Journal of Electromagnetic Waves and Applications, Vol. 12, 103-138,
1998
SHAPED BEAM ANTENNA SYNTHESIS PROBLEMS:
FEASIBILITY CRITERIA AND NEW STRATEGIES
T. Isernia, O. M. Bucci, and N. Fiorentino
Dipartimento di Ingegneria Elettronica Universita "Federico II" di Napoli
via Claudio 21, 80125, Naples, Italy
Abstract-After briefly reviewing the properties of the squared amplitudes
of radiated fields, a simple and effective necessary condition to test if a
source of given size and structure can radiate or not a power pattern lying
in a given mask is furnished. The criterion is shown to be also sufficient for
linear arrays, thus allowing to reduce the overall problem to the synthesis of
a nominai pattern.Then, new synthesis procedures, strongly relying on the
properties of squared amplitude distributions and on the quadraticity of the
operator relating the source to the power pattern, are introduced. They are
implemented and tested in the cases of linear and planar uniform arrays,
showing that thanks to a full exploitation of the properties of quadratic
operators and of those of squared amplitude distributions, they allow one to
achieve very efficient solutions to mask constrained synthesis problems.
l. INTRODUCTION
In its full generality, the antenna synthesis problem consists of designing a
radiating system fulfilling a given set of requirements concerning:
the far-field pattern (or patterns, in the case of scanning or reconfigurable beam antennas);
(b) the antenna structure an d geometry;
(c) the feeding system.
(a)
With reference to point (a), a quite natural and flexible way to state the
far-field specifications is requiring to realize a power pattern lying in a given
"mask." In fact, an antenna engineer is usually interested not in achieving
a precise nominai pattern, but in synthesizing an antenna satisfying given
performances indexes (f.i., gain, level of secondary lobes, beam shape, and
so on) within certain tolerance limits. All these requirements can be simply
summarized in terms of a mask.
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Isernia et al.
Concerning points (b) an d (c), in many instances, either the structure is
fixed in advance (i.e., an array, a reflector or a mixed antenna) leading to
synthesis procedures wherein one tries to meet the pattern specifications by
varying the source excitation, or, for a given source, the fulfillment of the
far-field requirements is sought by changing the geometrie and/or electromagnetic structure of the antenna. Although the antenna synthesis problem
can be considered as one of the most long-standing in electromagnetics (see
[1~3] fora partial references list) and also the above formulation is not new
[4, 5], only very recently it has been addressed in its full generality [1, 3, 6],
leading to development of viable and flexible computer codes.
Apart from very particular (and simple) cases, all the power synthesis
techniques are optimization procedures, which, in order to achieve the design
goals, rely either explicitly or implicitly on the minimization of a proper
functional of the set of parameters specifying the antenna structure and
excitation. Because of the inherent non linearity and non convexity of the
problem, in all practical instances such a functional has many local minima,
which can trap the algorithm, unless the starting point of the minimization
procedure, by chance or some good ansatz, happens to lie in their attraction
zones. As a consequence, the designer can be led to the wrong conclusion
that the specifications cannot be met. Even if the synthesis is successful, one
usually has no way to judge if the design could be improved (f.i., by using a
smaller number of feeds, or a smaller antenna), save that by repeating the
synthesis, with a proper constraints modification.
Notwithstanding its relevance, the trapping problem has been practically
neglected up to now. Only very recently, use of genetic algorithms [7] has
been proposed in order to avo id this drawback. However, such "global"
optimization approaches are extremely heavy from the computational point
of view, so that it is dubious that they can be applied even in the case of
moderately complicateci problems.
In the light of above arguments, it seems clear and natural that the first
question an antenna engineer should address, in order to optimize the overall
synthesis procedure, is to understand whether or not, for an antenna with
given size and characteristics, a power pattern satisfying the given specifications does exist. Then, an adequate synthesis procedure, possibly optimized
with respect to numerica! efficiency, stable and robust against spurious solutions, should be devised.
To the best of our knowledge, simple and effective criteria to establish
a priori if a given radiating system can radiate or not a pattern lying in a
given mask have no t been proposed unt il very recently [8~ 10], an d only for
particular cases.
Shaped beam antenna synthesis problems
105
In this paper it is shown how the mathematical properties of squared
amplitude distributions of radiated fìelds (Sect. 2) can be usefully exploited
in order to ascertain "a priori" the feasibility of a given power synthesis
problem and to achieve design solutions as efficient as possible.
Thanks to proper finite dimensionai representations of squared amplitude
distributions (Sect. 2), we furnish in Sect. 3 a simple criterion to test for such
feasibility. The problem is shown to be equivalent to establish if a system
of linear inequalities admits a solution. As it is possible to answer this
question, through standard routines [11], in a simple, accurate and speedy
way, it turns out that the criterion can be of great usefulness in practical
applications, as it avoids possible expensive and unsuccessful synthesis trials.
The generality of the criterion with respect to the kind of source, its
capability to deal with additional constraints on the power pattern and its
"sharpness" are discussed in full detail. In particular, it is shown that the
developed existence criterion is both necessary and sufficient in the case
of uniform linear arrays (Sect. 4). On the other side, the criterion is only
necessary in case of 2-D arrays (Sect. 5) and generic sources (Sect. 6), and
sufficient (but not necessary) in the case of 2-D factorable masks and in all
cases which can be traced back to 1-D problems.
As in the feasible cases the existence criterion furnishes a pattern which
lies inside the given mask, in all circumstance wherein the existence criterion is sufficient the designer can do the subsequent synthesis procedure
searching for a surely synthetizable nominai pattern. The knowledge of the
pattern to look for and the properties of 1-D polynomials allow to achieve,
in such "sufficient" cases the most efficient solution to power pattern mask
constrained synthesis problems. In all other cases, i.e., when the existence
criterion is just necessary, above mentioned representations of squared amplitude distributions, together with the linear disequacies which express the
mask constraints, allow to identify the smallest convex set containing all
feasible patterns complying with the specifìcations. Then the "generalized
projections" approach [1, 4, 5] can be applied to this set and to the range
of the quadratic operator [12] relating the real and imaginary part of the
source excitation 1 to the power pattern. The main advantage of this choice
comes from the geometrica! properties of the two involved sets which are
convex and "quasi convex" [12] respectively. This knowledge suggests a
simple strategy to avoid trapping of the solution procedure into subsidiary
minima, so that the proposed formulation comes out to be very effective
even in such "non sufficient" cases. The resulting algorithm is basically a
two step iterated procedure, the fìrst one being a standard (globally solvl
In the case of fixed antenna structure. For the generai case, see Sect. 6.
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Isernia et al.
able) Quadratic Programming (QP) problem [13], and the other one being
very similar to a Phase Retrieval (PR) problem [14].
In Sections 7-8 the developed procedure is implemented and tested for the
case of array antennas. Sect. 7 deals with linear arrays, showing how it is
effectively possible single out the most efficient solution. Then, algorithms
for the case of 2-D uniform arrays are developed, implemented and tested in
Sect. 8. Numerica! efficiency of the overall synthesis procedures is optimized
exploiting an appropriate metric in the space of squared amplitude distributions and an alternative (approximate) implementation of the QP step. The
developed examples compare favorably with those reported in literature,
fulfilling the same specifications on the power pattern with less elements
or tighter constraints with the same number of elements. Conclusions and
suggestion for future work are collected in Sect. 9.
2. REPRESENTATION OF SQUARED AMPLITUDE FIELD
DISTRIBUTIONS
In all practical instances, electromagnetic fields can be considered as belonging to a finite dimensionai space. This happens not only when the
source has by itself a finite number of degrees of freedom, such as, f.i., in
the case of an array antenna, but also when arbitrary radiating systems
are considered. In the generai case, the number of degrees of freedom of
the field is defined as the minimum number of independent parameters required for its representation within a given accuracy. It turns out that, for
large sources (in terms of wavelength), this number is scarcely dependent on
the required accuracy, and essentially dictated by the source size [15, 16].
In particular, fields radiated from sources of bounded energy enclosed in a
sphere of radius a can be effectively (i.e., in an efficient and not redundant
way) approximated with bandlimited functions of bandwidth slightly larger
than {3a [15]. Accordingly, each far-field component can be represented in
a sampling series, i.e., [17]:
E( O, qy) ~ E(O, qy)DM(O)
M
Mn
+L { DM(O- On) LE(On,f/Jn,m)DMn(fjJ- f/Jn,m)
n=l
m=-Mn
Mn
- DM(O + On) L E(On, f/Jn,m)DMn (qy + 7r- f/Jn,m)}
m=-Mn
where
(l)
Shaped beam antenna synthesis problems
27m
107
21rm
()n = 2M + l' <Pn,m = 2M + l , M ~ {3a, M n ~ {3a sin ()n
and
D
M
(x)-
1x)
sin (2M+
2
(2M+ l) sin (x/2)
(2)
is the Dirichlet sampling function.
As a consequence, squared amplitude distributions can be represented by
bandlimited functions with a double band. And so straightforward generalization of (1) furnishes, with reference to an arbitrary squared component
of the far-field of a generic 3D source:
IE(O,</J)I 2 = P(O,</J) ~ P(O,<f;)DM(())
2M
2Mn
+L { D2M(()- ()n) L
n=1
P(()n, cPn,m)D2Mn (</J- cPn,m)
m=-2Mn
2Mn
- D2M(() +()n)
L
P(On, cPn,m)D2Mn (c/J + 1 f - cPn,m)}
m=-2Mn
(3)
where ()n = 4 'fJ~ 1 , <Pn,m = 4 ~~ 1 , and P(Bn, cPn,m) denote the corresponding sample of IE(B,c/JW.
Whenever additional information on the source is available, representations (1) or (3) may be redundant, and other, more specialized representations can be used. F.i., in the case of a linear (equispaced) array of N
antennas lying along the z axis, one may easily get for the squared amplitude of the array factor:
N-1
P(u) =co+ L
p=l
N-1
[cpcos (pu) + spsin (pu)] =
L
PpD2N-1(u- up)
p=l-N
(4)
wherein u = {3dcos (}, up = 2 ~7r_ 1 p, Pp = P(up) and d is the distance
between two adjacent antennas. Strictly speaking, (4) represent a non redundant representation of squared amplitude distributions only for d ~ ~ 2 .
However, it is well known that the invisible part of the spectrum must be
2
Otherwise, discrete prolate spheroidal functions [18] should be used.
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Isernia et al.
controlled in order to avoid superdirectivity, so that it makes sense using (4)
even for spacings Iess than haifa waveiength. In a simiiar fashion, in case
of a planar equispaced array, with N x M eiements lying on the x-y plane,
we have:
M-1
P(u,v) =coo+
L
[coqcos(qv)+soqsin(qv)]
q=l
N-1
+ L [cpo cos (pu) + Bpo sin (pu)]
p= l
+
N-1
M-1
p=l
q=-M+l
L L
[epq cos (pu + qv) + Bpq sin (pu + qv)]
q,<O
N-1
M-1
L L
PpqD2N-l(u-up)D2M-l(v-vq)
(5)
p=l-N q=l-M
wherein u
=
f3dx sin() cos <P, v
= /3dy sin() cos <P,
Up
= 2 ~7f_ 1 p,
Vq
= 2}J7r_ 1 q ,
Ppq = P(up, vq) and dx and dy denotes the distance between two adjacent
eiements aiong x direction and y directions respectiveiy, and the same warnings as before do appiy. 3 Above resuits can be summarized introducing the
generic finite dimensionai representation:
T
P(B, qy)
=L Dp\IIp(B, qy)
(6)
p= l
which, by a proper choice of T and Wp, is representative of (3), (4) or (5),
depending on the case at hand.
It must be noted right now that, in generai, not all functions expressibie
as in (6) correspond to squared amplitude distributions. This is because,
apart from the case (4), the dimensionality T ofthe representation is Iarger
than the number of (reai) degrees of freedom of the fieid. Accordingiy, in the
generai case, the set of all squared amplitude distributions is a (non linear)
variety embedded in the space spanned by the Wp functions. However,
above representations provides the smallest Iinear space containing the set
of all squared amplitude distributions.
3
Of course, the points such that (u/f3dx) 2
range even when dx, dy ~ ~.
+ (v/{3dy) 2 >
l belong to the invisible
Shaped beam antenna synthesis problems
109
3. NECESSARY EXISTENCE CONDITIONS
Exploiting (6) is now easy to show which conditions must be fulfilled in
order that a pattern lying in a given mask does exist. In fact, as (6), by
construction, is able to represent all possible patterns radiated from a given
classes of sources, a necessary condition for the existence of a field fulfilling
given constraints is that the following system of functionallinear inequalities
in the variables Dp is satisfied:
T
L Dp'llp(O, 4>) ::; u B(O, 4>)
p= l
T
(7)
L Dpwp(e, 4>) 2 LB(e, 4>)
p= l
wherein functions U B(O, 4>) and LB(B, 4>) denote the upper and lower
bound of the mask respectively. 4 lf we take into account the bandlimitedness of P( e, 4>) , equations (7) can be substituted with a sufficiently fine
discretization, so that (7) becomes:
T
LDp'llp(Oi,4>j)::; UB(Bi,4>j)
p= l
T
L Dpwp(ei, 4>1) 2 LB(ei, 4>1)
p= l
i.e., a system of ordinary linear inequalities in the Dp.
The solvability of a system of linear inequalities is a well known problem,
and it is equivalent to assess the existence of a "feasible point" for a "Linear
Programming" problem [13] .
To be fully satisfactory, an existence criterion should be numerically efficient, should be able to take into account as many kinds of constraints as
possible, and should be "sharp," or, in the ideal case, sufficient.
With respect to the first point, note that, in contrast to the full synthesis
problem, which is non linear, our existence criterion requires the solution of
a linear problem, which is a radical, strongly advantageous, difference. It is
worth noting that optimized programs for finding (if it exists) a solution to
4
When (4) or (5) are used, enforcing an adequately low value of UB in the invisible
part of the spectrum also allows to avoid superdirectivity.
110
Isernia et al.
a system of linear inequalities are present in the most common numerica!
libraries (see, f.i., [11]). Moreover, solving a system of linear inequalities is
much faster than solving a system of linear equations with the same number
of unknowns.
As a far as the second point, i.e., generality, is concerned, it is simple to
see that the developed approach is capable to take into account, in a simple
manner, any kind of constraint which is expressible in terms of a linear
functional of squared amplitude distributions. In fact, it is sufficient to add
such an inequality to system (8), with no conceptual complication. Note
that many constraints of interest, such as f.i., those on directivity, power
pattern slope, and nulls can be expressed as linear inequalities in the Dp .
Moreover, any kind of constraint which is convex with respect t o squared
amplitude distributions can be added to system (8). In fact, the introduction
of such kind of constraints does not impair the possibility of determining in
a simple fashion a feasible point for (8) [13].
As far as the third poi n t is concerned, t hat is the sharpness of the criterion,
the question amounts to establish, once a solution satisfying (8) has been
obtained, if we can effectively get a field corresponding to that solution.
Because, as it has been stressed before, T 2: 2C wherein C is the number of
complex degrees of freedom of the field, the set of (mathematically) feasible
patterns is generally only a subset of the space determined by (6), so that
fulfillment of conditions (8) is usually just necessary, but not sufficient, for
the existence of a pattern satisfying the constraints. Anyway, it does exist
at least a case, very important for the applications, wherein the criterion is
absolutely sharp (actually )3ufficient), i.e., the case of uniform linear arrays,
which we are going to examine in the next Section. Moreover, the criterion
is sufficient (but not necessary) in some planar arrays synthesis problems
(see Sect. 5), and has proved to be quite sharp in several other cases (see
Sect. 7).
Finally, as the representation of radiated fields through bandlimited functions automatically excludes superdirective sources [15], use of expansion
(3) in the system (8) allows to establish if given conditions on the power
pattern cannot be fulfilled using a source contained in a given sphere. Note
that the criterion is very generai and powerful, as it does not require to fix
in advance the structure of the source. Of course, when the structure of
the source is fixed in advance, use of more specialized expansion can lead to
sharper criteria.
In the following we will focus the attention on linear and planar arrays,
leaving to Sect. 6 the extension to arbitrary sources.
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Shaped beam antenna synthesis problems
4. THE LINEAR ARRAYS CASE
A real function such as (4) can be written as:
N-1
P(u) =
2:
with
(9)
p=-N+l
the star denoting complex conjugation.
Now, according to the Fejér-Riesz theorem [19], any non negative trigonometric polynomial such as (9) can be factorized as:
P(u) = F(u)F*(u)
(10)
wherein
N-l
F(u) =
2: Fpe_Jpu
(11)
p=O
which can be regarded as the array factor of an N element array. Therefore,
if it exist a real and positive function like (4) or (9) which lies in the assigned
mask, it does certainly exist a set of coefficients able to radiate that pattern.
Actually, because the factorization (10) is not unique, "fiipping" of the zeros
of F(u) lying outside the real axis of the complex u plane being allowed,
there exist 2No distinct sets of coefficients able to do it, wherein No is the
number of zeros of F( u) no t belonging to the real axis.
Therefore, in this case, the criterion is also sufficient, so that it is possible
to determine exactly if a pattern satisfying given requirements can be radiated or not from an array of given size. As an immediate consequence, once
the existence criterion is satisfied, the mask constrained linear array synthesis can be turned into the synthesis of a nominai, well identified, surely
synthesizable, pattern.
This synthesis strategy can be viewed as a modified version, simpler to
implement and to use, of the largely diffused "zero location" method [20],
largely adopted for the synthesis of shaped patterns. In fact, rather than
(adaptively) locating the zeros, the designer can shape the pattern by calling
a Linear Programming (LP) routine, such as f.i., the NAG-E04MBE [11],
and then using any program he likes for the synthesis of the power pattern.
Among other possibilities he can directly use the zeros of (9).
Let us explicitly note that the availability of 2No distinct solutions makes
it possible to introduce some kind of functional that quantify the "goodness"
of a solution (f.i., its smoothness, or dynamic range, etc.), and then look
for the optimum of this functional. This last problem can be seen as the
112
Isernia et al.
optimization of a functionai of No binary variabies, so that it is particuiarly
apt to be soived by using genetic aigorithms [21].
5. THE PLANAR ARRAYS CASE
As far as two dimensionai arrays are concerned, the main difference is that
factorization ruies anaiogous t o the one dimensionai case do no t ho Id for 2-D
polynomials. On the contrary, factorizable polynomials are a zero measure
subset of the set of polynomials in two variables [22]. This implies that even
ifa function of the kind (5) satisfies the constraints (8), because in generai
it cannot be factorized, it does not represent a physically feasibie squared
amplitude distribution, so that the existence criterion is just necessary, but
not sufficient.
Nevertheless, there exist at least two remarkabie exceptions. The first
one refers to power pattern masks which can be factorized as product of
two mask (one along each principal cut). In this case we can use for each
of the principal cuts the procedure of the previous Section. Note that in
this case the criterion is sufficient but not necessary. In fact, whiie it provides a possible solution to the synthesis problem, it Iooks for factorizabie
excitations, which are just a subset of all the possible ones. A similar reasoning applies to all cases wherein masks are such that the 2-D synthesis
problem can be reduced to a 1-D one through Bakianov [23] or McClellan
[24] transformations.
In the generai case, wherein sufficiency is not guaranteed, the criterion
can be used to discard those probiems which are certainiy unfeasibie. In the
"feasible" cases, the pattern furnished by the criterion will be quite certainly
not synthetizable. However, expioitation of representation (5) allows tostate
the power synthesis problem in a linear space as small as possibie, thus
drastically squeezing the set of patterns one should look for with respect to
the much larger set of all generic functions compatibie with the constraints.
In such a way, an effective synthesis procedure can be devised.
To this end, let us note that the operator which relates the unknown
excitations to the squared ampiitude distribution of radiated field can be
regarded as a quadratic operator, Q say, acting on the vector, g_, of the
reai and imaginary parts of the excitations coefficients [14]. The synthesis
procedure considered in Sect. 4 for the Iinear array case can be seen as a
way to find a solution to the quadratic operator equation:
Q(g_) =p
(12)
wherein P is the nominai power pattern furnished from the existence criterion. As we stressed before, in the planar array case, it is likely that
Shaped beam antenna synthesis problems
113
(12) does not admit a solution at all, so that the functional equation (12)
should be replaced with the minimization of some suitable distance between
the range of Q and the (nominai) pattern P. However, to fix in advance
a "nominai", non realizable, pattern is not convenient, as it unnecessarily
restricts the procedure. In fact, a power pattern fulfilling the constraints
is determined from the two conditions of being expressible as in (5) and
satisfying (8). Accordingly, instead of looking for a point of Q(gJ nearest
to the particular pattern P, it is natural to search for the element of Q(gJ
nearest to the set, F say, satisfying (5) and (8), i.e., to globally minimize
the functional:
T
<I>(g, D)= IIQ(g)-
L Dp'lip(B, 4>)11
2
subject to (8)
(13)
p= l
wherein D is the vector of the Dp coefficients.
From a geometrica! point of view, minimizing (13) amounts to find the
minimum distance between the smooth manifold Q(g), whose "quasiconvexity" properties have been extensively discussed in [12, 25], and the
convex set F . There are two main advantages of this formulation. The
first one is t hat F is (essentially) the smallest convex set containing all
the feasible power patterns complying with the mask, which has a positive
influence on the local minima problem. The second is that we can exploit
the available knowledge on the geometrica! properties of the two involved
sets in order to perform the minimization of <I> in an efficient and possibly
global fashion.
As a first point, note that such minimization can be easily performed
with a two steps iterated procedure, wherein at each step either the point
on the range of Q or the point of the set F is kept fixed. The advantage of
this procedure is that when we fix the point on the range of Q , a standard
QP globally solvable problem does result [13], while when we fix a point in
the set F , a quartic unconstrained functional, of the same kind of those
occurring in PR problems [26], has to be minimized. This minimization
procedure can be viewed as a particular case of the generalized projection
approach considered in [1]. However, it deals with sets narrower t han those
usual adopted (f.i., a small subset of all the functions compatible with the
constraints), which are "quasi convex" [12] and convex respectively. Accordingly, the approach should allow to avoid some of the "trapping" problems
occurring when generic non convex sets are used [1, 5, 27].
Moreover, the geometrica! characteristics of the involved sets suggest a
really simple and effective strategy to avoid stagnation of the synthesis procedure into residual local minima. To this end, let us first note that the
114
Isernia et al.
D
/
Figure l. A segment in the space of excitation coefficients maps into an
are of parabola in the space of square amplitude distributions: the range of
Q is made up by arcs of parabolas.
Figure 2. Concerning the technique to "escape" from a possible local
minimum.
quadratic operator Q(gJ maps segments of the space of the unknowns into
arcs of parabola in the space of the data [12, 25] (see Fig. 1). Then, let us
suppose that the minimization procedure got stuck into a local minimum,
Shaped beam antenna synthesis problems
115
so that the minimization procedure oscillates between a point of Q(gJ , say
IELI 2 (corresponding to the excitation f!L) and a point of the set F, say
PA. By the sake of simplicity, let us assume that the set F is constituted
by the single point P A . Moreover, le t us define as lE a 12 ( corresponding
to the excitation f!a) the point corresponding to the global minimum of
<I>. As IELI 2 , IEal 2 and PA define a plane in the space of squared amplitude distributions, the situation can be depicted as in Fig. 2 (however,
the parabola through IELI 2 and 1Eal 2 does not really need to lie in this
plane). The goal is to escape from the attraction point f!L jumping, let us
say, to the other branch of the parabola in order to (hopefully) fall into the
attraction region of f!G . As the PR step amounts to minimize the distance
between Q(g_) and the data point PA, a possible strategy is to enforce a
new objective point in the space of data in such a step. To this end, a point
of the same kind as PB of Fig. 2, lying beyond P A along the straight line
through lEL 12 and P A , is chosen. In this way, (see the dashed lines in
Fig. 2) IELI 2 is no more an attraction point for the functional to be minimized. As we have switched to the other branch of the parabola, if we take
again the usual succession of QP and PR steps, the minimization process
cannot be trapped into f!L(IELI 2 ) any more (see Fig. 2). Of course, it could
be necessary to apply this "escaping" procedure several times in order to
avoid other possible local minima. Similar reasonings apply when the set F
is not a single point.
6. EXTENSION TO GENERIC SOURCES
The similarity between (3) and (5) allow to extend the results relative to
planar arrays to 3-D sources (and planar apertures).
While the extension to the synthesis of planar apertures is straightforward, the case of 3-D sources enclosed in a sphere of radius a requires some
additional effort. To this end, note that because of expansion (3), it is stili
possible to expand each squared amplitude component in 2-D Fourier series
o n the (fictitious) interval [O :::; () :::; 2n, O :::; </> :::; 2n] provided that the
following equality constraints
P((),</>)= P(2n- e,</>+ n)
Fm (P(()n, </>))=O
Imi > 2Mn
(14)
(15)
are enforced, wherein Fm denotes the m-th Fourier harmonic. As conditions (14) and (15) are convex with respect to the Fourier coefficients of
squared amplitudes, these relationships can be (conceptually) added to the
116
Isernia et al.
system (8) without impairing convexity of the corresponding set of squared
amplitude distributions. It follows that both the existence criterion and the
QP step of the synthesis procedure, proposed in Sect. 5, can be realized with
straightforward modifications of the 2-D array case.
In the fixed geometry case, the same is true for the PR step, as the squared
field amplitudes are stili quadratically dependent on the unknowns. The only
difference (which, however, can be very significant from the computational
point of view) is that the operator Q(gJ is no more the squared amplitude
of a Fourier transform. Whenever the structure of the (fixed) source does
not fill the given sphere, expansions more specialized than (3) can allow a
sharper existence criterion. As dealing with a set of feasible power patterns
as small as possible also has a positive infiuence on the local minima problem, development of non redundant representations of squared amplitude
distributions as suggested in [14] or [28] would improve effectiveness of the
overall procedure.
In the variable geometry case, because the relationship between the radiated field and the parameters specifying the antenna geometry is strongly
non linear, the geometry of the set of all feasible power pattern is wildly
modified, loosing the "quasi convexity" property. To preserve as much as
possible this last property, and its benefit effect on the trapping problem,
we can split the overall synthesis procedure in two steps. In the first one,
we exploit the fundamental representation (1), which shows that each component of the radiated field can be seen as the array factor of a "virtual"
equispaced array, provided, again, that equality constraints analogous to
(14-15) are enforced. With reference to this point, recently developed nonredundant sampling expansions [29, 30], which also can take into account
the overall "shape" of the source, are particularly worth mentioning, as they
allow to obtain non redundant Fourier expansions for the fields and their
squared amplitudes. Accordingly, the overall synthesis procedure available
for equispaced planar arrays can be adopted to determine a complex field
meeting the required power specifications. If this step is unsuccessful, we
can be quite confident that these specifications cannot be fulfilled by an
antenna with the given size. Otherwise, we can proceed to the second step,
i.e., a field synthesis, to determine the antenna shape and excitation. In this
way a power pattern mask constrained problem can be reduced to an easier
field synthesis one. The overall design procedure is summarized in the fiow
chart in Fig. 3.
Shaped bea:m antenna synthesis problem~
117
introduce auxiliary Fourier Series
for thc ficld and thc
squarcd amplitude
r~tatiom
Feasibility Criterion
(including ali convex constrainu)
NO
Sinthesis p-occdure for the cquivalent 2-0
array problem (includiD8 lineat constraints
ofthe kind [14 ) e (IS))
Theproblem
is unfeasible
(nominai field)
NO
SOURCE GEOMETRY
SOURCE EXCITATIONS
Figure 3 . F low chart summarizing steps of the proposed approach to variable geometry power synthesis problems.
118
Isernia et al.
7. LINEAR ARRAY SYNTHESIS: IMPLEMENTATION AND
EXAMPLES
In the linear case both the number of unknowns (2N - l) and that
of inequalities (8) (depending on the adopted discretization step) are such
that the the existence criterion can be implemented, without any problem,
by using library routines.
With regard to the excitation synthesis, because the existence criterion
is necessary and sufficient, the question amounts to synthesize a nominai,
feasible power pattern. From an implementative point of view, two suitable
ways are the extraction and managing of the zeros of the squared amplitude
distributions or the use of a PR procedure. 5
Should the existence criterion be just necessary due to the introduction
of further constraints, the synthesis procedure of Sect. 5 can be adopted,
wherein a library routine (NAG-E04NCF [5]) can be used to implement the
QP step, as the computational time is stili acceptable.
To show the usefulness of the existence criterion, as well as of the synthesis
procedure suggested in Sect. 4, we have checked it against the synthesis of a
flat top pattern, with the mask shown in Fig. 4 (solid line). As a first step,
we have tested if we can obtain a pattern lying in the mask with an array
of 14 elements, half wavelength spaced. After a few second of computation
on an Alpha workstation, we got a negative answer, with a corresponding
"best" pattern reported in Fig. 4 (dashed line). Therefore, the designer
has now two possibilities, i.e., either slightly relaxing the requirements, or
increasing the number of elements of the array, which is the only choice in
the case of strict pattern constraints.
By using 15 elements, the existence criterion is satisfied and a possible
way to synthesize it is the use of a PR procedure. By using the PR procedure
described in [26], and some "zero-flipping," we got the pattern of Fig. 5 and
the excitations of Tab. l, which are sufficiently smooth to be realized. Let
us stress that even if no bounds have been enforced on the excitations, a
dynamic range of about 6 has been obtained, which is well acceptable.
Note also explicitly that the application of the existence criterion ensures
that we absolutely cannot find the same pattern, or anyway a pattern complying with the constraints, with a lower number of elements.
As a second example, in the following, the synthesis of a classica! pattern
of interest for radar applications is considered. The prescribed mask is shown
in Fig. 6 (soli d line) an d it has a cosec2 (e - 90°) x cos (e - 90°) behavior
in the shaped area (100° :S e :S 140°). Assuming d = ~, the minimum
number of antennas satisfying the existence criterion is 16. In Fig. 6 (dashed
5
Results of Sect. 4 imply that local minima do not exist in such 1-D PR problems.
Shaped beam antenna synthesis problems
IO
l
_l
l
o r-
'
l
-
-20
-
-30
-
l_
-
'
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
r-
l
l
'
-IO
l
'
'
'
:'
iD(dB)
119
l
l
l
l
\l
-
l
l
l
l
l
l
l
l
l
l
l
l
-
-
11
,',
l\
-40 l-'l
l ~ :
: l l
l
l
"'
l
l
Il
Il
\:: ::
l
r~:
-50
l
-3
::::
::d \:l
f
l
1 1
Il
'
l
-2
.
\:\q
"i \Al !-l
\::l: l 'J
l,,
l::
l ,,
,.
\ :: f, : ~ :
"v
l
-l
o
\t
1\i
2
l
i'
~
.! r
3
u
Figure 4. Synthesis of a flat top beam: prescribed mask (solid line) and
power pattern resulting from the existence criterion (dashed line).
line) is reported the synthesized pattern by using a PR procedure. The
aperture distribution (see Tab. 2) has a dynamic range of about 16. We
stress that, while using the same number of antennas as in [20], much tighter
constraints are met.
A third example refers to the case wherein further constraints are enforced, making impossible to rely on the zeros of P in order to compute
excitations coefficients, so that we must turn to the synthesis procedure suggested in Sect. 5. The additional constraint is the enforcement to zero of
the 11-th excitation coefficient in the flat top example. By a simple alternate projections procedure, we obtain the (unsatisfactory) result of Fig. 7.
Therefore, we are lead to apply our "global minimization strategy" by properly modifying, every t ime a stagnati on occur, the nominai pattern to be
pursued in the PR steps. After a few escaping from local minima, the pattern of Fig. 8 is finally achieved. According to the Tab. 3, where excitation
coefficients are reported, we can observe excitations exhibiting a dynamic
range slightly larger than the one seen in example l. The proposed synthesis
120
Isernia et al.
10
l
_l
o-
l
l
l
l\
l
l
l
l
l
l
l
l
-IO i-
-
l
l
l
iD(dB)
l_
' '
'
l
l
-
l
l
l
l
l
l
-20
l
l
l
l
-
-
-30 i-
~ \ ; \ :1
\ f\ l\ :\ :'
f\ !
\1\:\:\ ;~
: \ : ~ : \:
\: ~~ :: \
,,,,
-40 :--\l
l
•
-50
, 1
,,
\:.. ~1 l: \l :t \1 ,l
,,
,,,,
,,,,
l l
Il
-l
~
-3
-2
-1
,,
~J
l
ft
l
o
2
3
u
Figure 5. Synthesis of a flat top beam: prescribed mask (solid line) and
synthesized pattern (dashed line).
procedure is effectively capable to identify the globally optimum. Needless
to say, the result is the most efficient possible one.
8. PLANAR ARRAY SYNTHESIS: IMPLEMENTATION AND
EXAMPLES
In the 2-D case, the number of unknowns and constraints grow quadratically with the size of the array, so that both the existence criterion and the
synthesis strategy need to be optimized from a numerica! point of view.
As far as the feasibility problem is concerned, note that it can be stated
as the search of an intersection point between two sets, i.e., the set X of
real functions expressible as in (5) and the subset Y of the space L 2 of
square integrable real functions lying in the given mask. It can be easily
verified that both X and Y are convex sets. Then the problem of finding
an intersection point can be solved, in a simple and effective way, exploiting
Shaped beam antenna synthesis problems
n
la.!
l
4.745E-02
0.1183
0.1776
0.1878
0.1976
0.2444
0.2259
0.1269
0.1381
0.1885
0.1410
7.057E-02
8.452E-02
7.612E-02
3.746E-02
2
3
4
s
6
7
8
9
10
11
12
13
14
15
121
La.
(rad)
0.7859
0.7384
0.6071
0.2636
-0.3645
-0.7874
-0.8201
-0.2842
1.0793 .
1.5967
1.9898
2.9895
-2.1826
-1.8360
- 1.6683
Table l. Synthesis of a fiat top beam: excitation coefficients achieved by a
PR procedure and a zero-fiipping operation.
the alternate projections procedure. In fact, it is well known that for convex
sets, whenever X n Y =f. 0, this procedures converges to an intersection
point [31]. Should the intersection be empty, the alternating projections
will oscillate between the two points of minimal mutuai distance. A detailed
description of the required projectors is given in Appendix A.
In the synthesis procedure, which amounts to iteratively salve PR and QP
problems, the PR steps are already numerically efficient by virtue of proper
use of Fast Fourier Transform (FFT) procedures and numerically optimized
codes [26]. As far as the QP steps are concerned, the idea is once again to
exploit the knowledge of the geometry of the involved sets.
To this end, let us first note that the convex set (8) is the intersection
between the two above defined sets X and Y . Furthermore we note that
the arcs of parabolas, spanning the range of the operator Q, belong to the
set X. Accordingly, the situation can be pictorially represented as in Fig. 9.
The QP step has to perform the projection onto the set X n Y from a
given point A of Q(g_) (let B be this projection). In the situation depicted
in Fig. 9, as well as in many other cases, the QP step can be performed,
starting from A, by alternatively projecting on the sets Y and X. In such
122
Isernia et al.
o
-10
-20
-30 f------------,----1
-2
-1
o
2
3
u
Figure 6. Synthesis of a cosecant beam: prescribed mask (solid line) and
synthesized pattern (dashed line).
a way we get large computing time savings. While above reasoning cannot
be considered a proof, many computational examples we have performed
strongly support such a kind of procedure with respect to a standard QP
step.
A second important point we must consider is the possible slow convergency of the minimization procedure. In fact, the functional (representative
of the squared distance between the set F and the manifold Q(g) ) can
decrease very slowly. In the alternating projection framework this situation
is known as the "tunneling problem" [32]. As in PR problems [33], such a
slow convergence can be traced back to the fact that the functional to be
minimized, representing the distance in L 2 between a function belonging to
the mask and one that is external to mask itself, weighs "absolute" errors,
thus strongly emphasizing the relevance of the high level zones of the mask.
As a consequence, only after that the constraints in the high level regions
have been substantially fulfilled, the procedure can operate on the low level
ones, with a corresponding slow down of the convergence. A simple way to
Shaped beam antenna synthesis problems
n
123
Lan
lanl
-1.3834
-0.5624
-0.2948
4.093E-02
0.8420
1.2767
1.5314
2.2522
2.9073
-3.1235
-2.5058.
-1.4922
-0.3466
0.8142
1.8492
1.8268
1.974E-02
2.228E-02
1.942E-02
3.620E-02
4.711E-02
4.274E-02
6.367E-02
8.720E-02
8.166E-02
8.792E-02
0.1401
0.1741
0.1547
9.762E-02
4.033E-02
1.061E-02
2
3
4
s
6
7
8
9
IO
Il
12
13
14
lS
16
(rad)
Table 2. Synthesis of a cosecant beam: excitation coefficients achieved by
a PR procedure and a zero-fiipping operation.
alleviate this problem is to modify the functional to be minimized, in order
to weigh both high and low level zones. On the basis of results in [14, 25],
an effective choice is to divide each contribution to functional <I> by some
kind of "average mask," i.e.,
Fw(B,
1) =
U B(B, 1); LB(O, 1)
(16)
Then the functional to minimize becomes
8(Q, D)=
Il Q(Q)-
L-~=l
Dp\I!p(B, 1) 11
12
F~ (0, 1)
2
subject
to
(8)
(17)
wherein Il · Il denotes the (properly discretized) L 2 norm.
From a geometrie point of view the introduction of a weight function can
be seen as an appropriate change of metric in the space of the data, which
"widens" the tunnels, with a corresponding faster convergence toward the
minimum. The introduction of such weights makes it necessary to modify
both the PR step (which is trivial) and the QP step, which introduces some
124
Isernia et al.
IO
l
l
_l
o-
'
/
-IO 1-
l
l
' ' '
P(dB)
-20
l
l_
-
'
\
-
-
-
-30 1....
l
-40
,
.... - , , /
-
,,
\
''
'
., ' :'
'' '
: ~~ l
': t'l'
\: \~
l
li
-3
l
-2
-
v~·o l
l
j
-50
/',~ ...... , ....' \
l,,
l
l
'l
l
l
l
-l
o
l
l
3
u
Figure 7. Synthesis of a flat top beam with 11-th coefficient enforced to zero
and without application of the "escaping technique" application: prescribed
mask (solid line) and synthesized pattern (dashed line).
difficulties when it is realized using alternate projections. This last point is
dealt with in Appendix B.
To show usefulness and validity of the proposed implementations of the
existence criterion and the synthesis procedure in 2-D case, in the following
we consider the synthesis of a flat top pattern. The prescribed mask has an
almost triangular section which is reported in Fig. 10.
First of all we have examined if an array of 14 x 14 half wavelength
spaced elements can radiate or nota power pattern which satisfies the given
constraints. After about half a minute of computation on an Alpha station
the existence criterion gives a negative answer. Assuming that the constraints are strictly imposed, we (slightly) enlarge to 15 x 15 the number
of elements. Now the existence criterion gives a satisfactory result which is
shown in Fig. 11. The next step consists of performing the synthesis with
this number of elements. Applying the proposed algorithm together with
the technique to "escape" from local minima, we are able to find the pattern
in Fig. 12.
125
Shaped beam antenna synthesis problems
10
l
_l
l
o f-
l
l
l
:
P(dB)
'
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
-
-20 1-
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
'
l
-lO
l_
'
'
l
l
l
l
-
-
-
-30 1-
l\ : \ : ~ 1\ :
1\/
-40 ~\:\:l
1 1 1 1 1 1 r ''
~J ~ / \ / y
•
-50
1 l1
1 l1
l'l
-l
u
-2
Il
Il
l l
Il
l
l l
\f l : \ l \ :_J,
~
,, ,,
'
""
~"
l l
l l
1 l
\-'
Il
Il
Il
Il
Il
Il
Il
-3
t1
Il
l l
l
-1
o
2
13
u
Figure 8. Synthesis of a flat top beam with 11-th coefficient enforced to
zero and applying the "escaping technique": prescribed mask (solid line)
an d synthesized pattern (dashed line) _
It is worth to note that we are practically able to synthesize the desired
pattern with a number of elements which, according to the existence criterion, cannot be lower. This example also shows the practical usefulness of
the existence criterion in the "non sufficient" cases.
As a further check of the effectiveness of the proposed approach, we have
considered a 2-D pencil beam pattern synthesis which can be treated by the
"Tseng-Cheng" formulas [23, 34]. The mask meets the following constraints:
side lobe level (SLL) = 20dB; direction of scan Bo = O(uo = O, vo = O;
angular beamwidth both in (} and c/> of about 26° _ The existence criterion
tell us that we need at least 10 x 10 elements (half a wavelength spaced
each other) in arder to satisfy these constraints. The result of the synthesis
strategy with this (minimum) number of elements is reported in Fig. 13. In
Fig. 14 our result is compared with the Tseng-Cheng one.
Excitation constraints cannot be directly managed by our approach, but
when they can be translated into convex constraints on the power pattern
126
Isernia et al.
n
la.J
La. (rad)
l
2
3
4
4.380E-02
0.1109
0.1529
9.843E-02
6.281E-02
0.2496
0.3382
0.2782
0.1313
.I.743E-02
O.OOOE+OO
5.092E-02
9.119E-02
8.014E-02
4.014E-02
0.8468
0.8964
0.9454
0.9946
-2.0980
-2.0489
- 1.9998
- 1.9507
- 1.9018
- 1.8534
O.OOOE+OO
- 1.7545
- 1.7052
- 1.6561
- 1.6072
s
6
7
8
9
lO
11
12
13
14
15
Table 3. Synthesis of a flat-top beam with 11-th coefficient enforced to zero:
excitation coefficients achieved by the proposed synthesis procedure.
Figure 9. On the possibility to salve in an alternative faster manner the
QP problem.
Shaped beam antenna synthesis problems
127
rr.--------------,
v
-n L----------~
UTT
""'11'
Figure 10. Synthesis of a trianguiar contoured beam: the prescribed mask
A
B
c
-0.5dB :S Pds :S 0.5dB
'PdB :S 0.5dB
PdB:::; -30dB
(see Sect. 3). Accordingiy, we must resort to more generai and fl.exibie
approaches, as, f.i., the generaiized projection procedure considered in [1].
However, the method deveioped in this paper can stili be very usefui to avoid,
or at Ieast mitigate the trapping probiem. In fact, it is likeiy that if the
pattern specification can be oniy marginally satisfied in the unconstrained
case, no soiution will be possibie in presence of excitation constraints which
are not aiready satisfied. Viceversa, if in the unconstrained case a soiution
well inside the mask does exist, 6 it is likeiy that the constrained case will
be solvabie, as one is allowed to modify this pattern in order to satisfy the
additionai constraints. Moreover, the unconstrained solution corresponding
to this pattern can be in the attraction zone of a soiution of the constrained
problem, so that it can be used as a good starting point for the above
mentioned more generai procedure.
As preliminary check of the viability of such an approach we show in
Fig. 15 the pattern obtained by a more traditionai "alternating projections"
procedure starting from previous results of Fig. 12 and prescribing an excitations dynamic range of 20. lt can be noted that the obtained pattern
is oniy slightly worse than the one in Fig. 12, while the dynamic range has
6
A possible way to obtain a pattern which is (well) inside the polytope determined
from the constraints is to make them more and more stringent, until no solution is possible.
128
Isernia et al.
(a)
lO
LI
l
l
o r-
f-'
1'(dll)
-10
l
1/
r-
l
l
l.
-
~
-
l
-
. - zo 1-
- )()
~
o
l
..$()
-3
-2
-l
o
v
l
1\ 1'\
l\
v
_l
2
3
(b)
Figure 11. Sy nthesis of a t riangular contoured beam: "powcr pattern"
d elivercd from t he existence criterion with 15 x 15 elemcnts: (a) 3-D re~
resentation, (b) wors t eu t.
129
Shaped beam antenna synth esis pro blems
(a)
IO
o
J
l
l
l
l
-
~
(
~ (dB)
-IO
l
i\
l-
-
-20 l-
- 30
"
-.o ~
-$O
.
~
f\
~l
l1
-3
-2
l
l
-l
o
2
3
u
(b)
Figure 12. Synthesis of a triangular contoured beam: the synthesized pattern when using 15 x 15 elemcnts and without dynamic constraints: (a)
3-D representation , (b) worst cut .
l :SO
Isernia et al.
Figure 13. Synthesized pencil beam (3-0 representation). Sidelobes are at
-20dB with respect to the maximum.
been reduced from 130 to 20. Note that no similar pattern can be obtained
when starting the alternating projection technique from a generic first guess.
9. CONCLUSIONS
A new point of view to mask constrained antenna synthesis has been presented. The developed approach starts from the idea that feasibility of the
assigned synthesis problem should be discussed first. Therefore, simple and
effective feasibility criteria have been developed which allow to avoid possible
expensive and unsuccessful synthesis trials.
In addition to that, t he developed criteria furnish plausible power patterns. lt follows that the subsequent synthesis can be done by looking for
a nominai pattern. When t he criteria are also sufficient, this simply ends
the overall synthesis procedure. This happens in all cases wherein the problem is equivalent to the power pattern synthesis of a uniform linear array,
so that the synthesis procedure presented in Sect. 4, which is very easy to
implement, gives in a very fast manner ali possible "zero flipping" solutions.
Shaped beam antenna synthesis problems
131
Figure 14. Comparison between the synthesized pattern and the TsengCheng pattern (section v = O) .
When the criteria are just necessary so that their fulfillment does not
guarantee the feasibility of the nominai pattern, the problem can be stated
as that of finding the (globally) minimum distance between a convex set of
functions fulfilling the pattern constraints and the range of a quadratic operator. The proposed minimization procedure, which amounts to iteratively
solve PR and QP problems, strongly resembles those based on alternating
projections onto non convex sets, but exploits completely different, "narrower" and "smoother" sets. Moreover, knowledge of the geometrica! properties of the involved sets allows to devise some suitable strategies in order
to escape from the local minima of the distance between the two involved
sets.
The numerica! results, as compared with those available in the literature,
show that the developed approach is able to solve mask constrained synthesis problems with the least number of elements, or minimum antenna
sizes. Moreover, a t variance of other approaches, the suggested procedure
does not need any "good starting guess," in order to be effective. It can be
concluded that the presented approach allows to recognize power pattern
configurations which are otherwise diffi.cult to reach, thus improving the
132
Ise rnia e t al.
(a)
10
LI
l
l
l
l
l
l_
-
o r-
l
P (dB)
l
- 20
- 30
- 40
\
l
-
r-
~\
r1
~
l
l
l
l
o
~:
2
3
"
(b)
Figure 15. Synthesis of a triangular contoured beam: the synthesized pattern when using 15 x 15 elements and a dynamic range constraint of 20:
(a) 3-D representation, (b) worst cut.
Shaped beam antenna synthesis problems
133
design of radiating systems when "mask" constraints on the power pattern
are enforcedo The same comment applies when additional feasibility constraints on the source excitations are presento In fact, use of the proposed
procedure as a preliminary step ofmore fiexible approaches (such as [l]) has
also been shown effective in reducing the trapping problemso
APPENDIX A
In this Appendix we discuss the realization of the projections onto the sets
X and Y considered in Secto 80
Projection onto Y
If x E X we need to determine Py(x), that is the projection of x onto Y o
Denoting again with LB(u, v) and U B(u, v) the lower and upper bounds
of the mask, it results [1]
LB
Py(x) =
{
x
UB
for x::; LB
for LB ::; x ::; U B
for x~ UB
(Aol)
For the equispaced planar arrays the functions of the set X can be represented in exact way through a finite number of samples [(2N -l) x (2M -l)]
equispaced on the u-v piane (see (5))0 However, the projection (Aol) should
be performed for each different u and v valueo An approximate solution
amounts to take a fine discretization of the u-v region of interest (much
finer than in (5))0 Therefore, before performing (Aol) one needs to interpolate the square amplitude samples in (5) up to N out x M out sampling
pointso It is convenient to choose both N out and M aut as power of 2
Accordingly, the projection of x onto Y can be performed through the
sequence of steps
l.
DFT of the (2N- l) x (2M- l) samples of x;
20
Zero padding (ZP) up to N aut x M aut;
30 Inverse FFT;
40 Projection on the mask (see (Aol))o
Projection onto X
For y E Y, the problem is that to determine Px(Y), ioeo the projection
of y onto the set X o In the Fourier domain this projection consists of
taking the spectrum of y, truncating it to the proper band and imposing
the hermitian conditiono Then, if F is the Fourier transform operator, we
134
Isernia et al.
can write
Px(y)
= F- 1 {Psp [F(y)]}
(A.2)
wherein Psp realizes the two operations of truncating and imposing the hermitian condition on the spectrum. However, because the functions of Y are
themselves real, and the spectrum truncation does not alter the symmetry
of this last one, the second of the two above operations become unnecessary.
Therefore Px (y) c an be performed as follows
l.
FFT of the Nout x Mout samples of y;
2.
Extraction ofthe (2N-l) x (2M-l) centrai harmonics from the Noutx
M aut harmonics found a t the step l;
3.
Inverse DFT of the sequence achieved at the step 2.
so that
Px(y)
= DFT- 1 {Ex
(A.3)
[FFT(y)]}
wherein Ex indicates the extraction operator that performs the second step.
APPENDIX B
In this Appendix we show how to implement the QP step, and in particular
the projection onto the set X when the QP step itself is realized through
the alternate projections in the weighted norm defined by (17). The goal is
finding the Fourier harmonics, A, of the unknown function Px (y) such t hat
8(A)
=Il
FFT-1 ~~(A)]- y 112
(B.l)
is minimum.
From (B .l), one gets
88
(FFT- 1 [ZP(8A)] FFT- 1 [ZP(A)]-
=
2
1/2
Fw
'
1/2
y)
Fw
(B.2)
As the extraction operator Ex, considered in Appendix A, is the adjoint
of ZP, we have
1
88 = (8A,2ExFFT { FFT- [::(A)]-
y})
(B.3)
wherein the second factor of the scalar product can be recognized as the
gradient of e' i.e.,
Shaped beam antenna synthesis problems
135
Ve= 2ExF FT { F FT-l [~:(A)] -
y}
(B.4)
lmposing the stationarity of e' i.e., ve= o' one gets
ExFFT { FFT-l
[~:(A)]-
y}
= ExFFT
{:w}
(B.5)
so that the projection onta the set X can be obtained solving (B.5) with
respect to the harmonics A (then, the corresponding field can be easily
computed). Note that (B.5) is a system of linear equations with an equal
number ( (2N- l) x (2M- l)) of unknowns and data. Note also that the
matrix implicitly defined in the right-hand member can be inverted once for
all and then stored to perform projections with different y.
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Ovidio M. Bucci graduateci summa cum laude in Electronic Engineering
at the University of Naples in 1966. Assistant Professar at the Istituto Universitario Navale since 1967, he became Associate Professar (1970) and Full
Professar (1976) of Electromagnetic Fields at the "Federico II" University.
From 1984 to 1986 and in 1993 he has been the Head of the Electronic Engineering Department at the University of Naples and he is now the vice
Rector of the University. His recent scientific activities include high performance antennas analysis and synthesis, near field to far field techniques and
EM inverse problems. Professar Bucci is a Fellow of the IEEE AP Society.
Nunzio Fiorentino graduateci summa cum laude at the "Federico II" University in 1996. After cooperating for some time with the Applied Electromagnetics group at the "Federico II" University of Naples, he is now with
Ericsson Telecomunicazioni SpA, Italy.
Tommaso Isernia graduateci summa cum laude at the "Federico II" University of Naples, and earned his Ph.D. degree in 1992. Since 1988, he has
cooperateci with the Applied Electromagnetics group of the same university
as a Ph.D. student (1988-1991) and as an Assistant Professar (1992-present).
He presently teaches the "Antenne" course. Tommaso Isernia was the winner of the "G. Barzilai" Award of the Italian Electromagnetics Society in
138
Isernia et al.
1994, and is a member of the Electromagnetics Academy since 1996. His scientific interests include phase retrieval problems, antenna diagnostics and
synthesis, and microwave tomography.