Abstract

We describe a theoretical procedure for analyzing astronomical phased arrays with overlapping beams and apply the procedure to simulate a simple example. We demonstrate the effect of overlapping beams on the number of degrees of freedom of the array and on the ability of the array to recover a source. We show that the best images are obtained using overlapping beams, contrary to common practice, and show how the dynamic range of a phased array directly affects the image quality.

© 2008 Optical Society of America

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References

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  1. R. Braun, “The concept of the square kilometer array interferometer,” in Proceedings of High Sensitivity Radio Astronomy, N.Jackson and R.J.Davies, eds. (Cambridge U. Press, 1997), pp. 260-268.
  2. A. van Ardenne, A. Smolders, and G. Hampson, “Active adaptive antennas for radio astronomy: results of the R & D program towards the square kilometer array,” Proc. SPIE 4014, 420-433 (2000).
    [CrossRef]
  3. A. Ardenne, P. Wilkinson, P. Patel, and J. Vaate, “Electronic multi-beam radio astronomy concept: embrace a demonstrator for the European SKA program,” Exp. Astron. 17, 65-77 (2004).
    [CrossRef]
  4. N. E. Kassim, T. J. W. Lazio, P. S. Ray, P. C. Crane, B. C. Hicks, K. P. Stewart, A. S. Cohen, and W. M. Lane, “The low-frequency array (LOFAR): opening a new window on the universe,” Planet. Space Sci. 52, 1543-1549 (2004).
    [CrossRef]
  5. I. Daubechies, A. Grossmann, and Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. 27, 1271-1283 (1986).
    [CrossRef]
  6. I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961-1005 (1990).
    [CrossRef]
  7. A. Refregier, “Shapelets: a method for image analysis,” Mon. Not. R. Astron. Soc. 338, 35-47 (2003).
    [CrossRef]
  8. R. H. Berry, M. P. Hobson, and S. Withington, “Modal decomposition of astronomical images with application to shapelets,” Mon. Not. R. Astron. Soc. 354, 199-211 (2004).
    [CrossRef]
  9. R. Masset and A. Refregier, “Polar shapelets,” Mon. Not. R. Astron. Soc. 363, 197-210 (2005).
    [CrossRef]
  10. S. Withington, G. Saklatvala, and M. P. Hobson, “Theoretical analysis of astronomical phased arrays,” J. Opt. A, Pure Appl. Opt. 10, 015304 (2007).
    [CrossRef]
  11. L. Milner and M. Parker, “A broadband 8-18 GHz 4-input 4-output Butler matrix,” Proc. SPIE 6414, 641406 (2007).
    [CrossRef]
  12. S. W. Ellingson, “Efficient multibeam synthesis with interference nulling for large arrays,” IEEE Trans. Antennas Propag. 51, 503-511 (2003).
    [CrossRef]
  13. C. Lee, D. Leigh, K. Ryall, H. Miyashita, and K. Hirata, “Very fast subarray position calculation for minimizing sidelobes in sparse linear phased arrays,” in Proceedings of the European Conference on Antennas and Propagation (EuCAP, 2006), Vol. 606, pp. 80-87. Available online at http://www.merl.com/reports/docs/TR2006-022.pdf.
  14. E. H. Moore, “On the reciprocal of the general algebraic matrix,” Bull. Am. Math. Soc. 26, 394-395 (1920).
  15. R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406-413 (1955).
    [CrossRef]
  16. R. Penrose, “On best approximate solutions of linear equations,” Proc. Cambridge Philos. Soc. 52, 17-19 (1955).
    [CrossRef]
  17. P. G. Casazza, “The art of frame theory,” Taiwan. J. Math. 4, 129-201 (2000).
  18. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
    [CrossRef]

2007 (2)

S. Withington, G. Saklatvala, and M. P. Hobson, “Theoretical analysis of astronomical phased arrays,” J. Opt. A, Pure Appl. Opt. 10, 015304 (2007).
[CrossRef]

L. Milner and M. Parker, “A broadband 8-18 GHz 4-input 4-output Butler matrix,” Proc. SPIE 6414, 641406 (2007).
[CrossRef]

2005 (1)

R. Masset and A. Refregier, “Polar shapelets,” Mon. Not. R. Astron. Soc. 363, 197-210 (2005).
[CrossRef]

2004 (3)

R. H. Berry, M. P. Hobson, and S. Withington, “Modal decomposition of astronomical images with application to shapelets,” Mon. Not. R. Astron. Soc. 354, 199-211 (2004).
[CrossRef]

A. Ardenne, P. Wilkinson, P. Patel, and J. Vaate, “Electronic multi-beam radio astronomy concept: embrace a demonstrator for the European SKA program,” Exp. Astron. 17, 65-77 (2004).
[CrossRef]

N. E. Kassim, T. J. W. Lazio, P. S. Ray, P. C. Crane, B. C. Hicks, K. P. Stewart, A. S. Cohen, and W. M. Lane, “The low-frequency array (LOFAR): opening a new window on the universe,” Planet. Space Sci. 52, 1543-1549 (2004).
[CrossRef]

2003 (2)

S. W. Ellingson, “Efficient multibeam synthesis with interference nulling for large arrays,” IEEE Trans. Antennas Propag. 51, 503-511 (2003).
[CrossRef]

A. Refregier, “Shapelets: a method for image analysis,” Mon. Not. R. Astron. Soc. 338, 35-47 (2003).
[CrossRef]

2000 (2)

P. G. Casazza, “The art of frame theory,” Taiwan. J. Math. 4, 129-201 (2000).

A. van Ardenne, A. Smolders, and G. Hampson, “Active adaptive antennas for radio astronomy: results of the R & D program towards the square kilometer array,” Proc. SPIE 4014, 420-433 (2000).
[CrossRef]

1990 (1)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961-1005 (1990).
[CrossRef]

1986 (1)

I. Daubechies, A. Grossmann, and Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. 27, 1271-1283 (1986).
[CrossRef]

1955 (2)

R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406-413 (1955).
[CrossRef]

R. Penrose, “On best approximate solutions of linear equations,” Proc. Cambridge Philos. Soc. 52, 17-19 (1955).
[CrossRef]

1920 (1)

E. H. Moore, “On the reciprocal of the general algebraic matrix,” Bull. Am. Math. Soc. 26, 394-395 (1920).

Bull. Am. Math. Soc. (1)

E. H. Moore, “On the reciprocal of the general algebraic matrix,” Bull. Am. Math. Soc. 26, 394-395 (1920).

Exp. Astron. (1)

A. Ardenne, P. Wilkinson, P. Patel, and J. Vaate, “Electronic multi-beam radio astronomy concept: embrace a demonstrator for the European SKA program,” Exp. Astron. 17, 65-77 (2004).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

S. W. Ellingson, “Efficient multibeam synthesis with interference nulling for large arrays,” IEEE Trans. Antennas Propag. 51, 503-511 (2003).
[CrossRef]

IEEE Trans. Inf. Theory (1)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961-1005 (1990).
[CrossRef]

J. Math. Phys. (1)

I. Daubechies, A. Grossmann, and Y. Meyer, “Painless nonorthogonal expansions,” J. Math. Phys. 27, 1271-1283 (1986).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

S. Withington, G. Saklatvala, and M. P. Hobson, “Theoretical analysis of astronomical phased arrays,” J. Opt. A, Pure Appl. Opt. 10, 015304 (2007).
[CrossRef]

Mon. Not. R. Astron. Soc. (3)

A. Refregier, “Shapelets: a method for image analysis,” Mon. Not. R. Astron. Soc. 338, 35-47 (2003).
[CrossRef]

R. H. Berry, M. P. Hobson, and S. Withington, “Modal decomposition of astronomical images with application to shapelets,” Mon. Not. R. Astron. Soc. 354, 199-211 (2004).
[CrossRef]

R. Masset and A. Refregier, “Polar shapelets,” Mon. Not. R. Astron. Soc. 363, 197-210 (2005).
[CrossRef]

Planet. Space Sci. (1)

N. E. Kassim, T. J. W. Lazio, P. S. Ray, P. C. Crane, B. C. Hicks, K. P. Stewart, A. S. Cohen, and W. M. Lane, “The low-frequency array (LOFAR): opening a new window on the universe,” Planet. Space Sci. 52, 1543-1549 (2004).
[CrossRef]

Proc. Cambridge Philos. Soc. (2)

R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406-413 (1955).
[CrossRef]

R. Penrose, “On best approximate solutions of linear equations,” Proc. Cambridge Philos. Soc. 52, 17-19 (1955).
[CrossRef]

Proc. SPIE (2)

L. Milner and M. Parker, “A broadband 8-18 GHz 4-input 4-output Butler matrix,” Proc. SPIE 6414, 641406 (2007).
[CrossRef]

A. van Ardenne, A. Smolders, and G. Hampson, “Active adaptive antennas for radio astronomy: results of the R & D program towards the square kilometer array,” Proc. SPIE 4014, 420-433 (2000).
[CrossRef]

Taiwan. J. Math. (1)

P. G. Casazza, “The art of frame theory,” Taiwan. J. Math. 4, 129-201 (2000).

Other (3)

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

C. Lee, D. Leigh, K. Ryall, H. Miyashita, and K. Hirata, “Very fast subarray position calculation for minimizing sidelobes in sparse linear phased arrays,” in Proceedings of the European Conference on Antennas and Propagation (EuCAP, 2006), Vol. 606, pp. 80-87. Available online at http://www.merl.com/reports/docs/TR2006-022.pdf.

R. Braun, “The concept of the square kilometer array interferometer,” in Proceedings of High Sensitivity Radio Astronomy, N.Jackson and R.J.Davies, eds. (Cambridge U. Press, 1997), pp. 260-268.

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Figures (26)

Fig. 1
Fig. 1

Positions of the sythesized beams on the sky.

Fig. 2
Fig. 2

Linear gray-scale plots showing the magnitude, as a function of Ω ̂ x C and Ω ̂ y C , of the primary beam (left) and test source (right) used in the subsequent simulations.

Fig. 3
Fig. 3

Linear gray-scale plots showing the magnitude, as a function of Ω ̂ x C and Ω ̂ y C , of the synthesized beams of an array with 5 × 5 horns and a total size of 10 × 10 C wavelengths.

Fig. 4
Fig. 4

Linear gray-scale plots showing the magnitude, as a function of Ω ̂ x C and Ω ̂ y C , of the dual beams of an array with 5 × 5 horns and a total size of 10 × 10 C wavelengths.

Fig. 5
Fig. 5

Linear gray-scale plots showing the magnitude, as a function of Ω ̂ x C and Ω ̂ y C , of the eigenfields of an array with 5 × 5 horns and a total size of 10 × 10 C wavelengths.

Fig. 6
Fig. 6

Singular values in decibels of an array with 5 × 5 horns and a total size of 10 × 10 C wavelengths.

Fig. 7
Fig. 7

Linear gray-scale plots showing the magnitude of a reconstructed coherent source (left) and the intensity of a reconstructed incoherent source (right), as a function of Ω ̂ x C and Ω ̂ y C , for an array with 5 × 5 horns and a total size of 10 × 10 C wavelengths.

Fig. 8
Fig. 8

Linear gray-scale plots showing the magnitude, as a function of Ω ̂ x C and Ω ̂ y C , of the synthesized beams of an array with 5 × 5 horns, a total size of 10 × 10 C wavelengths, and uniform primary beams.

Fig. 9
Fig. 9

Linear gray-scale plots showing the magnitude, as a function of Ω ̂ x C and Ω ̂ y C , of the dual beams of an array with 5 × 5 horns, a total size of 10 × 10 C wavelengths, and uniform primary beams.

Fig. 10
Fig. 10

Linear gray-scale plots showing the magnitude, as a function of Ω ̂ x C and Ω ̂ y C , of the eigenfields of an array with 5 × 5 horns, a total size of 10 × 10 C wavelengths, and uniform primary beams.

Fig. 11
Fig. 11

Singular values in decibels of an array with 5 × 5 horns, a total size of 10 × 10 C wavelengths, and uniform primary beams.

Fig. 12
Fig. 12

Linear gray-scale plots showing the magnitude, as a function of Ω ̂ x C and Ω ̂ y C , of the synthesized beams of an array with 4 × 4 horns and a total size of 10 × 10 C wavelengths.

Fig. 13
Fig. 13

Singular values in decibels of an array with 4 × 4 horns and a total size of 10 × 10 C wavelengths.

Fig. 14
Fig. 14

Linear gray-scale plots showing the magnitude, as a function of Ω ̂ x C and Ω ̂ y C , of the synthesized beams of an array with 5 × 5 horns and a total size of 5 × 5 C wavelengths.

Fig. 15
Fig. 15

Linear gray-scale plots showing the magnitude, as a function of Ω ̂ x C and Ω ̂ y C , of the synthesized beams of an array with 5 × 5 horns and a total size of 5 × 5 C wavelengths.

Fig. 16
Fig. 16

Singular values in decibels of an array with 5 × 5 horns and a total size of 5 × 5 C wavelengths.

Fig. 17
Fig. 17

Linear gray-scale plots showing the magnitude of a reconstructed coherent source (left) and the intensity of a reconstructed incoherent source (right), as a function of Ω ̂ x C and Ω ̂ y C , for an array with 5 × 5 horns and a total size of 5 × 5 C wavelengths.

Fig. 18
Fig. 18

Linear gray-scale plots showing the magnitude, as a function of Ω ̂ x C and Ω ̂ y C , of the synthesized beams of an array with 5 × 5 horns and a total size of 20 × 20 C wavelengths.

Fig. 19
Fig. 19

Linear gray-scale plots showing the magnitude, as a function of Ω ̂ x C and Ω ̂ y C , of the dual beams of an array with 5 × 5 horns and a total size of 20 × 20 C wavelengths.

Fig. 20
Fig. 20

Singular values in decibels of an array with 5 × 5 horns and a total size of 20 × 20 C wavelengths.

Fig. 21
Fig. 21

Linear gray-scale plots showing the magnitude of a reconstructed coherent source (left) and the intensity of a reconstructed incoherent source (right), as a function of Ω ̂ x C and Ω ̂ y C , for an array with 5 × 5 horns and a total size of 20 × 20 C wavelengths.

Fig. 22
Fig. 22

Linear gray-scale plots showing the magnitude, as a function of Ω ̂ x C and Ω ̂ y C , of the synthesized beams of an array with 5 × 5 horns and a total size of 40 × 40 C wavelengths.

Fig. 23
Fig. 23

Linear gray-scale plots showing the magnitude, as a function of Ω ̂ x C and Ω ̂ y C , of the dual beams of an array with 5 × 5 horns and a total size of 40 × 40 C wavelengths.

Fig. 24
Fig. 24

Singular values in decibels of an array with 5 × 5 horns and a total size of 40 × 40 C wavelengths.

Fig. 25
Fig. 25

Linear gray-scale plots showing the magnitude of a reconstructed coherent source (left) and the intensity of a reconstructed incoherent source (right), as a function of Ω ̂ x C and Ω ̂ y C , for an array with 5 × 5 horns and a total size of 40 × 40 C wavelengths.

Fig. 26
Fig. 26

Linear gray-scale plots showing the magnitude of a reconstructed coherent source (left) and the intensity of a reconstructed incoherent source (right), as a function of Ω ̂ x C and Ω ̂ y C , for an array with 5 × 5 horns, a total size of 5 × 5 C wavelengths, and a dynamic range of 50 dB .

Equations (23)

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y m = A ϕ m * ( Ω ̂ ) E ( Ω ̂ ) d Ω ,
z p = m = 1 M S p m y m , p = 1 P ,
ψ p ( Ω ̂ ) = m = 1 M S p m * ϕ m ( Ω ̂ ) ,
z p = A ψ p * ( Ω ̂ ) E ( Ω ̂ ) d Ω .
z p z p * = A A ψ p * ( Ω ̂ 1 ) ψ p ( Ω ̂ 2 ) Γ ( Ω ̂ 1 , Ω ̂ 2 ) d Ω 1 d Ω 2 ,
Γ ( Ω ̂ 1 , Ω ̂ 2 ) E ( Ω ̂ 1 ) E * ( Ω ̂ 2 )
ψ p ( Ω ̂ ) = n N U p n * σ n V n ( Ω ̂ ) ,
A V n * ( Ω ̂ ) V n ( Ω ̂ ) d Ω = δ n n ,
p P U p n * U p n = δ n n .
ψ ̃ p ( Ω ̂ ) = n N U p n * σ n 1 V n ( Ω ̂ ) .
E ̃ ( Ω ̂ ) = p P z p ψ ̃ p ( Ω ̂ ) .
Γ ̃ ( Ω ̂ 1 , Ω ̂ 2 ) = p , p P z p z p * ψ ̃ p ( Ω ̂ ) ψ ̃ p ( Ω ̂ ) .
ϕ m ( Ω ̂ ) = e i b m Ω ̂ ϕ ( Ω ̂ ) ,
S p m = e i α ̂ p b m ,
ψ p ( Ω ̂ ) = m = 1 M e i b m ( α ̂ p Ω ̂ ) ϕ ( Ω ̂ ) ,
z p = A m = 1 M e i b m ( α ̂ p Ω ̂ ) ϕ * ( Ω ̂ ) E ( Ω ̂ ) d Ω ,
z p ϕ * ( α ̂ p ) E ( α ̂ p ) .
Ω ̂ C Ω ̂ ,
α ̂ C α ̂ ,
b 1 C b ,
ψ p ( Ω ̂ ) sinc ( B x ( Ω ̂ x α ̂ p x ) 2 ) sinc ( B y ( Ω ̂ y α ̂ p y ) 2 ) ϕ ( Ω ̂ ) .
b p = n p 1 Δ b 1 e 1 + n p 2 Δ b 2 e 2 ,
ψ p ( Ω ̂ ) ϕ ( Ω ̂ ) = ψ p ( Ω ̂ + m 1 2 π Δ b 1 e ̃ 1 + m 2 ( 2 π Δ b 2 ) e ̃ 2 ) ϕ ( Ω ̂ + m 1 2 π Δ b 1 e ̃ 1 + m 2 ( 2 π Δ b 2 ) e ̃ 2 )

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