Synthesis of Thinned Planar Circular and Square Arrays Using Density Tapering

Synthesis of Thinned Planar Circular and Square Arrays Using Density TaperingWill P.M.N.Keizer,Member,IEEEAbstract—Two new hybrid density taper methods for designing thinned planar arrays are presented.Both methods make use of a model taper to allocate for a number of rings across the array aperture the amount of turned ON elements per ring.One method applies a modified version of the iterative Fourier transform(IFT) method tofind the optimum turned ON element distribution for each ring to get a prescribed low sidelobe level.The second method gets suitable turned ON element locations by distributing these el-ements in a uniform random way over each ring.The purpose of the second method is tofind the optimum model taper since it is computational faster than thefirst ing the model taper determined by the second method,thefirst methodfinds then the optimum locations of the turned ON element positions.Both new methods can deal with the synthesis of thinned planar arrays of any size,any aperture shape and any degree of thinning.A wide range of numerical examples covering very small up to extremely large arrays will demonstrate the effectiveness of the two proposed methods for both circular and square aperture shapes.Index Terms—Array antennas,density taper,low sidelobes, thinned arrays.I.I NTRODUCTIONA RRAY thinning involves the removal(turning off andconnected to matched load)of radiating elements from an array antenna.The main motivation to use thinning is a re-duction in cost and weight while maintaining nearly the same narrow beamwidth as for afilled array of equal size.A second distinct advantage is that when the turned ON elements operate with equal amplitude,lower sidelobes can be obtained as for the samefilled array illuminated with uniform weighting.For these reasons thinned array antennas are over more than40years in use with large,high-performance itary phased array radar systems such as Pave Paws and Cobra Dane[1],HAPDAR [2],and Sea-Based X-Band(SBX)Radar[3].In recent years,a variety of different methods[4]–[12]for the synthesis of large thinned planar arrays have been published. Three of these methods[4]–[6]make use of the density taper approach that employs a reference or model amplitude taper to synthesize thinned planar circular arrays.The purpose of the model amplitude taper is to derive a turned ON elementManuscript received May13,2012;revised January01,2013;accepted May 01,2013.Date of publication June07,2013;date of current version April03, 2014.The author,retired,was with the TNO Physics and Electronics Laboratory, 2597AK,The Hague,Netherlands.He is now at2343JH Oegstgeest,The Netherlands(e-mail:willkeizer@).Color versions of one or more of thefigures in this paper are available online at .Digital Object Identifier10.1109/TAP.2013.2267194distribution across the aperture,when illuminated with equal amplitudes,has to produce a pattern comparable to that of the filled array when illuminated by the model taper.The density taper methods[4]and[6]rely on a deterministic technique tofind the turned ON element distribution while method[5] applies a statistical approach.All these three density taper methods are non-iterative procedures.The iterative Fourier transform(IFT)method[7]applies a forward two-dimen-sional fast Fourier transform(FFT)on the array factor of the thinned array after it’s matching to the user peak sidelobe level (SLL)requirement in order to get the element excitations. Subsequently,the turned ON element positions are found by selecting the largest element amplitudes.Various nature inspired stochastic global optimization methods like ant colony optimization(ACO)[8],the genetic algorithm(GA)[9],[10], and the hybrid approach HSPSO[11]based on the combination of the two-dimensional difference sets of Hadamard type(HS) and particle swarm optimization(PSO)have been applied successfully for the thinning of medium sized planar arrays in order to get low sidelobes.The thinning method[12]is an optimization technique based on a multistage decision(MSD) procedure and is only applicable to circular arrays using an equilateral triangular element grid.None of these techniques, except the IFT and ACO method,allow easily setting of the number of turned ON elements[8].This paper presents two new methods both derived from the taper density approach proposed by Willey[4]except for the way the positions of the turned ON elements is determined. Identical to Willey’s approach the two proposed methods di-vide the planar array aperture into a number of annular rings and using a suitable model taper,the number of turned ON elements is calculated for each ring.Thefirst method applies a modified IFT technique to select the optimum positions of the turned ON elements of each ring,the second one distributes these elements in a uniform random way over each ring.While recent global optimization methods[8]–[12]were re-stricted to medium sized planar arrays with less than8000el-ements in total andfill factors(ratio of the number of turned ON elements versus its total number)greater than40%,the two new methods can deal with arrays of any size,anyfill factor and any aperture shape.A wide range of numerical examples cov-ering very small arrays,144element positions,up to extremely large ones,879696element positions,will demonstrate the ef-fectiveness of two new edfill factors ranged from 5.2%to63.4%.Both circular and square arrays will be consid-ered.Where possible,the results of both new methods will be compared with those of published theoretical thinned array de-signs described in[6]–[12].0018-926X©2013IEEE.Personal use is permitted,but republication/redistribution requires IEEE permission.See /publications_standards/publications/rights/index.html for more information.II.T WO N EW M ETHODS U SING D ENSITY T APERING Using the Willey density taper approach,the circular array aperture is divided in annular rings of equal width.Next, the normalized array illumination function is summed over all element positions of each annular ring.Turned ON elements are then assigned to each annular ring and their number per ring follows from(1) where is the maximum value ofEquation(1)is an attempt to approximate the desired weighting.The total number of turned ON elements with the aperture divided in rings is the equal to. If is the total number of element positions across the aperture then the elementfill factor accounting for the degree offilling,follows from(2) In this case,one talks about natural thinning and represents the naturalfill factor that is fully governed by the selected model taper.In general,system requirements dictate afill factor that is different from the one given by(2).Suchfill factor can be obtained by multiplying in the right-hand side of(2)with a scale factor,a positive constant value.Thefill factor adapted to the system requirements using the scale factor is given by(3) For a value not equal to one,the number of turned ON ele-ments in ring is equal to and the total number of turned ON elements of the whole array follows from(4) All presented thinned array synthesis results are based on freely chosenfill factors using(2)and(3)to calculate the number of turned ON elements per ring.With Willey’s approach the width of the annular rings is equal to the element spacing of the square element grid.For the two new methods the ring width depends on the aperture radius and the selected number of rings by the user.The two new methods to be described differ only in the way they compute the locations of the turned ON elements for each ring and also differ from Willey’s deterministic procedure. A.Formulation of the IFT Density Taper MethodThefirst new method and the main subject of this paper,con-cerns a modified version of the IFT method[7]for selecting optimum positions of the turned ON elements of each ring.The original IFT method was initially applied for the synthesis of thinned large planar array without applying a model taper and the associated division of the aperture in annular rings.The IFT method is an iterative technique that links through forward and inverse FFTs the element excitations with the array factor AF and vice versa.It uses the property that the array factor of a planar array with elements arranged in a square grid at distance along columns and rows,can be written as a discrete Fourier transform(DFT)(5) where is the complex excitation of element,the wavenumber,the wavelength at which the array op-erates,and are the di-rection cosines,and are the angularfield positions.Since (5)represents afinite double Fourier series that relates the ele-ment excitations of the planar array to its AF through a discrete inverse2-D Fourier transform,a discrete forward2-D Fourier transform applied on AF will yield the element excita-tions.The original IFT method gets the positions of the turned ON elements by selecting the largest element amplitudes for the whole aperture after applying a forward2-D FFT on the array factor of the thinned array when matched to the user peak sidelobe level(SLL)requirements.The new,modified IFT method called the IFT density taper(IFTDT)technique follows basically the same route butfinds the turned ON elements of each ring by selecting the largest amplitudes of ring.Implementation of the IFTDT technique using the model taper,proceeds for a single run as follows: 1)Start the synthesis using a random initialization withequal0/1probability for the element excitations. 2)Compute AF from using a point2-Dinverse FFT withby applying zero padding,with defined as the smallest power of two that is greater or equal to thevalue of.3)Adapt AF to the prescribed peak SLL requirements.4)Compute from the adapted AF using apoint2-D forward FFT.5)Truncate from samples to samplesrelated to the columns and rows of the aperture. 6)In case of an aperture with a non-rectangular shape,makethe samples of located outside the aperture equal to zero.7)Set for the samples of belonging to ring theones with the largest amplitudes equal to one(ON) and make the other samples of belonging to ringequal to zero(OFF).Repeat this for all rings.8)Compare the ON element distribution of the presentiteration with that of the previous one,when differentproceed with step9),otherwise exit the iteration loop,stop the synthesis and go to step10.9)Store the newly obtained turned ON element distributionfollowed by a new iteration by repeating steps2)through9)until the peak SLL requirements are met or themaximum number of iterations is reached.10)Select from the stored element distributions of the turnedON elements the one with the lowest maximum SLL.KEIZER:SYNTHESIS OF THINNED PLANAR CIRCULAR AND SQUARE ARRAYS USING DENSITY TAPERING1557Since the IFTDT method starts the synthesis with equal prob-ability0/1for all element excitations,each new syn-thesis run will reveal a different candidate thinned element dis-tribution.With a sufficient number of runs,one is able to get the global optimum thinned element distribution.It turned out that 50runs were adequate to meet this goal.Like the IFT method, the IFTDT method is used as a multi-run approach.The use of a model taper along with a set of annular rings provides the possibility to control the distribution of the turned ON elements across the aperture.However,for very large arrays with diameters andfill factors20%adverse clustering of turned ON elements in one or two quadrants can arise with the IFTDT method causing distortion of the main beam shape.To avoid this,each annular ring is separated into four equal sized subsectors,one per quadrant.The number of turned ON ele-ments of a subsector is quarter that of the associated ring.By using subsectors,step7of the IFTDT method is applied to each of the four sub-sectors of ring in order to select turned ON samples per subsector.B.Random Density Taper MethodThe second method gets suitable turned ON element locations by dispersing these elements in a uniform random way over each ring.The uniform nature of the random distribution guarantees quasi-circular symmetry of the turned ON elements and there-fore also that of the far-field.Moreover,this type of distribution avoids the occurrence of grating lobes.The random positioning of turned ON elements from in total elements across a ring sector can easily be arranged with the MATLAB func-tion when programming of the new random den-sity taper(RDT)method is done in MATLAB.The functionreturns a row vector containing unique integers selected randomly from1:N.The advantage of the RDT method,a non-iterative one,is its high computational speed.This method is therefore very suited to evaluate the performance of suitable model tapers regarding maximum SLL,3dB beamwidth and directivity.After the most suitable model taper is found satisfying these three design re-quirements,the optimum location of the turned ON elements is subsequently determined with the IFTDT method.All model tapers used by the IFTDT method for the thinned array designs described in this paper were selected by the RDT method.Like the IFTDT method,the RDT one is used as a multi-run approach due to the random determination of the location of the turned ON elements.C.Other Array Shapes Than CircularThinning of planar arrays of any shape is feasible with both the IFTDT and RDT method.Rectangular planar arrays can be thinned by applying rectangular instead of annular rings.Ellip-tical rings will be used for apertures having an elliptical aperture shape.D.Mutual CouplingThe mutual coupling behavior of a thinned array with all turned OFF elements connected to a matched load,will not be different from that of itsfilled counterpart,[13,pp.13.24].The embedded element pattern of the thinned array is therefore iden-tical to that of thefilled array with all elements excited.Besides the embedded element pattern is independent of the array ex-citation,whether the array is thinned,tapered or uniformly il-luminated.The concept of an embedded element pattern that applies alike to every element is only valid when an isolating, matched feed is used and edge effects can be ignored.For nearly all thinned planar arrays edges effects are hardly relevant since their apertures near the periphery are barely populated by turned ON elements.This circumstance ensures there is no need to take mutual coupling into account during the synthesis of a thinned array.Such a synthesis can therefore be performed by consid-ering only the array factor,a conclusion supported by electro-magneticfield simulations carried out on a thinned200-element linear array[10].E.Beam Scanning and Invisible SpaceThe evaluation of the performance of the IFTDT and RDT method was carried out with arrays having mainly the elements arranged in a square grid spaced apart.Due to this element spacing,the array factors of these arrays cover in space the area defined by and occu-pies therefore a substantial part of invisible space defined by.When scanning the main beam or raising the operating frequency,sidelobesfirst located in invisible space will enter visible space and can therefore degrade the maximum SLL.To prevent this,all sidelobes both in vis-ible and invisible space have to obey the same maximum SLL requirement during the thinned array synthesis.All thinned array designs achieved with the IFTDT and the RDT method meet this SLL condition.III.C OMPUTATIONAL D ETAILSThe obtained IFTDT results were for each design derived from50independent runs.Since the number of iterations per run varied between5and50dependent on the array design,the total number of the far-field calculations ranged therefore be-tween250and2500evaluations for the IFTDT thinned array designs.While the number of evaluations for the RDT method is equal to the number of independent runs this method is there-fore a factor5to50faster than the IFTDT method for the same number of runs.In[7],it was reported that for the IFT method the peak SLL requirement used during the thinned array synthesis had to be about5–7dB lower than the actual optimum value obtained from the synthesis.The same condition applies to the IFTDT method.Since the RDT method predicts quite well within a few dBs the maximum SLL value obtainable with the IFTDT method,finding the exact peak SLL requirement is therefore not a time consuming ually,a few runs with the IFTDT method suffice to identify the right SLL requirement.To assure a good numerical accuracy all here presented max-imum SLL results realized with the IFTDT or RDT method were finally derived from array factors computed with a resolution where.IV.T HINNED A RRAY S YNTHESIS R ESULTSThe effectiveness the two methods is demonstrated by a number of representative planar thinned array designs that1558IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,VOL.62,NO.4,APRIL2014puted far-field pattern of the thinned circular array with a40%fill factor for equal amplitude illumination of the turned ON elements.(a)Prin-cipal u-cut.(b)Main and peak sidelobe distribution of the far-field in visible u-v space.comprise very small array sizes,less than200element posi-tions up to extremely large ones,with almost900000element positions and havingfill factors from5.2%to63.35%.Circular and square arrays were considered.In all cases the elements had an isotropic embedded element pattern which also applies to all published designs mentioned in this paper.The results obtained with the IFTDT method are compared with those of the RDT method and with previous published data of compa-rable theoretical designs.Far-field properties of most frequent concern to the array designer are the array directivity,peak SLL and3-dB beamwidth.The comparison of the thinned array synthesis capabilities of the IFTDT and RDT method is based on these three properties.All designs realized by the IFTDT and RDT methods were obtained with50independent runs. A.Thinned Planar Circular ArraysFor nearly all thinned arrays presented in this paper,the ele-ments are arranged in a square lattice apart.Fig.1(a)shows the-cut of the far-field of a circular planar array with diameter thinned by the IFTDT method.The40%fill factor results in3116turned ON elements.The applied model taper was a circular Taylor distribution with dB and (number of equal amplitude sidelobes adjacent to the main beam)[14].The synthesis was performed with the aperture divided into12annular sectors all split up in four subsectors. Fig.1(b)shows the corresponding peak sidelobe distribution of the far-field produced by the design of Fig.1(a).The result of Fig.1is the outcome of50independent synthesis runs and represents the run with the lowest peak SLL equal to33dB.Fig.2.Convergence curves of the peak SLL during the IFTDT synthesis of the thinned array of Fig.1for runs#22and#2using50iterations.Fig.3.Histograms of the maximum SLLs of the far-field patterns produced by the50independent runs for the circular array with a40%fill factor using both the IFTDT and RDT method.(a)IFTDT method.(b)RDT method. Fig.2shows the convergence rate of the maximum SLL for the best and worst runs out of the50independent runs when al-lowing50iterations per run.The performance of the50runs with respect to their peak SLL values is demonstrated by the peak SLL distribution of Fig.3(a).One can see that38runs have a peak SLL value between32and33dB and11runs a peak SLL between31and32dB.The two SLL values in the top corner at the left of Fig.3(a)represent the peak SLL value of the best run with,lowest peak SLL value,and that of the worst run having the highest peak SLL.Fig.3(b)shows the histogram of the peak SLL values of50independent runs when the thinning of the same array was arranged with the RDT method using the same model taper and number of rings as for the IFTDT design of Fig.1.The best RDT synthesis run features a peak SLL of 29.7dB,which is1.5dB higher than that of the worst run of the IFTDT design.Fig.4depicts the u-cut of the far-field result obtained with the IFTDT method applied on a circular array with a diameter for a25%fill factor.As model taper a circular Taylor distribution with dB and was used jointlyKEIZER:SYNTHESIS OF THINNED PLANAR CIRCULAR AND SQUARE ARRAYS USING DENSITY TAPERING1559puted far-field pattern of the thinned circular array with a 25%fill factor for equal amplitude illumination of the turned ON elements.(a)Principal u-cut.(b)Main and peak sidelobe distribution of whole far-field in visible u-vspaceFig.5.Histogram of the peak SLLs of the far-field patterns produced by the50independent runs for thecircular array with a 25%fill factor.with 42ring sectors all split up in four subsectors.This thinnedarray design features a 46.3dB maximum SLL.The maximum SLL score of the 50independent runs to get this design is shown in Fig.5.The maximum SLL values of these 50runs are within 2.6dB equal to each other.Numerous circular arrays were thinned using both the IFTDT and RDT method.Table I lists a selection of the results obtained with both methods comprising maximum SLL,directivity and 3-dB beamwidth,along with the used model tapers,a circular Taylor distribution,and the number of ring sectors.Like the IFTDT results,the listed RDT ones represent designs with the lowest peak SLL results out of 50independent runs.As one can notice,the RDT results have always a higher maximum SLL than comparable IFTDT results.In nearly all cases the dif-ference in maximum SLL of the two methods varies between 1.2dB and 3.8dB.On the other hand,comparable designs fea-ture almost identical directivities and 3-dB beamwidths.The re-sults of Table I indicate that for arrays with the same diameter their 3dB beamwidth is mainly determined by the SLL value of the used Taylor model taper and not by the fill factor.See the 3-dB beamwidth results of the arrays with diameters 50–.Remarkable is that large arrays with very small fill factors can still have very low sidelobes;the diameter array with a 5.2%fill factor features a 35.9dB maximum SLL.Only for this particular design the IFTDT and RDT methods perform equally.The diameter design,the only one withelement spacing,is with respect to its diameter and number of turned ON elements representative for the thinned array antenna of the SBX radar [3].By comparing the results for circular thinned array designs obtained with the IFTDT method,Table I,and IFT designs pub-lished in [7]it can be concluded that nearly all IFT designs have larger maximum SLL values than those of comparable IFTDT rgest differences are noted for arrays with diame-ters .The array with a diameter and 40%fill factor synthesized with the IFT method features a maximum SLL of 32.7dB and the IFTDT design 37.7dB.Barely any differ-ence is noted between the 3-dB beamwidth and directivity of comparable designs realized by both methods.For thinned ar-rays with diameters ,the IFTDT method outperforms fully the IFT one with respect to maximum SLL.Other relevant numerical results pertaining to large thinned circular arrays are presented in [6]and were obtained with a de-terministic density taper (DDT)method;a non-iterative single run approach.This method places the turned ON elements in non-uniformly spaced concentric rings using a square element grid with spacing.Table II gives the comparison between the results of [6]and those obtained with the IFTDT method for comparable thinned circular array con figurations.One can see that the maximum SLL values of the DDT designs exceed those of the IFTDT designs with at least 3.5dB while the differences in directivity and 3-dB beamwidth are marginal.If the reported SLL results obtained with the DDT method also pertain to in-visible u-v space is not known.Table III presents a limited comparison of thinned array synthesis results obtained with the MSD method [12],the GA based method [10]and those realized with the two new pro-posed methods.Two circular arrays were considered,one with a diameter of and 53.65%fill factor [12]the other array has a diameter of and 64.24%fill factor [10].The MSD design has an equilateral triangular element grid while the GA design uses a square grid with spacing.No information was provided for the MSD and GA designs about their 3-dB beamwidth and directivity.The GA design was obtained by global optimization of a random initialization,the MSD design by local optimization using as input a statistical turned ON element distribution derived from a density taper.Table III illustrates that the IFTDT designs outperform re-garding maximum SLL the thinned array designs realized by the MSD and GA methods,with 1.8dB for the MSD design and 1.5dB for the GA design,while the RDT designs show compa-rable performance.The maximum SLL result for the GA design pertains only to visible u-v space while the maximum SLL re-sult for the MSD design includes the whole area of AF in u-v1560IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,VOL.62,NO.4,APRIL 2014TABLE IC OMPARISON OF THE P ERFORMANCE OF T HINNED C IRCULAR A RRAYS S YNTHESIZED W ITH THE IFTDT AND RDT METHODSTABLE IIC OMPARISON OF THE P ERFORMANCE OF T HINNED C IRCULAR A RRAYSS YNTHESIZED W ITH THE IFTDT AND DDT [6]METHODSspace and therefore also its part located in invisible space;see Fig.3of [12].As described in [15],the IFT method and therefore also the IFTDT method is able to handle triangular element grids.The associated distribution of the turned ON elements across the aperture is depicted in Fig.6.The strong concentration of these elements at the array border is due to the severe reverse ta-pering near the array periphery of the used model taper,a cir-cular Taylor distribution with dB and .The large number of ring sectors favors too the high concentration of turned ON elements at the aperture border and ensures that the 3-dB beamwidth of the design of Fig.6is only a factor 1.14larger than that of the same array fully filled and uniformly illuminated.This example illustrates that by selecting an appropriate model taper it is possible to synthesize a thinnedarray that combines a low maximum SLL with an optimum 3-dB beamwidth,a unique feature not provided by the IFT method [7]and the global optimization methods [8]–[12].B.Thinned Planar Square ArraysTo investigate the capabilities of the IFTDT and RDT method for thinning square arrays,both methods were used on the same small arrays with high fill factors 40%for which theoretical SLL results have been reported in [8]and [11].Table IV present the results obtained with the IFTDT and RDT thinning methods applied to several of these arrays all featuring square ele-ment grid spacing.The array size is indicated by the numbers of rows and columns and listed in the first column of Table IV that provides also information on the used model tapers,linear Taylors,and used number of ring sectors.The maximum SLL results obtained with the IFTDT method for the square arrays of Table IV outperform those of the RDT ones with values ranging between 1.7dB and 4.2dB.Table V compares the maximum SLL results obtained with the IFTDT method for the square ar-rays listed in Table IV with those obtained with the ACO method [8]and the HSPSO approach [11].The HSPSO and ACO de-signs were all obtained by local optimization using a determin-istic element distribution as input in order to speed up conver-gence.The maximum SLL results for the two ACO designs in Table V are estimations derived from Fig.1in [8].The compar-ison in Table V is restricted to maximum SLL values since in [8]and [11]no information was provided on the 3-dB beamwidth and directivity of their thinned array designs.Fig.7shows the normalized far-field of the thinned square array obtained with IFTDT method comprising 24rows and 24columns and featuring a 44%fill factor,same con figuration as。