# Pattern Synthesis of Time-Modulated Sparse Array by an OPM-CVX Algorithm

Authors: Lei Liang; Yachao Jiang; Jialing Liu; Hailin Li; Jianjiang Zhou
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper addresses the constrained multiobjective optimization problem of time-modulated sparse arrays. The synthesis objective is to find an optimal element arrangement and associated excitation strategy of sparse arrays, which realize the balance of radiation power and sideband suppression performance with minimum number of elements, and suppress side lobe level simultaneously. A novel hybrid algorithm based on orthogonal perturbation method and convex optimization (OPM-CVX) for the synthesis of time-modulated sparse antenna array is presented in this paper. In order to satisfy the main lobe beamforming and side lobe suppression of sparse arrays, the proposed method optimizes element positions with minimum array numbers by orthogonal perturbation method and optimizes excitations of array element with dynamic range ratio constraint by convex optimization. Furthermore, a trapezoidal pulse time-modulated switching function is proposed to find the balance of radiation power and sideband suppression performance. The numerical results indicate that the proposed algorithm can be an effective approach for synthesis problems of time-modulated sparse arrays.


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## Body

## 1. Introduction

The antenna arrays have diverse structures and flexible beam control. It is widely used in radar early warning, meteorology and hydrology, and wireless communication systems [1–3]. Compared with single antenna, antenna arrays make it easier to implement pencil beams, flat top beams, multibeams, and low side lobe patterns. The pencil beam can overcome multipath interference and cochannel interference. Lei [4] optimized the array element positions through an iterative convex optimization algorithm and realized pencil beam and low side lobe in linear and planar arrays. The shaped beam is mainly used to meet the requirements of the radiation area and the intensity of the signal in the specified direction. For example, the flat top beam can ensure the uniformity of the power density at the receiving set, which is significant for improving the reception efficiency and simplifying the design of the rectifier circuit. Cao [5] and Qi [6] synthesized flat top beam with linear array and rectangular planar array, respectively. They effectively controlled the main beam ripple and obtained low side lobe level. In addition, Qi also synthesized a planar array of 10000 (100 × 100) elements with half wavelength spacing and verified the applicability of synthesizing shaped beam of large array. However, the cost of the antenna system is increased by the half-wavelength equidistant arrangement.The sparse array is an important branch in the field of antenna arrays. Compared with uniform antenna arrays, the spacing of array element in sparse array is usually not equal and greater than half wavelength. Under the same array aperture, the sparsely distributed array can obtain better radiation performance with fewer array elements. Morabito [7] synthesized flat top and cosecant patterns using rectangular array with the least number of array elements. Compared with the array with uniform spacing of half wavelength, 58% of the array elements were saved. Shen [8] and Yan [9] both constrained the aperture of sparse array and synthesized flat top beam and pencil beam. Shen [8] used the unitary matrix pencil method to solve the excitation and element position of linear array, which improves the matching accuracy of shaped beam and reduces the computational complexity. However, the dynamic range ratio (DRR) was still greater than 10. Yan [9] adopted the perturbed compressive sampling (PCS) algorithm to optimize the excitation and element position at the same time, which improves the matching accuracy of pattern and obtains a large DRR too. A lot of achievements have been made in the pattern synthesis of sparse arrays. However, the large DRR will increase the implementation difficulty and hardware cost of the feeding network.The time modulation technology controls the working state of the array element through the RF switch to obtain the equivalent excitation coefficient so that the array element can also obtain good performance of pattern synthesis under the uniform excitation. Poli [10] synthesized a pencil beam pattern and suppressed sideband radiation by optimizing the equivalent excitation of the time-modulated linear array. Alberto Reyna [11] synthesized rectangular and circular sparse arrays with the same aperture size and obtained flat top beam pattern and better feeding network efficiency, but the sideband level was high. Patra [12] synthesized a flat top beam pattern in time-modulated sparse linear array. A combination of static excitation, unequal element spacing, and time is used to obtain a side lobe level of −40.49 dB and a sideband level of −26.46 dB. Through the abovementioned analysis, in addition to obtaining the target pattern, the time-modulated array needs to suppress the sideband radiation caused by the RF switching action. Many scholars have carried out in-depth research on the optimization algorithms of time-modulated arrays and sparse arrays.The pattern synthesis algorithms of sparse array mainly include evolutionary algorithm, numerical algorithm, and convex optimization algorithm. There are many widely used evolutionary algorithms. D'Urso [13] proposed simulated annealing algorithm to optimize the switch-on duration time, which realized the low side lobe of sparse array with equal spacing and effectively suppressed the sideband radiation. With the constraint of exactly matching the desired pattern of main band, Poli [14] applied the pulse shifting technique to time-modulated arrays and used particle swarm optimization algorithm to suppress the side lobe level and sideband level effectively. However, with the increase of the optimization dimension, the computational complexity of the evolutionary algorithm increases exponentially, the optimization efficiency decreases accordingly, and it is easy to fall into a local optimal solution.The deterministic numerical methods are widely used with stable optimization solutions. Gassab [15] proposed an efficient mathematical method for pattern synthesis of time-modulated linear array and obtained lower side lobe and sideband than the antenna array optimized by differential evolution algorithm. However, only the switch-on duration time of array element was optimized, while the position of an array element was fixed. Yepes [16] proposed a new hybrid technique to synthesize different shaped beams (narrow beam, flat top beam, and square cosecant beam) for linear array. This technique uses a deterministic algorithm to optimize the nonuniform amplitude excitation and antenna element position and then uses an iterative optimization strategy to calculate the phase of the excitation. Although the deterministic iterative algorithm has high operational stability, its optimization effect largely depends on the initial state of the array. Therefore, joint optimization of parameters such as the position and excitation of an array element can usually achieve better optimization result.Convex optimization algorithm is applied to pattern synthesis with the advantages of its efficient optimization process and the global optimal result. Echeveste [17] proposed a flat top beam synthesis methodology for planar array by transforming a nonconvex synthesis problem to a convex optimization scheme with conjugate symmetric excitation weights. Under the constraint of the minimum spacing of array elements, Yan [18] proposed a hybrid algorithm of PCS and convex optimization to synthesis a flat top beam of planar array. This method solved the problems of the small element spacing using PCS algorithm and the long or even invalid solving time for the large-scale array using convex optimization algorithm. The above studies all involved transforming a nonconvex problem to a convex problem. This is the key to the application of convex optimization algorithm to pattern synthesis.Combined with the abovementioned analysis, this paper uses the hybrid algorithm of OPM-CVX to minimize the number of array elements and suppress side lobe and sideband. This algorithm achieves joint optimization of array element excitation and position under the constraint of desired pattern. However, multivariate optimization is a multiobjective multiconstrained nonlinear optimization problem [19]. Taking this issue into consideration, the optimization model is disassembled into linear optimization problems about the number of elements, shaped beam, side lobe, and the balance between sideband and power. The iterative strategy is adopted to achieve the above objectives step by step.This paper consists of five sections. Some key studies in the field of time-modulated sparse arrays are reviewed in Section1 and the problem to be addressed by the present study is pointed out. In Section 2, the formula deduction of Fourier coefficient of trapezoidal pulse switching function and the model of optimization are provided. Section 3 gives a step by step solution to the optimization model and describes the optimization process. In Section 4, the effectiveness of this hybrid algorithm is shown by detailed simulation results. Finally, Section 5 presents the conclusions.


## 2. Shaped Beam Pattern Synthesis of Time-Modulated Sparse Antenna Array

### 2.1. Time-Modulated Sparse Antenna Array

For planar antenna arrays withN identical array elements, as shown in Figure 1, the electric field intensity of far field [20] is given by(1)Eθ,φ=∑n=1NInejkxnu+ynvej2πf0t,where In is the excitation coefficient of the nthelement located at xn,yn and θn and φn are pitch and azimuth angle, respectively. The radiation direction of the array is θ,φ, u=sinθcosφ, and v=sinθsinφ. k=2π/λ=2πf0/c is the propagation coefficient of the free space with the wavelength λ, where f0 is the carrier frequency.Figure 1
Planar arrays with square lattices and square boundaries.In time-modulated antenna arrays [21], the nth array element is only controlled by Unt, which is the function of the high speed RF switches. The electric field intensity is redefined as(2)Eθ,φ=∑n=1NInUntej2πf0tejkxnu+ynv.The expansion form of time-modulated switching functionUnt in frequency domain can be expressed as(3)Unt=∑m=−∞+∞amnej2πmfnt,m=0,±1,±2,…,where fn=1/Tn,Tn are the time-modulated fundamental frequency and period of the array elements separately and amn is the Fourier coefficient of the mth sideband. Equation (2) can be rewritten as(4)Eθ,φ=∑n=1N∑m=−∞+∞Inamnej2πf0+mfntejkxnu+ynv,where the electric field intensity of the antenna array at the mth sideband is(5)Emθ,φ=∑n=1NInamnej2πf0+mfntejkxnu+ynv.The power of main bandP0 and the total power of sideband PSR can be expressed as(6)P0=∫2π2π∫0πE0θ,φ2sinθdθdφ,(7)PSR=∑m=−∞,m≠0m=∞∫02π∫0πEmθ,φ2sinθdθdφ,where E0θ,φ is the electric field intensity of main band and Emθ,φ is the electric field intensity of the mth sideband.The average power is the sum of the integral of power in all directions. According to equations (6) and (7), it can be defined as(8)Pav=P0+PSR4π.The directivity of time-modulated antenna array can be expressed as [22](9)D=4πmaxE0θ,φ2P0+PSR.


### 2.2. Trapezoidal Pulse Time-Modulated Switching Function

Under the variable aperture sizes (VAS) modulation mode [23], the rectangular pulse switching function is usually used for time-modulated antenna array, as shown in Figure 2(a).Figure 2
Time-pulse shaping. (a) Rectangular pulse. (b) Trapezoidal pulse.As shown in Figure2(b), the trapezoidal pulse switching function is obtained as follows:(10)Unt=tτa,0<t≤τa,1,τa≤t≤τn−τa,τn−tτa,τn−τa≤t≤τn,0,else.The Fourier coefficient of the trapezoidal pulse switching function is (for details, see Appendix A)(11)amn=τn−τafnsincπmfnτasincπmfnτn−τae−jπmfnτn,0<2τa≤τn≤Tn.While the Fourier coefficient of the rectangular pulse switching functionUn′t=1,0≤t≤τn0,else is(12)amn′=τnfnsincπmfnτne−jπmfnτn,0<τn≤Tn.Compared with rectangular pulse switching function, trapezoidal pulse switching function has better sideband suppression performance as its Fourier coefficientamn has an extra sincπmfnτa component.A switch has only two state of ON and OFF, so in order to form trapezoidal pulse in reality, each element can be equipped with a radio frequency (RF) variable gain amplifier (VGA). The periodic trapezoidal pulse of periodTn governs the digitally controlled VGA instead of RF switches to transmit signal [24]. Compared with conventional time-modulated array based on RF switches, the power consumption is acceptable and the complexity of hardware structure is approximate [25]. Moreover, the VGAs guarantees flexibility and advanced dynamic control.


### 2.3. The Mathematical Model of Shaped Beam Pattern Synthesis

For side lobe and sideband suppression of sparse arrays based on time-modulated technology, the objective of this paper is to find an optimal solution for shaped beam pattern synthesis of sparse array. That is to optimize the array element position and excitation coefficient to make the pattern achieve the following goals: (a) minimize the peak side lobe level (PSLL) and peak sideband level (PSBL) under the constraint of main band; (b) minimize the number of elements; (c) make the dynamic range ratio (DRR) meet the constraint conditions. The abovementioned optimization problem can be expressed as(13)minmaxθ,φ∈ΩsE0θ,φmaxEmθ,φ,m≠0Ns.t.E0θ,φ−Edθ,φ2<ε1,θ,φ∈ΩcminIn=1,maxIn≤ε2,where N is the number of array elements, E0θ,φ is the electric field intensity of main lobe, Ωcand Ωs are the main lobe area and the side lobe area of main band, respectively, Emθ,φ is the electric field intensity of sideband, Edθ,φ is the desired shaped beam pattern of main band, ε1 is the maximum relaxed coefficient of shaped beam pattern, and ε2 is the maximum amplitude of the excitation coefficient.


## 2.1. Time-Modulated Sparse Antenna Array

For planar antenna arrays withN identical array elements, as shown in Figure 1, the electric field intensity of far field [20] is given by(1)Eθ,φ=∑n=1NInejkxnu+ynvej2πf0t,where In is the excitation coefficient of the nthelement located at xn,yn and θn and φn are pitch and azimuth angle, respectively. The radiation direction of the array is θ,φ, u=sinθcosφ, and v=sinθsinφ. k=2π/λ=2πf0/c is the propagation coefficient of the free space with the wavelength λ, where f0 is the carrier frequency.Figure 1
Planar arrays with square lattices and square boundaries.In time-modulated antenna arrays [21], the nth array element is only controlled by Unt, which is the function of the high speed RF switches. The electric field intensity is redefined as(2)Eθ,φ=∑n=1NInUntej2πf0tejkxnu+ynv.The expansion form of time-modulated switching functionUnt in frequency domain can be expressed as(3)Unt=∑m=−∞+∞amnej2πmfnt,m=0,±1,±2,…,where fn=1/Tn,Tn are the time-modulated fundamental frequency and period of the array elements separately and amn is the Fourier coefficient of the mth sideband. Equation (2) can be rewritten as(4)Eθ,φ=∑n=1N∑m=−∞+∞Inamnej2πf0+mfntejkxnu+ynv,where the electric field intensity of the antenna array at the mth sideband is(5)Emθ,φ=∑n=1NInamnej2πf0+mfntejkxnu+ynv.The power of main bandP0 and the total power of sideband PSR can be expressed as(6)P0=∫2π2π∫0πE0θ,φ2sinθdθdφ,(7)PSR=∑m=−∞,m≠0m=∞∫02π∫0πEmθ,φ2sinθdθdφ,where E0θ,φ is the electric field intensity of main band and Emθ,φ is the electric field intensity of the mth sideband.The average power is the sum of the integral of power in all directions. According to equations (6) and (7), it can be defined as(8)Pav=P0+PSR4π.The directivity of time-modulated antenna array can be expressed as [22](9)D=4πmaxE0θ,φ2P0+PSR.


## 2.2. Trapezoidal Pulse Time-Modulated Switching Function

Under the variable aperture sizes (VAS) modulation mode [23], the rectangular pulse switching function is usually used for time-modulated antenna array, as shown in Figure 2(a).Figure 2
Time-pulse shaping. (a) Rectangular pulse. (b) Trapezoidal pulse.As shown in Figure2(b), the trapezoidal pulse switching function is obtained as follows:(10)Unt=tτa,0<t≤τa,1,τa≤t≤τn−τa,τn−tτa,τn−τa≤t≤τn,0,else.The Fourier coefficient of the trapezoidal pulse switching function is (for details, see Appendix A)(11)amn=τn−τafnsincπmfnτasincπmfnτn−τae−jπmfnτn,0<2τa≤τn≤Tn.While the Fourier coefficient of the rectangular pulse switching functionUn′t=1,0≤t≤τn0,else is(12)amn′=τnfnsincπmfnτne−jπmfnτn,0<τn≤Tn.Compared with rectangular pulse switching function, trapezoidal pulse switching function has better sideband suppression performance as its Fourier coefficientamn has an extra sincπmfnτa component.A switch has only two state of ON and OFF, so in order to form trapezoidal pulse in reality, each element can be equipped with a radio frequency (RF) variable gain amplifier (VGA). The periodic trapezoidal pulse of periodTn governs the digitally controlled VGA instead of RF switches to transmit signal [24]. Compared with conventional time-modulated array based on RF switches, the power consumption is acceptable and the complexity of hardware structure is approximate [25]. Moreover, the VGAs guarantees flexibility and advanced dynamic control.


## 2.3. The Mathematical Model of Shaped Beam Pattern Synthesis

For side lobe and sideband suppression of sparse arrays based on time-modulated technology, the objective of this paper is to find an optimal solution for shaped beam pattern synthesis of sparse array. That is to optimize the array element position and excitation coefficient to make the pattern achieve the following goals: (a) minimize the peak side lobe level (PSLL) and peak sideband level (PSBL) under the constraint of main band; (b) minimize the number of elements; (c) make the dynamic range ratio (DRR) meet the constraint conditions. The abovementioned optimization problem can be expressed as(13)minmaxθ,φ∈ΩsE0θ,φmaxEmθ,φ,m≠0Ns.t.E0θ,φ−Edθ,φ2<ε1,θ,φ∈ΩcminIn=1,maxIn≤ε2,where N is the number of array elements, E0θ,φ is the electric field intensity of main lobe, Ωcand Ωs are the main lobe area and the side lobe area of main band, respectively, Emθ,φ is the electric field intensity of sideband, Edθ,φ is the desired shaped beam pattern of main band, ε1 is the maximum relaxed coefficient of shaped beam pattern, and ε2 is the maximum amplitude of the excitation coefficient.


## 3. Hybrid Algorithm Based on Orthogonal Perturbation Method and Convex Optimization

In order to meet the optimization objective of equation (13), this section decomposes equation (13) into four problems and uses different methods to optimize them, respectively: (a) under the constraints of the side lobe and sideband, the minimum number of elements is achieved by iteratively decreasing; (b) the suppression of sideband level is achieved by nonuniform periodic modulation and trapezoidal pulse switching function; the nonuniform periodic modulation can effectively suppress the PSBL, and the trapezoidal pulse switching function can further reduce the sideband level and sideband power radiation; (c) the array element position is optimized by the orthogonal perturbation method [26–28], and the optimized array realizes the sparse layout; (d) the equivalent excitation of array element is obtained by the convex optimization, and the joint optimization of equivalent excitation and position minimizes the PSLL. At the end of this section, the algorithms above are summarized and explained, and the general flowchart is given.


### 3.1. Minimum NumberN of Array Elements

According to the optimization objective of equation (13), the number of array elements needs to be minimized under the constraints of the side lobe and sideband. Under the condition that the array aperture remains invariant, the number of array elements is initialized toN0, and the spacing of array elements meets λ/2. N0 can be written as(14)N0=ceil2Lxλ⋅ceil2Lyλ,where Lx,Ly is the aperture size of the planar array. In this paper, the minimum number of array elements is obtained by gradually reducing the array elements while the side lobe and sideband still meet the constraints.


### 3.2. Sideband Suppression Based on Trapezoidal Pulse Switching Function

According to the optimization objective of equation (13), the maximum value of the sideband electric field intensity Emθ,φ,m≠0 of the antenna array is required to be minimum. From equation (11), the amplitude of amn decreases with the increase of m in the function sinc⋅. That is, the sideband level decreases as the sideband order increases, so the sideband level of the array at m=1 is the highest [18]. This paper uses a nonuniform period time-modulated method [29], so the sideband energy of the array elements cannot be superimposed. Therefore, the PSBL of the array element is the PSBL of the array. Minimizing the PSBL is equivalent to(15)minmaxn=1,2,…Na1n.Under the condition thatτn and fn are known, a1n is only determined by τa in equation (11). According to the pattern synthesis algorithm of OPM-CVX proposed in Section 3.3, the best coefficients Ina0n can be obtained under the constraints of the main lobe and side lobe. That is, Ina0n=Inτn−τafn is known when m=0. The value range of τa is 0,τn/2. When τa=0, the trapezoidal pulse switching function becomes the rectangular pulse switching function. The sideband suppression performance is the worst, while the radiated power of sidebands is the largest. When τa=τn/2, the trapezoidal pulse switching function becomes a triangular pulse switching function, the sideband suppression performance is the best, while the radiated power of sidebands is the lowest. In this paper, the best rise/fall time τa of trapezoidal pulse and the best static excitation In are solved under the constraint of DRR minIn=1,maxIn≤ε2 in equation (13) so that the radiated power of the main band and the suppression effect of the sideband are balanced. Under the constraints of minIn=1,maxIn≤ε2, this paper initializes the In of the array element firstly, and then design the value of τa under the balance requirement of main band radiated power and sideband suppression.


### 3.3. Orthogonal Perturbation Method for Optimal Location

According to the optimization objective of equation (13), the mean square error between the main lobe of the actual pattern and the main lobe of the desired pattern is required to be less than the set value, which can be expressed as(16)δ=14π∫Ωc∫ΩcE0θ,φ−Edθ,φ2sinθdθdφ<ε1,where δ is the mean square error.From equation (4), the electric field intensity of main band can be rewritten as(17)E0θ,φ=∑n=1NIna0nejkxnu+ynv=WTB,where B=b1,…,bNT, bn=ejkxnu+ynv is the steering vector of the antenna array, W=w1,…,wNT, wn=Ina0n is the excitation vector of the antenna array, and the carrier frequency common factor ej2πf0t of main band is ignored.Under the condition that the excitation vectorW is given, this paper uses orthogonal perturbation method to solve the optimal steering vector Bbest, so as to obtain the optimal array layout. The idea is as follows: according to the total differential equation, when the excitations of array elements are known and the element position disturbance is small, the variation of equation (17) can be written as(18)ΔE=WTΔB=∑n=1NwnΔB=∑n=1NwnΔxn∂bn∂xn+Δyn∂bn∂yn=WeΔSTB′,where ΔS=Δx1,…,ΔxN,Δy1,…,ΔyN is the perturbation of position information, S=x1,…,xN,y1,…,yN. B′=∂b1/∂x1,…,∂bN/∂xN,∂b1/∂y1,…,∂bN/∂yN is the direction of position perturbation, We=diagw1,…,wN,w1,…,wN, Q is a set of standard orthogonal basis constructed by using Gram–Schmidt orthogonal process, which makes B′=CQ. C is the coefficient of orthogonal base, then equation (18) can be rewritten as(19)ΔE=CTWeΔSTQ.AssumeΔSc=CTWeΔS, after the element position disturbance, and equation (17) is converted to(20)E0θ,φ=WTB+ΔE=WTB+ΔScTQ.Define the inner product of any two column vectorsx,y as(21)x,y=14π∫02π∫0πxθ,φyHθ,φsinθdθdφ.Take equations (20) and (21) into (16), then(22)δ=14π∫02π∫0πWTB−Edθ,φ+ΔScTQ2sinθdθdφ=WTB−Edθ,φ,WTB−Edθ,φ+WTB−Edθ,φ,ΔScTQ+ΔScTQ,WTB−Edθ,φ+ΔScTQ,ΔScTQ.According to the method for calculating the derivative of scalar function to complex vector [30], the partial derivative to ΔSc is zero, that is,(23)∂δ∂ΔSc=Q,ΔE+ΔSc=0.The iterative process is introduced by the differenceΔEi so that the actual pattern can constantly approach the desired pattern. According to ΔS=We−1C−TΔSc, on the i+1 iteration, the position is updated to(24)Si+1=Si+ΔSi+1=Si−We−1Ci−TQi,ΔEi.Therefore, when the excitation vectorWis determined, the optimal element position Sbest can be solved by the orthogonal perturbation method.


### 3.4. Convex Optimization for Optimal Excitation Vector

According to the optimization objective of (13), the maximum value of the side lobe electric field intensity E0θ,φ,θ,φ∈Ωs of the antenna array is required to be minimum. For a given array layout, that is, when the steering vector of antenna array is known, the problem of solving minθ,φ∈ΩsmaxWTB is a convex problem [31]. The best excitation vector Wbest can be solved with an available tool, such as CVX [4].


### 3.5. The Procedure of Proposed Algorithm

For time-modulated sparse planar array, the step by step implementation method is proposed to achieve the objectives mentioned in Section3.1–3.4. The four objectives are minimizing the number of array elements, achieving sideband suppression and radiant power balance design, realizing shaped beam of main lobe, and suppressing side lobe level, respectively. This paper is to find the optimal array layout and excitation and modulated timing sequence to meet the abovementioned four objectives. Figure 3 is the flowchart of the proposed algorithm.Figure 3
Flowchart of the proposed algorithm.Figure3 shows the implementation process of the minimum number of array elements, sideband suppression and radiant power balance design, shaped beam of main lobe, and side lobe level constraint. The optimization process of the array starts with the initialization of the shaped beam pattern, the PSLL and PSBL, the initialization of the number N, positionxn,yn and the excitation amplitude In of the array element, the initialization of the switch-on duration timeτn, the rise/fall timeτa, and the modulation frequencyfn. The initial array layout is distributed at equal intervals of λ/2. The array elements are gradually reduced to find the minimum number of array elements that meets the sideband level constraint. The solving steps of the algorithm are (1) update the excitation amplitude according to the element's equivalent excitation obtained by the optimization of the main band; (2) update the rise/fall time τa; (3) calculate the PSBL of the array using the nonuniform period modulation method; (4) reduce the number of array elements iteratively and make N=N−1. When the sideband constraint cannot be satisfied, output the result of N=N+1.In Step (1), the OPM-CVX algorithm is used to iteratively update the position and equivalent excitation of the array element. The specific process is (a) construct orthogonal basis by Gram–Schmidt orthogonal process; (b) update the perturbation location parameters; (c) adopt convex algorithm to optimize the excitation coefficient and reduce PSLL with the position information; (d) calculate the actual electric field intensity according to the updated position and excitation; (e) loop iteratively until the actual electric field intensity meets the constraints of the shaped beam pattern and side lobe.


## 3.1. Minimum NumberN of Array Elements

According to the optimization objective of equation (13), the number of array elements needs to be minimized under the constraints of the side lobe and sideband. Under the condition that the array aperture remains invariant, the number of array elements is initialized toN0, and the spacing of array elements meets λ/2. N0 can be written as(14)N0=ceil2Lxλ⋅ceil2Lyλ,where Lx,Ly is the aperture size of the planar array. In this paper, the minimum number of array elements is obtained by gradually reducing the array elements while the side lobe and sideband still meet the constraints.


## 3.2. Sideband Suppression Based on Trapezoidal Pulse Switching Function

According to the optimization objective of equation (13), the maximum value of the sideband electric field intensity Emθ,φ,m≠0 of the antenna array is required to be minimum. From equation (11), the amplitude of amn decreases with the increase of m in the function sinc⋅. That is, the sideband level decreases as the sideband order increases, so the sideband level of the array at m=1 is the highest [18]. This paper uses a nonuniform period time-modulated method [29], so the sideband energy of the array elements cannot be superimposed. Therefore, the PSBL of the array element is the PSBL of the array. Minimizing the PSBL is equivalent to(15)minmaxn=1,2,…Na1n.Under the condition thatτn and fn are known, a1n is only determined by τa in equation (11). According to the pattern synthesis algorithm of OPM-CVX proposed in Section 3.3, the best coefficients Ina0n can be obtained under the constraints of the main lobe and side lobe. That is, Ina0n=Inτn−τafn is known when m=0. The value range of τa is 0,τn/2. When τa=0, the trapezoidal pulse switching function becomes the rectangular pulse switching function. The sideband suppression performance is the worst, while the radiated power of sidebands is the largest. When τa=τn/2, the trapezoidal pulse switching function becomes a triangular pulse switching function, the sideband suppression performance is the best, while the radiated power of sidebands is the lowest. In this paper, the best rise/fall time τa of trapezoidal pulse and the best static excitation In are solved under the constraint of DRR minIn=1,maxIn≤ε2 in equation (13) so that the radiated power of the main band and the suppression effect of the sideband are balanced. Under the constraints of minIn=1,maxIn≤ε2, this paper initializes the In of the array element firstly, and then design the value of τa under the balance requirement of main band radiated power and sideband suppression.


## 3.3. Orthogonal Perturbation Method for Optimal Location

According to the optimization objective of equation (13), the mean square error between the main lobe of the actual pattern and the main lobe of the desired pattern is required to be less than the set value, which can be expressed as(16)δ=14π∫Ωc∫ΩcE0θ,φ−Edθ,φ2sinθdθdφ<ε1,where δ is the mean square error.From equation (4), the electric field intensity of main band can be rewritten as(17)E0θ,φ=∑n=1NIna0nejkxnu+ynv=WTB,where B=b1,…,bNT, bn=ejkxnu+ynv is the steering vector of the antenna array, W=w1,…,wNT, wn=Ina0n is the excitation vector of the antenna array, and the carrier frequency common factor ej2πf0t of main band is ignored.Under the condition that the excitation vectorW is given, this paper uses orthogonal perturbation method to solve the optimal steering vector Bbest, so as to obtain the optimal array layout. The idea is as follows: according to the total differential equation, when the excitations of array elements are known and the element position disturbance is small, the variation of equation (17) can be written as(18)ΔE=WTΔB=∑n=1NwnΔB=∑n=1NwnΔxn∂bn∂xn+Δyn∂bn∂yn=WeΔSTB′,where ΔS=Δx1,…,ΔxN,Δy1,…,ΔyN is the perturbation of position information, S=x1,…,xN,y1,…,yN. B′=∂b1/∂x1,…,∂bN/∂xN,∂b1/∂y1,…,∂bN/∂yN is the direction of position perturbation, We=diagw1,…,wN,w1,…,wN, Q is a set of standard orthogonal basis constructed by using Gram–Schmidt orthogonal process, which makes B′=CQ. C is the coefficient of orthogonal base, then equation (18) can be rewritten as(19)ΔE=CTWeΔSTQ.AssumeΔSc=CTWeΔS, after the element position disturbance, and equation (17) is converted to(20)E0θ,φ=WTB+ΔE=WTB+ΔScTQ.Define the inner product of any two column vectorsx,y as(21)x,y=14π∫02π∫0πxθ,φyHθ,φsinθdθdφ.Take equations (20) and (21) into (16), then(22)δ=14π∫02π∫0πWTB−Edθ,φ+ΔScTQ2sinθdθdφ=WTB−Edθ,φ,WTB−Edθ,φ+WTB−Edθ,φ,ΔScTQ+ΔScTQ,WTB−Edθ,φ+ΔScTQ,ΔScTQ.According to the method for calculating the derivative of scalar function to complex vector [30], the partial derivative to ΔSc is zero, that is,(23)∂δ∂ΔSc=Q,ΔE+ΔSc=0.The iterative process is introduced by the differenceΔEi so that the actual pattern can constantly approach the desired pattern. According to ΔS=We−1C−TΔSc, on the i+1 iteration, the position is updated to(24)Si+1=Si+ΔSi+1=Si−We−1Ci−TQi,ΔEi.Therefore, when the excitation vectorWis determined, the optimal element position Sbest can be solved by the orthogonal perturbation method.


## 3.4. Convex Optimization for Optimal Excitation Vector

According to the optimization objective of (13), the maximum value of the side lobe electric field intensity E0θ,φ,θ,φ∈Ωs of the antenna array is required to be minimum. For a given array layout, that is, when the steering vector of antenna array is known, the problem of solving minθ,φ∈ΩsmaxWTB is a convex problem [31]. The best excitation vector Wbest can be solved with an available tool, such as CVX [4].


## 3.5. The Procedure of Proposed Algorithm

For time-modulated sparse planar array, the step by step implementation method is proposed to achieve the objectives mentioned in Section3.1–3.4. The four objectives are minimizing the number of array elements, achieving sideband suppression and radiant power balance design, realizing shaped beam of main lobe, and suppressing side lobe level, respectively. This paper is to find the optimal array layout and excitation and modulated timing sequence to meet the abovementioned four objectives. Figure 3 is the flowchart of the proposed algorithm.Figure 3
Flowchart of the proposed algorithm.Figure3 shows the implementation process of the minimum number of array elements, sideband suppression and radiant power balance design, shaped beam of main lobe, and side lobe level constraint. The optimization process of the array starts with the initialization of the shaped beam pattern, the PSLL and PSBL, the initialization of the number N, positionxn,yn and the excitation amplitude In of the array element, the initialization of the switch-on duration timeτn, the rise/fall timeτa, and the modulation frequencyfn. The initial array layout is distributed at equal intervals of λ/2. The array elements are gradually reduced to find the minimum number of array elements that meets the sideband level constraint. The solving steps of the algorithm are (1) update the excitation amplitude according to the element's equivalent excitation obtained by the optimization of the main band; (2) update the rise/fall time τa; (3) calculate the PSBL of the array using the nonuniform period modulation method; (4) reduce the number of array elements iteratively and make N=N−1. When the sideband constraint cannot be satisfied, output the result of N=N+1.In Step (1), the OPM-CVX algorithm is used to iteratively update the position and equivalent excitation of the array element. The specific process is (a) construct orthogonal basis by Gram–Schmidt orthogonal process; (b) update the perturbation location parameters; (c) adopt convex algorithm to optimize the excitation coefficient and reduce PSLL with the position information; (d) calculate the actual electric field intensity according to the updated position and excitation; (e) loop iteratively until the actual electric field intensity meets the constraints of the shaped beam pattern and side lobe.


## 4. Simulation Results

In this section, a variety of simulations and comparisons are presented in order to illustrate the advantages of the proposed hybrid algorithm. The simulations in this paper are achieved by a PC with CPU model i7-6500U, main frequency 2.5 GHz, and RAM of 8.00 GB.


### 4.1. Side Lobe and Sideband Suppression

Under the constraint that the main lobe pattern is consistent, two examples are presented in comparison with the results of pencil beam and flat top beam pattern obtained in [12, 15] to demonstrate the effectiveness of the proposed algorithm in suppressing the PSBL and PSLL. In the first example, A linear array of 9-element with initial equal spacing of 0.5λ is studied. In the case that the main lobe pattern is consistent with the one in [15], the objective is to minimize the PSLL and PSBL. Similar simulations have been previously carried out by the efficient mathematical method (EMM) in [15]. The optimized PSLL and PSBL were −22 dB and −15.2 dB, respectively. With the 0.2° sampling interval of θ, the optimization process takes 2 iterations and 0.784 seconds. Under the condition that number of array elements and the array aperture are the same, the PSLL and PSBL optimized by proposed algorithm are −22.42 dB and −15.6 dB, respectively, which are lower than the values in [15]. The main band and the first sideband patterns, the histogram of the excitation amplitude of array elements, are shown in Figures 4 and 5, respectively. Table 1 shows the specific value of values of position and excitation. Because the positions and excitations of the elements are centrosymmetric, only 5 sets of data are listed in Table 1. Table 2 shows the comparison of the parameters optimized by the proposed algorithm and the EMM in [15].Figure 4
Comparison between the designed array and the one in [15].Figure 5
Excitation amplitude optimized by OPM-CVX.Table 1
Position and excitation of array elements.
Element No.
5
6
7
8
9
Positionλ
0
0.4829
0.9688
1.4724
2.0000
Excitation
0.9976
1
0.9733
0.5656
0.3546Table 2
Performance comparison between the designed array and the actual one in [15].
Method
Number of Elements
PSLL (dB)
PSBL (dB)
Aperture Sizeλ
Time consumed (s)
EMM in [15]
9
−22
−15.2
4
—
OPM-CVX
9
−22.42
−15.6
4
0.784The second example studies a linear array of 30 elements with initial equal spacing of0.6λ. The flat top beam pattern with a main lobe width of 60° is set as the desired pattern. The objective is to minimize the PSLL and PSBL. Patra et al. [12] used DE algorithm to synthesize the same array and obtained the PSLL of −26.46 dB and the PSBL of −40.49 dB, respectively. In this paper, the PSBL of −26.66 dB and the PSLL of −47.51 dB are obtained by OPM-CVX algorithm. The optimized ripple of flat top beam is 0.965 dB, slightly higher than 0.7224 dB in [12]. Figure 6 shows the synthesized pattern by OPM-CVX and DE. Figure 7 shows the amplitude and position of array elements by OPM-CVX.Figure 6
Comparison between the designed array and the one in [12].Figure 7
Excitation amplitude and position optimized by OPM-CVX.With the 0.2° sampling interval ofθ, the optimization process takes 2 iterations and 0.806 seconds. The spacing of the two elements closest to the centre of the array is 0.2004λ, and the spacing of other elements is greater than 0.5λ. The minimum equivalent excitation is 0.011. The detailed comparison of parameters is shown in Table 3. By comparison, the proposed algorithm obtains a smaller aperture size and a better flat top beam under the equal amplitude in-phase excitation.Table 3
Performance comparison between the designed array and the actual one in [12].
Algorithm
Number of Elements
PSLL (dB)
PSBL (dB)
MAX Ripple (dB)
DRR
Apertureλ
DE in [12]
30
−40.4932
−26.4655
0.7224
6.2289
19.5904
OPM-CVX
30
−47.51
−26.66
0.965
1
17.652The comparison results with the pattern in [12, 15] shows that the OPM-CVX algorithm with nonuniform period time-modulated mode can obtain better effect of pattern synthesis.


### 4.2. Minimization of the Number of Array Elements

In order to demonstrate the effectiveness of the OPM-CVX algorithm in reducing the number of the array elements, two examples are presented in comparison with the results in [18, 21, 32, 33]. The first example studies a planar array of 100 (10 × 10) elements with equal spacing of 0.5λ. The optimization objective is to reduce the element number and suppress the PSBL under the side lobe constraint of −20 dB. Similar simulations have been previously carried out in [21, 32], but only the array elements with an equal spacing of 0.5λ were considered. Under the aperture constraint of 5λ×5λ, the number of array elements synthesized by the proposed algorithm is 64 (8 × 8). The PSBL is −42.27 dB, lower than −31.8 dB in [21] and −34.69 dB in [32]. Figure 8 shows the main band pattern. Figure 9 shows the first sideband pattern, and it is essentially the first sideband pattern of a certain array element. Because the array element is isotropic, the electric field intensity is consistent in the θ and φ directions.Figure 8
Designed array of 8 × 8.Figure 9
First sideband of 8 × 8.With the 1° and 2° sampling interval ofθ and φ, The optimization process takes 30 iterations and 449.8 seconds. Compared with the linear array above, the planar array increases the number of array elements and the complexity of the array, so the time consumption and the number of iterations in the optimization process are increased. The comparison of parameters is shown in Table 4. The element positions are shown in Figure 10 and listed in Table 5. The minimum element spacing of x direction and y direction is 0.6877λ, and the maximum element spacing of x direction and y direction is 0.7089λ. The optimized aperture of 8 × 8 planar array is 4.9007λ×4.8986λ.Table 4
Performance comparison between the designed array and the actual one in [21, 32].
Algorithm
PSLL (dB)
PSBL (dB)
Number of elements
Positionλ
Apertureλ
PSO in [21]
−19.6
−31.8
76
0.5
5 × 5
DE in [32]
−19.86
−34.69
80
0.5
5 × 5
OPM-CVX
−19.90
−42.27
64
Table5
4.9007 × 4.8986Figure 10
Element position of 8 × 8 planar array.Table 5
Optimized element position.
y
1
λ
y
2
λ
y
3
λ
y
4
λ
y
5
λ
y
6
λ
y
7
λ
y
8
λ
x
1
λ
(−2.45 −2.45)
(−2.45 −1.75)
(−2.45 −1.04)
(−2.45 −0.35)
(−2.45 0.34)
(−2.45 1.04)
(−2.45 1.74)
(−2.45 2.45)
x
2
λ
(−1.74 −2.45)
(−1.74 −1.75)
(−1.75 −1.06)
(−1.75 −0.36)
(−1.75 0.34)
(−1.75 1.05)
(−1.75 1.75)
(−1.74 2.45)
x
3
λ
(−1.04 −2.45)
(−1.04 −1.75)
(−1.04 −1.05)
(−1.05 −0.35)
(−1.05 0.34)
(−1.06 1.05)
(−1.05 1.75)
(−1.04 2.45)
x
4
λ
(−0.34 −2.45)
(−0.34 −1.74)
(−0.34 −1.04)
(−0.35 −0.34)
(−0.36 0.35)
(−0.36 1.05)
(−0.36 1.74)
(−0.35 2.45)
x
5
λ
(0.35 −2.45)
(0.36 −1.74)
(0.35 −1.04)
(0.35 −0.34)
(0.35 0.36)
(0.34 1.05)
(0.35 1.74)
(0.34 2.45)
x
6
λ
(1.04 −2.45)
(1.05 −1.75)
(1.05 −1.04)
(1.05 −0.34)
(1.05 0.35)
(1.04 1.05)
(1.05 1.75)
(1.04 2.45)
x
7
λ
(1.74 −2.45)
(1.75 −1.75)
(1.75 −1.05)
(1.75 −0.35)
(1.75 0.34)
(1.75 1.05)
(1.75 1.75)
(1.74 2.45)
x
8
λ
(2.45 −2.45)
(2.45 −1.74)
(2.45 −1.04)
(2.45 −0.34)
(2.45 0.35)
(2.45 1.04)
(2.45 1.74)
(2.45 2.45)The second example studies a planar array with 400 (20 × 20) elements. The elements are arranged with square lattices and square boundaries. The optimization objective is to minimize the number of elements under the side lobe constraint of −42 dB. Similar simulations have been previously carried out in [18, 33]. The comparison of two-dimensional pattern between the proposed algorithm and the PCS-CVX in [18] is shown in Figure 11. The optimized PSLL is −43.17 dB, and the optimized element number is 324, which is 10% less than that in [18].Figure 11
Comparison of 2D normalized pattern with the one in [18].With the 1° and 2° sampling interval ofθ and φ, The optimization process takes 12 iterations and 1981.8 seconds. Compared with the 8 × 8 planar array, the increase of array elements further increases the computational complexity and time consumption. The minimum element spacing of x direction and y direction is 0.8446λ, and the maximum element spacing of x direction and y direction is 0.8507λ. The optimized aperture of 18×18 planar array is 14.4499λ×14.4501λ. The element positions are shown in Figure 12. The detailed comparison of parameters is shown in Table 6.Figure 12
Element position of 18 × 18 planar array.Table 6
Performance comparison between the designed array and the actual one in [18, 33].
Algorithm
Gain (dB)
Number of elements
PSLL (dB)
PSBL (dB)
d
min λ
Apertureλ
[18]
29.63
358
−42
—
0.6295
15.6 × 15.6
[33]
—
496
−41
—
<0.5
10 × 10
OPM-CVX
32.00
324
−43.17
−49.39
0.847
14.4499 × 14.4501Through the experiments compared with [18, 21, 32, 33], the nonuniform period time-modulated mode and OPM-CVX algorithm can effectively reduce the element number and suppress the PSBL, with the constraints of the main lobe response and the side lobe level.


### 4.3. Analysis of the Rise/Fall Time, Sideband, and Power

Two examples are presented to verify the effect of the rise/fall time of the trapezoidal pulse switching function on the sideband performance and to achieve the power balance between sideband and main band under the constraint of DRR. The first example shows the relationship between the sideband level and the rise/fall time. The same 9-element linear array as that in Section4.1 is adopted. From the synthesis results of the 9-element linear array, the normalized minimum switch-on duration time is τn=0.3546. According to the value range τa∈0,τn/2, we can get τa≤0.1773. Figure 13 shows the first 20 sideband curves corresponding to different τa. The sideband level in this figure decreases as τa increases. Table 7 shows the relationship between the power of the main band and the sideband level when τa is different. As τa increases, the sideband level decreases faster. The power of main band is obtained by equation (6).Figure 13
Comparison of the first 20 order sidebands under differentτa.Table 7
Relationship betweenτa, sideband level, and power of main band.
τ
a
Power of main band (dB)
1st SBL (dB)
2nd SBL (dB)
3rd SBL (dB)
4th SBL (dB)
5th SBL (dB)
6th SBL (dB)
0
70.5526
−26.7594
−34.6214
−37.8957
−38.9044
−44.2325
−42.6294
0.04
69.7273
−26.7823
−34.7130
−38.1024
−39.2733
−44.8117
−43.4686
0.08
68.8667
−27.4881
−36.0300
−39.3719
−41.4599
−47.6923
−46.8514
0.12
67.9685
−28.2907
−37.6815
−41.1580
−44.7104
−52.3954
−53.3080
0.16
67.0302
−29.1995
−39.7258
−43.5518
−49.4283
−60.4416
−72.3277As shown in Figure14, the higher harmonic reduces faster when applying trapezoidal pulse and the power reduction is higher as τa increase, until the electric field intensity goes to zero. Table 8 shows the power of main band, the power of the first 8 sidebands and the directivity of 9-element linear array, which are calculated by equations (6), (7), and (9), respectively. Comparing τa=0.16 with τa=0, the reduction of the first 8 sideband power turns out to be around 21.73%. Meanwhile, the decrease of directivity is very small. Therefore, the increase of τa reduces the power of the main band but improves the radiation efficiency. According to reference [22], since the directivity is almost constant in Table 8, the gain of antenna array is improved.Figure 14
Power comparison of the main band and the first 8 order sidebands under differentτa.Table 8
Power of the first 8 sidebands and main band under differentτa.
τ
a
0
0.04
0.08
0.12
0.16
Power of main band (dB)
70.55
69.73
68.87
67.97
67.03
Power of first 8 sidebands (dB)
156.74
134.92
100.51
79.33
60.00
Radiation efficiency of main band (%)
31.04
34.07
40.66
46.14
52.77
Directivity
27.19
27.16
27.13
27.10
27.04In the second example, the same 64 (8 × 8) elements planar array as that in Section4.2 is adopted. From the synthesis results of the 64 (8 × 8) elements planar array, the normalized minimum switch-on duration time is τn=0.03. Under the condition that the power of the main band is constant, changing the DRR from 1 to 8, then the rise/fall time τaof the trapezoidal switching function can be set to a maximum value of 0.12. Figure 15 shows the first 20 sidebands optimized by the rectangular switching function, trapezoidal switching function when τa=0.12. It can be seen that the sideband suppression effect of the trapezoidal pulse switching function is better than the rectangular pulse switching function and the DE algorithm in [32].Figure 15
Comparison of the first 20 order sidebands optimized by OPM-CVX and DE.As shown in Figure16 and Table 9, the power variation trend of the main band and sidebands of the planar array are the same as that of the linear array. Comparing τa=0.12 with τa=0, the reduction in the first 8 sideband power turns out to be around 5.35%. According to reference [22], because of the tiny decrease of directivity and the large increase of main band radiation efficiency, the gain of antenna array is improved.Figure 16
Power comparison of the main band and the first 8 sidebands under differentτa.Table 9
Power of the first 8 sideband and main band under differentτa.
τ
a
0
0.06
0.12
Power of main band (dB)
109.71
108.62
107.53
Power of first 8 sidebands (dB)
546.50
500.22
379.79
Radiation efficiency of main band (%)
16.72
17.84
22.07
Directivity
22.42
22.27
22.07According to the simulations of linear array and planar array, the trapezoidal pulse switching function has better sideband suppression effect than the rectangular pulse switching function. The sideband power decreases faster than the main band power asτa increases, and the radiation efficiency of the main band is improved. Therefore, the increase of τacan effectively suppress the sideband radiation and improve the radiation efficiency and gain of the main band. In addition, the power loss of the main band caused by the increase of τacan be reduced by increasing the DRR.


## 4.1. Side Lobe and Sideband Suppression

Under the constraint that the main lobe pattern is consistent, two examples are presented in comparison with the results of pencil beam and flat top beam pattern obtained in [12, 15] to demonstrate the effectiveness of the proposed algorithm in suppressing the PSBL and PSLL. In the first example, A linear array of 9-element with initial equal spacing of 0.5λ is studied. In the case that the main lobe pattern is consistent with the one in [15], the objective is to minimize the PSLL and PSBL. Similar simulations have been previously carried out by the efficient mathematical method (EMM) in [15]. The optimized PSLL and PSBL were −22 dB and −15.2 dB, respectively. With the 0.2° sampling interval of θ, the optimization process takes 2 iterations and 0.784 seconds. Under the condition that number of array elements and the array aperture are the same, the PSLL and PSBL optimized by proposed algorithm are −22.42 dB and −15.6 dB, respectively, which are lower than the values in [15]. The main band and the first sideband patterns, the histogram of the excitation amplitude of array elements, are shown in Figures 4 and 5, respectively. Table 1 shows the specific value of values of position and excitation. Because the positions and excitations of the elements are centrosymmetric, only 5 sets of data are listed in Table 1. Table 2 shows the comparison of the parameters optimized by the proposed algorithm and the EMM in [15].Figure 4
Comparison between the designed array and the one in [15].Figure 5
Excitation amplitude optimized by OPM-CVX.Table 1
Position and excitation of array elements.
Element No.
5
6
7
8
9
Positionλ
0
0.4829
0.9688
1.4724
2.0000
Excitation
0.9976
1
0.9733
0.5656
0.3546Table 2
Performance comparison between the designed array and the actual one in [15].
Method
Number of Elements
PSLL (dB)
PSBL (dB)
Aperture Sizeλ
Time consumed (s)
EMM in [15]
9
−22
−15.2
4
—
OPM-CVX
9
−22.42
−15.6
4
0.784The second example studies a linear array of 30 elements with initial equal spacing of0.6λ. The flat top beam pattern with a main lobe width of 60° is set as the desired pattern. The objective is to minimize the PSLL and PSBL. Patra et al. [12] used DE algorithm to synthesize the same array and obtained the PSLL of −26.46 dB and the PSBL of −40.49 dB, respectively. In this paper, the PSBL of −26.66 dB and the PSLL of −47.51 dB are obtained by OPM-CVX algorithm. The optimized ripple of flat top beam is 0.965 dB, slightly higher than 0.7224 dB in [12]. Figure 6 shows the synthesized pattern by OPM-CVX and DE. Figure 7 shows the amplitude and position of array elements by OPM-CVX.Figure 6
Comparison between the designed array and the one in [12].Figure 7
Excitation amplitude and position optimized by OPM-CVX.With the 0.2° sampling interval ofθ, the optimization process takes 2 iterations and 0.806 seconds. The spacing of the two elements closest to the centre of the array is 0.2004λ, and the spacing of other elements is greater than 0.5λ. The minimum equivalent excitation is 0.011. The detailed comparison of parameters is shown in Table 3. By comparison, the proposed algorithm obtains a smaller aperture size and a better flat top beam under the equal amplitude in-phase excitation.Table 3
Performance comparison between the designed array and the actual one in [12].
Algorithm
Number of Elements
PSLL (dB)
PSBL (dB)
MAX Ripple (dB)
DRR
Apertureλ
DE in [12]
30
−40.4932
−26.4655
0.7224
6.2289
19.5904
OPM-CVX
30
−47.51
−26.66
0.965
1
17.652The comparison results with the pattern in [12, 15] shows that the OPM-CVX algorithm with nonuniform period time-modulated mode can obtain better effect of pattern synthesis.


## 4.2. Minimization of the Number of Array Elements

In order to demonstrate the effectiveness of the OPM-CVX algorithm in reducing the number of the array elements, two examples are presented in comparison with the results in [18, 21, 32, 33]. The first example studies a planar array of 100 (10 × 10) elements with equal spacing of 0.5λ. The optimization objective is to reduce the element number and suppress the PSBL under the side lobe constraint of −20 dB. Similar simulations have been previously carried out in [21, 32], but only the array elements with an equal spacing of 0.5λ were considered. Under the aperture constraint of 5λ×5λ, the number of array elements synthesized by the proposed algorithm is 64 (8 × 8). The PSBL is −42.27 dB, lower than −31.8 dB in [21] and −34.69 dB in [32]. Figure 8 shows the main band pattern. Figure 9 shows the first sideband pattern, and it is essentially the first sideband pattern of a certain array element. Because the array element is isotropic, the electric field intensity is consistent in the θ and φ directions.Figure 8
Designed array of 8 × 8.Figure 9
First sideband of 8 × 8.With the 1° and 2° sampling interval ofθ and φ, The optimization process takes 30 iterations and 449.8 seconds. Compared with the linear array above, the planar array increases the number of array elements and the complexity of the array, so the time consumption and the number of iterations in the optimization process are increased. The comparison of parameters is shown in Table 4. The element positions are shown in Figure 10 and listed in Table 5. The minimum element spacing of x direction and y direction is 0.6877λ, and the maximum element spacing of x direction and y direction is 0.7089λ. The optimized aperture of 8 × 8 planar array is 4.9007λ×4.8986λ.Table 4
Performance comparison between the designed array and the actual one in [21, 32].
Algorithm
PSLL (dB)
PSBL (dB)
Number of elements
Positionλ
Apertureλ
PSO in [21]
−19.6
−31.8
76
0.5
5 × 5
DE in [32]
−19.86
−34.69
80
0.5
5 × 5
OPM-CVX
−19.90
−42.27
64
Table5
4.9007 × 4.8986Figure 10
Element position of 8 × 8 planar array.Table 5
Optimized element position.
y
1
λ
y
2
λ
y
3
λ
y
4
λ
y
5
λ
y
6
λ
y
7
λ
y
8
λ
x
1
λ
(−2.45 −2.45)
(−2.45 −1.75)
(−2.45 −1.04)
(−2.45 −0.35)
(−2.45 0.34)
(−2.45 1.04)
(−2.45 1.74)
(−2.45 2.45)
x
2
λ
(−1.74 −2.45)
(−1.74 −1.75)
(−1.75 −1.06)
(−1.75 −0.36)
(−1.75 0.34)
(−1.75 1.05)
(−1.75 1.75)
(−1.74 2.45)
x
3
λ
(−1.04 −2.45)
(−1.04 −1.75)
(−1.04 −1.05)
(−1.05 −0.35)
(−1.05 0.34)
(−1.06 1.05)
(−1.05 1.75)
(−1.04 2.45)
x
4
λ
(−0.34 −2.45)
(−0.34 −1.74)
(−0.34 −1.04)
(−0.35 −0.34)
(−0.36 0.35)
(−0.36 1.05)
(−0.36 1.74)
(−0.35 2.45)
x
5
λ
(0.35 −2.45)
(0.36 −1.74)
(0.35 −1.04)
(0.35 −0.34)
(0.35 0.36)
(0.34 1.05)
(0.35 1.74)
(0.34 2.45)
x
6
λ
(1.04 −2.45)
(1.05 −1.75)
(1.05 −1.04)
(1.05 −0.34)
(1.05 0.35)
(1.04 1.05)
(1.05 1.75)
(1.04 2.45)
x
7
λ
(1.74 −2.45)
(1.75 −1.75)
(1.75 −1.05)
(1.75 −0.35)
(1.75 0.34)
(1.75 1.05)
(1.75 1.75)
(1.74 2.45)
x
8
λ
(2.45 −2.45)
(2.45 −1.74)
(2.45 −1.04)
(2.45 −0.34)
(2.45 0.35)
(2.45 1.04)
(2.45 1.74)
(2.45 2.45)The second example studies a planar array with 400 (20 × 20) elements. The elements are arranged with square lattices and square boundaries. The optimization objective is to minimize the number of elements under the side lobe constraint of −42 dB. Similar simulations have been previously carried out in [18, 33]. The comparison of two-dimensional pattern between the proposed algorithm and the PCS-CVX in [18] is shown in Figure 11. The optimized PSLL is −43.17 dB, and the optimized element number is 324, which is 10% less than that in [18].Figure 11
Comparison of 2D normalized pattern with the one in [18].With the 1° and 2° sampling interval ofθ and φ, The optimization process takes 12 iterations and 1981.8 seconds. Compared with the 8 × 8 planar array, the increase of array elements further increases the computational complexity and time consumption. The minimum element spacing of x direction and y direction is 0.8446λ, and the maximum element spacing of x direction and y direction is 0.8507λ. The optimized aperture of 18×18 planar array is 14.4499λ×14.4501λ. The element positions are shown in Figure 12. The detailed comparison of parameters is shown in Table 6.Figure 12
Element position of 18 × 18 planar array.Table 6
Performance comparison between the designed array and the actual one in [18, 33].
Algorithm
Gain (dB)
Number of elements
PSLL (dB)
PSBL (dB)
d
min λ
Apertureλ
[18]
29.63
358
−42
—
0.6295
15.6 × 15.6
[33]
—
496
−41
—
<0.5
10 × 10
OPM-CVX
32.00
324
−43.17
−49.39
0.847
14.4499 × 14.4501Through the experiments compared with [18, 21, 32, 33], the nonuniform period time-modulated mode and OPM-CVX algorithm can effectively reduce the element number and suppress the PSBL, with the constraints of the main lobe response and the side lobe level.


## 4.3. Analysis of the Rise/Fall Time, Sideband, and Power

Two examples are presented to verify the effect of the rise/fall time of the trapezoidal pulse switching function on the sideband performance and to achieve the power balance between sideband and main band under the constraint of DRR. The first example shows the relationship between the sideband level and the rise/fall time. The same 9-element linear array as that in Section4.1 is adopted. From the synthesis results of the 9-element linear array, the normalized minimum switch-on duration time is τn=0.3546. According to the value range τa∈0,τn/2, we can get τa≤0.1773. Figure 13 shows the first 20 sideband curves corresponding to different τa. The sideband level in this figure decreases as τa increases. Table 7 shows the relationship between the power of the main band and the sideband level when τa is different. As τa increases, the sideband level decreases faster. The power of main band is obtained by equation (6).Figure 13
Comparison of the first 20 order sidebands under differentτa.Table 7
Relationship betweenτa, sideband level, and power of main band.
τ
a
Power of main band (dB)
1st SBL (dB)
2nd SBL (dB)
3rd SBL (dB)
4th SBL (dB)
5th SBL (dB)
6th SBL (dB)
0
70.5526
−26.7594
−34.6214
−37.8957
−38.9044
−44.2325
−42.6294
0.04
69.7273
−26.7823
−34.7130
−38.1024
−39.2733
−44.8117
−43.4686
0.08
68.8667
−27.4881
−36.0300
−39.3719
−41.4599
−47.6923
−46.8514
0.12
67.9685
−28.2907
−37.6815
−41.1580
−44.7104
−52.3954
−53.3080
0.16
67.0302
−29.1995
−39.7258
−43.5518
−49.4283
−60.4416
−72.3277As shown in Figure14, the higher harmonic reduces faster when applying trapezoidal pulse and the power reduction is higher as τa increase, until the electric field intensity goes to zero. Table 8 shows the power of main band, the power of the first 8 sidebands and the directivity of 9-element linear array, which are calculated by equations (6), (7), and (9), respectively. Comparing τa=0.16 with τa=0, the reduction of the first 8 sideband power turns out to be around 21.73%. Meanwhile, the decrease of directivity is very small. Therefore, the increase of τa reduces the power of the main band but improves the radiation efficiency. According to reference [22], since the directivity is almost constant in Table 8, the gain of antenna array is improved.Figure 14
Power comparison of the main band and the first 8 order sidebands under differentτa.Table 8
Power of the first 8 sidebands and main band under differentτa.
τ
a
0
0.04
0.08
0.12
0.16
Power of main band (dB)
70.55
69.73
68.87
67.97
67.03
Power of first 8 sidebands (dB)
156.74
134.92
100.51
79.33
60.00
Radiation efficiency of main band (%)
31.04
34.07
40.66
46.14
52.77
Directivity
27.19
27.16
27.13
27.10
27.04In the second example, the same 64 (8 × 8) elements planar array as that in Section4.2 is adopted. From the synthesis results of the 64 (8 × 8) elements planar array, the normalized minimum switch-on duration time is τn=0.03. Under the condition that the power of the main band is constant, changing the DRR from 1 to 8, then the rise/fall time τaof the trapezoidal switching function can be set to a maximum value of 0.12. Figure 15 shows the first 20 sidebands optimized by the rectangular switching function, trapezoidal switching function when τa=0.12. It can be seen that the sideband suppression effect of the trapezoidal pulse switching function is better than the rectangular pulse switching function and the DE algorithm in [32].Figure 15
Comparison of the first 20 order sidebands optimized by OPM-CVX and DE.As shown in Figure16 and Table 9, the power variation trend of the main band and sidebands of the planar array are the same as that of the linear array. Comparing τa=0.12 with τa=0, the reduction in the first 8 sideband power turns out to be around 5.35%. According to reference [22], because of the tiny decrease of directivity and the large increase of main band radiation efficiency, the gain of antenna array is improved.Figure 16
Power comparison of the main band and the first 8 sidebands under differentτa.Table 9
Power of the first 8 sideband and main band under differentτa.
τ
a
0
0.06
0.12
Power of main band (dB)
109.71
108.62
107.53
Power of first 8 sidebands (dB)
546.50
500.22
379.79
Radiation efficiency of main band (%)
16.72
17.84
22.07
Directivity
22.42
22.27
22.07According to the simulations of linear array and planar array, the trapezoidal pulse switching function has better sideband suppression effect than the rectangular pulse switching function. The sideband power decreases faster than the main band power asτa increases, and the radiation efficiency of the main band is improved. Therefore, the increase of τacan effectively suppress the sideband radiation and improve the radiation efficiency and gain of the main band. In addition, the power loss of the main band caused by the increase of τacan be reduced by increasing the DRR.


## 5. Conclusion

In time-modulated antenna array, the high side lobe will lower the radiation performance of the main lobe, and the redundant sideband will reduce the radiation efficiency of the main band. In order to solve these problems, a hybrid OPM-CVX algorithm is proposed to suppress the side lobe of sparse array, and the nonuniform periodic time-modulated mode and trapezoidal pulse switching function are used to suppress sideband. In addition, under the constraint of the shaped beam of main lobe, the minimum array elements, low side lobe, and sideband are realized step by step in the way of loop iteration strategy. The simulation results of linear and planar arrays show that the proposed algorithm can achieve shaped beam of main lobe and low side lobe with fewer array elements. The algorithm can achieve the balanced design of sideband suppression and the main band power by adjusting the DRR of the antenna array and the rise/fall time of the trapezoidal pulse switching function. Compared with other commonly used algorithms, the proposed algorithm can improve the performance of the pattern of time-modulated sparse array and have important research value and practicability.



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*Source: 5491921-2020-04-14.xml*