TELKOM
NIKA
, Vol.14, No
.4, Dece
mbe
r
2016, pp. 12
17~121
9
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v14i4.4789
1217
Re
cei
v
ed
Jul
y
22, 201
6; Revi
sed Aug
u
st
31, 2016; Accepted Sept
em
ber 14, 20
16
Editorial
Power Synthesis of Mask-Constrained Shaped Beams
Through Maximally-S
parse P
l
anar Arrays
Andre
a
Fran
cesco M
o
rab
i
to
Univers
i
t
y
of Regg
io Ca
la
bria,
DIIES Department, Regg
io
Cala
bri
a
, Ital
y
Cons
orzio N
a
zi
ona
le Interu
niv
e
rsitario
per le
T
e
lecomunic
a
z
i
oni, Parm
a, Ital
y
e-mail: a
ndre
a
.morabit
o
@u
nir
c
.it
A
b
st
r
a
ct
A new
appro
a
c
h to the opti
m
a
l
synthes
is
of pl
an
ar arr
a
ys abl
e to ra
diate
mask-c
o
nstrain
e
d
shap
ed b
e
a
m
s
by expl
oiti
ng t
he
min
i
mu
m n
u
mber of ra
di
ating e
l
e
m
ents is
prese
n
ted. By
taking
adva
n
ta
ge
from b
o
th th
e recent the
o
ry of
Co
mpr
e
ssive
Sensi
ng
a
nd t
he
mu
ltipl
i
city
of equ
ival
ent s
o
luti
ons av
ai
la
bl
e
for the gen
era
t
ion of an u
n
i
que sh
ap
ed-b
e
a
m
pow
er p
a
ttern, the synthesis res
u
lts e
x
treme
l
y fast and
effective. In particul
a
r, the overall d
e
si
gn i
s
reduc
e
d
to a Conv
ex Progra
m
min
g
opti
m
i
z
at
io
n, w
i
th
the
inh
e
rent a
d
van
t
ages in ter
m
s
of solutio
n
s
’
o
p
t
ima
lity an
d co
mp
utatio
nal
bu
rden.
Ke
y
w
ords
: Array anten
nas, c
o
mpress
ed se
nsin
g, pow
er synthesis, sha
p
ed be
a
m
s.
Copy
right
©
2016 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
Maximally-sparse arrays,
i.e., array anten
nas
able to fulfill as
signed radiation
requi
rem
ents by recu
rri
ng
to the minimum po
ssi
bl
e numb
e
r of
element
s, re
pre
s
ent a top
i
c of
high inte
re
st in many ap
plicatio
ns
(e.
g
., phase
d
a
rray rada
rs,
satellite com
m
unication
s, and
multiple-i
nput
-multiple
-
outp
u
t system
s). In fact,
they
may offer rel
e
vant advant
age
s in te
rm
s of
size, weig
ht, co
st, and bea
m forming net
work’
s
co
mpl
e
xity [1-6].
Re
cently, in t
he case of
maximally-sp
a
rs
e lin
ear a
rray
s
radiatin
g sh
ape
d b
e
a
ms, the
approa
ch
introdu
ced
in [1]
outpeformed
all techniq
u
e
s
publi
s
h
ed i
n
[2-5]. Thi
s
ha
s b
een
po
ssi
b
le
by jointly exploiting the the
o
ry of Comp
ressive Se
n
s
i
ng (CS) [7] a
nd the m
u
ltipl
i
city of equiva
lent
solutions generally available to generat
e
a si
ngle shaped power pattern
[8],[9].
Moreover,
by
avoiding exploitation
of Global Optm
ization
(G
O) algorith
m
s
and relying
only to Co
n
v
ex
Programmin
g
(CP
)
tool
s,
the a
pproa
ch i
n
[1
] p
r
ovided
relev
ant adva
n
ta
ges in te
rm
s of
comp
utationa
l times.
Due to the
relevan
c
e of the outcome
s achieve
d
by
the pro
c
ed
u
r
e in [1], the pre
s
ent
work i
s
aime
d at extendi
n
g
it to the
ca
se
of pl
an
ar
arrays. To
prese
n
t the
re
sulting ap
proa
ch, it
is wo
rth first recalli
ng the p
r
ocedu
re in [1
], which is
co
mposed by the followin
g
st
eps:
1.
Assig
n
the
p
o
we
r ma
sk, i
.
e., the lowe
r and
upp
er
b
ound fu
nctio
n
s
req
u
ire
d
t
o
sh
ape
a
s
desi
r
ed the
ra
diated po
we
r pattern;
2.
Synthesize, b
y
means of t
he app
roa
c
h
pre
s
ent
e
d
in
[9], a ‘virtual’ linear a
r
ray radiatin
g a
power pattern fulfilling the mask of
step 1.
Notabl
y, such a
st
ep provides
a number of
different po
ssible field solut
i
ons [8];
3.
Identify, amongst all th
e
different far-field di
stributio
ns
comin
g
o
u
t from ste
p
2, the one
leadin
g
to the minimum
ℓ
1
-no
r
m of the ‘virtual’ array excita
tion
s. By virtue of th
e theory in
[7], this far-field distrib
u
tion
is the one a
s
so
ciated to m
a
ximum amo
unt of ‘sparsit
y
’;
4.
Synthesize a
‘new’
equi
spa
c
ed li
nea
r a
r
ray by
perfo
rming a
CS-b
ase
d
re
co
nst
r
uction. T
h
is
step
con
s
i
s
ts in solving t
he follo
wi
ng
CP problem
s in the un
kn
own
b
=[
b
1
,…,
b
N
], whic
h
rep
r
e
s
ent
s the vector of th
e ‘new’ a
r
ray’s excitation
s:
Minim
i
ze:
‖
‖
≔
|
|
(1)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 14, No. 4, Dece
mb
er 201
6 : 1217 – 121
9
1218
subj
ect to:
|
|
g
α
∀
|
|
∀
(2)
with:
β
α
(3)
Whe
r
ein
N
is the numbe
r of element
s,
f
r
is the far field comi
ng o
u
t from step
2,
α
is the
elevation an
gle with
resp
ect
to bo
re
si
ght,
β
is the
wave
numb
e
r
, an
d
x
=[
x
1
,…
,
x
N
] is the
vector contai
ning the ‘ne
w
’ array’
s equi
spa
c
e
d
locations. By so d
o
ing, while th
e obje
c
tive
function (1) will
induce
a minimization of
the
active element
s
number, convex
constraints
(2)
will ensure that the synthesi
z
ed power pattern:
a.
Fulfills an user-define
d
up
per bo
und co
nstr
ai
nt in the sidelo
b
e
s
region de
note
d
with
τ
1
(
g
bein
g
an a
r
bitra
r
y real a
nd po
sitive function
);
b.
Fits the refe
rence field
f
r
with a preci
s
io
n
ε
in the regi
on den
oted b
y
τ
2
;
5.
Furthe
r re
du
ce the overall
numbe
r of array elements
by:
a.
Disca
r
din
g
the element
s h
a
ving a negli
g
ible excitatio
n
amplitude;
b.
Re
cursively substituting ea
ch co
uple of
remainin
g el
ements
who
s
e distan
ce is lower
than a
thre
sh
old
σ
with
a
singl
e ele
m
e
n
t pla
c
ed i
n
t
he mid
d
le
poi
nt between t
hem a
n
d
excited with t
he sum of the
two origin
al excitation
s;
6.
Perform
a
ref
i
nement
of th
e solution
by
m
ean
s of a
local
optimi
z
ation p
r
o
c
ed
ure: id
entify
slight
shifts o
n
a
rray
locations an
d ex
citations
so as to recover
f
r
om poss
ible los
s
es on the
radiatio
n pe
rforma
nce indu
ced by ste
p
5
.
In orde
r to exented the
above pro
c
edure to the
planar a
r
ra
ys ca
se, we
tried to
synthe
size a
beam
having
a flat-to
p
b
e
havior
alon
g
the a
z
imuth
angle
an
d a
squ
a
re
-co
s
e
c
ant
behavio
r al
on
g the
elevati
on a
ngle
(se
e
Fig
u
re
1
)
,
wh
i
c
h i
s
of in
terest
in
rad
a
r
a
pplication
s
as
well as
for
ra
dio-b
a
se stati
ons.
In
so do
ing,
we
to
ok as a refe
ren
c
e
the
pattern gene
rated
fro
m
the facto
r
ization of the fi
elds
re
spe
c
ti
vely depi
cte
d
in blue
col
o
r in
sub
p
lots (b)
and
(c) of
Figure 1.
(a)
(b)
(c
)
Figure 1. Power p
a
ttern of
the array vie
w
in the sp
ect
r
al plan
e [sub
plot (a)], and
plot of the main
cuts al
ong th
e ordin
a
te [su
bplot (a
)] and
the abscissa
[subpl
ot (b)]. The refe
ren
c
e and
synthe
sized d
i
stributio
ns a
r
e respe
c
tive
ly depicte
d
in b
l
ue col
o
r an
d red colo
r
Then, the pro
c
ed
ure in [1] has b
een ext
ende
d to the two-dime
nsio
nal ca
se a
s
follows.
First,
we fa
ctorize
d
the separate one
-dimen
sion
al
solutio
n
s a
c
h
i
eved by app
lying step
s 1
-
6
above to ea
ch of the reference po
wer p
a
ttern’s m
a
in
cuts.
Second,
we discarded th
o
s
e ele
m
ent
s having an ex
citation am
pli
t
ude lower th
an
υ
=0.0
4 an
d
achi
eved the
array layout d
epicte
d
in Fig
u
re 2
[
s
ub
plot
(a), bl
ue
circl
e
s], whi
c
h i
s
comp
osed
by 94 elemen
ts.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Power Synthesi
s of Mask-Con
strai
ned
Shaped Beam
s Through...
(Andrea F
r
ancesco Morabito)
1219
Third,
by a
d
a
p
ting
step
6
a
bove to
the t
w
o-dime
nsio
nal
ca
se
(in
such
a
way to
jo
i
n
tl
y
refine
bo
th
the
x
an
d
y
a
r
ray lo
cation
s), we fin
a
lly a
c
hieved th
e
la
yout depi
cted
in Fig
u
re 2
[subplot
(a
),
red dot
s] and
the corre
s
p
o
n
d
ing optimal
excita
tion
s de
picted in Fig
u
r
e 2 [su
bplot (b)].
The a
c
hi
eved
power
pattern is
sh
own in
Figu
re 1,
wh
erein
θ
a
nd
ϕ
re
sp
ec
tive
ly r
e
pr
e
s
e
n
t
th
e
elevation and
azimuth angl
es. As it can be se
e
n
, a very good ag
reement is a
c
hieved betwe
en
the referen
c
e
field and the main cut
s
of the synthe
si
ze
d pattern.
Notably, the
overall
app
ro
ach
allo
wed
saving the
58
% of the el
e
m
ents with
re
spe
c
t to
a
fully-popul
ate
d
array with
an unifo
rm
/
2
sp
aci
ng, a
nd the 3
4
% of element
s
with re
sp
ect t
o
a
simple
facto
r
i
z
ation
of the
solutio
n
s i
n
[2]. This
circu
m
stan
ce
i
s
coherent with the
fact
that, as
long a
s
a pat
tern is fa
ctorable, the ele
m
ents’ n
u
mb
er re
du
ction i
n
the plan
ar
array is ro
ug
hly
doubl
ed with
respe
c
t to the one experi
e
n
c
ed in
the two unde
rlying
one-dime
nsio
nal arrays.
(a)
(b)
Figure 2. Subplot(a
): Array
la
yout achi
eved by factori
z
ing the one
-di
m
ensi
onal
ref
e
ren
c
e
solutio
n
s a
n
d
discardi
ng th
e element
s h
a
ving a
negli
g
ible excitatio
n
amplitude.
Location
s
achi
eved bef
ore (blue
circl
e
s) a
nd after
(re
d dot
s) the
refineme
n
t step. Subplot (b): Fianl
excitation am
plitude
s
Referen
ces
[1]
Morabit
o
AF
, Lag
an
à AR, Sorbe
llo G, Isernia
T
.
Mask-constrai
ned
po
w
e
r s
y
nthes
is
of maximal
l
y
sparse
lin
ear
arra
ys thro
ug
h
a com
p
ressi
ve-sens
ing-
driv
en strate
g
y
.
J
ourn
a
l of
Elec
troma
g
n
e
ti
c
W
a
ves and Ap
plicati
ons
. 2
0
1
5
; 29(10): 1
384
-139
6.
[2]
Oliveri G, Carli
n
M, Massa A.
Compl
e
x-
w
e
ig
ht s
parse li
ne
a
r
arra
y
s
y
nt
hes
is b
y
B
a
yesi
an
Compress
iv
e
Sampli
ng.
IEEE Transactions
on Ante
nnas and Pr
opagation
. 2012; (5): 230
9-23
26.
[3]
F
u
chs B. Synthesis of spars
e
arra
ys
w
i
th fo
cuse
d or sha
ped b
eam patt
e
rn via seq
u
e
n
tial co
nve
x
optimiz
ations.
IEEE Transacti
ons on Ant
enn
as and Pro
p
a
g
a
tion
. 20
12; 60
(7): 3499-
35
03
.
[4]
Nai
SE, Ser
W
,
Yu Z
L
, C
h
en
H. Be
ampa
ttern s
y
nth
e
sis
for l
i
ne
ar
an
d
pl
anar
arr
a
ys
w
i
th
a
n
tenn
a
selecti
on b
y
c
onve
x
optimiz
ation.
IEEE Transactions
on Ant
ennas and Propagation
.
201
0; 58(
12):
392
3-39
30.
[5]
Liu
Y, Ni
e Z
P
,
Liu
QH. A n
e
w
method
for th
e
s
y
nt
hesis
of
no
n-un
iform l
i
ne
a
r
arra
ys
w
i
th
sh
ape
d p
o
w
e
r
patterns.
Progr
ess in Electrom
agnetic Research
. 201
0; 10
7: 349-3
63.
[6]
Bucci OM, Isernia T
,
Morabit
o
AF
, Perna S
,
Pincher
a D.
Density
and
el
ement-si
z
e
tap
e
rin
g
for the
desi
gn of arra
ys w
i
th a redu
ced nu
mber of
control p
o
ints
and h
i
gh
efficiency
. Proce
e
d
i
ngs of t
h
e
F
ourth Europ
e
an Co
nfere
n
ce
on Anten
nas a
nd Prop
ag
ation
.
Barcelon
a. 20
10.
[7]
Can
dès EJ,
W
a
kin MB. An Introducti
on
T
o
Compres
s
ive Sam
p
li
ng
.
IEEE Signal
Processi
n
g
Maga
z
i
ne
. 2008; 25: 21-30.
[8]
Morabit
o
AF
, Iserni
a T
,
Di Donato
L. Opti
mal s
y
nth
e
sis of
phas
e-o
n
l
y
reconfi
gura
b
l
e
line
a
r
sp
arse
arra
ys havi
n
g
uniform-
ampl
itude
e
x
cit
a
tions
.
Progress
In E
l
ectro
m
a
gnetic
s Res
earch
. 20
12;
1
24: 40
5
-
423.
[9]
Isernia T
,
Bucci OM, F
i
orenti
no N. S
h
a
ped
beam
ante
n
n
a
s
y
nthesis
pro
b
lems: feas
ib
ili
t
y
criteri
a
an
d
ne
w
strateg
i
es.
Journa
l of Ele
c
troma
g
n
e
tic W
a
ves and Ap
plicati
ons
. 1
9
9
8
; 12: 103-
137.
Evaluation Warning : The document was created with Spire.PDF for Python.