n ) ,
h ( x ; ξ ) = p h ( x ; ξ p ) sinc 2 W v ( ξ - ξ p ) ,
f ( ξ ) = k f ( ξ k ) sinc 2 W f ( ξ - ξ k ) ,
G ( f x ) = 1 4 W f W v k { f ( ξ k ) p H x ( f x ; ξ p ) [ rect ( f x 2 W v ) × exp ( - j 2 π f x ξ p ) ] * [ rect ( f x 2 W f ) exp ( - j 2 π f x ξ k ] } .