An innovative approach based on a tree-searching algorithm for the optimal matching of independently optimum sum and difference excitations

UNIVERSITY

OF TRENTO

DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE

38123 Povo – Trento (Italy), Via Sommarive 14

http://www.disi.unitn.it

AN INNOVATIVE APPROACH BASED ON A TREE-SEARCHING

ALGORITHM FOR THE OPTIMAL MATCHING OF

INDEPENDENTLY OPTIMUM SUM AND DIFFERENCE

EXCITATIONS

L. Manica, P. Rocca, A. Martini, and Andrea Massa

January 2008

gorithm for the Optimal Mathing of Independently

Op-timum Sum and Dierene Exitations

L. Mania, P. Roa, A. Martini, and A. Massa, Member, IEEE

Department of Information and Communiation Tehnology,

University of Trento,Via Sommarive 14,38050 Trento - Italy

Tel. +39 0461 882057,Fax +39 0461 882093

E-mail: andrea.massaing.unitn.it,

{lua.mania, paolo.roa,anna.martini}dit.unitn.it

gorithm for the Optimal Mathing of Independently

Op-timum Sum and Dierene Exitations

L. Mania, P. Roa, A. Martini, and A. Massa

Abstrat

An innovative approah for the optimal mathing of independently optimum sum

and dierene patterns through sub-arrayed monopulse linear arrays is presented.

Byexploitingtherelationshipbetweentheindependentlyoptimalsumanddierene

exitations, the set of possible solutions is onsiderably redued and the synthesis

problem is reast as the searh of thebest solution in a non-omplete binary tree.

Towards thisend, afastresolution algorithm thatexploitsthepreseneof elements

more suitable to hange sub-arraymembership is presented. The results ofa set of

numerialexperimentsarereportedinordertovalidatetheproposedapproah

point-ing out its eetiveness also in omparison with state-of-the-art optimal mathing

tehniques.

Keywords: LinearArrays,MonopulseAntennas,SumandDierenePatternSynthesis,

A traking radar system using the monopulse tehnique [1℄ an be realized through an

antenna array able to generate two dierent patterns, namely the dierene patternand

the sum pattern. These patterns are required to satisfy some onstraints as narrow

beamwidth,lowsidelobelevel(

SLL

)andhighdiretivity. Inpartiular,asfarasthesum pattern is onerned, there is the need of maximizingthe gain. On the other hand, the

more ritial issues to be addressed dealing with dierene patterns are onerned with

both therst nullbeamwidthand the normalized diereneslopeonboresightdiretion,

sine they are stronglyrelated tothe sensitivity of the radar(i.e., to the angularerror).

Theoptimalexitationoeientsforthesumandthedierenepatternsanbe

indepen-dently omputed by using analytial methods as desribed in [2℄ and in [3℄, respetively.

Nevertheless, the implementation of two independent feed networks is generally

una-eptable beause of the osts, the oupied physial spae, the iruit omplexity and

the arising interferenes. Thus, it isneessary to nd asuitableompromise between the

feednetwork omplexityand the losenessof the synthesized sum anddierenepatterns

to the optimal ones. Sine the sum pattern is used in both signal transmission and

re-eption, the most ommon way to solve the problem onsists in generating an optimal

sum patternand a sub-optimal dierene pattern[4℄, the latter synthesized by applying

a sub-arrayingtehnique. Aordingly, the synthesis isaimed at optimizingpre-speied

sub-array layouts by sinthesizing sub-array and radiating element weights, but not the

atual beamformingnetwork.

In suh a framework, several approahes for dening how the elements ould be grouped

and the sub-arrays weights omputed have been proposed. As far as linear arrays are

onerned, MNamara proposed in [4℄ the Exitation Mathing method (

EMM

) aimed at determining a best ompromise dierene pattern lose as muh as possible to the

optimumintheDolph-Chebyshevsense[5℄(i.e.,narrowestrstnullbeamwidthandlargest

normalized dierene slope on the boresight for a speied sidelobe level). Towards this

end, for eah possible grouping, the orresponding sub-arrays oeients are iteratively

proessisextremelytime-expensiveduetotheexhaustiveevaluations. Moreover, beause

of the ill-onditioningof the matrix system, large arrays annotbe easilymanaged.

Inordertooverometheill-onditioningand relatedissues,optimizationapproaheshave

been widely used [6℄[7℄[8℄[9℄[10℄. Although suh tehniques allows a signiant

advane-mentin the framework of sum-dierenepatternsynthesis, they are stilltime-onsuming

when dealing with large arrays. As a matter of fat, even though the solution spae is

sampledwitheientsearhingriteria,thedimension ofthesolutionspaeisvery large.

In order to overome suh drawbaks allowing an eetive hoie of the array elements

groupingaswellasafastandsimplesolutionproedure,thispaperproposesaninnovative

approah that, likewise[4℄ and unlike [6℄[7℄[8℄[9℄[10℄, is aimed atobtaininga ompromise

dierenepatternoptimumintheDolph-Chebyshevsense[5℄startingfromtheobservation

that the sub-arraying is not blind. As a matter of fat, it an be guided by onsidering

similaritypropertiesamongthe arrayelements, thussigniantlyreduing the dimension

of the solution spae. Starting from suh an idea and by representing eah solution by

meansofapathinanon-ompletebinary tree,thesynthesis problemisthenreastasthe

searhingoftheminimal-ostpathfromthe roottotheleafsofthesolutiontree. Ingraph

theory,atree isagraphdened asanon-emptyniteset ofvertiesornodes inwhihany

twonodesareonnetedbyexatlyonepath. Thenodesarelabeledsuhthatthereisonly

one node alled the root of the tree,and the remainingnodes are partitionedinsubtrees.

Inour ase,sine thetreeiseitheremptyoreahnodehas notmorethan twosubtrees, it

is abinary tree. Aordingly, eah node ofa binarytree has either(I) nohildren,or(ii)

one left/righthild (i.e., non-omplete binary tree), or(iii) a left hild and a right hild

(i.e., omplete binary tree), eah hild being the root of a binary tree alled a subtree

[11℄[12℄. In order to solve the problem at hand, thus eiently exploring the solution

tree, suitable ost funtions or metris are dened and an innovative algorithm for the

exploration of the solution spae is dened by exploiting the loseness (to a sub-array)

property of some elements, alledborder elements, of the array.

one (Set. 2.1) as well as the tree struture (Set. 2.2) and the algorithmfor eetively

exploring the solution spae (Set. 2.3). In Setion 3, the results of seleted numerial

experimentsarereportedandomparedwiththosefromstate-of-the-artoptimalmathing

solutions. Conlusions and future possible trendsare drawn inSetion 4.

2 Mathematial Formulation

Letusonsideralinearuniformarrayof

N

= 2

M

elements

{ξm

;

m

=

−M, ...,

−

1

,

1

, ..., M

}

. Following a sub-optimal strategy, the sum pattern is generated by means of the

sym-metri set of the real optimal

(1)

exitations

A

opt

=

{αm

;

m

= 1

, ..., M

}

[2℄[13℄, while

the dierene pattern is dened through an anti-symmetri real exitation set

B

=

{bm

=

−b−m

;

m

= 1

, ..., M

}

[5℄. Thanks to suh symmetry properties, one half of the elementsof the array

S

=

{ξm

;

m

= 1

, ..., M

}

isdesriptiveof the wholearray.

Groupingoperationyieldstoasub-arrayongurationmathematiallydesribedinterms

of the grouping vetor

C

=

{cm

;

m

= 1

, . . . , M

}

,

cm

∈

[1

, Q

]

being the sub-array index of the

m

-th element ofthe array [7℄. Suessively, a weight oeient

wq

is assoiated to eah sub-array,

q

= 1

, ..., Q

, and, as a onsequene, the sub-optimal dierene exitation set isgiven by

B

=

{bm

=

wmqαm

;

m

= 1

, ..., M

;

q

= 1

, ..., Q}

(1)

where

wmq

=

δcmqwq

(

δcmq

= 1

if

cm

=

q

,

δcmq

= 0

otherwise) is the weight assoiated to the

m

-tharray element belongingto the

q

-thsub-array.

Aordingly, the original problem is reast as the denition of a sub-array onguration

C

and theorrespondingsetof weights

W

=

{wq

;

q

= 1

, ..., Q}

suhthatthe sub-optimal dierenepattern

B

isasloseaspossibletothe optimalone,

B

opt

=

{βm

;

m

= 1

, ..., M

}

.

Towards this end, let us formally proeed as follows. Firstly, two dierent metris are

dened inorder toquantify the loseness of the sub-optimalsolution tothe optimalone.

Then, exploiting some properties of the sub-array ongurations, a non-ompletebinary

(1)

algorithm for a fast searh of the lowest ost path in the binary tree is presented for

dening the best sub-optimal solution of the problemin hand.

2.1 Denition of the Solution-Metri

In order tond the optimalsolution,letus denea suitableost funtionor metri that

quanties the loseness of every andidate/trialsolution

C

t

tothe optimal one,

Ψ

{C

t

}

=

M

X

m

=1

[

vm

−

dm

{C

t

}

]

2

,

(2)

where

vm

and

dm

are referene and estimated parameters, respetively. The estimated parameters

dm

{C

t

}

are dened as the arithmeti mean of the referene parameters

vm

related to the array elements belonging to the same sub-array. As far as the referene

parameters

V

=

{vm

;

m

= 1

, ..., M

}

and the sub-arrays weights

W

=

{wq

;

q

= 1

, ..., Q}

are onerned, they are dened aording to two dierent strategies, namely the Gain

Sorting (

GS

)algorithmand the Residual Error (

RES

) algorithm. Conerningthe

GS

tehnique, the refereneparameters

v

(

GS

)

m

areset totheoptimal gains

v

(

GS

)

m

=

βm

αm

,

m

= 1

, . . . , M,

(3)

while the sub-array weights are assumed tobe equalto the omputed gains

d

(

GS

)

m

w

(

GS

)

q

=

δcmqd

(

GS

)

m

n

C

(

t

ess

)

o

,

q

= 1

, ..., Q, m

= 1

, . . . , M.

(4)

Conerning the

RES

algorithm, the referene parameters are equal to the the so-alled optimal residual errors

v

(

RES

)

m

given by

v

(

RES

)

m

=

αm

−

βm

βm

,

m

= 1

, . . . , M.

(5) Aordingly, sine

βm

αm

=

1

1+

v

(

m

RES

)

, m

= 1

, . . . , M,

terms of the omputed residual errors

d

(

RES

)

m

as follows

w

(

RES

)

q

=

1

1 +

δcmqd

(

m

RES

)

n

C

(

t

ess

)

o

,

q

= 1

, . . . , Q, m

= 1

, . . . , M.

(6)

2.2 Denition of the Solution-Tree

In general,thetotal numberof sub-arrayongurations isequalto

T

=

Q

M

sineeah of

themmightbeexpressed asasequeneof

M

digitsina

Q

-basednotationsystem. Without any lossof information,suhanumberanbereduedby onsideringonlytheadmissible

(or reliable) solutions, i.e., grouping where there are no empty sub-arrays. Moreover, let

us observe that if an equivalene relationship

(2)

among sub-array ongurations holds

true, it isonvenient to onsider just one sub-array ongurationfor eah set (instead of

the wholeset), therefore obtaininga set of non-redundant solutions.

Now, let us sort the known referene parameters

{vm

;

m

= 1

, ..., M

}

[omputed aord-ing to either the

GS

(3) or the

RES

algorithm (5)℄ for obtaining a ordered list

L

=

{lm

;

m

= 1

, ..., M

}

, where

li

≤

li

+1

, i

= 1

, ..., M

−

1

,

l

1

=

minm

{vm}

, and

lM

=

maxm

{vm}

. Sine the ost funtion is minimized provided that elements belonging to eah sub-array are onseutive elements of the ordered list

L

(see Appendix A for a detailed proof),the solutionspae an be further redued tothe so-alledessential

solu-tion spae

ℜ

(

ess

)

omposed by allowed solutions. Consequently, the dimension

T

of the

solution spae turns out to be redued from

T

=

Q

M

up to

T

(

ess

)

=









M

−

1

Q

−

1









(see

Appendix B for adetailedproof)and the essentialsolutionspae

ℜ

(

ess

)

an beformally

represented bymeansof thenon-ompletebinarytreedepited inFigure1. In partiular,

eah omplete path in the tree odes an allowed sub-array onguration

C

(

ess

)

t

∈ ℜ

(

ess

)

and the positive integer

q

inside eah node at the

lm

-th level indiates that the array element identied by

lm

is a member of the

q

-thsub-array. Thanks to this formulation,

(2)

A sub-arrayonguration

C

i

is equivalentto theonguration

C

j

whenit is possibleto obtain theonefromtheotherjust usingadierentnumberingforthesame

c

m

oeients. Asanexample,the sub-arrayonguration

C

i

=

{

1

,

2

,

3

,

3

,

2

,

3

,

2

,

1

}

isequivalentto

C

the originalminimizationproblem(i.e.,

C

opt

=

arg

{mint

=1

,...,T

[Ψ (

C

t

)]

}

)isreastasthat of nding the optimalpath in the solution tree.

2.3 Tree-Searhing Proedure

Although the set of andidate solutions has been onsiderably redued by limiting the

solutionspaetotheessential spae,itsdimension

T

(

ess

)

beomesverylargewhen

M

≫

Q

and an exhaustive searhing would be omputationally expensive. In order to overome

suh a drawbak, let us observe that only some elements of the list

L

are andidate to hangetheirsub-arraymembershipwithoutviolatingthesortingonditionof theallowed

sub-array ongurations,

n

C

(

t

ess

)

;

t

= 1

, ..., T

(

ess

)

o

[see Eq. (14) - Appendix B℄. These

elements,referredto asborder elements,satisfy the followingproperty: anarray element

relatedto

lm

isaborderelement ifoneoftheelementswhoselistvalueis

lm−

1

or/and

lm

+1

belongs toa dierent sub-array. Therefore, the aggregation

Copt

∈ ℜ

(

ess

)

minimizingthe

ost funtion

Ψ

is found startingfrom an initialpath randomlyhosen amongthe set of paths in the solutiontree and iteratively updating the andidate solutionjust modifying

the membershipof the border elements. More indetail, the iterative proedure (

k

being the iterationindex) onsists ofthe following steps.

•

Step

0

- Initialization. Initialize the iteration ounter (

k

= 0

) and the sequene index (

m

= 0

). Randomly generate a trialpath in the solution tree orresponding to a andidate sub-arrays onguration

C

(0)

∈ ℜ

(

ess

)

. Set the optimal path to

C

(

opt

k

)

k

k

=0

=

C

(0)

.

•

Step

1

- Cost Funtion Evaluation. Compute the ost funtion value of the urrent andidate path

C

(

k

)

by means of (2),

Ψ

(

k

)

= Ψ

n

C

(

k

)

o

. Compare the ost

of the aggregation

C

(

k

)

to the best ost funtion value attainedat any iterationup

to the urrent one,

Ψ

(

k−

1)

opt

=

minh

=1

,...,k−

1

Ψ

n

C

(

h

)

o

and update the optimaltrial

solution

C

(

k

)

opt

=

C

(

k

)

if

Ψ

n

C

(

k

)

o

<

Ψ

n

C

(

opt

k−

1)

o

.

•

Step

2

- Convergene Chek. If the termination riterion, based on a maxi-mum numberof iterations

K

or onastationaryondition for the tnessvalue(i.e.,

K

window

Ψ

(

k

−

1)

opt

−

P

Kwindow

j

=1

Ψ

(

j

)

opt

Ψ

(

opt

k

)

≤

η

,

Kwindow

and

η

being a xed number of iterations and a xed numerialthreshold, respetively), is satisedthen set

C

opt

=

C

(

k

)

opt

and

stop the minimizationproess. Otherwise, goto Step

3

.

•

Step

3

- Iteration Updating. Update the iteration index (

k

←

k

+ 1

) and reset the sequene index (

m

= 0

).

•

Step

4

-Sequene Updating. Updatethesequene index(

m

←

m

+ 1

). If

m > M

then go toStep

3

else goto Step

5

.

•

Step

5

- Aggregation Updating. If the array element related to

l

(

k

)

m

is a

bor-der element belonging to the

q

-th sub-array then dene a new grouping

C

(

k,m

)

by

aggregating suh an element to the

(

q

−

1)

-th sub-array [if the array element or-responding to

l

(

k

)

m−

1

is a member of the

(

q

−

1)

-th sub-array℄ or to the

(

q

+ 1)

-th sub-array [if the array element orrespondingto

l

(

k

)

m

+1

isa memberof the

(

q

+ 1)

-th sub-array℄. If

Ψ

(

k,m

)

= Ψ

n

C

(

k,m

)

o

<

Ψ

n

C

(

k

)

o

thenset

C

(

k

)

=

C

(

k,m

)

and gotoStep

1

. Otherwise, gotoStep

4

.

3 Numerial Simulations and Results

Inordertoassessthe eetivenessoftheproposedmethod,anexhaustivesetofnumerial

experiments has been performed and some representative results will be shown in the

following.

For a quantitative evaluation, a set of beam pattern indexes has been dened and

om-puted. More in detail, (a) the pattern mathing

∆

that quanties the distane between the synthesized sub-optimal patternand the optimalone

∆ =

R

π

0

|AF

(

ψ

)

|

opt

n

− |AF

(

ψ

)

|

rec

n

dψ

R

π

0

|AF

(

ψ

)

|

opt

n

dψ

,

(7)

where

ψ

= (2

πd/λ

)

sinθ

,

θ

∈

[0

, π/

2]

, (

λ

and

d

being the free-spae wavelength and the inter-element spaing, respetively),

|AF

(

ψ

)

|

opt

n

and

|AF

(

ψ

)

|

rec

optimalandgeneratedarraypatterns,respetively;(b)themainlobesbeamwidth

BW

and ()the powerslope

Pslo

thatgivesomeindiationsonthe slopeonthe boresightdiretion

Pslo

= 2

×

"

max

ψ

(

|AF

(

ψ

)

|

n

)

×

ψmax

−

Z

ψmax

0

|AF

(

ψ

)

|

n

dψ

#

,

(8)

ψmax

being the angular position of the maximum inthe array pattern; (d) the sidelobes power

Psll

Psll

=

Z

π

ψ

1

|AF

(

ψ

)

|

n

dψ,

(9)

where

ψ

1

is the angular position of the rst null in the dierenebeam pattern.

The remainingof this setionisorganized asfollows. Firstly,some experiments aimedat

showing the asymptotibehaviourof the proposedsolutionare presented (Set. 3.1) and

a omparative study is arried out (Set. 3.2). Furthermore, some experiments devoted

at showing the potentialities of the proposed solution in dealing with large arrays are

disussed inSet. 3.3. Finally,the omputationalissues are analyzed(Set. 3.4).

3.1 Asymptoti Behavior Analysis

In order to assess that inreasing the number of sub-arrays

Q

the synthesized dierene patterns get loser and loser to the optimal one, let us onsider a linear array of

N

=

2

×

M

= 20

elements haraterized by a

d

=

λ

2

inter-element spaing. The optimal sum

pattern exitations,

{αm, m

= 1

, ..., M

}

, have been xed tothat of the linear Villeneuve pattern [13℄ with

n

= 4

and

25

dB

sidelobe ratio (Fig. 2 - Villeneuve, 1984), while the optimal dierene weights

{βm, m

= 1

, ..., M

}

, have been hosen equal to those of a Zolotarevdierenepattern[5℄withasidelobelevel

SLL

=

−

30

dB

(Fig. 24-MNamara, 1993). Then,

Q

has been varied between

2

and

M

and both

GS

and

RES

tehniques have been applied. For sake of spae, seleted results onerned with

Q

= 3

,

Q

= 6

,and

Q

= 9

are reported interms of diereneexitations[Fig. 2(a)-

GS

approah;Fig. 2(b) -

RES

approah℄. As expeted, the oeients obtained with both the

GS

and

RES

onvergetothe optimalonesand, startingfrom

Q

= 6

,the dierenesbetweengenerated and referene dierenepatterns turn out to be smallerand smaller.

For omparison purposes and in the framework of synthesis tehniques aimed at

deter-mining the best ompromise dierene pattern as lose as possible to the optimal one,

let us onsider the

EMM

by MNamara [4℄ as referene

(3)

. As far as the test ases are

onerned,thesamebenhmarkinvestigatedin[4℄hasbeentakenintoaount. Thearray

geometry and the optimal sum exitations was as in Set. 3.1, while the optimal

dier-ene exitationvetor

B

opt

has beenhosen forgeneratingamodiedZolotarevdierene

pattern with

n

= 4

,

ε

= 3

and asideloberatio of

25

dB

[3℄.

The rst test ase deals with a uniform sub-arraying over the antenna with

Q

= 5

. The valuesofthesub-arraysweightsoptimizedwiththe

GS

andthe

RES

are

W

GS

=

{

₀

.

2951

,

0

.

8847

,

1

.

1885

,

1

.

3994

,

1

.

4878

}

and

W

RES

=

{

0

.

3411

,

0

.

8915

,

1

.

1193

,

1

.

4016

,

1

.

4881

}

, respetively. Moreover, the synthesized dierene patterns are shown in Figure 3, while

the omputedbeam-patternindexesare reportedinTableI.The advantagesontheuse of

thetree-basedapproahesareevident,asonrmedbythevaluesofboththe

SLL

(almost

4

dB

belowthelevelahievedbythe

EMM

,

SLL

EM M

=

−

17

.

00

dB

vs.

SLL

GS

=

−

21

.

00

and

SLL

RES

=

−

20

.

50

) and the pattern mathing index (

∆

EM M

∆

RES

≃

1

.

4

and

∆

EM M

∆

GS

≃

1

.

5

- Tab. I). Moreover, it is worth noting that, thanks to the struture of the solutiontree,

the dimension of the essential spae redues to

T

(

ess

)

= 1

(sine

l

1

and

l

2

belong to the rst sub-array,

l

3

and

l

4

to the seond one, and so on), thusallowinga signiant saving of omputational resoures. As a matter of fat, the

EMM

requires the solution of an overdetermined system of linear equations in orrespondene with any possible uniform

grouping[4℄, i.e., anumberof

T

= 945

evaluations.

Seond and thirdtest asesonsider non-uniformsub-arraying. The formeronguration

is an example of the limited number of sub-arrays (

Q

= 3

) that might be used with a small monopulse antenna. The latter has the same number of sub-arrays as that of

the rst onguration (

Q

= 5

). The tree-based algorithms have been applied and the followingsub-arrayongurations havebeendetermined. Inpartiularthesamegrouping

(3)

Noomparisonwith optimization-basedproedures(i.e.,[6℄[7℄[8℄[9℄[10℄)havebeenreportedsine theyareaimedatminimizingapatternparameter(e.g.,the

SLL

)andnotatbettermathinganoptimal dierenepattern.

C

GS, RES

opt

=

{

1

,

2

,

3

,

3

,

4

,

5

,

5

,

5

,

4

,

3

}

has been synthesized when

Q

= 5

, while

C

GS

opt

=

{

1

,

1

,

2

,

2

,

3

,

3

,

3

,

3

,

3

,

2

}

and

C

RES

opt

=

{

1

,

2

,

3

,

3

,

3

,

3

,

3

,

3

,

3

,

3

}

have been obtained for

Q

= 3

.

The obtained beam patterns are shown in Fig. 4 and the orresponding values of the pattern indexes are reported in Tab. II. As it an be notied, the

GS

and

RES

improve the performanes ofthe

EMM

inmathing the optimaldierenepattern as pointed out by the behavior of the global mathing index

∆

(

∆

EM M

∆

GS

k

Q

=3

= 1

.

33

and

∆

EM M

∆

RES

k

Q

=3

= 1

.

42

;

∆

EM M

∆

GS

k

Q

=5

= 1

.

63

and

∆

EM M

∆

RES

k

Q

=5

= 1

.

68

). Conerning the smaller

onguration, itisfurtheronrmed(asalreadypointedout inSetion3.1)the exibility

and reliability of the

GS

algorithm in dealing also with omplex ases where a limited numberofsub-arraysistakenintoaount. Asamatteroffat,for

Q

= 3

the

GS

givesthe best performanes getting the highest sidelobe ratio of

SLL

= 18

.

63

dB

and synthesizing a main lobe very lose to the optimal one, i.e.,

B

GS

w

=

B

w

opt

= 0

.

3735

and

P

GS

slo

= 0

.

1800

vs.

P

opt

slo

= 0

.

1802

.

3.3 Large Arrays Analysis

This setion is aimed at analyzing the performanes of the proposed tree-based

teh-niques when dealing with large arrays. As far as the optimal setup is onerned, sum

{αm, m

= 1

, ..., M

}

anddierene

{βm, m

= 1

, ..., M

}

optimalexitationshavebeen ho-sen to generate a Dolph-Chebyshev pattern [15℄ with

SLL

=

−

25

dB

and a Zolotarev pattern [5℄with

SLL

=

−

30

dB

, respetively.

As arstexperiment,alineararrayof

N

= 200

elementswith

λ/

2

spainghas beenused byonsideringvarioussub-arrayingongurations. Figure5shows the optimaldierene

pattern (i.e., the synthesis target) and the patterns obtained when

Q

= 4

and

Q

= 6

by using both

GS

and

RES

. For ompleteness, the values of the synthesized dierene exitations are displayed in Figure 6. It is worth noting that the

GS

algorithm outper-formsthe

RES

. Asamatteroffat,althoughbothapproahes satisfatorilyapproximate the optimal main lobe harateristis in terms of both

BW

and

Pslo

, the solutions om-putedwiththe gain-basedlogipresenthighersideloberatios(

SLL

GS

k

Q

=4

=

−

21

.

90

and

SLL

GS

k

Q

=6

=

−

25

.

13

tothe

RES

approah(

SLL

RES

k

Q

=4

=

−

10

.

10

and

SLL

RES

k

Q

=6

=

−

19

.

95

),respetively.

Moreover, the overall mathing performanes turn out signiantly inreased as further

onrmed by the values of

∆

(

∆

RES

∆

GS

k

Q

=4

≃

3

.

77

and

∆

RES

∆

GS

k

Q

=6

≃

2

.

47

).

Thelasttestase(andseondexperimentdealingwithlargestrutures)isonernedwith

a linear array of

N

= 2

×

M

= 500

elements (

d

=

λ/

2

). As a representative example, the ase of

Q

= 4

is reported and analyzed (Tab. III). The arising beam patterns allow one to drawn similar onlusions to those from the previous senario, sine one again

the eetiveness of the

GS

tehnique in dealing with a limited number of sub-arrays is pointedout. As amatter of fat,the ratio between the mathing indexesturns out quite

large and equalto

∆

RES

∆

GS

k

Q

=4

≃

4

.

1

(Tab. III).On the other hand,itisworth notingthat

unliketree-basedproeduresthe

EMM

isnotreliableindealingwithlargearrayssineit requiresthenumerialproessingofoverdeterminedlinearsystems,whoseill-onditioning

get worse when the ratio

M

Q

grows.

3.4 Computational Issues

Now, let us analyze the omputational osts of the tree-based approahes, providing

a omparison with the

EMM

, as well. Towards this end, let us rstly onsider the dependene ofthedimension ofthesolutionspaeonthenumberof elementsofthe array

M

. As a representative ase, let us analyze the behavior of

T

and

T

(

ess

)

when

Q

= 3

(

K

= 100

and

η

= 10

−

3

) (Fig. 7). As it an be observed, the dimension of the solution

spae

T

of the

EMM

grows exponentially with

M

, while, as expeted [see Appendix A℄,

T

(

ess

)

shows apolynomial behavior. Obviously, the same behaviorholds true alsofor

dierent values of

Q

(Fig. 7).

On the other hand, the omputational eetiveness of the Tree-Searhing proedure in

samplingthesolutionspaeisfurtherpointedoutfromtheevaluationofthe

CP U

-time,

t

, needed for reahing the onvergene (Fig. 8). As a matter of fat,

maxQ

{tQ}

= 70 [

sec

]

(

kopt

= 90

) in orrespondene with the largest array (

M

= 250

), while

maxQ

{tQ}

=

12

.

8 [

sec

]

(

kopt

= 8

) and

maxQ

{tQ}

= 2

.

3 [

sec

]

(

kopt

= 4

) when

M

= 100

and

M

= 50

, respetively.

Inthispaper,aninnovativeapproahforthesynthesisofsub-arrayedmonopulseantennas

by mathing independently-optimum sum and dierene exitations has been proposed.

By exploiting some properties of the sub-array ongurations, the problem of nding a

best ompromise dierene pattern by grouping array elements has been reast as the

searh of the optimum, in terms of either the

GS

or the

RES

logi, path inside a non-omplete binary tree. Towardsthis purpose, a fastresolution algorithmhas been dened

and assessed by meansof several numerialexperiments.

Conerningthe methodologialnoveltiesof this work,the mainontributionisonerned

with the following issues: (a) an appropriate denition of the solution spae; (b) an

original and innovativeformulationof the sum-diereneproblemin terms ofa searh in

a non-omplete binary tree; () a simple and fast solution proedure based on swapping

operations amongborder elementsand ost funtion evaluations.

Moreover, the main features of the proposed tree-based tehniques are the following: (i)

a redution of the dimensionality

T

(

ess

)

of the synthesis problem,by exploitingthe

infor-mationontentofindependentlyoptimalsumand diereneexitations; (ii)asigniant

redution of the omputationalburden, by applying afast solution algorithmfor

explor-ing the solution tree (i.e., sampling the solution spae); (iii) the apability to deal with

large-arrays synthesis in aneetive and reliable way.

Beause of the favorable trade-o between omplexity/osts and eetiveness, the

pro-posedtree-based strategy seemsa promisingtooltobe furtheranalyzed and extended to

other geometries and synthesis problems. Towards this purpose, further methodologial

studieswillbeorientedintwodierentdiretions: (I)improvingthesolutionproedureby

developingaustomizedombinatorialapproah,thusfurtherreduingtheomputational

osts as well as improving the onvergene rate; (II) re-formulating the sum/dierene

The authors wish to thank Prof. T. Isernia and Dr. M. Donelli for useful disussions

and suggestions. This work has been partially supported in Italy by the Progettazione

di un Livello Fisio 'Intelligente' per Reti Mobili ad Elevata Riongurabilità, Progetto

di Rieradi Interesse Nazionale -MIUR Projet COFIN 2005099984.

Appendix A

Thisappendix isaimedatprovingthat,given

Q

sub-arrays,the valueoftheost funtion (2) is minimum provided that the elements belonging to eah sub-array are onseutive

elements of the ordered list

L

=

{lm

;

m

= 1

, . . . , M

;

lm

≤

lm

+1

}

. With referene to a set of elements

V

=

{vm

;

m

= 1

, ..., M

}

be to be divided in

Q

sub-sets, the thesis to beproved is that the partitionminimizingthe ost funtion(2) isa ontiguous partition

(i.e., if two elements

vi

and

vn

belong to the same lass and

vi

< vj

< vn

, then element

vj

is assigned to the same subset of elements). Towards this end, the proof follows the guidelines reported in[16℄.

Letusonsider anon-ontiguous partition

P

Q

=

{Vq

;

q

= 1

, ..., Q}

of theset

V

and three elements

vi, vj

, vn

suhthat

vi

< vj

< vn

. Letelements

vi

and

vn

belongtoasubsetwith meanvalue

dr

andlet

vj

belongtoadierentsubsethavingmeanvalue

ds

. Whateverthe values of

dr

and

ds

, atleast one the following statementsholds true































|vj

−

ds| ≥ |vj

−

dr|

>

0

,

|vi

−

dr| ≥ |vi

−

ds|

>

0

,

|vn

−

dr| ≥ |vn

−

ds|

>

0

.

(10)

Letusdenotewith

vt

theelementsatisfying(10)anditsownsubsetas

V

k

=

{vk

;

k

= 1

, ..., Nk}

. Moreover, let us refer to the other subset as

V

h

=

{vh

;

h

= 1

, ..., Nh}

. Aordingly, the ost funtion(2) assoiated to the partition

P

Q

may be written as:

Ψ =

M

X

m

=1

v

m

2

−

Nk

·

d

2

k

−

Nh

·

d

2

h

−

Q

X

q

=1;

q6

=

h,k

Nq

·

d

2

q

(11)

Nq

and

dq

being the number of elements and the mean value of the

q

-th sub-array, re-spetively.

Now, let us onsider a new partition

P

(1)

Q

obtained by moving the element

vt

from the subset

V

k

to the subset

V

h

. We obtain two new subsets

V

(1)

k

=

V

k

\ {vt}

and

V

(1)

h

=

V

k

∪ {vt}

(4)

with mean values equal to

d

(1)

k

=

Nkdk−vt

N

k

−

1

and

d

(1)

h

=

Nhdh

+

vt

N

h

+1

,

respetively.

Aordingly, the ost funtion assoiated tothe partition

P

(1)

Q

an be written as:

Ψ

(1)

=

M

X

m

=1

v

m

2

−

(

Nkdk

−

vt

)

2

Nk

−

1

−

(

Nhdh

−

vt

)

2

Nh

−

1

−

Q

X

q

=1;

q6

=

h,k

Nqd

2

q

.

(12)

Now, by subtrating (12) from (11), aftersome manipulations,it turns out that

Ψ

−

Ψ

(1)

=

Nk

Nk

−

1

(

vt

−

dk

)

2

−

Nh

Nh

+ 1

(

vt

−

dh

)

2

.

(13) Aording to (10),

Ψ

>

Ψ

(1)

and it an be onluded that for every non-ontiguous

partition we an nd another one with the same number of subsets, but with a smaller

ost. Hene, the partitionminimizingthe ost funtion (2)is aontiguous partition.

Appendix B

This setionisdevoted atquantifyingthe dimension

T

(

ess

)

of theessential solutionspae

ℜ

(

ess

)

=

n

C

(

t

ess

)

;

t

= 1

, ..., T

(

ess

)

o

, thuspointingout the omputationalsaving allowed by

the proposed approah ompared to exhaustive or global sampling solution proedures.

More in detail, the aim is that of determining the number

T

(

ess

)

of andidate solutions

or, inanequivalent fashion, the number of allowed paths in the solutiontree.

Generally speaking, sine a sub-array onguration

C

an be mathematially desribed by asequene of

M

digits of a

Q

-symbolsalphabet, the wholenumberof aggregations is equal to

T

=

Q

M

.

Thanks tothe equivalene relationship, the set of andidate solutions

an be limited to the number of paths in a omplete binary tree of depth

M

, thus the

(4)

We expliitly note that the new partition

P

(1)

Q

has the same number of subsets as

P

Q

. As a matteroffat,aordingto(10),theelement

v

t

annotbeequaltothemeanvalue

d

k

andthus,

V

k

has ardinalitygreaterthanone. Itfollowsthatthesub-set

V

(1)

number of non-redundant solutions results

T

= 2

M

−

1

.

Moreover, by taking into aount

only admissible (i.e.,groupingwhere there isatleast one elementineah sub-array)and

allowed (i.e., sorted aggregations) omplete sequenes, the set of solution an be further

redued. With referene to the ordered list

L

=

{lm

;

m

= 1

, . . . , M

;

lm

≤

lm

+1

}

, the allowed pathsare mathematiallydesribed as

C

(

t

ess

)

=

n

c

(

t,m

ess

)

c

(

ess

)

t,m

≤

c

(

ess

)

t,m

+1

, c

(

ess

)

t,

1

= 1

, c

(

ess

)

t,M

=

Q

o

,

t

= 1

, ..., T

(

ess

)

,

(14) where

c

(

ess

)

m

denotes the sub-array number to whih the

m

-th element

lm

of the ordered list

L

belongs.

In order todetermine the essentialdimension

T

(

ess

)

=

T

(

ess

)

(

Q, M

)

ofthe solution spae,

let us onsider the reursive nature of the binary solution tree and, as a referene

ex-ample, the ase

Q

= 2

. In suh a situation, the grouping vetor

C

(

ess

)

t

is a sequene of

M

symbols fromthe set

{

1

,

2

}

that satises the followingonstraints: (a)

c

(

ess

)

t,

1

= 1

, (b)

c

(

t,M

ess

)

= 2

,and ()if

c

(

ess

)

t,m

= 2

then

c

(

ess

)

t,m

+1

=

c

(

ess

)

t,M

= 2

. Thus, eah possiblesolution

C

(

ess

)

t

ismadeup ofasub-sequeneofonseutivesymbols1followedby asub-sequeneof

sym-bols2. Aordingly,the trialsolutions

C

(

ess

)

t

,

t

= 1

, ..., T

(

ess

)

,are obtainedbymovingthe

startingpointofthe sub-sequene ofsymbols

2

from

m

= 2

(being

c

1

= 1

)up to

m

=

M

,

T

(

ess

)

(

Q, M

)

k

Q

=2

=









M

−

1

1









=

M

−

1

.

(15)

Asfarasthease

Q

= 3

isonerned,similaronsiderationsholdtrue. Inpartiular,eah allowed trialsolution

C

(

ess

)

t

endswithasub-sequeneofsuessivesymbols

3

. Thenumber of elements of suh a sub-sequene rangesfrom 1 to

M

−

2

, leading to a omplementary sub-sequene of symbols1 and 2 of length

M

−

i

. Aordingly,

T

(

ess

)

(

Q, M

)

k

Q

=3

=

M

−

2

X

i

=1

T

(

ess

)

(

Q, M

−

i

)

k

Q

=2

(16)

Generalizing, sine the smallest and largest number of ourrenes of the symbol

Q

in a sequene is

1

and

M

−

(

Q

−

1)

,respetively,the essentialdimension ofthe solutionspae

when a

M

elementsarray is partitionedinto

Q

sub-arrays isequal to

T

(

ess

)

(

Q, M

) =

M−

(

Q−

1)

X

i

=1

T

(

ess

)

(

Q

−

1

, M

−

i

) =









M

−

1

Q

−

1









.

(17) Referenes

[1℄ M. I. Skolnik,Radar Handbook.New York: MGraw-Hill,1990.

[2℄ T. T. Taylor, Design of line-soure antennas for narrow beam-width and low side lobes, Trans.IRE, AP-3,16-28, 1955.

[3℄ D. A. MNamara, Disrete

n

¯

-distributions for dierene patterns, Eletron. Lett., 22(6), 303-304,1986.

[4℄ D. A.MNamara,Synthesis ofsub-arrayed monopulselineararrays through math-ingofindependentlyoptimumsumanddiereneexitations, IEEPro.H,135(5),

371-374, 1988.

[5℄ D. A.MNamara,Diret synthesisof optimumdierenepatternsfordisretelinear arrays using Zolotarev distribution, IEE Pro. H, 140(6),445-450, 1993.

[6℄ F. Ares, S. R. Rengarajan, J. A. Rodriguez, and E. Moreno, Optimal ompromise among sum and dierene patterns through sub-arraying, Pro. IEEE Antennas

Propagat. Symp., 1142-1145,1996.

[7℄ S.Caorsi,A.Massa, M.Pastorino,and A.Randazzo, Optimizationofthe dierene patterns formonopulseantennasbyahybridreal/integer-odeddierentialevolution

method, IEEE Trans. Antennas Propagat., 53(1), 372-376,2005.

[8℄ P.Lopez,J.A.Rodriguez,F.Ares,andE.Moreno,Subarrayweightingfordierene patterns of monopulse antennas: Joint optimization of subarray ongurations and

eetive hybrid approah, IEEE Trans. Antennas Propagat., 55(3),750-759,2007.

[10℄ M. D'Urso, T. Isernia, and E. F. Meliado' An eetive hybrid approah for the optimalsynthesis ofmonopulse antennas, IEEE Trans. Antennas Propagat.,55(4),

1059-1066, 2007.

[11℄ R. Otter, The number of trees,,Annals of Mathematis, 49(3),Jul. 1948.

[12℄ D.B.West,IntrodutiontoGraphTheory.EnglewoodClis,NJ:Prentie-Hall,2000.

[13℄ A. T. Villeneuve,Taylorpatterns fordisretearrays,IEEE Trans.Antennas Prop-agat., 32,1089-1093, 1984.

[14℄ C. A. Balanis,Antenna Theory: Analysisand Design.New York: Wiley,1982.

[15℄ C. L. Dolph, A urrent distributionoptimizes for broadside arrays whih optimizes the relationship between beam width and sidelobe level, Pro. IRE, 34, 335-348,

1946.

[16℄ W.D.Fisher,Ongroupingofmaximumhomogeneity, AmerianStatistialJournal, 789-798, 1958.

•

Figure 1. Solution-Tree struture representing the essential solutionspae

ℜ

(

ess

)

.

•

Figure 2. Asymptoti Behavior (

M

= 10

,

d

=

λ

2

) - Sum

{αm

;

m

= 1

, ..., M

}

and dierene

{βm

;

m

= 1

, ..., M

}

optimal exitations. Compromise dierene oe-ients

{bm

;

m

= 1

, ..., M

}

for dierentvalues of

Q

when (a) the

GS

algorithmand (b) the

RES

algorithmare applied.

•

Figure 3. - Uniform sub-arraying (

M

= 10

,

d

=

λ

2

,

Q

= 5

) - Referene optimum and normalized dierene patterns obtained by means of the

EMM

, the

GS

, and the

RES

approahes.

•

Figure 4. Non-uniform sub-arraying (

M

= 10

,

d

=

λ

2

) - Referene optimum and

normalized dierene patterns obtained by means of the

EMM

, the

GS

, and the

RES

approahes when (a)

Q

= 3

and (b)

Q

= 5

.

•

Figure 5. Large Arrays (

M

= 100

,

d

=

λ

2

) - Referene optimum and normalized

dierene patterns obtained by means of the

GS

and

RES

tehniques when

Q

= 4

and

Q

= 6

.

•

Figure 6. Large Arrays (

M

= 100

,

d

=

λ

2

) - Dierene exitations determined by

the tree-based tehniques when

Q

= 4

(a) and

Q

= 6

(b).

•

Figure7. ComputationalAnalysis-ComputationalAnalysis -Behaviorof

T

versus

M

when the tree-based searhing isapplied [

T

=

T

(

ess

)

℄.

•

Figure 8. Computational Analysis - Behavior of

t

versus

M

for dierent values of

Q

(

GS

Approah).

•

Table I. Uniform sub-arraying (

M

= 10

,

d

=

λ

2

,

Q

= 5

) -Beam patternindexes.

•

Table II. Non-uniform sub-arraying (

M

= 10

,

d

=

λ

2

,

Q

= 3

,

5

) - Beam pattern indexes.

•

Table III. Large Arrays (

M

= 250

,

d

=

λ

l

1

l

2

l

3

l

4

l

m

l

m+1

l

M−1

l

M

Q

Q

Q

Q

"Non−Admissible" Path

"Allowed" Path

"Forbidden" Path

1

1

2

2

2

3

1

1

2

2

3

3

2

3

4

1

1

2

2

3

2

2

2

3

q−1

q−1

q

q

q+1

q

Q

2

2

3

3

2

2

1

1

2

Q−1

Q−1

Q

Q−1

Q−1

Q−1

q

q−1

q−1

"Admissible" Paths

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1

2

3

4

5

6

7

8

9

10

α

m

,

β

m

, b

m

Element Index, m

[Villeneuve, 1984]

[McNamara, 1993]

Difference Pattern (Q=3)

Difference Pattern (Q=6)

Difference Pattern (Q=9)

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1

2

3

4

5

6

7

8

9

10

α

m

,

β

m

, b

m

Element Index, m

[Villeneuve, 1984]

[McNamara, 1993]

Difference Pattern (Q=3)

Difference Pattern (Q=6)

Difference Pattern (Q=9)

(b)

-60

-50

-40

-30

-20

-10

0

0

0.2

0.4

0.6

0.8

1

P

RL

[dB]

ψ/π

[McNamara, 1986]

GS

RES

[McNamara, 1988]

-60

-50

-40

-30

-20

-10

0

0

0.2

0.4

0.6

0.8

1

P

RL

[dB]

ψ/π

[McNamara, 1986]

GS

RES

[McNamara, 1988]

(a)

-60

-50

-40

-30

-20

-10

0

0

0.2

0.4

0.6

0.8

1

P

RL

[dB]

ψ/π

[McNamara, 1986]

GS

RES

[McNamara, 1988]

(b)

-60

-50

-40

-30

-20

-10

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P

RL

[dB]

ψ/π

[McNamara, 1993]

GS - Q=4

RES - Q=4

GS - Q=6

RES - Q=6

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0

20

40

60

80

100

b

m

,

α

m

Element Index, m

Difference Excitations - GS

Difference Excitations - RES

Sum Excitations

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0

20

40

60

80

100

b

m

,

α

m

Element Index, m

Difference Excitations - GS

Difference Excitations - RES

Sum Excitations

(b)

10

25

10

20

10

15

10

10

10

5

10

0

0

50

100

150

200

250

10

250

10

200

10

150

10

100

10

50

10

0

T

(ess)

T

M

T

(ess)

- Q=3

T

(ess)

- Q=4

T

(ess)

- Q=6

T

(ess)

- Q=8

T

(ess)

- Q=10

T - Q=3

0

10

20

30

40

50

60

70

80

0

50

100

150

200

250

t [sec]

M

Q=3

Q=4

Q=6

Q=8

Q=10

Approach

P

slo

B

W

P

sll

max

{SLL}

∆

EM M

[4 ℄

0

.

1970 0

.

3610 0

.

1038

−

17

.

00

0

.

4015

GS

0

.

1811 0

.

3784 0

.

1082

−

21

.

10

0

.

2633

RES

0

.

1805 0

.

3735 0

.

1160

−

20

.

50

0

.

2831

Optimal

[3 ℄

0

.

1802 0

.

3735 0

.

0598

−

25

.

00

−

ab. I -L. Mania et al., An Inno v ativ e Approa h Based on ... 30

Q

= 3

Q

= 5

EM M

[4 ℄

GS

RES

EM M

[4 ℄

GS

RES

Optimal

[5 ℄

P

slo

0

.

2117

0

.

1800

0

.

1822

0

.

2000

0

.

1806

0

.

1805

0

.

1802

B

W

0

.

3745

0

.

3735

0

.

3930

0

.

3854

0

.

3735

0

.

3735

0

.

3735

P

sll

0

.

1798

0

.

1054

0

.

1365

0

.

0950

0

.

0823

0

.

0827

0

.

0598

max

{SLL}

−

14

.

70

−

18

.

63

−

17

.

00

−

23

.

40

−

23

.

00

−

23

.

00

−

25

.

00

∆

0

.

5438

0

.

4073

0

.

3829

0

.

2562

0

.

1571

0

.

1517

−

ab. I I -L. Mania et al., An Inno v ativ e Approa h Based on ... 31

GS

RES

Optimal Dif f erence

[5 ℄

P

slo

0

.

0066

0

.

0064

0

.

0066

B

W

0

.

0148

0

.

0158

0

.

0151

P

sll

0

.

0868

0

.

1797

0

.

0824

max

{

SLL

} −

18

.

00

−

10

.

05

−

30

.

00

∆

0

.

2921

1

.

1934

−