Synthesis of Large Ultra-wideband Sparse Circular Planar Arrays Based on Rotationally Symmetric Structure

: This article aims to improve the synthesis efﬁciency and radiation performance of large ultra-wideband (UWB) rotationally symmetric sparse circular planar arrays by using a modiﬁed differential evolution algorithm (MDEA). In the proposed MDEA, we adopt a new encoding mechanism in which an individual represents an element position expressed in polar coordinates. Importantly, such an encoding mechanism can facilitate the multiplication calculation for the array factor in the individual being updated while making it easier to meet the given minimum element spacing constraint. Moreover, to cater to the new encoding mechanism, some low-dimensional evolution operators are introduced to avoid the prematurity. In particular, the UWB rotationally symmetric sparse planar array synthesis problem is transformed into the sidelobe suppression problem of the array pattern at the highest frequency under the given array aperture, and the minimum spacing constraints is used to guarantee enough space to place physical UWB antenna elements. Two synthesis examples of UWB sparse planar arrays based on rotationally symmetric structures are presented. The results show that the peak sidelobe level (PSL) obtained by the proposed MDEA is signiﬁcantly lower than the results obtained by some existing algorithms in an acceptable CPU time, which proves the effectiveness and superiority of the proposed MDEA.


Introduction
Ultra-wideband (UWB) antenna arrays have been broadly used in a variety of practical applications such as modern radar, radio astronomy, and wireless communication [1][2][3].In particular, UWB arrays are different from narrowband arrays, and they need to consider the grating lobe suppression problem in a wide frequency band, which greatly increases the difficulty in the array synthesis.Thus, the grating lobe suppression for UWB arrays is a critical problem.For uniformly spaced arrays, we can avoid grating lobes by using electrically small spacings, such as half a wavelength, at the highest frequency over the whole bandwidth.However, to achieve good impedance matching and radiation efficiency at a low frequency band, the application of some tightly coupled array techniques [4,5] is usually required.Meanwhile, such small element spacings usually need a large number of elements to obtain a relatively narrower beamwidth, which will greatly increase the weight and cost of the antenna system.As is known, the element position is also a degree of array freedom; thus, non-uniformly spaced array layouts can be applied to avoid the appearance of grating lobes.Compared to the uniformly spaced array, it requires fewer antenna elements for achieving an equivalent or similar radiation pattern performance, which significantly saves the cost of the antenna system.However, determining a method to optimize the element position distribution to meet the practical application requirement is a vital problem.
Over the past few decades, a variety of non-uniformly spaced array synthesis methods have been reported in succession.For examples, an analytical method [6], stochastic optimization method [7], matrix pencil method [8], iterative convex optimization [9], compressive sensing technique [10], and other hybrid method [11].Noticeably, most of these methods focus on the narrowband array case.Certainly, some UWB sparse array synthesis methods have been presented in [12][13][14], but most are limited to linear array and small-scale array.Obviously, it is more difficult for large UWB sparse planar array synthesis, but it was noticed that the usage of some special geometries will contribute to the suppression of grating lobes.For example, in [15], the aperiodic tiling array was synthesized by using genetic algorithm perturbation, and it obtained a good UWB performance.In [16], the perturbed Peano-Gosper (PG) array combined with recursion technique showed a low sidelobe and grating lobe level over a wide frequency band.As another special geometry, the rotationally symmetric structure also possesses the property of suppressing grating lobe and sidelobe in a wide frequency band [17].In [18], by defining the projection operators, the rotationally symmetric sparse circular planar array synthesis problem was converted into an optimization problem with box constraints, and then the improved harmony search algorithm was used to solve this problem.Although the obtained radiation pattern showed a relatively lower PSL over the wide bandwidths, the performance and computing efficiency of the pattern need to be further improved.
In this article, we introduce a modified differential evolution algorithm (MDEA) to efficiently synthesize a large UWB sparse circular planar array based on rotationally symmetric structure.Different from various stochastic optimization methods with traditional encoding mechanism, the proposed MDEA adopts a new encoding mechanism in which an individual represents an antenna element position in polar coordinates.In each individual updating, such an encoding mechanism contributes to greatly reduce the number of multiplication calculations for the array factor and facilitates meeting the minimum element spacing constraint.Moreover, to match the new encoding mechanism, some low-dimensional evolution operators are proposed to enhance its global search ability.Two examples of UWB sparse planar array synthesis based on the rotationally symmetric structure are carried out to verify the effectiveness and efficiency of the proposed MDEA.The obtained results show that the proposed MDEA has certain advantages on the obtained pattern performance and computational efficiency compared with some advanced algorithms.
This research is further organized as follows: In Section 2, the synthesis problem of UWB rotationally symmetric sparse circular planar array is formulated, and the proposed MDEA is given in detail.In Section 3, numerical experiments are carried out, and the synthesis results are analyzed.In Section 4, conclusions are drawn.

UWB Rotationally Symmetric Sparse Circular Planar Array Synthesis Problem
Consider a sparse circular planar array based on rotationally symmetric structure.It is constitutive of M folds and N isotropic radiating elements in the xy plane, as shown in Figure 1.Assume that this array works over a wide frequency band [ f L , f H ], and its bandwidth can be defined as ( f H / f L ):1.At a certain frequency f ∈ [ f L , f H ], the array factor can be written as where j = √ −1, c denotes the propagation velocity in free space, and K = N/M denotes the element number in each fold and it is required to be a positive integer.
where (θ 0 , φ 0 ) represents the beam's pointing direction.AF k (θ, φ; f ) denotes the subarray factor generated by the k-element in the first fold and its Electronics 2023, 12, 4833 3 of 11 (M − 1) rotationally symmetric elements.For the convenience of the problem description, some notations are given as follows: Electronics 2023, 12, x FOR PEER REVIEW 3 of 11 sin  sin  − sin  0 sin  0 , where ( 0 ,  0 ) represents the beam's pointing direction.AF  (, ; ) denotes the subarray factor generated by the -element in the first fold and its ( − 1) rotationally symmetric elements.For the convenience of the problem description, some notations are given as follows: As we all know, it is more difficult for the UWB array to be synthesized than the narrow-band case due to the consideration of the array pattern over the whole frequency band [ L ,  H ]. When the mutual coupling effect is not considered, the peak sidelobe level (PSL) of the array factor at  H is the largest PSL over the whole frequency band, varying from  L to  H [14].Moreover, it is very critical and difficult for UWB sparse planar array synthesis to meet the minimum element spacing constraint.Due to the rotationally symmetric sparse planar array concerned in this article, this constraint can be transformed into the minimum element spacing constraint on 2K elements in the previous two folds.Thus where Θ SL denotes the sidelobe region in (, ) plane,  const represents the required minimum element spacing. denotes the radius of the circular array aperture,  and ℎ denote the -th and ℎ-th element in the first fold, respectively.The subscripts  and  represent the -th and -th fold in the rotationally symmetric circular planar array.For an arbitrary index  and  ∈ {1, 2} ,  (,ℎ) =   cos   −  ℎ cos  ℎ and  (,ℎ) =   sin   −  ℎ sin  ℎ denote the position difference value of the -th and ℎ-th element in and -axis, respectively.As we all know, it is more difficult for the UWB array to be synthesized than the narrow-band case due to the consideration of the array pattern over the whole frequency band [ f L , f H ]. When the mutual coupling effect is not considered, the peak sidelobe level (PSL) of the array factor at f H is the largest PSL over the whole frequency band, varying from f L to f H [14].Moreover, it is very critical and difficult for UWB sparse planar array synthesis to meet the minimum element spacing constraint.Due to the rotationally symmetric sparse planar array concerned in this article, this constraint can be transformed into the minimum element spacing constraint on 2K elements in the previous two folds.Thus, UWB rotationally symmetric sparse circular planar array synthesis can be expressed as minimizing the PSL at f H by optimizing element positions under the constraints on a circular aperture with radius R and minimum element spacing.In other word, min r,ψ where Θ SL denotes the sidelobe region in (u, v) plane, d const represents the required mini- mum element spacing.R denotes the radius of the circular array aperture, g and h denote the g-th and h-th element in the first fold, respectively.The subscripts d and l represent the d-th and l-th fold in the rotationally symmetric circular planar array.For an arbitrary index d and l ∈ {1, 2}, x (gd, hl) = r g cos ψ gd − r h cos ψ hl and y (gd, hl) = r g sin ψ gd − r h sin ψ hl denote the position difference value of the g-th and h-th element in xand y-axis, respectively.

Modified Differential Evolution Algorithm (MDEA)
As can be seen, Problem (3) describes the synthesis of rotationally symmetric sparse circular planar array with the purpose of minimizing the PSL under the required array aperture and minimum element spacing constraints.Due to the usage of the rotationally symmetric structure, the minimum spacing constraint for the whole array can be simplified to this constraint for 2K elements in the first two folds.
To solve this problem, we introduce a modified differential evolution algorithm (MDEA) by using a new encoding mechanism and some low-dimensional evolution operators.In the following section, we introduce these improvements in detail based on the basic differential evolution algorithm.

New Encoding Mechanism
When using a stochastic optimization method to solve Problem (3), the traditional encoding mechanism is that each individual represents an array layout, denoted as where the subscript i ∈ {1, 2, • • • , NP} represents the i-individual in the population comprising NP individuals, and r k and ϕ k (k ∈ {1, 2, • • • , K}) denote the polar diameter and the angle of the k-element position in polar coordinates, respectively.Figure 2a shows the traditional encoding mechanism.Notably, Problem (3) also involves the constraint on the minimum element spacing, which is a very important and difficult problem in sparse planar array synthesis due to the consideration of the placement of physical UWB element.Thus, the candidate individuals generated by evolution operators need to be checked to satisfy the minimum element spacing constraint.When considering large-scale and even medium-scale arrays, it is very difficult for the candidate individuals generated by evolution operators in a stochastic optimization algorithm to satisfy the given element spacing constraint, which easily results in algorithm stagnation.Moreover, the evaluation of the large 2D array pattern is required in each iteration, which will cost a considerable amount of time.
When using a stochastic optimization method to solve Problem (3), the traditional encoding mechanism is that each individual represents an array layout, denoted as  ⃗  = [ 1 ,  1 ,  2 ,  2 , ⋯ ,   ,   ], where the subscript  ∈ {1, 2, ⋯ , NP} represents the -individual in the population comprising NP individuals, and   and   ( ∈ {1, 2, ⋯ , }) denote the polar diameter and the angle of the -element position in polar coordinates, respectively.Figure 2a shows the traditional encoding mechanism.Notably, Problem (3) also involves the constraint on the minimum element spacing, which is a very important and difficult problem in sparse planar array synthesis due to the consideration of the placement of physical UWB element.Thus, the candidate individuals generated by evolution operators need to be checked to satisfy the minimum element spacing constraint.When considering large-scale and even medium-scale arrays, it is very difficult for the candidate individuals generated by evolution operators in a stochastic optimization algorithm to satisfy the given element spacing constraint, which easily results in algorithm stagnation.Moreover, the evaluation of the large 2D array pattern is required in each iteration, which will cost a considerable amount of time.
To overcome this problem, we adopt a modified differential evolution algorithm with new encoding mechanism and some low-dimensional evolution operators.This encoding mechanism has been used to optimize a wind farm layout in [19] and array element positions in [20].Particularly, in the new encoding mechanism, an individual denotes an element position expressed by the polar diameter and angle in polar coordinates, i.e.,  ⃗  = [  ,   ] ( = 1, 2, ⋯ , ).To overcome this problem, we adopt a modified differential evolution algorithm with new encoding mechanism and some low-dimensional evolution operators.This encoding mechanism has been used to optimize a wind farm layout in [19] and array element positions in [20].Particularly, in the new encoding mechanism, an individual denotes an element position expressed by the polar diameter and angle in polar coordinates, i.e., 2b shows this new encoding mechanism that the whole population expressed as } represents an array layout.Thus, each individual is always kept at two dimensions, and NP is equal to K.Moreover, the variation in an individual only causes a change in one element position in each fold of the whole array.When the individual has been updated, only one element and its (M − 1) rotationally symmetric elements for the contribution to the array pattern has been changed, which greatly saves time for the evaluation of large 2D array pattern.In addition, candidate individuals in such an encoding mechanism have more possibility to satisfy the given minimum spacing constraint.

Low-Dimensional Evolution Operators
In the proposed MDEA, we adopt new encoding mechanism in which the variable dimension of each individual in the population is always two.To better match this mechanism, some low-dimensional evolution operators including the mutation, crossover, replacement, and selection are introduced to achieve the evolution of the population.Particularly, these operators are different from traditional DE.Due to the usage of a new encoding mechanism, NP is equal to the element number K in each fold of rotationally symmetric array.

•
Mutation: For the individual → X i , the candidate individual → V i can be generated by using the following mutation operator: where F > 0 denotes the scaling factor, where i denotes the i-th individual in the population, m∈{1,2}, rand(0, 1) represents a random number ranging from 0 to 1, and CR denotes the crossover constant.When rand(0, 1) < CR, we randomly select one dimension of a candidate individual as a trial individual and remain unchanged for another dimension.Otherwise, we remain the current individual.

•
Replacement: When mutation and crossover operators are used, K offspring individuals are generated to make up an offspring population Q.To improve the global search ability of the proposed algorithm, each individual → U i in the offspring population Q will randomly select an individual in the population P to replace, and the generated new population is expressed as S. In particular, the minimum element spacing constraint is considered, thus we need to check the new population S whether meets meets this constraint, S will compete with the original population P.

•
Selection: According to the UWB array synthesis problem concerned in this study, the fitness function is set as the PSL of the array pattern at the highest frequency.That is, It can be observed that, the population P determines the fitness function value, which is greatly different from the traditional DE using a certain individual to achieve the fitness function value.When an individual changes, we need to assess its fitness function value generated by whole population, and select the better one with the lower PSL to retain by the following selection operator:

The Proposed MDEA Procedure
Firstly, we randomly select K element positions from a uniformly spaced planar array with the spacing of half a wavelength at the lowest frequency as the initial population P. Obviously, this can quickly and simply generate an initial population that meets the given minimum spacing and array aperture constraints.Then, we calculate the contribution of each element in the first fold and the corresponding (M − 1) rotationally symmetric elements to the array pattern and store them.Moreover, the PSL for the initial population P is calculated by Equation (6).At each iteration, for the individual → X i , the mutation and crossover operators in Equations ( 4) and ( 5) are applied to generate a trial individual → U i , which will generate an offspring population Q.Then, the replacement operator is used to obtain the new population S.Moreover, the selection operator is performed on the population P and S.Meanwhile, we check the obtained array layout to determine whether it meets the minimum element spacing constraint in this process.When the termination condition is met, the final population with the lowest PSL is the output.The proposed MDEA procedure is detailed in Algorithm 1.

Algorithm 1
The proposed MDEA synthesis procedure for large UWB sparse planar array based on the rotationally symmetric structure 1.
Set some parameters including K, F, CR and MaxFEs, d const ; 2.
Randomly select K element positions from a uniformly spaced planar array with the element spacing of half a wavelength at the lowest frequency as the initial population, represents an element position expressed by the polar radius and angle in polar coordinates; 3.
Calculate the PSL of the population P by Equation ( 6) and store the contribution of each element in the first fold and the corresponding (M − 1) rotationally symmetric elements to the array pattern.

Generate the candidate individual
→ V i by using the mutation operator in Equation ( 4) and obtain the trial individual → U i by using the crossover operator in Equation (5); 6.
Perform the replacement operator for the offspring population Q, the obtained population is expressed as S; 7.
Check the new population S whether meets the required minimum element spacing constraint; if meet, the selection operator in Equation ( 6) is applied into the population S and P. Otherwise, the population P keeps unchanged.Then, the one with the lower PSL is retained, and FEs = FEs + 1; 8.
Return the final population P.

Numerical Experiments
In this section, two examples of ultra-wideband (UWB) rotationally symmetric sparse circular planar array synthesis with different bandwidths and aperture sizes are conducted to validate the effectiveness of the proposed MDEA.In all examples, the scaling factor is set as F = 0.5, and the crossover constant is set as CR = 0.9.Moreover, NP is equal to K. All simulations are executed by using MATLAB (R2016a) software installed on a computer with an Intel(R) Core(TM) i7-1165G7 @ 2.80 GHz and 8.00 GB of RAM.

UWB Sparse Planar Array with 600 Elements and 5:1 Bandwidth
Firstly, we consider synthesizing a UWB sparse planar array with 600 elements and 5:1 bandwidth.Moreover, this array is bounded with a circular area of radius R = 12 λ L = 60 λ H where λ H and λ L denotes the wavelength at the highest and lowest frequency, respectively.In addition, the minimum element spacing of 0.5 λ L is required.
In [15], a Penrose tiling array with 551 elements and 5:1 bandwidth was synthesized by using the genetic algorithm perturbation technique.Moreover, the obtained PSLs at the highest and lowest frequency were −10.35 dB and −16.64 dB, respectively.In [17], the covariance matrix adaption evolutionary strategy was used to synthesize this example with rotationally symmetric structure, and the obtained PSL at the highest frequency was −18.85 dB.However, the optimization time for this example was 240 h on six 2.4 GHz intel Xeon processor cores.Recently, an improved differential evolution algorithm in [20] was introduced to synthesize a randomly sparse circular planar array with the same element number and bandwidth as this example.Finally, the obtained PSLs at the highest and lowest frequency were −18.35 dB and −18.21 dB, respectively.
In this article, we use the proposed MDEA to synthesize the above-mentioned example, but the array possesses rotationally symmetric structure with M folds.As introduced in the problem description section, the UWB array synthesis problem can be converted as minimizing the PSL of the array factor at the highest frequency.In the proposed MDEA, some parameters are set as follows: MaxFEs = 20, 000, d const = 2.5 λ H , M = 15.Under these parameter settings, the proposed MDEA is performed five times, then the obtained lowest PSL is −20.12 dB. Figure 3 shows the corresponding array layout, and it can be checked that the obtained minimum element spacing is 2.5 λ H , which meets the given requirement.Figure 4a,b show the array pattern at the highest frequency f H and cuts at φ = 0 • , 30 • , 60 • , 90 • , respectively.Compared −10.58 dB in [15], −18.85 dB in [17], and −18.35 dB in [20], the obtained lowest PSL in MDEA is −20.12 dB, which shows the advantage of 9.54 dB, 1.24 dB, and 1.77 dB, respectively.Figure 5a,b show the array pattern at the lowest frequency f L and cuts at φ = 0 • , 30 60 H where  H and  L denotes the wavelength at the highest and lowest frequency, respectively.In addition, the minimum element spacing of 0.5 L is required.In [15], a Penrose tiling array with 551 elements and 5:1 bandwidth was synthesized by using the genetic algorithm perturbation technique.Moreover, the obtained PSLs at the highest and lowest frequency were −10.35 dB and −16.64 dB, respectively.In [17], the covariance matrix adaption evolutionary strategy was used to synthesize this example with rotationally symmetric structure, and the obtained PSL at the highest frequency was −18.85 dB.However, the optimization time for this example was 240 h on six 2.4 GHz intel Xeon processor cores.Recently, an improved differential evolution algorithm in [20] was introduced to synthesize a randomly sparse circular planar array with the same element number and bandwidth as this example.Finally, the obtained PSLs at the highest and lowest frequency were −18.35 dB and −18.21 dB, respectively.
In this article, we use the proposed MDEA to synthesize the above-mentioned example, but the array possesses rotationally symmetric structure with  folds.As introduced in the problem description section, the UWB array synthesis problem can be converted as minimizing the PSL of the array factor at the highest frequency.In the proposed MDEA, some parameters are set as follows: MaxFEs = 20,000 ,  const = 2.5 H ,  = 15 .Under these parameter settings, the proposed MDEA is performed five times, then the obtained lowest PSL is −20.12 dB. Figure 3 shows the corresponding array layout, and it can be checked that the obtained minimum element spacing is 2.5 H , which meets the given requirement.Figure 4a,b show the array pattern at the highest frequency  H and cuts at ϕ = [0 °, 30 °, 60 °, 90 °] , respectively.Compared −10.58 dB in [15], −18.85 dB in [17], and −18.35 dB in [20], the obtained lowest PSL in MDEA is −20.12 dB, which shows the advantage of 9.54 dB, 1.24 dB, and 1.77 dB, respectively.Figure 5a,b show the array pattern at the lowest frequency  L and cuts at ϕ = [0 °, 30 °, 60 °, 90 °], respectively.The synthesis results verify the effectiveness of the proposed MDEA.

2000-Element Square Kilometer Array (SKA) Operating over 70 MHz to 450 MHz
In the second example, we synthesize a large-scale square kilometer array (SKA) considering a low frequency part ranged from 70 MHz to 450 MHz, which is a new generation of radio telescopes.Especially, this array has as many as 2000 elements distributed over a large circular area.For a UWB planar array with such a large number of elements and wide bandwidth, it is very difficult to achieve the array synthesis.In [21], a non-iteration method was introduced by applying simple controlled randomization and the space tapering of the aperture distribution to synthesize large SKA arrays for different geometries.When restricting the array to a circular aperture with a radius of 28.3 λ L and considering the randomization case with large d const , the obtained PSL was −17 dB over the whole bandwidth (70 MHz to 450 MHz).Moreover, the half-power beamwidth (HPBW) of the array pattern at 70 MHz was 1.1 • .

2000-Element Square Kilometer Array (SKA) Operating over 70 MHz to 450 MHz
In the second example, we synthesize a large-scale square kilometer array (SKA) considering a low frequency part ranged from 70 MHz to 450 MHz, which is a new generation of radio telescopes.Especially, this array has as many as 2000 elements distributed over a large circular area.For a UWB planar array with such a large number of elements and wide bandwidth, it is very difficult to achieve the array synthesis.In [21], a non-iteration method was introduced by applying simple controlled randomization and the space tapering of the aperture distribution to synthesize large SKA arrays for different geometries.When restricting the array to a circular aperture with a radius of 28.3 L and considering the randomization case with large  const , the obtained PSL was −17 dB over the whole bandwidth (70 MHz to 450 MHz).Moreover, the half-power beamwidth (HPBW) of the array pattern at 70 MHz was 1.1 °.
In this example, the proposed MDEA is applied to synthesize a UWB sparse planar array with some of the same constraints as the above-mentioned one in [21].In the proposed MDEA, the parameters are set as follows:  const = 3.21 H , MaxFEs = 20,000, and  = 25.As a result, it takes 60.9 h to accomplish the synthesis for this example.The obtained lowest PSL at 450 MHz is −19.46 dB, and the corresponding SKA array layout with 2000 elements is shown in Figure 6.It can be seen that this array layout meets the minimum element spacing constraint.At 450 MHz, the corresponding PSL for the optimized array in Figure 6 is −19.46 dB, which is 2.46 dB lower than −17 dB in [21].Figure 7a,b show the array pattern at 450 MHz, and its cuts at ϕ = [0 °, 30 °, 60 °, 90 °], respectively.At 70 MHz,

2000-Element Square Kilometer Array (SKA) Operating over 70 MHz to 450 MHz
In the second example, we synthesize a large-scale square kilometer array (SKA) considering a low frequency part ranged from 70 MHz to 450 MHz, which is a new generation of radio telescopes.Especially, this array has as many as 2000 elements distributed over a large circular area.For a UWB planar array with such a large number of elements and wide bandwidth, it is very difficult to achieve the array synthesis.In [21], a non-iteration method was introduced by applying simple controlled randomization and the space tapering of the aperture distribution to synthesize large SKA arrays for different geometries.When restricting the array to a circular aperture with a radius of 28.3 L and considering the randomization case with large  const , the obtained PSL was −17 dB over the whole bandwidth (70 MHz to 450 MHz).Moreover, the half-power beamwidth (HPBW) of the array pattern at 70 MHz was 1.1 °.
In this example, the proposed MDEA is applied to synthesize a UWB sparse planar array with some of the same constraints as the above-mentioned one in [21].In the proposed MDEA, the parameters are set as follows:  const = 3.21 H , MaxFEs = 20,000, and  = 25.As a result, it takes 60.9 h to accomplish the synthesis for this example.The obtained lowest PSL at 450 MHz is −19.46 dB, and the corresponding SKA array layout with 2000 elements is shown in Figure 6.It can be seen that this array layout meets the minimum element spacing constraint.At 450 MHz, the corresponding PSL for the optimized array in Figure 6 is −19.46 dB, which is 2.46 dB lower than −17 dB in [21].Figure 7a,b show the array pattern at 450 MHz, and its cuts at ϕ = [0 °, 30 °, 60 °, 90 °], respectively.At 70 MHz, In this example, the proposed MDEA is applied to synthesize a UWB sparse planar array with some of the same constraints as the above-mentioned one in [21].In the proposed MDEA, the parameters are set as follows: d const = 3.21 λ H , MaxFEs = 20, 000, and M = 25.As a result, it takes 60.9 h to accomplish the synthesis for this example.The obtained lowest PSL at 450 MHz is −19.46 dB, and the corresponding SKA array layout with 2000 elements is shown in Figure 6.It can be seen that this array layout meets the minimum element spacing constraint.At 450 MHz, the corresponding PSL for the optimized array in Figure 6 is −19.46 dB, which is 2.46 dB lower than −17 dB in [21].Figure 7a,b show the array pattern at 450 MHz, and its cuts at φ = [0 • , 30 • ] are shown in Figure 8.Meanwhile, the HPBW of 1.1 • is nearly same as 1.1 • in [21].
From the synthesis results, it can be included that, although the non-iteration algorithm in [21] does not take much time, the lack of the optimization results in a poor radiation performance.The proposed MDEA shows a 2.46 dB PSL improvement than that in [21] while nearly keeping the same beamwidth.Moreover, the cost time of the proposed MDEA is only 60.9 h for such large UWB planar array.Therefore, this approach can be regarded as an efficient method in stochastic optimization methods for such arrays with large size and wide bandwidth.

Conclusions
In this article, we have introduced a modified differential evolution algorithm denoted as MDEA to achieve the synthesis of large UWB sparse circular planar arrays based on the rotationally symmetric structure.In particular, the proposed MDEA adopts a new encoding mechanism in which an individual represents an element position, which can greatly reduce the calculation time for the array factor in the individual being updating and facilitate the

Figure 1 .
Figure 1.Rotationally symmetric sparse circular planar array with N elements.
Figure 2b shows this new encoding mechanism that the whole population expressed as  = {[ 1 ,  1 ], [ 2 ,  2 ], ⋯ , [  ,   ]} represents an array layout.Thus, each individual is always kept at two dimensions, and NP is equal to .Moreover, the variation in an individual only causes a change in one element position in each fold of the whole array.When the individual has been updated, only one element and its ( − 1) rotationally symmetric elements for the contribution to the array pattern has been changed, which greatly saves time for the evaluation of large 2D array pattern.In addition, candidate individuals in such an encoding mechanism have more possibility to satisfy the given minimum spacing constraint.(a) (b)

Figure 2 .
Figure 2. Different encoding mechanism in stochastic optimization algorithm.(a) Traditional encoding mechanism, (b) new encoding mechanism.

→X
min denotes an individual closest to → X i , and → X i − → X min represents the opposite operator.In particular, such an operator distances the generated individual from the individual → X i , which has a greater probability of satisfying the given minimum element spacing constraint and contributes to the enhancement of its global search ability.• Crossover: In the new encoding mechanism, the variable dimension of the individual is always two.Faced with such a low-dimension individual, we should greatly exploit each dimension to enhance the local search ability of the proposed algorithm.Therefore, for the candidate individual → V i and → X i , a trial individual → U i is generated by the following crossover operator: