main beam representation in non-regular arrays christophe craeye 1, david gonzalez- ovejero 1, eloy...
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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, 20112d calibration Workshop, Algarve, Sep 26, 2011
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2 parts
Next: analysis of aperture arrays with mutual coupling
Preliminary: Patterns of apertures – a review
Algarve meeting, 2011
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br
Algarve meeting, 2011
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Fourier-Bessel series
Algarve meeting, 2011
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Zernike series
NB: the Zernike function is a special case of the Jacobi Function
Algarve meeting, 2011
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Zernike functions
Picture from Wikipedia
CALIM 2011
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Radiation pattern
Hankel transform
Algarve meeting, 2011
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F.T. of Bessel
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FT of Zernike
Algarve meeting, 2011
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Sparse polynomial
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Context: SKA AA-lo
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
Type of element
Bowtie
Spiral
Log-periodic
CALIM 2011
Non-regular: max effective area with min nb. elts w/o grating lobes.
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Problem statement
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
Goal: pattern representation for all modes of operation at station level.
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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
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Outline
1.Limits of traditional coupling correction
2.Array factorization
3.Array factors: series representations
4.Reduction through projection
5.Scanning
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Z L
To get voltages in uncoupled case: multiply voltage vector to the left by matrix CALIM 2011
Embedded element pattern
Impedance isolated elt
Array impedance matrix
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Z L
After correction, we are back to original problem, with (zoomable, shiftable) array factor
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Embedded element pattern
Gupta, I., and A. Ksienski (1983), Effect of mutual coupling on the performance of adaptive arrays, IEEE Trans. Antennas Propag., 31(5), 785–791.
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Mutual coupling correction
Half-wave dipole
Isolated Embedded Corrected
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CALIM 2011
Mutual coupling correction
Bowtie antenna
Isolated Embedded Corrected
l=1.2 m, =3.5 m
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CALIM 2011
Mutual coupling correction
Bowtie antenna
Isolated Embedded Corrected
l=1.2 m, =1.5 m
SINGLE MODE assumption not valid for l < /2 ~
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Macro Basis Functions (Suter & Mosig, MOTL, 2000,
MBF 1 MBF 2
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Multiple-mode approach
cf. also Vecchi, Mittra, Maaskant,…)
MBF 3 MBF 4
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C111 C211 C311
C122 C222 C322
MBF 1
MBF 2
Array factorisation
Coefficients for IDENTICAL current distribution
…
Z L
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Antenna index
s n
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C111 C211 C311
C122 C222 C322
MBF 1
MBF 2
Array factorisation
Coefficients for IDENTICAL current distribution
…
Z L
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Antenna index
s n
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23
Example array
- Distance to ground plane = λ0/4.
- No dielectric.
- Array radius = 30λ0. - Number of elements = 1000.
λ0
λ0
Z = 200Ω
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24-50 0 50
-60
-50
-40
-30
-20
-10
(º)
dBW
EEP's meanSingle element patternError
E-plane
~ 35 dB
H-plane
-50 0 50
-60
-50
-40
-30
-20
-10
(º)
dBW
EEP's meanSingle element patternError
2
sin10 ),(),(log10 glemean EEe
Random configuration
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Random arrangement
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25-50 0 50
-60
-50
-40
-30
-20
-10
(º)
dBW
EEP's meanSingle element patternError
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Quasi-random arrangement
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Radius of Influence
10 elements100 elements1000 elements
H-plane
22
22
10
),(),(max
),(),(),(),(
log10
),(
fullv
fullh
iv
fullv
ih
fullh
i
EE
EEEE
e
ICEAA 2011
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Aperture sampling (1)
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Aperture sampling (2)
Define a local density (several definitions possible)
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Aperture scanning (1)
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Aperture scanning (2)
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Patterns versus size of array
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Coherent & incoherent regimes
Number of sidelobes in “coherent” regime
~0.3
~
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Aperture field representation
b
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Pattern representation
Angle from broadside
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Polynomial decomposition
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Fourier-Bessel decomposition
=
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Fourier-Bessel decomposition
=
A. Aghasi, H. Amindavar, E.L. Miller and J. Rashed-Mohassel,“Flat-top footprint pattern synthesis through the design of arbitrarilyplanar-shaped apertures,” IEEE Trans. Antennas Propagat.,Vol. 58, no.8, pp. 2539-2551, Aug. 2010.
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Zernike-Bessel decomposition
Y. Rahmat-Samii and V. Galindo-Israel, “Shaped reflector antennaanalysis using the Jacobi-Bessel series,” IEEE Trans.Antennas Propagat., Vol. 28, no.4, pp. 425-435, Jul. 1980.
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Polynomial Fourier-Bessel
Zernike-Bessel
Fast functions
Good 1st order
Good 1st order
weaker at low orders
weak direct convergence
fast direct convergence
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Array factor
CALIM 2011
with apodization
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Array factor
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with apodization
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Apodization function w(r) extracted
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Approximate array factor extracted
20 % error on amplitudes/4 error on positions (at 300 MHz)
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Residual error with Z-B approach
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Residual error with Z-B approach
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Residual error with Z-B approach
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Residual error with Z-B approach
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Residual error with Z-B approach
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Residual error with Z-B approach
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Residual error with Z-B approach
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Residual error with Z-B approach
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CALIM 2011
Residual error with Z-B approach
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Density function
The number of terms tells the “resolution” with which density is observed CALIM 2011
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Array of wideband dipoles
Algarve meeting, 2011
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AF convergence
Over just main beam
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Main beam + 1st sidelobe
AF convergence
Algarve meeting, 2011
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AF convergence
Project on 1, 2, 3 MBF patterns at most
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Pattern projections
Power radiated by MBF pattern 0
NB: can be projected on more than 1 (orthogonal) MBFs
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Full pattern
Algarve meeting, 2011
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Error w/o MC
Algarve meeting, 2011
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Error after pattern projection
Only 1 pattern used here (pattern of “primary”)Algarve meeting, 2011
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l =0.0
Algarve meeting, 2011
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l =0.1
Algarve meeting, 2011
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l =0.2
Algarve meeting, 2011
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l =0.3
Algarve meeting, 2011
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l =0.4
Algarve meeting, 2011
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l =0.5
Algarve meeting, 2011
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l =0.6
Algarve meeting, 2011
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Variation of maximum
Real Imaginary
Algarve meeting, 2011
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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