1. Main beam representation in non-regular arrays
Christophe Craeye1, David Gonzalez-Ovejero1, Eloy de Lera Acedo2, Nima Razavi Ghods2, Paul Alexander2
1Université catholique de Louvain, ICTEAM Institute
2University of Cambridge, Cavendish Laboratory
CALIM 2011 Workshop, Manchester, July 25-29, 2011
2d calibration Workshop, Algarve, Sep 26, 2011

2. 2 parts Preliminary: Patterns of apertures – a review
Next: analysis of aperture arrays with mutual coupling
Algarve meeting, 2011

3. a b r
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4. Fourier-Bessel series
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5. Zernike series Algarve meeting, 2011
NB: the Zernike function is a special case of the Jacobi Function
Algarve meeting, 2011

6. CALIM 2011
Zernike functions
Picture from Wikipedia

7. Radiation pattern
Hankel transform
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8. F.T. of Bessel
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9. FT of Zernike
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10. Sparse polynomial
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11. Context: SKA AA-lo CALIM 2011
Type of element
Bowtie
Spiral
Log-periodic
Non-regular: max effective area with min nb. elts w/o grating lobes.
Parameter
Specification
Low frequency
70 MHz
High frequency
450 MHz
Nyquist sampling frequency
100 MHz
Number of stations
50 => 250
Antennas per station
10.000
CALIM 2011

12. Problem statement
Goal: pattern representation for all modes of operation
at station level.
Too many antennas vs. number of calibration sources
Calibrate the main beam and first few sidelobes
Suppress far unwanted sources using
interferometric methods (open).
Compact representations of patterns, inspired from radiation from apertures, including effects of mutual coupling
Algarve meeting, 2011

13. Specific to SKA AAlo Fairly circular stations (hexagonal would be OK)
Relatively dense
Weak amplitude tapering – some space tapering
Irregular => all EEP’s very different
Even positions are not 100 % reliable (within a few cm)
Correlation matrix not available
Nb. of beam coefficients << nb. Antennas
Restrict to main beam and first few sidelobes
Even assuming identical EEP’s and find 1 amplitude coefficient per antenna is way too many coefficients
Algarve meeting, 2011

14. Outline Limits of traditional coupling correction Array factorization
Array factors: series representations
Reduction through projection
Scanning
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15. Embedded element pattern
CALIM 2011
Embedded element pattern
Z L
Impedance isolated elt
Array impedance matrix
To get voltages in uncoupled case: multiply voltage vector to the left by matrix

16. Embedded element pattern
CALIM 2011
Embedded element pattern
Z L
After correction, we are back to original problem, with (zoomable, shiftable) array factor
Gupta, I., and A. Ksienski (1983), Effect of mutual coupling on the performance of adaptive arrays, IEEE Trans. Antennas Propag., 31(5), 785–791.

17. Isolated Embedded Corrected
CALIM 2011
Mutual coupling correction
Half-wave dipole
Isolated Embedded Corrected

18. Isolated Embedded Corrected
CALIM 2011
Mutual coupling correction
Bowtie antenna
l=1.2 m, l=3.5 m
Isolated Embedded Corrected

19. Mutual coupling correction
CALIM 2011
Mutual coupling correction
Bowtie antenna
l=1.2 m, l=1.5 m
Isolated Embedded Corrected
SINGLE MODE assumption not valid for l < l/2 ~

20. Multiple-mode approach
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Multiple-mode approach
Macro Basis Functions (Suter & Mosig, MOTL, 2000,
cf. also Vecchi, Mittra, Maaskant,…)
a b g d
MBF 1
MBF 2
MBF 3
MBF 4

21. Array factorisation … CALIM 2011 Z L s n Antenna index C111 C211 C311
MBF 1
MBF 2
Coefficients for IDENTICAL current distribution …

22. Array factorisation … CALIM 2011 Z L s n Antenna index C111 C211 C311
MBF 1
MBF 2
Coefficients for IDENTICAL current distribution …

23. Example array CALIM 2011 - Array radius = 30λ0.
- Number of elements = 1000. λ0
- Distance to ground plane = λ0/4.
- No dielectric. λ0
Z = 200Ω

24. Random arrangement CALIM 2011 Random configuration ~ 35 dB E-plane
H-plane

25. Quasi-random arrangement
CALIM 2011
Quasi-random arrangement

26. Radius of Influence H-plane ICEAA 2011 1000 elements 100 elements

27. CALIM 2011
Aperture sampling (1)

28. (several definitions possible)
CALIM 2011
Aperture sampling (2)
Define a local density
(several definitions possible)

29. CALIM 2011
Aperture scanning (1)

30. CALIM 2011
Aperture scanning (2)

31. Patterns versus size of array
CALIM 2011
Patterns versus size of array

32. Coherent & incoherent regimes
CALIM 2011
Coherent & incoherent regimes ~ ~
0.3
Number of sidelobes in “coherent” regime

33. Aperture field representation
CALIM 2011
Aperture field representation b a

34. Pattern representation
CALIM 2011
Pattern representation
Angle from broadside

35. Polynomial decomposition
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Polynomial decomposition

36. Fourier-Bessel decomposition
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Fourier-Bessel decomposition =

37. Fourier-Bessel decomposition
CALIM 2011
Fourier-Bessel decomposition =
A. Aghasi, H. Amindavar, E.L. Miller and J. Rashed-Mohassel,
“Flat-top footprint pattern synthesis through the design of arbitrarily
planar-shaped apertures,” IEEE Trans. Antennas Propagat.,
Vol. 58, no.8, pp , Aug

38. Zernike-Bessel decomposition
CALIM 2011
Zernike-Bessel decomposition
Y. Rahmat-Samii and V. Galindo-Israel, “Shaped reflector antenna
analysis using the Jacobi-Bessel series,” IEEE Trans.
Antennas Propagat., Vol. 28, no.4, pp , Jul

39. CALIM 2011
Polynomial
Fourier-Bessel
Zernike-Bessel
Fast functions
Good 1st order
weaker at low orders
weak direct convergence
fast direct convergence

40. CALIM 2011
Array factor
with apodization

41. CALIM 2011
Array factor
with apodization

42. Apodization function w(r) extracted
CALIM 2011
Apodization function w(r) extracted

43. Approximate array factor extracted
CALIM 2011
Approximate array factor extracted
20 % error on amplitudes
l/4 error on positions (at 300 MHz)

44. Residual error with Z-B approach
CALIM 2011
Residual error with Z-B approach

45. Residual error with Z-B approach
CALIM 2011
Residual error with Z-B approach

46. Residual error with Z-B approach
CALIM 2011
Residual error with Z-B approach

47. Residual error with Z-B approach
CALIM 2011
Residual error with Z-B approach

48. Residual error with Z-B approach
CALIM 2011
Residual error with Z-B approach

49. Residual error with Z-B approach
CALIM 2011
Residual error with Z-B approach

50. Residual error with Z-B approach
CALIM 2011
Residual error with Z-B approach

51. Residual error with Z-B approach
CALIM 2011
Residual error with Z-B approach

52. Residual error with Z-B approach
CALIM 2011
Residual error with Z-B approach

53. CALIM 2011
Density function
The number of terms tells the “resolution” with which density is observed

54. Array of wideband dipoles
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55. AF convergence Over just main beam
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56. AF convergence Main beam + 1st sidelobe
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57. Project on 1, 2, 3 MBF patterns at most
AF convergence
Project on 1, 2, 3 MBF patterns at most
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58. Pattern projections Power radiated by MBF pattern 0
NB: can be projected on more than 1 (orthogonal) MBFs

59. Full pattern
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60. Error w/o MC
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61. Error after pattern projection
Only 1 pattern used here (pattern of “primary”)
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62. l =0.0
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63. l =0.1
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64. l =0.2
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65. l =0.3
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66. l =0.4
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67. l =0.5
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68. l =0.6
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69. Variation of maximum
Real
Imaginary
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70. Conclusion Representation based on array factorization
Projection of patterns of MBF on 1 or 2 of them
Representation of array factors with functions use for apertures (done here with 1 array factor)
Array factor slowly varying when shifted upon scanning
(to be confirmed with more elements and other elt types)
Algarve meeting, 2011