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:

(a) the far-field pattern (or patterns, in the case of scanning or reconfig-urable beam antennas);

(b) the antenna structure an d geometry; (c) the feeding system.

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.

104 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 ha ve no t been proposed un t 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, i t 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 solv-

l In the case of fixed antenna structure. For the generai case, see Sect. 6.

106 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

M n

- DM(O + On) L E(On, f/Jn,m)DMn (qy + 7r- f/Jn,m)} (l) m=-Mn

where

Shaped beam antenna synthesis problems 107

27m 21rm ()n = 2M + l' <Pn,m = 2M + l , M ~ {3a, M n ~ {3a sin ()n

an d sin (2M+1x)

D (x)- 2

M - (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 P(()n, cPn,m)D2Mn (</J- cPn,m) n=1 m=-2Mn

2M n

- D2M(() +()n) L P(On, cPn,m)D2Mn (c/J + 1f- 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 N-1

P(u) =co+ L [cpcos (pu) + spsin (pu)] = L PpD2N-1(u- up) (4) p=l p=l-N

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, i t is well known that the invisible part of the spectrum must be

2 Otherwise, discrete prolate spheroidal functions [18] should be used.

108 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

+ L L [epq cos (pu + qv) + Bpq sin (pu + qv)] p=l q=-M+l

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, V q = 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 + (v/{3dy) 2 > l belong to the invisible

range even when dx, dy ~ ~.

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

L Dpwp(e, 4>) 2 LB(e, 4>) p= l

(7)

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 ha t is the sharpness of the criterio n, 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.

Shaped beam antenna synthesis problems 111

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 trigono­

metric 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 c an 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 i t 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 o ne 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 2

<I>(g, D)= IIQ(g)- L Dp'lip(B, 4>)11 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 o ne is t ha t 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, i t 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 l E 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 1Eal2 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 lE L 12 and P A , is chosen. In this way, (see the dashed lines in Fig. 2) IELI2 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~

introduce auxiliary Fourier Series r~tatiom for thc ficld and thc

squarcd amplitude

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))

(nominai field)

SOURCE GEOMETRY SOURCE EXCITATIONS

NO

Theproblem is unfeasible

117

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 i t 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 119

IO _l l l l l l l_

o r- - ' ' -' ' l

' l

' l

: l l

iD(dB)

' l l l l \ l r- l l -

l l -IO

l l l l l l l l l l l l l l l l l l - l -

l -20

-30 - -' Il l ,. . " Il

11 ,', l " \ :: f, : ~ : l \ l ~ : v \:\q "i \Al !-l l-' l : l l l l l,, \::l: l 'J l \:: :: :::: \: ' l:: f ::d l l i l ,,

Il 1 1 ~ l '

r~: l l \t 1\i . ! r l

-40

-50 -3 -2 -l o 2 3

u

Figure 4. Synthesis of a flat top beam: prescribed mask (solid line) and power pattern resulting from the existence criterio n ( 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 o n 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

10 _l l l

o -l l l l iD(dB) l l l l

i-l l -IO l l l l l l l l l

- l l l

-20

-30 i-

~ \ ; \ :1 f\ ! :--\ : \ : ~ : \:

l l , 1 ,, ,,

\: ~~ :: \ • Il ,, ,,

~ -l

-40

-50 -3 -2 -1

l

'

l o u

l

' '

l\

Isernia et al.

l l_

-

-

-

-

\ f\ l\ :\ :' \1\:\:\ ;~ \: ~ : \ : \ l .. 1 l l t 1 ,

l l ,, ~J ,, ,, ,, ,, ft l 2 3

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 L2 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 121

n la.! La. (rad)

l 4.745E-02 0.7859 2 0.1183 0.7384 3 0.1776 0.6071 4 0.1878 0.2636 s 0.1976 -0.3645 6 0.2444 -0.7874 7 0.2259 -0.8201 8 0.1269 -0.2842 9 0.1381 1.0793 . 10 0.1885 1.5967 11 0.1410 1.9898 12 7.057E-02 2.9895 13 8.452E-02 -2.1826 14 7.612E-02 -1.8360 15 3.746E-02 - 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

o

-10

-20

-30 f------------,----1

-2 -1 o u

Isernia et al.

2 3

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

2 3 4 s 6 7 8 9 IO Il

12 13

14

lS 16

lanl 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

123

Lan (rad)

-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

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) 112

subject to (8) (17) F~12 (0, 1)

wherein Il · Il denotes the (properly discretized) L2 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

IO _l l

o -

P(dB)

-IO 1-

-20 -

-30 1-

.... , ,, ' l .... -,,/ \ ' ., ' : l l ' - ' l

: t'l \: \~ j

-40

-50 li l -3 -2

l l

' ' '

/

-l l o u

' '

l

\

Isernia et al.

l l_

-

-

-

-' /',~ ...... , .... '\ ' '

l,,

l l

' l

' ' -: ~~ l v~· o l

l l l

3

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.

Shaped beam antenna synthesis problems

10 _l l

o f-

P(dB)

-lO -

-20 1-

-30 1-

l\ : \ : ~ 1\ : ~\:\:l 1\/ 1 1 1 1 1 1 r ''

~J ~ / \ / y • Il

Il Il

-40

-50 -l

Il Il

u -3 -2

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

-1

'

l

l o u

125

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 l l l

-

1 l1 t1 Il Il 1 l1 Il Il l l'l l l l l l l

\f l : \ l \ :_J, ~ l l l l 1 l ,, ,, \-'

Il ' Il

" " " ~ 1-2 3

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 4.380E-02 0.8468 2 0.1109 0.8964 3 0.1529 0.9454 4 9.843E-02 0.9946 s 6.281E-02 -2.0980 6 0.2496 -2.0489 7 0.3382 - 1.9998 8 0.2782 - 1.9507 9 0.1313 - 1.9018 lO .I.743E-02 - 1.8534 11 O.OOOE+OO O.OOOE+OO 12 5.092E-02 - 1.7545 13 9.119E-02 - 1.7052 14 8.014E-02 - 1.6561 15 4.014E-02 - 1.6072

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----------~ ""'11' UTT

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

lO LI l l

o r- f-'

1'(dll) 1/ -10 r-

. - zo 1-

- )()

o

..$() l - 3 -2 -l

(a)

~ l

l

o v

(b)

l

~ l _l

Isernia et al.

l l.

-

-

-

1\ 1'\ l\

v

2 3

Figure 11. Synthesis of a t riangular contoured beam: "powcr pattern" delivercd from the existence criterion with 15 x 15 elemcnts: (a) 3-D re~ resentation, (b) wors t eu t.

Shaped beam antenna synthesis pro blems

IO J l

o ~

~(dB)

-IO l-

-20 l-

" f\ ~

- 30

-.o

-$O ~l l 1 - 3 -2

l

(

-l

(a)

l

l o u

(b)

129

l l

-

i\ -

-

. ~

l

2 3

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, the developed criteria furnish plausible power pat­terns. lt follows that the subsequent synthesis can be done by looking for a nominai pattern. When the 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

10 LI l l

o r-P (dB) l

l

- l

- 20 r-

~ \ ~ - 30

- 40

r1 l

(a)

l

l o "

(b)

l l

\

Isernia e t al.

l_

-

-

l ~: l

2 3

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 P x (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 [FFT(y)]} (A.3)

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 P x (y) such t ha t

8(A) =Il FFT-1 ~~(A)]- y 112 (B.l)

is minimum. From (B .l), o ne gets

88 = (FFT- 1 [ZP(8A)] FFT-1 [ZP(A)]- y)

2 1/2 ' 1/2 Fw Fw (B.2)

As the extraction operator Ex, considered in Appendix A, is the adjoint of ZP, we have

88 = (8A,2ExFFT { FFT-1 [::(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.

REFERENCES

l. Bucci, O. M., G. D'Elia, G. Mazzarella, and G. Panariello, "Antenna pattern synthesis: a new generai approach," Proc. of the IEEE, Vol. 82, 358-371, 1994.

2. Hall, P. F., and S. J. Vetterlein, "Review of radio frequency beam techniques for scanned and multiple beam antennas," IEE Proc., Pt. H, Vol. 137, 293-303, 1990.

3. Duan, D. W., and Y. Rahmat-Samii, "A generalized diffraction synthe­sis technique for high performance reflector antennas," IEEE Trans. on AP, Vol. 43, 27-40, l 995.

4. Poulton, G. T., "Antenna power pattern synthesis using method of successive projections," Electron. Lett., Vol. 22, 1042-1043, 1986.

5. Bucci, O. M., G. Franceschetti, G. Mazzarella, and G. Panariello, "ln­tersection approach to array pattern synthesis," IEE Proc., Pt. H, Vol. 137, 349-357, 1990.

6. Bucci, O. M., G. D'Elia, and R. Romito, "Synthesis technique for scan­ning an d/ or reconfigurable beam reflector antennas with phase-only control," IEE Proc. on Microw. Antennas and Propagation, Vol. 143, 402-412, 1996.

7. Randy, Haupt L., "An Introduction to Genetic Algorithms for Elec­tromagnetics," IEEE Antennas and Propagation Magazine, Vol. 37, 1995.

8. Isernia, T., "Problemi di sintesi in potenza: criteri di esistenza e nuove strategie," Atti X Riunione di Elettromagnetismo applicato, Cesena, ltaly, 533-536, 1994.

136 Isernia et al.

9. Hay, S. G., and G. T. Poulton, "On the existence of aperture antennas with directivity pattern within desired bounds," Proc. of 1995 URSI Symposium on Electromagnetic Theory, S. Petersburg, Russia, 215-217, 1995.

10. Hay, S. G., and G. T. Poulton, "On the existence of nonsuperdirective aperture antennas with directivity patterns within desired bounds," Radio Science, Vol. 31, 1671-1679, 1996.

11. Numerica! Algorithrns Group (NAG) FORTRAN LIBRARY Manual, Section E04. Oxford, UK.

12. Isernia, T., G. Leone, and R. Pierri, "New approach to antenna testing from near field phaseless data: the cylindrical scanning," IEE Proc., Pt. H, Vol. 139, 363-368, 1992.

13. Fletcher, R., Practical methods of optimization, New York, Wiley, 1990. 14. Isernia, T., G. Leone, and R. Pierri, "Phase Retrieval of Radiated

Fields," Inverse Problems, Vol. 11, 183-203,1995. 15. Bucci, O. M., and G. Franceschetti, "On the spatial bandwidth of

scattered fields," IEEE Trans. on AP, Vol. 35, 1445-1455, 1987. 16. Bucci, O. M., and G. Franceschetti, "On the degrees of freedom of

scattered fields," IEEE Trans. on AP, Vol. 37, 918-926, 1989. 17. Bucci, O. M., C. Gennarelli, and C. Savarese, "Optimal interpolation of

radiated fields over a sphere," IEEE Trans.on AP, Vol. 39, 1633-1643, 1991.

18. Slepian, D., "Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty-V: The discrete case," The Bell System Technical Journal, Vol. 57, 1371-1430, 1978.

19. Fejér, L., and F. Riesz, " Uber einige funktionen theorethishe Ungle­ichungen," Math. Z., Vol. 11, 305-314, 1921.

20. Orchard, M., R. S. Elliot, and G. J. Stern, "Optimising the synthesis of shaped beam antenna patterns," IEE Proc., Pt. H, Vol. 132, 63-68, 1985.

21. Kaufmann, Foundation of Genetic Algorithms 2, Whitley L. D., 1992. 22. Bruck, Y., and L. Sodin, "On the ambiguity of the image reconstruction

problem," Optics Communications, Vol. 30, 305-310, 1979. 23. Tseng, F., and D. Cheng, "Optimum scannable planar arrays with an

invariant side lobe level," Proc. of the IEEE, Vol. 56, 1771-1778, 1968. 24. McClellan, J. H., and T. W. Parks, "A unified approach to the design of

optimum FIR linear phase digitai filters, IEEE Trans. Circuit Theory, Vol. 20, 697-701, 1973.

25. Isernia, T., G. Leone, and R. Pierri, "Phaseless near field techniques: uniqueness conditions and attainment of the solution," Journal of Elec­tromagnetic Waves and Applications, Vol. 8, 889-908, 1994.

26. Isernia, T., G. Leone, and R. Pierri, "Radiation pattern evaluation from near-field intensities on planes," IEEE Trans. on AP, Vol. 44, 701-710, 1996.

Shaped beam antenna synthesis problems 137

27. Bucci, O. M., G. Mazzarella, and G. Panariello, "Reconfigurable arrays by phase-only control," IEEE Trans. an AP, Vol. 39, 919-925, 1991.

28. Isernia, T., G. Leone, and R. Pierri, "Phaseless near field techniques: Formulation of the problem and Field Properties," Joumal of Electro­magnetic Waves and Applications, Vol. 9, 871-888,1994

29. Bucci, O. M., C. Gennarelli, and C. Savarese, "Non redundant rep­resentations of electromagnetic fields," Proc. JINA 94, Nice, France, 1994.

30. Bucci, O. M., and G. D'Elia, "Advanced sampling tecniques in elec­tromagnetics," Review of Radio Science, W.R. Ross Stone et al. Eds., Oxford University Press, 1996.

31. Gubin, L. G., B. T. Polyak, andE. V. Raik, "The method ofprojections for finding the common point of convex sets," USSR Comput.Math.and Math.Phys., Vol. 7, 1-24, 1967.

32. Barakat, R., and G. Newsam, "Algorithms for reconstruction of par­tially known, bandlimited Fourier pairs from noisy data," Joumal of the Optical Society of America, Pt. A, Vol. 2, 2027-2039, 1985.

33. Isernia, T., G. Leone, R. Pierri, and F. Soldovieri, "On the local minima in phase reconstruction algorithms," Radio Science, Vol. 31, 1887-1899, 1996.

34. Kim, Y. U., and R. S. Elliot, "Extension of the Tseng-Cheng pattern synthesis technique," Joumal of Electromagnetic Waves and Applica­tions, Vol. 2, 255-268, 1988.

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" A ward 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.