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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 3, MARCH 2007
Solving Some Array Synthesis Problems by Means
of an Effective Hybrid Approach
Michele D’Urso and Tommaso Isernia, Member, IEEE
Abstract—Most global optimization based synthesis procedures
act in essentially the same way on all the involved variables. In
this paper, it is theoretically discussed and numerically emphasized
how explicit exploitation of the convexity of the problem with respect to a part of the degrees of freedom (when available) allows
to achieve increased performances with respect to previous results,
and to develop new interesting design solutions. Examples are presented for the synthesis of both sum and difference patterns by
using nonuniformly spaced arrays and for the so called “optimal
compromise amongst sum and difference patterns.” Finally, the
proposed approach is exploited in developing effective design solutions (based on the concept of interleaved arrays) to the problem
of achieving very flexible antennas while using simple feeding networks. As an example, array antennas capable to radiate two different beams with individually steerable patterns, or to perform
jammer rejection at the physical layer or even to realize a “flat-top”
beam pattern are synthesized and presented.
Index Terms—Array antenna synthesis, hybrid optimization,
interleaved array, simulated annealing.
I. INTRODUCTION AND RATIONALE
N THE LAST years, a large interest has been devoted to
the exploitation of global optimization techniques in electromagnetic design. Techniques as different as simulated annealing (SA) [1], [2], genetic algorithms (GAs) [3] and particle
swarm optimization (PSO) [4], [5] have been widely applied to
different problems as the design of absorbers, lens, several devices, inverse scattering problems and many other applications
[6]–[11]. Not surprisingly, global optimization techniques have
been also applied to the synthesis of arrays, including the design
of sparse arrays [12]–[17] and the so-called ‘optimal compromise amongst sum and difference patterns’ problem [18]–[20].
On the other side, the diffused (and justified) enthusiasm
for a wide range of ‘physically inspired’ optimization techniques (such as SA, GA, PSO, and many others) has induced
to neglect, in a number of cases, some characteristics of the
problems at hand which may be instead very useful in the global
optimization of the considered devices. Global optimization
techniques are in fact actually limited in their performances
from the fact that the computational burden raises very rapidly
with the number of unknowns. As a consequence, a suboptimal
solution (rather than the globally optimal one) will be gen-
I
Manuscript received March 31, 2006; revised October 5, 2006.
M. D’Urso is with the DIET, Università Federico II di Napoli, I-80125 Napoli,
Italy (e-mail: [email protected]).
T. Isernia is with the DIMET, Università Mediterranea di Reggio Calabria,
I-89060 Reggio Calabria, Italy (e-mail: [email protected]).
Digital Object Identifier 10.1109/TAP.2007.891554
erally achieved in large scale applications in the (necessarily
limited) time one has at his disposal. Such a circumstance,
which is very well known to researchers involved in inverse
scattering and nonlinear inverse problems in general, is usually
underestimated in design problems, also on the basis that
synthesis problems, opposite to imaging problems, may have
many different satisfactory solutions, (so that a suboptimal
solution can be anyway a good solution). On the other side, as
we have already shown in [21], [22], the (suboptimal) solutions
obtained in such a way (which are anyway much better than
the ones one would obtain by means of local optimization) can
be significantly worse than the actually (globally) optimal solutions. For example, recent results achieved in the synthesis of
pencil beams by means of sparse arrays [21] and in the optimal
compromise amongst sum and difference patterns problem
[22] show that usual implementation of global optimization
based synthesis procedures may keep significantly far from the
actually globally optimal solutions, with merit figures on the
side lobe level (SLL) which can be worse than alternative (and
better) solutions by more than 15 decibels.
Starting from such a consideration, in the following we resume in a unitary fashion the approach underlying [21] and [22],
and discuss in detail this recently introduced approach to array
synthesis which exploits convexity of the problem with respect
to a part of the degrees of freedom (when available). Such a
strategy allows to achieve increased performances with respect
to previous synthesis techniques in a number of classical design
problems, and to find effective solutions to new interesting design problems.
The paper is organized as follows. In Section II, the essentials of the proposed strategy are presented, and the underlying
rationale is motivated with the help of a simple yet significant
two-dimensional optimization problem. In Section III, an already available synthesis procedure proposed for the synthesis
of pencil beams by means of linear nonuniformly spaced arrays
is extended to the synthesis of difference patterns and to the
synthesis of pencil beams by means of sparse linear and planar
arrays, respectively. Very recent results on the optimal compromise amongst sum and difference patterns are also briefly
reviewed. Finally, in Section IV, the effectiveness of the proposed hybrid procedure allows to get effective design solutions,
by means of the concept of interleaved arrays, to the problem
of realizing a very flexible array antenna. Notwithstanding the
simplicity of the feeding network, this latter is capable to radiate a dual beam with individually steerable patterns or to perform jammer rejection at the physical layer or even to achieve a
flat-top pattern. Some conclusions follow.
0018-926X/$25.00 © 2007 IEEE
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D’URSO AND ISERNIA: SOLVING SOME ARRAY SYNTHESIS PROBLEMS
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II. A POSSIBLE HYBRID APPROACH FOR
SOME SYNTHESIS PROBLEMS
In a number of previous contributions [23]–[26] it has been
shown how a class of synthesis problems concerning fixed geometry arrays can be dealt with as “convex programming” (CP)
problems. As a matter of fact, unless nonconvex constraints are
enforced on the complex excitations of the array, both the synthesis of pencil beams [23]–[25] and of difference patterns [26]
subject to arbitrary upper bounds for the sidelobes (or even for
the near fields) can be formulated as CP problems, so that no
global optimization procedure is needed in order to get the globally optimal solutions. It has also to be noted that in case of uniformly spaced arrays (with fixed spacing) these problem can be
even reduced to linear programming (LP) problems (see [26]).
On the other side, many other synthesis problems (such as
for instance phase-only synthesis, synthesis of shaped beams,
synthesis of nonuniformly spaced arrays with unknown spacings, as well as many others) are not convex, so that it makes
sense in these latter cases to exploit global optimization procedures. In the following, we focus our attention on a peculiar
class of synthesis problems which are convex with respect to a
subset of the variables (for each value of the other ones), while
being not-convex with respect to all the remaining ones. This
class of problems include a number of important applications
(see Sections III and IV).
In all these cases, let us denote by
a first subset of variables to be determined, and by
a second subset of variables to be
optimized. Moreover, let us suppose the synthesis problem at
hand can be formulated as
(1.1)
(1.2)
Finally, let us suppose that both the objective function and
, are convex with respect
the constraint functions
to
for each fixed value of . Then, opposite to the simple
approach of using global optimization acting simultaneously on
all variables, let us consider a hybrid optimization procedure
trying to take advantage from the convexity of the problem with
respect to a part of the unknowns. To this end, let us define the
, defined as
auxiliary function
(2)
is the convex set defined by the
wherein, for each fixed
is a function of
intersection of the constraints (1.2). Note
the second subset of variables. While not being generally availcan be comable in an analytical fashion, the function
puted as the solution of a CP problem. Its solution will provide
for the given
not only the (unique) minimum value of
value of , but even, as a by-product, one of the values of
where such a minimum is achieved.1
Then, the overall problem can be conveniently formulated as
. As a matter of
the global optimization of the function
fact, such a strategy allows the number of unknowns dealt with
in the global optimization process to be reduced with respect
to the simpler and largely adopted solution of performing the
global optimization simultaneously on all variables. This latter
unknowns while only
variables are
would deal with
involved in the global optimization process of our approach. As
a consequence, global optimization tools will have to deal with a
reduced number of unknowns, thus saving computational times
or even finding better solutions with respect to previous approaches. In particular, much better performances with respect
to alternative approaches have been found in both cases wherein
is a significant part of the overall number of unknowns (given
) [21], [22] and in all those cases wherein
by
is very large [22] (according to the fact that the that computational complexity of global optimization procedures grows very
rapidly with the number of unknowns [1]–[5]).
In order to provide a simple yet significant example, let us
consider the case of an array whose elements locations and excitations are given by and , respectively. Then, a possible
expression for
, amounting to minimize the difference
with reamongst the field at broadside and in a direction at
spect to array axis, is
(3)
where is the wave number and the effective height. It can be
is convex with
checked that for any fixed value of
respect to , while being nonconvex with respect to . Then,
the rationale of the proposed procedure can be better explained
, obtained by expressing
by looking to a 2-D cut of
the excitations and locations as linear functions of single
,
variables, say and , respectively.2 The behavior of
which is obviously convex with respect to and not convex
and
with respect to , is depicted in Fig. 1(a) for
, while the behavior corresponding to the auxiliary
, which emphasizes its reduced dynamics with refunction
spect to
is in Fig. 1(b).
or
Let us now compare the problems of minimizing
by means of global optimization techniques such as SA or
GA. When using SA [1], [2], because of the random nature of
variations undertaken at each iteration, it can be argued that the
is by far more convenient than the
global optimization of
. To this end, first note that while is automatione of
, even
cally optimized at each iteration when dealing with
assuming one has reached the optimal value of , further iterations are needed to reach the optimal value of when dealing
. Also, these latter iterations may lead away from
with
. Last, but not least, as
has a much reduced dynamic with
1Note the minimum can be achieved in a single point or even in a convex
subset of the entire convex set.
2In particular, it has been chosen M = N = 6; h(=5) = 1:0; h (=2 =
1;
= [1; 1 + j(1 + x); 1 + j (1
x ); 1 +
1 :1
j1:5, with j =
j(1 + x); 1 + j(1
x); 1 + j(1 x)] and = [0; 0:12 + 0:6y; 3:78 +
0:6y; 5:1; 6:48 + 0:24y; 8:52 + 0:06y ].
0 0
p0 X
0
0
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Y
0
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 3, MARCH 2007
each generation), could result in an excessively large computational burden. Therefore, simulated annealing [1] will be considered in the following.
In summary, the simple reasoning above and the existing literature support the conclusion that computational complexity of
global optimization techniques grows (rapidly) with the number
of unknowns (see also [1]–[4]). In any actual case this implies
that for a very large number of unknowns “global” optimization
techniques will be unable to actually find the global optimum
in the limited time one has at his disposal, so that “suboptimal”
solution will be actually found. It follows that notwithstanding
the draw-back discussed above, an approach based on the minirather than of
has much more chances
mization of
to find the globally optimal solution (or anyway a better solution) in the limited time one has at his disposal.
III. REVIEW AND EXTENSIONS OF PREVIOUS RESULTS
In order to illustrate by means of examples the above introduced approach, in this section we briefly review some results
already achieved in the synthesis of pencil beams by means of
sparse linear arrays, and extend it to the synthesis of pencil
beams by sparse planar arrays, and to the synthesis of difference patterns. A very recent result in the synthesis of a system
capable to produce optimal sum and difference patterns contemporarily is also briefly recalled.
A. Pencil Beams by Means of Sparse Linear Arrays
In order to exemplify the approach above, let us discuss the
problem already considered in [21] to determine the excitations
and locations of a linear array in such a way to maximize the
field in a given direction while keeping the sidelobes below a
given arbitrary mask. By defining
(4)
Fig. 1. Normalized behavior of (a) F (x; y ) and (b) f (y ).
respect to
, it is easier for the random variations to cross
the mountains in order to come to the region of attraction of the
global optimum.
In GA [3] similar comments as above apply. In fact, the reduced dynamics and the reduced space to explore will make it
rather than of
.
easier to find the global optimum of
On the other side, if a larger population is used for , two
draw-backs arise. First, an enlargement of the population gives
rise per se to a heavier computational burden. Second, to fully
exploit the genes of a very large population [3], a large number
of generations have to be considered [3].
On the other side, the proposed approach also has a drawback. In fact, as it is computed through the solution of an auxilis
iary CP problem, the evaluation of the objective function
,
usually by far more cumbersome than the evaluation of
which has also to be taken into account. In particular, such a circumstance affects the choice of the global optimization one has
to adopt. GA based procedures [2], which require computation
of the objective function for each element of the population (at
where
and
are the excitations of the array elements,
(5)
are their unknown locations, it can be
where
easily shown (see [21]) that the synthesis problem can be reduced to a CP for any fixed set of locations, so that the overall
formulation proposed in Section II can be applied. In particular,
the objective function is given in such a case by
(6)
where
is the angular position of the main beam while conare given by a sufficiently fine discretizastraints
tion of the constraint on the sidelobes level, plus additional constraints on locations of the different element of the arrays (see
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D’URSO AND ISERNIA: SOLVING SOME ARRAY SYNTHESIS PROBLEMS
753
[21] for more details). Finally, following [21], the constraint
has to be considered as well.
It is worth to note that in standard benchmark problems used
in the literature the resulting synthesis procedure largely outperforms previous approaches. In particular, by using the hybrid approach described in [21], it is possible to achieve a SLL
which can be as low as 6 dB below the result in [14], [16], and
[27], wherein global optimization is performed simultaneously
on both the sets of unknowns (weights and locations), by using
a SA [14], [16] or a GA [27] based scheme, respectively.
In order to make the proposed hybrid approach more convincing, its computational cost has been compared with the case
wherein the global optimization acts on all the involved unknowns (as in [14], [16], and [27]). As, to the best of our efforts, we were not capable to achieve the same performances as
above by using different optimization schemes, and the single
iteration of the hybrid approach is more time-consuming than
, the final performances have
the simple evaluation of
been compared for a fixed CPU time. In our experience, for the
same benchmark problem considered above and in [14], [16],
and [27], the final SLL obtained by using the proposed hybrid
approach is 6 dB better than the one obtained, in the same CPU
time, by simultaneously optimizing with a simulated annealing
based procedure all the involved unknowns.
TABLE I
SYNTHESIZED LOCATIONS AND EXCITATIONS OF THE ARRAY OF FIG. 2
B. Difference Patterns by Means of Sparse Linear Arrays
As for any fixed geometry array the optimal synthesis of difference patterns can be again formulated as a CP problem [26],
a straightforward extension of the problem considered in Section III-A can be given to the determination of excitations and
locations of the elements of a linear array such to get a difference pattern with the maximum possible slope in the target direction while being bounded by an arbitrary mask elsewhere.
The overall unknowns can be still subdivided as in (4) and (5),
and the constraints
are exactly of the same type as
above, while the objective function is given herein by
(7)
where
represents the direction wherein the null has to be
located. In this case, according to the general theory in [26], the
has to be added.
additional constraint
As a test of performances one can achieve, let us consider a
synthesis problem with the same number of elements
and the same overall dimension
usually adopted as a
benchmark problem for the corresponding “sum pattern” case
[14], [16], and [27]. The power mask and the beam width
have been fixed as in [14]. The initial temperature of our
was the
hybrid SA-based procedure was set to
scaling factor for the temperatures and 25 was chosen as the
number of cycles to be performed at each temperature. In every
cycle, a new set of locations was randomly generated and the
MATLAB subroutine FMINCON [28] was used to solve the
CP problem involved, which also furnish, as a by-product, the
optimal excitations corresponding to the given set of locations.
Fig. 2 Symmetrically constrained difference beam pattern synthesized by
means of the hybrid technique using a 25-element linear array with an aperture
length of 50 , corresponding to a SLL = 13 dB.
0
The synthesized array and the resulting pattern are resumed in
Table I and Fig. 2.
It is interesting to note that notwithstanding the fact we are
dealing indeed with the synthesis of a difference pattern, a SLL
only slightly lower than the one in [14], [16] (where a sum pattern is synthesized) is indeed achieved.
C. Pencil Beams by Means of Sparse Planar Arrays
Another straightforward generalization of results in
Section III-A is given by the optimal synthesis of sum patterns
for planar arrays with unknown locations. The approach is
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identical to the one in Section III-A but for the fact that the set
has now to contain a couple of real numbers
of unknowns
for each element of the array, as both coordinates of the plane
have to be determined for each array element.
As example of performances one can achieve, let us first compare the effectiveness of the hybrid approach herein proposed
with the one proposed in [29]. To this end, let us consider the
problem of synthesizing a sparse planar array of elements located in a hexagonal cell included in a square of side 5 . In [29],
dB has been achieved with
. By using
an
the same beam-width of the main lobe as in [29], and starting
from a randomly defined spatial location of the array elements,
we ran our hybrid optimization approach. In particular, only
elements were used, the initial temperature of the hybrid
was the scaling
SA-based procedure was set to
factor for the temperature and only 10 cycles were performed
at each temperature. In every cycle, a new set of locations was
generated and the MATLAB subroutine FMINCON [28] was
used to solve the CP problem involved, which, as above, also
furnish the optimal excitations corresponding to the given set
of locations. Fig. 3(a) shows the final beam pattern and the
optimized locations. Notwithstanding the small number of cycles performed at each temperature, and the fact that the procedure in [29] exploits a number of antennas 50% larger, we get
dB, which is only slightly worse, and which
a
can be eventually improved by acting on the number of cycles.
As it can be seen, a proper exploitation of the convexity of the
problem with respect to a part of the unknowns allows to obtain
design solutions better than the ones which are achieved when
(the same kind of) global optimization procedures acts on all the
involved unknowns.
As a second example, let us consider the problem of synthesizing a planar array containing a maximum of 63 elements
. In
not-uniformly located on a rectangular lattice of
particular, the aim is to optimize the locations and weights of
the elements such to maximize the beam pattern in the broadside direction, while the beam width of the main lobe has been
fixed as in [30]. Starting from a randomly defined spatial loracation of the array elements, and by only using
diating elements, the hybrid method herein proposed is able to
dB. All the parameters of the proposed
achieve a
hybrid procedure have been fixed as before. Fig. 3(b) shows
the final beam pattern and the optimized locations. Note that
by using the GA based procedure in [30], at the same working
dB has been achieved (decreased
conditions, a
dB when using
elements [30]).
to
Therefore, even if a low number of SA cycles has been considered in the numerical procedure, a proper exploitation of the
convexity of the problem with respect to a part of unknown can
allow to significantly improve the final results. In particular, the
reduction of the number of unknowns involved in the global optimization scheme allows largely better results to be achieved as
compared to [30].
D. Optimal Compromise Amongst Sum and Difference
Patterns
Fig. 3. (a) Beam power pattern synthesized by means of the hybrid technique
with N = 41 corresponding to a SLL = 16:5 dB. The final locations are
also reported. (b) The achieved result by using a 29-element sparse planar array
producing a SLL = 19 dB.
0
0
cation of the feeding network (one network for the pencil beam
and another one for the difference pattern) it has been suggested
since from [31], [32] to adopt a feeding network of the kind depicted in Fig. 4. In such a scheme, the basic architecture of the
feeding network is organized in such a way to optimize the sum
pattern. At the same time, the array is subdivided into subarrays,
and a proper clustering into subarrays, and a proper choice of the
weights of the subarrays, allow to get a “good” difference pattern. Obviously, the larger the number of subarrays, the better
the difference pattern.
As we are going to show, also this kind of synthesis problems
can be dealt with by means of the hybrid approach we proposed
in Section II. In such a case (see [22] for more details)
In radar applications, both sum and difference patterns are
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(8)
D’URSO AND ISERNIA: SOLVING SOME ARRAY SYNTHESIS PROBLEMS
755
Fig. 4. Schematic diagram of a linear array partitioned into subarray (from
[18]).
wherein
is the number of adopted subarrays, and
the (complex) weights associated to the subarrays. Moreover
(9)
is an array of integer unknowns, with
, wherein
means that the th element of the array belongs to the th
means that the th element does not
subarray, whereas
contribute to the difference pattern.
By leaving the interested reader to [22] for more details, let
us just note herein that our proposed design procedure largely
outperforms previously known synthesis methods (based on the
use of global optimization procedures acting simultaneously on
all variables). By fixing the same parameters for the SA scheme
as before and by considering only 25 cycles to be performed
at each temperature, it turns out that the SLL reduces by more
15 dB in a benchmark [19] problem involving 20 antennas and
8 subarrays. In particular, the solid line in Fig. 5(a) shows the
result one achieves by using the hybrid approach herein proposed while the dotted one the result achieved in [19]. The re[see
duction of the SLL becomes of 10 dB in the case
subarrays are used
solid line, Fig. 5(b)] and 5 dB when
[see dash-dot line, Fig. 5(b)], according to the reduction of the
number of convex unknowns (i.e., the (complex) weight of the
subarrays) with respect to the total number of unknowns. Note
that also in these two last cases, by using the approach in [19],
dB has been achieved by using the same values
a
for . Table II shows the synthesized subarray weights.
IV. INTERLEAVED ARRAYS AS SIMPLE AND
VERSATILE ANTENNAS
Encouraged from the very good results above, we next considered the problem to synthesize arrays such to have at the same
time a good degree of flexibility and a very simple feeding network. To this end, the concept of interleaved arrays, recently
brought to our attention by a paper of R. L. Haupt [33], was
considered. Roughly speaking, an interleaved array is an array
Fig. 5. (a) Difference patterns obtained by using the proposed hybrid optimization (solid line) and the method in [19] (dash-dot line), with = 8 and = 20.
(b) Results for = 6 (sold line) and = 4 (dash-dot line).
R
R
R
N
TABLE II
SUBARRAYS WEIGHTS FOR THE CONFIGURATIONS IN FIG. 5
partitioned into subarrays, whereas each subarray is dedicated
to a different function (for example, each subarray radiates a
different field). In [33], several interesting examples are shown
wherein by properly aggregating the different elements of the
overall arrays into two subarrays, (and properly phasing the unitary excitations) different functions (such as sum and difference
pattern) are obtained.
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such to minimize the sidelobe level of two interleaved arrays.
The final beam patterns achieved in [33] correspond to a
dB. In order to have some similarity while dealing with
simple constraints we enforced the conditions
Fig. 6. Schematic diagram of the interleaved array concept (from [33]).
(12)
In the following, to further extend capabilities, the concept
of interleaved arrays is exploited in a different fashion, as both
the clustering into subarrays and the excitations of the different
antennas constituting the overall system are degrees of freedom
of our synthesis problems. Note that by virtue of the approach
proposed above, provided sum and difference patterns are of
interest, our global optimization problem has exactly the same
number and kind of unknowns (that is, aggregations) that in the
problem considered in [33].
As a basic problem, we considered the synthesis of a system
such to radiate two independent (and independently steerable)
pencil beams by means of a feeding network as simple as possible. Moreover, each individual beam has to fulfil given constraints on the sidelobes (in order, for example, to reduce crosstalk within given bounds). By using the above recalled concept
of interleaved arrays, the very simple architecture of Fig. 6 can
be considered, whereas the excitation of each element of the
overall system, as well as its belonging to the first or second
subarray, have to be determined. Again, such a problem can be
dealt with in an effective fashion by the approach described in
Section II. As a matter of fact, it suffices to fix
(10)
where is a binary variable such to be zero if the th element
belongs to the first subarray, and it is equal to 1 if the th element
belongs to the second subarray. Then
(11)
contains the (complex) excitations of the different elements. According to the general strategy in Section II, the constraint functions
contain all the constraints we have to consider
in the optimization problem. In these case, besides enforcing for
each beam the conditions already considered in Section III-A
(but for the ones on locations), they have to include and additional constraint enforcing an (at least approximate) equality for
the optima of the two beams (i.e., one looks for two beam patterns with the same value in the target direction).
As a test case, we considered one of the array configurations
adopted in [33] constituted by 60 total elements
spaced. In
that paper, wherein the (nonconvex) condition
(or turn
OFF the th element) is considered, the goal of the synthesis was
to use a GA based procedure to optimize the placement of the
elements on an assigned aperture and the phases of excitations
which are clearly convex, thus allowing to be considered in a
simple way in the proposed approach. Of course, the additional
degrees of freedom on the excitation amplitudes with respect to
[33] is expected to allow better performances with respect the
previous approach [33].
The initial temperature of our hybrid SA-based procedure
was set to
was the scaling factor for the
temperatures and 100 was chosen as the number of cycles to be
performed at each temperature. In every cycle, a new clustering
was randomly generated and the MATLAB subroutine
FGOALATTAIN [28] was used to solve the CP problem
involved, which furnish, as before, the optimal excitations
corresponding to the given clustering for both the interleaved
arrays. Note that, different from previous examples, due to the
multiobjective nature of the optimization problem (wherein
two different cost functions, corresponding to the real parts of
the array factors in the target direction and subject to mutual
dependence are optimized) the function FGOALATTAIN has
to be preferred to the FMINCON function, which is the one we
adopted for optimizing scalar cost functions (i.e., in all the
other cases of this paper). The synthesized array beam patterns
are shown in Fig. 7 (solid line), where the results obtained in
[33] are also reported (dotted line). As expected, the additional
degrees of freedom on the excitation amplitudes allow better
performances to be obtained with respect to [33]. In
dB)
particular, a pattern with a lower SLL (equal to
is obtained. It is also interesting to note that the final
clustering of the elements achieved by using the hybrid
method herein proposed, which is
, is quite different from the one in [33]. Needless to
say, a proper phasing of the two subarrays allows them to be
steered in an independent fashion.
It is interesting to note that the elementary bricks, as synthesized above, also may serve to achieve, by means of a very
simple feeding network, an array performing (in the overall) a
jammer rejection at the physical layer. For example, by phasing
the two above 30-elements interleaved arrays in such a way that
and
they are focused into two different directions, say =
[see Fig. 8(a)], and subtracting the second from the
first one (with a proper weight pre-computed off-line) it is possible to suppress an interfering signal coming from the direc[see Fig. 8(b)]. Note that as the two elementary
tion
bricks fields are super-imposed with rather different weights
(the second beam pattern is subtracted from the first one by reducing its level of about 16 dB) there is indeed a very small effect [not visible in the scale of Fig. 8(b)] on the maximum level
of sidelobes when comparing Fig. 8. Obviously, initial consideration of more subarrays would allow rejection of more than one
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D’URSO AND ISERNIA: SOLVING SOME ARRAY SYNTHESIS PROBLEMS
757
Fig. 7. Beam power patterns (solid line) for a 60-elements linear array optimized by the hybrid approach producing a SLL = 16:6 dB for (a) b = 1
and (b) b = 0. The result achieved in [33] is also reported (dotted line).
Fig. 8. (a) Beam power patterns of Fig. 7 pointing in two different directions:
# = =2 and # = =10. (b) Beam pattern obtained by subtracting the two
patterns of Fig. 8 (a) such to reject an interfering signal coming from # = =10.
jammer, and division of the overall array into subarrays having
a different number of elements is also worth to be investigated.
Finally, also note that a proper combination of two (or more)
independent patterns synthesized as above, provided a proper
shaping of the individual patterns has been performed, also allows easy reconfiguration of the individually steerable pencil
beams into a single shaped beams, with essentially no additional
need or complication of the very simple feeding network. For
example, the sum of the two patterns of Fig. 9(a), obtained by
optimizing two 10-elements interleaved arrays pointing towards
two different directions slightly different from the broadside
, allows the flat-top pattern of Fig. 9(b)
one,
to be achieved. Note the final pattern is characterized by a ripple
dB. Furthermore, as expected, the SLL in Fig. 9(b)
equal to
is higher than that in Fig. 9(a).
It is worth to note that initial consideration of P subarrays
would allow reconfiguration by a multibeam antenna into a wide
flat top pattern, or even to hybrid patterns having some pencil
beams together with a shaped one. Of course, because of its
expected deterioration, particular care has to be given in these
cases to the SLL of the final pattern, which requires both a
proper design of the single beams and a very careful combination of the elementary bricks. Also these ideas are worth to be
further analyzed.
0
V. FINAL REMARKS
An approach to the synthesis of array already proposed from
the authors for specific problems [21], [22] has been herein
formalized in a unitary framework (in such a way to include
a large number of possible applications), and thoroughly discussed. The approach takes advantage from the convexity of
the problem with respect to a part of the unknowns (whenever
available) to drastically reduce the computational complexity
of the global optimization issues involved in the overall synthesis problem. Such a reduction then gives rise to much faster
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758
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 3, MARCH 2007
exhibit good performances, thus leaving room for the realization
of very flexible antennas with very simple feeding networks.
While being a criticism to a “blind” exploitation of global
optimization procedures, this contribution enlarges indeed in the
whole the scope of global optimization in synthesis problems,
and sets new interesting challenges in this area. As a matter
of fact, the proposed exploitation of “hidden” convexity allows
to deal in an actually globally optimal fashion with problems
having an overall number of unknowns (much) larger than it
has been done up to now. Such a circumstance opens the way
to effective solutions of new exciting design problems (see for
example Section IV).
A number of interesting methodological issues also arises,
such as the most appropriate choice of the global optimizer for
this kind of problems (see the discussion at the end of Section II)
and the question of whether and how the exploitation of convexity (or “quasi-convexity”) can be of help in other synthesis
problems.
ACKNOWLEDGMENT
The authors would like to thank Prof. O. M. Bucci for interesting discussions. T. Isernia would like to thank Prof. A. Massa
for discussions leading the authors to Fig. 1 and relative comments. Thanks are also due to Dr. F. Di Rosa for the considerable help in the numerical simulations.
REFERENCES
Fig. 9. (a) Beam power patterns for two interleaved 10-elements linear array
optimized with the hybrid approach producing a SLL = 41 dB. (b) Flat top
beam pattern obtained by summing the two patterns of Fig. 9(a), with a SLL =
35 dB and a ripple of 0:2.
0
0
6
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approaches.
After proving the effectiveness of the proposed approach by
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by nonuniformly spaced planar arrays (both with unknown locations to be determined). Obviously, the approach can be extended in a straightforward fashion to conformal arrays with unknown locations of the elements. As shown in [21] and [26], the
hybrid approach herein discussed also can accommodate wideband requirements. Then, taking advantage from the effectiveness of the proposed hybrid scheme, the challenging problem of
realizing very flexible antennas by means of the concept of ‘interleaved arrays’ has been dealt with. The synthesized structures
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Michele D’Urso was born in Cercola, Italy, in 1976.
He received the Telecommunication Engineering
Master degree (summa cum laude) and the Ph.D.
degree from the University “Federico II” of Naples,
Italy, in 2002 and 2006, respectively.
After graduation, he joined the Applied Electromagnetic Group at the Department for Electronics
and Telecommunication Engineering of the University “Federico II” in Naples, first as an Associate
Researcher and then as a Ph.D. Student, from 2003 to
2005. From September 2004 to January 2005, he was
an intern in the Mathematics and Modeling Department of Schlumberger-Doll
Research, Ridgefield, CT, under the supervision of Prof. Habashy. Here, he
worked on electromagnetic modeling for cross-well tomography and marine
electrical prospecting. He is currently an Associate Researcher at the University
“Federico II” of Naples. His scientific interests are devoted to forward and
inverse electromagnetic scattering methods, phase retrieval problems and array
antenna synthesis.
Dr. D’Urso was awarded the G. Barzilai Award of the Italian Electromagnetic
Society (SIEM) in 2004.
Tommaso Isernia (M’92) received the M.S. degree
(summa cum laude) in electronic engineering from
the University of Naples “Federico II,” Naples, Italy,
in 1988.
From 1992 to 1998, he was a Researcher and,
from 1998 to 2003, an Associate Professor at the
Università di Napoli Federico II. He is now a Full
Professor of electromagnetic fields at the Università
Mediterranea di Reggio Calabria, Italy, wherein
he also serves as a member of the Consiglio di
Amministrazione. His research interests cover
inverse problems in electromagnetics, with particular emphasis on phase
retrieval, inverse scattering and antenna synthesis problems. More recently, he
is also engaged in effective solution procedures for forward scattering problems
as applied to very large scale problems and photonic crystal devices.
Prof. Isernia was awarded the G. Barzilai Award of the Italian Electromagnetic Society (SIEM) in 1994.
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