S S symmetry

Article An Efficient Approach for Sidelobe Level Reduction Based on Recursive Sequential Damping

Yasser Albagory * and Fahad Alraddady

Department of Computer Engineering, College of Computers and Information Technology, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia; [email protected] * Correspondence: [email protected]

Abstract: Recently, antenna array radiation pattern synthesis and adaptation has become an essential requirement for most wireless communication systems. Therefore, this paper proposes a new recur- sive sidelobe level (SLL) reduction algorithm using a sidelobe sequential damping (SSD) approach based on pattern subtraction, where the sidelobes are sequentially reduced to the optimum required levels with near-symmetrical distribution. The proposed SSD algorithm is demonstrated, and its performance is analyzed, including SLL reduction and convergence behavior, mainlobe scanning, processing speed, and performance under mutual coupling effects for uniform linear and planar arrays. In addition, the SSD performance is compared with both conventional tapering windows and optimization techniques, where the simulation results show that the proposed SSD approach has superior maximum and average SLL performances and lower processing speeds. In addition, the SSD is found to have a constant SLL convergence profile that is independent on the array size, working effectively on any uniform array geometry with interelement spacing less than one wavelength, and deep SLL levels of less than −70 dB can be achieved relative to the mainlobe level, especially for symmetrical arrays.




 Keywords: antenna arrays; sidelobe level reduction; tapered beamforming; optimization techniques

Citation: Albagory, Y.; Alraddady, F. An Efficient Approach for Sidelobe Level Reduction Based on Recursive 1. Introduction Sequential Damping. Symmetry 2021, 13, 480. https://doi.org/10.3390/ 1.1. Background and Motivation sym13030480 Antenna arrays and beamforming systems have become essential parts for most recent wireless communication systems and play important roles in the provision of high capaci- Received: 28 February 2021 ties and data rates. Signals can be received more efficiently, and the system performance Accepted: 12 March 2021 can be enhanced greatly by using beamforming techniques with antenna arrays such as in Published: 15 March 2021 multiple-input multiple-out (MIMO) mobile systems, radar, sonar, satellite, and many other recent applications, such as wireless sensors and medical networks [1–8]. To improve the Publisher’s Note: MDPI stays neutral system capacity and provide higher data rates for users, it is necessary to utilize millimeter with regard to jurisdictional claims in wave frequencies such as in the fifth-generation (5G) mobile systems and networks beyond published maps and institutional affil- that, where larger bandwidths are available. However, the millimeter wave frequencies iations. are characterized by high atmospheric propagation losses and require being boosted by efficient antenna array systems [9]. One of the most important capabilities of antenna arrays is the capability to concentrate the radiation pattern in the desired directions and control the level of unwanted radiations such as the sidelobes, which usually collect the Copyright: © 2021 by the authors. unwanted interfering signals. Therefore, providing a high-gain mainlobe while reduc- Licensee MDPI, Basel, Switzerland. ing the sidelobe levels (SLLs) is one of the most important requirements for achieving a This article is an open access article higher signal-to-interference ratio and spectral efficiency. There are many SLL reduction distributed under the terms and techniques in the literature, including simple constant tapering windows [10–15], opti- conditions of the Creative Commons mized tapering windows [16–18], and many other efficient optimization techniques [19–29]. Attribution (CC BY) license (https:// These techniques differ in complexity, ranging from simple straightforward feeding as in creativecommons.org/licenses/by/ the tapered beamforming techniques to more sophisticated optimization techniques that 4.0/).

Symmetry 2021, 13, 480. https://doi.org/10.3390/sym13030480 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 480 2 of 22

optimize the array parameters to achieve the required optimum levels. Tapering window functions can provide very deep sidelobe levels but suffer from the lack of adaptability and may result in very wide beamwidths. Optimized tapering windows, such as Dolph- Chebyshev, Taylor, and Kaiser windows [16–18,30], are efficient at reducing SLLs, especially for small-sized arrays. The Dolph-Chebyshev window provides the smallest beamwidth at a certain SLL among other tapered window techniques [30]. On the other hand, swarm optimization techniques are effective for designing antenna arrays of any size [29] without the constraints found in [16,17]. For nonuniform linear antenna arrays, genetic algorithms (GAs) [28] can be used for SLL reduction, while particle swarm optimization (PSO) [20] is used for optimizing circular arrays and concentric ring arrays. Other revolutionary techniques optimize the element currents as well as the element separation distances, such as the firefly algorithm (FA) [22]. Additionally, some techniques focus on suppressing the highest SLL as the main objective, such as the flower pollination algorithm (FPA) [19], cuckoo search algorithm (CS) [23], invasive weed optimization (IWO), biogeography-based optimization (BBO), bat algorithm (BA), whale optimization algorithm (WOA), atom search optimization (ASO) [24–27], and many other algorithms. Most of these techniques are used for optimizing the radiation patterns of linear and circular antenna arrays, such as FPA, BBO, and IWO, while CS optimizes large planar arrays [29]. For concentric ring arrays, PSO, CS, and FA were examined and proved their efficient SLL reduction. Many comparisons have been conducted between these optimization algorithms, such as in [27–29], where the ASO, WOA, FPA, and BA provided much lower SLLs for the same array size compared with the BBO, PSO, IWO, CS, and FA techniques where, for a 16 element uniform linear array (ULA), the lowest SLL of −41 dB relative to the mainlobe level was provided by the ASO algorithm [27], while the highest SLL of −17 dB was obtained using PSO [27,29]. In addition, chicken swarm optimization (CSO) [31] has been developed for simple opti- mization applications, including SLL reduction. However, it is not suitable for complex optimization problems due to its exploitation ability. The grey wolf optimizer (GWO) [32] is another optimization technique based on a nature-inspired metaheuristic algorithm. In the GWO, the social hierarchy and hunting behavior of gray wolves inspired the optimization process, which is applied in solving many optimization problems, including antenna array pattern synthesis. However, the work done in [32] to improve the SLL level optimized the element locations, which limited the flexibility of application in different communication scenarios. Other optimization techniques, such as the differential search algorithm [33], Taguchi method [34], and backtracking search optimization [35], have been applied to linear array optimization. However, providing a deep SLL requires a larger number of iterations and hence a longer processing time to achieve the optimum solution, which is not suitable for real-time communication scenarios. Additionally, some techniques provide efficient solutions for SLL reduction in specific array geometries while failing in others. For example, tapering window techniques are straightforward for all ULAs, while they can be applied without an optimum solution for planar or concentric ring arrays such as in [13], where the design of the Dolph-Chebyshev window to provide a −80 dB SLL for a linear array provided −40 dB in a uniform concentric ring array. In addition, these fast conventional tapering windows produce constant SLL profiles for a fixed array size and suffer from the lack of adaptation.

1.2. Paper Contribution To provide a general and array configuration-independent SLL reduction technique with a faster convergence time and adaptive beampattern generation, this paper proposes a recursive, adaptable SLL reduction and control technique by sidelobe sequential damping (SSD), which is performed successively on the local maximum SLL in the radiation pattern. The proposed algorithm starts by finding the maximum SLL, then cancelling it, determining the new resulting weighted vector, proceeding to the resulting next maximum SLL, and so on. The process is continued until the acceptable SLL is reached or the required shape of the radiation pattern is achieved. The proposed technique has adaptation and SLL Symmetry 2021, 13, x FOR PEER REVIEW 3 of 22

required shape of the radiation pattern is achieved. The proposed technique has adapta- tion and SLL control capabilities, as in most optimization techniques, but with a faster convergence time, an ability to work on any array configuration with interelement spacing less than one wavelength in distance, and being able to effectively work under mutual coupling. An important feature of the proposed SSD technique is that the number of iter- ations to find the optimum weights is independent on the array size, and the processing time can be sped up by finding the best angular sampling step sizes. The comparison with various conventional tapering windows shows that the proposed SSD algorithm provides much deeper SLL levels at the same beamwidth and provides a lower average SLL level

Symmetry 2021, 13, 480 compared with the Dolph-Chebyshev technique, while the comparison with optimization 3 of 22 techniques shows the capability of the SSD algorithm to provide much deeper SLLs with lower average values at a slightly wider beamwidth.

1.3. Papercontrol Organization capabilities, as in most optimization techniques, but with a faster convergence Sectiontime, 2 models an ability the to proposed work on any technique, array configuration with its performance with interelement analyzed spacing in Section less than one 3, while inwavelength Section 4, a in performance distance, and comparison being able is to performed effectively between work under the proposed mutual coupling.SSD An techniqueimportant and various feature windows of the proposedand optimization SSD technique techniques. is that In the Section number 5, ofthe iterations impact of to find the mutual couplingoptimum on weights the SSD is algorithm independent is disc onussed, the array while size, Section and the 6 processingdiscusses the time applica- can be sped up tion of SSDby on finding two-dimensional the best angular planar sampling arrays step with sizes. the presence The comparison of mutual with coupling. various Sec- conventional tion 7 discussestapering the windows operational shows constraints that the and proposed limitations SSD algorithmof the proposed provides SSD, much and deeperfi- SLL levels at the same beamwidth and provides a lower average SLL level compared with the nally, Section 8 concludes the paper. Dolph-Chebyshev technique, while the comparison with optimization techniques shows 2. Modellingthe capabilityof the Proposed of the SSD algorithmTechnique to provide much deeper SLLs with lower average values at a slightly wider beamwidth. Consider the geometry shown in Figure 1a,b of uniform arrays formed by isotropic antenna elements,1.3. Paper with Organization uniform element separation of less than a wavelength in distance and where it isSection essential2 models to form the only proposed one mainlobe technique, in the with radiation its performance pattern. analyzed in Section3 , Let uswhile start in with Section the 4ULA, a performance configuration comparison shown in Figure is performed 1a and assume between that the the proposed an- SSD tenna elementstechnique are weighted and various by a windows vector 𝑊 and, which optimization is initially techniques.set to have unity In Section amplitude5, the impact coefficientsof with mutual linear coupling phase shifts on the to SSDsteer algorithmthe mainlobe is discussed,toward the whiledesired Section direction6 discusses 𝜃, the as in the delay-and-sumapplication of SSD beamformer. on two-dimensional planar arrays with the presence of mutual coupling. In general,Section the7 discusses array factor, the operational𝐴(𝜃), can be constraints written as andfollows: limitations of the proposed SSD, and finally, Section8 concludes the𝐴 paper.(𝜃) =𝑊𝑆(𝜃) (1) where 𝑆(𝜃)2. Modelling is the array of steering the Proposed vector SSDand 𝐻 Technique is the Hermitian operator. For delay-and- sum beamforming,Consider we set the 𝑊=𝑆(𝜃 geometry), shown and the in array Figure factor1a,b is of given uniform by arrays formed by isotropic antenna elements, with uniform element separation of less than a wavelength in distance 𝐴(𝜃) =𝑆 (𝜃)𝑆(𝜃) (2) and where it is essential to form only one mainlobe in the radiation pattern.

(a) (b)

Figure 1. UniformFigure 1. arrayUniform configurations array configurations for (a) a forone-dimensional (a) a one-dimensional array and array (b and) a two-dimensional (b) a two-dimensional array. array. Let us start with the ULA configuration shown in Figure1a and assume that the antenna elements are weighted by a vector W, which is initially set to have unity amplitude coefficients with linear phase shifts to steer the mainlobe toward the desired direction θo, as in the delay-and-sum beamformer. In general, the array factor, A(θ), can be written as follows:

A(θ) = WHS(θ) (1)

where S(θ) is the array steering vector and H is the Hermitian operator. For delay-and-sum beamforming, we set W = S(θo), and the array factor is given by

H A(θ) = S (θo)S(θ) (2) Symmetry 2021, 13, 480 4 of 22

The sidelobe directions, θi can be obtained by finding the local maxima of A(θ), at which the partial derivative with θ is equal to zero as follows:

∂A(θ) ∂A(θ) | = = 0, subject to | = + < 0, i = 0, 1, 2, . . . , I (3) ∂θ θ θi ∂θ θ θi δθ

where δθ is an infinitesimal angular increment. The set of sidelobe directions θi is obtained by arranging the corresponding array factor values at θ = θi in a descending order and excluding the first element in this set, which corresponds to the mainlobe as follows:   A(θo)  A(θ1)     :  A =   (4) Peaks  A(θ )   i   :  A(θI )

where APeaks contains the radiation peaks, which include the mainlobe and all sidelobes, and the sidelobe levels vector ASide is therefore given by   A(θ1)  A(θ2)     :  A =   (5) Side  A(θ )   i   :  A(θI )

and the corresponding sidelobe directions θSide are given by   θ1  θ2     :  θ =   (6) Side  θ   i   :  θI

with the maximum sidelobe gain A(θ1) and direction θ1. For some uniform array structures, the sidelobe peaks may have an infinite or multiple number of directions with the same level, and in this case, only one peak with any of the directions is selected and inserted into Equation (4) while the others are neglected. The technique in the first iteration will provide perturbation for the array pattern, which eases subsequent sidelobe damping. To cancel the maximum sidelobe, a secondary mainlobe is formed with the same level and phase as A(θ1) and the same direction as θ1. Then, the array factor of this secondary mainlobe is subtracted from the original array factor as follows:

 H H 1 ∗ Ar(θ) = S (θo)S(θ) − A (θ1)S(θ1) S(θ) (7) |A(θ)|max or  H 1 ∗ Ar(θ) = S(θo) − A (θ1)S(θ1) S(θ) (8) |A(θ)|max

where Ar(θ) is the damped sidelobe array factor with only one sidelobe suppressed and |A(θ)|max is the maximum value of the array factor. The operation can be repeated to sequentially suppress the sidelobes, where in each cycle, the highest sidelobe is canceled out. After the sidelobe suppression, the overall Symmetry 2021, 13, 480 5 of 22

gain is affected, and the previously suppressed sidelobes may appear diminished, so the actual effect is the damping of the sidelobes rather than nulling them out. If the number Symmetry 2021, 13, x FOR PEER REVIEW 5 of 22 of damping cycles is K, then the array factor at the kth damping cycle can be determined according to the following recursive equation:

1 ∗  k−1  k−1 Ak(θ) = Ak−1(θ) − Ak−1 θ1 S θ1 , k = 1, 2, . . . , K (9) |A∗k−1(θ)| 𝐴(𝜃) = 𝐴(𝜃) − 𝐴(𝜃 max)𝑆(𝜃 ),𝑘 =1,2,…,𝐾 (9) |𝐴(𝜃)| H where A0(θ) = S (θo)S(θ) is the array gain, corresponding to the uniform feeding case where 𝐴(𝜃) =𝑆 (𝜃)𝑆(𝜃) is the array gain, 0corresponding to the uniform feeding case with the highest sidelobe direction at θo = θ1. with the highest sidelobe direction at 𝜃 =𝜃 . The SSD weighting vector, WSSD(θo), after K damping cycles is given by The SSD weighting vector, 𝑊(𝜃), after 𝐾 damping cycles is given by K  1     ( ) = ( ) − 1 ∗ k k W𝑊SSD (θ𝜃o ) =𝑆S (θ𝜃o ) −∑ A𝐴k∗ (𝜃θ1)𝑆(𝜃S θ1) (10) |Ak(θ)| (10) k=1 |𝐴(𝜃)|max TheThe proposed proposed technique technique can can be be extended extended for for two-dimensional two-dimensional arrays arrays of of a asize size NMNM, , shownshown in in Figure Figure 1b1b as as follows: follows: K  11     = − ∗ k k k k WSSD𝑊(θ(o𝜃,O,∅o)) =𝑆S(θ(o𝜃,O,∅o)) −∑ A𝐴k (𝜃θ1,,∅O1)𝑆(𝜃S θ1,∅, O )1 (11)(11) ||A𝐴k((𝜃,θ, O ∅))|max k=1

wherewhere ((𝜃θo,∅, O)o ) isis the the mainlobe directiondirection andandK Kis is the the number number of of the the damping damping cycles cycles for for SLL SLLreduction reduction in thein the two two dimensions dimensions(θ, O(𝜃,). ∅). SSDSSD can can be beperformed performed to achieve to achieve a certain a certain SLL by SLL starting by starting with a withcertain a certainsmaller smallervalue ofvalue 𝐾 and of updatingK and updating it until the it until required the required SLL is achieved. SLL is achieved. The flowchart The flowchart in Figure in2 demon- Figure2 stratesdemonstrates the iterative the iterativeSSD, where SSD, the where initial the conditions initial conditions are first aredefined first defined and then and the then side- the lobessidelobes are sequentially are sequentially reduced reduced in a one-by-one in a one-by-one fashion fashion to achieve to achieve the required the required SLL. SLL.

FigureFigure 2. 2.FlowchartFlowchart representation representation for forthe the recursive recursive sidelobe sidelobe sequential sequential damping damping (SSD) (SSD) sidelobe sidelobe levellevel (SLL) (SLL) reduction reduction algorithm. algorithm.

The damping process is examined as shown in Figure 3 for several damping cycles, using the ULA of 20 elements spaced at a 𝜆/2 distance. In Figure 3a, the two highest side- lobes are damped using two damping cycles, and the resulting array weighting profile is shown in Figure 3b. The operation starts to null the highest sidelobe on the left of the

Symmetry 2021, 13, 480 6 of 22

Symmetry 2021, 13, x FOR PEER REVIEW 6 of 22 The damping process is examined as shown in Figure3 for several damping cycles, using the ULA of 20 elements spaced at a λ/2 distance. In Figure3a, the two highest sidelobes are damped using two damping cycles, and the resulting array weighting profile mainlobe,is shown inthen Figure null3 theb. Theother operation sidelobe startson the to right null, and the highestthe overall sidelobe result onis mainly the left nulling of the thatmainlobe, on the then right null while the damping other sidelobe the previousl on the right,y canceled and the out overall one on result the left. is mainly Therefore, nulling it isthat expected on the rightafter a while large damping number of the damping previously cycles canceled that many out one sidelobes on the left.are damped Therefore, while it is theexpected last one after is canceled a large number out. In Figure of damping 3c, the cycles number that of many damping sidelobes cycles are is increased dampedwhile to 50 andthe lastthe oneSLL isis canceledreduced out.to −36 In dB Figure, and3c, the the final number weighting of damping profile cycles is shown is increased in Figure to 3 50d. Furtherand the increasing SLL is reduced the damping to −36 dB, cycles and will the finalreduce weighting the SLL as profile shown is shownin Figure in Figure3e, where3d. 퐾Further= 10,000 increasing and the the SLL damping is greatly cycles reduced will reduceto −65 the dB. SLL as shown in Figure3e, where K = 10,000 and the SLL is greatly reduced to −65 dB.

(a) (b)

(c) (d)

Figure 3. Cont.

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(e) (f)

FigureFigure 3.3. SLLSLL reductionreduction by SSD at different damping cycles cycles.. ( (aa)) 퐾K = 22 withwith (b)) the weighting profile profile at K퐾 == 2,(, (c) K퐾==5050 withwith ((dd)) the the weighting weighting profileprofile at atK 퐾==50,50 and, and (e ()eK) 퐾= 10,000= 10,000 with with (f) the(f) the weighting weighting profile profile at K at= 퐾 10,000= 10 .,000.

3.3. SSDSSD PerformancePerformance AnalysisAnalysis 3.1.3.1. SSDSSD DampingDamping BehaviorBehavior FigureFigure4 4demonstrates demonstrates the the SLL SLL reduction reduction behavior behavior with with the the number number of damping of damping cycles cy- forcles a for 20 a element 20 element ULA ULA where where there there was wa as great a great and and fast fast reduction reduction in in the the SLL, SLL,especially especially atat the beginning of of the the SSD SSD da dampingmping operation. operation. The The SLL SLL drop droppedped to toless less than than −40−40 dBdB at atvalues values of of퐾K<<100100, while, while at at 퐾K≈≈300300, ,it it drop droppedped to to −−5050 dB,dB, and further increasesincreases in K퐾 hadhad lessless ofof anan effecteffect onon decreasingdecreasing thethe SLL.SLL. TheThe dampingdamping curvecurve almostalmost hadhad aa decayingdecaying exponentialexponential behavior, where increasing K퐾 fromfrom 10001000 toto 10,00010,000 reducedreduced thethe SLLSLL onlyonly byby approximatelyapproximately 10 dB. dB. This This exponential exponential decaying decaying behavior behavior wa wass due due to tothe the splitting splitting of the of thecanceled canceled sidelobe sidelobe peak peak into intotwo tworeduced reduced peaks peaks in every in every damping damping cycle cyclewith some with someinter- interactionaction with withthe neighboring the neighboring peaks peaks,, which which may mayhave haveresult resulteded in slightly in slightly increasing increasing their theirlevels. levels. By proceeding By proceeding with more with damping more damping cycles, cycles, the sidelobes the sidelobes were decreased were decreased in magni- in magnitude and became nearly equal in levels, and the possibility of interaction between tude and became nearly equal in levels, and the possibility of interaction between the sub- the subtracted peak from the overall pattern increased with the other nearest peaks, which tracted peak from the overall pattern increased with the other nearest peaks, which slowed down the SLL reduction process. However, increases in the damping cycles were slowed down the SLL reduction process. However, increases in the damping cycles were always converging to produce lower SLLs, but they required longer processing times due always converging to produce lower SLLs, but they required longer processing times due to the interaction between the newly formed sidelobes and the existing ones. On the other to the interaction between the newly formed sidelobes and the existing ones. On the other hand, the SLL damping profile at different array sizes is shown in Figure5, where the hand, the SLL damping profile at different array sizes is shown in Figure 5, where the decaying exponential convergence behavior of the average SLL values had the same rate or decaying exponential convergence behavior of the average SLL values had the same rate profile. For array sizes larger than 10 elements, the SLL reduction by the recursive SSD had or profile. For array sizes larger than 10 elements, the SLL reduction by the recursive SSD the same convergence profile and almost had the same average values. This interesting had the same convergence profile and almost had the same average values. This interest- feature made the proposed SSD technique independent of the array size, where for any arraying feature size, we ma couldde the obtain proposed the same SSD SLLtechnique at the sameindepe numberndent o off dampingthe array cyclessize, whereK. for any array size, we could obtain the same SLL at the same number of damping cycles 퐾.

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FigureFigure 4. 4. RecursiveRecursive SSD SSD damping damping performance. performance.

FigureFigure 5. 5. VariationVariation ofof SSDSSD dampingdamping withwith thethe numbernumber ofof dampingdamping cyclescycles atat differentdifferent arrayarrayarray sizes.sizes.sizes.

3.2.3.2. SSD SSD Behavior Behavior at at Different Angular Angular Step Step Sizes Sizes ToTo speedspeed upup thethe recursiverecursive SSDSSD algorithm,algorithm, wewe shouldshould findfindfind thethe directionsdirections atat whichwhich thethe radiationradiation peaks exist (θ𝜃i),), andand thisthis includesincludes determiningdetermining the array factor A𝐴(𝜃)(θ) andand thenthen radiation peaks exist (𝜃), and this includes determining the array factor 𝐴(𝜃) and then findingfinding its peaks and their corresponding θ𝜃i,, otherwiseotherwise solvingsolving thethe differentialdifferential equationequation finding its peaks and their corresponding 𝜃, otherwise solving the differential equation in (3), which may be mathematically complicated, especially for large ULA sizes and inin (3),(3), whichwhich maymay bebe mathematicallymathematically compliccomplicated,ated, especiallyespecially forfor largelarge ULAULA sizessizes andand two-two- two-dimensional arrays with a large number of damping cycles. The samples of A(θ) at the dimensionaldimensional arraysarrays withwith aa largelarge numbernumber ofof dampingdamping cycles.cycles. TheThe samplessamples ofof 𝐴(𝜃)𝐴(𝜃) atat thethe angular steps or increments ∆θ affected the accuracy of finding θ and the SLL reduction angularangular stepssteps oror incrementsincrements ∆𝜃∆𝜃 affected affected thethe accuracyaccuracy ofof findingfinding 𝜃𝜃i and and thethe SLLSLL reductionreduction performance. As shown in Figure6, for a ULA of 20 elements, an angular resolution or step performance.performance. AsAs shownshown inin FigureFigure 6,6, forfor aa ULULAA ofof 2020 elements,elements, anan angularangular resolutionresolution oror size greater than 1◦ resulted in some higher spikes in the damping curve, which resulted stepstep sizesize greatergreater thanthan 1°1° resulted resulted inin somesome higherhigher spikesspikes inin thethe dampingdamping curve,curve, whichwhich re-re- from inaccurate directions of the sidelobes, and a retrodictive effect on the SLL reduction sultedsulted fromfrom inaccurateinaccurate directionsdirections ofof thethe sidelosidelobes,bes, andand aa retrodictiveretrodictive effecteffect onon thethe SLLSLL re-re- process occurred, while for the angular resolution ≤1◦, the damping behavior was nominal ductionduction processprocess occurred,occurred, whilewhile forfor thethe angularangular resolutionresolution ≤1°≤1°,, thethe dampingdamping behaviorbehavior and converging. However, the overall damping of the SSD converged in a broad sense, and waswas nominalnominal andand converging.converging. However,However, thethe ovoverallerall dampingdamping ofof thethe SSDSSD convergedconverged inin aa the recursive SSD still worked, but on a rough SLL estimation and in a reduced fashion. In broadbroad sense,sense, andand thethe recursiverecursive SSDSSD stillstill workworked,ed, butbut onon aa roughrough SLLSLL estimationestimation andand inin aa addition, the flowchart presented in Figure2 provides a solution for the cases where large reducedreduced fashion.fashion. InIn addition,addition, thethe flowchartflowchart prpresentedesented inin FigureFigure 22 providesprovides aa solutionsolution forfor angularthe cases step where sizes large were angular used, where step sizes the process were used, ended where at the the accepted process SLL ended regardless at the ac- of thethe roughcases where SLL estimation large angular and reduction.step sizes were used, where the process ended at the ac- ceptedcepted SLLSLL regardlessregardless ofof thethe roughrough SLLSLL estimationestimation andand reduction.reduction.

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FigureFigure 6.6. SSDSSD dampingdamping behaviorbehavior atat variablevariable angularangular step step sizes. sizes.

TheThe processingprocessing timetime variationvariation withwith thethe samplingsampling angular angular step step sizes sizes for forfor 200 200 damping damping cyclescycles atat differentdifferdifferentent arrayarray sizessizes isis shownshown inin FigureFigureFigure7 ,77, where, where it it varied varievariedd with with exponentially exponentially decayingdecaying profiles profileprofiless withwith the the angular angular step step size sizesize and and reduced reducereducedd greatlygreatly with with the the array array size. size. Therefore,Therefore, choosing choosing the the proper proper value value of of the the angular angularangular step step size size is is very very important important to toto achieve achieve a lowera lowelower processingr processing time time and and smoother smoother convergence convergence to to the the desired desired SLL. SLL. s d n secon i me i ng t i ocess r P

FigureFigure 7.7. SSDSSD processingprocessing timetime variationvariation withwith angularangularangular step step sizes sizes at at variable variable array array sizes. sizes.

AA roughrough estimationestimation forfor thethe maximummaximum samplingsamplingsampling angularangular stepstep sizesize couldccouldould bebe usedused asas oneone thirdthird ofof thethe mainlobemainlobe beamwidth,beamwidthbeamwidth,, whichwhich ensuredensureensuredd findingfinding thethe sidelobesidelobe peakspeaks inin thethe providedprovided arrayarray factorfactor samples.samples. ForFor thethe ULA,ULA,ULA, thethe maximummaximum samplingsampling angularangular stepstep sizesize ∆𝜃 ∆∆θ휃max푚푎푥can can be be considered considered as as follows: follows: 29 ∆∆𝜃휃 ≈29,, degrees (12) ∆θ 푚푎푥≈ 푁 , degrees (12)(12) max N where 푁𝑁 is is thethe arrayarray size.size. ThereforeTherefore,, thethe maximummaximum angular step size wawass adapted with wherethe arrayarrayN is sizesize, the, array and this size. empirical Therefore, equation the maximum wawass considered angular step as size thethe wasmaximum adapted applicable with the arrayvalue size, for ∆∆𝜃 and휃,, as this shown empirical in Figure equation 88,, which was consideredhelphelpeded inin finding as the maximumfewerfewer samplessamples applicable ofof 퐴𝐴(𝜃)(휃 value) and ( ) forspeeding∆θ, as shownup the processing in Figure8, time. which The helped curves in findingshownshown in fewer this figure samples correspond of A θ and to 200 speeding damp- upinging the cyclescycles, processing, as previously time. The considered curves shown in Figure in this 7.figure For twotwo-dimensional correspond-dimensional to 200 arrays, damping N wawas cycles,s con- as previously considered in Figure7. For two-dimensional arrays, N was considered as the sidered as the largest number of elements in one of the two directions of the array. Com- largest number of elements in one of the two directions of the array. Compared to Figure7 , pared to Figure 7, the processing time ccouldould be decreased to less than one sixth of its value the processing time could be decreased to less than one sixth of its value at ∆θ = 0.1◦, at ∆∆𝜃휃 = 0 =. 0.1°1°,, esespeciallypecially for large array sizes. especially for large array sizes.

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Figure 8. Maximum angular step size with the number of elements in the array. FigureFigure 8. 8.MaximumMaximum angular angular step step size size with with the the number number of ofelements elements in in the the array. array. 3.3. Impact of SSD on the Beamwidth 3.3.3.3. Impact Impact of ofSSD SSD on on th thee Beamwidth Beamwidth Generally, SLL reduction resulted in an increase in the beamwidth, and so did the Generally,Generally, SLL SLL reduction reduction result resulteded in in an an increase increase in in the the beamwidth beamwidth,, and and so so d didid the the proposed SSD technique, as shown in Figure 9 for a 20 element ULA. During the first 100 proposedproposed SSD SSD technique technique,, as asshow shownn in Figure in Figure 9 for9 for a 20 a element 20 element ULA. ULA. During During the first the 100 first 100damping damping cycles cycles,, the beamwidth the beamwidth increase increasedd rapidly rapidly where where the SLL the decrease SLL decreasedd rapidly, rapidly, while damping cycles, the beamwidth increased rapidly where the SLL decreased rapidly, while whileafter after퐾 = 100K =, the100 ,beamwidth the beamwidth increase increasedd slightly, slightly, where where the variation the variation was was less lessthan than 1° 1for◦ after 퐾 = 100, the beamwidth increased slightly, where the variation was less than 1° for foran anincrease increase in in퐾 Kfromfrom 100 100 to to 1000. 1000. an increase in 퐾 from 100 to 1000.

FigureFigure 9.9. BeamwidthBeamwidth variationvariation withwith thethe numbernumber ofof dampingdamping cyclescycles andand highesthighest SLL.SLL. Figure 9. Beamwidth variation with the number of damping cycles and highest SLL. 3.4.3.4. SSDSSD PerformancePerformance withwith InterelementInterelement SpacingSpacing 3.4. SSD Performance with Interelement Spacing AsAs thethe proposedproposed SSDSSD algorithmalgorithm dependeddepended onon beambeam spacespace processing,processing, inin whichwhich thethe As the proposed SSD algorithm depended on beam space processing, in which the sidelobessidelobes werewere removedremoved sequentially,sequentially, it it would would thereforetherefore be be efficientefficient only only forfor interelementinterelement sidelobes were removed sequentially, it would therefore be efficient only for interelement spacingspacing valuesvalues thatthat werewere lessless thanthan oneone wavelengthwavelength λ휆 duedue to to the the appearance appearance ofof secondarysecondary spacing values that were less than one wavelength 휆 due to the appearance of secondary majormajor lobeslobes inin thethe visiblevisible region.region. TheThe SSDSSD performanceperformance withwith somesome typicaltypical valuesvalues ofof thethe majorinterelement lobes in spacingthe visible distance region. is The shown SSD in performance Figure 10, where with thesome proposed typical values algorithm of the still interelement spacing distance is shown in Figure 10, where the proposed algorithm still interelementworked efficiently spacingat distance a very lowis shown spacing in Figure of λ/4 10as, where well as the a very proposed large spacingalgorithm of still0.9λ . worked efficiently at a very low spacing of 휆/4 as well as a very large spacing of 0.9휆. workThised performance efficiently at consistency a very low can spacing be interpreted of 휆/4 as as well the SSDas a alsovery beinglarge efficientspacing forof 0 wider.9휆. This performance consistency can be interpreted as the SSD also being efficient for wider Thisfrequency performance bands. consistency Furthermore, can if be the interpreted interelement as the spacing SSD werealso being incorporated efficient infor the wider array frequency bands. Furthermore, if the interelement spacing were incorporated in the array frequencysynthesis bands. process, Furthermore, the problem if of the increased interelement beamwidth spacing could were be incorporated solved while in maintaining the array synthesis process, the problem of increased beamwidth could be solved while maintain- synthesisthe required process, SLL. the problem of increased beamwidth could be solved while maintain- ing the required SLL. ing the required SLL.

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Figure 10. Performance of SSD at different interelement spacings. Figure 10. PerformancePerformance of SSD at differe differentnt interelement spacings.

3.5.3.5. SSDSSD SSD PerformancePerformance Performance withwith with thethe the MainlobeMainlobe DirectionDirection − TheThe SSDSSD SSD algorithmalgorithm algorithm waswas was runrun run forfor for obtainingobtaining obtaining aa a−−505050 dBdB dB SLLSLL SLL forfor for thethe the 2020 20elementelement element arrayarray array asas 20° ◦ shownshownas shown inin FigureFigure in Figure 1111 atat11 tenten at tenvariousvarious various mainlobemainlobe mainlobe directionsdirections directions withwith with 20° 20 steps, steps,steps, andand and itit showedshowed it showed in-in- dependencyindependency with with the the mainlobe mainlobe direction. direction. The The SLL SLL converged converged during during the the recursive recursive SSD dependency with the◦ mainlobe◦ direction. The SLL converged during the recursive SSD processprocess even atat θ𝜃o==0° 0 or or 180 180°. This. This robust robust performance performance helped helped form form scanning scanning beams beams with process even at 𝜃 =0° or 180°. This robust performance helped form scanning beams very low sidelobe levels, which is essential in many practical applications. withwith veryvery lowlow sidelobesidelobe levels,levels, whichwhich isis essentialessential inin manymany practicalpractical applications.applications.

FigureFigure 11. 11. PerformancePerformance of of SSD SSD at at various various mainlobe mainlobe directions. directions. Figure 11. Performance of SSD at various mainlobe directions. 4. SSD Performance Comparisons and Discussions 4. SSD Performance Comparisons and Discussions 4.4.1. SSD Comparisons Performance with Comparisons Conventional Tapering and Discussions Windows 4.1. Comparisons with Conventional Tapering Windows 4.1. ComparisonsThere are many with Conv well-knownentional andTapering efficient Windows tapering windows [9] used for SLL reduc- tion,ThereThere such asareare triangular manymany well-knownwell-known (Bartlett), andand Hamming, efficientefficient Hanning, taperingtapering windowswindows Dolph-Chebyshev, [9][9] usedused forfor Kaiser, SLLSLL reduc-reduc- Black- tion,tion,man, suchsuch Cosine-square, asas triangulartriangular and (Bartlett),(Bartlett), Gaussian HamminHammin windowsg,g, Hanning,Hanning, [13]. Among Dolph-Chebyshev,Dolph-Chebyshev, these windows, Kaiser,Kaiser, the Dolph-Black-Black- man,man,Chebyshev Cosine-square,Cosine-square, and Gaussian andand GaussianGaussian ones provide windowswindows the lowest [13].[13]. AmongAmong sidelobe thesethese levels windows,windows, with a smaller thethe Dolph-Che-Dolph-Che- increase byshevbyshevin the beamwidth. andand GaussianGaussian Therefore, onesones provideprovide the proposed thethe lowestlowest SSD sidelobesidelobe technique levelslevels was withwith compared aa smallersmaller with increaseincrease these two inin thetheefficient beamwidth.beamwidth. tapering Therefore,Therefore, windows the asthe shown proposedproposed in Figure SSDSSD techniquetechnique12a,b. The waswas comparison comparedcompared was withwith based thesethese on two pro-two efficientefficientviding a radiationtaperingtapering windows patternwindows that asas had shownshown the same inin FiFi beamwidthguregure 12a,b.12a,b. from TheThe thecomparisoncomparison proposed waswas SSD techniquebasedbased onon providingprovidingand the benchmarking aa radiationradiation patternpattern windows. thatthat Thehadhad resultsthethe samesame shown beamwidthbeamwidth in Figure fromfrom 12 a,b thethe investigated proposedproposed SSDSSD the tech-tech- SLL niqueniqueperformance andand thethe improvement benchmarkingbenchmarking obtained windwindows.ows. by TheThe the resultsresults proposed shownshown SSD, inin where FigureFigure the 12a,b12a,b two investigatedinvestigated benchmarking thethe SLLSLLwindows performanceperformance provided improvementimprovement SLL levels at leastobtainedobtained 2 dB higherbyby thethe thanproposedproposed that achieved SSD,SSD, wherewhere by the thethe proposed twotwo bench-bench- SSD markingmarkingtechnique, windowswindows although providedprovided the Dolph-Chebyshev SLLSLL levelslevels atat leastleast window 22 dBdB higherhigher provided thanthan the thatthat narrowest achievedachieved beamwidth byby thethe pro-pro- posedposedcompared SSDSSD with technique,technique, any other althoughalthough window thethe at Dolph-Dolph- the sameChebyshevChebyshev SLL level windowwindow [9]. The Gaussianprovidedprovided window, thethe narrowestnarrowest on the beamwidthbeamwidth comparedcompared withwith anyany otherother windowwindow atat thethe samesame SLLSLL levellevel [9].[9]. TheThe GaussianGaussian

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window, on the other hand, provided an SLL level that was higher than the proposed SSD othertechnique hand, by provided 2.83 dB at an the SLL same level beamwidth, that was as higher shown than in Figure the proposed 12b. SSD technique by 2.83 dB at the same beamwidth, as shown in Figure 12b.

(a) (b)

FigureFigure 12. SLL 12. comparisonSLL comparison between between the the proposed proposed SSD SSD techniquetechnique (blue curves) curves) and and conv conventionalentional windows windows (red (redcurves) curves) at at the same beamwidth. (a) A comparison with the Dolph-Chebyshev window. (b) A comparison with the Gaussian window. the same beamwidth. (a) A comparison with the Dolph-Chebyshev window. (b) A comparison with the Gaussian window. In addition, the property of constant levels of all sidelobes resulting from the Dolph- In addition, the property of constant levels of all sidelobes resulting from the Dolph- Chebyshev window was relaxed in the proposed SSD technique, where the average SLL Chebyshev window was relaxed in the proposed SSD technique, where the average SLL obtained from the Dolph-Chebyshev window was −53.38 dB while that of the SSD tech- − obtainednique was from −54.4 the dB, Dolph-Chebyshev which was an advantage window of 1 was dB less53.38 for the dB proposed while that technique. of the SSD tech- nique wasTable− 54.41 summarizes dB, which the was SLL an performance advantage ofcomparison 1 dB less forbetween the proposed the SSD technique technique. withTable well-known1 summarizes efficient the windows SLL performance at the same beamwidth. comparison These between windows the included SSD technique the withtriangular, well-known Hamming, efficient Hanning, windows Blackman, at the sameDolph-Chebyshev, beamwidth. Kaiser, These windowsGaussian, and included Kai- the triangular,ser windows. Hamming, The comparison Hanning, results Blackman, in this Dolph-Chebyshev, table show that the Kaiser, SLL obtained Gaussian, from and the Kaiser windows.SSD algorithm The comparison at the same beamwidth results in this was table lower show than any that window, the SLL where obtained the improve- from the SSD algorithmment in the at the SLL same was approximately beamwidth was 12 lowerdB compared than any with window, the cosine where square the window. improvement in the SLL was approximately 12 dB compared with the cosine square window. Table 1. SLL level comparison between conventional windows and the SSD algorithm at the same beamwidth. Table 1. SLL level comparison between conventional windows and the SSD algorithm at the same beamwidth. Relative Reduction Benchmarking Window SSD Beamwidth in the Maximum Window SLL (dB) SLL (dB) Relative Reduction(Degrees) Benchmarking Window SLL SSD SLL (dB) Beamwidth in the Maximum TriangularWindow window −26.39(dB) −32.32SLL (dB) 6.32 7.12° (Degrees) SLL (dB) Hamming −27.89 −30.5 2.61 7.14° ◦ TriangularHanning window −27.626.39 −30.6 −32.323 6.32 7.14° 7.12 HammingBlackman −−37.7427.89 −44.83 −30.57.09 2.61 8.06° 7.14◦ Dolph-ChebyshevHanning −5027.6 −52 −30.62 3 8.38° 7.14◦ Kaiser −35.96 −46.55 10.59 8.2° Blackman −37.74 −44.83 7.09 8.06◦ Gaussian −37.19 −40.02 2.83 7.74° − − ◦ Dolph-ChebyshevCosine square −2850 −40 5212 2 7.74° 8.38 Kaiser −35.96 −46.55 10.59 8.2◦ 4.2. ComparisonsGaussian with Optimization−37.19 Techniques− 40.02 2.83 7.74◦ In this section, the proposed SSD SLL reduction technique is compared to some of Cosine square −28 −40 12 7.74◦ the most efficient and recent techniques, such as FPA, WOA, ASO, IWO, and CS. These

4.2. Comparisons with Optimization Techniques In this section, the proposed SSD SLL reduction technique is compared to some of the most efficient and recent techniques, such as FPA, WOA, ASO, IWO, and CS. These techniques were tested and compared in [27,29] for ULAs of 16 and 32 antenna elements Symmetry 2021, 13, x FOR PEER REVIEW 13 of 22

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techniques were tested and compared in [27,29] for ULAs of 16 and 32 antenna elements and were examined with the same design parameters presented in these three references forand the were purpose examined of comparison with the samewith SSD. design For parameters the 16 element presented ULA, the inthese workthree in [27] references showed thatfor thethe purposeFPA, WOA, of comparison and ASO withoutperformed SSD. For thethe 16PSO element and BA ULA, algorithms, the work with in [27 the] showed ASO optionthat the being FPA, the WOA, best. and Therefore, ASO outperformed we compared the the PSO proposed and BA SSD algorithms, algorithm with with the these ASO threeoption efficient being thetechniques. best. Therefore, On the weother compared hand, for the the proposed 32 ULA SSDarrays, algorithm the work with in these[29] showedthree efficient that the techniques. IWO and CS On methods the other outperformed hand, for the FA, 32 ULABBO and arrays, PSO. the Therefore work in [we29] alsoshowed compared that the the IWO performance and CS methods of the propos outperformeded SSD FA,algorithm BBO and with PSO. the Therefore ASO, IWO, we and also CScompared algorithms. the performanceTherefore, comparisons of the proposed betw SSDeen the algorithm proposed with SSD the algorithm ASO, IWO, with and the CS PSO,algorithms. FA, BA, Therefore,and BBO algorithms comparisons were between not needed. the proposed Starting with SSD algorithmthe case where with the the ULA PSO, wasFA, formed BA, and by BBO 16 elements, algorithms four were values not needed.for the number Starting of with damping the case cycles where were the examined, ULA was whichformed were by 50, 16 elements,100, 150, and four 200, values and the for SLL the numberresulting of from damping each was cycles compared were examined, with the mentionedwhich were optimization 50, 100, 150, techniques and 200, and as thepresented SLL resulting in Figure from 13a–d. each was compared with the mentioned optimization techniques as presented in Figure 13a–d.

(a) (b)

(c) (d)

FigureFigure 13. 13. SLLSLL comparison comparison between between the the proposed proposed SSD technique (black(black curves) andand differentdifferent optimizationoptimizationtechniques techniques in in a a16 16 element element uniform uniform linear linear array array (ULA) (ULA) at at ( a()a)K 𝐾=50= 50, (,b ()bK) 𝐾= 100, = (100c,) (Kc) =𝐾150, = and150, and (d) (Kd)= 𝐾200. = 200.

The SSD performance showed lower SLL levels compared with the three efficient techniques, especially at K ≥ 100. However, at K = 50, the SSD algorithm had better average SLLs than any of the other techniques, as shown in Figure 14. Increasing the value of K would improve the SSD performance in terms of both the maximum and average SLLs,

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TheThe SSDSSD performanceperformance showedshowed lowerlower SLLSLL levelevelsls comparedcompared withwith thethe threethree efficientefficient techniques,techniques, especiallyespecially atat 𝐾𝐾 ≥ ≥ 100.100. However,However, atat 𝐾=50𝐾=50,, thethe SSDSSD algorithmalgorithm hadhad betterbetter av-av- Symmetry 2021, 13, 480 14 of 22 erageerage SLLsSLLs thanthan anyany ofof thethe otherother techniques,techniques, asas shownshown inin FigureFigure 14.14. IncreasingIncreasing thethe valuevalue ofof 𝐾𝐾 wouldwould improveimprove thethe SSDSSD performanceperformance inin termsterms ofof bothboth thethe maximummaximum andand averageaverage SLLs,SLLs, wherewhere thethe maximummaximum SLLSLL ofof −−4545 dBdB waswas lowerlower thanthan thethe achievedachieved ASOASO SLLSLL byby 44 dB,dB, whereandand thethe the averageaverage maximum SLLSLL SLL waswas of − −−455353 dB dB,dB, was whichwhich lower waswas than alsoalso the achieved lowerlower thanthan ASO thatthat SLL corresponding bycorresponding 4 dB, and the toto ASOASO − byaverageby 1212 dBdB SLL atat was𝐾𝐾 = =53 200200 dB,.. TheThe which processingprocessing was also timetime lower consumedconsumed than that corresponding byby thethe SSDSSD algorithmalgorithm to ASO by waswas 12 dB alsoalso muchmuch at K = 200. The processing time consumed by the SSD algorithm was also much smaller smallersmaller thanthan allall thesethese techniques,techniques, wherewhere forfor thethe 1616 elementelement ULA,ULA, thethe proposedproposed SSDSSD algo-algo- than all these techniques, where for the 16 element ULA, the proposed SSD algorithm rithm required only one second at 𝐾𝐾 = = 200200, while the CPU processing time for ASO in rithmrequired required only one only second one at secondK = 200 at, while the CPU, while processing the CPU time processing for ASO in time [27] wasfor ASO in [27]approximately[27] waswas approximatelyapproximately 27 s when using 2727 ss thewhenwhen same usingusing platform thethe samesame specification. platformplatform However, specification.specification. the improved However,However, thethe performanceimprovedimproved performanceperformance of the SLL achieved ofof thethe bySLLSLL SSD achievedachieved compromised byby SSDSSD slightly compromisedcompromised wider beams, slightlyslightly especially widerwider at beams,beams, especiallylargerespecially numbers atat largerlarger of damping numbersnumbers cycles, ofof dampingdamping as shown cyclescycles in Figure,, asas 15 shownshown, where inin an FigureFigure increase 15,15, of wherewhere less than anan increaseincrease ofoneof lessless degree thanthan resulted oneone degreedegree from SSDresultedresulted compared fromfrom with SSDSSD the comparedcompared ASO at K withwith= 200. thethe ASOASO atat 𝐾𝐾 = = 200200..

FigureFigure 14. 14.14. Maximum MaximumMaximum and andand average averageaverage SLL SLLSLL comparison comparisoncomparison between betwbetweeneen the the proposedthe proposedproposed SSD SSD techniqueSSD techniquetechnique (black (black(black curves)curves) and andand different differentdifferent optimization optimizationoptimization techniques tetechniqueschniques for a for 16for element aa 1616 elementelement ULA at ULAULAK = at50,at 𝐾=50 100,𝐾=50 150,,, and100100 200.,, 150150,, andand 200200..

FigureFigure 15. 15.15. Mainlobe MainlobeMainlobe beamwidth beamwidthbeamwidth comparison comparisoncomparison between betweenbetween the th proposedthee proposedproposed SSD SSD techniqueSSD techniquetechnique and different andand differentdifferent optimizationoptimization techniques techniquestechniques for forfor a 16 aa element1616 elementelement ULA ULAULA at K =atat 50,𝐾=50𝐾=50 100, 150,,, 100100 and,, 150150 200.,, andand 200200..

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On the other hand, the performance improvement in both the maximum and average SLL Onvalues the otherobtained hand, from the performancethe SSD algorithm improvement came with in both a price the paidmaximum in a slightly and average larger SLLbeamwidth valuesOn the obtained (less other than hand, from 1°) thecomparedthe performanceSSD algorithm with the improvement camethree withoptimization a inprice both paid techniques, the in maximum a slightly as shown and larger av- in beamwidthFigureerage SLL 15. Therefore, values(less than obtained if1° the) compared slightly from the increased with SSD the algorithm threebeamwidth optimization came could with be atechniques, pricetolerated paid foras in shownachieving a slightly in ◦ Figurebetterlarger 15.SLL beamwidth Therefore, performance, (less if the thanthen slightly 1the) comparedSSDincreased would beamwidth with be the the best three could choice optimization be among tolerated the techniques, forother achieving optimi- as betterzationshown SLL techniques. in Figureperformance, 15 . Therefore, then the if SSD the slightly would be increased the best beamwidth choice among could the be other tolerated optimi- for zationachievingThe techniques. normalized better SLL performance,coefficients of then the three the SSD opti wouldmization be thetechniques, best choice along among with the the other uni- formoptimizationThe feeding normalized techniques.coefficients, coefficients are displayed of the three in Fi optiguremization 16a for techniques,a 16 element along linear with array, the while uni- formthe four feedingThe cases normalized coefficients, of the SSD coefficients coefficienare displayed ofts are the in displayed three Figure optimization 16a in forFigure a 16 16b. techniques,element linear along array, with while the theuniform four cases feeding of the coefficients, SSD coefficien are displayedts are displayed in Figure in 16 Figurea for a 16b. 16 element linear array, while the four cases of the SSD coefficients are displayed in Figure 16b.

(a) (b) (a) (b) Figure 16. Normalized antenna coefficients for a 16 element ULA. (a) Optimization benchmarking techniques and (b) the FigureFigureSSD algorithm 16. 16. NormalizedNormalized at different antenna antenna values coefficients coefficients for 𝐾. for for a a 16 16 element element ULA. ULA. ( (aa)) Optimization Optimization benchmarking benchmarking techniques techniques and and ( (bb)) the the SSDSSD algorithm algorithm at at different different values values for for 𝐾K. The comparison was also performed for a larger 32 element ULA between the pro- posedTheThe SSD comparison comparison algorithm was wasand also alsothe performedthree performed most forefficie for a alarger largernt techniques 32 32 element element in [27,29],ULA ULA between between which thewere the pro- pro-the posedposedASO, IWO,SSD SSD algorithm and CS algorithms, andand thethe three three as mostshown most efficient efficiein Figuresnt techniques techniques 17–19. in As [27in expected, ,[27,29],29], which which SSD were provided were the ASO, the a ASO,much-improvedIWO, IWO, and CSand algorithms, CS maximum algorithms, as and shown as average shown in Figures SLL in Figures performance—as 17–19 .17–19. As expected, As expected,shown SSD in provided SSDFigure provided 18—com- a much- a much-improvedparedimproved with maximum all of maximum the three and average techniques,and average SLL even performance—asSLL performance—asat lower values shown of shown 𝐾 in. Moreover, Figure in Figure 18—compared the 18—com- increase paredwithin beamwidth allwith of all the of by three the the three techniques,SSD techniques,technique even was even at less lower at thanlower values 0.5° values compared of Kof. 𝐾 Moreover,. Moreover, with the the most the increase increase efficient in ◦ inASObeamwidth beamwidth technique. by by the the SSD SSDtechnique technique waswas lessless than 0.50.5° compared compared with the most most efficient efficient ASOASO technique. technique.

(a) (b) (a) (b) Figure 17. Cont.

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(c) (d) (c) (d) (c) (d) Figure 17. 17. SLLSLL comparison comparison between between the the proposed proposed SSD SSD technique technique (bla (blackck curves) curves) and different and different efficient efficient optimization optimization tech- Figure 17. SLL comparison between the proposed SSD technique (black curves) and different efficient optimization tech- Figureniques 17.for SLLa 32 comparisonelement ULA between at (a) 𝐾=50 the proposed, (b) 𝐾 SSD = 100 technique, (c) 𝐾 =(bla 150,ck and curves) (d) 𝐾 and = different200. efficient optimization tech- techniquesniques for a for 32 aelement 32 element ULA ULA at (a at) 𝐾=50 (a) K =, 50,(b) (𝐾b) K == 100100,, (c) (c𝐾) K = 150,150, and and (d ()d 𝐾) K = = 200.200. niques for a 32 element ULA at (a) 𝐾=50, (b) 𝐾 = 100, (c) 𝐾 = 150, and (d) 𝐾 = 200.

Figure 18. Maximum and average SLL comparison between the proposed SSD technique (black Figure 18. MaximumMaximum and and average average SLL SLL comparison comparison betw betweeneen the the proposed proposed SSD SSD technique technique (black (black Figurecurves) 18. and Maximum different andoptimization average SLL techniques comparison for a betw32 elementeen the ULA proposed at 𝐾=50 SSD ,technique 100, 150 ,(black and 200 . curves) and different optimization techniques for a 32 element ULA at 𝐾=50= , 100, 150, and 200. curves)curves) and and different different optimization optimization te techniqueschniques for a 32 element ULA at K𝐾=5050,, 100,100 150,, 150 and, and 200. 200.

Figure 19. Mainlobe beamwidth comparison between the proposed SSD technique and different Figure 19. Mainlobe beamwidth comparison between the proposed SSD technique and different FigureoptimizationFigure 19. 19. MainlobeMainlobe techniques beamwidth beamwidth for a 32 comparisonelement comparison ULA between betweenat 𝐾=50 th thee, proposed100 proposed, 150 ,SSD and SSD technique200 technique. and and different different optimization techniques for a 32 element ULA at 𝐾=50, 100, 150, and 200. optimizationoptimization techniques techniques for for a a 32 32 element element ULA ULA at at K𝐾=50= 50,, 100,100, 150, 150 and, and 200. 200.

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Figure 19 displays the beamwidth values of the different compared techniques at dif- ferent values of 퐾 where there was a slight increase, which was less than 1° in the case of theFigure SSD algorithm. 19 displays Deep the SLL beamwidth levels as valueslow as of−44 the dB different with an comparedaverage of techniques−52 dB could at ◦ differentbe achieved, values which of K wewherere much there better was a than slight the increase, other three which algorithms was less than, but 1at ina price the case of a of0.8° the incr SSDease algorithm. in beamwidth. Deep SLL levels as low as −44 dB with an average of −52 dB could be achieved,The normalized which were array much coefficients better than for the 32 other element three ULA algorithms, are also but displayed at a price Figure of a ◦ 0.820a increasefor the benchmar in beamwidth.king algorithms, while Figure 20b displays the normalized SSD co- efficientsThe normalized at different array values coefficients of 퐾, where for thethe 32 curves element bec ULAome aresteeper also displayedat larger valuesFigure of 20 퐾a , forincreasing the benchmarking the tapering algorithms, profile and while result Figureing in20 breducing displays the the SLL normalized to deeper SSD levels. coefficients at different values of K, where the curves become steeper at larger values of K, increasing the tapering profile and resulting in reducing the SLL to deeper levels.

(a) (b)

Figure 20. ImpactImpact of of mutual mutual coupling coupling on the SSD performance for a 16 element ULA at (a) K퐾==100100 and and ( b(b) )K 퐾==1000.1000.

5.5. MutualMutual CouplingCoupling EffectsEffects onon thethe SSDSSD PerformancePerformance TheThe previousprevious analysis for the the SSD SSD technique technique wa wass performed performed on on isotropic isotropic antenna antenna el- elements,ements, which which is is now now done done using using practical antenna models along withwith thethe inclusioninclusion ofof mutualmutual coupling coupling effects. effects. For For a 16a 16 element element ULA ULA using using half-wave half-wave dipoles dipoles as antenna as antenna elements ele- operatingments operating at 5 GHz, at 5 Figure GHz, 21Figure displays 21 displays the resulting the result normalizeding normalized radiation radiation patterns patterns at two numbersat two numbers of damping of damping cycles, namelycycles, namely 100 and 100 1000. and The 1000. two The sets two of curvessets of incurves this figurein this includefigure include the isotropic the isotropic isolated isolated antenna a elements,ntenna elements, practical practical isolated elements isolated elements (i.e., without (i.e., mutualwithout coupling mutual coupling effects), and effects) practical, and antennapractical elements antenna withelements the effect with ofthe mutual effect of coupling. mutual Forcoupling. both values For both of the values number of ofthe damping number cycles,of damping the normalized cycles, the power normalized patterns power displayed pat- veryterns deep display SLLsed usingvery deep the SSDSLLs algorithm, using the SSD where algorithm the impact, where of the the mutual impact couplingof the mutual just appearedcoupling tojust increase appear ined the to increase mainlobe in at the very mainlobe low radiation at very levels low radiation below − 30levels dB. However,below −30 thedB. maximumHowever, the and maximum average SLLs and average were still SLLs very we acceptablere still very and acceptable sometimes and belowsometimes the isotropicbelow the elements isotropic case. elements case.

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(a) (b) (a) (b) Figure 21. Impact of mutual coupling on the SSD performance for a 16 element ULA at (a) 𝐾 = 100 and (b) 𝐾 = 1000. FigureFigure 21. 21. ImpactImpact of of mutual mutual coupling coupling on on the the SSD SSD performance performance for for a a16 16 element element ULA ULA at at (a (a) )𝐾K = =100 100 andand ((bb)) K𝐾= 1000. = 1000. 6. SSD for Two-Dimensional Planar Arrays 6. SSD SSDAs for for mentioned Two-Dimensional Two-Dimensional in Section Planar 3, the Arrays application of the proposed SSD technique was inde- pendentAs mentioned of the array in in Section Sectionconfiguration 33,, thethe applicationapplication and therefore ofof thethe could proposedproposed be applied SSDSSD for techniquetechnique any array was geometry inde- pendentif there of was the onlyarray one configuration configuration mainlobe inand the therefore visible regioncould beof applied the radiation for any pattern array geometry with some ifif relatively there waswas loweronlyonly one onesidelobes mainlobe mainlobe to inguarantee in the the visible visibl SSD regione convergence.region of theof the radiation Theradiation radiation pattern pattern withpattern with some of some rela-a uni- relativelytivelyformly lower fed lower sidelobestwo-dimensional sidelobes to guarantee to guarantee planar SSD array SSD convergence. hadconvergence. plenty The of radiationThesidelobes radiation pattern compared pattern of a to uniformly of the a uni-ULA, formlyfedand two-dimensional therefore fed two-dimensional the SSD planar damping array planar hadprocess array plenty hadrequired of plenty sidelobes more of sidelobes compareddamping compared cycles to the ULA,to achieve to andthe ULA, there-the re- andforequired therefore the SSD SLL. damping theThe SSD process damping process follow required processed the morerecursiverequired damping moreprocess cyclesdamping in Eq touation achieve cycles (11) to the achieveand required could the SLL.be re- ex- quiredTheamined process SLL. for The followed a planar process array the follow recursive of 20ed ×the process20 recursive half-wavelength in Equation process in (11)dipole Eq anduation elements could (11) be asand examined shown could in be forFigure ex- a aminedplanar22a, operating array for a ofplanar20 at× 5 array 20GHzhalf-wavelength ofwith 20 a × beam 20 half-wavelength formed dipole in elements the broadside dipole as shown elements direction in Figure as shown(90°, 22a, 0°) operatingin. TheFigure nor- ◦ ◦ 22a,atmalized 5 GHzoperating with power at a beam 5levels GHz formed forwith the a inbeamuniform the formed broadside feeding in the directioncase broadside are shown (90 ,direction 0 in). Figure The normalized(90°, 22b, 0°) and. The the power nor- high- malizedlevelsest SLL for power was the − uniform 13.4levels dB for relative feeding the uniform to case the aremainlobe feeding shown case level. in are Figure The shown improvement 22b, in andFigure the 22b,in highest the and SLL the SLL appeared high- was − estclearly13.4 SLL dB was in relative Figure −13.4 dB to22c therelative with mainlobe the to effectthe level. mainlobe of mu Thetual improvementlevel. coupling, The improvement where in the the SLL SLL in appeared thewas SLL reduced appeared clearly to in−30 − clearlyFiguredB at in 22𝐾 Figurec with = 100 22c the, while with effect it the dropped of effect mutual of to coupling,mu lesstual than coupling, where−47 dB the whereat SLL𝐾 the was = SLL1000 reduced, where was reduced to100030 sidelobes to dB −30 at K = − K = dBhad at100 𝐾been, while =suppressed 100, itwhile dropped it from dropped to lessthe power thanto less pattern.47 than dB at−47 dB 1000at 𝐾, where = 1000 1000, where sidelobes 1000 sidelobes had been suppressed from the power pattern. had been suppressed from the power pattern.

(a) (b) (a) (b) Figure 22. Cont.

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(c) (d)

FigureFigure 22.22. Two-dimensional planar array configuration configuration formed formed by by 20× 20 dipole dipole antennasantennas withwith mutualmutual coupling effects. ((aa)) ArrayArray configuration.configuration. ((b)) NormalizedNormalized powerpower patternpattern withwith uniformuniform feeding.feeding. ((c)) NormalizedNormalized powerpower patternpattern withwith SSDSSD feedingfeeding atat K𝐾= 100. = 100. ( d(d)) Normalized Normalized power power pattern pattern with with SSD SSD feeding feeding at atK 𝐾= 1000. = 1000.

7. SSD Operational Constraints and Limitations Discussion In this section, the operational constraints and limitations of the proposed SSD algo- rithm are explored andand discussed.discussed. These constraintsconstraints include thethe operationaloperational interelementinterelement spacing and the impact of antenna elementelement failures.failures. The SSD algorithm was designed mainly for antenna arrays of any configurationconfiguration thatthat had symmetrical and uniform antennaantenna distributions with interelement spacings thatthat were less than one wavelength. These antenna arraysarrays produceproduce oneone mainlobemainlobe andand aa setset ofof smaller sidelobes, which allows the SSDSSD toto sequentiallysequentially reducereduce thethe sidelobe.sidelobe. IfIf thethe interelementinterelement separation were were a a distance distance of of more more than than on onee wavelength, wavelength, the the SSD SSD could could not not work work as re- as requiredquired due due to to the the appearance appearance of of secondary secondary major major lobes lobes in in the radiation pattern, whichwhich may have led to thethe suppressionsuppression ofof thethe requiredrequired mainlobe.mainlobe. Therefore, in thisthis case,case, therethere would bebe nono convergence,convergence, and and the the secondary secondar majory major lobes lobes would would suppress suppress each each other other in the in patternthe pattern alternately. alternately. The The SSD SSD can can be modified be modifi toed select to select only only the sidelobesthe sidelobes and and exclude exclude the secondarythe secondary major major lobes lobes in Equation in Equation (4), (4), so it so will it will work work effectively, effectively, but but without without suppression suppres- orsion reduction or reduction of the of secondary the secondary major lobes.major However,lobes. However, practically, practically, we almost we built almost arrays built with ar- antennarays with elements antenna separated elements byseparated a distance by thata distance was less that than was one less wavelength—specifically than one wavelength— aroundspecifically half around the wavelength half the wavelength value—so the value—so SSD could the work SSD fine.could On work the fine. other On hand, the ifother the interelementhand, if the interelement spacing is very spacing small, is the very SSD small, still efficientlythe SSD still works, efficiently as shown works, previously as shown in Figurepreviously 10, even in Figure at a one-quarter 10, even at a wavelength one-quarter interelement wavelength separation,interelement which separation, is practically which difficultis practically to implement difficult to due implement to the increase due to the of increase mutual couplingof mutual effects coupling and effects the physical and the antennaphysical dimensions.antenna dimensions. The second operational operational constraint constraint for for the the SSD SSD algorithm algorithm is isthe the impact impact of ofantenna antenna el- elementement failure failure on on the the SLL SLL damping damping performance. performance. If all If antenna all antenna elements elements are operating are operating nor- normally,mally, the theSSD SSD performs performs efficiently, efficiently, but if but one if or one more or moreantenna antenna elements elements fail or are fail turned or are turnedoff in the off array, in the then array, the then symmetry the symmetry structure structure is perturbed, is perturbed, which affects which the affects SSD the SLL SSD re- SLL reduction performance as the radiation pattern changes. Figure 23 demonstrates the duction performance as the radiation pattern changes. Figure 23 demonstrates the impact impact of one antenna element’s complete failure at different locations in a 16 element ULA, of one antenna element’s complete failure at different locations in a 16 element ULA, where it is noticed that the shutdown or malfunctioning antenna elements at the array ends where it is noticed that the shutdown or malfunctioning antenna elements at the array will not affect the SSD damping behavior, and the SSD still works fine. On the other hand, ends will not affect the SSD damping behavior, and the SSD still works fine. On the other if the failed element is located at the array’s center, then the SSD may fail completely to hand, if the failed element is located at the array’s center, then the SSD may fail completely reduce the SLL, and it is recommended in this case that the number of damping cycles is to reduce the SLL, and it is recommended in this case that the number of damping cycles reduced to the value that gives the best SLL level. The element failure can be expressed in the array steering vector as a zero element, which indicates the absence of that element.

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Symmetry 2021, 13, 480 20 of 22 is reduced to the value that gives the best SLL level. The element failure can be expressed in the array steering vector as a zero element, whichwhich indicatesindicates thethe absenceabsence ofof thatthat element.element.

Figure 23. TheThe impact of one antenna element failure onon thethe SDDSDD performanceperformance atat differentdifferent locationsloca- tionsin a 16 in element a 16 element ULA. ULA.

The absence of multiple elements in the array can be handled and tolerated by the SSD algorithm, except in the case where the malfunction element is located at the array’s center. FigureFigure 2424 displays displays this this scenario scenario for for some some of theof the antenna antenna elements elements that failedthat failed to work to workin two in cases. two Thecases. first The case first includes case includes elements el 2,ements 3, 8, 13, 2, and3, 8, 14, 13, where and 14, element where 8 representselement 8 representsthe central the element, central which element, seriously which impacts seriously the SSDimpacts convergence the SSD convergence behavior. On behavior. the other Onhand, the the other symmetry hand, the or near-symmetricalsymmetry or near-symme distributiontrical of faileddistribution antenna of elements, failed antenna excluding ele- ments,the central excluding antenna, the can central be handled antenna, by the can proposed be handledhandled SSD byby algorithm, thethe proposedproposed which appearsSSDSSD algorithm,algorithm, to have whichthe robust appears convergence to have the behavior. robust convergence behavior.

Figure 24. TheThe impact impact of of multiple multiple antenna antenna element element fa failuresiluresilures onon on thethe the SDDSDD SDD performanceperformance performance atat differentdifferent at different locationslocations inin aa 16 16 element element ULA. ULA.

We cancan concludeconclude from from this this discussion discussion that that the thethe proposed proposedproposed SSD SSDSSD algorithm algorithmalgorithm has robusthashas robustrobust SLL SLLreduction reduction performance performance under under abnormal abnormal operating operating conditions, conditions, which which include include one or one more or moreantenna antenna element element failing failing except except for the for case the where case where the malfunctioning the malfunctioning element element is located is lo- at catedthe array’s at the center. array’s center.

8. Conclusions Antenna arrays became essential in many recent communication systems to achieve high-quality connectivity performance and higher capacities through proper spatial filter- ing. Therefore, in this paper, spatial filtering has been achieved by a sequential sidelobe damping algorithm (SSD), where the sidelobes are sequentially damped for sidelobe level (SLL) reduction. The proposed algorithm can be applied for any uniform array geometry,

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and the SLL can be set to the required level according to the number of the damping cycles. Additionally, the algorithm has been thoroughly analyzed, and the processing time, convergence, and performance with practical scenarios have been investigated, show- ing independent and very fast convergence times compared with many of the current optimization techniques. In addition, comparisons between the proposed SSD with the well-known windows showed that the SSD achieved lower SLLs at the same beamwidth, while comparisons with many of the recent optimization techniques such as FPA, PSO, WOA, BA, ASO, IWO, FA, CS, and BBO indicated that the proposed SSD can provide lower SLLs at a slightly wider beamwidth, but at a much higher speed of processing and convergence, which is essential in real-time communication scenarios, in addition to the capability of application in any uniform array configuration.

Author Contributions: Conceptualization, Y.A. and F.A.; methodology, Y.A.; software, Y.A.; valida- tion, F.A., and Y.A.; formal analysis, Y.A.; investigation, F.A.; writing—original draft preparation, Y.A.; writing—review and editing, F.A.; visualization, Y.A.; funding acquisition, Y.A. All authors have read and agreed to the published version of the manuscript. Funding: Taif University Researchers Supporting Project number (TURSP-2020/161), Taif University, Taif, Saudi Arabia. Acknowledgments: The authors would like to thank Taif University Researchers Supporting Project number (TURSP-2020/161), Taif University, Taif, Saudi Arabia for supporting this work. Conflicts of Interest: The authors declare no conflict of interest.

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