Abstract

An optimization of passive interferometric circular arrays for redundant spacing calibration (RSC) is advanced to eliminate phase errors of the array system. The principle of RSC is presented to solve corresponded constraints for passive interferometric circular arrays. The simulated annealing algorithm (SAA) is introduced to settle the array optimization with a criterion of maximizing the distance between u-v points. The optimized circular arrays with element numbers of 8 to 16 antennas are laid out, and RSC is used for the optimized ten-element passive interferometric circular array.

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2009

P. Armand, J. Benoist, E. Bousquet, L. Delage, S. Olivier, and F. Reynaud, “Optimization of a one dimensional hypertelescope for a direct imaging in astronomy,” Eur. J. Oper. Res. 195(2), 519–527 (2009).
[CrossRef]

R. J. Eastwood, A. M. Johnson, and A. H. Greenaway, “Calculation and correction of piston phase aberration in synthesis imaging,” J. Opt. Soc. Am. A 26(1), 195–205 (2009).
[CrossRef]

2008

2006

2005

Y. Li, B. Braker, F. Schlottau, D. Gu, M. Colice, and K. H. Wagner, “Broadband RF imaging and spectrum analysis using spatial-spectral hole-burning in an inhomogeneously broadened absorber,” in Photonic Applications in Nonlinear Optics, Nanophotonics, and Microwave Photonics Proc. SPIE 5971, 597122 (2005).
[CrossRef]

I. Tcherniavski and M. Kahrizi, “Optimization of the optical sparse array configuration,” Opt. Eng. 44(10), 103201 (2005).
[CrossRef]

2004

Y. Su, R. D. Nan, B. Peng, N. Roddis, and J. F. Zhou, “Optimization of interferometric array configurations by “sieving” u - v points,” Astron. Astrophys. 414(1), 389–397 (2004).
[CrossRef]

2001

O. Guyon and F. Roddier, “Aperture rotation synthesis: Optimization of the (u, v)-plane coverage for a rotating phased array of telescopes,” The Astronomical Society of the Pacific 113(779), 98–104 (2001).
[CrossRef]

2000

L. Kogan, “Optimizing a large array configuration to minimize the sidelobes,” IEEE Trans. Antenn. Propag. 48(7), 1075–1078 (2000).
[CrossRef]

1999

1998

M. Peichl, H. Suess, M. Suess, and S. Kern, “Microwave imaging of the brightness temperature distribution of extended areas in the near and far field using two-dimensional aperture synthesis with high spatial resolution,” Radio Sci. 33(3), 781–801 (1998).
[CrossRef]

1996

1994

A. Lannes, E. Anterrieu, L. Koechlin, and ., “Concept of field-to-resolution ratio in aperture synthesis,” in Space Optics 1994: Earth Observation and Astronomy, M. G. Cerutti-Maori ed. Proc. SPIE 2209, 402–412 (1994).
[CrossRef]

1993

C. S. Ruf, “Numerical annealing of low-redundant linear arrays,” IEEE Trans. Antenn. Propag. 41(1), 85–90 (1993).
[CrossRef]

1990

A. H. Greenaway, “Self-calibrating dilute-aperture optics”, in Digital image synthesis and inverse optics Proc. SPIE 1351, 738–748 (1990).
[CrossRef]

A. Lannes, “Remarkable algebraic structures of phase closure imaging and their algorithmic implications in aperture synthesis,” J. Opt. Soc. Am. A 7(3), 500–512 (1990).
[CrossRef]

1989

T. J. Cornwell, “The applications of closure phase to astronomical imaging,” Science 245(4915), 263–269 (1989).
[CrossRef] [PubMed]

1988

C. S. Ruf, C. T. Swift, A. B. Tanner, and ., “Interferometric synthetic aperture microwave radiometry for the remote sensing of the earth,” IEEE Trans. On GRS 26, 597–611 (1988).

T. J. Cornwell, “A novel principle for optimization of the instantaneous Fourier plane coverage of correlation arrays,” IEEE Trans. Antenn. Propag. 36(8), 1165–1167 (1988).
[CrossRef]

1986

J. E. Baldwin, C. A. Haniff, C. D. Mackay, and P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320(6063), 595–597 (1986).
[CrossRef]

1983

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[CrossRef] [PubMed]

1982

J. E. Noordam and A. G. de Bruyn, “High dynamic range mapping of strong radio sources with application to 3C84,” Nature 299(5884), 597–600 (1982).
[CrossRef]

1973

1971

1968

A. T. Moffet, “Minimum-Redundancy Linear Arrays,” IEEE Trans. Antenn. Propag. 16(2), 172–175 (1968).
[CrossRef]

Anderton, R. N.

Anterrieu, E.

A. Lannes and E. Anterrieu, “Redundant spacing calibration: phase restoration methods,” J. Opt. Soc. Am. A 16(12), 2866–2879 (1999).
[CrossRef]

A. Lannes, E. Anterrieu, L. Koechlin, and ., “Concept of field-to-resolution ratio in aperture synthesis,” in Space Optics 1994: Earth Observation and Astronomy, M. G. Cerutti-Maori ed. Proc. SPIE 2209, 402–412 (1994).
[CrossRef]

Appleby, R.

Armand, P.

P. Armand, J. Benoist, E. Bousquet, L. Delage, S. Olivier, and F. Reynaud, “Optimization of a one dimensional hypertelescope for a direct imaging in astronomy,” Eur. J. Oper. Res. 195(2), 519–527 (2009).
[CrossRef]

Baldwin, J. E.

J. E. Baldwin, C. A. Haniff, C. D. Mackay, and P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320(6063), 595–597 (1986).
[CrossRef]

Bandyopadhyay, A.

Barat, R.

Benoist, J.

P. Armand, J. Benoist, E. Bousquet, L. Delage, S. Olivier, and F. Reynaud, “Optimization of a one dimensional hypertelescope for a direct imaging in astronomy,” Eur. J. Oper. Res. 195(2), 519–527 (2009).
[CrossRef]

Blanchard, P. M.

Bousquet, E.

P. Armand, J. Benoist, E. Bousquet, L. Delage, S. Olivier, and F. Reynaud, “Optimization of a one dimensional hypertelescope for a direct imaging in astronomy,” Eur. J. Oper. Res. 195(2), 519–527 (2009).
[CrossRef]

Braker, B.

Y. Li, B. Braker, F. Schlottau, D. Gu, M. Colice, and K. H. Wagner, “Broadband RF imaging and spectrum analysis using spatial-spectral hole-burning in an inhomogeneously broadened absorber,” in Photonic Applications in Nonlinear Optics, Nanophotonics, and Microwave Photonics Proc. SPIE 5971, 597122 (2005).
[CrossRef]

Cassaing, F.

Colice, M.

Y. Li, B. Braker, F. Schlottau, D. Gu, M. Colice, and K. H. Wagner, “Broadband RF imaging and spectrum analysis using spatial-spectral hole-burning in an inhomogeneously broadened absorber,” in Photonic Applications in Nonlinear Optics, Nanophotonics, and Microwave Photonics Proc. SPIE 5971, 597122 (2005).
[CrossRef]

Cornwell, T. J.

T. J. Cornwell, “The applications of closure phase to astronomical imaging,” Science 245(4915), 263–269 (1989).
[CrossRef] [PubMed]

T. J. Cornwell, “A novel principle for optimization of the instantaneous Fourier plane coverage of correlation arrays,” IEEE Trans. Antenn. Propag. 36(8), 1165–1167 (1988).
[CrossRef]

de Bruyn, A. G.

J. E. Noordam and A. G. de Bruyn, “High dynamic range mapping of strong radio sources with application to 3C84,” Nature 299(5884), 597–600 (1982).
[CrossRef]

Delage, L.

P. Armand, J. Benoist, E. Bousquet, L. Delage, S. Olivier, and F. Reynaud, “Optimization of a one dimensional hypertelescope for a direct imaging in astronomy,” Eur. J. Oper. Res. 195(2), 519–527 (2009).
[CrossRef]

Eastwood, R. J.

Federici, J. F.

Federici, M. D.

Gary, D.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[CrossRef] [PubMed]

Golay, M. J. E.

Goodman, J. W.

Greenaway, A. H.

Gu, D.

Y. Li, B. Braker, F. Schlottau, D. Gu, M. Colice, and K. H. Wagner, “Broadband RF imaging and spectrum analysis using spatial-spectral hole-burning in an inhomogeneously broadened absorber,” in Photonic Applications in Nonlinear Optics, Nanophotonics, and Microwave Photonics Proc. SPIE 5971, 597122 (2005).
[CrossRef]

Guyon, O.

O. Guyon and F. Roddier, “Aperture rotation synthesis: Optimization of the (u, v)-plane coverage for a rotating phased array of telescopes,” The Astronomical Society of the Pacific 113(779), 98–104 (2001).
[CrossRef]

Haniff, C. A.

J. E. Baldwin, C. A. Haniff, C. D. Mackay, and P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320(6063), 595–597 (1986).
[CrossRef]

Harvey, A. R.

Johnson, A. M.

Kahrizi, M.

Kern, S.

M. Peichl, H. Suess, M. Suess, and S. Kern, “Microwave imaging of the brightness temperature distribution of extended areas in the near and far field using two-dimensional aperture synthesis with high spatial resolution,” Radio Sci. 33(3), 781–801 (1998).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[CrossRef] [PubMed]

Koechlin, L.

A. Lannes, E. Anterrieu, L. Koechlin, and ., “Concept of field-to-resolution ratio in aperture synthesis,” in Space Optics 1994: Earth Observation and Astronomy, M. G. Cerutti-Maori ed. Proc. SPIE 2209, 402–412 (1994).
[CrossRef]

Kogan, L.

L. Kogan, “Optimizing a large array configuration to minimize the sidelobes,” IEEE Trans. Antenn. Propag. 48(7), 1075–1078 (2000).
[CrossRef]

Lannes, A.

Li, Y.

Y. Li, B. Braker, F. Schlottau, D. Gu, M. Colice, and K. H. Wagner, “Broadband RF imaging and spectrum analysis using spatial-spectral hole-burning in an inhomogeneously broadened absorber,” in Photonic Applications in Nonlinear Optics, Nanophotonics, and Microwave Photonics Proc. SPIE 5971, 597122 (2005).
[CrossRef]

Mackay, C. D.

J. E. Baldwin, C. A. Haniff, C. D. Mackay, and P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320(6063), 595–597 (1986).
[CrossRef]

Michalopoulou, Z.-H.

Moffet, A. T.

A. T. Moffet, “Minimum-Redundancy Linear Arrays,” IEEE Trans. Antenn. Propag. 16(2), 172–175 (1968).
[CrossRef]

Mugnier, L. M.

Nan, R. D.

Y. Su, R. D. Nan, B. Peng, N. Roddis, and J. F. Zhou, “Optimization of interferometric array configurations by “sieving” u - v points,” Astron. Astrophys. 414(1), 389–397 (2004).
[CrossRef]

Noordam, J. E.

J. E. Noordam and A. G. de Bruyn, “High dynamic range mapping of strong radio sources with application to 3C84,” Nature 299(5884), 597–600 (1982).
[CrossRef]

Olivier, S.

P. Armand, J. Benoist, E. Bousquet, L. Delage, S. Olivier, and F. Reynaud, “Optimization of a one dimensional hypertelescope for a direct imaging in astronomy,” Eur. J. Oper. Res. 195(2), 519–527 (2009).
[CrossRef]

Peichl, M.

M. Peichl, H. Suess, M. Suess, and S. Kern, “Microwave imaging of the brightness temperature distribution of extended areas in the near and far field using two-dimensional aperture synthesis with high spatial resolution,” Radio Sci. 33(3), 781–801 (1998).
[CrossRef]

Peng, B.

Y. Su, R. D. Nan, B. Peng, N. Roddis, and J. F. Zhou, “Optimization of interferometric array configurations by “sieving” u - v points,” Astron. Astrophys. 414(1), 389–397 (2004).
[CrossRef]

Reynaud, F.

P. Armand, J. Benoist, E. Bousquet, L. Delage, S. Olivier, and F. Reynaud, “Optimization of a one dimensional hypertelescope for a direct imaging in astronomy,” Eur. J. Oper. Res. 195(2), 519–527 (2009).
[CrossRef]

Rhodes, W. T.

Roddier, F.

O. Guyon and F. Roddier, “Aperture rotation synthesis: Optimization of the (u, v)-plane coverage for a rotating phased array of telescopes,” The Astronomical Society of the Pacific 113(779), 98–104 (2001).
[CrossRef]

Roddis, N.

Y. Su, R. D. Nan, B. Peng, N. Roddis, and J. F. Zhou, “Optimization of interferometric array configurations by “sieving” u - v points,” Astron. Astrophys. 414(1), 389–397 (2004).
[CrossRef]

Rousset, G.

Ruf, C. S.

C. S. Ruf, “Numerical annealing of low-redundant linear arrays,” IEEE Trans. Antenn. Propag. 41(1), 85–90 (1993).
[CrossRef]

C. S. Ruf, C. T. Swift, A. B. Tanner, and ., “Interferometric synthetic aperture microwave radiometry for the remote sensing of the earth,” IEEE Trans. On GRS 26, 597–611 (1988).

Schlottau, F.

Y. Li, B. Braker, F. Schlottau, D. Gu, M. Colice, and K. H. Wagner, “Broadband RF imaging and spectrum analysis using spatial-spectral hole-burning in an inhomogeneously broadened absorber,” in Photonic Applications in Nonlinear Optics, Nanophotonics, and Microwave Photonics Proc. SPIE 5971, 597122 (2005).
[CrossRef]

Schulkin, B.

Sengupta, A.

Stepanov, A.

Su, Y.

Y. Su, R. D. Nan, B. Peng, N. Roddis, and J. F. Zhou, “Optimization of interferometric array configurations by “sieving” u - v points,” Astron. Astrophys. 414(1), 389–397 (2004).
[CrossRef]

Suess, H.

M. Peichl, H. Suess, M. Suess, and S. Kern, “Microwave imaging of the brightness temperature distribution of extended areas in the near and far field using two-dimensional aperture synthesis with high spatial resolution,” Radio Sci. 33(3), 781–801 (1998).
[CrossRef]

Suess, M.

M. Peichl, H. Suess, M. Suess, and S. Kern, “Microwave imaging of the brightness temperature distribution of extended areas in the near and far field using two-dimensional aperture synthesis with high spatial resolution,” Radio Sci. 33(3), 781–801 (1998).
[CrossRef]

Swift, C. T.

C. S. Ruf, C. T. Swift, A. B. Tanner, and ., “Interferometric synthetic aperture microwave radiometry for the remote sensing of the earth,” IEEE Trans. On GRS 26, 597–611 (1988).

Tanner, A. B.

C. S. Ruf, C. T. Swift, A. B. Tanner, and ., “Interferometric synthetic aperture microwave radiometry for the remote sensing of the earth,” IEEE Trans. On GRS 26, 597–611 (1988).

Tcherniavski, I.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[CrossRef] [PubMed]

Wagner, K. H.

Y. Li, B. Braker, F. Schlottau, D. Gu, M. Colice, and K. H. Wagner, “Broadband RF imaging and spectrum analysis using spatial-spectral hole-burning in an inhomogeneously broadened absorber,” in Photonic Applications in Nonlinear Optics, Nanophotonics, and Microwave Photonics Proc. SPIE 5971, 597122 (2005).
[CrossRef]

Warner, P. J.

J. E. Baldwin, C. A. Haniff, C. D. Mackay, and P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320(6063), 595–597 (1986).
[CrossRef]

Webster, K.

Zhou, J. F.

Y. Su, R. D. Nan, B. Peng, N. Roddis, and J. F. Zhou, “Optimization of interferometric array configurations by “sieving” u - v points,” Astron. Astrophys. 414(1), 389–397 (2004).
[CrossRef]

Zimdars, D.

Appl. Opt.

Astron. Astrophys.

Y. Su, R. D. Nan, B. Peng, N. Roddis, and J. F. Zhou, “Optimization of interferometric array configurations by “sieving” u - v points,” Astron. Astrophys. 414(1), 389–397 (2004).
[CrossRef]

Eur. J. Oper. Res.

P. Armand, J. Benoist, E. Bousquet, L. Delage, S. Olivier, and F. Reynaud, “Optimization of a one dimensional hypertelescope for a direct imaging in astronomy,” Eur. J. Oper. Res. 195(2), 519–527 (2009).
[CrossRef]

IEEE Trans. Antenn. Propag.

C. S. Ruf, “Numerical annealing of low-redundant linear arrays,” IEEE Trans. Antenn. Propag. 41(1), 85–90 (1993).
[CrossRef]

L. Kogan, “Optimizing a large array configuration to minimize the sidelobes,” IEEE Trans. Antenn. Propag. 48(7), 1075–1078 (2000).
[CrossRef]

A. T. Moffet, “Minimum-Redundancy Linear Arrays,” IEEE Trans. Antenn. Propag. 16(2), 172–175 (1968).
[CrossRef]

T. J. Cornwell, “A novel principle for optimization of the instantaneous Fourier plane coverage of correlation arrays,” IEEE Trans. Antenn. Propag. 36(8), 1165–1167 (1988).
[CrossRef]

IEEE Trans. On GRS

C. S. Ruf, C. T. Swift, A. B. Tanner, and ., “Interferometric synthetic aperture microwave radiometry for the remote sensing of the earth,” IEEE Trans. On GRS 26, 597–611 (1988).

J. Lightwave Technol.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature

J. E. Noordam and A. G. de Bruyn, “High dynamic range mapping of strong radio sources with application to 3C84,” Nature 299(5884), 597–600 (1982).
[CrossRef]

J. E. Baldwin, C. A. Haniff, C. D. Mackay, and P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320(6063), 595–597 (1986).
[CrossRef]

Opt. Eng.

I. Tcherniavski and M. Kahrizi, “Optimization of the optical sparse array configuration,” Opt. Eng. 44(10), 103201 (2005).
[CrossRef]

Proc. SPIE

A. H. Greenaway, “Self-calibrating dilute-aperture optics”, in Digital image synthesis and inverse optics Proc. SPIE 1351, 738–748 (1990).
[CrossRef]

Y. Li, B. Braker, F. Schlottau, D. Gu, M. Colice, and K. H. Wagner, “Broadband RF imaging and spectrum analysis using spatial-spectral hole-burning in an inhomogeneously broadened absorber,” in Photonic Applications in Nonlinear Optics, Nanophotonics, and Microwave Photonics Proc. SPIE 5971, 597122 (2005).
[CrossRef]

A. Lannes, E. Anterrieu, L. Koechlin, and ., “Concept of field-to-resolution ratio in aperture synthesis,” in Space Optics 1994: Earth Observation and Astronomy, M. G. Cerutti-Maori ed. Proc. SPIE 2209, 402–412 (1994).
[CrossRef]

Radio Sci.

M. Peichl, H. Suess, M. Suess, and S. Kern, “Microwave imaging of the brightness temperature distribution of extended areas in the near and far field using two-dimensional aperture synthesis with high spatial resolution,” Radio Sci. 33(3), 781–801 (1998).
[CrossRef]

Science

T. J. Cornwell, “The applications of closure phase to astronomical imaging,” Science 245(4915), 263–269 (1989).
[CrossRef] [PubMed]

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[CrossRef] [PubMed]

The Astronomical Society of the Pacific

O. Guyon and F. Roddier, “Aperture rotation synthesis: Optimization of the (u, v)-plane coverage for a rotating phased array of telescopes,” The Astronomical Society of the Pacific 113(779), 98–104 (2001).
[CrossRef]

Other

P. J. M. Laarhoven, and E. H. L. Aarts, Simulated annealing: theory and applications. (D. Reidel Publishing Company, Dordrecht, (1987).

M. Faucherre, F. Merkle, and F. Vakili, “Beam combination in aperture synthesis from space: Field of view limitations and (u, v) plane coverage optimization” in New Technologies for Astronomy, J. P. Swings, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1130, 138–145 (1989).

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Figures (3)

Fig. 1
Fig. 1

The redundancy in a circular array. Point 1, 2, 3 and 4 are the elements’ position and O is the circle center.

Fig. 2
Fig. 2

u-v coverage of 8-element arrays (in first row) and 16-element arrays (in second row). (a) and (d) is with optimized arrays based on RSC; (b) and (e) is with optimized arrays without constraint from RSC; (c) and (f) is with uniformly distributed arrays.

Fig. 3
Fig. 3

Optimized circular array configuration based on RSC with ten elements and its u-v coverage figure.

Tables (3)

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Table 2 Optimized Positions of 8 ~16 elements array

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Table 3 Objective function values with different array configuration of 8 and 16 elements

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Table 1 The corresponded q with arrays of 8~16 elements

Equations (13)

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[ M p × 1 O ( r + 3 ) × 1 ] = [ I p × p B p × n R ( r + 3 ) × p G ( r + 3 ) × n ] × [ Φ p × 1 E n × 1 ] = A ( p + r + 3 ) × ( p + n ) × U ( p + n ) × 1
L = s p a n { A p + 1 , A p + 2 , , A p + n }
A ( p + r + 3 ) × ( p + n ) = [ A 1 , A 2 , , A p , A p + 1 , A p + 2 , , A p + n ]
U ( p + n ) × 1 = A ( p + n ) × ( p + n ) 1 × M ( p + n ) × 1
U ( p + n ) × 1 = [ Φ p × 1 E n × 1 ] = S ( p + n ) × ( p + n ) V ( p + n ) × ( p + r + 3 ) 1 D ( p + r + 3 ) × ( p + r + 3 ) T [ M p × 1 O ( r + 3 ) × 1 ]
A ( p + n ) × ( p + r + 3 ) = S ( p + n ) × ( p + n ) V ( p + n ) × ( p + r + 3 ) D ( p + r + 3 ) × ( p + r + 3 ) T
q = { ( n 1 ) / 2         if  n is even n / 2 1           if  n is odd
E ( ρ 1 , ρ 2 , ... , ρ n ) = i = 1 n j = 1 n log ( d i j )
d i j = { ( u i u j ) 2 + ( v i v j ) 2       ( u i u j ) 2 + ( v i v j ) 2  > ε   C c                                   ( u i u j ) 2 + ( v i v j ) 2   ε
[ M 45 × 1 O 10 × 1 ] = [ I 45 × 45 B 45 × 10 R 10 × 45 G 10 × 10 ] × [ Φ 45 × 1 E 10 × 1 ] = A 55 × 55 × U 55 × 1
[ M 45 × 1 O 10 × 1 ] = [ m 1 , 2 ( 1 )   m 1 , 3 ( 2 ) m 1 , 10 ( 9 )   m 2 , 3 ( 10 ) m 2 , 10 ( 18 ) m 9 , 10 ( 45 ) 45   0 0 10 ] T
= { 1 7 } = { { ( 1 , 2 ) ( 5 , 6 ) } , { ( 2 , 3 ) ( 6 , 7 ) } , { ( 3 , 4 ) ( 7 , 8 ) } , { ( 1 , 10 ) ( 5 , 10 ) } , { ( 2 , 10 ) ( 6 , 10 ) } , { ( 3 , 10 ) ( 7 , 10 ) } , { ( 4 , 10 ) ( 8 , 10 ) } }
[ Φ 45 × 1 E 10 × 1 ] = A 55 × 55 1 × [ M 45 × 1 O 10 × 1 ]

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