Optimal sub-arraying of compromise planar arrays through an innovative ACO-weighted procedure

UNIVERSITY

OF TRENTO

DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE

38123 Povo – Trento (Italy), Via Sommarive 14

http://www.disi.unitn.it

OPTIMAL SUB-ARRAYING OF COMPROMISE PLANAR ARRAYS

THROUGH AN INNOVATIVE ACO-WEIGHTED PROCEDURE

G. Oliveri and L. Poli

January 2011

an Innovative ACO-Weighted Pro edure

G. Oliveriand L. Poli

ELEDIA Resear h Group

Department of Information Engineeringand ComputerS ien e,

University of Trento,Via Sommarive 14,38123 Trento - Italy

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

E-mail: {gia omo.oliveri,lorenzo.poli}disi.unitn.it

an Innovative ACO-Weighted Pro edure

G. Oliveriand L. Poli

Abstra t

Inthispaper,thesynthesisofsub-arrayedmonopulseplanararraysprovidingan

op-timalsumpatternandbest ompromisedieren epatternsisaddressedbymeansof

an innovative lustering approa h based on the Ant Colony Optimizer. Exploiting

the similarity propertiesof optimal and independent sum and dieren e ex itation

sets,theproblemisreformulatedintoa ombinatorialonewherethedenitionofthe

sub-array onguration isobtained through the sear h of apath within aweighted

graph. Su ha weightingstrategy allows oneto ee tivelysamplethesolution spa e

avoiding bias towards sub-optimal solutions. The sub-array weight oe ients are

then determined in an optimal way by exploiting the onvexity of the problem at

hand by means of a onvex programming pro edure. Representative results are

reported to assess the ee tiveness of the weighted global optimization and its

ad-vantages overprevious implementations.

Key words: Sum and Dieren e Patterns Synthesis, Monopulse Antennas, Planar

Monopulseradarspresentseveraladvantagesoverothersear h-and-tra ksystems[1℄based

on oni als anorlobeswit hingapproa hes [2℄. Indeed, tra kingtheangularpositionsof

high-speedtargetsisenabledjustpro essingasinglepulsee ho(amonopulse). Moreover,

rangemeasurementsaregenerallymorereliablebe auseofe hosignalswithhigher

signal-to-noise ratios are dealt with, the sum beam being always dire ted towards the target.

Monopulse radars require the generation of one sum pattern and a ouple of

spatially-orthogonal dieren epatterns totra k targetsboth inazimuth and elevation [3℄. Several

implementationsexploit ree torsorlens antennas[2℄, even ifantennaarrays turnout to

bemore onvenient for te hnologi al(e.g., the main beam an beele troni ally steered),

implementative (e.g., heavy stru tures as ree tors are avoided), and appli ative (e.g.,

arrays anbemade onformaland installedonair rafts) reasons. However, the

omplex-ity of the underlyingbeamformingnetwork (

BF N

) must be properlytaken into a ount

sin e it unavoidably grows be ause of the need to generate more than one pattern and

touse alarge numberofelements. Toover omethese limitations,sub-arrayingstrategies

(e.g., sub-array weighting [4℄ and overlapped sub-arrays [5℄) as well as sharing ommon

weightsbetweenthesumanddieren e hannels[6℄[7℄havebeenproposed. Thesub-array

weighting te hnique has re eived the widest interest as onrmed by the large number of

publishedresear hworks[8℄-[17℄. Generally,theproblemisformulatedasthe synthesis of

an optimal sum beam and the best ompromise dieren e patterns groupingthe array

elements into suitably weighted sub-arrays. Towards this purpose, several optimization

strategies have been applied. More spe i ally, the Simulated Annealing (

SA

) has been

used in[8℄to ompute the sub-arrayweightsfora-priori xed elementgroupings, whilea

Geneti Algorithm(

GA

)[9℄andtwodierentimplementationsoftheDieren eEvolution

(

DE

) algorithm[10℄[13℄ have been adopted to determine both weights and subarraying.

Moreover, anee tive hybridmethodhas been proposed in[11℄ to exploit the onvexity

of the problem with respe t to the sub-array weights. Whether, on one hand, global

op-timization is mandatory to deal with the non- onvex part of the problem, on the other,

elements of the admissible sub-array ongurations. Su h a bottlene k has been

e- ientlysolvedin[18℄ by meansofanex itationmat hing strategywherethe sub-arraying

grouping is guided by the similarity properties between the ex itations providing the

sum pattern and a set of referen e ex itations generating an optimal (referen e)

dier-en e pattern. The dimensionof the solutionspa ehas beensigni antlyredu ed and the

nal partitioning has been obtained by hoosing

Q

− 1

ut points (

Q

being the number

of sub-arrays) in a sorted list of real values ea h one related to an antenna element. In

su h a way, the admissible set of sub-array ongurations grows polynomiallyversus the

numberof elements with a non-negligible redu tion of the solution spa e if ompared to

standard approa hes. Furthermore,the essential solutionspa e has been represented by

means of anon- omplete binary tree [18℄ and, su essively, through amore ompa t and

non-redundant dire t a y li graph (

DAG

) [19℄. By virtue of its hill limbing behavior

(mandatory for non- onvex fun tionals), the AntColonyOptimizer (

ACO

)[20℄has been

used to look forthe optimal sub-array onguration both withinthe solutiontree [21℄ as

well asin the

DAG

[22℄. Although the

ACO

has shown to outperform the ad-ho

deter-ministi method alled Border Element Method (

BEM

) in both linear [18℄ and planar

[19℄ problems, it still presents some ine ien ies when large-dimension problems as for

planarar hite tures. It isworth pointingout that thesedrawba ks donot dependonthe

representation of the solution spa e or its dimension, but mainly on the ontrol of the

evolution pro ess. Indeed, if all edges of the

DAG

have the same probability of being

hosenat the initialization,some paths(i.e., sub-arraying solutions)turn out havingless

probability of being explored, while other paths are privileged. Su h a bias is undesired

and unavoidably limits the potentialities of the approa h. On the other hand, although

the non- omplete binary tree [21℄ is not ae ted by su h a drawba k, it is not suitable

for synthesizing large arrays be ause of high omputational osts and memory storage

requirements. In thiswork, anewsynthesis approa hbasedonanedge-weightings heme

is proposed toguarantee ea h path of the

DAG

be explored with anequal probability.

as well. Se tion 3 is devoted to the numeri alanalysis aimed at des ribing the behavior

of the proposed approa h and assessing its advantages and enhan ed potentialities over

previous implementations. Eventually, on lusionsare drawn (Se t. 4).

2 Mathemati al Formulation

Let us onsider a monopulse planar array of

2M × 2N

elements displa ed on a regular

latti e with inter-element spa ing

d

x

and

d

y

along the

x

and

y

axes, respe tively. The

antenna aperture is subdivided intofour symmetri alquadrants whose outputsare

om-bined to generate the sum and dieren e mode signals (Fig. 1)for the estimation of the

o-boresightangle(

OBA

),namelythedire tionofthetargetwithrespe ttotheele tri al

axis (i.e., the boresightdire tion) of the antenna[2℄[3℄.

Thesummode,usedbothintransmission(i.e.,forthegenerationoftheradarpulsesaimed

atsensingthe surroundingenvironment) andinre eption (i.e.,for dete tingthe presen e

and range of a target through a monopulse omparator), is obtained by summing the

signal from the four quadrants in phase. Under the assumption of quadrantalsymmetry

for the ex itations [24℄, the sum pattern an be expressed as follows

S (θ, φ) = 4

M

X

m=1

N

X

n=1

α

mn

cos

 2m − 1

2

ψ

x



cos

 2n − 1

2

ψ

y



(1)

where

α

mn

,

m

= 1, ..., M

,

n

= 1, ..., N

, are real ex itation weights. Moreover,

ψ

x

=

kd

x

sinθcosφ

,

ψ

y

= kd

y

sinθsinφ

,

k

=

2π

λ

is the free-spa e wavenumber,

λ

being the

wave-length.

The oupleofdieren emodesignalsusedtodeterminetheazimuthalandelevation

OBA

aregeneratedsumminginphasereversalpairsofquadrantsoftheoptimalex itations

β

mn

thataord adesireddieren epattern

D (θ, φ)

. More spe i ally,the followingdieren e

pattern issynthesized

D

az

(θ, φ) = 4j

M

X

m=1

N

X

n=1

β

mn

sin

 2m − 1

2

ψ

x



cos

 2n − 1

2

ψ

y



(2)

to tra k the target along the azimuthal plane [

D

az

_{(θ, φ) = D (θ, φ)}

℄, while the dieren e

pattern forthe elevationmode [

D

el

_{(θ, φ) = D θ, φ +}

π

2

℄ is given by

D

el

_{(θ, φ) = 4j}

M

X

m=1

N

X

n=1

β

mn

cos

 2m − 1

2

ψ

x



sin

 2n − 1

2

ψ

y



.

(3)

A ording to the sub-arraying strategy [4℄, the ex itations of the ompromise dieren e

patterns turn out tobe

b

m,n

= α

mn

Q

X

q=1

δ

cmnq

w

q

; m = 1, ..., M ; n = 1, ..., N ; q = 1, ..., Q

(4)

where

C

= {c

mn

; m = 1, ..., M; n = 1, ..., N}

with

c

mn

∈ [0, Q]

and

W

= {w

q

; q = 1, ..., Q}

arethedegreesoffreedomoftheproblemathand. Theyaretwosetsofintegervaluesthat

ode the elementgroupingandthe weights ofthe orresponding lusters, respe tively. In

(4),

δ

cmnq

isthe Krone ker delta fun tion dened as:

δ

cmnq

= 1

if the element belongs to

the

q

-th sub-array (i.e.,

c

mn

= q

)and

δ

cm,nq

= 0

, otherwise.

Following the guidelines des ribed in [18℄, given a set of independent ex itations

A

=

{α

mn

; m = 1, ..., M; n = 1, ..., N}

aording an optimal sum pattern, the solution of the

ompromisebetweensum and dieren epatterns isobtained by minimizingthefollowing

ost fun tion

Ψ (C) =

1

Γ

M

X

m=1

N

X

n=1

α

2

mn

(

g

mn

−

Q

X

q=1

δ

cmnq

w

q

(C)

)

2

(5) where

g

mn

,

βmn

αmn

,

m

= 1, ..., M

,

n

= 1, ..., N

is the set of optimal gains and

Γ ≤ M × N

isthe numberofradiating/a tiveelementsinea hquadrant. Equation(5)denes a'least

square' problem and its solution (i.e., the partition that minimizes the ost fun tion) is

a ontiguous partition whose As for the the unknown weighting ve tor

W

an be it is

analyti ally omputed for ea h trialsub-array onguration

C

as follows

w

q

(C) =

P

M

m=1

δ

cmnq

α

mn

β

mn

P

M

m=1

δ

cmnq

α

2

mn

.

(6)

weighted arithmeti mean of the orresponding

g

mn

values. In order to determine the

optimal sub-array ongurations

C

opt

, Eq. (5) is optimized a ording to the following

pro edure:

•

Step 1 - Contiguous Partition Method (CPM). Exploiting the theory in [23℄

forthe denitionof ontiguouspartitionsleast-squaregroupingofreal-valued

quan-tities,

C

opt

is obtained by hoosing

Q

subsets of the optimal gains

g

mn

sorted on a

line [19℄. Towards this end, a list

L

of

Γ

referen e parameters is generated setting

l

1

= min

m,n

{g

mn

}

and

l

Γ

= max

m,n

{g

mn

}

. In su h a way, the number of

admis-sible sub-array ongurations (or ontiguous partitions) belonging to the so- alled

essential solution spa e

ℜ

(ess) (1)

amountsto

U

(ess)

₌







Γ − 1

Q

− 1







.

•

Step 2 - Solution Spa e Representation. Thanks to the sorted list dened at

Step 1, the solutions in

ℜ

(ess)

are oded into a Dire t A y li Graph (

DAG

) [28℄.

The graph

G (Γ, Q, Ψ)

represented inFig. 2 is hara terizedby:



Q

rows ea h one ontaining

V

= (Γ − Q + 1)

vertexes,

V

being the maximum

number of elements that an be grouped in asub-array;

 a maximum depth

Γ

equal to the number of levels of the

DAG

and to the

dimension of the list

L

aswell asthe numberof vertexes along ea h

r

-thpath

P

_r

,

r

= 1, ..., U

(ess)

in

G

;

 asuitability fun tion

Ψ

(5)aimed atevaluatingthe goodness ofea h path

P

r

,

r

= 1, ..., U

(ess)

.

Thelevelsofthe

DAG

mapone-to-onetheelementsin

L

. Avertex

v

q,lq

,

q

= 1, ..., Q

,

l

q

= q, ..., q + V − 1

is identied by its row index,

q

, and the depth index,

l

q

.

Moreover, itsargument,

arg v

q,lq

= q

, indi atesthe sub-array membershipof ea h

array element of the list

L

. A path

P

of

Γ

vertexes and

Γ − 1

edges odes a trial

solution

C

. As shown inFig. 2,

e

+

q,lq

istheedge (ifpresent) onne tingthevertexes

(1)

Essentialwithrespe ttothesolutionspa ewhi h anbesampledusingstandardglobaloptimizers

whosedimensionis

U

= Q

Γ

v

q,lq

and

v

q,lq+1

onthe same row of the

DAG

, while

e

−

q,lq

is the edge (if admissible)

between the vertexes

v

q,lq

and

v

q+1,lq

ontwodierent rows of the

DAG

;

•

Step 3 - Edge Weighting. In [22℄, the

ACO

was used to explore the

DAG

for

identifying the best sub-array onguration

C

opt

. Sin e the quantity of pheromone

τ

_q,lq

±

(0)

,

q

= 1, ..., Q

,

l

q

= q, ..., q + V − 1

was uniformly set, the edges

e

±

q,lq

(0)

,

q

= 1, ..., Q

,

l

q

= q, ..., q + V − 1

have at the initializationthe same probability of

being explored. Be ause ofthe

DAG

stru ture and the value of theratio

V

Q

, su h a

hoi eae tsinanon-negligiblewaythe

ACO

-basedsamplingofthe

DAG

. Indeed,

someedgespathshaveahigherprobabilityofbeingsampledsin ethevertexes ould

belong to a dierent number of paths. As representative examples, the

DAG

s of

the ases (

Γ = 8

,

Q

= 3

)and (

Γ = 8

,

Q

= 6

),both having

U

(ess)

_{= 21}

, are reported

in Fig. 3(a) and Fig. 3(b), respe tively. Bysakeof larity,the numberof solutions

to whi h edge belongs tois indi ated.

In orderA properedge weighting s heme ishere adopted toassure auniform

prob-ability of samplingto ea h solution/pathand toallowan unbiased sear h aproper

edge weighting s heme isne essary. Towards this end, The ee t isthat of

in reas-ing/redu ing the level of pheromoneon ea hedge is in reased/redu ed

proportion-ally tothe number ofdierent ontiguouspartitiondened through that edge. Let

us observe that the number of paths leaving the root vertex

v

1,1

orresponds to

the dimension of the whole solution spa e

Ω

1,1

= U

(ess)

₌







Γ − 1

Q

− 1







, whilethose

departingfromthevertex

v

1,2

[Fig. 4(a)℄and

v

2,2

[Fig. 4(b)℄are

Ω

1,2

=







Γ − 2

Q

− 1







and

Ω

2,2

=







Γ − 2

Q

− 2







, namely the number of path through

G (Γ − 1, Q, Ψ)

and

availablefrom the generi vertex

v

q,lq

is equalto

Ω

q,lq

=







Γ − l

q

Q

− q







.

(7)

Therefore, the edge-weighting s heme is applied at the initialization (

j

= 0

) as follows: A ordingly,the levelof pheromone onedge

e

+

q,lq

isset to

τ

_q,lq

+

(j) =

Ω

q,lq+1

Ω

q,lq

(8)

while onthe edge

e

−

q,lq

τ

_q,lq

−

(j) =

Ω

q+1,lq

Ω

q,lq

.

(9)

It is worth noting that

Ω

q,lq

= Ω

q,lq+1

+ Ω

q+1,lq

.

•

Step4 -

DAG ACO

-Sampling. Iteratively,the

ACO

[20℄[25℄exploresthe

DAG

to

nd

C

opt

. Ea h ant of the olony

A

(j) = {a

t

(j) ; t = 1, ..., T }

,

T

being the olony

dimension,samplesthegraphstartingfromtheroot

v

1,1

and hoosingthenextedge

with probability

η

±

_q,lq

(j) =

τ

±

q,lq

(j)

τ

_q,lq

+

(j) + τ

−

q,lq

(j)

, q

= 1, ..., Q; l

q

= q, ..., q + V − 1.

(10)

The set of vertexes visited by an ant,

a

t

(j)

, from the root to the end of the graph

odes a path

P

t

(j) =

v

q,lq

; q = 1, ..., Q; l

q

= 1, ..., Γ

of

Γ

vertexes omposed by

Γ − 1

edgesthatidentiesatrialsub-array onguration

C

t

(j) = arg {P

t

(j)}

. The

optimality of ea h trial solution is quantied by the value of the ost fun tion in

orresponden e withthe orresponding subarray onguration,

Ψ (C

t

(j))

. Su h an

informationisexploitedtoupdatethe pheromonelevelonthe edgesof the

DAG

as

τ

_q,lq

±

(j + 1) ← (1 − ρ)

"

τ

_q,lq

±

(j) +

T

X

t=1

H

× Ψ

min

j

Ψ (C

t

(j))

#

(11)

where either

e

+

q,lq

or

e

−

q,lq

∈ P

t

(j)

and

Ψ

min

j

= min

t=1,...,T

{Ψ (C

t

(j))}

. Moreover,

ρ

∈ (0, 1]

and

H

are positive indexes that ontrol the pheromone evaporation and

depositionontheedgesofthe

DAG

. Thealgorithmstopswhenamaximumnumber

of iterations

J

max

is rea hed or the minimization of the ost fun tion rea hes a

stationary point (

j

= J

stat

), then

C

opt

hosen as

C

opt

_{= arg [min}

_j

_min

_t

_{{Ψ (C}

_t

_{(j))}] .}

(12)

3 Numeri al Results

A set of numeri alexperimentshas been arried outtopointout the potentialities ofthe

proposed approa h aswell asits improvementsover previousimplementations.

The rstexample dealswith thesynthesis ofasmallarray inordertodetailina

ompar-ative fashion the behavior of the edge-weighted approa h versus the uniform te hnique

[22℄. The array elementsare lo ated ona regular latti ewith

M

= N = 3

(

d

x

= d

y

=

λ

2

)

and belong toa ir ularsupportof radius

R

= 1.5λ

su h that the resultingarrangement

is omposed by

Γ = 32

radiators(

8

forea hquadrant). The ex itationsofthe sum mode

(Fig. 5) have been hosen to aord a Taylor pattern with

SLL

= −35 dB

and

n

¯

= 6

[24℄. As far asthe referen edieren e beam

D (θ, φ)

is on erned, aBaylisspatternwith

SLL

= −30 dB

and

n

¯

= 7

[24℄ has been used by setting the ex itationdistribution as in

Fig. 6.

The ompromise dieren e beam has been synthesized varying the numberof sub-arrays

in the range

Q

∈ [2, 6]

to analyze the performan e of the proposed method. First, the

Γ

optimalgains have been omputed and the list

L

generated (Fig. 7) a ording to the

CP M

.

Figure8showsthevaluesofthe ostfun tionforthebestsolutionsfoundbytheproposed

weighted-graph

ACO

-based (

W G

− ACO

) approa h and the

ACO

version in [22℄ when

a ording to the out omes from [22℄:

T

= 0.1 × U

(ess)

with a minimum value equal to

T

min

= 5

toexploit the ooperative behaviorof the olony,

J

max

= 1000

,

H

= 1

, and

ρ

=

0.05

. Itisworthnotingthatbothmethodsndtheglobaloptimumwhen

Q

issmallerthan

Γ

(e.g.,

Q

= {2, 3, 4}

) as onrmed by the plot in Fig. 9(a) that shows the ost fun tion

values for all the solutions belonging to

ℜ

(ess)

(

Γ = 8

,

Q

= {2, 3, 4}

). Nevertheless, the

bare

ACO

doesnotrea htheglobalsolutionwhen

Q

≃ Γ

(

Γ = 8

,

Q

= {5, 6}

)sin eitgets

stu kinalo alminimum[Fig. 9(b)℄. Asamatteroffa t,

Ψ

opt

W G−ACO





Q=5

= 5.023×10

−4

vs.

Ψ

opt

_ACO





Q=5

= 5.438×10

−4

and

Ψ

opt

W G−ACO





Q=6

= 1.685×10

−4

vs.

Ψ

opt

ACO





Q=6

= 4.965×10

−4

.

The orrespondingpathswithinthe

DAG

areasfollows:

P

opt

W G−ACO





Q=5

= {11123445}

vs.

P

opt

_ACO





Q=5

= {12234445}

and

P

opt

W G−ACO





Q=6

= {12234556}

vs.

P

opt

ACO





Q=6

= {11123456}

.

Let usnoti e that,despite the dimension of the solutionspa e doesnot vary from

Q

= 3

up to

Q

= 6

(see Tab. I),the uniform

ACO

isable toget thebest ompromisesolution

only in the former ase, while sub-optimal solutions are found otherwise. Su h a result

is not due tothe

DAG

representation of the solution spa e, but on the  ontrollevel of

the

ACO

[25℄(i.e., ontrolparameters,initialization riteria, onstraints,andtermination

onditions)whi hexploitsthepheromoneupdateme hanism tosamplethesolutionspa e

lookingfortheglobaloptimum. Asamatteroffa t,stillkeepingthesame

ACO

stru ture

presentedin[22℄,butinitializingthepheromonelevelsthroughtheweightedapproa h,the

reliability of the

DAG

sampling has been improved. As an illustrativeexample, Figure

10 gives a representation of the relative amount of pheromone on the edges of the

DAG

for the ase (

Γ = 8

,

Q

= 3

) [Fig. 10(a)℄ and the ase (

Γ = 8

,

Q

= 6

) [Fig. 10(b)℄.

More in detail,the thi kness of the segments between twovertexes isproportionaltothe

amount of pheromone on the orresponding edge. Moreover, the dotted lines indi ate

obliged hoi es when onlythe orresponding path is admissible.

The ine ien ies of the uniform-weight approa h is more evident when

U

(ess)

grows as

pointedout bythe plots of

Ψ

opt

inFig. 11. Thetest ase is here on erned with alatti e

of dimension

2M × 2N = 20 × 20

, a ir ularboundary

R

= 5.0λ

inradius,and a number

of a tiveelementsforea hquadrant equalto

Γ = 75

. Thenumberof sub-arrayshas been

toaordaTaylorpatternwith

SLL

= −35 dB

and

n

¯

= 6

[24℄,whilereferen eex itations

was used togenerate a Bayliss patternwith

SLL

= −30 dB

and

n

¯

= 7

[24℄. Con erning

the parameters of the

ACO

, the same setting of the previous experiment has been used

also introdu ing a maximum threshold

T

max

= 1000

(when

T

= 0.1 × U

(ess)

_{> T}

max

) on

the numberof ants forea h iterationtolimitthe omputationaltime. As expe ted (Fig.

11),theweightedapproa halwaysoutperformsthepreviousimplementationand,forea h

example (i.e.,

U

ess

value or

Q

value),solutions withlower ost fun tion values have been

determined.

As far as the omputational issues are on erned, let us onsider that the

CP U

-time required to omplete an

ACO

iteration is the same for both the weighted and uniform s heme. It is also worth noti ing that the improvements from the weighted s heme are

not on erned with the onvergen e speed, but rely in a more reliable sear h of the

optimal solution. For ompleteness and as a representative example, the ase

Q

= 5

needs

J

stat

= 85

iterations of the

W G

− ACO

(i.e., a total

CP U

-time of

16.34

se ) to samplethesolutionspa eofdimension

U

(ess)

_{= 1150626}

,whiletheuniformapproa hwith

thesame

ACO

parametersettingstopsafter

J

stat

= 122

iterations(i.e.,atotal

CP U

-time of

23.28

se ).

3.1 The Hybrid Extension

Although the

W G

− ACO

proved its ee tiveness, the omputation of the sub-array

weights through (6) does not guarantee the retrieval of the global optimum solution.

Moreover, itdoes notallowtoset onstraintsonthe desiredradiationpatternina dire t

fashion [26℄[27℄. The hybrid method in [11℄[29℄ over omes su h a limitation. On e

C

opt

was dened by meansof the

W G

− ACO

,the optimalweights

W

opt

an be omputed by

means of a onvex programming (

CP

) strategy[30℄, aimedat minimizing

Φ (W) = −Im

 ∂D (θ, φ)

∂γ



γ={θ,φ}











θ

= θ

₀

φ

= φ

0

(13)

to the maximizethe slope along the boresight dire tion

(θ

0

, φ

0

)

of the dieren e pattern

D (θ, φ)

, subje t to

Re

n

∂D(θ,φ)

∂γ

o

γ={θ,φ}









θ

= θ

0

φ

= φ

0

= 0

,

D (θ

0

, φ

0

) = 0

, and

|D (θ, φ)|

2

≤

M (θ, φ)

,

M (θ, φ)

being a positive upper bound fun tion on the power radiated in the

sideloberegion. Moreover,

Re { }

and

Im { }

indi atethe real andimaginarypart,

respe -tively. Furthermore,

θ

∈

0,

π

2



and

φ

∈ [0, 2π]

.

Toshowthebehaviorofthehybridmethod(

H

− W G − ACO

),anarraywith

M

= N = 5

elements lo ated on a square grid with uniform spa ing

d

=

λ

2

is used as ben hmark

geometry. Theapertureradiushasbeenset to

R

= 2.5λ

su hthat

Γ = 19

. Thesamesum

patternof the previousexampleshas been kept, whilethe referen edieren eex itations

β

mn

[Figs. 12(a)-(b)℄havebeen generatedby applyingthepro edurein[30℄ tosynthesize

the optimal dieren e pattern

D(θ, φ)

with

SLL

ref

_{= −25 dB}

shown in Fig. 12( ). In

order to design the ompromise dieren e pattern,

Q

= 5

sub-arrays have been used for

ea h quadrant.

The array lustering found by the

W G

− ACO

when exploring the solution graph with

T

= 30

ants is shown in Fig. 13(a). Su essively, the onvex programming pro edure

has been applied by onstraining the pattern to the same mask used to determine the

optimal dieren epattern[Fig. 12( )℄. The values

W

opt

are then given inTab. II, while

the orresponding pattern is shown in Fig. 13(b). For omparison, the same synthesis

problem has been addressed with the hybrid-

BEM

(

H

− BEM

) approa h [19℄ and the

resultsare reportedinFig. 13andTab. II, aswell. For ompleteness, Figure14plotsthe

levelof the se ondary lobenormalized tothe maximum of the powerpattern for ea h

φ

- ut,

φ

∈ [0 : 80

o

_]

[Fig. 14(a)℄and thesideloberatiodened as

SLR

(φ) =

SLL(φ)

max0

≤θ< π

₂

[D(θ,φ)]

,

φ

∈ [0 : 80

o

_]

[Fig. 14(b)℄. As it an beobserved, the

H

− W G − ACO

solution improves

that obtained with the

H

− BEM

interms of maximum

SLL

(

SLL

H−BEM

= −21.3 dB

vs.

SLL

H

−W G−ACO

= −25.4 dB

)and

SLR

value, whi hturnsout tobesmallerinalarge

part (i.e., almost

90

%) of the angular range. The reliability of the new hybridization in

better mat hing the referen e pattern

D (θ, φ)

[Fig. 12( )℄ is further pointed out in Fig.

15wherethemismat hindex

Ξ (θ, φ)

,





D

dB

(θ, φ) − D

dB

H

(θ, φ)



Finally,inorder togivesome indi ationsonthe omputational ostsof the hybrid

ACO

-based approa h, let us onsider that sampling the solution spa e of dimension

U

(ess)

₌

3060

requires

133 ACO

iterationsand

11CP

iterationswhen using the

H

− W G − ACO

[i.e.,

5.8 × 10

−2

se (

W G

− AGO

) and

850

se (

CP

)℄, while the

H

− BEM

performs

22

BEM

iterationsand

17 CP

iterations[i.e.,

<

10

−6

se (

BEM

) and

1370

se (

CP

)℄.

4 Con lusions

In this work, an edge weighting te hnique has been proposed for the ee tive

ACO

-based sampling of the graph ar hite ture oding the admissible lustering ongurations

ofasub-arrayed monopulseplanararray. Theadvantagesofthe

ACO

indealingwiththe

non- onvexityoftheproblemathandandtoexploregraphrepresentationsofthesolution

spa ehavebeenfurtherandbetterexploitedforenablingthesynthesisoflarge-s aleplanar

arrangements. Representativeresultshavedemonstratedtheenhan ementofthesynthesis

performan e with respe t to previous methods (e.g.,

BEM

) and implementations (i.e.,

[1℄ M. I. Skolnik,Radar Handbook (3rd Edition). USA: M Graw-Hill,2008.

[2℄ S. M. Sherman, Monopulse Prin iplesand Te hniques.USA: Arte hHouse, 1984.

[3℄ D. K.Barton, Monopulse Radar. USA: Arte h House, 1977.

[4℄ D. A.M Namara,Synthesis ofsub-arrayed monopulselineararrays through

mat h-ing of independently optimum sum and dieren e ex itations, IEE Pro . H

Mi- rowaves Antennas Propag., vol.135, no. 5,pp. 371-374, O t. 1988.

[5℄ T.-S. Lee and T.-Y. Dai, Optimum beamformers for monopulse angle estimation

using overlapping subarrays, IEEE Trans. Antennas Propag., vol. 42, no. 5, pp.

651-657, May 1994.

[6℄ T.-S. Lee and T.-K. Tseng, Subarray-synthesized low-side-lobe sum and dieren e

patterns with partial ommonweights, IEEE Trans.AntennasPropag.,vol.41,no.

6, pp. 791-800,Jun. 1993.

[7℄ A. F. Morabito and P. Ro a, Optimal synthesis of sum and dieren e patterns

witharbitrarysidelobessubje tto ommonex itations onstraints, IEEEAntennas

Wireless Propag. Lett., vol.9, pp. 623-626,2010.

[8℄ F. Ares, S. R. Rengarajan, J. A. Rodriguez, and E. Moreno, Optimal ompromise

among sum and dieren e patterns, J.Ele tromag. Waves Appl., vol. 10,pp.

1143-1555, 1996.

[9℄ P.Lopez,J.A.Rodriguez,F.Ares,andE.Moreno,Subarrayweightingfordieren e

patterns of monopulse antennas: Joint optimization of subarray ongurations and

weights, IEEE Trans.Antennas Propag.,vol.49, no. 11,pp. 1606-1608, Nov. 2001.

[10℄ S.Caorsi,A.Massa, M.Pastorino,and A.Randazzo, Optimizationofthe dieren e

patterns formonopulseantennasbyahybridreal/integer- odeddierentialevolution

optimal synthesis of monopulse antennas, IEEE Trans. Antennas Propag., vol. 55,

no. 4,pp. 1059-1066, Apr. 2007.

[12℄ P.Ro a, L.Mani a,A.Martini, andA. Massa,Synthesis oflargemonopulse linear

arraysthrough atree-basedoptimalex itationsmat hing, IEEEAntennasWireless

Propag.Lett.,vol. 6,pp. 436-439, 2007.

[13℄ Y. Chen, S. Yang, and Z. Nie, The appli ation of a modied dierential evolution

strategy tosomearray patternsynthesis problems, IEEE Trans.Antennas Propag.,

vol. 56,no. 7, pp. 1919-1927, Jul.2008.

[14℄ L. Mani a, P. Ro a, and A. Massa, On the synthesis of sub-arrayed planar array

antennas for tra king radar appli ations, IEEE Antennas Wireless Propag. Lett.,

vol.7, pp. 599-602,2008.

[15℄ L. Mani a, P. Ro a, and A. Massa, An ex itation mat hing pro edure for

sub-arrayed monopulse arrays with maximum dire tivity, IET Radar, Sonar &

Naviga-tion, vol.3, no. 1, pp. 42-48,Feb. 2009.

[16℄ P. Ro a, L. Mani a,and A. Massa, Dire tivity optimizationin planarsub-arrayed

monopulse antenna, PIER L, vol. 4,pp. 1-7, 2008.

[17℄ P. Ro a, L. Mani a, M. Pastorino, and A. Massa, Boresight slope optimization of

sub-arrayed lineararraysthroughthe ontiguouspartitionmethod, IEEE Antennas

Wireless Propagat. Lett., vol. 8,pp. 253- 257, 2008.

[18℄ L. Mani a, P. Ro a, A. Martini, and A. Massa, An innovative approa h based on

a tree-sear hing algorithmfor the optimalmat hing of independently optimum sum

and dieren eex itations, IEEE Trans.AntennasPropag.,vol.56,no.1,pp. 58-66,

Jan. 2008.

[19℄ L.Mani a,P.Ro a,M.Benedetti,andA.Massa, Afastgraph-sear hingalgorithm

enabling the e ient synthesis of sub-arrayed planar monopulse antennas, IEEE

of ooperating agents, IEEE Trans. Syst. Man and Cybern. B , vol. 26, no. 1, pp.

29-41, Feb. 1996.

[21℄ P. Ro a, L. Mani a, F. Stringari,and A. Massa, Ant olony optimization for

tree-sear hing based synthesis of monopulse array antenna, Ele tron. Lett., vol. 44, no.

13, pp.783-785,Jun. 2008.

[22℄ P. Ro a, L. Mani a, and A. Massa, An improved ex itation mat hing method

based onanant olony optimizationforsuboptimal-free lustering insum-dieren e

ompromisesynthesis, IEEETrans.AntennasPropag., vol.57,no.8,pp.2297-2306,

Aug. 2009.

[23℄ W.D.Fisher,Ongroupingofmaximumhomogeneity, Ameri anStatisti alJournal,

pp. 789-798,De . 1958.

[24℄ R. S.Elliott, Antenna Theory and Design.Wiley-Inters ien e IEEE Press, 2003.

[25℄ P. Ro a, M. Benedetti, M. Donelli, D. Fran es hini, and A. Massa, Evolutionary

OptimizationasApplied toInverse S attering Problems, InverseProblems, vol.25,

p. 1-41, 2009.

[26℄ P. Ro a, L. Mani a, and A. Massa, Synthesis of monopulse antennas through the

iterative ontiguous partitionmethod, Ele tron. Lett., vol. 43, no. 16,pp. 854-856,

Aug. 2007.

[27℄ P. Ro a, L. Mani a, A. Martini, and A. Massa, Compromise sum-dieren e

opti-mization through the iterative ontiguouspartition method, IET Mi rowaves,

An-tennas & Propagat.,vol. 3,no. 2, pp. 348-361,2009.

[28℄ P.Ro a, L.Mani a,and A.Massa, Anee tivemat hingmethodfor thesynthesis

of optimal ompromisebetween sumanddieren epatterns inplanararrays, PIER

of sub-arrayed monopulse linear arrays, IEEE Trans. Antennas Propagat., vol. 57,

no. 1,pp. 280-283, Jan. 2009.

[30℄ O. Bu i, M. D'Urso, and T. Isernia, Optimalsynthesis of dieren e patterns

sub-je t toarbitrary sidelobe bounds by using arbitrary array antennas, IEE Pro .

•

Figure 1. Sket h of asub-arrayed monopulse array antenna.

•

Figure 2.

DAG

representation of the solution spa e.

•

Figure 3.

DAG

Analysis (

Γ = 8

,

Q

= {3, 6}

) -Numberof trialsolutionstowhi h

the

DAG

edges belong to when (a)

Γ = 8

,

Q

= 3

and (b)

Γ = 8

,

Q

= 6

.

•

Figure 4. Edge WeightingApproa h -

DAG

regionsadmissiblefrom(a)thevertex

v

1,1

and (b)the vertex

v

2,2

.

•

Figure 5.

W G

− ACO

Numeri al Results (

M

= N = 3

,

Γ = 8

,

Q

∈ [2, 6]

)

-Ex itations of the optimalsum pattern (Taylor,

SLL

= −35 dB

,

n

¯

= 6

[24℄).

•

Figure 6.

W G

− ACO

Numeri al Results (

M

= N = 3

,

Γ = 8

,

Q

∈ [2, 6]

)

-Ex itations of the referen e dieren e pattern (Bayliss,

SLL

ref

_{= −30 dB}

,

n

¯

= 7

[24℄): (a) amplitudes and (b)phase weights.

•

Figure 7.

W G

− ACO

Numeri al Results (

M

= N = 3

,

Γ = 8

, Taylor -

SLL

=

−35 dB

-

n

¯

= 6

, Bayliss -

SLL

ref

_{= −30 dB}

-

n

¯

= 7

)- List

L

ofthe sortedoptimal

gains.

•

Figure 8. Comparative Assessment (

M

= N = 3

,

Γ = 8

,

Q

∈ [2, 6]

, Taylor

-SLL

= −35 dB

-

n

¯

= 6

, Bayliss -

SLL

ref

_{= −30 dB}

-

n

¯

= 7

) - Cost fun tion

values in orresponden e with the optimal solutions found by the

ACO

and the

W G

− ACO

.

•

Figure 9. Comparative Assessment (

M

= N = 3

,

Γ = 8

,

Q

∈ [2, 6]

, Taylor

-SLL

= −35 dB

-

n

¯

= 6

, Bayliss -

SLL

ref

_{= −30 dB}

-

n

¯

= 7

) -Cost fun tionvalues

ofthe solutions odedwithinthe

DAG

when (a)

Q

∈ [2, 5]

and(b)

Q

= {5, 6}

. The

ACO

and the

W G

− ACO

solutionare denoted by with a ir le.

•

Figure 10.

W G

− ACO

Numeri al Results (

Γ = 8

,

Q

= {3, 6}

) - Edge weighting

approa h as applied to the

DAG

sampling when (a)

Γ = 8

,

Q

= 3

and (b)

Γ = 8

,

•

Figure 11. Comparative Assessment (

M

= N = 20

,

Γ = 75

,

Q

∈ [2, 20]

, Taylor

-

SLL

= −35 dB

-

n

¯

= 6

, Bayliss -

SLL

ref

_{= −30 dB ¯}

_n

_{= 7}

) - Cost fun tion

values in orresponden e with the optimal solutions found by the

ACO

and the

W G

− ACO

versus (a) the dimension of the solution spa e,

U

(ess)

, and (b) the

number of sub-arrays,

Q

.

•

Figure 12. Hybrid Extension (

M

= N = 5

,

Γ = 19

, Referen e dieren e [30℄

-SLL

ref

= −25 dB

)-Referen edieren eex itations: (a)amplitudes and(b)phase

weights. Power patternof the referen emode ( ).

•

Figure 13. Hybrid Extension (

M

= N = 5

,

Γ = 19

, Referen e dieren e [30℄

-SLL

ref

_{= −25 dB}

,

Q

= 5

) - Plots of (a)( ) the sub-array ongurations and of

(b)(d)the relativepowerpatterndeterminedwith (a)(b) the

H

− W G − ACO

and

( )(d)the

H

− BEM

.

•

Figure 14. Hybrid Extension (

M

= N = 5

,

Γ = 19

, Referen e dieren e [30℄

-SLL

ref

_{= −25 dB}

,

Q

= 5

)-Plots of (a) the

SLL

and (b)the

SLR

of the solutions

found by the

H

− W G − ACO

and the

H

− BEM

.

•

Figure 15. Hybrid Extension (

M

= N = 5

,

Γ = 19

, Referen e dieren e [30℄

-SLL

ref

_{= −25 dB}

,

Q

= 5

) - Plot of the mismat h fun tion

Ξ(θ, φ)

when applying

the (a)

H

− W G − ACO

and the (b)

H

− BEM

.

TABLE CAPTIONS

•

Table I.

W G

− ACO

Numeri al Results (

M

= N = 3

,

Γ = 8

,

Q

∈ [2, 6]

)

-Dimension of the solution spa e

U

(ess)

.

•

Table II. Hybrid Approa h (

M

= N = 5

,

Γ = 19 × 4

,

Q

= 5

) - Values of the

y

x

(1,1)

(m,n)

(M,1)

(1,N)

(−1,1)

(−m,n)

(1,−1)

(m,−n)

(−1,N)

(M,−1)

(1,−N)

(−1,−N)

(−1,−1)

(−M,−1)

(−M,1)

(−m,−n)

v

_1,1

v

_1,2

v

_2,2

v

_2,3

v

_1,V

v

_1,V-1

v

_2,V

v

_2,V+1

v

Q,Q+1

Q-1

v

Q-1,

v

_Q-1,Q

v

Q-1,

Q+V-2

v

Q-1,

Q+V-1

e

+

1,1

e

+

1,2

e

+

1,V-2

e

+

1,V-1

e

+

2,2

e

+

2,3

e

+

2,V-1

e

+

2,V

e

−

1,1

e

−

1,2

e

−

1,V-1

e

−

1,V

e

−

2,2

e

−

2,3

e

−

2,V

e

−

2,V+1

e

+

_Q,Q

e

+

_Q,Q+1

e

+

_Q,Q+V-2

v

Q,

Q+V-1

v

Q,

Q+V

e

+

_Q,Q+V-1

e

−

_Q-2,Q-2

e

−

_Q-2,Q-1

e

−

_Q-2,Q+V-1

e

−

_Q-2,Q+V

e

−

Q-1,Q-1

e

+

Q-1,Q-1

e

+

Q-1,Q

e

+

Q-1,Q+V-2

e

+

Q-1,Q+V-1

e

−

Q-1,Q+V-2

e

−

Q-1,Q+V-1

e

−

Q-1,Q

v

Q,Q

v

= 1

v

= 2

v

= V

q

= 1

q

= 2

q

= Q

l

Γ

l

2

l

1

2 -G. Oliv eri and L. P oli , Optimal Sub-Arra ying of Compromise Planar ... 22

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

1

1

1

1

1

1

5

4

3

2

1

10

6

3

1

15

1

1

1

1

1

4

5

6

3

2

1

(a)

4

4

4

3

3

3

5

5

5

6

6

6

1

1

1

2

2

2

5

15

1

1

10

4

5

1

3

1

1

4

1

6

3

3

1

1

1

1

1

2

1

1

1

6

1

(b)

v

1,1

v

1,2

v

2,2

v

2,3

v

1,V

v

1,V-1

v

2,V

v

2,V+1

v

Q,Q+1

v

Q,

Q+V-2

Q-1

v

Q-1,

v

_Q-1,Q

v

Q,

Q+V-1

v

Q-1,

Q+V-2

v

Q-1,

Q+V-1

v

Q,Q

(a)

v

1,1

v

1,2

v

2,2

v

2,3

v

1,V

v

1,V-1

v

2,V

v

2,V+1

v

Q,Q+1

v

Q,

Q+V-2

Q-1

v

Q-1,

v

_Q-1,Q

v

Q,

Q+V-1

v

Q-1,

Q+V-2

v

Q-1,

Q+V-1

v

Q,Q

(b)

-3

-2

-1

1

2

3

m

3

2

1

-1

-2

-3

n

0

0.2

0.4

0.6

0.8

1

α

mn

-3

-2

-1

1

2

3

m

3

2

1

-1

-2

-3

n

0

0.2

0.4

0.6

0.8

1

|

β

_mn

|

(a)

-3

-2

-1

1

2

3

m

3

2

1

-1

-2

-3

n

0

π

/4

π

/2

3π

/4

π

arg(

β

_mn

)

(b)

l

₁

l

₂

l

₃

l

₄

l

₅

l

₆

l

₇

l

_Γ

=l

₈

10

-4

10

-3

10

-2

2

3

4

5

6

Cost Function Value,

Ψ

opt

Number of Subarrays, Q

ACO-WG

ACO

10

-3

10

-2

10

-1

0

U

(ess)

/4

U

(ess)

/2

3U

(ess)

/4

U

(ess)

Cost Function Value,

Ψ

(C

r

)

Solution Index, r

Ψ

opt

WG-ACO

Ψ

opt

ACO

Q=2

Q=3

Q=4

(a)

10

-4

10

-3

10

-2

10

-1

0

U

(ess)

/4

U

(ess)

/2

3U

(ess)

/4

U

(ess)

Cost Function Value,

Ψ

(C

r

)

Solution Index, r

Ψ

opt

WG-ACO

Ψ

opt

ACO

Q=5

Q=6

(b)

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

(a)

1

1

1

2

2

2

3

3

3

4

4

4

5

5

5

6

6

6

(b)

10

-5

10

-4

10

-3

10

-2

10

-1

10

0

10

3

10

6

10

9

10

12

10

15

10

18

Cost Function Value,

Ψ

U

ess

WG-ACO

ACO

(a)

10

-5

10

-4

10

-3

10

-2

10

-1

2

4

6

8

10

12

14

16

18

20

Cost Function Value,

Ψ

Sub-array Number, Q

WG-ACO

ACO

(b)

-5

-4

-3

-2

-1

1

2

3

4

5

m

-5

-4

-3

-2

-1

1

2

3

4

5

n

0

0.2

0.4

0.6

0.8

1

|

β

mn

|

(a)

-5

-4

-3

-2

-1

1

2

3

4

5

m

-5

-4

-3

-2

-1

1

2

3

4

5

n

0

π

/4

π

/2

3π

/4

π

arg(

β

_mn

)

(b)

−1

v

1

1

u

−

1

0

Relative P ower

[dB]

− 60

(c)

−1

v

1

q=1

q=4

q=2

q=3

q=Q=5

1

u

−

1

0

Relative P ower

[dB]

− 60

(a)

(b)

−1

v

1

q=1

q=4

q=2

q=3

q=Q=5

1

u

−

1

0

Relative P ower

[dB]

− 60

(c)

(d)

-36

-32

-28

-24

-20

0

10

20

30

40

50

60

70

80

SLL(

φ

) [dB]

φ

[deg]

H-WG-ACO

H-BEM

Reference

-30

-26

-22

-18

-14

-10

0

10

20

30

40

50

60

70

80

SLR(

φ

) [dB]

φ

[deg]

H-WG-ACO

H-BEM

Bayliss

−1

v

1

1

u

−

1

32

Ξ(θ, φ)

0

(a)

−1

v

1

1

u

−

1

32

Ξ(θ, φ)

0

(b)

Γ

8

Q

2

3

4

5

6

U

(ess)

7

21

35

35

21

w

1

w

2

w

3

w

4

w

5

H

− W G − ACO 1.0942 2.0305 2.9870 4.5573 5.6723

H

− BEM

1.0488 2.7605 4.2845 4.8999 5.5077