730
IEEE TRANSACTIONS
ON
ANTENNAS AND PROPAGATION,
VOL.
44,
NO.
5,
MAY
1996
F ac al Design
o
Mul iband
and Low Side-Lobe A ays
Ca les Puen e-Balia da, Membe ,
IEEE,
and Ra ael
Pous,
Membe ,
IEEE
Abs ac -
Mos a ay ac o design echniques a e highly
dependen
on
he ope a ing wa eleng h.
In
his pape , a no el
echnique based
on
ac al s uc u es is desc ibed o mul iband
ope a ion. The analysis is ocused in wo di e en app oaches:
he ac al spa ial a angemen o a ay elemen s and he ac al
design o a ay ac o s. Al hough he pa e ns
o
ac al a ays
show some in e es ing simila i y p ope ies
a
se e al bands, he
di ec i i y is no held cons an h ough he bands. Ne e heless,
such s uc u es ha e been shown o be use ul o designing low
side-lobe a ays wi h equally weigh ed cu en elemen s. On he
o he hand, he ac al a ay ac o s p esen ed do keep he
same shape a se e al bands because hey a e designed as sel -
simila cu es. The a ays ha would syn hesize such pa e ns
p esen
a
cha ac e is ic powe -law cu en dis ibu ion analogous
o he spec al dis ibu ion o he bandlimi ed ac al Weie s ass
unc ion.
I.
INTRODUCTION
HE main p oblem wi h he design o wideband o
equency-independen adia ing sys ems is ha once
he sys em is designed o ma ch i s cha ac e is ic size o
he ope a ing wa eleng h, i s pa ame e s will change when
ope a ing a di e en wa eleng hs. Fo single an ennas, his
cha ac e is ic size is he leng h o he an enna. Fo a ays, he
spacing be ween elemen s is also impo an .
This cons ain has been p esen in he de elopmen
o
all
cu en equency-independen an ennas. Angles, cones, and
spi als a e examples o some shapes ha ha e success ully
been used o design equency-independen an ennas
[
11-[4].
All o hem ha e a common cha ac e is ic: hei shape is
in a ian unde a scaling ans o ma ion o , in o he wo ds,
hei shape ela i e o all wa eleng hs is cons an . Simila ly,
log-pe iodic dipole a ays
[5]
a e designed wi h dipoles o
se e al leng hs and
a
a iable spacing be ween elemen s in
such a way ha he s uc u e keeps he same shape unde
some scaling ans o ma ions as well.
F ac als a e sel -simila s uc u es. This means ha hei
shape emains he same unde a change o scale. They a e said
o
possess no cha ac e is ic size 261-[8]. Hence, i hey could
be
used
in
he
design
o
adia ing sys ems, one would
expec
o hem a mul iwa eleng h ope a ion. This pape analyzes wo
possible app oaches o he ac al design o mul i equency
a ays. Fi s , ac al spacing be ween a ay elemen s is ana-
lyzed (Sec ion
11).
Second, a ac al design o a ay pa e ns is
Manusc ip ecei ed May 16, 1994; e ised Augus 21, 1995. This
wo k
was
suppo ed by he Spanish Commission
o
Science and Technology unde
G an MAT95-1038-C02-02.
The au ho s a e wi h he Depa men
o
Signal Theo y and Communica ions,
Poly echnic Uni e si y
o
Ca alonia, Ba celona, Spain
0807
1.
Publishe I em Iden i ie
S
0018-926X(96)03232-2.
0018-926)3/96$05
in oduced and he ela i e cu en dis ibu ion o such a ays
is de i ed (Sec ion
111).
Al hough he ac al a ays analyzed
in Sec ion
I1
p esen some in e es ing simila i y p ope ies
( he a ay ac o has a simila shape a se e al equencies),
he di ec i i y and he main-lobe wid h is no held cons an
wi h equency due o ac al unca ion. Ne e heless, such
a ac al dis ibu ion o he a ay elemen s ha e shown some
o he in e es ing p ope ies: hey can be used o syn hesize low
side-lobe adia ion pa e ns wi h a uni o m cu en dis ibu ion
o elemen s wi hin he a ay. Tha is, some app oxima ions
o common low side-lobe adia ion pa e ns, like hose co -
esponding o iangula and binomial dis ibu ions, can be
achie ed by placing a se o equal ampli ude a ay elemen s
acco ding o he same algo i hm used o gene a e ac ally
spaced a ays.
The undesi ed changes in he pa e n pa ame e s o he
analyzed ac al a ays has led he esea ch o he design o
he ac al pa e ns in Sec ion
111.
In
his sec ion, he design
is ocused on he de ini ion
o
he a ay ac o . A well-known
se o ac al cu es, he Koch cu es, a e used o desc ibe
pa e ns ha keep he same shape a di e en scales. When
he isible ange o he a ay ac o is modi ied by means o
a change in he a ay ope a ing equency, he a ay adia es
h ough a scaled e sion
o
he whole a ay ac o wi h he
same di ec i i y and lobe p o ile. The a ays ha gene a e such
ac al pa ems ha e been shown o ha e powe -law cu en
dis ibu ions which p esen some in e es ing scaling p ope ies
as well
[9]-[ll].
11.
FRACTAL
ARRAYS
A ay ac o s a e highly dependen on he ope a ing wa e-
leng h. An inc ease in he ope a ing equency
is
ansla ed
in o an expansion o he isible ange. This means ha g a ing
lobes will appea in he adia ion pa e n a high enough
equencies. This is a s ong incon enience because al hough
many di e en echniques based on an ideal cu en eeding
o each a ay elemen allow
us
o syn hesize a ay ac o s
wi h a desi ed di ec i i y o side-lobe a io
(SLR),
he design
is equency dependen . Mos o hese echniques assume
a
uni o m spacing be ween elemen s which becomes he main
bandwid h limi ing ac o .
In his sec ion, a nonuni o m ac al spacing o he elemen
dis ibu ion o an an enna a ay will be analyzed. An ideal
cu en eeding sys em o each a ay elemen is assumed, as
usual, in a ay heo y. Kim and Jagga d [lo] i s p oposed a
nonuni o m andom ac al spacing o imp o ing he
SLR
o andom a ays in 1986. Also, in 1992, Jagga d
e
al.
8.00
0
1996
IEEE
PUENTE-BALIARDA
AND
POUS:
FRACTAL
DESIGN
OF
MULTIBAND
AND
LOW
SIDE-LOBE
ARRAYS
73
1
showed how he di ac ion pa e n om a iadic Can o
a ge p esen ed some in e es ing sel -simila i y p ope ies wi h
inc easing g ow h s ages. He e, a de e minis ic ac al a ay
designed by placing he a ay elemen s a he poin s o a
gene alized Can o se is in oduced.
Al hough o he ac al s uc u es could be used o designing
a ays, his pa icula one has been chosen o he analysis
because i p o ides a simple, well-known ac al se o disc e e
poin s ha can desc ibe a linea a ay. A bandlimi ed
[9]
e sion o he ac al s uc u e can be cons uc ed i e a ing
se e al con olu ions which leads o an analy ical exp ession
o he a ay ac o , p o iding a compac way
o
de i ing
he p ope ies o he esul ing a ay. One o he cha ac e is ic
ea u es o he Can o se and o he ac al s uc u es is ha
hey con ain an in ini e numbe o subse s a di e en scales
which a e all iden ical o he whole se . Thus, i an a ay
is buil by placing he elemen s a he poin s o he Can o
se , one could expec hese smalle subs uc u es o adia e a
sho e wa eleng hs in he same way ha he whole s uc u e
adia es a longe wa eleng hs.
To analyze he beha io o an a ay based on such a ac al,
le
us
i s poin ou an al e na i e p ocedu e o gene a e he
Can o se . The p ocedu e s a s by aking wo del a unc ions,
spaced a dis ance d in he
z
axis, as he basic s uc u e (usually
known as
gene a o
in ac al e minology
[8],
[9]).
Then,
he gene a o is scaled by a ac o o h ee o ob ain ano he
s uc u e composed by wo del a unc ions spaced
d/3.
I one
con ol es hese wo s uc u es, a se o ou del a unc ions
will be placed a he poin s
o
a Can o se cons uc ed wi h
only wo i e a ions. I can be seen ha his con olu ion could
be i e a ed an in ini e numbe o imes o ob ain he comple e
se . Tha is, i we call
(z)
he wo del a unc ion gene a o ,
he whole Can o se
c(z)
can be w i en as
c(z)
'.
'
j(z)
*
(3
.
2)
*
(9
.
z)
.
'.
*
(3'L
'
z)
.
'.
.
(1)
I should also be no ed ha his p ocedu e can be ca ied
ou wi h a di e en gene a o (wi h an a bi a y numbe
o
del a unc ions) and wi h a di e en scaling ac o
S.
In his
case, a gene alized Can o s uc u e (he ea e , Can o a ay)
can be de ined as ollows:
whe e
X
symbolizes he con olu ion ope a o . The s uc u e
jus de ined is a u he gene aliza ion o he Can o ba
in oduced by Sun and Jagga d in
[16]
and co e s om he
o iginal Can o se o he iangula Can o a ay and mo e
sophis ica ed s uc u es.
The a ay ac o co esponding o
~(z)
can be w i en in
e ms o he Fou ie ans o m o he gene a o
F($)
as
C($)
=
. .
'
S2
.
F(S$)
'
s
.
F(6$)
.
F($J)
(3)
whe e
!41/
is de ined as usual in a ay heo y as
$
=
kd
cos
H
+
p
(4)
wi h
d
being he spacing o he gene a o a ay,
H
he angle
be ween he di ec ion o p opaga ion and he axis o he
a ay,
/3
he p og essi e phase-shi o he gene a o a ay,
and
k
=
27 /X
he wa e numbe . This al e na i e way o
gene a ing he ac al and de i ing i s a ay ac o can gi e a
physical insigh on he modula ion e ec s on he pa e ns i s
sugges ed in
[
111.
Tha is, he esul ing a ay ac o
(3)
can
be ob ained by epea edly modula ing (mul iplying) he a ay
ac o o he gene a o wi h a scaled e sion o i sel .
I can be seen ha a equency change by a ac o o
implies a p opo ional scaling o bo h he
$
pa ame e and he
a ay ac o
C($).
Tha is,
C($)
becomes
=
i
F(i!).
n=--00
Now, i he equency shi
is aken o be
sp
hen he a ay
ac o will be
C(SP4)
=
11
F
(2)
n=
-cc
CO
=
11
F(&)
n=--w
=
F($)
m=-m
which implies ha he a ay ac o gene a ed by he Can o
s uc u e is a log-pe iodic
(LF')
unc ion wi h a log-pe iod
5.
Tha is, in loga i hmic scale
The main conclusion de i ed om
(7)
is ha he in ini e
Can o a ay would ha e he same a ay ac o a an in ini e
numbe o bands (which is a ema kable p ope y no sha ed
by uni o m spaced a ays, e en when in ini e). Be o e going
any u he in ou conclusions wo impo an ac s should be
poin ed ou . Fi s , his would be a mul iband sys em and no a
equency-independen sys em, since
(7)
only implies ha he
beha io will be he same a se e al bands spaced by a ac o
o
6,
bu does no imply a equency-independen beha io
wi hin each band; howe e , a mul iband beha io would be a
signi ican imp o emen o an a ay design in applica ions,
such as equency hopping schemes in ada and sp ead
spec um communica ion sys ems. Second, his p ope y would
apply only o he in ini e a ay. The co esponding bandlimi ed
ealiza ion o he ac al s uc u e will hold he simila i y
p ope ies h ough as many bands as i e a ions used in he
132
IEEE
TRANSACTIONS ON ANTENNAS AND PROPAGATION,
VOL.
44,
NO.
5,
MAY
1996
2,
,
,
Can o
A ay
-
N=Z,
del a=3.
M=6
,
,
,
1
0
508
0
E06
L
a,
W
x
-
F
z04
02
"
-TOO
-80
-60
-40 -20
0
20
40
60
80 100
Z
(wa eleng hs)
Fig.
1.
Can o a ay based on he classical Can o se . The a ay has
64
elemen s and i
is
cons uc ed om a wo-elemen gene a o and a
log
pe iod
6
=
3.
Can o A ay Fac o
-
N=2, del a=3,
M=6
~~
1'
I
05
0
(a)
O
-3 -2
1
2
3
is kep almos cons an a di e en wa eleng hs. This kind
o simila i y is analogous o he simila i y a se e al ac al
g ow h s ages o he di ac ion pa e ns shown in
1111
o a
ixed wa eleng h. One can explain his esul by no icing ha
each ime he wa eleng h is inc eased, he closes elemen s
collapse in o an equi alen single elemen om he adia ion
poin o iew. Hence, a each longe wa eleng h he equi alen
a ay e ec i ely loses one i e a ion o g ow h s age. I he
ac al we e ideal, i would keep exac ly he same shape a e
collapsing, bu since i is a bandlimi ed ac al, he a ay and
i s adia ion pa e ns emain simila bu no equal, a each
equency. Toge he wi h he a ay's high seconda y lobes, his
implies an incon enien ea u e on he pe o mance o he a -
ay: he main lobe wid h inc eases as he equency is educed.
The p oblem o he high seconda y lobes o he pa e n
is ela ed o an a ay cha ac e is ic known in ac al heo y
as
lacuna i y.
A ac al s uc u e is said o p esen high
lacuna i y when i has la ge gaps be ween he di e en ac al
subs uc u es
[6],
[7]. In e ms o he a ay o Fig.
1,
he
gaps be ween suba ays a e oo la ge compa ed o he smalles
ope a ing wa eleng h which has been chosen o ma ch wice
he dis ance be ween wo elemen s
o
he smalles subs uc u e.
To analyze he e ec o he lacuna i y and he log pe iod in
he pa e n con o ma ion, he analysis will be pa icula ized o
he simples case-Can o a ays cons uc ed om a uni o m
elemen ampli ude gene a o .
05
0
-
-1
-08
-06
-04 02 0
02 04
06 08
1
a,
U
0
LL
b05-
$0
-
-03
-02
01
02
03
-01
(c)O
a
-0
1
-008
-006
004
-002
(d)O
002 004 006 008
0
1
-003 -002
-0
01
0 001
002
003
PSI
(e)
Fig.
2.
A ay ac o o he Can o a ay on Fig.
2.
The a ay ac o
is
plo ed
o
i e ope a ing wa eleng hs: (a)
XO
=
6/2,
(b)
XI
=
3x0,
(c)
A2
=
9x0,
(d)
A3
=
27x0,
and (e)
A4
=
81x0.
The simila lobe s uc u e o he pa e ns
a hose equencies can be no iced.
cons uc ion p ocedu e, bu no h ough an in ini e se o bands.
The example in Figs.
1
and
2
illus a es his ac .
The pa e ns on Fig.
2
p esen some in e es ing ea u es.
They
look
simila in he sense ha hey show a simila
dis ibu ion o he main seconda y lobes and ha he
SLR
A.
One Pa icula Case
o
he Gene alized Can o
A ay:
The
Uni o m
Gene a o
In
he ollowing, he analysis will be ocused on he pa ic-
ula case o he gene a o unc ion
which ep esen s a se o N-del a unc ions o equal ampli ude,
spaced a dis ance
d
and cen e ed a he o igin. Acco ding o
(3),
he a ay ac o o a Can o a ay gene a ed a e
M
i e a ions wi h he abo e gene a o can be w i en as
whe e
N
is he numbe o elemen s o he gene a o in
(8)
and 6 is he log pe iod. As i will be shown in his sec ion,
he a io
6/N
is s ongly ela ed o he lacuna i y
o
he ac al
and de ines some
o
he p ope ies o he a ay ac o . Also,
his a io can be ela ed o he ac al dimension
D
which can
be calcula ed as [6], [9],
[lo],
and
[12]
Equa ion
(10)
gi es a ac al dimension o D
=
0.63 o he
case p esen ed in Fig.
2.
A his poin , he analysis will be
ocused in wo cases, when
6
becomes close o uni y and
when
6
is la ge han
N.
PUENTE-BALIARDA AND POUS: FRACTAL DESIGN OF MULTIBAND AND LOW SIDE-LOBE ARRAYS
~
733
Can o
A ay
-
M=6,
N=2.
del a=l
1
OL
-2
-1
5
1
-0
5
0
0.5
1
15
2
Z
(wa eleng hs)
Fig.
3.
Nea -binomial a ay gene a ed a e six i e a ions om a wo-elemen
gene a o and a log-pe iod
5
=
1.1.
Al hough he a ay has a uni-
o m-ampli ude dis ibu ion o elemen s, i s pa e n is close o ha
o
he
binomial a ay.
The Binomial A ay as a Pa icula Case
o
he Can o
A ay:
The pa icula case 6
=
1
ep esen s he con olu ion
o
M
equal dis ibu ions. In his case,
(9)
can be ew i en as
Thus, he binomial dis ibu ion can be ob ained by aking a
gene a o wi h only wo elemen s, i.e.,
N
=
2 and a log-
pe iod
S
=
1.
A la ge numbe o elemen s in he gene a o will
esul in a iangula dis ibu ion o
M
=
2 and a dis ibu ion
ha ends o a Gaussian shape o inc easing alues o M
(cen al limi heo em). In his la e case
(N
>
2),
he SLR
in dB dec eases linea ly wi h he numbe o i e a ions, which
is a con enien me hod o designing low-side lobe a ays.
Ne e heless, his me hod has a g ea incon enience- he
dynamic ange o he elemen ampli udes wi hin he a ay is
so
la ge ha small e o s in he eeding ne wo k change he
weigh o smalles elemen s, dis o ing he inal pa e n.
Uni o m-Ampli ude, Nonuni o mly-Spaced A ays o Nea -
Binomial Pa e n Design:
Now le us ake a gene a o wi h
wo elemen s and chose a log pe iod close o
1,
o ins ance
6
=
1.1.
One can expec he co esponding pa e n o look
e y simila o he binomial one since he exp ession o he
a ay ac o
(9)
will be almos he same. Hence, al hough in
his case he a ay loses i s mul iband p ope ies, i p esen s
a e y in e es ing low side-lobe a ay ac o . Also, he e is a
undamen al di e ence be ween he shape o his a ay and
he binomial one. In he
h
=
1.1
case, none
o
he elemen s o
he M con ol ing suba ays o e lap, hus ha ing a inal a ay
wi h a uni o m-ampli ude dis ibu ion o e
2M
nonuni o mly-
spaced elemen s (Fig.
3).
The main ea u e o his a ay is ha al hough he pa e n is
e y close o he binomial one (SLR
<
65
dB), he ampli ude
dis ibu ion o he elemen s is uni o m which g ea ly simpli ies
he eeding ne wo k. A uni o m cu en ampli ude dis ibu ion
h ough all he a ay elemen s can be ob ained by means
o
a combina ion o
X/4
and
X/2
ansmission lines, ega dless
o
mu ual coupling e ec s
[13].
Also, i can be seen ha his
scheme could be epea ed o gene a e e y close pa e ns o any
o
he amily desc ibed in
(I
11,
bu wi h uni o m dis ibu ions
o elemen s.
The g aph in Fig.
4
can be used o he design o such nea -
binomial, low side-lobe a ays. I ep esen s he SLR on he
a ay ac o o se e al log pe iods and numbe o i e a ions
(M).
I is in e es ing o ema k ha he side lobes a e g ea ly
educed o log pe iods below
1.5.
Also, he SLR ends o he
same le el ega dless o
M
when he log pe iod is o e
6
=
2,
which is a logical esul i one no ices ha abo e his alue
he pa e ns become sel -simila a each g ow h s age
[ll].
When
6
is below
2
he SLR is lowe o a highe M and he
di ec i i y o he a ay inc eases because he o al leng h o he
a ay is la ge as well. Ne e heless, oo many i e a ions will
make he a ay s uc u e dense and some elemen s will be
placed e y close o each o he , which can make he physical
implemen a ion
o
he a ay complica ed.
The Uni o m A ay as a Pa icula Case
o
he Can o
A ay:
The case unde s udy
is
now
S
=
N.
Again, by aking
he gene al exp ession in
(9),
he ollowing exp ession can be
de i ed:
sin
(NM
;)
which is he a ay ac o o a uni o m dis ibu ion o
NM
elemen s. The cons uc ion o his uni o m a ay as a con o-
lu ion o M-uni o m dis ibu ions a di e en scales can be
seen again as a pa icula case o he p ocess desc ibed in
(2).
The scaling pa ame e is such ha a each i e a ion, he
sepa a ion be ween suba ays
11s
equal o he spacing be ween
wo elemen s o hose suba ays. In con as o he cases
p esen ed be o e, he SLR hle e does no change wi h he
numbe o i e a ions, always keeping i s alue a ound
13
dB.
Op imum Lucuna i y
o
Sel -simila ,
Low
Side-Lobe F ac-
al-Can o A ays:
Le us now analyze he ela ionship be-
ween he lacuna i y o he ac al s uc u e and he SLR
o
he pa e ns o a mo e gene al case. As poin ed ou in
(3),
he Can o a ay ac o s can be unde s ood as a p oduc o
M
subpa e ns a di e en scales. Each ime he s uc u e is
con ol ed wi h he nex wide scaled gene a o , he pa e n
is mul iplied wi h a comp essed e sion
o
he gene a o ’s
a ay ac o . By choosing a log pe iod e y close o
6
=
1,
as shown p e iously, he comp ession becomes e y sligh a
IEEE TRANSACTIONS ON ANTENNAS AND. PROPAGATION,
VOL.
44,
NO.
5,
MAY
1996
Rela i e SLR o se e al i e a ion numbe s
(M)
-
Can o A ay
wi h
N=2 Op imum lacuna i y o SLR p o ec ion. del a/N=l
2
~
(N=5.
del a=6)
12
14
16
18 2 22 24
26
28
/I/,
12
14
16
18 2 22 24
26
28
Log-pe iod
6
Fig.
4.
Side-lohe a io
(SLR)
as a unc ion o he
log-pe iod
5
o he
wo-elemen gene a o
(N
=
2)
Can o a ay. The g aph
is
plo ed o se e al
i e a ion numbe s
(M).
No ice ha o
6
=
2
we ha e a uni o m may wi h a
SLR
a ound 13 dB, and ha he
SLR
is g ea ly educed o small log pe iods.
each i e a ion, esul ing in a p oduc o M-a ay ac o s e y
simila o he p oduc exp essed in
(1 1).
Howe e , i
S
is made
la ge enough, he comp ession will be such ha many g a ing
lobes o he la ge gene a o will appea wi hin he isible
ange. The p oblem becomes specially impo an when
6
>
N
and he a ay becomes sel -simila a se e al bands. When
b
=
N
he main g a ing lobe
o
a gene a o 's a ay ac o is
placed
on
op
o
he i s null
o
he p e ious gene a o 's a ay
ac o . Equa ion
(12)
demons a es ha he p oduc o hese
wo sinc unc ions esul s in a na owe sinc unc ion as well.
Ne e heless, when he a io
SIN
is made la ge han uni y, he
g a ing lobes appea wi hin he main lobe o he p e ious a ay
ac o
o
he i e a ion. I hese lobes all nea he maximum
o
his p e ious a ay ac o , hey will be g ea ly enhanced. In
his case, he a ay p esen s a highly lacuna dis ibu ion and,
consequen ly, la ge seconda y lobes.
One can es ablish an uppe bound o sa ely design ac al-
Can o
a ays
wi hou inc easing he side-lobe le el abo e
-
13
dB. By choosing a a io
SIN
=
1.2 (Fig.
5),
he i s g a ing
lobe is placed a a
$I
poin wi hin he main lobe, such ha
when weigh ed by he ampli ude o his main lobe, he g a ing
lobe is educed o he le el o he i s seconda y lobe (i.e., 13
dB). The e o e, one could s a e ha Can o a ays should be
designed wi h a lacuna i y such ha
s
-
<
1.2
N
o equi alen ly wi h a ac al dimension
I can be no iced ha
D
ends o uni y o a la ge numbe
o
gene a o elemen s
N.
Tha makes sense when one akes in o
accoun ha o a oid high seconda y lobes he gaps be ween
-3
-2
1
0
1
2
3
PSI
'
'
'1
O:;
-3
-2
Psi
Fig.
5.
The
maximum a io
SIN
ha does no inc ease he side-lobe le el
wi h espec o he uni o m case is
SIN
=
1.2.
In such a case, he i s g a ing
lobe
is loca ed a a poin whe e he main lobe has decayed 13
dB,
wi h
espec
o
he maximum.
subs uc u es mus ha e a limi ed size ela i e o he sho es
wa eleng h, ega dless
o
he o al numbe o elemen s wi hin
he a ay. The e o e, o a la ge
N
he elemen s o he a ay
end o ill a s aigh line mo e densely (one dimension) and
he ac al dimension ends o one. This is consis en wi h he
esul s shown in [9] o he ac al andom a ay.
B.
Fu he SLR Reduc ion: T iangula
and Highe O de Gene a o s
The de elopmen in Sec ion 11-A was based on a uni o m
ampli ude gene a o . Al hough his migh be he op imal choice
o di ec i i y conside a ions, o he shapes could be used o
imp o e he
SLR.
Ins ead o con ol ing M-uni o m dis ibu-
ions
o
di e en scales, M- iangula dis ibu ions could be
used. Since a iangula dis ibu ion
(x)
can be w i en as
he con olu ion o wo iden ical uni o m dis ibu ions
U(.),
a
iangula Can o a ay
c(z)
could be gene a ed as ollows:
M-1
c(2)
=
x
(z
.
P)
n=O
The co esponding a ay ac o
TC($I)
can be w i en now as
which implies ha he
SLR
has been doubled wi h espec
o
he uni o m Can o se
C($).
Equa ion
(16)
also implies ha
he iangula Can o a ay is equi alen o he con olu ion
o

PUENTE-BALIARDA
AND
POUS: FRACTAL DESIGN
OF
MULTIBAND
AND
LOW
SIDE-LOBE
ARRAYS
135
T iangula Can o a ay
-
N.3, del a.3,
M=6
I,
FO
6
504
1
202
-150
-100
-50
0
50
100
150
2
(wa eleng hs)
(a)
U.
$0
2
:n
-3
-2
1
0
1
2
3
PSI
(b)
Fig.
6.
T iangula Can o a ay gene a ed wi h six i e a ions: (a) he log
pe iod is
5
=
3
and he gene a o is a h ee-elemen iangula a ay and (b)
he co esponding pa e n is he squa ed e sion o ha in Fig. 2(a) ( he SLR
is doubled in a
dB
scale).
wo equally-uni o m Can o a ays, an al e na i e p ocedu e
o i s cons uc ion algo i hm. The shape and he pa em o
his iangula Can o a ay a e shown in Fig.
6.
Al hough his cons uc ion scheme may be use ul o
SLR
imp o emen , i again in oduces he incon enience
o nonuni o n-ampli ude dis ibu ion o he elemen s. Again,
his p oblem could be sol ed by subs i u ing he iangula
gene a o by a ac al (Can o ) gene a o ha would app oach
he same basic pa e n. Howe e , bo h solu ions ha e a g ea
incon enience- he la ge numbe o elemen s o he a ay.
111.
FRACTAL
RADIATION
PATTERNS
In
he p e ious sec ion, a ac al analysis and design o
se e al a ay ac o s has been de eloped. F ac al-elemen
dis ibu ions ha e shown o be use ul o designing low side-
lobe a ay ac o s wi h a uni o m-ampli ude dis ibu ion o
elemen s. On he o he hand, hese ac al a ays ha e shown
some in e es ing simila i y p ope ies a se e al wa eleng hs,
howe e , hese simila i y p ope ies do no sa is y some o he
equi emen s one would desi e o a equency-independen
a ay: he di ec i i y and he main-lobe wid h a e no held
cons an a each band. In gene al, one would like o ha e an
a ay ac o which had he same shape a di e en scales o
keep he same adia ing pa ame e s a se e al wa eleng hs.
This leads o he app oach p esen ed in his sec ion- he
design o ac al a ay ac o s.
The pa e ns designed in his sec ion a e based
on
a amily
o
sel -simila cu es known as Koch cu es
[6],
[7],
[151.
The pa em-cons uc ion algo i hm is qui e simila
o
ha o
he Koch cu es, bu
is
modi ied o p o ide a unc ional o m.
The shape and he p inciple o wo k o hese kind o a ay
ac o s is summa ized in Fig.
7.
Koch
Pa e n
P=kd
p og essi e
phase
Fig.
7.
The Koch-a ay ac o . The cu e keeps i s simila i y p ope ies a
six di e en scales (i has been cons uc ed wi h six i e a ions,
M
=
6).
By
adding a p og essi e phase
13
=
kd, he isible ange is always cen e ed a
a seconda y lobe ha has he same shape as he o al pa e n. The equency
change by a ac o
5
=
1/3
educes he isible ange a ound his simila
subpa em.
The main ea u e o his pa em is ha each lobe o he
cu e is equal o he whole pa e n. When he a ay adia es
a a longe wa eleng h, he isible ange
is
educed and only
a ac ion o he whole a ay ac o appea s in he adia ion
pa e n. Thus, i we we e able o design an a ay wi h an
a ay ac o as he one in Fig.
7,
and i he isible ange could
be educed a ound one o he seconda y lobes, he esul ing
isible pa em would be he same as he o iginal one. The
isible ange can be cen e edl o any a bi a y poin o he
?,b
domain by adding a p og essi e phase
i
(4)
o he phase
equi ed o each elemen o syn hesize he co esponding
a ay ac o . I can be seen hal i one akes such a p og essi e
phase o be
he isible ange will co e he in e al
(0,
2kd}
a any
equency. Hence, o he pa icula case o Fig.
7,
a equency
educ ion by a ac o
o
(5)"
would educe he isible ange
a ound a seconda y lobe which has he same shape as he
whole pa e n.
In
o he wo ds, we would ha e an a ay ac o
wi h he same adia ion pa ame e s
o
a se o bands spaced
a ac o
o
i.
I is also in e es ing o poin ou ha al hough
he p og essi e phase in
(17)
is usually in ended o end i e
a ays, he Koch pa e ns a e designed he e o adia e in he
b oadside di ec ion.
I should be no iced ha , al hough he a ays jus desc ibed
would ha e a simila adia ion pa em a se e al bands, he
pa em magni ude is educed when he ope a ing wa eleng h
is inc eased. Tha means ha ix he same cu en dis ibu ion,
he elec ic- ield in ensi y is educed a lowe bands o , in
o he wo ds, nei he he adia ion esis ance no he adia ion
e iciency a e held cons an h ough each band. This is an
136
1.
8
0.8-
2
0.6.
a
c
0
x
0.4.
0.2.
IEEE
TRANSACTIONS ON ANTENNAS AND PROPAGATION,
VOL.
44,
NO.
5,
MAY
1996
A ay Fac o
1
Cu en
Dis ibu ion
iz-
u
-1
3
-1
n
2
-2
._
+
0
-2
g
-3
5
-3
c
L
-4
0-
Koch
A ay
M=4,6=5,
a=l
0
Koch
Pa e n
M=4,6=5,
a=
1
h
g
-5
5
-10
1
I
8
0.8
I?
0.6
$
0.4
._
0
2
-15
-20
E
-25
5
-30
._
%
m
-
0
-2
L
3
-35
w
2Olog(Z+)
0
-40
0123456
0
5
10
15
20
25
30
35
1-
:
0.8-
I
0
L??
0.6.
Z
0.4.
2
0.2.
0-
=-
Koch
Pa e n
M=5,6=3,
u=2.5
-0
-20
._
0
-30
2
-40
g
-10
+
._
z
-50
E
-60
-80
-
-70
5
-90
0-1
00
A A
-110
iz-
s
s
-1
5
-1
-z
-2
._
Q
c
g
-2
E
-3
3
-4
a,
-3
-4
2Olog(Z+)
Fig.
8.
is plo ed
o
each pa e n
on
he le column.
Koch pa e ns
a e
con o med wi h a ays cons uc ed by in e lea ing hype bolic dis ibu ions. The igh -hand side
o
he a ay cu en dis ibu ion
in insic cons ain
o
such an a ay design which should be
aced in he physical implemen a ion
o
he a ay.
Once he ac al-a ay ac o has been de ined o ha e a
mul iband beha io , he ela i e cu en dis ibu ion be ween
elemen s ha would gene a e such a ac al pa e n has o
be de i ed. This dis ibu ion can be nume ically compu ed
by aking he in e se Fou ie ans o m (IFT)
o
he Koch-
a ay ac o . Fig.
8
shows se e al con igu a ions
o
Koch
PUENTE-BALIARDA
AND
POUS:
FRACTAL
DESIGN
OF
MULTIBAND
AND
LOW SIDE-LOBE
ARRAYS
131
Blackman Koch a ay
be o e
unca ion
IO0
10.'
10
-400
-200
0
200 400
Fig.
9.
Cu en dis ibu ion o he Blackman-Koch a ay loga i hmic scale.
The main cons uc ion pa ame e s a e
A
=
6,
5
=
3,
and
cy
=
4.
The
educ ion ac o
CY
=
4 has been chosen he e o imp o e he
SLR
wi h espec
o he p e ious case
on
Fig.
7.
See he co esponding pa em on he bo om
case o Fig.
8.
a ays wi h he co esponding cu en dis ibu ion. Again,
S
is he log pe iod o band a io,
M
is he numbe o i e a ions
(and he numbe o bands), and
a
is
an
ampli ude weigh
ac o ha adds an ex a deg ee o eedom in he pa e n
design as i is discussed in he nex subsec ion. Only hal
o he igh -side o he a ay cu en dis ibu ion is displayed
since i is symme ical a ound i s cen al elemen . The main
cha ac e is ic o such cu en dis ibu ions is hei powe -law-
like shape; his ea u e is sha ed by many ac al unc ions,
such as he bandlimi ed Weie s ass unc ion
[9].
The sel -
simila i y p ope ies o hese a ay pa e ns a e based
on
he
scaling p ope ies o he Fou ie ans o m (FT) which s a e
ha i
a
unc ion such as a powe -law unc ion is sel -simila ,
so
will be i s coun e pa in he spec al domain.
In
an enna
heo y e minology, ha means ha a cu en dis ibu ion (z)
which holds he p ope y
will ha e
a
sel -simila pa e n
F($),
ha is
F($)
=
.
F(S
'
$).
(19)
This beha io is qui e di e en om cu en equency-
independen an ennas. Such an ennas a e based on an ac i e
egion o he an enna ha changes i s size wi h equency.
On
he o he hand, he ac al a ays in oduced in his pape
assume a cu en dis ibu ion ha does no change wi h
equency bu has
a
scale-independen shape.
Ano he impo an issue conce ning he compu ed a ay
s uc u e is ha i equi es
a
la ge numbe o elemen s
(36
=
729
o
he a ay
on
Fig.
7).
In gene al, he numbe
o
elemen s
N
and he numbe o bands
M
in he Koch a ay a e ela ed as
The numbe
M
o i e a ions used o cons uc he cu e
de e mines he numbe o i nes he cu e will look simila
unde
a
6
ac o scaling ans o ma ion. In o he wo ds,
M
is
he numbe o bands o log pe iods in which he a ay will
ha e a simila pa e n. Hence, he e is a ade-o be ween he
size o he a ay and he numbe o ope a ing bands. O cou se,
he e a ises wha can be an in insic limi a ion o hese a ays:
he numbe o elemen s g ows exponen ially wi h he numbe
o log-pe iodic bands. Since all di e en Koch pa e ns ha
ha e been analyzed
[
141
ha e he common cha ac e is ic o
concen a ing he mos impo .an cu en con ibu ion a ound
he cen al elemen , one could hink ha he numbe o
elemen s could be educed by me ely unca ing he a ay a i s
ips. Howe e , i can be eadily seen ha his p ocedu e would
limi he mul iband beha io o he a ay.
An
a ay unca ion
is equi alen o a spa ial windowing o he s uc u e, which is
equi alen o low-pass il e ing he pa e n in he
4
domain.
The e o e, he a ay ac o is smoo hed and he pa e n loses i s
cha ac e is ic lobe s uc u e which is he base o i s mul iband
beha io
[14].
A deepe analysis o he Koch a ay s uc u e
will help in bo h unde s anding i s beha io and educing he
numbe o elemen s.
Analysis
o
he A ay Elemen Dis ibu ion o Koch-Pa e n
Con o ma ion:
A key poin o unde s anding he a ay cu -
en dis ibu ion de i ed om he ac al pa e ns is he Koch-
pa em cons uc ion algo i hm i sel . Le
us
ake a pe iodic
pulse ain in he spa ial- equency domain
$
and scale i s
wid h by a ac o
b
and i s ampli ude by
a
ac o o
aS.
A e
i e a ing his scheme M imes, he M- esul ing pa e ns a e
added, ob aining
a
Koch pa e n such
as
he one in Fig.
7.
In
pa icula , o he pa e n on Fiig.
7,
a ec angula pulse and a
log-pe iod
S
=
3,
and an ampli ude ac o
a
=
1
was chosen
o gene a ing he pa e n wi h
M
=
6
i e a ions.
The analy ical exp ession o each gene a ing pulse ain
can be w i en as
M
whe e F($) is he single pullse unc ion, which, in gene al
could be aken o ha e any a bi a y shape such
as
a
ec an-
gula window o
a
Blackman window, and
l/l~
is he pe iod
o he pulse ain. F om
(21),
he analy ical exp ession o he
Koch pa e n
K($)
a e adding he M-scaled pulse ains is
Se e al combina ions o
a,
6,
and
M
a e essayed
in
he
a ays o Fig.
S.
A ec angula gene a ing pulse is chosen
on
he i s h ee examples, and a Blackman window
on
he las
one. Once
an
exp ession o he Koch pa e n has been de i ed,
an
exp ession o he Koch-a ay elemen dis ibu ion
k(z)
can
738
IEEE
TR
be easily ound by aking he IFT o (22)
The ain o del a unc ions in
(23)
samples he cu en
dis ibu ion a he disc e e se o poin s
z
=
n
. d .
Sp
whe e
he a ay elemen s a e loca ed. Hence, aking in o accoun ha
$T
=kd
2nd
x
-
~
-
one can w i e
,
M-1
A
k(z)
=
~
p=o
2nd
1
Pm
.
(nd)
'
S(z
-
n
'
d
'
Sp)
n=--oo
which gi es an insigh in o he shape o he esul ing a ay; he
Koch a ay is a supe posi ion o
M
a ays ha ha e he same
elemen dis ibu ion bu a wide spacing be ween elemen s,
depending on he i e a ion s age o which hey belong. Tha
is, he elemen s a e uni o mly spaced wi hin he same a ay,
bu he spacing changes a each suba ay by a ac o Sp. When
he a ays a e added, some elemen s migh all a he same
posi ion as o he elemen s om he o he a ays; in such a
case, he esul is a single elemen whose weigh is he sum
o he weigh s
o
all he elemen s ha would all a ha poin .
In pa icula , i can be seen ha all he a ays ha e a common
elemen a
z=n.
d.
SM-l.
(26)
Equa ion (25) can gi e an insigh on he powe -law shape
o he cu en dis ibu ion ha gene a es he Koch pa e n in
Fig. 7. Fo his pa icula case, he squa ed pulse gene a o
has an in e se ans o m
which gi es he shape
o
all
he
M
suba ays
ha con o m he
Koch a ay. The weigh o each elemen can be easily ound
in ou sampling (27) a
z
=
m
.
d.
I can be seen ha o he
case we ha e'been s udying
(S
=
3)
lo
m
=
3"
.ANSACTIONS ON ANTENNAS
AND
PROPAGATION, VOL.
44,
NO.
5,
MAY
1996
which is, in absolu e alue, a powe -law (hype bolic) unc ion
o he index elemen m. Ano he impo an p ope y can be
seen i we ealize ha (28) is null o hose
m
such ha
m
=
SW
(29)
wi h
W
being an in ege . F om his p ope y, one concludes
ha all he suba ays con ibu e o he weigh o he cen al
elemen , bu hey do no o e lap a any o he poin since
he nulls o each a ay a e illed by an elemen o an a ay
co esponding o he nex i e a ion s ages. The e o e, he global
a ay ob ained a e M i e a ions can be seen as an a ay
composed by in e lea ing he elemen s o M-equal a ays a
M-di e en scales. The esul o he pa icula case
6
=
3
and a squa e pulse gene a o , is an equally-spaced a ay wi h
a hype bolic dis ibu ion o he elemen -cu en magni udes.
Two impo an conclusions can be de i ed om he analysis
o he gene alized Koch-a ay
k(z)
and he pa icula case we
ha e jus s udied. Fi s , he shape o he M-supe imposed sub-
a ays depends on he shape o he pulse gene a o . Second, he
supe posi ion o he suba ays migh esul in he con luence
o
many elemen s in a single loca ion o migh esul in an
in e lea ing
o
he elemen s. As will be shown in he ollowing
subsec ion, bo h conclusions will help in he educ ion o he
numbe o elemen s o he Koch a ay.
The Blackman-Koch A ay: A Fu he
SimpliJica ion
o
he
F ac al-Pa e n A ay:
Since he a ay cu en dis ibu ion
is basically a supe posi ion o he in e se ans o ms o he
pulse gene a o , i should be chosen a pulse gene a o wi h
a low side-lobe le el ans o m o allow a be e unca ion
o
he Koch a ays jus shown. The Blackman window is
cha ac e ized o ha ing low side-lobes in he ans o med
domain. The e o e, one could chose a ain o Blackman pulses
o gene a e he Koch pa e ns ins ead o he ec angula ones.
The esul s
o
applying such a echnique a e shown in he
bo om case in Fig.
8.
I can be no iced ha he pa e n esul s in a smoo he
shape ha keeps he same simila p ope ies o he Koch-a ay
ac o o Fig.
7.
The main ad an age o his pa e n is ha he
a ay- ela i e cu en dis ibu ion has lowe side-lobes and a
be e con inemen a ound he cen al elemen s (Fig. 9). Also,
he loga i hmic plo o he cu en dis ibu ion e eals some
impo an isola ed cu en peaks well beyond he cen e o he
a ay. One should expec a signi ican con ibu ion o hese
isola ed elemen s
o
he global-pa e n con o ma ion. Thus,
ins ead o
jus
unca ing he ips o he a ay, a h eshold
le el can be se o disce n which elemen s a e impo an in
he pa e n syn hesis and which a e no . The esul is ha
he a ay s uc u e can be educed o only he 75 elemen s
(as opposed
o
729) wi h a highe cu en con ibu ion and
s ill keep i s sel -simila beha io a i e bands h ough a
whole
81
:
1
equency ange. The esul ing a ay is
no
longe
a uni o mly-spaced a ay since he main 75 cu en elemen s
a e no placed oge he nea he midpoin
o
he a ay. Thus,
some elemen s a e placed u he om he o igin han in
he unca ion scheme which means ha as e a ia ions will
appea in he dual domain ( he pa e n domain). This explains
why his scheme can be e keep he mul iband beha io in a
la ge numbe o bands han he unca ion scheme: he u he