A New Perspective in the Synthesis of Reconfigurable Linear or Circularly Symmetric Array Antennas

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1 A New Perspective in the Synthesis of Reconfigurable Linear or Circularly Symmetric Array Antennas Andrea Francesco Morabito, Antonia Rita Laganà, and Tommaso Isernia DIMET, LEMMA Research Group University of Reggio Calabria Mediterranea Via Graziella, Loc. Feo di Vito, I Reggio Calabria, Italy {andrea.morabito, antonia.lagana, Abstract - An innovative and general point of view is presented to the optimal, mask-constrained power pattern synthesis of array antennas able to dynamically reconfigure their radiation behaviour by just modifying the phase of excitations or other suitable parameters. The perspective leads to an effective design approach able to deal in an effective fashion with easily-reconfigurable arrays, including phase-only reconfigurable isophoric linear sparse arrays, arrays lying on a regular lattice, and circularly symmetric arrays. By taking advantage from basic results achieved in the optimal synthesis of pencil beams and shaped beams, the approach guarantees the achievement of high performance without recurring to global optimization schemes. Keywords - Antenna synthesis; Reconfigurable arrays. I. INTRODUCTION The synthesis of easily-reconfigurable antennas is a classical problem in Electromagnetics [1]-[8]. As a matter of fact, several different architectures and strategies have been proposed aimed at synthesizing easily reconfigurable beam patterns by using a simplified Beam Forming Network (BFN) [1]-[8]. These solutions include the chance of sub-arraying the antennas for one of the radiation modalities (so that the BFN is not entirely duplicated) [1], the possibility of using completely-common excitations in a portion of the antenna aperture [2], the phase-only control [4],[6]-[8], and other solutions. Amongst the different possibilities, phase-only reconfigurability seems to be the more interesting one, as it allows the use of a single power-divider network wherein just the phases differ among the different excitation sets providing the various radiation patterns [1],[4],[6]-[8]. For a number of years, the scientific literature on the synthesis of such kind of radiating systems has been mostly related to the exploitation of global optimization procedures, or to the application of the so called alternating projection technique [4]. Because of the limitations of the latter (see [3]-[8] for a more in depth discussion), different and innovative procedures have been recently proposed for both cases of phase-only reconfigurable arrays lying on a regular grid (where the common amplitudes and the different phases are the unknowns of the problem) [6],[8], or onto a non-uniform grid to be determined (while excitation amplitudes are assumed to be constant and fixed in advance, giving rise to the so-called isophoric architectures) [7]. In this paper, by taking advantage of the lessons learned from [6]-[8], we develop a general and effective strategy for the optimal synthesis of reconfigurable arrays which is valid in both cases of linear and circularly symmetric architectures, as well as for generic linear sources or circular sources radiating circularly symmetric patterns. To this end, we first recall in Sect. II some general properties which hold true in the synthesis of shaped and pencil beams in both cases of (discrete and continuous) linear and circularly symmetric sources. Then, in order to achieve hopefully optimal solutions to the class of problems at hand, the proposed strategy is introduced and discussed in Sect. III. Finally, the developed approach is exemplified and tested in a number of cases of applicative interest in Sect IV. Conclusions follow. II. THE STARTING BRICKS In order to better introduce the proposed approach, let us recall some theoretical results available in the separate synthesis of the different patterns amongst which one wants to reconfigure the source. First, note that, for any arbitrary fixed-geometry array, the optimal synthesis of a pencil beam subject to arbitrary sidelobe levels (SLL) constraints can be formulated as a Convex Programming (CP) problem [9]. In fact, all constraints on the sidelobes are convex in terms of the excitations constraints, and the cost function to be optimized turns out to be convex in the case

2 wherein one wants to maximize the field in the target direction as well as in the case one wants to optimize the directivity for a given minimum SLL (i.e., for a given minimum separation amongst the main lobe and sidelobes) [9]. As a consequence of convexity, in both cases one is able to find the globally optimal solution without making resort to global optimization techniques. Notably, these approaches and results can be extended in an easy fashion to the case where a linear or circularly symmetric continuous are looked for [10]. In fact, in both cases the only residual task is to counteract possible superdirectivity problems, which can be managed by using suitable finite dimensional expansions for the source, or by proper regularization techniques. As a second fundamental result, note that, in the canonical case of linear uniformly-spaced arrays, the optimal power pattern synthesis of a shaped beam subject to both upper and lower bounds can be performed in a globally optimal fashion without recurring to global optimization procedures, and admits a number of equivalent solutions [11]. In fact, once a power pattern mask has been fixed, a suitable square-amplitude array factor distribution can be determined as the solution of a Linear Programming (LP) problem [11]. Also, once the power pattern distribution fulfilling the imposed constraints has been identified, one can determine a number of different array excitation sets all generating it. In fact, due to well-known properties of the real and positive trigonometric polynomials, the so called zeroflipping procedure allows one to achieve 2 N different equivalent solutions, being N the number of zeroes of the Schelkunoff polynomial underlying the power pattern and not lying on the unit circle [11]. Very interestingly, such procedure can be extended in an easy fashion to the case of one-dimensional continuous sources [12],[13]. In fact, the field radiated by an arbitrary, non-superdirective source can be approximated with negligible error (in the visible range) with that of an equispaced array having an inter-element spacing slightly smaller than half a wavelength. As a consequence, all results in [11] can be applied to the case of line-source distributions, provided that, in order to avoid superdirective solutions, suitable constraints are enforced in the invisible part of the source spectrum [12],[13]. Finally, all these results concerning the synthesis of shaped beams can be extended to all cases wherein the source can be reduced to one-dimensional polynomials, which include planar arrays with factorable patterns, planar arrays with peculiar symmetries [14], as well as circularly symmetric sources [15],[16]. III. THE PROPOSED APPROACH TO SYNTHESIS In any reconfigurable array, one can subdivide the degrees of freedom at his/her disposal in the synthesis in two different classes, i.e., completely free parameters, say y, whose value can vary with pattern reconfiguration; constrained parameters, say x, whose value has to be the same with varying patterns. Constrained parameters will be for instance the common array excitation amplitudes in case usual arrays on pre-defined grids are used [6],[8], or the common locations in case isophoric sparse arrays are considered [7], or the common tail in the problem considered in [2], and so on. The completely-free parameters will be the phase distributions for the first two cases, the core excitations for the problem in [2], or other possible parameters in other kind of reconfigurable structures. If we assume that some optimal procedure exists for the optimal (independent) synthesis of the different reference sources, which of course seems to be reasonable, then the problem will admit an interesting representation in terms of the x variables. In fact, as a multiplicity of continuous reference sources fulfilling the given constraints on each shaped beam exists (see Sect. II), the application of standard synthesis procedure implies that a multiplicity of solutions also will exist, for each pattern, in terms of the x variables (see Fig. 1). Fig. 1. Representation in the space of x variables of the unique solution of a pencil beam synthesis problem (green point), of three equivalent solutions for a first shaped beam (red points), and of three different solutions for a second shaped beam (blue points). Then, it order to guarantee a minimal degradation of the pattern when reconfiguring amongst the different field modalities, one has to identify, for each mode, the solution which is the closest to the solutions of the other modes. More precisely, one has to identify an hypersphere of minimal dimension such to contain at least a solution for each of the patterns to be generated (see the yellow set of Fig. 1). In fact, this seems to be a very good starting point in order to trade-off the quality of the different patterns to be generated. However, as at the end of the synthesis procedure one needs to have a single value for the x variables, while the various solutions x 1, x 2, x p are different each from the other, some further action is required.

3 In this respect, it is useful to note that each single point of Fig. 1 will become a (small) connected set when relaxing the initial constraints on the patterns (see Fig. 2). Fig. 2. Representation in the space of x variables of the effect of a slight relaxation on the radiation requirements initially imposed to identify the solutions depicted in Fig. 1. Hence, a common solution can be found by means of a local optimization procedure which looks for the intersection of the different sets (see Fig. 2) or even picking the centroid of the yellow hypersphere in Fig. 1. Obviously, some unavoidable degradation of one or more of the independently optimal patterns is implied. In particular, the amount of degradation will depend on the radius of the hypersphere: the smaller the radius of the hypersphere, the smaller the degradation. Hence, provided one is able to find the optimal hypersphere, the procedure guarantees the minimal possible worsening with respect to the independently optimal patterns, thus resulting in an optimal design. Notably, as one commutes amongst the different solutions of a single shaped beam by means of zero flipping operations (see Sect. II), the search for the optimal hypersphere can be formulated as an optimization over binary variables, which can be effectively performed by means of genetic algorithms [17]. IV. APPLICATIONS A number of different reconfigurable arrays can be effectively designed by means of the suggested strategy. A first obvious application concerns the synthesis of linear arrays in case the amplitudes (common to all modes) and phases (different for each mode) of the excitations are the degrees of freedom of the problem. In such a case, the axes of Fig. 1 will correspond to the amplitude of excitations. More details on the solution of such a canonical problem can be found in [6],[8]. Notably, the approach can be very easily extended to the case of factorable patterns, i.e., to the case wherein the excitations a nm of a planar array lying on a rectangular grid are such that a nm =a n a m, {a n } and {a m } being the excitations pertaining to two auxiliary linear arrays determining the field shape on the two principal cuts in the spectral plane. A second, less obvious application concerns the synthesis of phase-only reconfigurable isophoric linear arrays. In such a case, all the excitation amplitudes are unitary, and the degrees of freedom of the problem are the elements locations (which must be common to all radiation modalities) and the excitation phases. In order to find the optimal solution of this problem, by applying the general strategy of Sect. III we have devised the following four-steps procedure: 1. continuous reference sources corresponding to the desired patterns are identified by the synthesis techniques recalled in Sect. II; 2. the fast density taper procedures [12],[13],[18] provide, for each line-source distribution, a uniform-amplitude aperiodic array; 3. the selection procedure is performed by reasoning in terms of array elements locations. In practice, the selection follows the same guidelines as in Fig. 1, where the x variables will denote the array elements locations; 4. local optimization refines the needed trade-off amongst the quality of the different patterns. More details on the solution of such a problem, supported by interesting numerical examples, can be found in [7]. Notably, by virtue of the fact that the synthesis of circularly symmetric shaped beams also admits a multiplicity of solutions (see Sect. II and [15],[16]), the above procedures can also be applied to the synthesis of arrays having a circular symmetry (or nearly so). In fact, as a third example, we can consider the case of an uniformly-spaced ring array where one wants to commute amongst the different patterns by simply changing the phase of the excitations (while the amplitudes, which are the other degrees of freedom of synthesis problem, are instead common to all radiation modalities). By assuming, without any loss in generality, that the radiated field must be reconfigured just between a pencil and a shaped beam, a possible design procedure can be summarized as follows: 1. By means of the approach in [10], synthesize the optimal, real and positive continuous aperture source fulfilling the constraints of the pencil beam modality; 2. By exploiting the synthesis technique developed in [15],[16], identify the multiplicity of complex, circular aperture fields fulfilling the shaped beam power mask, and choose, amongst them, the closest, in terms of amplitudes, to the source coming out from step 1;

4 3. Identify a new, real and positive aperture field distribution as the amplitude of the complex source provided by step 2; 4. Determine the reconfigurable array s layout and excitations by sampling, onto a suitable ringsymmetric uniform grid, the continuous aperture distributions coming out from steps 2 and 3. By so doing, and provided the spacing amongst the elements is adequate, the reconfigurable array will be certainly able to radiate a circularly symmetric shaped beam fulfilling the initial mask. Differently, the fulfillment of the pencil beam constraints will be not guaranteed: the larger the distance between the amplitudes of the sources coming out from steps 1 and 2, the larger the pencil beam s degradation. Of course, a different strategy implying some alternative weighted worsening of the both patterns is possible. As an example of performances, let us report the results achieved in the synthesis of a phase-only reconfigurable ring array providing, in the shaped beam modality, an uniform coverage of the Earth from a geostationary satellite. In particular, Fig. 3 reports the pencil and shaped power patterns synthesized in steps 1 and 2, which fulfill the following masks: (for the pencil beam case) a maximum Peak Sidelobe Level (PSL) lower than db for θ 3.5, being θ the aperture elevation angle; (for the shaped beam case) a maximum ripple not larger than ±0.5 db for θ 9, and a maximum Peak Sidelobe Level (PSL) lower than -20 db for θ 14. Fig.3. Reference power pattern distributions: (blue line) shaped beam modality; (red line) pencil beam modality. The aperture radius is equal to 11λ, being λ the wavelength in free space. The circularly symmetric continuous aperture sources coming back from steps 1 and 2 are shown in Fig. 4, wherein a very similar behaviour can be noted in terms of amplitude distribution [see Fig. 4 ]. The sources amplitude distributions given by steps 2 and 3, and corresponding to the reconfigurable array s excitation amplitudes, are depicted in Fig. 5. The excitations obtained by using these amplitudes and the phase distributions of Fig. 4, as sampled onto a ring array wherein the average spacings amongst the element is 0.5λ, give rise to the patterns shown in Fig. 5. As it can be seen, only a slight degradation in terms of beamwidth is experienced on the pencil beam, while the PSL results even better than expected. Fig. 4. Circularly symmetric aperture fields radiating the power pattern of Fig. 3 and coming out from the step 3 of the synthesis procedure: amplitude; phase. Blue and red colors respectively correspond to the shaped and pencil radiation modalities. Fig. 5. Final solutions of the synthesis problem: amplitude of the circularly symmetric source (giving the reconfigurable array s excitation amplitudes); power patterns corresponding to the sources having the amplitudes of Fig. 6 and the phases of Fig. 5. As an interesting characteristic, it has to be noted that the overall proposed strategy is not limited to classical reconfigurable arrays where just phases can be changed when commuting from one pattern to another. As an example, let us consider the case where just the amplitudes of a part of the array are common to the different modalities, while in other parts both amplitudes and phases can be commuted. In particular, a common core architecture can been achieved by executing the above procedure with the following modifications: perform the search of step 2 by considering the distance between the different sources amplitudes only in the spatial range [0, R] λ; substitute step 3 with a convex optimization identifying a real and positive continuous source fulfilling the pencil beam constraints and enforcing the equality of the amplitudes of the final sources in the spatial range [0, R] λ. Amongst the different possibilities, such a procedure can be used to find the maximum value of R such that the pencil beam s requirements can be still fulfilled, or such that the worsening of the patterns is still acceptable.

5 In the present case, by starting from the reference sources of Fig. 5, the above criterion allowed us to establish that for R=6λ all the initial constraints can be fulfilled. The result is represented in figures 6 and 7, which respectively show the the reconfigurable sources distributions and power patterns. In particular, Fig. 7 depicts a comparison between the reference radiation patterns and the reconfigurable ones. Fig 6. Final solutions for the common core architecture: amplitude and phase of the circularly symmetric source giving the reconfigurable array s excitations. The red and blue colors are referred to the pencil and shaped beam modalities, respectively. Fig. 7. Power patterns of the sources of Fig. 7 (green lines) compared with the reference solutions (magenta lines): pencil beam modality; shaped beam modality. No degradation with respect to the reference solutions is experienced in both the modalities. V. CONCLUSIONS An innovative, general and effective perspective to the optimal synthesis of reconfigurable (array) antennas has been presented, discussed and checked. Present efforts are devoted to the optimal synthesis of reconfigurable isophoric ring arrays, as well as, by taking advantage from the lessons learned from the last example of Sect. IV, to a further possible exploitation of the proposed perspective. As a matter of fact, once the beams of interest have been established, a careful observation and study of Fig. 1 (or, better to say, of the properties of the different solutions for the different patterns) can even allow one to establish optimal architectures for the reconfigurable array. In fact, by observing the distribution of the different solutions in all the x space (as well as on the different coordinates of such a space), one can establish for instance for which elements a phase reconfiguration will suffice, and for which elements an amplitude reconfiguration is instead needed. ACKNOWLEDGEMENTS Useful discussions with A. Massa, P. Rocca and O.M. Bucci are gratefully acknowledged. REFERENCES [1] R. J. Mailloux, Phased Array Antenna Handbook (2nd Ed.), Norwood, MA: Artech House, [2] A. F. Morabito and P. Rocca, Optimal synthesis of sum and difference patterns with arbitrary sidelobes subject to common excitations constraints, IEEE Antennas and Wireless Propagation Letters, vol. 9, pp , [3] G. K. Mahanti, A. Chakraborty, and S. Das, Phase-only and amplitude-phase only synthesis of dual-beam pattern linear antenna arrays using floating-point genetic algorithms, Progress In Electromagnetics Research, PIER 68, pp , [4] O. M. Bucci, G. Mazzarella, and G. 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