# Shared Aperture Multibeam Forming of Time-Modulated Linear Array

Authors: Li-juan Sun; Zhen-kai Zhang; Hamid Esmaeili Najafabadi
Journal: Complexity
 (2019)

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## Abstract

A novel technique is proposed in this paper for shared aperture multibeam forming in a complex time-modulated linear array. First, a uniform line array is interleaved randomly to form two sparse array subarrays. Subsequently, the theory of time modulation for linear arrays is applied in the constructed subarrays. In the meantime, the switch-on time sequences for each element of the two subarrays are optimized by an optimized differential evolution (DE) algorithm, i.e., the scaling factor of the sinusoidal iterative chaotic system and the adaptive crossover probability factor are used to enhance the diversity of the population. Lastly, the feasibility of the new technique is verified by the comparison between this technique and the basic multibeam algorithm in a shared aperture and the algorithm of iterative FFT. The results of simulations confirm that the proposed algorithm can form more desired beams, and it is superior to other similar approaches.


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## Body

## 1. Introduction

In recent years, electromagnetic environments have been more complicated due to the increased number and complexity of antennas, and they have been vital to many fields [1, 2]. Shared aperture technology is one of the feasible approaches to achieve multifunction application of the array antenna [3, 4]. This technology enables two or more antenna subarrays to occupy one aperture by space-division multiplexing, in which each subarray can perform different functions. This method can not only reduce the number of antennas, but also effectively avoid electromagnetic compatibility problems between the antenna units. Recently, some promising results in sharing aperture antennas have been achieved. In [5–8], the differential set theory was applied to implement the sparsely interlaced array of multi-subarrays, in which the pattern of each subarray is quite different and requires further optimization. In [9–11], using the Fourier transform between the array element excitation and the pattern, a multi-subarray interleaved shared aperture design method was proposed based on spectral energy matching of the subarray antenna pattern, in which the energy of the antenna pattern distributes to each subarray evenly. However, the yielded subarray pattern exhibits a higher peak sidelobe level (PSL), so it cannot flexibly control beam pointing. The literature [12, 13] discussed the method to reduce the PSL in linear antenna arrays by genetic algorithms (GA), whereas it cannot be applicable to the shared aperture multibeam forming. The literature [14–17] also employed intelligent algorithms to optimize the location of array elements. Though the above-shared aperture methods can implement sparsely interleaved multi-subarray, there are also unsolved problems (e.g., the large difference in the pattern of each subarray, high PSL, and inflexible beam steering).In this paper, a new shared aperture method is proposed based on the time-modulated linear array (TMLA) for shared aperture multibeam forming. TMLA has been adopted as a means of synthesizing low/ultralow sidelobe patterns exhibiting a simple system structure and low cost. At present, the research based on TMLA is primarily about sideband suppression, pattern synthesis, and harmonic beamforming compared with conventional antenna arrays. The TMLA introduces a time variable into the conventional array antenna to control its performance. Each element is connected to a Radio Frequency (RF) switch. The excitation of the antenna array element and aperture excitation distribution is controlled by controlling the working state of the RF switch in the access array to achieve harmonic beamforming. The amplitude of each harmonic component is consistent with the harmonic distribution ofsin⋅ function, and the magnitudes of low-order harmonic components are relatively large. Accordingly, it is more suitable to select only low-order harmonic components for beamforming. In this study, the first harmonic is selected for beamforming. Traditional shared aperture can only create two beams while forming two subarrays. In contrast, the shared aperture based on the TMLA can create five different desired beams using the random sparse interleaving. By optimizing the control time sequence by an optimized differential evolution algorithm, the PSL of each beam can be further reduced. In recent years, numerous studies have been published on the pattern synthesis of time-modulated array antennas. The literature [18–20] studied the synthesis of the TMLA patterns. The synthesis of simultaneous and multiple harmonic beams for applications (e.g., beam steering) has been already investigated [21, 22] to enhance the reliability of the communication system. The literature [23–26] presented pattern synthesis and steerable energy patterns in a two-dimensional planar time modulation array. Besides, in recent works, novel timed arrays of linear polarised Vivaldi antennas have been proposed [27]. Novel techniques for circularly polarised UWB-scanned patterns were explored with low sidelobes in [28]. Unlike [18–26], the present study can create beams at more than two different desired directions. Thus, the TMLA is adopted to solve the shared aperture multibeam forming problems for the first time.The rest of this paper is organized as follows. In Section2, the mathematical model of the time-modulated linear array is presented. In Section 3, the theory of the novel shared aperture multibeam forming technique is explained, and the optimized DE algorithm is presented. In Section 4, simulations are performed to assess the performance of the proposed method. Lastly, the conclusion is drawn in Section 5.


## 2. Mathematical Model of Time-Modulated Linear Array

A uniform linear array (ULA) withN elements is considered, in which the element spacing d is 1/2 wavelength, and the array elements are ideal omnidirectional units. The radiation pattern is expressed as(1)Fθ=∑n=0N−1Anejnβdsinθ,where An denotes the static excitation amplitude of the n-th element; β=2π/λ the wavenumber; d the element spacing; and θ the beam scanning angle. According to TMLA, when all N elements are time modulated, equation (1) can be rewritten as(2)Fθ=∑n=0N−1AnUntejnβdsinθ,where Unt denotes the periodic signal of the n-th element. The structure of time-modulated antenna array is shown in Figure 1.Figure 1
Basic structure for time modulation array.In this paper, the array is controlled by the RF switch connected to each array element. The waveform of the periodic control timing of then-th array element is shown in Figure 2, and its mathematical expression is(3)Unt=1,ton≤t≤ton+td,0,otherwise,where ton denotes the instant time of the switch opening and td the duration of the switch closure time.Figure 2
Control timing of then-th array element.TheUnt denotes a gate function, which can be expanded to a Fourier series according to the nature of the periodic functions as expressed below:(4)Unt=∑k=−∞∞cnkej2πkfmt,where Tm denotes the time modulation period; fm=1/Tm modulation frequency; and cnk the complex Fourier coefficient of k-th harmonic, which is expressed as(5)cnk=sinπkfmtdπke−jkπfm2ton,n+td,n,k≠0,fmtd,k=0.Equation (5) suggests that the phase is always zero when k=0, i.e., the fundamental component always points to 0°. When k≠0, Fourier coefficients of the positive k-th harmonic component and the negative k-th harmonic component are symmetric with respect to the array axis. Accordingly, the pattern of the TMLA is expressed as(6)Fkθ=∑n=0N−1Ansinπkfmtde−jkπ2ton,n+td,nejnβdsinθ.Equation (6) shows that the k-th harmonic component generated by TMLA is a complex number. The excitation of the harmonic component is associated with the duration of the opening of the RF switch and the harmonic order k, while its phase is only associated with the central moment of the opening and closing times. Thus, when beamforming is performed by TMLA, it is appropriate to select lower-order harmonic components. The positive and negative first harmonics are taken only to point to the desired direction. Thus, the fundamental and positive and negative first harmonic components of the pattern are calculated as follows:(7)F0θ=∑n=0N−1AnTdnejnβdsinθ,(8)F1θ=∑n=0N−1AnsinπTdn×e−jπ2Tsn+Tdn×ejnβdsinθ,(9)F−1θ=∑n=0N−1AnsinπTdn×ejπ2Tsn+Tdn×ejnβdsinθ,where Tsn=ton,n/Tm,Tdn=td,n/Tm. By scrutinizing (7)–(9), it can be observed that since the range of the sinx function is 0,1, the range of sinπTdn is also 0,1, suggesting that Tdn varies in 0,0.5. To make the beam phase shift vary in −π,π, Tdn should adopt a value in −0.5,0.5. For the first harmonic, Gn = Ancn1=AnsinπTdn, where Gn denotes the amplitude of the beam pattern, and it is assumed that Gn is a given random sequence in [0, 1]. Besides, denote the direction of the first harmonic component by θ0. Subsequently, the control sequence is expressed as(10)Tdn=1π⋅arcsinGnAn,(11)Tsn=12nβdsinθ0π−Tdn.From (6), it can be observed that the excitation and phase synthesis of the harmonic components can be obtained by adjusting the control sequences. Accordingly, the harmonic excitation of the TMLA can be utilized for the beamforming. The proposed TMLA-based shared aperture technique is based on sparse interleaving to yield two subarrays. By controlling the control sequences of the two subarrays, each subarray can yield beams with different directions.In the meantime, the PSL of the antenna pattern is expressed as(12)PSL0=maxF0θθ∈S0,PSL1=maxF1θθ∈S1,PSL−1=maxF−1θθ∈S−1,PSL2=maxF2θθ∈S2,PSL−2=maxF−2θθ∈S−2,where F0θ,F1θ,F−1θ,F2θ, and F−2θ denote the fundamental wave of the subarray 1, the positive first-order harmonics of the subarray 1, the negative first-order harmonics of the subarray 1, the positive first-order harmonics of the subarray 2, and the negative first-order harmonics of the subarray 2, respectively. Besides, S0,S1,S−1,S2, and S−2 refer to the sidelobe regions of the corresponding beams, respectively. The following objective function is proposed as(13)fx=αPSL0−SLL+βPSL1−SLL+γPSL2−SLL,where PSL0, PSL1, and PSL2 denote the PSL values for the fundamental, subarray 1, and subarray 2, respectively. SLL represents the desired value for the PSL. Also, α, β, and γ are the corresponding weight coefficient values. The multibeam PSL formed by this method is higher and will further be optimized by the optimized differential evolution (DE) algorithm.


## 3. Shared Aperture Multibeam Forming Optimization

In this section, the theory of the novel shared aperture multibeam forming technique is presented, and the DE algorithm is optimized here.


### 3.1. Differential Evolution Algorithm

#### 3.1.1. Standard Differential Evolution Algorithm

DE refers to a population-based global optimization algorithm. It adopts a random search strategy and uses differential information between different individuals to guide the evolution of the entire population. Besides, it employs the greedy selection mechanism to control the evolution direction and search for the optimal solution that satisfies the conditions. The advantage of the DE algorithm is that its control parameters are relatively small, simple to implement, reliable, and fast [29, 30]. The use of differential information of individual populations to achieve variation exhibits better stability. In the generation process of the offspring, the greedy competition mechanism is adopted to ensure the evolution of the population [31–34]. The DE method is selected as the optimization algorithm for its advantage of simplicity and efficiency, as it can exploit the greedy selection mechanism to control the evolution direction and search for the optimal solution that satisfies the conditions. Furthermore, this method can be optimized straightforwardly. Without the loss of generality, the solution for a given D-dimensional continuous optimization problem is as follows:(14)minfxs.t.x∈S⊂RD,where D denotes the individual dimension, i.e., the number of elements and S is the search domain. The algorithm flow is presented as follows:(1)
Set parameters, which include the population sizeNP, the maximum iteration number GM and the mutation factor F, and the hybridization probability CR; an initial population is yielded, and the running algebra is set as G=0.(2)
The element excitation amplitude is initialized to a random number within 0-1, and the control sequencesTdn and Tsn are calculated by equations (10) and (11). Set the duration Tdn as the optimization variable; the initial population is given by(15)XG=x1G,x2G,x3G,…,xNPG,whereG denotes the number of iterations; the variation range of X is controlled at 0–0.5; the i-th vector in the group is(16)xiG=xi,1G,xi,2G,xi,3G,…,xi,DG.To make the initial population cover the entire search spaceRD, it is initialized as(17)xi,j0=xminj+rand0,1xmaxj−xminj,wherexminj and xmaxj denote the extrema of the optimization variable. Also, rand(0, 1) denotes a random number in [0, 1].(3)
Two individuals different from the individual are randomly selected in the initial population; the difference vector is multiplied by the mutation operatorF and then added to the mutant individual:(18)xv,iG=xr1G+Fxr2G−xr3G,where r1, r2, and r3 represent three different numbers in [1, Np].(4)
The cross operation of the vector after the mutation and the parent vector can enhance the diversity of the population. Common cross equations are presented as follows:(19)xu,iG+1=xv,iG,rand≤CR,xiG,otherwise,where rand is a random number.(5)
The selection operation follows the “greedy” search strategy and adopts individuals with small adaptation values as descendants, which is expressed as(20)xiG+1=xu,iG,fxu,iG≤fxiG,xiG,otherwise,wheref(x) denotes the objective function, which can be calculated by the above equations (12) and (13). Lastly, the above operations are repeated until the best solution is selected.


#### 3.1.2. Optimized Differential Evolution Algorithm

The DE algorithm has downsides, e.g., slow convergence rate and easy to fall into local optimum. Especially in the late stage of its evolution, the population diversity and the convergence speed decrease, which leads to the algorithm falling into local optimum [35]. In this study, an optimized DE algorithm is proposed, exploiting the scaling factor of the sinusoidal iterative chaotic system and the adaptive crossover probability factor (CR) to enhance the diversity of the population. The sinusoidal chaotic random sequence [36] refers to a variable that fluctuates within the interval [0, 1]. It exhibits randomness, ergodicity, aperiodicity, and extreme sensitivity to initial values, making it suitable for generating nonrepeating pseudorandom numbers in the range [0, 1]. When the chaotic random sequence is employed for the population initialization and optimization search, the algorithm is capable of verifiably avoiding from falling into the optimal local solution.The scaling factor based on a sinusoidal iterative chaotic system is given by(21)Fn=sin2carcsinFn−1,where c denotes the weight coefficient, generally chosen from the interval [0, 3]; also, the initial value of the mutation operator is chosen randomly, i.e., F0∈rand0,1.The adaptive CR is expressed as(22)CRG=CRl+Ch−CRlGm−GGm,where CRh and CRl denote the maximum and minimum values of CR, respectively. After numerous experiments, CRl = 0.1 and CRh = 0.9 are applicable to most optimization problems. funmax and funmin are the maximum and minimum values of the objective function, respectively. Gm indicates the maximum number of iterations, and G the current number of iterations. The optimized DE algorithm flow chart is shown in Figure 3.Figure 3
The optimized DE algorithm flow chart.


### 3.2. Shared Aperture Multibeam Forming Design Algorithm

A shared aperture sparsely interleaved array is capable of reducing the number of antennas, saving space, and reducing load and manufacturing costs. In this study, the TMLA is employed to form more beams from multiple subarrays based on the interleaved shared aperture. The combination of shared aperture technology and the TMLA can yield more beams based on the same antenna resources and enhance the efficiency of the antenna utilization. The following is the locational distribution of the array elements with sparsely interleaved apertures, which is expressed asSubarray 1: 100110001011010110100101010110010101010110100110000101010110Subarray 2: 011001110100101001011010101001101010101001011001111010101001where 1 denotes an array of elements, 0 means free here. A uniform linear array is considered with N elements. N elements are chosen randomly from 0 to 1, where 0 denotes the elements of subarray 1, and 1 the array elements of subarray 2. After the two subarrays are formed by random sparse interleaving, the excitations for the two subarrays are initialized to the number Gn in [0, 1] randomly. The proposed algorithm for multiple beamforming is presented in Algorithm 1.Algorithm 1: Shared aperture multibeam forming for time-modulated arrays based on optimized DE algorithm.
Input: The array number N, the number of samples L, target directions, the static excitation amplitude Ak, the parameters of the optimized DE algorithm Np,Gm, CR, and F.
Initialization: Randomly and sparsely interlace the line arrays of the N array elements into two subarrays, and initialize the excitation of the two subarrays.
CalculateF0θ,F1θ,F−1θ,F2θ, and F−2θ using (7)–(9). Then Tsn and Tdn can be calculated by (10) and (11). Lastly, the objective function value can be obtained by (11)–(13).
Iteration: While
G < Gm
Adaptively updating the value of CR andF using (21) and (22)
Generating variant individuals
Cross operation
Select the mutated individuals based on the objective function
IffxG_next<fxG
record the optimal value off and xG_next, then update
end loop.
End for
Output: Final weight vector x1,x2 and f.


### 3.3. Multibeam Forming Based on Iterative FFT Algorithm

The iterative FFT algorithm exploiting the feature of the relationship between the uniform linear array element excitation and the Fourier transform converts the problem to determine the location of the subarray element into the problem to select the excitation energy of the array element. This method, based on the principle of density-weighted matrix and cross-selection, ascertain the location of the sparse subarray unit and achieve the uniform distribution of spectral energy of the subarray. In such a way, multibeam forming based on sparse interlaced arrays can be achieved.Considering a linear array as shown in equation (1), the discrete inverse Fourier transform (DIFT) can be expressed as(23)ft=1M∑n=0M−1Fnejntω0,where M denotes the period and ω0=2π/M is the sampling interval. It can be observed from equations (1) and (23) that there is a Fourier transform relationship between the excitation A of the array element and the array factor F.Since there exists a Fourier transform relationship between the excitation and the pattern of the array antenna element, the fast Fourier transform (FFT) method can be adopted to convert the unit excitation and the pattern, significantly reducing the calculation amount and saving computing time. Taking the antenna design withT subarrays interlaced as an example, the specific steps are presented as follows:(1)
Set the array excitation valueI to 1 according to a uniform linear array with sparse rate R=1/T random sparse array elements N.(2)
Perform an inverse FFT ofQ point on I to calculate P.(3)
Determine the value of the sidelobe region ofP, so the point value on the region larger than the constraint sidelobe value equals to the constraint sidelobe value, and the values at other points remain unchanged.(4)
Perform an FFT conversion of theQ point on the corrected P to yield a new excitation value I.(5)
Intercept the firstN values in I, yield a new excitation matrix, arrange them from large to small, and yield a new excitation vector.(6)
The excitation at the locationIri,Iri+T,…,Ir+T×K corresponds to the location on I as the location where the i-th subarray unit is located, and set the excitation value at the location as 1.(7)
Perform an inverse FFT on the excitation vector of the new subarray 1, determine whether the edge lobe peak of the updated antenna pattern changes with respect to the sidelobe peak at the previous iteration, and then proceeds to the next calculation; otherwise, output the result.(8)
The excitation sequence of the subarray 1 is entered as the new input value into the next iteration. Steps (2) to (8) are repeated until the newly generated antenna pattern sidelobe peaks no longer change, and output the optimized result.


## 3.1. Differential Evolution Algorithm

### 3.1.1. Standard Differential Evolution Algorithm

DE refers to a population-based global optimization algorithm. It adopts a random search strategy and uses differential information between different individuals to guide the evolution of the entire population. Besides, it employs the greedy selection mechanism to control the evolution direction and search for the optimal solution that satisfies the conditions. The advantage of the DE algorithm is that its control parameters are relatively small, simple to implement, reliable, and fast [29, 30]. The use of differential information of individual populations to achieve variation exhibits better stability. In the generation process of the offspring, the greedy competition mechanism is adopted to ensure the evolution of the population [31–34]. The DE method is selected as the optimization algorithm for its advantage of simplicity and efficiency, as it can exploit the greedy selection mechanism to control the evolution direction and search for the optimal solution that satisfies the conditions. Furthermore, this method can be optimized straightforwardly. Without the loss of generality, the solution for a given D-dimensional continuous optimization problem is as follows:(14)minfxs.t.x∈S⊂RD,where D denotes the individual dimension, i.e., the number of elements and S is the search domain. The algorithm flow is presented as follows:(1)
Set parameters, which include the population sizeNP, the maximum iteration number GM and the mutation factor F, and the hybridization probability CR; an initial population is yielded, and the running algebra is set as G=0.(2)
The element excitation amplitude is initialized to a random number within 0-1, and the control sequencesTdn and Tsn are calculated by equations (10) and (11). Set the duration Tdn as the optimization variable; the initial population is given by(15)XG=x1G,x2G,x3G,…,xNPG,whereG denotes the number of iterations; the variation range of X is controlled at 0–0.5; the i-th vector in the group is(16)xiG=xi,1G,xi,2G,xi,3G,…,xi,DG.To make the initial population cover the entire search spaceRD, it is initialized as(17)xi,j0=xminj+rand0,1xmaxj−xminj,wherexminj and xmaxj denote the extrema of the optimization variable. Also, rand(0, 1) denotes a random number in [0, 1].(3)
Two individuals different from the individual are randomly selected in the initial population; the difference vector is multiplied by the mutation operatorF and then added to the mutant individual:(18)xv,iG=xr1G+Fxr2G−xr3G,where r1, r2, and r3 represent three different numbers in [1, Np].(4)
The cross operation of the vector after the mutation and the parent vector can enhance the diversity of the population. Common cross equations are presented as follows:(19)xu,iG+1=xv,iG,rand≤CR,xiG,otherwise,where rand is a random number.(5)
The selection operation follows the “greedy” search strategy and adopts individuals with small adaptation values as descendants, which is expressed as(20)xiG+1=xu,iG,fxu,iG≤fxiG,xiG,otherwise,wheref(x) denotes the objective function, which can be calculated by the above equations (12) and (13). Lastly, the above operations are repeated until the best solution is selected.


### 3.1.2. Optimized Differential Evolution Algorithm

The DE algorithm has downsides, e.g., slow convergence rate and easy to fall into local optimum. Especially in the late stage of its evolution, the population diversity and the convergence speed decrease, which leads to the algorithm falling into local optimum [35]. In this study, an optimized DE algorithm is proposed, exploiting the scaling factor of the sinusoidal iterative chaotic system and the adaptive crossover probability factor (CR) to enhance the diversity of the population. The sinusoidal chaotic random sequence [36] refers to a variable that fluctuates within the interval [0, 1]. It exhibits randomness, ergodicity, aperiodicity, and extreme sensitivity to initial values, making it suitable for generating nonrepeating pseudorandom numbers in the range [0, 1]. When the chaotic random sequence is employed for the population initialization and optimization search, the algorithm is capable of verifiably avoiding from falling into the optimal local solution.The scaling factor based on a sinusoidal iterative chaotic system is given by(21)Fn=sin2carcsinFn−1,where c denotes the weight coefficient, generally chosen from the interval [0, 3]; also, the initial value of the mutation operator is chosen randomly, i.e., F0∈rand0,1.The adaptive CR is expressed as(22)CRG=CRl+Ch−CRlGm−GGm,where CRh and CRl denote the maximum and minimum values of CR, respectively. After numerous experiments, CRl = 0.1 and CRh = 0.9 are applicable to most optimization problems. funmax and funmin are the maximum and minimum values of the objective function, respectively. Gm indicates the maximum number of iterations, and G the current number of iterations. The optimized DE algorithm flow chart is shown in Figure 3.Figure 3
The optimized DE algorithm flow chart.


## 3.1.1. Standard Differential Evolution Algorithm

DE refers to a population-based global optimization algorithm. It adopts a random search strategy and uses differential information between different individuals to guide the evolution of the entire population. Besides, it employs the greedy selection mechanism to control the evolution direction and search for the optimal solution that satisfies the conditions. The advantage of the DE algorithm is that its control parameters are relatively small, simple to implement, reliable, and fast [29, 30]. The use of differential information of individual populations to achieve variation exhibits better stability. In the generation process of the offspring, the greedy competition mechanism is adopted to ensure the evolution of the population [31–34]. The DE method is selected as the optimization algorithm for its advantage of simplicity and efficiency, as it can exploit the greedy selection mechanism to control the evolution direction and search for the optimal solution that satisfies the conditions. Furthermore, this method can be optimized straightforwardly. Without the loss of generality, the solution for a given D-dimensional continuous optimization problem is as follows:(14)minfxs.t.x∈S⊂RD,where D denotes the individual dimension, i.e., the number of elements and S is the search domain. The algorithm flow is presented as follows:(1)
Set parameters, which include the population sizeNP, the maximum iteration number GM and the mutation factor F, and the hybridization probability CR; an initial population is yielded, and the running algebra is set as G=0.(2)
The element excitation amplitude is initialized to a random number within 0-1, and the control sequencesTdn and Tsn are calculated by equations (10) and (11). Set the duration Tdn as the optimization variable; the initial population is given by(15)XG=x1G,x2G,x3G,…,xNPG,whereG denotes the number of iterations; the variation range of X is controlled at 0–0.5; the i-th vector in the group is(16)xiG=xi,1G,xi,2G,xi,3G,…,xi,DG.To make the initial population cover the entire search spaceRD, it is initialized as(17)xi,j0=xminj+rand0,1xmaxj−xminj,wherexminj and xmaxj denote the extrema of the optimization variable. Also, rand(0, 1) denotes a random number in [0, 1].(3)
Two individuals different from the individual are randomly selected in the initial population; the difference vector is multiplied by the mutation operatorF and then added to the mutant individual:(18)xv,iG=xr1G+Fxr2G−xr3G,where r1, r2, and r3 represent three different numbers in [1, Np].(4)
The cross operation of the vector after the mutation and the parent vector can enhance the diversity of the population. Common cross equations are presented as follows:(19)xu,iG+1=xv,iG,rand≤CR,xiG,otherwise,where rand is a random number.(5)
The selection operation follows the “greedy” search strategy and adopts individuals with small adaptation values as descendants, which is expressed as(20)xiG+1=xu,iG,fxu,iG≤fxiG,xiG,otherwise,wheref(x) denotes the objective function, which can be calculated by the above equations (12) and (13). Lastly, the above operations are repeated until the best solution is selected.


## 3.1.2. Optimized Differential Evolution Algorithm

The DE algorithm has downsides, e.g., slow convergence rate and easy to fall into local optimum. Especially in the late stage of its evolution, the population diversity and the convergence speed decrease, which leads to the algorithm falling into local optimum [35]. In this study, an optimized DE algorithm is proposed, exploiting the scaling factor of the sinusoidal iterative chaotic system and the adaptive crossover probability factor (CR) to enhance the diversity of the population. The sinusoidal chaotic random sequence [36] refers to a variable that fluctuates within the interval [0, 1]. It exhibits randomness, ergodicity, aperiodicity, and extreme sensitivity to initial values, making it suitable for generating nonrepeating pseudorandom numbers in the range [0, 1]. When the chaotic random sequence is employed for the population initialization and optimization search, the algorithm is capable of verifiably avoiding from falling into the optimal local solution.The scaling factor based on a sinusoidal iterative chaotic system is given by(21)Fn=sin2carcsinFn−1,where c denotes the weight coefficient, generally chosen from the interval [0, 3]; also, the initial value of the mutation operator is chosen randomly, i.e., F0∈rand0,1.The adaptive CR is expressed as(22)CRG=CRl+Ch−CRlGm−GGm,where CRh and CRl denote the maximum and minimum values of CR, respectively. After numerous experiments, CRl = 0.1 and CRh = 0.9 are applicable to most optimization problems. funmax and funmin are the maximum and minimum values of the objective function, respectively. Gm indicates the maximum number of iterations, and G the current number of iterations. The optimized DE algorithm flow chart is shown in Figure 3.Figure 3
The optimized DE algorithm flow chart.


## 3.2. Shared Aperture Multibeam Forming Design Algorithm

A shared aperture sparsely interleaved array is capable of reducing the number of antennas, saving space, and reducing load and manufacturing costs. In this study, the TMLA is employed to form more beams from multiple subarrays based on the interleaved shared aperture. The combination of shared aperture technology and the TMLA can yield more beams based on the same antenna resources and enhance the efficiency of the antenna utilization. The following is the locational distribution of the array elements with sparsely interleaved apertures, which is expressed asSubarray 1: 100110001011010110100101010110010101010110100110000101010110Subarray 2: 011001110100101001011010101001101010101001011001111010101001where 1 denotes an array of elements, 0 means free here. A uniform linear array is considered with N elements. N elements are chosen randomly from 0 to 1, where 0 denotes the elements of subarray 1, and 1 the array elements of subarray 2. After the two subarrays are formed by random sparse interleaving, the excitations for the two subarrays are initialized to the number Gn in [0, 1] randomly. The proposed algorithm for multiple beamforming is presented in Algorithm 1.Algorithm 1: Shared aperture multibeam forming for time-modulated arrays based on optimized DE algorithm.
Input: The array number N, the number of samples L, target directions, the static excitation amplitude Ak, the parameters of the optimized DE algorithm Np,Gm, CR, and F.
Initialization: Randomly and sparsely interlace the line arrays of the N array elements into two subarrays, and initialize the excitation of the two subarrays.
CalculateF0θ,F1θ,F−1θ,F2θ, and F−2θ using (7)–(9). Then Tsn and Tdn can be calculated by (10) and (11). Lastly, the objective function value can be obtained by (11)–(13).
Iteration: While
G < Gm
Adaptively updating the value of CR andF using (21) and (22)
Generating variant individuals
Cross operation
Select the mutated individuals based on the objective function
IffxG_next<fxG
record the optimal value off and xG_next, then update
end loop.
End for
Output: Final weight vector x1,x2 and f.


## 3.3. Multibeam Forming Based on Iterative FFT Algorithm

The iterative FFT algorithm exploiting the feature of the relationship between the uniform linear array element excitation and the Fourier transform converts the problem to determine the location of the subarray element into the problem to select the excitation energy of the array element. This method, based on the principle of density-weighted matrix and cross-selection, ascertain the location of the sparse subarray unit and achieve the uniform distribution of spectral energy of the subarray. In such a way, multibeam forming based on sparse interlaced arrays can be achieved.Considering a linear array as shown in equation (1), the discrete inverse Fourier transform (DIFT) can be expressed as(23)ft=1M∑n=0M−1Fnejntω0,where M denotes the period and ω0=2π/M is the sampling interval. It can be observed from equations (1) and (23) that there is a Fourier transform relationship between the excitation A of the array element and the array factor F.Since there exists a Fourier transform relationship between the excitation and the pattern of the array antenna element, the fast Fourier transform (FFT) method can be adopted to convert the unit excitation and the pattern, significantly reducing the calculation amount and saving computing time. Taking the antenna design withT subarrays interlaced as an example, the specific steps are presented as follows:(1)
Set the array excitation valueI to 1 according to a uniform linear array with sparse rate R=1/T random sparse array elements N.(2)
Perform an inverse FFT ofQ point on I to calculate P.(3)
Determine the value of the sidelobe region ofP, so the point value on the region larger than the constraint sidelobe value equals to the constraint sidelobe value, and the values at other points remain unchanged.(4)
Perform an FFT conversion of theQ point on the corrected P to yield a new excitation value I.(5)
Intercept the firstN values in I, yield a new excitation matrix, arrange them from large to small, and yield a new excitation vector.(6)
The excitation at the locationIri,Iri+T,…,Ir+T×K corresponds to the location on I as the location where the i-th subarray unit is located, and set the excitation value at the location as 1.(7)
Perform an inverse FFT on the excitation vector of the new subarray 1, determine whether the edge lobe peak of the updated antenna pattern changes with respect to the sidelobe peak at the previous iteration, and then proceeds to the next calculation; otherwise, output the result.(8)
The excitation sequence of the subarray 1 is entered as the new input value into the next iteration. Steps (2) to (8) are repeated until the newly generated antenna pattern sidelobe peaks no longer change, and output the optimized result.


## 4. Experimental Simulation and Analysis

The number of elements of the antenna array is set to 60, the desired direction isθ1=±10° and θ2=±30°. Referring to the literature [37, 38], and after simulating repeatedly, it is found that the beam pointing is only associated with the opening time Tsn. The desired direction in this study is stable, so the decision variable is only Tdn. The specific parameter settings to optimize the DE algorithm are listed in Table 1.Table 1
The parameters to improve the DE algorithm.
Parameter
Significance
Value
NP
Number of initial population
50
x
max
Maximum velocity
0.5
x
min
Minimum velocity
0
F(1)
Initial value ofF
0.4
G
m
Maximum number of iteration
200
CRl
Minimum number of adaptive crossover operator
0.1
CRh
Maximum number of adaptive crossover operator
0.9


### 4.1. Simulation Experiment 1

In TMLA harmonic beamforming, the first two harmonic components are larger, while the other subharmonic components are smaller. In this study, only the first harmonic component is applied for multibeam forming. The second harmonic is primarily considered suppressed as much as possible to reduce power loss. Take subarray 1 as an example. The fundamental wave, first harmonic, and second harmonic component formed by the underlying algorithm are presented in Figure4. The patterns of the fully populated array and after optimization by the proposed algorithm are presented in Figures 5 and 6. Also, the gain differences between the second harmonic component and the first harmonic component are shown in Table 2. The second harmonic component and the first harmonic component have a gain difference of 6.03 dB in the underlying algorithm. In the fully populated array, the gain difference between the second and first harmonic components is 6.36 dB. After optimization by the proposed algorithm, the gain difference between the second and first harmonic components is increased to 18.03 dB. Thus, the proposed algorithm significantly suppresses the second harmonic.Figure 4
Array factor pattern fundamental wave and first two positive harmonics of subarray 1.Figure 5
Array factor pattern fundamental wave and first two positive harmonics of fully populated array.Figure 6
Optimized array factor pattern fundamental wave and first two positive harmonics of subarray 1.Table 2
The gain of different algorithm.
Element number
Gain1 (dB)
Gain2 (dB)
ΔGain (dB)
Subarray 1
29
19.60
13.57
6.03
Fully populated array
60
22.09
15.73
6.36
Improved DE subarray 1
31
16.71
−1.32
18.03To verify the feasibility of the proposed algorithm, the algorithm of this study is simulated and compared with the beam performance formed by the basic TMLA shared aperture. The fundamental wave is set to point to 0°, while the positive first harmonic of subarray 1 points to 10°, and the positive first harmonic of subarray 2 points to 30°. The simulation results are presented in Figures7–9.Figure 7
Fundamental patterns obtained by optimized DE and basic algorithm.Figure 8
The positive first harmonic patterns of subarray 1 obtained by optimized DE and basic algorithm.Figure 9
The positive first harmonic patterns of subarray 2 obtained by optimized DE and basic algorithm.The simulation diagram reveals that the PSL of the basic TMLA shared aperture multibeam forming fundamental wave is −10.18 dB, while the PSL of the positive first harmonic of the subarray 1 is −10.01 dB, and the PSL of the positive first harmonic of subarray 2 reaches −11.31 dB. After using the proposed optimized DE algorithm, the beam pattern performance becomes closer, and the fundamental wave and the two subarrays are positive. Besides, the PSL of the positive first harmonic of the fundamental and the two subarrays are further optimized. The PSL of the fundamental wave is downregulated to −13.78 dB, while the PSLs of the positive first harmonic of the subarray 1 and 2 to −12.89 dB and −13.65 dB, respectively. Besides, the −3 dB bandwidths of the preoptimized fundamental wave, the positive first harmonic of subarray 1, and subarray 2 are 1.66, 1.68, and 1.93 degrees, respectively. Though the PSL of each beam is reduced through optimization, the bandwidth variations are negligibly small. In particular, they are 1.96, 2.04, and 2.22 degrees, only upregulated by 0.30, 0.36, and 0.29 degrees, respectively. Accordingly, no additional constraint bandwidth is imposed in this study. It can be concluded that the antenna pattern performance is considerably improved by applying the algorithm proposed in this study and using the duration of the switch openings of the two subarrays as optimization variables.The iterative curve of the control sequences for the two subarrays optimized by the proposed algorithm is plotted in Figure10. It can be observed that the optimized update variant mutation operator and the scaling factor of the sinusoidal iterative chaotic system can enhance the diversity of the population. Thus, the objective function value is very useful.Figure 10
Iterative curve of subarray 1 and subarray 2 optimized variables.The location of the subarray element after optimization by the algorithm is as follows:Subarray 1: 100110101011010100100101010101011101010110100110110101010010Subarray 2: 011001010100101011011010101010100010101001011001001010101101


### 4.2. Simulation Experiment 2

To further verify the feasibility of the proposed algorithm, the proposed algorithm is compared with the iterative FFT algorithm. The algorithm in this study can yield more beams simultaneously than the iterative FFT. The fundamental wave to point was set to 0°, while the positive first harmonic of subarray 1 points to 10°, and the positive first harmonic of subarray 2 points to 30°. The beam performance of the two methods is compared, and the simulation results are shown in Figures11–13.Figure 11
Fundamental patterns obtained by optimized DE and FFT algorithm.Figure 12
The positive first harmonic patterns of subarray 1 obtained by optimized DE and FFT algorithm.Figure 13
The positive first harmonic patterns of subarray 2 obtained by optimized DE and FFT algorithm.Compared with the iterative FFT algorithm, the proposed algorithm exhibits the following two advantages. First, in the case of a certain number of array elements, the algorithm can form more beams while the beam directions are flexible. Second, the PSL of the formed multibeam is lower, and the pattern performance is superior to the iterative FFT performance. The results of simulations 11, 12, and 13 suggest that the PSL of the beam pointing to 0° formed by the iterative FFT algorithm is −10.47 dB, while it is −10.96 dB and −10.49 dB for 10° and 30°, respectively. However, beams formed by the algorithm in this study are better. The PSL of the fundamental wave formed is −13.48 dB, while the PSL of the positive first harmonic of subarrays 1 and 2 are −12.52 dB. These values are 3.01 dB, 1.56 dB, and 2.96 dB lower than the PSL of the beam corresponding to the iterative FFT algorithm, respectively. The above simulations reveal that the optimized DE algorithm has a better direction in the graph and has a faster convergence speed. The algorithm can adaptively adjust the mutation weight of the mutation operator according to the current state of the individual population, avoiding the early maturity attributed to the premature falling into the local optimum, and it exhibits better beam performance than other conventional methods.


## 4.1. Simulation Experiment 1

In TMLA harmonic beamforming, the first two harmonic components are larger, while the other subharmonic components are smaller. In this study, only the first harmonic component is applied for multibeam forming. The second harmonic is primarily considered suppressed as much as possible to reduce power loss. Take subarray 1 as an example. The fundamental wave, first harmonic, and second harmonic component formed by the underlying algorithm are presented in Figure4. The patterns of the fully populated array and after optimization by the proposed algorithm are presented in Figures 5 and 6. Also, the gain differences between the second harmonic component and the first harmonic component are shown in Table 2. The second harmonic component and the first harmonic component have a gain difference of 6.03 dB in the underlying algorithm. In the fully populated array, the gain difference between the second and first harmonic components is 6.36 dB. After optimization by the proposed algorithm, the gain difference between the second and first harmonic components is increased to 18.03 dB. Thus, the proposed algorithm significantly suppresses the second harmonic.Figure 4
Array factor pattern fundamental wave and first two positive harmonics of subarray 1.Figure 5
Array factor pattern fundamental wave and first two positive harmonics of fully populated array.Figure 6
Optimized array factor pattern fundamental wave and first two positive harmonics of subarray 1.Table 2
The gain of different algorithm.
Element number
Gain1 (dB)
Gain2 (dB)
ΔGain (dB)
Subarray 1
29
19.60
13.57
6.03
Fully populated array
60
22.09
15.73
6.36
Improved DE subarray 1
31
16.71
−1.32
18.03To verify the feasibility of the proposed algorithm, the algorithm of this study is simulated and compared with the beam performance formed by the basic TMLA shared aperture. The fundamental wave is set to point to 0°, while the positive first harmonic of subarray 1 points to 10°, and the positive first harmonic of subarray 2 points to 30°. The simulation results are presented in Figures7–9.Figure 7
Fundamental patterns obtained by optimized DE and basic algorithm.Figure 8
The positive first harmonic patterns of subarray 1 obtained by optimized DE and basic algorithm.Figure 9
The positive first harmonic patterns of subarray 2 obtained by optimized DE and basic algorithm.The simulation diagram reveals that the PSL of the basic TMLA shared aperture multibeam forming fundamental wave is −10.18 dB, while the PSL of the positive first harmonic of the subarray 1 is −10.01 dB, and the PSL of the positive first harmonic of subarray 2 reaches −11.31 dB. After using the proposed optimized DE algorithm, the beam pattern performance becomes closer, and the fundamental wave and the two subarrays are positive. Besides, the PSL of the positive first harmonic of the fundamental and the two subarrays are further optimized. The PSL of the fundamental wave is downregulated to −13.78 dB, while the PSLs of the positive first harmonic of the subarray 1 and 2 to −12.89 dB and −13.65 dB, respectively. Besides, the −3 dB bandwidths of the preoptimized fundamental wave, the positive first harmonic of subarray 1, and subarray 2 are 1.66, 1.68, and 1.93 degrees, respectively. Though the PSL of each beam is reduced through optimization, the bandwidth variations are negligibly small. In particular, they are 1.96, 2.04, and 2.22 degrees, only upregulated by 0.30, 0.36, and 0.29 degrees, respectively. Accordingly, no additional constraint bandwidth is imposed in this study. It can be concluded that the antenna pattern performance is considerably improved by applying the algorithm proposed in this study and using the duration of the switch openings of the two subarrays as optimization variables.The iterative curve of the control sequences for the two subarrays optimized by the proposed algorithm is plotted in Figure10. It can be observed that the optimized update variant mutation operator and the scaling factor of the sinusoidal iterative chaotic system can enhance the diversity of the population. Thus, the objective function value is very useful.Figure 10
Iterative curve of subarray 1 and subarray 2 optimized variables.The location of the subarray element after optimization by the algorithm is as follows:Subarray 1: 100110101011010100100101010101011101010110100110110101010010Subarray 2: 011001010100101011011010101010100010101001011001001010101101


## 4.2. Simulation Experiment 2

To further verify the feasibility of the proposed algorithm, the proposed algorithm is compared with the iterative FFT algorithm. The algorithm in this study can yield more beams simultaneously than the iterative FFT. The fundamental wave to point was set to 0°, while the positive first harmonic of subarray 1 points to 10°, and the positive first harmonic of subarray 2 points to 30°. The beam performance of the two methods is compared, and the simulation results are shown in Figures11–13.Figure 11
Fundamental patterns obtained by optimized DE and FFT algorithm.Figure 12
The positive first harmonic patterns of subarray 1 obtained by optimized DE and FFT algorithm.Figure 13
The positive first harmonic patterns of subarray 2 obtained by optimized DE and FFT algorithm.Compared with the iterative FFT algorithm, the proposed algorithm exhibits the following two advantages. First, in the case of a certain number of array elements, the algorithm can form more beams while the beam directions are flexible. Second, the PSL of the formed multibeam is lower, and the pattern performance is superior to the iterative FFT performance. The results of simulations 11, 12, and 13 suggest that the PSL of the beam pointing to 0° formed by the iterative FFT algorithm is −10.47 dB, while it is −10.96 dB and −10.49 dB for 10° and 30°, respectively. However, beams formed by the algorithm in this study are better. The PSL of the fundamental wave formed is −13.48 dB, while the PSL of the positive first harmonic of subarrays 1 and 2 are −12.52 dB. These values are 3.01 dB, 1.56 dB, and 2.96 dB lower than the PSL of the beam corresponding to the iterative FFT algorithm, respectively. The above simulations reveal that the optimized DE algorithm has a better direction in the graph and has a faster convergence speed. The algorithm can adaptively adjust the mutation weight of the mutation operator according to the current state of the individual population, avoiding the early maturity attributed to the premature falling into the local optimum, and it exhibits better beam performance than other conventional methods.


## 5. Conclusion

In this study, TMLA technology is employed to design the shared aperture multibeam forming method for the first time. More beams are formed after the shared apertures are sparsely interleaved to form two subarrays. The optimized DE algorithm, introducing the scaling factor of the sinusoidal iterative chaotic system and the adaptive cross-probability factor, is presented to optimize the control sequences of the two subarrays. The PSLs of the fundamental wave and the harmonics, formed by the two subarrays optimized by optimized DE, were revealed to be lower than those optimized by other algorithms. Simulation results confirm the feasibility of the proposed algorithm.The TMLA technology can achieve the beam control in the antenna array. In the meantime, it not only has the characteristics of simple system structure and low cost but also has a very practical significance for realizing miniaturization of the phased array system. Thus, TMLA will be further applied to the multibeam forming of the conformal array.



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*Source: 4101909-2019-11-26.xml*