A Generalized Hybrid Approach for the Synthesis of ... - IEEE Xplore

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 1, JANUARY 2012

A Generalized Hybrid Approach for the Synthesis of Uniform Amplitude Pencil Beam Ring-Arrays Ovidio Mario Bucci, Fellow, IEEE, and Daniele Pinchera, Member, IEEE

Abstract—A generalized hybrid algorithm for the synthesis of uniform amplitude ring-arrays is introduced. The method exploits the analytical properties of the field radiated by a circular array in order to get advantages from the use of convex programming techniques. The method, aimed to the synthesis of circularly symmetric patterns, is very flexible and allows the use of different kind of feeds as well as stepped excitations. The synthesis procedure is demonstrated in the case of high-directivity pencil beam patterns. Index Terms—Antenna arrays, genetic algorithms, optimization, quadratic programming.

I. INTRODUCTION

U

SE of active array antennas for satellite communications is attracting increasing interest as an alternative to reflector antennas [1]–[3]. In order to have a competitive solution, as compared to reflector antennas in terms of reliability, cost, and power efficiency, it is important to reduce as much as possible the number of required amplifiers and phase shifters (i.e., the control points). Furthermore, in order to achieve the maximum efficiency in the DC-RF conversion chain, all the amplifiers should work under the same optimal conditions, thus providing equi amplitude (isophoric) excitations. This last requirement is in contrast with the necessity to have a tapering of the array excitations, in order to satisfy the side lobe level (SLL) constraints on the synthesized beam, so that it is not possible to use uniformly spaced planar arrays. It is possible to identify at least three different architectures of non uniformly spaced arrays: thinned arrays [4]–[6], where the positions of the radiating elements are selected from a regular lattice, sparse arrays [7]–[18], where the non-regular element positions can be freely chosen within an assigned area, and clustered arrays [19], where the elements of an otherwise standard arrays are clustered into single entry sub-arrays. This work will be focused on the sparse array architecture but, as it will be shown in the following, the flexibility of the proposed method allows to obtain mixed architectures with clustered elements and element-size tapering [20].

Manuscript received January 27, 2011; revised May 18, 2011; accepted July 20, 2011. Date of publication October 03, 2011; date of current version January 05, 2012. This work was supported by the European Space Agency under Contract ESA/ESTEC 21689/08/NL/ST. O. M. Bucci is with the Dipartimento di Ingegneria Biomedica, Elettronica e delle Telecomunicazioni, Università Federico II di Napoli, I-80125 Napoli, Italy (e-mail: [email protected]). D. Pinchera is with the DAEIMI, via G. Di Biasio 43, Università di Cassino, Cassino, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2167931

Generally speaking, in the synthesis of a sparse planar isophoric array of identical elements, once the kind of radiating element has been chosen, the problem is to determine their positions in the available area, in such a way to satisfy, possibly in an optimal fashion, the design specifications. From the mathematical point of view, this amounts to perform a (possibly constrained) minimization of a properly chosen functional. This minimization can be done adopting different approaches. The simplest one consists in the direct synthesis of all the element positions such that the radiated field fulfills the given constraints. Due to the presence of the elements’ positions in a complex exponential, the relationship between the elements’ positions and the array factor is strongly non-linear; for this reason, the solution of this kind of problems (minimization of a strongly non-convex functional on a non-convex search space)1 by means of local optimization techniques becomes unaffordable, since the algorithm can get easily stuck in a local minimum of the functional. On the other hand, the use of global optimization algorithms [21], [22] could be problematic, since the number of unknowns is proportional to the total number of radiating elements used, and the computational effort of global optimization algorithms increases very rapidly with the number of unknowns [23], if the attainment of the optimum must be ensured within a prescribed precision. And so, in the case of arrays with a large number of elements a “brute force” procedure could be ineffective. A second possibility is the use of a deterministic density taper technique: in a first step a continuous source fulfilling in an optimal way the given constraint is synthesized; then the locations of the radiating elements is determined exploiting a proper density taper rule [4], [9], [10], [13], [17], [18], allowing to emulate “at the best” the continuous aperture distribution. Since the first step can be dealt with convex programming techniques and the second step is deterministic, the approach is very quick, but we do not have guarantees on the optimality of the obtained solution. Moreover, in the available density taper approaches, we cannot take explicitly into account the feed pattern. In this paper we present an effective strategy able to overcome above limitations, in the relevant case of circularly symmetric pencil beams. In this case it is quite natural to put the radiating elements equispaced along circles: in this way we have just to search for the ring radii, the number of feeds per ring and the relative orientation of the feed along the rings, achieving a strong reduction of the parameters describing the array. Furthermore, as it will be shown in the next Sections, a further computational improvement is obtained by reformulating the 1The non convexity of the search space is due to the presence of the nonoverlapping constraints on the elements.

0018-926X/$26.00 © 2011 IEEE

BUCCI AND PINCHERA: A GENERALIZED HYBRID APPROACH FOR THE SYNTHESIS OF UNIFORM AMPLITUDE PENCIL BEAM RING-ARRAYS

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problem in such a way to allow the use of a hybrid approach for the solution of a simplified ring-array synthesis problem, in order to exploit as much as possible its partial convexity. It has to be underlined that, in principle, we are not guaranteed that the optimal array layout should be a ring array, which, accordingly, could provide a sub-optimal solution. However, this geometry is certainly appealing, due its simplicity and to the strong reduction of the number of unknowns, which greatly improves the performance of the optimization algorithm. Once such a sub-optimal layout has been obtained, it could be refined by a further local optimization. Nevertheless, it will be shown that such a refinement is usually not worthy. In the following, first the mathematical framework for the problem will be illustrated, then the synthesis method will be presented and a number of numerical results will be given and discussed. Conclusions follow. II. STATEMENT OF THE PROBLEM As the beam can be conveniently steered to any desired direction by means of a linear phase shift, while scanning losses can be taken into account by a proper strengthening of the requirements on the far field, we can limit ourselves to consider the case of a (circular) spot beam pointing at boresight. Let us first consider a circular array of radius with identical equispaced isophoric elements, with common excitation (see Fig. 1). The corresponding array factor, , reads ( time convention)

(1) being

the free space propagation constant and the angular position of the element. By exploiting the well known relationship [24]

(2)

Fig. 1. Geometry of the circular array.

for values of well inside , which in the case of our interest, that is, large focusing antennas, is much larger than the beam-width, which is of the order of , being the radius of the whole antenna. Accordingly, in the angular sector around the bore-sight direction wherein the field is significant, higher order field harmonics are negligible and thus the array factor is practically independent. Accordingly, if we are interested in the design of a high gain circular antenna array (as in the case for satellite communications), we can simply use the term of (3) to obtain a good approximation for the array pattern in the region wherein the constraints are relevant, using the complete series (or expression (1)) for a successive validation of the results. Under this approximation, it is straightforward to obtain the expression of the pattern radiated by a set of circular arrays with radii , each one with a number of elements equal to , as

(1) can be recast as (5)

(3) As well known, Bessel functions of first kind go rapidly to zero when the argument is smaller than the order. Accordingly, even the first non-zero harmonic in (3) becomes negligible for

(4) wherein is the wavelength and the distance between the feeds. And so, all higher order field harmonics are negligible

where is the (copolar) pattern of the feeds on the k-th ring (considered uniform in ),2 and , so that a value of can be used to take into account the absence or presence of a feed in the center of the array. It has to be underlined that under approximation (5), the array factor is independent of , thus achieving a further reduction of the search space. From this point on we will thus search only for the set of the ring radii and the number of feeds per ring. 2Unless we deal with strongly asymmetric feeds, this assumption is likely to be satisfied in the angular region of interest. Anyway, if required, it can be removed by exploiting the azimuthal Fourier expansion of the feed pattern to generalize relation (5).

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This is still a strongly non linear, non convex, problem. However, relation (6) shows that if the radii and the element factors are fixed, the relationship between the radiated field and the excitations is linear. Accordingly, given the feeds and the rings geometry, the corresponding optimal excitations can be found by solving a simple, convex, quadratic programming problem—see [25] for more details. This circumstance suggests to extend to our case the Hybrid Approach (HA), proposed in [26], in order to reduce the number of unknowns to be dealt with a global optimization algorithm, thus improving its effectiveness. III. THE HYBRID APPROACH AND ITS IMPLEMENTATION Fig. 2. Pattern mask; the mask is uniform with the

variable.

The synthesis problem can be thus reduced to the search of the appropriate and (and, possibly, ) with in such a way to satisfy given constraints (pattern shape, directivity, physical dimensions of the array, non overlapping of the feeds and so on) in an optimal way. We have to underline that, while neglecting the dependence is quite justified in the region wherein the pattern constraints are relevant, so that the synthesis of the array layout can be safely performed under this approximation, higher order harmonics becomes significant far from the boresight, decreasing the directivity of the actual array with respect to that given by (5). Accordingly, this last expression, which physically corresponds to substitute the actual array with one consisting of continuous ring sources, provides an upper bound for the array performances. Obviously, such continuous rings source provides a more realistic and tight upper bound as compared to a continuous aperture antenna, often used as reference source, since it is closer to the actual array and can include some of the physical constraints of the design. Once each ring array is substituted with a continuous source, there is no reason to restrict its “excitation”, i.e., (see (5)) to be an integer. And so, substituting in place of in (5), we obtain

(6) In the case of satellite communications, the far field specifications are usually given in term of a minimal directivity to be achieved at the end of coverage (EOC) of the beam and of an upper bound mask for the relative (to the EOC value) power pattern outside the EOC (see Fig. 2). Accordingly, the continuous ring version of our problem can be stated as follows: “Find the rings radii, the corresponding excitations (and, possibly, feed types), such that the radiated pattern is equal to one (i.e., 0 dB) at the EOC, fulfills the mask constraints and maximizes the EOC directivity, i.e., minimizes the radiated power.”

In its essence, the idea lying at the basis of the HA is very simple: exploit the partial convexity of the overall synthesis problem with respect to some variables (in our case the rings excitations) to eliminate these variables from the overall cost function, by substituting in it their optimal values (for given values of the other variables). In such a way the global optimization algorithm must deal only with the “non convex” variables (rings radii and feed type, in our case), with a beneficial effect on its convergence, hence on the reliability of the obtained results. Of course, this requires the solution of a convex programming problem for each step of the global optimization: however, due to the availability of very efficient convex programming algorithms, the overall balance is favorable to the HA. The synthesis can be thus decomposed into two parts, a global optimization problem and a convex programming one; in particular the convex part will be called by the optimization algorithm within its cost function. Not withstanding its simplicity, the actual implementation of the HA is by no means trivial. Since we have divided the synthesis procedure in two parts we have to understand how the constraints on the search space influence the behavior of each one of them. A first constraints is that the feeds belonging to the same ring should not overlap: we can code this constraint only in the convex part of the hybrid algorithm. A second constraint is that the feeds of different rings should not overlap: this constraint could be coded in the evolutionary part of the hybrid algorithm. The same is true for the possible choice among various kind of feeds. These points influence both the choice and implementation of the global optimization algorithm and the implementation of the convex sub-problem, as detailed in the following. A. Choice and Implementation of the Global Algorithm While a “generally optimal” global search algorithm does not exists (“No-Free Lunch Theorems” [27]–[29]), if we have enough information on the problem we are going to solve we can exploit this information to choose and set up an algorithm tailored to the class of problems to be solved, in order to achieve a better convergence. This led us to choose a genetic algorithm [30] (GA) as a global optimizer. This choice has been driven by the fact that GAs are very flexible and tunable; furthermore, each set of radii needs to verify radial “non overlapping feeds” constraint, that is very easy to implement in a GA if the gene contains the coding of the radii as real numbers.


Taking into account the statement of the problem (see Section II), the natural cost function is obviously the total radiated power, or, equivalently, the inverse of the EOC directivity, . Actually, since we are not guaranteed that the considered set of rings are able to radiate a pattern that fits the mask, we should modify the definition of the cost function in order to take this issue into account. We would like the algorithm to search first for a field belonging to the mask: only when this specification has been met, the algorithm can search for the maximum directivity pattern. The mask-fitting constraint can be taken into account by introducing a second cost function, f. i., the norm, , of the field that outfits the mask. This would lead to a multi-objective problem, but since the “power cost” has to be taken into account only when the “mask-fitting cost” is zero (in practice, below a threshold), we can adopt the following stepped cost function: (7) is a threshold that has to be chosen according to the where numerical precision of the used computer. The cost is positive and equal to when the mask specification is not verified; it becomes negative, and equal to the opposite of the EOC directivity when the mask specification is verified. We are thus guaranteed that any layout with a pattern not fulfilling the mask would have a cost greater than one with pattern belonging to the mask, and so we are sure that the algorithm will try first to find a pattern fitting the mask, and then it will optimize for the directivity. It has to be underlined that the tournament selection scheme is robust with respect to the use of a cost function that can assume positive and negative numbers; this approach, in the large performed investigation campaign, has been proven to be very robust and effective. It should be also noted that the nature of the genetic algorithm is intrinsically parallel, thus allowing a great speed-up when properly programmed to exploit the presence of multicore/multi-threaded processors. B. The Implementation of the Convex Part of the Problem

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if a central element is present, and

(10) if there is no central element.3 Equation (9) assumes that we are putting a single element in the center of the array, while (10) assumes that we are putting the maximum possible number of feeds in the first ring. It is important to underline that both are convex constraints, so the overall problem remains convex. Once the values of satisfying the non overlapping constraint are obtained, the values of can be found as

(11) In some cases it could be useful to fix the overall number of feed of the final array. This specification could be also coded directly in the convex problem, by adding another linear constraint

(12) that obviously does not change the convex nature of the problem. It is important now to recall one of the main results of Section II. The angle for which the effect of the higher order harmonics in (3) needs to be taken into account is function of the distance between the feeds belonging to the same ring. This means that we could control the influence of higher harmonics by imposing a maximum inter-feed distance on each ring, in order to verify the condition (4) for a larger angular sector. This would lead to the following further convex constraints: (13) if a central element is present, and

As seen in Section II, the ring excitations , to be determined in the convex sub-problem, are, apart from a discretization, proportional the numbers of feed to put on the rings. Now, for each ring, we must satisfy the azimuthal non-overlapping constraint (i.e. non overlapping of the feeds belonging to the same ring). Assuming circular feeds, this implies that we cannot have for a ring of radius a number of circular feeds greater than (8) where is the feed radius of the elements on the -th ring. This constraint on the number of feeds can be translated into a constraint on the dynamic of the excitations, i.e. (9)

(14) imposes a if there isn’t any central element. The factor maximum element distance on each ring. From the geometrical viewpoint, (9), (13) or (10), (14) restrict the excitations, seen as function of the radii, to belong to an angular sector. A final observation is now in order. The optimal solution and the corresponding number of feeds per circle will be, in general, slightly different, since there is a quantization error, due to the discrete nature of the feeds number; for the synthesis it is thus preferable to calculate the optimal solution and then to compute the cost 3Equations (9) and (10) can be easily generalized to the case of feeds of non circular shape.

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TABLE I SUMMARY OF THE SPECIFICATIONS OF [2];

IS THE FREE SPACE WAVELENGTH FOR THE CENTRAL FREQUENCY OF THE EMPLOYED X-BAND

Fig. 4. a) Layout of the 13-ring UCAs array; b) layout of the same layout after a local optimization of the positions.

Fig. 3. The 12 radii found by 4 instances of the GA after a variable number of iterations. (a) GA after 100 iterations; (b) GA after 3000 iterations.

function using the corresponding ; in this way the global search algorithm would also optimize for a set of radii that minimizes the discretization error. IV. NUMERICAL EXAMPLES In order to assess the effectiveness and the flexibility of the proposed approach, in the following we present a number of numerical examples. To be definite, we will focus on a specific case of interest for the realization of an array antenna for satellite applications, able to radiate multiple pencil beams (see [2] for more details). The main requirements for the beam pattern are reported in Table I. All simulations have been performed assuming as feeds uniform circular apertures (UCA), possibly with variable radius. As a preliminary step, in order to verify the possibility to actually achieve, with good precision, the global optimum, an analysis of the performances of the implemented genetic algorithm has

been carried out. The considered problem is that of finding the optimal radii in the case of 12 continuous ring sources, using as feeds UCA of radius . Fig. 3(a) reports the values of the radii found by four different instances of the GA after 100 iterations, while Fig. 3(b) shows the radii found after 3000 iterations. While the solutions found after 100 iterations are slightly different, after 3000 iterations they are almost perfectly coincident, confirming that the absolute optimum has been reached. The continuous ring case directivity in the four cases of 100 iterations is in the interval 46.11 dBi–46.17 dBi, and the directivity after 3000 iterations is 46.19 dBi. This behavior has been confirmed in other cases, indicating that we have either a high number of close local minima or a relatively “flat” region surrounding the absolute minimum. And so if we are interested in finding the global optimum with high precision we have to use an high number of iterations (or a certain number of separate instances of the optimizer), whereas if we are interested in assessing the performance level of a certain geometry we could adopt relatively few iterations. It must also be noticed that the ring radii turn out to be linearly related to the radius index; this is probably due to the fact that we have not enforced any constraint on the overall number of feeds or on the dynamic of the auxiliary solution , so the ring array behaves similarly to a linear array. According to these results, unless differently stated, the number of iterations for the GA will be fixed to 100. A. Isophoric Arrays of Identical Elements As a first example, an isophoric array of 13 rings of identical UCAs of radius equal to has been synthesized. The obtained


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TABLE II PERFORMANCE SUMMARY

Fig. 5. a) Pattern of the 13-ring UCAs array; b) pattern of the same layout after a local optimization of the positions.

layout, consisting of 379 feeds, is shown in Fig. 4(a), whereas the -cuts of the corresponding directivity pattern are shown in Fig. 5(a). The EOC directivity is equal to 44.10 dBi and the maximum side lobe level on Earth is 20.02 dB. It is interesting to observe that, as expected, the -cuts are overlapped up to . It has to be underlined that the values of have been set in a random way, so it could be possible to obtain slightly better results by an optimal choice of these values. However, as said before, the amount of improvement is very marginal, so it has not been considered in this work. In order to judge the quality of the obtained solution, we recall that the optimal continuous aperture sources with a radius, which has an aperture efficiency of 0.8 dB and achieves an EOC directivity of 46.8 dBi [25], while that of the continuous ring array is 46.24 dBi, showing that the azimuthal discretization has a much higher impact that the radial one. This should have been expected, as the directivity decrease is just the reflex of the decrease of the aperture efficiency due to its discretization and sparsification. It is easy to realize that, if we neglect coupling effects, in the case of isophoric feeds with uniform aperture distribution, the aperture efficiency coincides with the filling factor, i.e., the ratio between the total surface occupied by the feeds and that of the continuous aperture, which, in our case, amounts to 3.7 dB, with a decrease of 2.9 dB with respect to the optimal continuous distribution, in good agreement with the actual EOC directivity decrease. To quantify the degree of optimality of our solution we can compare it with the result of a further local optimization of the feed position, allowing the feeds to move outside the circles [31]. The obtained layout is depicted in Fig. 4(b), and the corresponding pattern is shown in Fig. 5(b); the obtained EOC direc-

Fig. 6. Comparison of the performances for a variable

.

tivity is 44.5 dBi, so with an increase of only 0.4 dB.4 For sake of completeness a summary of the results is given in Table II, where we have added the case of a full populated layout of 698 non isophoric UCAs located on 15 rings (whose excitation have been chose to deliver the optimal directivity) and that of a layout of 382 elements obtained by means of the deterministic density taper approach (DA) presented in [18]. Note that the EOC directivity of the HA isophoric layout is only 1.55 dB lower than the full populated non-isophoric layout, which has almost a double number of control points. On the other side, the improvement over the deterministic approach is marginal, showing that in the case of isophoric array of identical elements the DA and HA are essentially equivalent. Accordingly, the main advantage of the HA is its flexibility, which allows to deal with a much larger class of architectures, as it will be shown in the following. In the example above, no constraints on the number of elements have been given, apart from the non-overlapping one, so that the total number of feeds is not fixed in advance, but derives from the discretization of the continuous excitations via (10) and (11). Because, as pointed out before, the EOC directivity is essentially related to the filling factor, which depends on the number of feeds, it is interesting to see as the EOC directivity varies with . This can be done by enforcing the constraint (12) in the convex part of the synthesis. Fig. 6 shows the behavior of both the continuous and discretized ring EOC directivities as a function of . 4This behavior has been confirmed in other cases analyzed: if the GA converged to a “good” solution, the successive local optimization typically improves the directivity of less than 0.5 dB.

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Since completely removing the constraints on the excitation amplitude would lead either to a non optimal use of the feeding amplifiers or to the necessity of adopting complex and inefficient beamforming networks, it could be interesting to see what happens if we group the elements into subarrays by means of 3-dB power dividers. When we use a power divider the intensity of the excitation on the two feeds is reduced of a factor , but the number of antennas doubles, thus resulting in an overall increase of the ring excitation of a factor. According to this consideration, if we have (15) when a central element is present and

Fig. 7. Performance comparison for the coefficient .

UCAs ring array by varying the

As can be seen, the continuous ring directivity decreases as the number of elements is increased; this is due to the fact that enforcing the constraint (12) does not allow the use of the optimal set of excitations. On the contrary the performances of the discretized case increase with the number of feeds (because of the reduction of the effect of the higher order harmonics) up to a value of , practically equal to that of the considered example, and then stays constant. This is a very interesting result, since it shows that, for a sparse isophoric array of identical elements, the EOC directivity cannot be increased beyond a certain level by simply increasing the number of feeds, because at the end this would induce a performance decrease, due to the impossibility to realize the proper tapering of the aperture distribution. The increase of the feeds number can be also obtained by enforcing constraints (13), (14). In Fig. 7 it is possible to see the effect of varying the coefficient . The figure is organized in 4 subplots: the first reports the directivity for the continuous ring case; the second that of the discretized ring case; in the third plot we report the side lobe level up to 16 , whereas in the fourth plot the overall number of elements is reported. It is interesting to see that introducing a lower limit for the excitation dynamic decreases significantly the performances in the continuous ring cases only for ; on the contrary the directivity in the discretized array increases with the decrease of , and values of in the range 1.5–3 are particularly interesting. B. Relaxing the Isophoricity Constraint The major drawback of the above ways for increasing the filling factors is the increase of the number of elements, which is in contrast with the request of a minimal number of control points. Furthermore, the analysis of the results reported in the previous sub-section indicates that the increase of the number of elements per ring forces the equivalent aperture distribution of the array to depart more and more from the optimum one, as testified by Figs. 6 and 7.

(16) when there isn’t a central element, we can put on the -th circle a number of feeds grouped in couples controlled by a 3-dB power divider equal to

(17) This procedure, in the following referred to as “stepped excitation”, can be used also in case of nested 3-dB dividers; in the case of a single 3-dB divider we will speak of 1-level stepped excitation, in the case of a double 3-dB divider we will speak of 2-level stepped excitation and so on. The use of power divider allows an higher number of feeds on each ring for a given number of control points, without the need of restricting the excitation dynamic by means of the coefficient . The drawbacks of this solution are the introduction of losses by the power dividers, and the reduction of the steering performance due to the grouping of the feeds. Fig. 8 shows a layout of 12 rings of UCAs in which we have introduced the possibility to have stepped excitations; a has been used in order to have the maximum number of feeds per ring when using the stepped excitations. The obtained EOC directivity is 44.84 dBi, but we are now using 508 feeds for an overall number of 342 control points, that is still a very large number. The directivity -cuts are plotted in Fig. 9. It is interesting to observe that, because of the increased number of feeds per ring, the -cuts coincide up to about . Of course, it would be possible to use a multistepped excitation, but while the directivity improvement turns out to be marginal, possibly canceled be the increased losses, scanning performances would be significantly worsened, due to the larger grouping of the elements. C. Using Feeds of Different Sizes According to (5) and (6), our model allow us to use a different kind of feed per ring. And so, we could try to exploit a kind of “size tapering” in order to improve the radiating characteristics


Fig. 8. Layout of the geometry employing stepped excitations; white feeds: no power divider employed; gray feeds: 3-dB power divider employed.

Fig. 9. Directivity plot -cuts for the layout employing stepped excitations.

of our array [20]. As a matter of fact, the previously reported results show that, in order to achieve the proper tapering of the equivalent aperture distribution, looser feeds or stepped excitation must be employed in the more external rings. This naturally suggest to use bigger feeds in these parts of the array. To this end, a first possibility could be to leave the user to identify which rings are to be filled with a kind of feed and which have to be filled with the other one. A more interesting possibility, which we adopt here, is that of leaving to the GA to choose both the ring radii and the type of feed for each radius. This would add a discrete variable vector to the search space, so the convergence speed would be only slightly influenced, with only minor modifications of the optimization code. Fig. 10 shows the result of the optimization of a 11 ring layout employing UCAs of radii equal to and . The EOC directivity obtained with this layout is 44.73 dBi, with a side lobe level on Earth of 20.72 dB and an overall number of feeds equal to 314. The directivity -cuts are reported in Fig. 11. As foreseen, the use of feeds of different sizes allows a significant increase of the EOC directivity, while the number of feeds is reduced.

Fig. 10. Layout of the geometry employing UCAs of

and

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radii.

Fig. 11. Directivity plot -cuts for the layout employing UCAs of radii.

and

Since both the use of a stepped excitation and of feeds of different sizes improve the array performances, with a minor impact on the complexity of the antenna, we could try to synthesize a layout employing both approaches, in order to achieve the minimum number of control points. Fig. 12 shows 10-ring layout employing two kind of UCAs, of radii and , and using stepped excitations when needed. The corresponding directivity -cuts are depicted in Fig. 13. The performances of this architecture are very good, achieving a side lobe level of 20.59 dB and an EOC directivity of 44.85 dBi, i.e. only 0.8 dB less than a fully populated array of about 700 elements (see Table II), using only 280 feeds and 230 control points. V. CONCLUSIONS A generalized hybrid approach for the synthesis of large sparse arrays has been presented. The method exploits the analytical properties of the field radiated by a circular array in order to split the synthesis in a global optimization procedure, on a strongly reduced search space, and a convex programming problem.

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possibly extended to the case of any shaped circularly symmetric pattern, with a proper modification of the convex part of the algorithm. REFERENCES

Fig. 12. Layout of the geometry employing stepped excitations and UCAs of and radii; white feeds: no power divider employed; gray feeds: 3-dB power divider employed.

Fig. 13. Directivity plot -cuts for the layout employing stepped excitations and radii. and UCAs of

The proposed synthesis method is able to achieve practically optimal solutions, and its flexibility allows to synthesize layouts with different kind of feeds per each ring, since the radiating element pattern is directly taken into account in the synthesis algorithm, and/or the use of stepped excitation. This allows a strong reduction of the number of control points, thus making the use of direct radiating arrays interesting for a larger class of problems. Furthermore, with a proper and relatively simple modification of the mathematical framework, the technique can be also extended to the case of feeds with patterns depending on the variable, f.i., square apertures or sub-arrays. The performances achievable, even with a small number of global optimization steps, are very good, so that the obtained solutions do not need to be further ameliorated by further local optimization which would move the feeds out of the rings. It can be also foreseen that the synthesis procedure, here demonstrated for the case of pencil beam patterns, could be

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Electronic Engineering, 1984–86 and 1989–90, Vice Rector of the University of Naples, 1994–2000, Director of the CNR Institute of Electromagnetic Environmental Sensing (IREA) 2001–2010. He is the author or coauthor of more than 370 scientific papers, mainly published on international scientific journals or proceedings of international conferences. His scientific interests include scattering from loaded surfaces, reflector and array antennas, efficient representations of electromagnetic fields, near-field far-field measurement techniques, inverse problems and noninvasive diagnostics, biological applications of nanoparticles and electromagnetic fields. Prof. Bucci is a Fellow of the IEEE since 1993 and a Member of the Academia Pontaniana. He was President of the National Research Group of Electromagnetism, of the MTT-AP Chapter of the Centre-South Italy Section of IEEE, and Director of the Interuniversity Research Centre on Microwaves and Antennas (CIRMA). He was a recipient of the International Award GUIDO DORSO for Scientific Research, 1996, and of the Presidential Gold Medal for Science and Culture, 1998, among others.

Daniele Pinchera (S’05–M’08) received the Dr. Eng. degree (summa cum laude) in telecommunication engineering and the Ph.D. in information and electronic engineering from the University of Cassino, in 2004 and 2008, respectively. He is currently working as a Postdoctoral Researcher for the Faculty of Engineering, University of Cassino. His current research interests are in the fields of smart antennas and MIMO systems, large array synthesis, compressed sensing, sensor networks and industrial and medical applications of microwaves.