Modular Design of Hexagonal Phased Arrays Through Diamond Tiles
IIEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 1
Modular Design of Hexagonal Phased Arraysthrough Diamond Tiles
Paolo Rocca,
Senior Member, IEEE , Nicola Anselmi,
Member, IEEE , Alessandro Polo,
Member, IEEE , andAndrea Massa,
Fellow, IEEE
Abstract —The modular design of planar phased array anten-nas with hexagonal apertures is addressed by means of innovativediamond-shaped tiling techniques. Both tiling configuration andsub-array coefficients are optimized to fit user-defined power-mask constraints on the radiation pattern. Towards this end,suitable surface-tiling mathematical theorems are customized tothe problem at hand to guarantee optimal performance in caseof low/medium-size arrays, while the computationally-hard tilingof large arrays is yielded thanks to an effective integer-coded GA -based exploration of the arising high-cardinality solutionspaces . By considering ideal as well as real array models, a setof representative benchmark problems is dealt with to assess theeffectiveness of the proposed architectures and tiling strategies.Moreover, comparisons with alternative tiling architectures arealso performed to show to the interested readers the advantagesand the potentialities of the diamond sub-arraying of hexagonalapertures.
Index Terms —Phased Array Antenna, Planar Array, Hexago-nal Array, Diamond Tiles, Irregular Tiling, Optimization.
I. I
NTRODUCTION N OWADAYS, a key factor driving the evolution of phasedarray is the cost reduction [1] since services and productsbased on such a technology are expected to be widely diffusedon the commercial market because of the constantly increasingdemand of 5G applications (e.g., high-speed mobile commu-nications, internet-of-things, and industry 4.0) and the boostof autonomous driving systems [2]. Indeed, phased arrays
Manuscript received March XX, 2019; revised October XX, 2019This work has been partially supported by the Italian Ministry of ForeignAffairs and International Cooperation, Directorate General for Cultural andEconomic Promotion and Innovation within the SNATCH Project (2017-2019)and by the Italian Ministry of Education, University, and Research withinthe Program "Smart cities and communities and Social Innovation" (CUP:E44G14000060008) for the Project "WATERTECH - Smart Community perlo Sviluppo e l’Applicazione di Tecnologie di Monitoraggio Innovative perle Reti di Distribuzione Idrica negli usi idropotabili ed agricoli" (Grant no.SCN_00489) and within the Program PRIN 2017 for the Project "CYBER-PHYSICAL ELECTROMAGNETIC VISION: Context-Aware Electromag-netic Sensing and Smart Reaction (EMvisioning)".P. Rocca, N. Anselmi, A. Polo, and A. Massa are with the ELEDIA@UniTN(DISI - University of Trento), Via Sommarive 9, 38123 Trento - Italy (e-mail:{paolo.rocca, nicola.anselmi.1, alessandro.polo.1, andrea.massa}@unitn.it).P. Rocca is also with the ELEDIA Research Center (ELEDIA@XIDIAN -Xidian University), 3P.O. Box 191, No.2 South Tabai Road, 710071 Xi’an,Shaanxi Province - China (e-mail: [email protected])A. Massa is also with the ELEDIA Research Center (ELEDIA@L2S -UMR 8506), 3 rue Joliot Curie, 91192 Gif-sur-Yvette - France (e-mail:[email protected])A. Massa is also with the ELEDIA Research Center (ELEDIA@UESTC -UESTC), School of Electronic Engineering, Chengdu 611731 - China (e-mail:[email protected])A. Massa is also with the ELEDIA Research Center (ELE-DIA@TSINGHUA - Tsinghua University), 30 Shuangqing Rd, 100084Haidian, Beijing - China (e-mail: [email protected]) are considered an enabling technology in these fields thanksto their capability of guaranteeing the necessary quality-of-service and a suitable level of safety and reliability. Onthe other hand, even though the continuous development ofelectronics (e.g., fast analog-to-digital converters and massivesystems-on-chip) and material science (e.g., artificial materialsand meta-materials) as well as the introduction of innovativemanufacturing processes (e.g., 3D printing and flexible elec-tronics), phased arrays are still far from being commercial-off-the-shelf (
COTS ) devices. In recent years, innovative un-conventional architectures have been studied and developed toprovide better cost/performance trade-offs [3]. For instance,sparse [4]-[7] and thinned [8]-[11] arrays have been proposedto reduce the number of transmit-receive modules (
TRM s) orradio-frequency ( RF ) chains that represent one of the maincosts in phased arrays of modern radar systems [1]. Thesesolutions have shown to be quite effective in yielding lowsidelobes for interference and noise rejection and to affordarbitrary pattern shapes. However, sparse and thinned arrayscannot be implemented with a modular architecture and eachdesign and manufacturing turns out to be ad-hoc . Furthermore,the final arrangement in sparse arrays is based on a non-uniform placement of the array elements within the antennaaperture and this implies an inefficient use of the space atdisposal and a consequent directivity loss. Otherwise, theantenna aperture is fully exploited when dealing with sub-arrayed arrays in which the array elements, arranged on aregular lattice, are grouped and controlled at the sub-arraylevel. The main issue of those unconventional architecturesis the unavoidable presence of quantization and grating lobes[12][13] that become more and more impacting when increas-ing the scanning angle from the broadside direction and/orenlarging the operational bandwidth [14][15]. To cope withthese drawbacks, several strategies have been presented for theoptimization of the sub-array configuration and the sub-arrayamplitudes and/or phase excitation coefficients. Fast local-search techniques like the Contiguous Partition Method ( CPM )[16][17][18] and the K-means [19] as well as global nature-inspired optimization algorithms [20][21][22][23] have beenprofitably proposed. All rely on the exploitation of unequal andarbitrary sub-array sizes and shapes to break the periodicity ofthe aperture illumination that causes the undesired quantizationand grating lobes. More recently, tiled arrays have been alsoadopted [24][25][26][27][28]. They are clustered architectureswhere the sub-array modules belong to a finite set of simpletile shapes. The irregularity of the sub-array configuration,which allows the reduction of the undesired lobes, is here
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 2 ( a ) ( b )( c ) ( e )( d ) Figure 1.
Illustrative Geometry ( N ℓ = 2 , N = 24 , Q = 12 ) - Sketch of ( a ) a hexagonal aperture on a honeycomb lattice structure, ( b ) a tiling configurationwith ( c ) the three diamond-shaped tiles (vertical, σ V , horizontal-left, σ L , and horizontal-right, σ R , orientations), and ( d ) the corresponding sub-arrayedbeam-forming network. A realistic model of the three tiles with two patch antenna elements ( e ). obtained by properly choosing suitable tile shapes and by op-timizing their disposition and orientation within the aperture.For instance, polyomino-shaped sub-arrays have been used in[24] and a customized Genetic Algorithm ( GA ) has been alsoproposed to maximize the aperture coverage as well as theoperational bandwidth for a given aperture size [26]. Thanks tothe exploitation of pre-defined sub-array shapes, tiled aperturesare simple and modular structures, thus potential candidatesolutions for synthesizing low-cost and large-scale/mass pro-duction phased arrays. Unfortunately, the literature on optimal-design methodologies for array tiling, which enable the fullcoverage of the array aperture with given tile shapes, is stillquite scarce. This is due to the mathematical complexity of theclustered array design [29] and the limited set of combinationsbetween tiles- and aperture-shapes [30][31][32][33] for whichan optimal solution has been found. In [34], a mathematicaltheory for clustering rectangular apertures with domino-shapedtiles has been customized to the design and optimal tilingof phased array antennas. Pertinent theorems from [35][36],which assure a complete tessellation of the antenna aperture and the generation of all possible tiling configurations, havebeen properly exploited. This work extends such a theoreticalframework to the modular design of phased arrays withhexagonal aperture through an irregular and optimal placementof diamond-shaped tiles. Starting from the mathematical the-ory on the tessellation of hexagonal surfaces, two strategies,one enumerative and one computational, are derived for theoptimization of the tiling configuration and the sub-arraycoefficients to fit user-defined power-mask constraints on thepower pattern radiated by regular hexagonal apertures. Theinterest in phased array with hexagonal aperture is high sincethey are more suitable, with respect to rectangular arrays,for scanning the mainlobe around a pointing direction dueto their six-fold symmetry and since they better fit circularshapes, as well. Moreover, the hexagon is one of the threeregular polygons, along with the square and the equilateraltriangle, that allows the perfect tessellation of the Euclideanplane and it can be easily combined to compose complexplanar/conformal structures (e.g., geodesic domes) [38][39].As for the main novelties of this research work over the This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 3
Figure 2.
Illustrative Geometry ( N ℓ = 2 , N = 24 ) - Black-and-whiterepresentation of the triangular unit cells composing the hexagonal arrayaperture A with external vertices, { v ( ext ) m ; m = 1 , ..., M }, internal vertices,{ v ( int ) l ; l = 1 , ..., L }, and pixel-edges, { e m → ( m ± ; m = 0 , ..., M − }. existing literature on the field and to the best of the authors’knowledge, they comprise: ( i ) the engineering exploitation ofmathematical theorems providing the conditions for the fullcoverage of hexagonal apertures with diamond-shaped tiles[40][41][42] and the knowledge of the total number of tilingconfigurations [43][44]; ( ii ) the introduction of an innovativeenumerative strategy, which is based on analytic rules [45],previously applied to the case of domino-tiled arrays [34] buthere derived for diamond tiles, to determine the optimal tilingconfiguration in case of low- and medium-size arrays; ( iii )the exploitation of an innovative customization of an Integer-coded GA ( IGA ), while a standard binary GA ( BGA ) has beenused in [34], for an effective exploration of high-cardinalitysolution spaces to enable the design of large arrays affordingmask-constrained power patterns.The rest of the paper is organized as follows. The mathematicalformulation of the tiling problem of hexagonal-aperture phasedarrays with diamond-shaped tiles is reported in Sect. 2. Section3 is devoted to the description of the synthesis strategiesfor low/medium and large arrays. The validation and thecomparative assessment of the proposed tiling methods arecarried out in Sect. 4 by means of a set of representativenumerical examples concerned with realistic array models,as well. Eventually, some conclusions and final remarks aredrawn (Sect. 5).II. M
ATHEMATICAL F ORMULATION
Let us consider a planar phased array defined over a regularhexagonal aperture A ( l i = ℓ , i = 1 , ..., , l i being the lengthof its i -th side) where N elementary radiators are displacedon the xy -plane according to the honeycomb lattice in Fig.1( a ), ρ being the side of its unit cell equal to an equilateraltriangle. With reference the st Quadrant ( x ≥ , y ≥ ),the values of the coordinates of the lattice points (i.e., thecandidate positions of the barycenters of the array elements)are x r = r × ρ (1) and y s = (cid:16) s − √ (cid:17) × ρ if ( r and s ) areboth even or odd (cid:16) s − √ (cid:17) × ρ otherwise (2)where ( r, s ) is a couple of integer indexes that univocallyidentifies a point of the lattice, r ( r ≥ ) and s ( s ≥ )being the column and the row-strip index, respectively [Fig.1( a )]. For the sake of symmetry, the positions of the otherbarycenters belonging to the other quadrants can be easilyinferred by means of mirroring operations with respect to the x and y axes: x − r = − x r and y s = y s ( nd Quadrant), x − r = − x r and y − s = − y s ( rd Quadrant), and x r = x r and y − s = − y s ( th Quadrant). The array antenna is composedby Q clusters, { σ q ; q = 1 , ..., Q }, each one grouping twoadjacent unit cells of A (i.e., Q = N ) that share one edge[Fig. 1( b )] so that the resulting tiles have a diamond shapeand three possible orientations: vertical, σ V , horizontal-left, σ L , and horizontal-right, σ R [Fig. 1( c )]. While the fabricationof a single tile and its rotation to obtain σ V , σ L , and σ R could be a mathematically viable solution, it is not admis-sible from an electromagnetic viewpoint, thus three differentprimitives/building-blocks, ( σ V , σ L , σ R ), are necessary. Forinstance, let the elementary radiator be a patch antenna withlinear or horizontal polarization. To avoid any polarizationissues on the radiated electromagnetic ( EM ) field, the threetiles sketched in Fig. 1( e ) must be implemented.For a given tiling configuration [e.g., Fig. 1( b )], the signaleither transmitted or received to/from each q -th ( q = 1 , ..., Q )tile is controlled by a TRM characterized by an amplitudecoefficient α q and a phase delay β q [Fig. 1( d )], while the EM field generated in far-field is given by E ( u, v ) = P N ℓ s = − N ℓ , s =0 P × N ℓ −| s | r = − (2 × N ℓ −| s | ) p rs ( u, v ) × P Qq =1 δ c rs q α q e j [ k ( x r u + y s v )+ β q ] (3)where p rs ( u, v ) is the embedded element pattern [12][13] ofthe ( r, s ) -th element of the honeycomb lattice of the array, N ℓ ( N ℓ , ℓρ ) is the number of unit cells that exist oneach side of A , and N ( s ) ( N ( s ) , × N ℓ − × | s | + 1 )is the total number of elements within the s -th row strip[i.e., P N ℓ s = − N ℓ , s =0 N ( s ) = N ]. Moreover, k = πλ is thewavenumber, λ being the wavelength, u = sin θ cos φ and v = sin θ sin φ are the direction cosines ( θ ∈ [0 : 90] [deg] and φ ∈ [0 : 360] [deg]). Furthermore, c is the membership vector( c = { c rs ; r = − (cid:16) N ( s ) − (cid:17) , ..., (cid:16) N ( s ) − (cid:17) ; s = − N ℓ , ..., N ℓ ; s = 0 } ) whose ( r, s ) -th entry is an integer value ( c rs ∈ [1 : Q ] ) that identifies the membership of the ( r, s ) -th elementof the array lattice to the q -th diamond tile (i.e., c rs = q if the ( r, s ) -th element belongs to the q -th tile), while δ c rs q is theKronecker delta ( δ c rs q = 1 if c rs = q and δ c rs q = 0 otherwise[ c rs = q ]).The degrees-of-freedom ( DoF s) of the tiled array architec-ture at hand, namely the sub-array configuration, c , and theamplitude, α ( α = { α q : q = 1 , ..., Q } ), and the phase, β ( β = { β q : q = 1 , ..., Q } ), excitation vectors, are definedwhen solving the following “ mask-closeness ” synthesis prob-lem: This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 4 ( a )( b )( c )( d )( e ) Figure 3.
EDM Synthesis ( N ℓ = 2 , N = 24 ) - Generation of the second( t = 2 ) tiling word from the first/minimal ( t = 1 ) one. HF values of theinternal vertices, { v ( int ) l ; l = 1 , ..., L }, and tiling words for ( a ) the minimaltiling c (1) (i.e., w (1) ) and ( e ) the second tiling c (2) (i.e., w (2) ) along with( b )( c )( d ) three intermediate solutions not satisfying ( b )( c ) and satisfying ( d )the tileability condition (9). Diamond-Tiling Hexagonal Array Synthesis - Givena fully-populated hexagonal array [Fig. 1( a )] fittinga user-defined power mask U ( u, v ) , find its optimalsub-arraying configuration c opt (i.e., the optimalarrangement of vertical, σ V , horizontal-left, σ L , andhorizontal-right, σ R , tiles fully covering the aperture A ) and the corresponding values of the sub-arrayamplitude, α opt , and phase, β opt , coefficients so thatthe cost function χ χ ( c , α , β ) , R Ω h | E ( u, v ) | − U ( u, v ) i ×H n | E ( u, v ) | − U ( u, v ) o dudv (4)is minimized, that is, the radiated pattern minimallyviolates the mask constraint, H {·} and Ω beingthe Heaviside function and the visible range ( Ω = (cid:8) ( u, v ) : u + v ≤ (cid:9) ), respectively.III. H EXAGONAL A RRAY D IAMOND -T ILING M ETHODOLOGIES
In order to address the “
Diamond-Tiling Hexagonal ArraySynthesis ” problem at hand (Sect. 2), two innovative designmethodologies are presented in the following. Unlike [34],where domino tiles and rectangular apertures with the arrayelements located on a rectangular lattice have been taken intoaccount, the proposed approaches, namely the EnumerativeDesign Method (
EDM ) and the Computational Design Method(
CDM ), are specific for the tiling of hexagonal arrays withdiamond-shaped sub-arrays. Both methods benefit from thetileability theorem (
Appendix I ) [40][41][42], which statesthe conditions for the full-coverage with diamond tiles ofhexagonal array apertures with radiating elements disposedon a honeycomb lattice. They are also based on the optimaltiling algorithm [45] that assures the exhaustive generationof all possible T tiling configurations, { c ( t ) , t = 1 , ..., T },( Appendix II ) fitting the full coverage condition. Therefore,when one has “enough” time and/or computational resourcesto generate the whole set of T admissible tiling configurationsand to check the optimality of each t -th ( t = 1 , ..., T ) solutionby computing (4), the EDM allows one to faithfully retrievethe optimal problem solution [i.e., the global minimum of (4)],which corresponds to the best tiled-array performance. Other-wise (i.e., if testing all T solutions becomes computationallyintractable), the CDM enables an effective/computationally-efficient sampling of the tiling-solution space by means ofa customized
IGA . It is worth pointing out that the integer-coding of the
DoF s is aimed at improving the efficiencyand the convergence rate of the GA , with respect to the BGA used in [34], but also to extend the applicability ofthe analytic schemata-driven tiling optimization [34] to largerarrays. Since the choice of which methodology to adopt is ofimportance for an antenna designer and the use of one or theother methodology should not be left to chance, the analyticrelationship derived from [43][44] for the cardinality T ofthe solution space ( Appendix II ) can be a-priori exploited toestimate the
CPU -time required by the
EDM [i.e., τ = T × ∆ t , ∆ t being the amount of time for generating a new trial solutionand to compute (4)] and its admissibility with the array sizeat hand. A. Enumerative Design Method
Given a fully diamond-tileable aperture A ( Appendix I ), the
EDM generates and evaluates all T tiling configurations, { c ( t ) ; t = 1 , ..., T }, each one characterized by a different spatialarrangement of the diamond tiles σ V , σ L , and σ R [Fig. 1( c )].Like the optimal tiling algorithm in [45], the EDM exploitsthe
Height Function ( HF ) h ( · ) [36], which is defined on the This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 5
Figure 4.
EDM Synthesis ( N ℓ = 2 , N = 24 ) - Sketch of the arrangementof vertical, σ V , horizontal-left, σ L , and horizontal-right, σ R , diamond-tilesfor the minimal tiling, c (1) . vertices of the check-board pattern in Fig. 2 and its values aredetermined ( Appendix III ) by considering the correspondingedges, to univocally encode the N -dimensional tiling vector c ( t ) into a smaller L -dimensional ( L , × N ℓ − × N ℓ + 1 , L ≪ N ) string of integer values, called tiling word w ( t ) ( t = 1 , ..., T ). The tiling words are iteratively yielded startingfrom the first one, called minimal tiling word , having all entriesequal to zero (i.e., w ( t ) (cid:12)(cid:12) t =1 = w (1) = { w (1) l = 0; l =1 , ..., L } ). More in detail, the generation of the second ( t = 2 )tiling word, w (2) , starts with the selection of the internal vertexwith the largest index ξ ∈ [1 : L ] that satisfies one of thefollowing conditions [Fig. 3( b ) - right plot, h (2) ] h ( t − ξ − ≥ h ( t − ξ ≤ h ( t − ξ +1 ξ ∈ [2 : L − h ( t − ξ ≤ h ( t − ξ +1 ξ = 1 h ( t − ξ ≤ h ( t − ξ − ξ = L. (5)Then, the first ξ − letters of w ( t − are copied into w ( t ) w ( t ) j = w ( t − j j = 1 , ..., ξ − , (6)while the ξ -th one is increased of one unity [Fig. 3( b ) - rightplot, w (2) ] w ( t ) ξ = w ( t − ξ + 1 . (7)The HF values for the first ξ internal vertices are computedas follows h ( t ) j = 3 w ( t ) j + h (1) j j = 1 , ..., ξ (8)where { h (1) l ; l = 1 , ..., L } are the HF values of the minimaltiling configuration c (1) (Fig. 4) computed as detailed in Appendix IV .If the condition (cid:12)(cid:12)(cid:12) h ( t ) j − h ( t ) i (cid:12)(cid:12)(cid:12) = { , } (9)is not verified for every couple of neighboring vertices v ( t ) i and v ( t ) j on ∂ ˜ A , ∂ ˜ A being the boundary of ˜ A , ˜ A being theportion of the aperture A in which the HF values of internalvertices have not been yet defined [Fig. 3( b ) - left plot], thesecond largest index ξ ∈ [1 : L ] satisfying (5) is taken intoaccount [Fig. 3( c ) - right plot, h (2) ] and the steps (6), (7),and (10) are carried out. Such a procedure is iterated untila value ξ ∈ [1 : L ] is found [Fig. 3( d ) - right plot, h (2) ] sothat (9) holds true [Fig. 3( d ) - left plot]. The diamond tilesare then placed in the region (cid:16) A − ˜ A (cid:17) to match the tilingconditions (18). The new complete tiling configuration c ( t ) Table I
Solution-Space Cardinality - N
UMBER OF COMPLETE TILINGCONFIGURATIONS , T , FOR A SET OF REPRESENTATIVE N - ELEMENTSARRAYS WHEN USING DOMINO OR DIAMOND CLUSTERS . Hexagonal Aperture Rectangular Aperture
N N ℓ T ( hex ) N x × N y T ( dom )
24 2 20 6 × × . ×
96 4 2 . × ×
12 8 . ×
150 5 2 . × ×
15 2 . ×
216 6 1 . × ×
18 2 . ×
294 7 3 . × ×
21 1 . × and the corresponding HF values [Fig. 3( e )] are determined bymeans of the Thurston ’s algorithm [36] (
Appendix V ). Finally,the missing letters ( j = ξ + 1 , ..., L ) of the new tiling word, w ( t ) , are computed as follows w ( t ) j = h ( t ) j − h (1) j . (10)The previous procedure for defining w (2) starting from w (1) isthen successively applied to generate w ( t ) from w ( t − until t = T .Once all tiling words, { w ( t ) ; t = 1 , ..., T } and the correspond-ing sub-array configurations, { c ( t ) ; t = 1 , ..., T }, have beengenerated, the optimal tiling, c opt , is selected among the whole T set as the diamond-tiled arrangement whose cost functionvalue χ ( t ) [ χ ( t ) , χ (cid:0) c ( t ) , α ( t ) , β ( t ) (cid:1) ] is minimum c opt = arg (cid:18) min t =1 ,...,T n χ ( t ) o(cid:19) (11)by setting the t -th sub-array vectors, α ( t ) and β ( t ) , to α ( t ) q β ( t ) q ! = P N ℓ s = − N ℓ , s =0 P (cid:18) N ( s ) − (cid:19) r = − (cid:16) N ( s ) − (cid:17) δ c rs q × (cid:18) α refrs β refrs (cid:19) (12) (cid:0) α refrs , β refrs (cid:1) being the ( r, s ) -th [ r = − (cid:16) N ( s ) − (cid:17) , ..., (cid:16) N ( s ) − (cid:17) ; s = − N ℓ , ..., N ℓ ; s = 0 ]reference amplitude and phase coefficients of a fully-populated array antenna (i.e., an array having an independent TRM with a dedicated amplifier and a phase shifter for eacharray element).
B. Computational Design Method
When the number of array elements N significantly grows, thesynthesis with the EDM is unaffordable since the cardinalityof the solution space T turns out to be very high (Tab. Iand Fig. 5) despite the use of L integer values (i.e., thedimension of a word, w , namely the number of word letters) tounivocally encode a trial tiling solution instead of N (i.e., thedimension of c ( t ) ) ( LN < ) thanks to the bijective relationshipbetween the HF values of the internal vertices and a trial tilingconfiguration (Fig. 3). The CDM exploits a novel GA -basedoptimization to enable the synthesis of large arrays through an This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 6 N u m be r o f T ili ng s , l og ( T ) Number of Elements, NDominoDiamond
Figure 5.
Solution-Space Cardinality - Number of complete tiling configu-rations, T , of an array composed by N elements clustered with diamond ordomino tiles. Table II Illustrative Example ( L = 37 ) - C HROMOSOME SEQUENCE USING ANINTEGER - CODING ( IGA ) AND A BINARY - CODING ( BGA ). ChromosomeIGA
BGA effective sampling of the solution space and the convergencetowards - or very close - to the global optimum. The mainnovelty with respect to the GA in [34] is the integer codingof the tiling words to considerably reduce the chromosomelength and, in turn, to map the original solution space ofdimension N into a L -sized smaller one still of cardinality T [ T = ( N ℓ ) L ] to speed up the convergence and to enablethe tiling of wider arrangements, as well. For example, let ussuppose the maximum value of a letter in a word be w max = 4 ( w max , max t =1 ,...,T ; l =1 ,...,L n w ( t ) l o ) and the word lengthis L = 37 , when applying the BGA , the chromosome [34]has a length equal to bits (Tab. II) since three bits areneeded to encode the set of admissible values of a letter (i.e., w ( t ) l = { , , , , } ). Differently, the chromosome lengthof the IGA is exactly equal to L (Tab. II). Furthermore, the IGA turns out to be more advantageous than the
BGA for thefollowing reasons: ( i ) the operation of coding the tiling wordsinto binary chromosomes and decoding the chromosomes intotiling words is avoided, thus saving CPU -time; ( ii ) the directuse of integer variables (i.e., the genes of the chromosome)allows one to a-priori and intrinsically handle constraints onthe admissible values of the word letters, which is not doablein the BGA . Indeed, it is known that the minimum value thata letter may assume is zero and it is that of the minimal tiling ( w ( t ) l (cid:12)(cid:12)(cid:12) t =1 = 0 ; l = 1 , ..., L ), while the maximum one is equalto that of the maximal tiling ( w ( t ) l (cid:12)(cid:12)(cid:12) t = T ; l = 1 , ..., L ). Thislatter corresponds to the “depth” of the internal vertex, namely,the minimum number of edges that, starting from an externalvertex on ∂A , one has to cross for reaching the internal vertex.For illustrative purposes, two representative examples of themaximal tiling for N ℓ = 2 and N ℓ = 5 are shown in Fig. 6.As for the IGA , it follows the algorithmic implementation of ( a )( b ) Figure 6.
CDM Synthesis - Maximal tiling , c ( T ) , for a hexagonal array with( a ) N ℓ = 2 , N = 24 and ( b ) N ℓ = 5 , N = 150 . the optimization method described in [34]. Thanks to the pos-sibility of a-priori and analytically defining the first/minimaltiling word w (1) and the last/maximal tiling word w ( T ) alongwith the constraint that w (1) l ≤ w ( t ) l ≤ w ( T ) l l = 1 , ..., L ; t = 2 , ..., T − , (13)the initial ( k = 0 ) population of P individuals/trial solutions isselected to have a set of words with the maximally-diversifiedchromosomal content. Towards this end, the procedure de-scribed in [34] has been adapted to integer-variables, whilea further correlation check on each couple of generated wordshas been performed for having more/different schemata inthe initial population. Successively ( k > ), standard GA operators, namely the roulette-wheel selection, the single-pointcrossover , and the mutation [46], properly customized to dealwith integer-coded chromosomes, are iteratively applied - untilthe convergence - to the initial trial solutions to generatenew populations. More in detail, for each new trial solu-tion/offspring, the HF values associated to the correspondingtiling word are computed and the condition (9) is verified, oth-erwise the IGA operators are re-applied until a new ( k → k +1 )admissible word is obtained. To preserve the best individualduring the iterative process, the new population undergoeselitism [46]. The IGA optimization loop is iterated until thegeneration of a tiling solution that completely fulfills the maskconstraints U ( u, v ) [i.e., χ ( c opt , α opt , β opt ) = 0 ] or whenthe maximum number of iterations K has been carried out orwhen the GA -stagnation condition (cid:12)(cid:12)(cid:12) K st χ ( best ) k − P K st j =1 χ ( best ) j (cid:12)(cid:12)(cid:12) χ ( best ) k ≤ γ st (14) This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 7 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2x/ λ -2-1.5-1-0.5 0 0.5 1 1.5 2 y / λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] ( a ) ( b )( c ) Figure 7.
Test Case 1 ( N ℓ = 4 , N = 96 , ρ = √ λ , T = 2 . × )- Plot of ( a ) the power mask, U ( u, v ) , ( b ) the distribution of the ref-erence amplitudes, α ref , and ( c ) the reference normalized power pattern, (cid:12)(cid:12) E ref ( u, v ) (cid:12)(cid:12) . arises ( k = K conv , k > K st ). In (14), χ ( best ) k = min p =1 ,..,P n χ ( p ) k o , while K st and γ st are a user-definednumber of iterations and a fixed numerical threshold, respec-tively. IV. N UMERICAL R ESULTS AND C OMPARATIVE A SSESSMENT
The behavior of the design methodologies presented in theprevious section is here analyzed. The
EDM is first appliedto the synthesis of hexagonal arrays with small/medium sizeapertures. Then, the
CDM is exploited for synthesizing largerarrays and the
IGA is compared with the
BGA [34] to highlightthe advantages of the integer coding. A comparative assess-ment between a diamond-tiled array and a domino-tiled arraycomposing a hexagonal array aperture is carried out and theantenna performance of a selected solution are analyzed byconsidering a realistic antenna model with mutual coupling,as well.
A. EDM for Small/Medium-Array Synthesis
The first numerical example deals with a hexagonal array with N ℓ = 4 and ρ = √ λ composed by N = 96 ideal/isotropicelements (i.e., p rs ( u, v ) = 1 , ∀ r , ∀ s ). The user-defined powermask U ( u, v ) , which mathematically codes the synthesisconstraints, is characterized by a sidelobe rejection outsidethe main lobe of − [dB], while the mainlobe region hasbeen centered in the visible domain [i.e., the mainlobe peak isexpected along broadside, ( u , v ) = (0 , ] with an extensionof . along u and v as shown in Fig. 7( a ). The referenceexcitation coefficients, used in (12), have been derived witha convex programming ( CP ) based approach [47] and theyare shown in Fig. 7( b ) along with the corresponding power Figure 8.
EDM Synthesis ( Test Case 1 : N ℓ = 4 , N = 96 , ρ = √ λ , T = 2 . × ) - Values of the cost function (4) for the whole set ofcomplete tiling configurations, { c ( t ) ; t = 1 , ..., T }, ordered from the worsttiling, c worst , to the best one, c opt . pattern [Fig. 7( c )], whose pattern features are reported in Tab.III. Only the amplitudes excitations, α ref , are shown in Fig.7 since the phase terms are all zero (i.e., β ref = ) becauseof the broadside steering.In order to generate the whole set of T = 2 . × tilingwords, { w ( t ) ; t = 1 , ..., T } and the corresponding sub-arrayconfigurations, { c ( t ) ; t = 1 , ..., T }, to compute the excitationcoefficients, { α ( t ) , β ( t ) ; t = 1 , ..., T }, and to evaluate thecost function (4) for each t -th trial solution, χ ( t ) , the EDM run for about hours and minutes on a . GHz
PCwith GB of RAM. The values of the cost function for thecomplete set of T tiling configurations are sorted in Fig. 8.The “best” and the “worst” cost function values are equal to χ opt = 2 . × − and χ worst = 2 . × − , respectively.Those values correspond to the sub-array layouts in Figs. 9( a )-9( b ) and Fig. 9( c ) for the best, c opt , and the worst, c worst [ c worst , arg (cid:0) max t =1 ,..,T (cid:8) χ (cid:0) c ( t ) (cid:1)(cid:9)(cid:1) ], solution, respec-tively. The solutions in Fig. 9( a ) and Fig. 9( b ) are both optimalsince the corresponding sub-array arrangements are equalexcept for a mirroring with respect to the x axis. Therefore, theradiated power patterns have the same behavior as confirmedby the values of the pattern indexes in Tab. III and illustratedby the power pattern cuts along the principal planes ( φ = 0 [deg] and φ = 90 [deg]) shown in Fig. 9( d ) and Fig. 9( e ),respectively. As it can be observed from Fig. 9( d ), the EDM pattern slightly violates the mask because of an incrementof the secondary lobes of . [dB] ( SLL opt = − . [dB] - Tab. III). For the sake of completeness, the sub-arrayconfiguration, the amplitude coefficients [Fig. 9( c )], and thecuts of the radiated power pattern [Figs. 9( d )-( e )] of the“worst” tiling solution are reported, as well. In this case, thesecondary lobes grow up to SLL worst = − . [dB] (Tab.III) with a deterioration of more than . [dB] with respect tothe reference solution and of about . [dB] as compared tothe best tilings. This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 8 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2x/ λ -2-1.5-1-0.5 0 0.5 1 1.5 2 y / λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -2 -1.5 -1 -0.5 0 0.5 1 1.5 2x/ λ -2-1.5-1-0.5 0 0.5 1 1.5 2 y / λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] ( a ) ( b ) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2x/ λ -2-1.5-1-0.5 0 0.5 1 1.5 2 y / λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] ( c ) -40-35-30-25-20-15-10-5 0-1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] v=sin( θ )sin( φ ) φ = 0 [deg]ReferenceEDM - best-1EDM - best-2EDM - worstPower Mask -40-35-30-25-20-15-10-5 0-1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] v=sin( θ )sin( φ ) φ = 90 [deg]ReferenceEDM - best-1EDM - best-2EDM - worstPower Mask ( d ) ( e ) Figure 9.
EDM Synthesis ( Test Case 1 : N ℓ = 4 , N = 96 , ρ = √ λ , T =2 . × ) - Plot of the synthesized sub-array amplitude coefficients for( a )( b ) the two optimal/best solutions ( c opt , α opt ) and ( c ) the worst solution( c worst , α worst ). Power pattern cuts along ( d ) the φ = 0 [deg] and ( e ) the φ = 90 [deg] planes. -4.5 -3 -1.5 0 1.5 3 4.5x/ λ -4-2 0 2 4 y / λ E x c i t a t i on A m p li t ude [ a r b i t r a r y un i t] ( a ) ( b )( c ) Figure 10.
Test Case 2 [ N ℓ = 10 , N = 600 , ρ = √ λ , T ≃ . × ; ( u , v ) = (0 . , . ] - Plot of ( a ) the power mask, U ( u, v ) , ( b ) the distri-bution of the reference amplitudes, α ref , and ( c ) the reference normalizedpower pattern, (cid:12)(cid:12) E ref ( u, v ) (cid:12)(cid:12) . Table III EDM Synthesis ( Test Case 1 : N ℓ = 4 , N = 96 , ρ = √ λ , T = 2 . × ) - R ADIATION INDEXES ( SLL , D , HP BW az , HP BW el ) AND COST FUNCTION VALUES ( χ ). SLL D HP BW az HP BW el χ [dB] [dBi] [deg] [deg] [ × − ] Reference − .
00 19 .
42 21 .
10 21 . − EDM − best − − .
04 19 .
39 21 .
10 21 .
08 2 . EDM − best − − .
04 19 .
39 21 .
10 21 .
08 2 . EDM − worst − .
43 19 .
40 21 .
16 20 .
88 26 . B. CDM for Large-Array Synthesis
The second example is devoted to the numerical assessment ofthe
CDM for the design of larger hexagonal arrays. Towardsthis aim, a regular array aperture with N ℓ = 10 , ρ = √ λ , and N = 600 elements has been considered. The mask U ( u, v ) has been chosen, also in this case, with a mainlobe regioncentered in the visible range, but having a reduced extensionof . along both u and v directions [Fig. 10( a )] as wellas a sidelobe rejection of − [dB]. The CP -synthesized[47] reference excitations of the fully-populated array and theradiated power pattern are given in Fig. 10( b ) and Fig. 10( c ),respectively. Because of the stochastic nature of the IGA , a setof
CDM optimizations has been run by setting the controlparameters as follows: p c = 0 . (crossover probability), p m = 0 . (mutation probability), P = 542 (populationsize) and K = 1000 (maximum number of iterations). Morespecifically, the values of p c and p m have been set accordingto [46], while P and K have been chosen following the samecriterion used in [34] so that P × K is about of thecardinality of the solution space equal to T ≃ . × .The cost function of the best individual of the populationversus the iteration index k ( k = 0 , ..., K ) is shown inFig. 11( a ) in correspondence with randomly-chosen ini-tialization seeds . As it can be observed, the optimization runlabeled as IGA -2 reached the lowest value of the cost function( χ ( IGA − k = K ≃ . × − - Tab. IV). The corresponding sub-array layout is reported in Fig. 12( a ) along with the radiatedpower pattern [Fig. 12( b )]. For the sake of clarity, the CDM power pattern cuts are compared in Fig. 12( c ) ( φ = 0 [deg])and Fig. 12( d ) ( φ = 90 [deg]) with the reference ones, whilethe corresponding radiation features are given in Tab. IV. It isworth pointing out that, despite the clustering of the radiatingelements, the SLL of the tiled solutions turns out to be only . [dB] above the peak sidelobe level of the referencesolution.To assess the advantages of exploiting a customized IGA ,the same set (i.e., using the same initial populations of trialsolutions of the
IGA ) of optimizations has been runwith the
BGA by setting the parameters of the evolutionaryoperators as in [34]. The behavior of the cost function ofthe
IGA and the
BGA simulations yielding the lowest χ To keep a clear visualization, but without altering the meaning of theresults also guaranteed by the random choice of the samples, no morecurves have been added to the plot. This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 9 χ ( k ) [ x - ] Iteration Index, kIGA-1IGA-2IGA-3 IGA-4IGA-5IGA-6 IGA-7IGA-8IGA-9 IGA-10 ( a ) χ ( k ) [ x - ] Iteration Index, k IGABGAk (BGA)conv =370k (IGA)conv =186 ( b ) Figure 11.
CDM Synthesis [ Test Case 2 - N ℓ = 10 , N = 600 , ρ = √ λ , T ≃ . × ; ( u , v ) = (0 . , . ] - Behavior of the optimal value ofthe cost function (4) versus the iteration index, k , ( a ) for randomly-choseninitialization seeds and ( b ) for the best simulations among sample runswhen using the IGA and the
BGA . values is compared in Fig. 11( b ). As one can notice, bothmethods converge to very small values of the cost function(i.e., χ ( IGA ) k = K ≃ . × − vs. χ ( BGA ) k = K ≃ . × − -Tab. IV), even though the IGA slightly outperforms the
BGA .The closeness of both tiling solutions to the pattern-mask isvisually confirmed by the plots of the power pattern cuts inFigs. 12( c )-( d ) where the IGA and
BGA curves essentiallyoverlap. However, it is worth highlighting that the number ofiterations to reach the convergence, k conv , is almost half forthe IGA . Indeed, k ( IGA ) conv = 186 vs k ( BGA ) conv = 370 [Fig. 11( b )]with a reduction of the CPU -time from τ ( BGA ) = 286 [sec]down to τ ( IGA ) = 50 [sec] (Tab. IV), which corresponds to acomputational saving of about
83 % .In the third example, the array aperture and the sidelobesuppression level of the previous case have been kept, but thepattern beam has been constrained to point along the direction ( u , v ) = (0 . , . as indicated by the mask U ( u, v ) inFig. 13( a ). Once CP -computed the amplitude [Fig. 10( b )] andthe phase [Fig. 13( b )] distributions of the reference fully-populated array, whose power pattern is given in Fig. 13( c ),the CDM has been applied to synthesize the tiled array. Thebest result among different
IGA -based optimizations isshown in Fig. 14. As expected, the layout of the diamondtiles within the aperture turns out to be different from that ofthe broadside-steered case [Fig. 14( a ) and Fig. 14( b ) vs. Fig. -4.5 -3 -1.5 0 1.5 3 4.5x/ λ -4-2 0 2 4 y / λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] ( a ) ( b ) -60-50-40-30-20-10 0-1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] v=sin( θ )sin( φ ) φ = 0 [deg] ReferenceCDM-IGACDM-BGAPower Mask -60-50-40-30-20-10 0-1 -0.5 0 0.5 1 N o r m a li z ed P o w e r P a tt e r n [ d B ] v=sin( θ )sin( φ ) φ = 90 [deg] ReferenceCDM-IGACDM-BGAPower Mask ( c ) ( d ) Figure 12.
CDM Synthesis [ Test Case 2 - N ℓ = 10 , N = 600 , ρ = √ λ , T ≃ . × ; ( u , v ) = (0 . , . ] - Plot of ( a ) the sub-arrayamplitude coefficients and the corresponding normalized power pattern: ( b )2D color map and cuts along ( c ) the φ = 0 [deg] and ( d ) the φ = 90 [deg]. -4.5 -3 -1.5 0 1.5 3 4.5x/ λ -4-2 0 2 4 y / λ - π - π /2 0 π /2 π E xc i t a t i on P ha s e [ deg ] ( a ) ( b )( c ) Figure 13.
Test Case 3 [ N ℓ = 10 , N = 600 , ρ = √ λ , T ≃ . × ; ( u , v ) = (0 . , . ] - Plot of ( a ) the power mask U ( u, v ) , ( b ) thedistribution of the reference phase coefficients, β ref , and ( c ) the referencepower pattern, (cid:12)(cid:12) E ref ( u, v ) (cid:12)(cid:12) . a )]. Concerning the power pattern [Fig. 14( c )], it faithfullymatches the reference one even though there is an incrementof the sidelobe level ( SLL
IGA = − . [dB] vs. SLL ref = − . [dB]) with respect to the second test case (Tab. IV)as also confirmed by the higher value of the cost function atthe convergence (i.e., χ ( IGA ) k = K (cid:12)(cid:12)(cid:12) ( u ,v )=(0 . , . = 1 . × − vs. χ ( IGA ) k = K (cid:12)(cid:12)(cid:12) ( u ,v )=(0 . , . = 4 . × − ). C. Comparison with Domino-Tiled Array
The fourth example is devoted to the comparison with thedomino-tiling architecture discussed in [34] where the radi-
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 10 -4.5 -3 -1.5 0 1.5 3 4.5x/ λ -4-2 0 2 4 y / λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -4 -2 0 2 4x/ λ -4-3-2-1 0 1 2 3 4 y / λ - π - π /2 0 π /2 π E xc i t a t i on P ha s e [ deg ] ( a ) ( b ) -60-50-40-30-20-10 0-1 -0.5 0 0.5 1 N o r m a li z ed P a tt e r n [ d B ] u=sin( θ )cos( φ ) φ = 0 [deg]ReferenceOTMPower Mask ( c ) ( d ) Figure 14.
CDM Synthesis [ Test Case 3 : N ℓ = 10 , N = 600 , ρ = √ λ , T ≃ . × ; ( u , v ) = (0 . , . ] - Plot of the sub-array ( a )amplitude and ( b ) phase coefficients, and the radiated normalized powerpattern: ( c ) 2D color map and ( d ) cut along the φ = 0 [deg] plane. ating elements are positioned on a square grid and clusteredinto dominoes. Towards this end, a regular ( ℓ = 6 λ ) hexagonalaperture A has been partitioned with equilateral triangularcells having ρ = 0 . λ ( → N ℓ = 10 ) yielding an array of N ( hex ) = 600 elements. The same aperture A has been alsocovered with domino tiles arranged on a square lattice withinter-element spacing equal to d ( dom ) x = d ( dom ) y = 0 . λ . Underthese conditions, the domino-tiled array turned out to be com-posed by N ( dom ) = 576 elements so that N ( dom ) ≈ N ( hex ) .As for the pattern mask U ( u, v ) , the size of the mainloberegion has been reduced to . along both u and v directionswith respect to the previous test case.By applying the CDM-IGA and [34]-
BGA for designing thediamond-tiled and the domino-tiled array, respectively, thearising optimal tiling arrangements and the synthesized dis-tributions of the sub-array amplitudes [Figs. 15( a )-( b )] andphases [Figs. 15( c )-( d )] are shown in Fig. 15 together with thecorresponding power patterns [Figs. 15( e )-( f )]. As expected,the domino-based solution [Fig. 15( f )] presents secondarylobes higher than those from the diamond partitioning ofthe aperture [Fig. 15( e )] because of the worse approximationof the reference excitations. Indeed, the values of the costfunction at convergence are χ ( hex ) = 1 . × − and χ ( dom ) = 1 . × − . Such an outcome is further highlightedby the cut of the power pattern along the φ = 0 [deg] plane inFig. 15( g ) as well as by the values of the sidelobe level (Tab.V). More specifically, it turns out that SLL ( dom ) = − . [dB], that is [dB] above the one of the diamond arrangement( SLL ( hex ) = − . [dB]).The next experiment has been performed to assess the per-formance of the optimized tiled-arrays while scanning. Bychoosing the sub-array configuration and the amplitude inFig. 15( a ) and Fig. 15( b ) as representative examples for thediamond-tiled and the domino-tiled solution, respectively, and -6 -4 -2 0 2 4 6x/ λ -6-4-2 0 2 4 6 y / λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] -6 -4 -2 0 2 4 6x/ λ -6-4-2 0 2 4 6 y / λ E xc i t a t i on A m p li t ude [ a r b i t r a r y un i t] ( a ) ( b ) -6 -4 -2 0 2 4 6x/ λ -6-4-2 0 2 4 6 y / λ - π - π /2 0 π /2 π E xc i t a t i on P ha s e [ deg ] -6 -4 -2 0 2 4 6x/ λ -6-4-2 0 2 4 6 y / λ - π - π /2 0 π /2 π E xc i t a t i on P ha s e [ deg ] ( c ) ( d )( e ) ( f ) -60-50-40-30-20-10 0-1 -0.5 0 0.5 1 N o r m a li z ed P a tt e r n [ d B ] u=sin( θ )cos( φ ) φ = 0 [deg]Diamond TilingDomino TilingPower Mask ( g ) Figure 15.
CDM Synthesis - Plot of the sub-array ( a )( b ) amplitude and( c )( d ) phase coefficients, the normalized power patterns - ( e )( f ) 2D color mapand ( g ) cut along the φ = 0 plane - for the optimal CDM solutions whenclustering the array aperture with ( a )( c )( e ) diamond-shaped tiles [ Test Case3 : N ℓ = 10 , N ( hex ) = 600 , ρ = 0 . λ , T ≃ . × ; ( u , v ) =(0 . , . ] and ( b )( d )( f ) domino-shaped tiles [ Test Case 3 : N ( dom ) = 576 , d ( dom ) x = d ( dom ) y = 0 . λ ; ( u , v ) = (0 . , . ]. by setting the sub-array phases, { β q ; q = 1 , ..., Q }, by meansof (12) with the reference phases set to β refrs = − k [ x r sin ( θ + θ γ ) cos ( φ + φ γ )+ y s sin ( θ + θ γ ) sin ( φ + φ γ )] (15)( r = − (cid:16) N ( s ) − (cid:17) , ..., (cid:16) N ( s ) − (cid:17) ; s = − N ℓ , ..., N ℓ ; s = 0 ),the behavior of the SLL and of the directivity, D, has beenanalyzed when scanning the beam around the pointing di-rection ( θ , φ ) = (30 , [deg] [i.e., ( u , v ) = (0 . , . ]and within the cone defined by the following angular ranges: ≤ φ γ < [deg] and − ≤ θ γ < [deg]. Withreference to the polar color maps in Fig. 16, it turns out that thedomino tiling shows good performance only along the φ = 0 [deg] plane [Figs. 16( b )-16( d )], while the diamond clustering This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 11
Table IV
CDM Synthesis [ Test Case 2 - N ℓ = 10 , N = 600 , ρ = √ λ , T ≃ . × ] - R ADIATION INDEXES ( SLL , D , HP BW az , HP BW el ), COSTFUNCTION VALUES ( χ ), AND COMPUTATIONAL TIME ( τ ) WHEN STEERING THE BEAM ALONG ( u , v ) = (0 . , . AND ( u , v ) = (0 . , . . SLL D HP BW az HP BW el χ τ [dB] [dBi] [deg] [deg] - [sec] Broadside Mainlobe - ( u , v ) = (0 . , . Reference − .
00 26 .
79 9 .
00 8 . − − IGA − .
74 26 .
77 9 .
00 8 .
93 4 . × − BGA − .
71 26 .
77 9 .
00 8 .
93 4 . × − Steered Mainlobe - ( u , v ) = (0 . , . Reference − .
00 26 .
13 10 . − − − IGA − .
57 26 .
12 10 . − . × − ( a ) ( b )( c ) ( d ) Figure 16.
CDM Synthesis [ Test Case 3 : N ℓ = 10 , N = 600 , ρ = √ λ , T ≃ . × ; ( u , v ) = (0 . , . ] - Plot of ( a )( b ) the SLL and ( c )( d )the D values of the patterns generated by the tiling configurations in ( a )( c )Fig. 15( a ) and Fig. 15( c ) and ( b )( d ) Fig. 15( b ) and Fig. 15( d ) when scanningthe mainlobe around the pointing direction ( θ , φ ) = (30 , [deg] withinthe cone { − ≤ θ γ < [deg]; ≤ φ γ < [deg]}. affords good values of the pattern features along three planesspaced of [deg] along the φ direction [Figs. 16( a )-16( c )].For instance, let us assume the maximum scan angle be equalto θ γ = 15 [deg] and let us evaluate the SLL and D over thewhole scanning cone ( ≤ φ γ <
360 [ deg ] ). The peak sidelobelevel of the diamond-tiled array is SLL ( hex ) = − . [dB], while SLL ( dom ) = − . [dB] for the domino-tiledone with a worsening of more than [dB]. Moreover, thedirectivity values turn out to be D ( hex ) > . [dBi] and D ( dom ) > . [dBi] within the same scanning cone.Finally, the performance of the optimized diamond-tiling inFigs. 15( a )-15( c ) have been validated against realistic radiatingelements [i.e., p rs ( u, v ) = 1 ], including mutual couplingeffects, as well. Towards this end, a coaxial-fed patch antennaresonating at f = 7 . [GHz] has been used as elementaryradiator, while the entire array structure (i.e., the dielectricsubstrate, the ground plane, the N = 600 elements arraytogether with the metallic coaxial connector) has been mod-elled with CST Microwave Studio [Fig. 17( a )]. The full-wave power pattern has been generated when pointing the beam at ( u , v ) = (0 . , . , as in Fig. 15, as well asalong the direction ( u , v ) = (0 . , . and in broadside ( u , v ) = (0 . , . . From the comparison of the realisticpattern with the ideal one (i.e., isotropic sources array) alongthe φ = 0 [deg] cut [Figs. 17( b )-17( d )] and the φ = 90 [deg] cut [Figs. 17( c )-17( e )], one can infer that the twopatterns are very similar with only some minor and almostnegligible deviations towards the boundaries of the visiblerange ( u = ± , v = ± ), thus highlighting the facts that thebeam of the patch antenna is broad and the mutual couplingeffects are not significant in the considered array structure [Fig.17( a )]. V. C ONCLUSIONS AND R EMARKS
The modular synthesis of hexagonal arrays with elementsdisposed on a regular honeycomb lattice and clustered indiamond-shaped tiles has been addressed. Tiling theoremshave been exploited to assess the perfect covering of the arrayaperture by means of diamond tiles as well as to a-priori determine the cardinality of the solution space of the full-aperture tiling configurations. Starting from such a theoreticalbasis, two tiling strategies have been presented to synthesizearray solutions fully covering the available antenna aperture.The former is based on an enumerative approach, while theother relies on a customized version of the
IGA .The key methodological advancements of the proposed re-search work with respect to the state-of-the-art literature canbe summarized in the following: • the modular design of phased array antennas with regularhexagonal apertures through diamond-shaped tiles; • the exploitation of mathematical theorems and algorithmsto define a theoretical framework for the clustering ofhexagonal phased arrays; • the formulation of the synthesis of hexagonal phasedarrays clustered with diamond tiles as a mask-constrainedpower pattern one; • the integer coding of the tiling words within the schematadriven optimization and the definition of a customizedinteger-coded GA -based tiling approach.A set of numerical examples, including both ideal-isotropicsources and real-directive antenna elements, has been reportedto assess the capabilities and the effectiveness of the proposed Each pattern has been normalized to its maximum in order to bettercompare the patterns shape.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 12
Table V
CDM Synthesis - R
ADIATION INDEXES ( SLL , D , HP BW az ), COST FUNCTION VALUES ( χ ), AND COMPUTATIONAL TIME ( τ ) WHEN CLUSTERING THEARRAY APERTURE WITH DIAMOND - SHAPED TILES [ Test Case 3 : N ℓ = 10 , N ( hex ) = 600 , ρ = 0 . λ , T ≃ . × ; ( u , v ) = (0 . , . ] AND ( b )( d )( f ) DOMINO - SHAPED TILES [ Test Case 3 : N ( dom ) = 576 , d ( dom ) x = d ( dom ) y = 0 . λ ; ( u , v ) = (0 . , . ]. SLL D HP BW az χ τ [dB] [dBi] [deg] - [sec] Domino Tiles
Reference Isotropic − .
00 29 .
52 6 . − − CDM Isotropic − .
20 28 .
62 7 .
56 1 . × − Diamond Tiles
Reference Isotropic − .
00 29 .
65 6 . − − CDM Isotropic − .
16 28 .
91 7 .
37 1 . × − CDM F ull − W ave − .
69 28 .
88 7 .
30 3 . × − − ( a ) -60-50-40-30-20-10 0-1 -0.5 0 0.5 1 N o r m a li z ed P a tt e r n [ d B ] u=sin( θ )cos( φ ) φ = 0 [deg]AnalyticFull-Wave -60-50-40-30-20-10 0-1 -0.5 0 0.5 1 N o r m a li z ed P a tt e r n [ d B ] v=sin( θ )sin( φ ) φ = 90 [deg]AnalyticFull-Wave ( b ) ( c ) -60-50-40-30-20-10 0-1 -0.5 0 0.5 1 N o r m a li z ed P a tt e r n [ d B ] u=sin( θ )cos( φ ) φ = 0 [deg]AnalyticFull-Wave -60-50-40-30-20-10 0-1 -0.5 0 0.5 1 N o r m a li z ed P a tt e r n [ d B ] v=sin( θ )sin( φ ) φ = 90 [deg]AnalyticFull-Wave ( d ) ( e ) Figure 17.
CDM Synthesis [ Test Case 3 : N ℓ = 10 , N = 600 , ρ = √ λ , T ≃ . × ] - Model of ( a ) the hexagonal array of coaxial-fed patchantennas and plot of the normalized power pattern cuts along the ( b )( d ) φ = 0 [deg] plane when ( b ) ( u , v ) = (0 . , . and ( d ) ( u , v ) = (0 . , . and along the ( c )( e ) φ = 90 [deg] plane when ( c ) ( u , v ) = (0 . , . and( e ) ( u , v ) = (0 . , . . synthesis strategies. From the numerical analysis, the follow-ing main outcomes can be drawn: • the EDM is efficient in generating all possible tilingconfigurations and to retrieve the optimal solution, orthe multiple optimal solutions in case of the symmetricaltilings, that guarantees the best admissible performance(i.e., the global minimum of the cost function at hand)when the size of the array is small/medium; • the exploitation of integer-coded chromosomes in the IGA implies a non negligible reduction of the computationalburden and the improvement of the convergence speed,with respect to the standard
BGA -based synthesis [34], for designing large arrays; • the radiation performance (e.g., the sidelobe control andthe peak directivity) of an hexagonal aperture partitionedwith diamond-shaped tiles are generally better than thosefrom an equivalent array clustered with domino sub-arrays when scanning the beam over a limited field ofview; • the power pattern generated by an ideal diamond-tiledarray is close to that from a realistic one that includesnon-isotropic elements and mutual coupling effects. Appendix I - T
ILEABILITY T HEOREM
An arbitrary hexagonal aperture A having sides of length l i , i = 1 , ..., is fully coverable with diamond tiles if and only ifthe opposite sides have the same dimension, namely l = l , l = l , and l = l [40][41][42]. Such a condition holdstrue for both regular [Fig. 18( a )] and non-regular [Fig. 18( b )]hexagons, while the aperture in Fig. 18( c ) is un-tileable since l = l , l = l , l = l , and the number of triangles N turnsout being odd. As for regular hexagonal apertures ( l i = ℓ ; i = 1 , ..., ), as those dealt with in this work, they alwayssatisfy the tileability condition (i.e., l = l , l = l , and l = l ) and they can be totally tileable with diamond-shapedtiles. Appendix II - C
ARDINALITY T HEOREM
By supposing a fully-tileable hexagonal aperture A [Figs.18( a )-( b )], the total number T of different tiling configurationsthat completely cover A is equal to [43] T = N l Y i =1 N l Y j =1 N l Y g =1 i + j + g − i + j + g − (16)where N l i , l i ρ ( i = { , , } ) is the number of equilateraltriangles (i.e., unit cells) along the i -th side of the hexagonalaperture.A set of cardinality values ( T ← T ( hex ) ) in correspondencewith different number of elements N is reported in Tab. I.For comparison purposes, the cardinality of the solution space( T ← T ( dom ) ) for the case of rectangular apertures having thesame number of elements N , but considering domino-shapedtiles as in [34], is reported, as well. As it can be inferred fromTab. I and observed in Fig. 5, T ( dom ) grows much faster than T ( hex ) when increasing N . This indicates a smaller solution This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 13 ( a ) ( b )( c ) Figure 18.
Illustrative Geometry - Examples of ( a ) regular tileable ( l i = ℓ ; i = 1 , ..., ), ( b ) irregular tileable ( l = l , l = l , l = l ), and ( c )irregular non-tileable ( l = l , l = l , l = l ) hexagonal apertures. space for hexagonal arrays, thus a higher complexity of thetiling problem at hand whether setting the same constraintsand requirements of the domino partitioning - because of thereduced number of admissible solutions - and the need of ad-hoc synthesis techniques. Appendix III - H
EIGHT F UNCTION C OMPUTATION
To define the height function, let us consider the lattice definedby the sets of vertices v and edges e of the N equilateraltriangles composing a regular hexagonal aperture A (Fig.2). The orientation of the edges is assumed clockwise forthe point-down triangles and counterclockwise for the point-up triangles, that are colored in black and white in Fig. 2,respectively. Moreover, the notation e i → g is used to indicatean edge connecting two adjacent vertices v i and v g , orientedfrom v i towards v g . As it can be inferred from Fig. 2, it turnsout that • a tile, whatever its orientation, namely vertical σ V ,horizontal-left σ L , or horizontal-right σ R , is obtainedby the union of two triangles sharing one side (i.e., adiamond tile is the combination of a black and a whitetriangle); • the triangles having one side on the boundary ∂A of theaperture A can generate only two different tile shapes.For example, the white triangles on the bottom of A withthe edges e → M and e M → M − on ∂A (Fig. 2) can beincluded into a σ L or a σ R tile, but not into a σ V tile.Differently, all other triangles within A , having all sidesshared with another triangle, can be potentially combinedinto one of the three admissible tiles in Fig. 1( c ).As for the HF values of the external vertices, v ( ext ) = n v ( ext ) m ∈ ∂A : m = 1 , ..., M o ( M , × N ℓ ), they are onlyfunction of the size of the hexagonal aperture and they donot depend on the t -th ( t = 1 , ..., T ) tiling configuration.Therefore, they are computed once. Towards this end, let us Figure 19.
Illustrative Geometry ( N ℓ = 2 , N = 24 ) - HF values, { h m ; m = 1 , ..., M } of the external vertices, { v ( ext ) m , m = 1 , ..., M }. order the external vertices clockwise starting from the bottom-left vertex of the aperture A (Fig. 2). The HF value of the firstvertex is set to zero, namely h = h (cid:16) v ( ext )1 (cid:17) = 0 , and thatof the others ( m = 2 , ..., M ) is iteratively computed (Fig. 19)according to the following rules h m +1 = h m + 1 if e m → m +1 h m +1 = h m − if e m +1 → m . (17)As for the internal vertices v ( int ) = n v ( int ) l : l = 1 , ..., L o ,they are indexed according to a raster order from bottom-leftto top-right as indicated in Fig. 2. Unlike the external vertices,the HF values of the internal vertices are function of the tilingconfiguration (i.e., h ( t ) l = h (cid:16) v ( int ) l (cid:17) = h (cid:0) c ( t ) (cid:1) , l = 1 , ..., L ).Each value h ( t ) l ( l = 1 , ..., L ) is computed by first selectinga neighboring vertex v g ∈ (cid:8) v ( ext ) , v ( int ) (cid:9) with HF valuealready assigned and then using the following four options if e l → g and e l → g ∈ ∂σ q ⇒ h ( t ) l = h ( t ) g + 1 if e g → l and e g → l ∈ ∂σ q ⇒ h ( t ) l = h ( t ) g − if e l → g and e l → g / ∈ ∂σ q ⇒ h ( t ) l = h ( t ) g + 2 if e g → l and e g → l / ∈ ∂σ q ⇒ h ( t ) l = h ( t ) g − (18)where ∂σ q is the boundary of a diamond tile ( q = 1 , ..., Q )inside A . In (18), the condition e l → g ∈ ∂σ q means that theedge e l → g belongs to the contour of a tile, while e l → g / ∈ ∂σ q indicates that the edge is covered by a tile. Appendix IV - M
INIMUM T ILING
Unlike the domino-tiling for rectangular apertures and thanksto the regularity of the hexagonal aperture A , the minimaltiling , c (1) , can be simply yielded by filling the three partsof A (i.e. the vertical, the horizontal-left, and the horizontal-right rhombus highlighted with different colors in Fig. 4) withvertical, σ V , horizontal-left, σ L , and horizontal-right, σ L , tiles,respectively (Fig. 4).By definition, the minimal tiling word is the first word, w (1) ,and it has all entries equal to zero ( w (1) l = 0 , l = 1 , ..., L ), w (1) = .The values of the HF of the minimum tiling , { h (1) l ; l =1 , ..., L }, are computed [Fig. 3( a ) - right plot, h (1) ] accordingto Appendix III . The first internal vertices to be considered will be those connected to theexternal ones v ( ext ) because the HF values of these latter can be computedonce known the array size without assuming any clustering configuration. This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 14
Appendix V - T
HURSTON ’ S A LGORITHM
The Thurston’s Algorithm [36] for the computation of the HF values of the vertices on ∂ ˜ A is applied according to thefollowing procedural steps: • Step 1.
Vertex Selection - Select the vertex on ∂ ˜ A withmaximum HF value. If multiple vertices have the samemaximum value, arbitrarily select one of them; • Step 2.
Tile Placement - Place one of the availablediamond tiles ( σ V or σ L or σ R ) so that the two verticeson ∂ ˜ A adjacent to the vertex selected in Step 1 are alsovertices of the diamond tile; • Step 3.
Aperture Boundary and HF Update - Completethe computation of HF on the vertices of the last placeddiamond tile according to the rules defined in (18) andupdate the aperture boundary ∂ ˜ A ← ∂ (cid:16) ˜ A − σ V/L/R (cid:17) by subtracting the area of the newly placed tile; • Step 4.
Termination Criterion - Stop if the aperture istotally covered and the HF is computed for all the internalvertices of A . Otherwise, go to the Step
CKNOWLEDGEMENTS
The authors thank Prof. A. D. Capobianco (University ofPadova, Italy) for providing the data from CST MicrowaveStudio full-wave simulations. A. Massa wishes to thank E.Vico for her never-ending inspiration, support, guidance, andhelp. R
EFERENCES[1] J. S. Herd and M. D. Conwey, “The evolution to modern phased arrayarchitectures,”
Proc. IEEE, vol. 104, no. 3, pp. 519-529, Mar. 2016.[2] “Special issue on antennas and propagation aspects of 5G communi-cations,”
IEEE Trans. Antennas Propag. , vol. 64, no. 6, p. 2588, Jun.2016.[3] P. Rocca, G. Oliveri, R. J. Mailloux, and A. Massa, “Unconventionalphased array architectures and design methodologies - A review,”
Proc.IEEE, vol. 104, no. 3, pp. 544-560, Mar. 2016.[4] B. Fuchs, “Synthesis of sparse arrays with focused or shaped beam-pattern via sequential convex optimizations,”
IEEE Trans. AntennasPropag. , vol. 60, no. 7, pp. 3499-3503, Jul. 2012.[5] F. Viani, G. Oliveri, and A. Massa, “Compressive sensing patternmatching techniques for synthesizing planar sparse arrays,”
IEEE Trans.Antennas Propag. , vol. 61, no. 9, pp. 4577-4587, Sep. 2013.[6] O. M. Bucci, T. Isernia, S. Perna, and D. Pinchera, “Isophoric sparsearrays ensuring global coverage in satellite communications ,” IEEETrans. Antennas Propag. , vol. 62, no. 4, pp. 1607-1618, Apr. 2014.[7] G. Oliveri, E. T. Bekele, F. Robol, and A. Massa, “Sparsening conformalarrays through a versatile BCS-based method,”
IEEE Trans. AntennasPropag. , vol. 62, no. 4, pp. 1681-1689, Apr. 2014.[8] G. Oliveri, L. Manica, and A. Massa, “ADS-Based guidelines for thinnedplanar arrays,”
IEEE Trans. Antennas Propag ., vol. 58, no. 6, pp. 1935-1948, Jun. 2010.[9] G. Oliveri, F. Caramanica, and A. Massa, “Hybrid ADS-based tech-niques for radio astronomy array design,"
IEEE Trans. AntennasPropag ., vol. 59, no. 6, pp. 1817-1827, Jun. 2011.[10] P. Rocca, R. L. Haupt, and A. Massa, “Interference suppression inuniform linear array through a dynamic thinning strategy,”
IEEE Trans.Antennas Propag. , vol. 59, no. 12, pp. 4525-4533, Dec. 2011.[11] F. Scattone, M. Ettorre, B. Fuchs, R. Sauleau and N. J. G. Fonseca,“Synthesis procedure for thinned leaky-wave-based arrays with reducednumber of elements,”
IEEE Trans. Antennas Propag ., vol. 64, no. 2, pp.582-590, Feb. 2016.[12] R. J. Mailloux,
Phased Array Antenna Handbook.
Boston, MA: ArtechHouse, 2005.[13] R. L. Haupt,
Antenna Arrays - A Computation Approach.
Hoboken, NJ,USA: Wiley, 2010. [14] R. J. Mailloux, “Array grating lobes due to periodic phase, amplitudeand time delay quantization,”
IEEE Trans. Antennas Propag. , vol. 32,no. 12, pp. 1364-1368, Dec. 1984.[15] R. L. Haupt, “Reducing grating lobes due to subarray amplitude taper-ing,”
IEEE Trans. Antennas Propag ., vol. 33, no. 8, pp. 846-850, Aug.1985.[16] L. Manica, P. Rocca, A. Martini, and A. Massa, “An innovative ap-proach based on a tree-searching algorithm for the optimal matching ofindependently optimum sum and difference excitations,”
IEEE Trans.Antennas Propag. , vol. 56, no. 1, pp. 58-66, Jan. 2008.[17] L. Manica, P. Rocca, and A. Massa, “Design of subarrayed linear andplanar array antennas with SLL control based on an excitation matchingapproach,”
IEEE Trans. Antennas Propag. , vol. 57, no. 6, pp. 1684-1691,Jun. 2009.[18] P. Rocca, L. Manica, R. Azaro, and A. Massa “A hybrid approach to thesynthesis of subarrayed monopulse linear arrays,”
IEEE Trans. AntennasPropag. , vol. 57, no. 1, pp. 280-283, Jan. 2009.[19] X. Yang, W. Xi, Y. Su, T. Zeng, T. Long, and T. K. Sarkar, “Optimizationof subarray partition for large planar phased array radar based onweighted K-means clustering method,”
IEEE Trans. Antennas Propag .,vol. 9, no. 8, pp. 1460-1468, Dec. 2015.[20] P. Lopez, J. A. Rodriguez, F. Ares, and E. Moreno, “Subarray weightingfor difference patterns of monopulse antennas: Joint optimization ofsubarray configurations and weights,”
IEEE Trans. Antennas Propag .,vol. 49, no. 11, pp. 1606-1608, Nov. 2001.[21] R. L. Haupt, "Optimized weighting of uniform subarrays of unequalsizes,"
IEEE Trans. Antennas Propag ., vol. 55, no. 4, pp. 1207-1210,Apr. 2007.[22] Y. Chen, S. Yang, and Z. Nie, “The application of a modified differentialevolution strategy to some array pattern synthesis problems,”
IEEETrans. Antennas Propagat. , vol. 56, no. 7, pp. 1919-1927, Jul. 2008.[23] P. Rocca, L. Manica, and A. Massa, “An improved excitation matchingmethod based on an ant colony optimization for suboptimal-free clus-tering in sum-difference compromise synthesis,”
IEEE Trans. AntennasPropag. , vol. 57, no. 8, pp. 2297-2306, Aug. 2009.[24] R. J. Mailloux, S. G. Santarelli, T. M. Roberts, and D. Luu, “Irregu-lar polyomino-shaped subarrays for space-based active arrays,”
Int. J.Antennas Propag. , vol. 2009, Article ID 956524, 2009.[25] J. Herd, S. Duffy, D. Carlson, M. Weber, G. Brigham, C. Weigand, andD. Cursio, “Low cost multifunction phased array radar concept,”
IEEEInt. Symp. Phased Array Syst. Tech ., Waltham, MA, USA, 12-15 Oct.2010, pp. 457-460.[26] P. Rocca, R. J. Mailloux, and G. Toso “GA-based optimization ofirregular sub-array layouts for wideband phased arrays design,”
IEEEAntennas Wireless Propag. Lett ., vol. 14, pp. 131-134, 2015.[27] G. Oliveri, M. Salucci, and A. Massa, “Synthesis of modular contigu-ously clustered linear arrays through a sparseness-regularized solver,”
IEEE Trans. Antennas Propag. , vol. 64, no. 10, pp. 4277-4287, Oct.2016.[28] J. Diao, J. W. Kunzler, and K. F. Warnick, “Sidelobe level and apertureefficiency optimization for tiled aperiodic array antennas,”
IEEE Trans.Antennas Propag. , vol. 65, no. 12, pp. 7083-7090, Jan. 2017.[29] R. M. Robinson, “Undecidability and nonperiodicity for tilings of theplane,”
Inventiones Mathematicae , vol. 12, pp. 177-209, 1971.[30] J. H. Conwey and J. C. Lagarias, “Tiling with polyominoes andcombinatorial group theory,”
J. Combinatorial Theory , Series A , vol.53, pp. 183-208, 1990.[31] R. A. Brualdi and T. H. Foregger, “Packing boxes with harmonic bricks,”
J. Combinatorial Theory , vol. 17, pp. 81-114, 1974.[32] D. Beauquier, M. Nivat, E. Remila, and M. Robson, “Tiling figures ofthe plane with two bars,”
Comput. Geometry , vol. 5, pp. 1-25, 1995.[33] S. Desreux, “An algorithm to generate exactly once every tiling withlozenges of a domain,”
Theoretical Comput. Sci ., no. 303, pp. 375-408,2003.[34] N. Anselmi, P. Rocca, and A. Massa, “Irregular phased array tiling bymeans of analytic schemata-driven optimization”
IEEE Trans. AntennasPropag ., vol. 65, no. 9, pp. 4495-4510, Sep. 2017.[35] P. Kasteleyn, “The statistics of dimers on a lattice I. The number ofdimer arrangements on a quadratic lattice,”
Physica , vol. 27, pp. 1209-1225, 1961.[36] W. P. Thurston, “Conway’s tiling groups,”
The American MathematicalMonthly , vol. 97, no. 8, pp. 757-773, Oct. 1990.[37] N. Anselmi, L. Poli, P. Rocca, and A. Massa, “Diamond tiling optimiza-tion for hexagonal shaped phased arrays,”
Proc. 12th European Conf.Antennas Propag. , London, UK, 9-13 April 2018.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561Copyright (c) 2020 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 15 [38] S. Liu, B. Tomasic, and J. Turtle, “The geodesic dome phased arrayantenna for satellite operations support - antenna resource management,”
IEEE Int. Symp. Antennas Propag. , Honolulu, HI, USA, 10-15 Jun. 2007,pp. 3161-3164.[39] H. Ahn, B. Tomasic, and, S. Liu, “Digital beamforming in a large confor-mal phased array antenna for satellite operations support - architecture,design, and development,”
IEEE Int. Symp. Phased Array Syst. Tech .,Waltham, MA, USA, 12-15 Oct. 2010, pp. 423-431.[40] D. Guy and C. Tomei, “The problem of the calissons,”
Amer. Math.Monthly, vol. 96, no. 5, pp. 429-431, 1989.[41] N. C. Saldanha and C. Tomei, “An overview of domino and Lozengetilings”,
Resenhas IME-USP, vol. 2, no. 2, pp. 239-252, 1995.[42] H. A. Helfgott and I. M. Gessel, “Enumeration of tilings of diamondsand hexagons with defects,”
Electron. J. Combin., vol. 6, 1999. [43] P. A. MacMahon,
Combinatory Analysis (vol. II) . Mineola, NY: DoverPublications, 2004.[44] J. Propp, “Enumeration of Tilings,”
Enumerative Combinatorics , CRCPress, U. Mass. Lowell, Aug. 2014.[45] S. Desreux and E. Remila, “An optimal algorithm to generate tilings,”
J. Discrete Alg ., no. 4, pp. 168-180, 2006.[46] P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa,“Evolutionary optimization as applied to inverse scattering problems,”
Inv. Prob. , vol. 24, pp. 1-41, 2009.[47] O. M. Bucci, L. Caccavale, T. Isernia, “Optimal far-field focusing ofuniformly spaced arrays subject to arbitrary upper bounds in nontargetdirections,”
IEEE Trans. Antennas Propag., vol. 50, no. 11, pp. 1539-1554, Nov. 2002.vol. 50, no. 11, pp. 1539-1554, Nov. 2002.