Innovative methodologies for the synthesis of large array antennas for communications and space applications.

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International Dotorate Shool in Information and

Communiation Tehnologies

DISI - University of Trento

Innovative methodologies for

the synthesis of large array antennas

for ommuniations and spae

appliations

Federio Caramania

Advisor:

Prof. Andrea Massa

University of Trento

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Modernommuniation andspae systemssuh assatelliteommuniation devies, radars, SAR

and radio astronomy interferometers are realized with large antenna arrays sine this kind of

radiatingsystems are able to generate radiation patterns with high diretivity and resolution. In

suh a framework onventional arrays with uniform inter-element spaing ould be not

satisfa-toryin termsof osts anddimensions. Aninteresting alternativeistoredue the array elements

obtaining the so alled thinned arrays. Large isophori thinned arrays have been exploited

be-ause of theiradvantages in terms of weight, onsumption, hardware omplexity, and osts over

theirlled ounterparts.

Unfortunately,thinninglargearraysredues the ontrol ofthe peaksidelobe level (PSL)anddoes

notgive automatially optimalspatial frequeny overage for orrelators. Firstof all the stateof

theartmethodologiesusedtooveromesuhlimitations,e.g.,randomandalgorithmiapproahes,

dynami programming and stohasti optimization algorithms suh as geneti algorithms,

sim-ulated annealing or partile swarm optimizers, are analyzed and desribed in the introdution.

Suessively, innovative guidelines for thesynthesis of largeradiating systems are proposed, and

disussed in order to point out advantages and limitations. In partiular, the following spei

issuesare addressed in this work:

1. Anewlassofanalytialretangularthinnedarrayswithlowpeaksidelobe level(PSL).The

proposed synthesis tehnique exploits binary sequenes derived from MFarland dierene

sets to design thinned layouts on a lattie of

P

×

P

(

P

+ 2)

positions for any prime

P

. The pattern features of the arising massively-thinned arrangements haraterized by only

P

×

(

P

+ 1)

ativeelementsaredisussedandtheresultsofanextensivenumerialanalysis are presented to assess advantagesand limitationsof the MFarland-based arrays.

2. A set of tehniques ispresented that is based on the exploitationof low orrelationAlmost

Dierene Sets (ADSs) sequenes to design orrelator arrays for radioastronomy

applia-tions. In partiular three approahes are disussed with dierent objetives and

perfor-manes. ADS-based analytial designs, GA-optimized arrangements, and PSO optimized

arrays are presented and applied to the synthesis of open-ended

Y

and

Cross

array ongurationstomaximize theoverage

u

−

v

ortominimizethe peaksidelobe level(PSL). Representative numerial resultsare illustrated topointout the features and performanes

of the proposed approahes, and to assess their eetiveness in omparison with

state-of-the-art design methodologies, as well. The presented analysis indiates that the proposed

approahes overome existing PSO-based orrelator arrays in terms of PSL ontrol (e.g.,

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im-3. A geneti algorithm (GA)-enhaned almostdierene set (ADS)-based methodology to

de-sign thinned planar arrays with low-peak sidelobe levels (PSLs). The method allows to

overome the limitationsof the standard ADS approah in termsof exibility and

perfor-mane. The numerial validation, arried out inthe far-eld andfornarrow-bandsignals,

points out that with aordable omputational eorts it is possible to design planar array

arrangements that outperform standard ADS-based designsas well asstandard GA design

approahes.

Keywords

[Planar Arrays, Thinned Arrays, Correlator Array Antenna, Dierene Sets, MFarland

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1 Struture of the Thesis 21

2 Introdution 23

2.1 Contextand Bakground . . . 23

3 State of the Art 29 3.1 Arrays for Communiation and Radio Astronomy - Introdution to the State-of-the-Art . . . 29

3.2 Random Arrays [6℄ . . . 35

3.2.1 Introdution . . . 35

3.2.2 Linear Random Array . . . 35

3.2.3 Planar Array . . . 39

3.2.4 Comparison between the Peak Sidelobe of the Random Array and AlgorithmiallyDesigned Aperiodi Arrays [12℄ . . . 40

3.2.4.1 Database . . . 40

3.2.4.2 Results . . . 40

3.3 StatistialRemoval (RandomRemoval)[4℄ . . . 41

3.3.2 Analysis of StatistialDensity-Tapered Arrays . . . 41

3.4 OptimizationAlgorithms Approah . . . 48

3.4.2 Geneti Algorithm[18℄ . . . 48

3.4.2.1 GA - Algorithm. . . 48

3.4.2.2 GA Optimizationfor the design of Linear Array . . . 50

3.4.2.3 GA Optimizationfor the design of Planar Array . . . 51

3.4.3 Simulated Annealing[38℄ . . . 52

3.4.3.1 SA - Algorithm . . . 53

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3.4.4.1 ACO - Algorithm . . . 55

3.4.4.2 Optimization Proedurefor Linear and Planar Arrays . . 57

3.5 Dierenes Sets [5℄[19℄ . . . 59

3.5.2 Notation . . . 60

3.5.3 Dierene Sets . . . 61

3.5.4 Dierene Sets, Autoorrelations,and Linear Arrays. . . 63

3.5.5 Linear Isophori Arrays . . . 63

3.5.6 Expeted Power Pattern of a Linear IsophoriArray . . . 66

3.5.7 Extension to PlanarArrays . . . 68

3.6 Almost Dierene Sets [22℄ . . . 72

3.6.2 Almost Dierene Sets -Denitions and Properties . . . 73

3.6.3 ADS-Based Linear Arrays - MathematialFormulation . . . 76

3.6.3.1 ADS-Based Innite Arrays. . . 76

3.6.3.2 ADS-Based FiniteArrays . . . 78

3.7 Basi Theory of Interferometry for Radio Astronomy [8℄[9℄[30℄[31℄ . . . 82

3.7.2 Problem Denition . . . 82

3.7.3 The U-V Coverage . . . 84

3.7.4 The Earth-Rotation Eet . . . 85

3.7.5 The Synthesized Beam . . . 86

3.7.6 Image Retrieval . . . 87

3.7.7 Basi Two-Elements Interferometer . . . 88

3.7.8 ComparisonbetweenConventionalSumArraysandCorrelatorArrays 91 3.8 PartileSwarm Optimization forRadio Astronomy [31℄ . . . 94

3.8.2 A Numerial Example: A UniformY-Shaped Array . . . 94

3.8.3 Optimization of Y-Shaped Arrays . . . 95

3.8.3.1 The Partile Swarm Optimization Tehnique . . . 95

3.8.3.2 Optimizing the U-V Coverage . . . 96

3.8.3.3 Optimizing the Synthesized Beam. . . 98

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4.1 Introdution . . . 103

4.2 Mathematial Formulation . . . 104

4.3 MFarland Array Synthesis Proedure . . . 106

4.4 Numerial Results and Disussion . . . 108

4.5 Appendix . . . 117

5 Hybrid ADS-Based Tehniques for Radio Astronomy Array Design 119 5.1 Introdution . . . 119

5.2 Mathematial Formulationand Problem Statement . . . 121

5.2.1 Problem A -Optimization of

S

T

(

u, v

)

. . . 123

5.2.2 Problem B - Optimization of the

u

−

v

Coverage in Snapshot Ob-servation . . . 123

5.2.3 Problem C - Optimization of the

u

−

v

Coverage in Traking Ob-servation . . . 123

5.3 ADS-Based Y-Shaped Correlator Arrays . . . 124

5.4 ADS-Based HybridMethodologies . . . 130

6 Hybrid Almost Dierene Set (ADS)-based Geneti Algorithm (GA) Method for Planar Array Thinning 143 6.1 Introdution . . . 143

6.2 Problem statement and mathematial formulation . . . 145

6.2.1 Problem I - PSLminimisation inarray synthesis . . . 150

6.2.2 Problem II - extension of the range of ADS appliability in array synthesis . . . 151

6.2.3 ProblemIII-denitionofageneralpurposeADSonstrution teh-nique forarray synthesis . . . 151

6.3 Numerial analysis . . . 152

6.3.1 Appliation toProblemI . . . 152

6.3.1.1 Array arrangement

P

×

Q

= 7

×

7

. . . 153

P

×

Q

= 11

×

11

. . . 155

P

×

Q

= 17

×

17

. . . 157

P

×

Q

= 23

×

23

. . . 159

6.3.1.5 Summary . . . 161

6.3.2 Appliation toProblemII . . . 162

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6.3.2.2

P

×

Q

= 6

×

6

Array Conguration . . . 163

6.3.2.3

P

×

Q

= 8

×

8

6.3.2.4

P

×

Q

= 12

×

12

6.3.2.5

P

×

Q

= 16

×

16

6.3.2.6 Summary . . . 171

6.3.2.7 ADSGA methodompared with [18℄ . . . 173

6.3.2.8

P

×

Q

= 10

×

20

6.3.2.9

P

×

Q

= 40

×

40

6.3.2.10 Summary . . . 178

6.3.3 Appliationto ProblemIII . . . 179

6.3.3.1

(36

,

32

,

28

,

23)

-ADS . . . 180

6.3.3.2

(60

,

6

,

0

,

29)

-ADS . . . 182

6.3.3.3

(64

,

59

,

54

,

43)

-ADS . . . 184

6.3.3.4

(100

,

5

,

0

,

79)

-ADS . . . 186

6.3.3.5

(144

,

137

,

130

,

101)

-ADS . . . 188

6.3.3.6

(192

,

184

,

176

,

135)

-ADS . . . 190

6.3.3.7

(196

,

7

,

0

,

153)

-ADS . . . 192

6.3.3.8

(225

,

8

,

0

,

168)

-ADS . . . 194

6.3.3.9 Summary . . . 196

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Table I. Linear Thinned Arrays based on Almost Dierene Sets - Examples of

ADSs and their desriptivefuntions.

TableII.RadioAstronomy-RadialElementDisplaementofOptimizedY-Shaped

Arrays (Unit: Kilometers).

Table III. MFarland Retangular Arrays (

P

≤

29

) - Features and Performane Indexes.

Table IV.

ADS

D

1

,

D

2

,

D

3

,and

D

4

and desriptive parameters.

Table V. Numerial results -

Y

ADS

Arrays [

P

= 18

,

Q

= 9

,

Λ = 4

,

r

= 13

℄

-Comparison of

ADS

-based

Y

-shaped arrays and some representative designs(bold

numbers identify optimized quantities).

Table V. Numerial results -

Y

ADS

Arrays [

P

= 18

,

Q

= 9

,

Λ = 4

,

r

= 13

℄

-Comparison of

ADS

-based

Y

-shaped arrays and some representative designs(bold

Table VI. Numerial results - Comparison of optimized

Y

-shaped arrays (bold

TableVII.Numerialresults -ComparisonamongoptimizedALMAonguration

(boldnumbers identify optimized quantities).

TableVIII.Numerialresults- ComparisonofoptimizedCross arrays (bold

num-bers identify optimized quantities).

Table IX. Properties of the ADS sequenes

TableX.ProblemI- PSLminimisationin array synthesis: Summaryof theresults

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standard GA methodology, we obtain a redution of PSL that goes from

1

.

73

[dB℄

to

0

.

24

[dB℄.

TableXI.ProblemI-PSLminimisationinarraysynthesis: Summaryoftheresults

obtained. Comparing the results of the new proposed ADSGA tehnique with the

standard GA methodology,the SPSO,the HSPSO [25℄ and DS[21℄,weobtain that

ADSGA is able to improvePSL performane also when

N

ˆ

6

=

N

ADS

.

Table XII. Problem I- PSL minimisation in array synthesis: Summary of the

results obtained. Comparing the results of the new proposed ADSGA tehnique

with the standard GA methodology, the SPSO, the HSPSO [25℄ and DS [21℄, we

obtain that ADSGA is able toimprovePSL performane alsowhen

N

ˆ

6

=

N

ADS

.

Table XIII. Problem II- extension of the range of ADS appliability: Summary

of the results obtained about thinning fator

ν

. Comparing the results of the new

proposed ADSGA tehnique with the standard GAmethodology and [18℄.

Table XIV. Problem II- extension of the range of ADS appliability: Summary of

the results obtained about mainlobedimension

BW

. Comparingthe results ofthe

new proposed ADSGA tehnique with the standard GA methodology and [18℄.

Table XV. Problem II- extension of the range of ADS appliability: Summary of

the resultsobtained. Comparingthe resultsofthenew proposedADSGA tehnique

with the standard GA methodology and [18℄. We obtain with ADSGA a redution

of PSL inboth examples.

Table XVI. Problem III- GA designedADS onstrution tehnique: Properties of

the ADS sequenes that have been designed by the proposed GA-basedtehniques.

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Figure 1. Introdution- Exampleof large reetor antenna.

Figure 2. Introdution -Example of onventional lledarray with path radiating

elements.

Figure 3. Introdution- Exampleof large irular thinnedarray.

Figure4. Introdution-TheVLA,anarrayof27elements,eaha25-mparaboloid,

is a Y-shaped array havingthree equiangular linear arms of 21km.

Figure 5. Introdution-

(

a

)

and

(

b

)

are examplesof radio maps.

Figure 6. Random Arrays - Examples of (a) a

50

×

50

elements square random array and (b) a

100

×

100

elements square randomarray.

Figure 7. RandomArrays- Patternof 70-wavelengthrandomarrayof 30isotropi

elements.

Figure 8. Random Arrays - Probabilisti estimator of peak sidelobe of random

array.

N

is the is number of array elements,

P SL/ML

is power ratio of peak

sidelobe to main lobe,

β

is probability or ondene level that nosidelobe exeeds

ordinate,

L

is array length,

λ

is wavelength,

θ

0

isbeam steeringangle.

Figure 8. StatistialArrays- Geometryof an

M

by

M

element arrayarranged on

a square grid. Angular oordinates are also shown.

Figure 9. Statistial Arrays - In

(

a

)

the solid urve is the omputed radiation

pattern of a statistially designed array naturally thinned using as a model the

30

dB

Taylor irular aperture distribution whose pattern is shown by the dashed

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Figure 10. StatistialArrays - In

(

a

)

there isthe omputed radiationpatternof a

statistially designed array using as a model the

25

dB

Taylor design but with

ap-proximately90perentofthe elementsremoved. In

(

b

)

theorrespondingloations

of the elements.

Figure 11. Thinned Arrays with Geneti Algorithms - Flow hart of a geneti

algorithm.

Figure 12. ThinnedArrayswithSimulatedAnnealing-Flow-hartofthe

optimiza-tion proedure.

Figure 13. Isophori Array -

(

a

)

Isophori linear array power pattern. Number

of elements

= 32

. Aperture size

= 62

half-wavelengths.

(

b

)

Random linear array

power pattern. Number of elements

= 32

. Aperturesize

= 62

half-wavelengths.

Figure 14. Isophori Array - Expeted power pattern of isophori array with

V

= 63

and

K

= 32

.

Figure 15. Isophori Array - Expeted power pattern of isophori planar array

with

V

=

V

x

V

y

= 15

×

17

half-waves and

K

= 128

elements. this exat pattern is realizablewith spatial hopping. Notepattern oorat

10 log

10

ρ

=

−

24

dB.

Figure 16. Linear Thinned Arrays based on AlmostDierene Sets-

Autoorrela-tion funtion

C

ADS

S

(

z

)

of

D

1

and

D

2

in Table I.

Figure 17. Linear Thinned Arrays based on Almost Dierene Sets - Normalized

P P

(

u

)

derived from the ADS derived from the ADS

D

4

(

D

4

=

D

(

σ

)

4

k

σ

=0

) and its

yli shifts

D

(

σ

)

4

(

σ

= 17

,

σ

= 24

). Number of elements:

N

= 45

-Aperture size:

22

λ

.

Figure 18. Linear Thinned Arrays based onAlmost Dierene Sets - Comparative

Assessment - Plots of the PSL bounds of the ADS-based nite arrays and of the

estimator of the PSL of the random arrays (RND - random array, RNL - random

lattie array) when

ν

= 0

.

489

versus

(

a

)

the array dimension,

N

, and

(

c

)

the index

η

. Normalizedgenerated from

D

opt

4

and estimated PSL values of the orresponding

random sequenes

(

b

)

.

Figure 19. Radio Astronomy - Coneptual sketh of a radio astronomial

mea-surement using a orrelator antenna array. The brightness distribution

I

(

l, m

)

in

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of its visibility

V

(

u, v

)

in the spatial frequeny domain. The sampling points are

determinedbyautoorrelatingthearrayonguration

f

(

x, y

)

inthespatialdomain.

Figure 20. Radio Astronomy - Relationship among antenna quantities for an

in-oherent eld.

Figure 21. Radio Astronomy - The geometry of an interferometer. The baseline

intersetsthe elestialsphereat

B

,whihhas delination

d

andtheloalhourangle

h

. The soure is at point

S

, with oordinates

δ

and

H

. The projetion of the

baseline on the intersetion of the plane

SOB

and a plane tangent to the elestial

sphere at

S

is

D

cos

θ

.

Figure 22. Radio Astronomy - Basiorrelator interferometersystem.

Figure 23. Radio Astronomy- Comparisonbetween the signal proessing shemes

of a 2-element:

(

a

)

sum array and

(

b

)

orrelator array.

Figure24. RadioAstronomy-

(

a

)

Originalsoureimagewiththevisibilityspeied

by the Gaussian funtion in (3.124).

(

b

)

Image retrieved by the uniform Y-shaped

array shown in Fig. 4

(

a

)

.

Figure 25. Radio Astronomy -

(

a

)

Conguration of the optimized 27-element

Y-shaped array (

Y

1

) for the maximum snapshot

u

−

v

overage.

(

b

)

Snapshot

u

−

v

overage of Y has 558s sampled grids.

(

a

)

Y-shaped array (

Y

2

) for the maximum traking

u

−

v

overage.

(

b

)

Traking

u

−

v

overage of

Y

2

has alling ratio of

86

.

5%

, asdened in (3.139).

(

a

)

Y-shaped array (

Y

2

)for thelowest SLL.

(

b

)

Synthesized beam ofY has apeakSLL of

−

20

.

3

dB.

Figure 28. Radio Astronomy - Comparison between a uniform array, a power-law

array(

α

= 1

.

7

)and theoptimizedarray

Y

3

forSLLsin8-hourtrakingobservations

with dierent souredelinations.

(

a

)

Original image of a Gaussian soure and

re-trieved images by

(

b

)

array

Y

1

,

(

c

)

array

Y

2

and

(

d

)

array

Y

3

. The best image is

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and (b)the assoiated (two-level) autoorrelationfuntion (

P

= 3

).

Figure 31. GA-Based MFarland Synthesis - Plots of (a) the PSL values of the

wholesetofMFarlandarraysand(b)evolutionofthePSLoftheGAsolutionduring

the iterative(

i

being theiterationindex) samplingofthe MFarlandsolutionspae.

Figure 32. MFarland Retangular Arrays - Behaviour of

∆(

η

)

versus

P

when

η

∈ {

0

.

7

,

0

.

8

,

0

.

9

,

1

.

0

}

.

Figure 33. GA-Based MFarland Synthesis - Evolution of the PSL of the GA

solutionduringtheiterative(

i

beingtheiterationindex)samplingoftheMFarland

solution spae when(a)

P

= 5

and (b)

P

= 7

.

Figure 34. GA-Based MFarland Synthesis - OptimalMFarland layouts (a), ()

and the orrespondingpower patterns (b), (d) when

P

= 5

(a),(b) and

P

= 7

(),

(d).

Figure 35. GA-Based MFarlandSynthesis -OptimalMFarlandlayouts (a)

P

=

11

and (b)

P

= 13

.

Figure 36. GA-Based MFarland Synthesis - Power patterns of the optimal

M-Farland layouts dedued for (a)

P

= 11

and (b)

P

= 13

.

Figure 37. Comparison with StandardGA-Thinned Retangular Arrays -Optimal

layout (a) and the orresponding power pattern (b) obtained by GA when

P

= 7

,

Q

= 63

and

K

= 56

.

Figure 38.

Y

-shaped Arrays [

P

= 18

,

Q

= 9

,

Λ = 4

,

r

= 13

, Equal-unequal

arms℄ - Plots of the arrangement (a) and assoiated

S

T

(

u, v

)

(b) for the array

Y

3

[31℄; optimal ADS geometry with equal () or unequal (e) arms, and assoiated

synthesized beams (d),(f).

Figure 39.

Y

ADS

Arrays [

P

= 18

,

Q

= 9

,

Λ = 4

,

r

= 13

, Equal-unequal arms℄

-Behavior of optimal(a) PSL, () , and (e)

ν

versus evaluated shift for ADS-based

Y

arrays, and omparison with referene designs from [31℄. Plots of (b) PSL, (d)

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Figure 40.

Y

ADS

Arrays [

P

= 18

,

Q

= 9

,

Λ = 4

,

r

= 13

- Behavior of (a)

B

versus PSL, (b)

ν

versus PSL, and ()

ν

versus for all

Y

ADS

arrays derived from

D

1

, and omparison with referene designsfrom [31℄.

Figure 41.

Y

ADS

Arrays [

P

= 18

,

Q

= 9

,

Λ = 4

,

r

= 13

-Behaviorof for

Ξ

all

Y

ADS

arrays derived from

D

1

,and omparison with referene designs from [31℄.

Figure 42. Problem A [Equal-unequal arms,

N

= 27

℄ - Synthesis results for the

GA and ADSGA approahes: (a)behavior of the optimal PSLversus the iteration

number

i

, and omparison with referene designs from [31℄, (b) optimal

Y

ADSGA

array arrangement, and () assoiated synthesized pattern.

Figure 43. Problem B [Equal-unequal arms,

N

= 27

RNDPSO and ADSPSO approahes: (a) optimal

Y

ADSP SO

array arrangement and

(b) assoiated

u

−

v

overage funtion.

Figure 44. Problem C [Equal-unequal arms,

N

= 27

RNDPSO and ADSPSO approahes: (a) optimal array arrangement and (b)

asso-iated traking

u

−

v

overage funtion.

Figure 45. Problem A [Equal-unequal arms,

N

= 27

RNDPSO and ADSPSO approahes: (a) Behavior of the optimal PSL versus the

iteration number

i

, and omparison with referene designs from [31℄, (b) optimal

Y

ADSP SO

array arrangement, and ()assoiated synthesized pattern.

Figure 46. Problem A -Behavior of the optimal PSLversus the iterationnumber

i

for the RNDGA, ADSGA, RNDPSO, and ADSPSO approahes for (a)

N

= 132

(equal and unequalarms) and (b)

N

= 270

(equal arms).

Figure 47. ALMA - Problem A [Equal-unequal arms,

N

= 63

℄ - Synthesis

re-sults for theADSPSO approah: (a)optimalarray arrangementand (b)assoiated

S

T

(

u, v

)

.

Figure 48. Cross arrays - Problem A [Equal-unequal arms,

N

= 60

℄ - Synthesis

results for the RNDGA, ADSGA, RNDPSO and ADSPSO approahes: (a)

behav-ior of the optimal PSL versus the iteration number

i

, (b) optimal ADSPSO array

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Figure 49. Example from [23℄ of Planar Array based on

D

opt

3

- ADS . Number of

elements:

P

×

Q

= 7

×

11

. Plots of the PSL bounds versus

η

=

t

P Q

−

1

(

P Q

= 77

,

ν

= 0

.

4805

)(a). Plotofthe normalized arrayfator (b) generatedfrom

D

opt

3

- ADS

array arrangement() (ourtesyfrom [23℄).

Figure50. Numerialvalidation-ProblemI-PSLminimisationinarraysynthesis:

Behaviour of the optimal tness value,

P SL

(

i

)

, against the number of iteration

number,

i

.

Power patterns

|

W

(

u, v

)

|

2

forADSGA (a) and forGA (b)approahes. () and(d)

show the orresponding array arrangements with ADSGA and GA-based methods,

respetively.

Figure 52. Numerialvalidation - Problem I - PSL minimisation in array

synthe-sis: Behaviourofthe optimaltness value,

P SL

(

i

)

, againstthe numberofiteration

number,

i

.

Power patterns

|

W

(

u, v

)

|

2

respetively.

P SL

(

i

)

number,

i

.

Power patterns

|

W

(

u, v

)

|

2

respetively.

P SL

(

i

)

number,

i

.

Power patterns

|

W

(

u, v

)

|

2

(17)

Graphial omparison of the PSL of dierent array ongurations (the side

P

on

the horizontal axis) for ADSGA an GA methodologies. We an observe that the

PSL improvement of the ADSGA method redues ompared with standard GA as

the dimension of the array inreases.

Figure 59. Numerial validation - Problem II - extension of the range of ADS

appliability: Behaviouroftheoptimaltnessvalue,

P SL

(

i

)

,againstthe numberof

iteration number,

i

.

appliability: Powerpatterns

|

W

(

u, v

)

|

2

forADSGA(a)andforGA(b)approahes.

()and(d)showtheorrespondingarrayarrangementswithADSGAandGA-based

methods, respetively.

P SL

(

i

)

iteration number,

i

.

|

W

(

u, v

)

|

2

P SL

(

i

)

iteration number,

i

.

|

W

(

u, v

)

|

2

P SL

(

i

)

iteration number,

i

.

|

W

(

u, v

)

|

2

(18)

Figure67. Numerialvalidation -ProblemII -PSL minimisationin array

synthe-sis: Graphial omparison of the PSL of dierent array ongurations (the side

P

onthe horizontalaxis)for ADSGAanGA methodologies. Wean observe thatthe

PSL improvement of the ADSGA method redues ompared with standard GA as

the dimension of the array inreases.

appliability: Graphial omparison of the PSL against the iteration

i

of ADSGA,

GAandHaupt[18℄approahesalongthetwomaindiretions

φ

= 0

°(a)and

φ

= 90

°

(b). Sliesoftheamplitudepatternobtainedafteroptimizationproedurealongthe

twomain diretions

φ

= 0

° () and

φ

= 90

° (d).

|

W

(

u, v

)

|

2

appliability: Graphial omparison of the PSL against the iteration

i

of ADSGA,

GAandHaupt[18℄approahesalongthetwomaindiretions

φ

= 0

°(a)and

φ

= 90

°

(b). Sliesoftheamplitudepatternobtainedafteroptimizationproedurealongthe

twomain diretions

φ

= 0

° () and

φ

= 90

° (d).

|

W

(

u, v

)

|

2

Figure 72. Numerial validation - Problem III - GA designed ADS onstrution

tehnique: (a)Behaviourofthe optimaltness,

F

P OP

,againsttheiterationnumber

i

, (b) Three-level autoorrelation funtion of the onvergene

(36

,

32

,

28

,

23)

-ADS

arrangement,() Final2D ADS layout.

tehnique: Plotof thepowerpatternassoiatedto theantenna arraybuilt withthe

(19)

tehnique: (a) Behaviour of the optimal tness,

F

P OP

, against the iteration num-ber

i

, (b)Three-level autoorrelationfuntion of the onvergene

(60

,

6

,

0

,

29)

-ADS

tehnique: Plotofthe powerpatternassoiatedtothe antenna arraybuilt withthe

(60

,

6

,

0

,

29)

-ADS arrangement.

tehnique: (a)Behaviouroftheoptimaltness,

F

P OP

i

(64

,

59

,

54

,

43)

-ADS

(64

,

59

,

54

,

43)

-ADS arrangement.

F

P OP

i

(100

,

5

,

0

,

79)

-ADS

(100

,

5

,

0

,

79)

-ADS arrangement.

F

P OP

i

(144

,

137

,

130

,

101)

-ADS arrangement,() Final 2D ADS layout.

(144

,

137

,

130

,

101)

-ADS arrangement.

F

P OP

i

(192

,

184

,

176

,

135)

(20)

(192

,

184

,

176

,

135)

-ADS arrangement.

F

P OP

i

(196

,

7

,

0

,

153)

-ADS

(196

,

7

,

0

,

153)

-ADS arrangement.

F

P OP

i

(225

,

8

,

0

,

168)

-ADS

(21)

Struture of the Thesis

This hapter desribeshow the Thesis is organized.

First of all, Chapter 2 presents an overview of the Thesis, pointing out the ontext

ofthe thinnedantenna arrays forommuniation and radioastronomy,the problemthat

have been onsidered and abrief analysis of the solutionsproposed inliterature.

Chapter3desribessomeofthemostsigniativeandrelevanttehniquesinthe

state-of-the-art,to design thinnedarrays for ommuniation and radio astronomy. The aimis

to present the basis and bakground of the work arried out in this Thesis during the

researh ativity developed during my PhD and make a omparative assessment with

methodologiesproposed inthis Thesis.

Chapter4dealswithanewlassofretangularthinnedarrayswithlowandontrolled

peak side lobe level (PSL). These arrays are based on MFarland Dierene Sets (DSs),

that likewise two-dimensional DSs exhibit a two-level autoorrelation funtion, and on

a suitable synthesis proedure based on Geneti Algorithm (GA) optimization. GA has

been exploited due to the extremely large number of admissible MFarland sequenes.

This methodology allows to obtain massively-thinned arrangements with a retangular

shape that exhibit dierenttotal main beam widths(TMBWs) inazimuth and elevation

and lowPSL.

Chapter5. In thishapter,inordertodesignorrelatorarraysforradioastronomy

ap-pliationsaset of hybridtehniquesis introduedand numerialvalidated. Thesehybrid

(22)

dioastronomyAlmost DiereneSets (ADSs)sequenes, thatare haraterizedby almost

ideal autoorrelation properties, are exploited with stohasti optimization algorithms

suh as genetialgorithms(GAs)and partile swarm optimizers (PSOs).

Chapter6proposesaGA-enhanedADStehnique(ADSGA)forthe synthesisof

pla-narantennaarraysforommuniationappliationsandshowsthatthedevelopedADSGA

hybridtehnique allows to overome the limitationsrelated tothe use of ADS sequenes

and obtainoptimal performane.

Chapter 7 onludes the Thesis. In partiular the main results are summarized, the

openproblems andfuture researhdiretionsinthe exploitationofthe proposed

(23)

Introdution

2.1 Context and Bakground

Thereare manypratialways toexploitantennaarrays. Antennaarraysarewidely used

bothiniviland militaryappliations. Inommuniationandbroadastengineeringthey

are used in TLC systems suh as TV and radio transmitters, for example in AM or FM

broadastradiostationstoenhanesignal. Arraysarelargelyutilizedinwarships,airraft

radarsystemsandmissilere-ontrolsystems. Otherusesaresonar,weatherresearhand

biomedial (e.g. radiotherapy) appliations [1℄[2℄. Another partiular kind of framework

whereantennaarraysan beveryusefulisrepresented byspaeappliations,e.g. satellite

ommuniationsystemsandradioastronomy. Theradiatingsystemsoftheseappliations

have some ommon requirements: high resolution (the term "resolution" is used in the

sense of Rayleigh and is proportional to the beamwidth), high gain, low sidelobe level

[3℄and, for radio astronomy appliations, optimal overage in spatial frequeny domain.

In ommuniation and spae appliations, steerable reetors are one of the most useful

kinds of antennas. Reetors have a diameter that an be equal up to

100

m but they

(24)

For these reasons, the attention has turned to very large arrays with a number of

radiating elements from two up to hundreds or thousands. For onventionally designed

arrays where all elements are uniformly spaed an upper limit exists to the spaing, if

the grating lobes are not permitted to appear in the visible region. In this ase we

have traditional lled arrays that have anelement plaed in every loationof a uniform

lattie with half-wavelength spaingbetween the lattie points. As a result the required

numberofelements,beingproportionaltotheaperturedimensioninwavelength, beomes

astronomially large if a beamwidth onthe order of minute of ar is desired[3℄.

Figure 2. Introdution-Exampleofonventional lledarraywithpathradiatingelements.

Mostofthereentinvestigationsonarrayswithnon-uniformlyspaedelementsshowed

the possibility ofreduing the numberof radiatingelementsand optimizingthe design of

arrays. An unequally spaed, thinned array may beused to:

(25)

appearane of gratinglobes

3. ahievedesirable radiationpatterns withoutamplitude taper aross the aperture.

Thinninganarraymeansturningosomeelementsinauniformlyspaedorperiodiarray

to reate a desired amplitude density aross the aperture [4℄. An element onneted to

the feednetwork ison, and an elementonneted to amathed ordummyload is o.

When thinned arrays have fewer than half of the elements of their lled ounterparts,

they are alled massively thinned arrays. In this researh proposalwe are not interested

in amplitude tapering tehniques sine these methodologies have a higher omplexity

and ost [5℄. We have to remember that thinning is normally aompanied by loss of

sidelobe ontrol, for this reason, thinned arrays are synthesized in aording to one or

more optimization riteria. For example, optimization of the beam pattern means to

ahievethe minimum PSL inthe entire visiblerange orthe maximum gain [3℄[4℄[6℄.

Figure3. Introdution-Exampleoflargeirularthinnedarray.

In this senario large thinned arrays allow us to obtain the following advantages:

better performane with respet to reetor antenna, inreased operational robustness,

implementationostsavingandmoreprogrammatiexibility. Eahofthesetopisis

dis-ussedfurtherinthefollowingparagraphs. Forlargerantennas,thebeamwidthnaturally

isnarrower. Asaresult,antenna-pointingerrorbeomesmoreritial. Tostaywithinthe

main beam and inur minimalloss, antenna pointing has to be more preise. Yet this is

diulttoahievefor largerstrutures. Withanarrayongurationofsmallerantennas,

(26)

degradation,anoptimalgainan beahieved. Arrayingalsoallowsaninreaseineetive

aperturebeyond the present apabilityfor supporting amissionatatime ofneed. Inthe

past, the Voyager Mission relied on arraying to inrease its data return during Uranus

and Neptuneenounters inthelate1980s. TheGalileoMission providesanotherexample

in whih arraying was used to inrease the siene data return by a fator of 3. (When

ombinedwith otherimprovements,suh asabetteroding sheme,amoreeient data

ompressionandaredutionofsystemnoisetemperature,atotalimprovementofafator

of 10was atually realized)[7℄. Arrayingan inrease system operability. Firstly, higher

resoure utilization an be ahieved. In the ase of an array the set an be partitioned

intomany subsetssupportingdierentmissionssimultaneously,everyonetailored

aord-ing to the link requirements. So doing, resoure utilization an be enhaned. Seondly,

arraying oers high system availability and maintenane exibility. Let us suppose an

array built with 10 perent spare elements. The regular preventive maintenane an be

done on a rotating basis while allowing the system to be fully funtional at all times.

Thirdly, the ost of spareomponents would besmaller. Instead of having tosupply the

system with100 perent sparesin orderto makeit fully funtionalaroundthe lok,the

array oersan option of furnishingspares at a frationallevel. Equally important isthe

operationalrobustness againstfailures. Witha singleresoure, failuretendsto bringthe

systemdown. Withanarray,failureinanarrayelementdegradessystemperformanebut

doesnot result in a servie shutdown [7℄. In partiular, thinnedarrays an be projeted

to have a ertain amount of redundant radiatingelements in order to guaranteeing PSL

ontrolin presene of one or multiple failures.

A ost saving is realized fromthe fat that smaller antennas, beause of their weight

andsize,areeasiertobuildandmove. Thefabriationproessanbeautomatedtoredue

theost. Itisoftenapproximatedthattheantennaonstrutionostisproportionaltothe

antennavolume. Thereeptionapability,however, isproportionaltotheantennasurfae

area. Note, however, thatantenna onstrutionisonlyapart ofthe overalllife yleost

for the entire system deployment and operations. To alulate the atual savings, one

needs to aount for the ost of the extra eletronis required atmultiple array elements

and the ost related tothe inrease insystem omplexity[7℄. One of the most important

qualityof thinnedarrays istheredued numberofantennas: withfewradiatingelements

weankeepunderontrolthePSL,satisfyingthetehnialrequirements,andalsoinrease

the ost saving. Arraying oers a programmati exibility beause additional elements

an be inrementally added to inrease the total aperture at the time of mission need.

(27)

tothe existing failitiesthat supportongoingoperations.

In onlusion thinnedarrays seem to be suitable tosatisfy the previous requirements

typialof ommuniation systems and improveperformane.

Radio interferometers and synthesis arrays, whih are basially ensembles of two

el-ement interferometers, are used to make measurements of the ne angular detail in the

deepradioemissionfromthe sky. Theangularresolutionofsingleradioantennasis

insuf-ient formany astronomialpurposes. Pratialonsiderationslimitthe resolution toa

fewtensofarseonds. Forexample,thebeamwidthofa

100

mdiameterantennaat

7

mm

wavelength is approximately17arse. In the optial range the diration limitof large

telesopes (diameter-8 m) is about 0.015 arse, but the angular resolution ahievable

fromthe ground by onventional tehniques is limitedto about one arseby turbulene

in the troposphere. For progress in astronomy it is partiularly important to measure

the positionsof radio soureswith suientauray toallowidentiationwith objets

deteted in the optial and other parts of the eletromagneti spetrum. It is also very

importanttobeabletomeasure parameterssuhasintensity,polarization,andfrequeny

spetrum with similar angular resolution in both the radio and optial domains. Radio

interferometryenables suh studies to be made. Preise measurement of the angular

po-sitions of stars and other osmi objets is the onern of astrometry. This inludesthe

study of the small hanges in elestial positions attributable to the parallax introdued

by the earth's orbitalmotion, aswell asthose resulting fromthe intrinsi motions ofthe

objets. Suh measurements are an essential step in the establishment of the distane

sale of the universe. Radio tehniques provide anauray of the order of arse orless

forthe relativepositions of objets losely spaed inangle.

Compared with ommuniation systems, to obtain optimal performane, namely a

high-sensitiveand high-resolutionmeasurementof radio soures, a uniforminter-element

spaingoftheradiatingelementsisnotthebestsolution. WeneednotonlyalowPSLbut

alsooverage ofspatial frequeny domainasuniform aspossible. If the spatialdomainis

notuniformlysampledtheradiosoureisnotorretlyreoveredandspuriousartifatsare

presents. Anon-uniformlyspaedorrelatorarray,asshownin[8℄[9℄,givesthepossibility

(28)

arrayhavingthreeequiangularlineararmsof21km.

(a) (b)

Figure 5. Introdution-

(

a

)

and

(

b

)

areexamplesofradiomapsobtainedwithradioastronomy

(29)

State of the Art

3.1 Arrays for Communiation and Radio Astronomy

-Introdution to the State-of-the-Art

In the framework of arrays for ommuniations, radar and spae appliations, Skolnik

proposed one of the rst examples of thinning large antenna arrays. In his work [4℄

he desribes statistially designed density-tapered arrays. With the usual method for

designingdiretiveantennaswithlowsidelobes,thereeived(orradiated)energyisgreater

atthe entre than at the edges [4℄. The idea proposed in [4℄is the following: the density

of elements loated within the aperture is made proportional to the amplitude of the

apertureilluminationofonventionallledarrays(designedwithTaylororDolphmethods

[10℄[11℄). In otherwords,the signalateahelementofthe array isofequalamplitude but

the spaing between adjaent elements diers. The seletion of the element loations is

performed statistiallyby utilizingthe amplitude illuminationas the probability density

funtion for speifying the loation of elements (for this reason it is also alled spae

taper) [4℄. Statistially designed density-tapered arrays are useful when the number of

elementsislargeandwhenitisnotpratialtoemployanamplitudetapertoahievelow

sidelobes. Adensitytaperhasadvantagesoveranamplitudetaperinertainappliations.

Transmitting arrays, for example, with individual power ampliers at eah element are

easier to design and to build and more eient to operate if eah amplier delivers full

ratedpower[4℄. The density-tapered array permitsthe system designer toemploy

equal-power ampliersat eah element and stillahieve low sidelobes. Reeiving antennas an

also benet from density tapering. In onlusion, this tehnique is to be onsidered for

thedesignoflargearray antennaswhere goodsidelobesareimportantand whereitisnot

(30)

randomly loated. In arandom array, the loationof eahradiatingelementis arandom

variabledrawnfromapopulationdesribedbyaprobabilitydensityfuntion(e.g.uniform

pdf). Sinean a-prioridesription of arandom array an onlybe given statistially, itis

logial to seek an estimatorof the peak sidelobe in terms of a probability or ondene

level that the predited value will not be exeeded. Steinberg obtained a probabilisti

estimator of the peak sidelobe of uniform random array with equally weighted elements.

This theoretialresultwastestedby measurementofthepeaksidelobeofseveral hundred

Monte Carlo omputer-simulated randomarrays [6℄.

During the1960'smany thinningalgorithmswere reated. The methodologiestothin

arrays fall into the following ategories: algorithmi-spei aperiodi designs;

random-element loations hosen at random; random removal-holes hosen at random; dynami

programming-quasi-trial-and- error. In [6℄, Steinberg ompared algorithmi design of

thinned aperiodi arrays tested by omputer simulations with random arrays. The

dis-tributionis ompared tothat of a set of 170 randomarrays [6℄[6℄. Both distributionsare

found tobenearly lognormal withthe same average and medianvalues. They markedly

dier intheir standard deviations. However, the standard deviation ofthe randomarray

distribution isapproximatelyhalfthatof the algorithmigroup. The authorshowed that

algorithmiallythinned arrays rarely oer enough ontrolof the far radiation patternto

be superior to random arrays. Furthermore the ompatness of the random distribution

almost guarantees against seletion of a random array with atastrophially large peak

sidelobes. The onlyproedurethat givessuperior performane isdynami

programming-quasi trial-and-errormethodof sidelobeontrol, a highlyonstrained approah. More in

detail, the rst element is loated at random. The seond loation is that whih gives

the best ombination. The third loation is that whih gives the best trio based on the

xed loationsof the rst two elements, et. Despite dynami random design method is

ommonlyonsideredasthereferene strategyforthesynthesisofthinnedarraysbeause

of its simpliity (does not require any omputational proedure), its good performane

(quasi trial-and-errormethod gives aslight improvement) and exibility [6℄[6℄.

Inordertoimproveperformane ofthinnedarraysrespettorandomarrays,dierent

ways have been used. The rst is based on the use of optimization algorithms and the

seond on partiularkind of ombinatorialsequenes.

Assuming,likeinthe previousmethodologies,thenumberof radiatorsisanite

num-ber and eah radiator an have two values on and o (thinning may also be thought of

(31)

bit), the number of possible ombinations, where

Q

is the number of array elements, is

2

Q

. Thinning a large array for low sidelobes involves heking a rather large number of

possibilitiesin order to nd the best thinned aperture. Exhaustive heking of all

possi-bleelementombinationsisonly pratialforsmall arrays [13℄. Optimizationalgorithms

represent an alternative to exhaustive searh. Most optimization methods (inluding

down-hillsimplex, Powell's method,and onjugate gradient)are not well suitedfor

thin-ning arrays. They an only optimize a few ontinuous variables and get stuk in loal

minima[14℄. Also,thesemethodsweredeveloped forontinuous parameters,whereasthe

array-thinningprobleminvolvesdisreteparameters. Thedynamiprogrammingmethod

an optimize a large parameter set (many elements), but it is also vulnerable to loal

minima[15℄. Simulatedannealing and geneti algorithms(GA) [14℄[16℄[17℄ are

optimiza-tionmethodsthat are well suitedfor thinningarrays. There isno limittothe number of

variables that an be optimized. Although quiteslow, these algorithmsan handle very

large arrays. These methods are global sine they have random omponents that test

for solutions outside the urrent minimum, while the algorithm onverges. The global

natureofthe algorithmsandthelakofderivativeinformationauseaveryslowonverge

ompared to other non-global methods. If the array is symmetri, then the number of

possibilitiesis substantially smaller and the GA onverges faster.

In [18℄,Haupt presents anexample of thinningstrategybased onGenetiAlgorithms

(GAs)used tond a thinnedarray that produesthe lowest PSLallowingus toimprove

the performane of large arrays. A Geneti Algorithm is a global method for

optimiza-tioninspiredbythe NaturalSeletionPriniplewhosemainoneptsareompetitionand

adaptability [14℄. The paper [18℄ shows that the on/o struture of the thinned array

(linearorplanar)isodiedintothehromosomesof theGA. Afterenodingthe

param-etersin binary strings alled genes,GA performs the geneti operationsof reprodution,

rossover,naturalseletion,and mutationtoarriveattheoptimumsolution. Duringeah

iteration,the trial solution provides by the GA is given in input to the tness funtion.

Thetnessisdenedin[18℄asthePSLandthe purposeofthe GAistondout thearray

ongurationminimizing this funtion. A geneti algorithm an be used to numerially

optimizeboth linear and planar arrays and arrives at better thinning ongurations for

arrays than previous optimization attempts orstatistial attempts. Previous methods of

array thinning used statistialmethods may fail to produe an optimum thinning while

the geneti algorithm searhes in a smart way for the best thinning that produes low

sidelobes[18℄.

(32)

om-thinnedlinearandplanararraysthatshowswell-behavedsidelobesinspiteofthethinning.

The Geneti searh algorithmsan obtain better performane but this method isnot

appropriate for very large orvery highly thinnedarrays and the improvements that this

methodologyoersare diulttopredita-priori. Ratherthan usingasearhalgorithm,

the approahin[5℄[19℄attaksdiretlythesidelobeontrolproblembyapplyingthe

prop-erties of Dierene Sets (DSs) [2℄, tothe plaement of antennaelementswithin aregular

lattie. In partiular Leeper uses the lass of Cyli-Dierene Sets (CDS) sequenes as

funtion that desribes the position of ative elements in arrays [20℄. The property that

makesCDS aneetive presription for the designof the thinnedarray isthat the

auto-orrelation of CDS(and generallyallkindof DSs)is atwo-valued funtion. Itis possible

todemonstrate[5℄thatthiskindofautoorrelationallowsontrollingthe PSLofanarray

built with CDS geometry. The CDS method guarantees more eetive suboptimal array

synthesis in terms of PSL with respet to random elements plaement. 2D-CDSs have

similar autoorrelationproperty of 1DCDSs [2℄[5℄[19℄[20℄.

ThedeterministiplaementsofDSreate anisophoriarray(isophori means

uni-form weight) withattendant uniformity ofspatial overage. The uniformity onsistently

produes, with no searhing required, a redution in PSL when ompared torandom

el-ement plaement. More speially, inany linear array of aperturehalf-wavelengths, the

Nyquist samplingtheorem shows that the array power patternan be ompletely

deter-mined from uniformly spaed samples of the pattern. In an isophori array, the

even-numbered samples will neessarily be loked to a onstant value less than

1

/K

times

the main-beam peak, where

K

is the number of elements in the thinned array. While

the odd-numbered samples are not so onstrained, the net eet is to produe patterns

with muh lower PSL than are typial with ut-and-try random plaement. Obviously,

isophori arrays an be planaras welllinear [5℄.

In[21℄,Kopilovihsuggestsanothermethodforsynthesizingaplanaraperiodithinned

arrayantennawithalowpeaksidelobelevel. InsteadofusingthepreviousCDS,Kopilovih

shows theimplementationofombinatorialonstrutionsallednon-Cylidierenesets.

Themostimportantlassofthenon-Cyli2D-DSsisrepresentedbythesetsofHadamard

type (HDSs). In the same way of the previous Leeper method, Kopilovih uses the fat

that when the elements of an equi-amplitude array antenna are arranged aording to

a DS law, its pattern takes onstant value in the net of uniformly loated spae points

in the sidelobe region, and this value is less than

1

/K

, where

K

is the ative element

(33)

lengths, the desribed method omits suh a onstraint. Based on suh sets, retangular

andsquare aperiodiroughly half-lledarrayantennas an bebuilt. Kopilovihuses this

strategy to obtain square array antennas, with the element number in the array up to

300.

The denition of binary sequenes of length with suitable autoorrelation properties,

for whih DSs are not available, has been arefully investigated in information theory

and ombinatorialmathematis. It has been found that it isoften possible to determine

sequenes witha three-levelautoorrelationfuntionby taking intoaount theso-alled

almost dierene sets (ADSs) [22℄[23℄. ADSs are a researh topi of great interest in

ombinatorial theory with important appliations in ryptography and oding theory.

Moreover, althoughADS generation tehniques are stillsubjet of researh, large

olle-tions of these sets are already available. In suh a framework, the whole lass of ADSs

seem to be a good andidate for enlarging the set of admissible analyti ongurations

with respet to the DS ase. From this viewpoint, ADSs allow to obtain low PSL and

preditableresults in avery eetive. Withrespet to DSs, ADSs have the advantage of

havinga larger set of admissiblesequenes [22℄[23℄.

Finally, the last approah desribed to improve large arrays performane is based on

mergingthe ombinatorialand stohasti methods inorder totakeadvantage from their

goodharateristisand to ompensate for their drawbaks [5℄.

One of the rst attempts to exploit this idea was developed by Caorsi et al. [24℄.

The ripples formation aused by CDS ould be orreted in some way by GA searh

apabilities,whilethe uniform spatialoverageof CDS-optimizedarrays ouldbe helpful

tospeed up the onvergene of the genetiproedure. One possible way of implementing

thisapproah istoonsider CDSbased arrays asa-prioriknowledge tobeinserted inthe

geneti searh proess in order to improve its eieny. To this end, the steps aimed

at transferring good CDS-derived shemata into the GA population are the following.

At the initialization step, the GA population is omposed by a seleted CDS

D

0

and

V

yli shifts of the

D

0

dierene set, while the remaining hromosomes of the initial

population are randomly mutated yli shifts. Moreover, during the iterative loop of

the GA, the mutation ours in order to introdue new unexplored solutions into the

searh spae. In order tokeep higherorder CDS-derived shemata,trialsolutionshaving

binaryongurationsbelongingtohigherordershemataaremutatedonlyinhromosome

positionsout oftheshemataloations[24℄. Thesemehanismsare aimedatonstraining

(34)

InthesamewayDonellietal. makeuseofahybridtehniquebasedonHDSandbinary

PSO [25℄[26℄. PSO is a stohasti multiple agents optimization algorithm extensively

applied in the framework of antenna array optimization [25℄[26℄[27℄. By imitating the

soial behaviour of groups of inset and animals in their food searhing ativities, PSO

is based upon the ooperation among partiles. The ensemble of the partiles, referred

to as swarm, explores the solution spae to nd out the best position (i.e., the optimum

of a suitably dened ost funtion). HSs-based arrays generate the initialtrialsolutions

of this hybrid method that then is optimized by binary PSO. Integrating the HS-based

methoddeveloped by Kopilovih[21℄ withPSO optimizationstrategygivesanimportant

improvement inthinned array performane.

In the framework of the antenna array for spae systems, we have a partiular

appli-ation where the previous synthesis tehniques were applied. Arrays are used in radio

astronomy to estimate the brilliane [9℄[29℄[30℄. Astronomers are interested in designing

orrelator arrays that properly sample the spatial distribution they observe. The design

of orrelator (also known as interferometri) arrays is essentially an optimal sampling

problem [9℄[29℄[30℄ in whih the positions of the antennas are hosen in order to ensure

optimal performane regarding all possible observation situations (soure positions and

durations of observation),sienti purposes (single eld imaging,astrometry, detetion,

...) and onstraints (ost,ground omposition andpratiability,operationof the

instru-ment, ...) [31℄[32℄. In order to obtain suh features, high performane orrelator arrays

have to show either a maximal overage in the spatial frequeny (or

u

−

v

) domain, or a minimum peak sidelobelevel(PSL) in the angular(or

l

−

m

) domain [8℄[31℄. Towards this end, many dierentdesignprinipleshavebeenproposed,inludingminimum

redun-dany [33℄,pseudo-randomness[34℄, powerlaws[35℄,dierenesetarrangements[36℄,and

minimization of the holes in the sampling [37℄. Ruf in [16℄ uses simulated annealing to

optimize low-redundany linear arrays whileJin [31℄ makes use of PSO. Well-established

optimization based sum-array design tehniques annot be diretlyapplied, sine, unlike

in traditionalsum arrays, the responses in both the

u

−

v

and the

l

−

m

domains have to be onsidered inthe design proedure [31℄. As a onsequene, design tehniques have

(35)

3.2.1 Introdution

The ost of a large phased array whih is designed primarily for high angular resolution

rather than for weak signal detetion may be redued manifold through thinning, i.e.,

reduing the number of elements in the aperture below that of the lled array in whih

the inter element spaing is nominallyone half-wavelength. Inreasing the inter element

spaing has another salutary eet; a separation of a few wavelengths redues mutual

oupling to negligible proportions. Thinning, therefore, is attrative from both points

of view. But these benets are not free of penalty. Unless the element loations are

randomized or made otherwise non periodi, grating lobes appear. Also, the redution

in the number of elements redues the designer's ontrol of the radiation pattern in the

sideloberegion,whih inturn inuenesthe levelof thelargest,orpeak, sidelobe. In this

hapter the peak sidelobe of random arrays is studied (N.B.: The random array ([6℄)

is haraterized by element loations hosen by some random proess. Conversely in a

statistial array ([4℄) aonventional lled array isdesigned and agiven fration ofthe

elementsis removed atrandom).

3.2.2 Linear Random Array

Consideranarrayof

N

unit,isotropiand monohromatiradiatorsatloations

x

n

. The

x

n

are hosen froma set of independentrandom variablesdesribed by some rst

proba-bility density distribution, initiallyassumed to be uniform over the interval

[

−

L/

2

, L/

2]

where

L

is the array length. It is assumed that eah element,irrespetive of itsloation,

is properly phased so that a main lobe of maximum strength is formed at

θ

0

, whih is measured from the normal tothe array. The redued angularvariable

u

= sin

θ

−

sin

θ

0

, ontains the beam steering information. The omplex far-eld radiationpattern

f

(

u

)

is

the Fourier transform of the urrent density. Sine the latter is a set of delta funtions,

f

(

u

)

is proportionalto the sum of unit vetors having phase angles

kx

n

u

,

k

= 2

π/λ

be-ing the wavenumber assoiated with the wavelength

λ

. The array fator is the Fourier

transform of the urrent density

i

(

x

)

. The urrent density

i

(

x

)

of a random array of

N

equally exited isotropi elements is the sum of delta funtions at the loations

x

n

and the omplex far-eld radiation pattern beomes

f

(

u

) =

F

(

N

X

n=0

δ

(

x

−

x

n)

)

=

N

X

n=0

(36)

f

(

u

) =

P

N

n=0

cos (

kx

n

u

) +

j

P

N

n=0

sin (

kx

n

u

)

=

a

(

u

) +

b

(

u

)

(3.2)

Sine

u

is dened overthe interval

[

−

1

,

1]

, it follows that

|

f

(

−

u

)

|

=

|

f

(

u

)

|

. Therefore, it

is suient toonsider the radiationpattern

|

f

(

u

)

|

onlyoverthe interval

[0

,

1]

.

The radiation pattern

f

(

u

)

as given by (3.2), is a omplex random proess. For the

speial ase where element loationsare independent and uniformly distributed overthe

interval

[

−

L/

2

, L/

2]

, the expeted values of the proesses

a

(

u

)

and

b

(

u

)

are

E

{

a

(

u

)

}

=

N

sin(

πuL/λ

)

πuL/λ

=

Nsinc

(

uL/λ

)

(3.3)

and

E

{

b

(

u

)

}

= 0

(3.4)

The proess

a

(

u

)

and

b

(

u

)

,for agiven value of

u

,are sums of

N

independent, identially

distributed random variables. When

N

is large, the entral the entral limit theorem

justies approximating

a

(

u

)

and

b

(

u

)

as Gaussian random variables. The mean of

a

(

u

)

,

asgivenby(3.3),isapproximatelyzerofor

u

greaterthanafewbeamwidths(thenominal

beamwidthis

λ/L

). Furthermore,forimagingproblemsinwhihhighangularresolutionis

demanded,

λ/L

≪

1

. Thusinmostofthesideloberegion,thetwoorthogonalomponents

of

f

(

u

)

are approximatelyzero-mean wide sense stationary Gaussianrandom proesses.

Foragiven

u

,themagnitudeof

f

(

u

)

isknown tobeRayleighdistributed[?℄. Let

us denote the magnitude pattern as

A

(

u

)∆

|

f

(

u

)

|

. The probability density funtion of

A

(

u

)

willbegiven by [6℄

p

(

A

) =

2

A

N

exp

−

A

2

/N

(3.5)

It follows that the meansquare value

A

2

, whih is the average sidelobe power level,

is

N

(and onsequently the rms amplitudeis

√

N

). The power ratio of the average

sidelobe to the main lobe is

N/N

2

= 1

/N

. The average is

A

=

p

πN/

2

. Hene, the

varianeis

σ

2

=

A

2

−

A

2

=

N

(1

−

π/

4)

.

The integral [6℄

α

=

Z

∞

A

0

p

(

A

)

dA

= exp

−

A

2

/N

(3.6)

istheprobabilitythatthemagnitudeofanarbitrarysampleoftheradiationpattern,away

from the region of the main lobe, exeeds some threshold

A

0

. Its omplement,

1

−

α

, is the probabilitythat suh asampleisless than

A

0

. If

n

independentsamplesare taken [6℄

β

=

1

−

exp

−

A

2

₀

/N

n

(37)

is the probability that none exeeds

A

0

. From (3.3),

A

2

0

=

−

N

ln 1

−

β

1/n

. It is

on-venient to normalize this expression to

N

, the average sidelobe level, and to give the

dimensionlesspowerratio

A

2

0

/N

a new symbol,

B

. Thus [6℄

B

=

−

ln 1

−

β

1/n

≈

ln (

n

)

−

ln ln

β

−

1

(3.8)

B

may beinterpreted asa statistialestimatorof the powerratio of the peak-to-average

sidelobe of a set of

n

independent samples.

B

is a ondene level; it is the probability

thatnoneof

n

independentsamplesofthe sidelobepowerpatternexeedsthemeanvalue

by the fator

B

.

n

is an array parameter, whih is a funtion of all the relevant array

properties other than

N

. It is proportional to the number of sidelobes in the visible

region. It maybe alulated in several ways. An interesting method utilizes the Nyquist

sampling theorem. The omplex radiation pattern of a random array is suh a

band-limited funtion, the limit being due to the nite length of the array. The far-eld

omplexradiationpattern

f

(

u

)

is relatedtothe radiatingelement positionsaording to

(3.1). From (3.1)wean denethe expressionfor thepower pattern ofanarrayof unit

radiators

f

(

u

)

f

⋆

(

u

) =

N

X

m=0

N

X

n=0

exp (

jk

(

x

n

−

x

m)

u

)

(3.9)

Thevisibledomainis

−

1

−

sin

θ

0

≤

u

≤

1

−

sin

θ

0

. Thelengthofthenon-redundantportion is

1 +

|

sin

θ

0

|

. Consequently, the number of independent samples needed to speify the omplex radiation pattern is

2 (

L/λ

) (1 +

|

sin

θ

0

|

)

. Half this number may be assoiated with the amplitude of the array fator and half with its phase. Therefore, the power

patternis uniquely speiedby [6℄

n

=

L

λ

(1 +

|

sin

θ

0

|

)

(3.10)

independent samples, the average angular interval between samples being

λ/L

.

n

is

dominated by the length of the array in units of wavelength and seondarily inuened

by the beam steering angle.

Equations (3.8) and (3.10), however, are insuient to provide an unbiased estimate

of the peak sidelobe. The probability is zero that any nite set of samples of

a power pattern falls exatly upon the rest of the largest sidelobe. Hene

suh estimation is downward b