Pattern Synthesis of Unequally Spaced Linear Arrays ... - IEEE Xplore

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 65, NO. 8, AUGUST 2017

Pattern Synthesis of Unequally Spaced Linear Arrays Including Mutual Coupling Using Iterative FFT via Virtual Active Element Pattern Expansion Yanhui Liu, Member, IEEE, Xin Huang, Kai Da Xu, Member, IEEE, Zhengyong Song, Shiwen Yang, Senior Member, IEEE, and Qing Huo Liu, Fellow, IEEE

Abstract— A virtual active element pattern (AEP) expansion method is presented in which each AEP in an unequally spaced array is considered to be the pattern radiated by a subarray of some equally spaced virtual elements. With the help of this method, the pattern of an unequally spaced array including mutual coupling can be efficiently evaluated by fast Fourier transform (FFT). By incorporating this idea into the iterative Fourier transform procedure, we develop a novel iterative synthesis method, which can apply the iterative FFT to efficiently synthesize unequally spaced arrays including mutual coupling. Different excitation constraints, such as phase-only control and amplitude-phase optimization with a prescribed dynamic range ratio, can be easily added into the proposed synthesis procedure. A set of synthesis examples for different antenna arrays with pencil and shaped beam patterns are provided to validate the effectiveness and advantages of the proposed method. Index Terms— Antenna array, fast Fourier transform (FFT), mutual coupling, radiation pattern synthesis, virtual active element pattern expansion (VAEPE).

I. I NTRODUCTION

A

NTENNA arrays have a wide range of applications in radar, sonar, and communication systems. Many numerical pattern synthesis methods have been developed over the past years, such as the stochastic optimization algorithms [1]–[7], convex optimization (CO) techniques [8]–[10], alternating projection method [11], iterative Fourier transform (IFT) methods [12]–[15], and some other synthesis techniques [16]–[19]. Among them, the IFT is one of the most efficient iterative synthesis methods. In this method, the

Manuscript received November 23, 2015; revised December 7, 2016; accepted May 12, 2017. Date of publication May 25, 2017; date of current version August 2, 2017. This work was supported in part by the National Natural Science Foundation of China under Grant 61301009, Grant 61601390, and Grant 41390453, in part by the Sino-Foreign Cooperation Project of Fujian Province under Grant 2017I0017, and in part by the Fundamental Research Funds for the Central Universities under Grant 20720160081. (Corresponding author: Kai Da Xu.) Y. Liu, X. Huang, K. D. Xu, and Z. Song are with the Department of Electronic Science, Institute of Electromagnetics and Acoustics, Xiamen University, Xiamen 361005, China (e-mail: [email protected]). S. Yang is with the School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China. Q. H. Liu is with the Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2017.2708081

Fourier transform relationship between the excitation distribution and the far-field pattern is employed, and then, the excitations of an equally spaced array can be efficiently obtained by successively applying the fast Fourier transform (FFT) and inverse FFT (IFFT). Recently, the IFT method has been modified to synthesize some thinned antenna arrays [20]–[22]. Despite the success of the IFT method and its modifications, they can only deal with the case of equally spaced arrays without considering coupling effects. In [23], the FFT used in the IFT is replaced with the nonuniform FFT (NUFFT), and an iterative NUFFT synthesis technique is developed. This method can successfully handle the array factor of an unequally spaced array, but it still cannot consider the mutual coupling between practical elements in the synthesis. This significantly limits the synthesis performance of the iterative NUFFT method, especially when a pattern with low sidelobe level (SLL) or some nulls is considered. As is well known, the array’s mutual coupling can be considered by using the active element pattern (AEP), which is defined as the radiation pattern when one element is excited and all the others are connected to matching loads [24]. However, since the AEPs usually differ from each other, the FFT cannot be directly applied to the evaluation of array pattern. More recently, we presented an AEP expansion (AEPE) method in which the AEP of one element is approximated as the radiated pattern by exciting this element and its neighboring ones [25], [26]. In this way, the pattern of an equally spaced array including coupling effect can be efficiently calculated by using the FFT. However, since the AEPE method utilizes a subarray of neighboring elements at real positions to expand the AEP, it is useful only for an equally spaced array when followed by the FFT. Due to this limitation, only focused beam patterns of equally spaced linear arrays are considered in [25] and [26], and more general cases, such as shaped pattern requirements and unequally spaced arrays, have not been discussed. In this paper, a virtual AEPE (VAEPE) method is presented in which each AEP for an unequally spaced array is considered as the pattern radiated by exciting several equally spaced virtual elements surrounding the real element position. This method can be considered to be a more generalized version of the original AEPE. With the VAEPE method, the

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LIU et al.: PATTERN SYNTHESIS OF UNEQUALLY SPACED LINEAR ARRAYS INCLUDING MUTUAL COUPLING

Fig. 1.

where [x] denotes the nearest integer to x, cn,q is the qth virtual element excitation for the nth real element, and Q is an odd positive integer, which represents the number of virtual elements for approximating the AEP of each real element. d = (x N−1 − x 0 )/(r N − 1) is the element spacing of the virtual array, and r is the parameter used to control the virtual spacing d. Usually, we choose r , such that d ≤ 0.5λ for the virtual array. gvep (u), called the virtual element pattern, is assumed to be the same for all the virtual elements. To reduce the approximation error of (2), we choose gvep(u) as the average over all the phase-adjusted AEPs. That is

Schematic of the VAEPE method.

pattern of an unequally spaced array including mutual coupling can be efficiently evaluated by the FFT. Incorporating this method into the IFT procedure, we develop a novel efficient iterative synthesis method, which can iteratively apply the FFT and IFFT to synthesize the pattern of an unequally spaced array including mutual coupling. In addition, some excitation constraints, such as phase-only control and amplitude-phase optimization with a prescribed dynamic range ratio (DRR), can be easily incorporated into the proposed synthesis procedure. A set of examples for synthesizing focused and shaped patterns with different antenna elements are provided to validate the efficiency, effectiveness, and robustness of the proposed method. II. F ORMULATION AND A LGORITHM Let us consider a linear array with N elements, which may be unequally placed at the positions x = [x 0 , x 1 , . . . , x N−1 ]T . The array’s far-field array pattern can be expressed as the following: N−1 wn gn (u)e jβ xn u (1) f (u) = n=0

where u = sin(θ ), and β = 2π/λ is the wavenumber in free space, where λ is the wavelength. wn is the complex excitation of the nth element, e jβ xn u is the position difference phase term, and gn (u) is the phased-adjusted AEP of the nth element (the coordinate origin is located at each element, and the position-related phase term is extracted) [24]. The AEP is the array’s radiation pattern when only one element is excited and all the others are connected to matching loads. Due to the presence of mutual coupling in the unequally spaced antenna array, all the AEPs are different from each other. In [25], we present an AEPE method for an equally spaced linear array. In this method, each AEP is approximated as the radiation pattern by appropriately exciting the neighboring elements, so that the FFT can be used to speed up the calculation of the array pattern, but only for equally spaced arrays as mentioned previously. Now, we develop a VAEPE method suitable for unequally spaced arrays. The idea is to consider each AEP to be the radiation pattern of a virtual equally spaced subarray surrounding the real element, as shown in Fig. 1. Mathematically, that is Q−1

gn (u)e jβ xn u = gvep (u)

q=− Q−1 2

xn

cn,q e jβ (

d

+q )du

gvep (u) =

(2)

N−1 1 gn (u). N

(3)

n=0

In addition, the virtual excitation coefficients cn,q in (2) can be obtained by minimizing the following objective function: min G En cn − gn 22

(4)

cn

where

⎡ ⎢ ⎢ G=⎢ ⎣

gvep(u 1 )

⎢ En = ⎢ ⎣

O gvep(u 2 )

..

.

⎤ ⎥ ⎥ ⎥ ⎦

gvep (u k )

O

⎡

A. Virtual Active Element Pattern Expansion Method

2

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e

jβ([ xdn ]− Q−1 2 )du 1

e

jβ([ xdn ]− Q−1 2 )du k

.. .

··· e .. .

jβ([ xdn ]+ Q−1 2 )du 1

··· e T

jβ([ xdn ]+ Q−1 2 )du k

.. .

(5)

⎤ ⎥ ⎥ ⎦

(6)

cn = cn,− Q−1 , . . . , cn, Q−1

(7)

gn = [gn (u 1 )e jβ xn u 1 , . . . , gn (u k )e jβ xn u k ]T .

(8)

2

2

The solution to (4) can be analytically obtained by cn = (ZnH Zn )−1 ZnH gn

(9)

where Zn = G En , and the superscript “H ” denotes the conjugate transpose of a matrix. B. Fast Calculation of the Array Pattern via VAEPE FFT By integrating (2) into (1), we can obtain f (u) = gvep (u)e− jβ

Q−1 2 du

L−1

vl e jβldu

(10)

l=0

where L = [r N] + Q − 1 denotes the total number of virtual elements, and vl denotes the lth composite excitation of the virtual array. v = [v 0 , v 1 , . . . , v L−1 ]T is given by v = Cw

(11)

where w = [w0 , w1 , · · · , w N−1 ]T , and C is a sparse matrix whose (l, n)th element is given by ⎧ x Q−1 n ⎨ − and |q| ≤ cn,q , for q = l − Q−1 2 [C]l,n = d 2 ⎩ 0, others. (12)

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By sampling (10) with u = mu = mλ/(Md), where m = −M/2, · · · , M/2 − 1 (M ≥ L), we obtain f (mu ) = gm

L−1

vl e j 2πlm/M

(13)

l=0 = g (m )e − j π(Q−1)m/M . By extending the where gm vep u sequence {vl ; l = 0, 1, · · · , L − 1} to be M-length with the zero-padding technique, we can apply the M-point IFFT to speed up the calculation of the above-mentioned summation. In such way, the array pattern including the mutual coupling for an unequally spaced array can be efficiently calculated. Moreover, the problem of finding the excitations from the array pattern can also be solved in an efficient way. From (13), we can obtain its inverse transform

vl =

M/2−1 m=−M/2

f (mu ) − j 2πlm/M e . gm

(14)

Clearly, the above-mentioned summation can be calculated by the M-point FFT. Once the virtual excitations vl (l = 0, 1, · · · , L − 1) are obtained, the real excitations for the original array can be obtained by w = (C H C)−1 C H v, which is the least-square solution to (11). C. Proposed VAEPE-IFT Synthesis Procedure The proposed synthesis procedure is developed by combining the VAEPE method and the idea of the original IFT [12]–[15]. The new procedure can synthesize a focused or shaped power pattern under some given masks for an unequally spaced linear array while maintaining the efficiency of the IFT method. In addition, some constraints on the excitations can be easily added into this procedure. One can perform the amplitude-only optimization, phase-only optimization, and amplitude-phase optimization with a prescribed DRR limitation. The detailed description of this procedure is given as follows. 1) Give a prescribed pattern: defining the main-lobe region, the sidelobe region, and sidelobe upper bound USL (u). If a shaped pattern is considered, one needs to also define the lower bound L ML (u) and upper bound UML (u) in the shaped region. 2) Choose the parameters d (by controlling the parameter r ) and Q for the virtual array, and calculate the matrix C and (C H C)−1 . 3) Initialize the excitation w randomly (or choose the initial w according to the excitation constraints if required). 4) Calculate the virtual excitation v = Cw, and compute the array pattern f (mu ) by applying the M-point IFFT of v in (13). 5) Check if the obtained pattern f (mu ) meet the requirement. If yes, the synthesis procedure stops. If not, make the following pattern adjustment. a) Focused pattern: adjust the sidelobes whose magnitudes are greater than USL (u) to be lower than this bound, and keep their phases unchanged. b) Shaped pattern: in addition to adjusting the sidelobe points, we also adjust the pattern magnitudes

Fig. 2. 32-element unequally spaced microstrip antenna array (a substrate with r = 2.2 and a thickness of 1.57 mm is used). (a) Element positions in [4, Table I]. (b) Antenna structure.

in main-lobe region if they are unsatisfied. That is, adjusting the magnitude of the point that is less than L ML (u) or larger than UML (u) to be within the two bounds. 6) Compute v from the adjusted f (mu ) by using the M-point FFT in (14). Note that only L results from the M outputs are needed. 7) Calculate real array excitations w = (C H C)−1 C H v, and adjust them to meet the possible excitation constraints, such as phase-only control and amplitude-phase optimization with a prescribed DRR constraint. 8) Go back to Step 4) before the allowed number of iterations is reached. III. N UMERICAL R ESULTS A. Phase-Only Optimization of an Unequally Spaced Microstrip Antenna Array As the first example, we intend to reduce the SLL by only optimizing the excitation phases of a 32-element microstrip antenna array whose element positions are given in [4, Table I]. In [4], all the elements are assumed to be isotropic, and the array with 32 elements is obtained by position-phase optimization using the differential evolution algorithm (DEA). The synthesized pattern in [4] has the peak SSL (PSLL) of −23.34 dB. The corresponding element positions are unequally but symmetrically distributed (the element spacing varies from 0.5λ to 1λ), as shown in Fig. 2(a). At first, we adopt these positions to design the geometry of a 32-element microstrip antenna array. The antenna structure is shown in Fig. 2(b). The central frequency of this array is 10 GHz. All the AEPs can be obtained by using the High Frequency Structure Simulator software [27]. The array pattern using the excitation phases given in [4] is shown in Fig. 3. We can see that the PSLL of this pattern reaches −17.85 dB that is much higher than the one with ideally isotropic elements, due to the mutual coupling between the microstrip antenna elements.


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TABLE I A CHIEVABLE PSLLs AND T IME C OSTS OF D IFFERENT M ETHODS FOR S YNTHESIZING S EVERAL D IPOLE A RRAYS W ITH VARYING S IZES ( THE E LEMENT P OSITIONS FOR E ACH A RRAY A RE G ENERATED R ANDOMLY BUT THE I NTERELEMENT S PACINGS R ESTRICTED TO BE W ITHIN [0.4∼1]λ)

Fig. 3. Patterns obtained by using real AEPs with the excitations synthesized by the proposed VAEPE-IFT and the DEA, and the pattern with the excitations given in [4, Table I] for the 32-element microstrip antenna array. The result denoted by VAEPE-IFT with approximated AEP is calculated by (13).

To reduce the SLL, we apply the proposed VAEPE-IFT method to optimize the excitation phases of this antenna array. The initial values are set to be zeros. For the virtual array, we set d = [0.45, 0.4, 0.35, 0.3]λ (which is obtained at r = [1.49, 1.67, 1.91, 2.22]) and Q = [1 : 2 : 19] to study their effect on the accuracy of the proposed VAEPE method. To do so, we define the following normalized error: N−1 gn − G En cn 22 . (15) = n=0 N−1 2 n=0 gn 2 Fig. 4 shows the error versus Q for different d values. As can be seen, for a fixed d, a larger Q gives a smaller error as expected. For a fixed Q, d = 0.35λ gives much better accuracy than that with d = 0.4 or 0.45λ. However, further reducing the value of d has no considerable improvement any more on the accuracy but leads to a larger number of virtual elements with increased computational cost. In this example, we choose d = 0.35λ and Q = 9 for better balance between accuracy and efficiency. That is, the virtual subarray used for approximating one AEP has nine elements with the spacing of d = 0.35λ. Fig. 5 shows the comparison between the approximated AEPs and the original ones for some of elements. As can be seen, the approximated AEPs are nearly the same as the original ones. With the chosen parameters, the proposed VAEPE-IFT takes only 0.75 s to obtain the phase-only synthesis result (on the computer with Intel Core i5-2400 [email protected] GHz).

Fig. 4.

Approximation error versus Q for different d values.

The synthesized real and approximated patterns are shown in Fig. 3 for comparison [the approximated pattern given in (13) is just obtained from the synthesis procedure output, and the real pattern defined in (1) is calculated by summation of all the real AEPs with the synthesized excitations]. We can see that they are almost the same. This further demonstrates the accuracy of the proposed method. Note that the obtained PSLL is now reduced to be −21.01 dB. For further comparison, we also apply the DEA method including the mutual coupling by using the AEPs to reoptimize the excitation phases for this array. The population size is set to be ten times of the optimization variables and the maximum number of generation is 2000. Since synthesis results obtained from different runs of DEA may not be identical due to the randomness, we run the DEA optimization five times on the same computer. The achievable lowest PSLL among five runs is −21.18 dB, and the average is −20.95 dB. The pattern with the lowest PSLL by the DEA is also shown in Fig. 3 for comparison. The obtained SLL performance is very close to that by the proposed method. However, the average CPU time for each run of the DEA optimization takes 174.8 s, more than 200 times of that required by the proposed method. Fig. 6 shows the excitation phase distributions obtained by the VAEPE-IFT and the DEA, as well as the original one given in [4]. B. Amplitude-Phase Synthesis of a Flat-Top Pattern In the second example, we are going to perform amplitudephase synthesis of a flat-top power pattern. This pattern was

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Fig. 7.

Positions in [18, Table I] used for the 12-element dipole array.

Fig. 5. Original AEPs (solid line) and the approximated ones (dashed line) by the VAEPE (with d = 0.35λ and Q = 9) for the [1, 10, 24, 32]th elements in the 32-element microstrip antenna array.

Fig. 8. Pattern shown in [18, Fig. 2(a)] (with isotropic elements), and the patterns obtained by using real AEPs with the excitations in [18] as well as the excitations synthesized by the proposed VAEPE-IFT. The result denoted by VAEPE-IFT with approximated AEP is calculated by (13) for the 12-element dipole array.

Fig. 6. Excitation phases synthesized by the proposed VAEPE-IFT and the DEA for the 32-element microstrip antenna array, and the phases given in [4, Table I].

synthesized in [18] by combining CO and forward–backward matrix pencil method with 12 optimized element positions and excitations. The element positions were given in [18, Table I] and are now shown in Fig. 7 of this paper. The obtained pattern in [18] completely meets its prescribed upper and lower bounds, as is replotted in Fig. 8. However, the hybrid method in [18] used the isotropic elements without mutual coupling effect, and consequently, the synthesized pattern may deteriorate significantly for a practical array. Now, assume that all the elements are dipole antennas working at the central frequency of 1 GHz. Each dipole element is designed to have a length of 144 mm and a radius of 1.5 mm. Due to the mutual coupling between the dipole elements, the array pattern deviates from the original especially at the low SLL region as shown in Fig. 8 (significant deterioration happens at two −40dB nulls). Now, we apply the proposed method to reoptimize the excitation amplitudes and phases for this array. We adopt the same parameters d = 0.35λ and Q = 9. The comparison between the original AEPs and the approximated ones for some of elements is shown in Fig. 9. In this example, a DRR constraint of DRR ≤ 3.2 is used. The synthesized real and approximated patterns [defined in (1) and (13), respectively]

Fig. 9. Original AEPs (solid line) and the approximated ones (dashed line) by the VAEPE (with d = 0.35λ and Q = 9) for the [1, 4, 11, 12]th elements in the 12-element dipole array.

by the proposed method are also shown in Fig. 8. As can be seen, the two patterns agree very well, and the current result including mutual coupling is satisfactory. Fig. 10 shows the synthesized excitation amplitudes and phases by the proposed method and those used in [18]. In this example, the DRR is slightly reduced, from previous 3.6 to current 3.2. C. Amplitude-Phase Synthesis of a Cosecant Shaped Pattern In the third example, we consider to perform the amplitudephase synthesis of a cosecant shaped pattern, which was obtained in [9] by using 13 isotropic elements with optimized


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Fig. 12. Pattern shown in [9, Fig. 4(a)] (with isotropic elements), and the patterns obtained by using real AEPs with the excitations in [9] as well as the excitations synthesized by the proposed VAEPE-IFT. The result denoted by VAEPE-IFT with approximated AEP is calculated by (13) for the 13-element microstrip antenna array.

Fig. 10. Excitations synthesized by the proposed VAEPE-IFT and those given in [18, Table I]. (a) Excitation amplitudes. (b) Excitation phases.

Fig. 13. Original AEPs (solid line) and the approximated ones (dashed line) by the VAEPE (with d = 0.35λ and Q = 9) for the [1, 6, 10, 13]th elements in the 13-element microstrip antenna array.

Q = 9 is applied to reoptimize the excitation amplitudes and phases for this array. The AEPs and the approximated ones of this array are almost the same, as shown in Fig. 13. The obtained approximated and real array’s patterns are plotted in Fig. 12. As can be seen, they agree very well, and meet the prescribed pattern bounds again. Fig. 14 shows the synthesized excitation amplitudes and phases by the proposed method and those given in [9]. In this example, the DRR is significantly reduced from about 9.6 to 4. Fig. 11. 13-element unequally spaced microstrip antenna array (a substrate with r = 4.4 and a thickness of 1.57 mm is used). (a) Element positions given in [9]. (b) Antenna structure.

D. Comparative Study on Accuracy and Efficiency Performance

positions and excitations. Now, assume that this array is implemented with 13 microstrip antenna elements resonating at 2.45 GHz. The element antenna positions and structure are shown in Fig. 11(a) and (b), respectively. Similarly, the practical array’s pattern deviates very much from the one with ideal elements used in [9], as shown in Fig. 12. Now, the proposed method with the same parameters d = 0.35λ and

In the last example, we perform a comparative study on both accuracy and efficiency performance of the proposed VAEPE-IFT and other advanced methods, including genetic algorithm (GA) [1], dynamic differential evolution (DDE) [6], CO [8], and the iterative NUFFT [23]. Unequally spaced dipole arrays with different apertures are considered. The number of elements is 8, 16, 32, 64, and 128, respectively, and each dipole element is 48 mm in length and 0.5 mm in

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Fig. 14. Excitations synthesized by the proposed VAEPE-IFT and those shown in [9]. (a) Excitation amplitudes. (b) Excitation phases.

radius, working at the central frequency of 3 GHz. The element positions for each array size are generated randomly, but the interelement spacings are controlled to be within [0.4∼1]λ. Note that the element positions for the same array size are identical for different methods. Assume that the synthesis problem is to reduce the PSLL of a focused beam pattern for each array by simultaneously optimizing the excitation amplitudes and phases. The pattern sampling interval (u) and the required first null beamwidth for each array are listed in Table I. For the GA, the population size is set to be eight times of the optimization variables and the maximum number of generations is 500. The DDE used in [6] is different from the conventional DEA. In the DDE, once the generated trial individual is better than the target individual, it will immediately replace the target individual in the following evolution instead of in the next generation. The dynamic updating of population leads to a larger virtual population and quicker response to change of population status than the conventional DEA. Nevertheless, both the DDE and GA need to calculate the direct summation of the array pattern given by (1) for each individual in every generation. For the proposed VAEPE-IFT, we still adopt the parameters d = 0.35λ and Q = 9 to approximate each AEP. Note that the iterative NUFFT cannot synthesize the practical array pattern with

mutual coupling, and the excitations are obtained by only synthesizing the corresponding array factor for each case. Table I shows the achievable PSLLs and the time costs of these methods for all test cases (note that for either the GA or the DDE, the PSLL result is chosen as the lowest one among five runs, and the time cost is calculated as the average over different runs). As can be seen, the CO method achieves the lowest PSLL for each array case, since it can provide the optimal solution to this unconstrained optimization problem. For the proposed method, the obtained PSLLs are very close to the lowest values given by the CO for all test cases (the differences are less than 0.5 dB in most cases). The stochastic GA and DDE have similar PSLLs as the proposed method in the cases with 8 and 16 elements. However, when a large array with increased number of elements is considered, both the GA and DDE show worse accuracy performance than the proposed method probably, because they have not found the best solution yet within the specified maximum number of iterations (better results could be obtained at the cost of even more time cost). The iterative NUFFT gives the highest PSLL for all test cases, due to its failure to including the mutual coupling in the synthesis procedure. Besides, we can check the time costs of these synthesis methods. Clearly, the GA and DDE are very time-consuming due to a large number of recalculations of the direction summation in (1), while the iterative NUFFT and the proposed VAEPE-FFT are the most efficient. Compared with the CO method, the proposed method is still much more efficient when a large array is considered. For example, for the case of 128 elements, the CO method takes 23.83 s while the proposed method requires only 1.92 s. Therefore, the proposed method would be more preferable than the CO method for large array cases. In addition, it should be mentioned that the proposed method can deal with more complicated synthesis problems, such as phase-only optimization, DRR constraint, and main-lobe shape control, while the CO method cannot be directly applicable in this situation due to the nonconvex property in these problems. IV. C ONCLUSION We have presented a VAEPE method with which the pattern for an unequally spaced linear array including mutual coupling can be efficiently calculated by using the FFT. By combining this idea with the IFT procedure, we develop a novel efficient synthesis method, which can successively apply the FFT and IFFT to perform pattern synthesis for unequally spaced arrays including mutual coupling. Some useful constraints can be easily incorporated into this method to control the SSL, pattern shape, and the excitation distribution. A set of synthesis examples have been studied, including phaseonly optimization of a microstrip antenna array, amplitudephase optimization with the DRR constraint for flat-top and cosecant shaped patterns, and low sidelobe pattern synthesis of dipole arrays with different sizes. The synthesis results have proved the efficiency, accuracy, and robustness of the proposed method. The comparisons with some other synthesis techniques have also been included in these examples. Finally, we note that the proposed VAEPE-IFT method can be considered a significant generalization of the iterative FFT


and NUFFT synthesis methods, and would be very useful for more general pattern synthesis cases in practice. R EFERENCES [1] K.-K. Yan and Y. Lu, “Sidelobe reduction in array-pattern synthesis using genetic algorithm,” IEEE Trans. Antennas Propag., vol. 45, no. 7, pp. 1117–1122, Jul. 1997. [2] M. M. Khodier and C. G. Christodoulou, “Linear array geometry synthesis with minimum sidelobe level and null control using particle swarm optimization,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2674–2679, Aug. 2005. [3] A. Trucco, “Thinning and weighting of large planar arrays by simulated annealing,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 46, no. 2, pp. 347–355, Mar. 1999. [4] D. G. Kurup, M. Himdi, and A. Rydberg, “Synthesis of uniform amplitude unequally spaced antenna arrays using the differential evolution algorithm,” IEEE Trans. Antennas Propag. Mag., vol. 51, no. 9, pp. 2201–2217, Sep. 2003. [5] S. Yang and Z. Nie, “Mutual coupling compensation in time modulated linear antenna arrays,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 4182–4185, Dec. 2005. [6] C. Lin, A. Qing, and Q. Feng, “Synthesis of unequally spaced antenna arrays by using differential evolution,” IEEE Trans. Antennas Propag., vol. 58, no. 8, pp. 2553–2561, Aug. 2010. [7] S. Karimkashi and A. A. Kishk, “Invasive weed optimization and its features in electromagnetics,” IEEE Trans. Antennas Propag., vol. 58, no. 4, pp. 1269–1278, Apr. 2010. [8] H. Lebret and S. Boyd, “Antenna array pattern synthesis via convex optimization,” IEEE Trans. Signal Process., vol. 45, no. 3, pp. 526–532, Mar. 1997. [9] B. Fuchs, “Synthesis of sparse arrays with focused or shaped beampattern via sequential convex optimizations,” IEEE Trans. Antennas Propag., vol. 60, no. 7, pp. 3499–3503, Jul. 2012. [10] P. Rocca, N. Anselmi, and A. Massa, “Optimal synthesis of robust beamformer weights exploiting interval analysis and convex optimization,” IEEE Trans. Antennas Propag., vol. 62, no. 7, pp. 3603–3612, Jul. 2014. [11] J. L. A. Quijano and G. Vecchi, “Alternating adaptive projections in antenna synthesis,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 727–737, Mar. 2010. [12] W. P. M. N. Keizer, “Fast low-sidelobe synthesis for large planar array antennas utilizing successive fast Fourier transforms of the array factor,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 715–722, Mar. 2007. [13] W. P. M. N. Keizer, “Low-sidelobe pattern synthesis using iterative Fourier techniques coded in MATLAB,” IEEE Trans. Antennas Propag. Mag., vol. 51, no. 2, pp. 137–150, Apr. 2009. [14] W. P. M. N. Keizer, “Low sidelobe phased array pattern synthesis with compensation for errors due to quantized tapering,” IEEE Trans. Antennas Propag., vol. 59, no. 12, pp. 4520–4524, Dec. 2011. [15] K. Yang, Z. Zhao, and Q. H. Liu, “An iterative FFT based flat-top footprint pattern synthesis method with planar array,” J. Electromagn. Waves Appl., vol. 26, pp. 1956–1966, Oct. 2012. [16] P. Y. Zhou and M. A. Ingram, “Pattern synthesis for arbitrary arrays using an adaptive array method,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 862–869, May 1999. [17] Y. Liu, Z. Nie, and Q. H. Liu, “Reducing the number of elements in a linear antenna array by the matrix pencil method,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2955–2962, Sep. 2008. [18] Y. Liu, Z.-P. Nie, and Q. H. Liu, “A new method for the synthesis of non-uniform linear arrays with shaped power patterns (invited paper),” Prog. Electromagn. Res., vol. 107, pp. 349–363, 2010. [19] A. F. Morabito, A. Massa, P. Rocca, and T. Isernia, “An effective approach to the synthesis of phase-only reconfigurable linear arrays,” IEEE Trans. Antennas Propag., vol. 60, no. 8, pp. 3622–3631, Aug. 2012. [20] W. P. M. N. Keizer, “Linear array thinning using iterative FFT techniques,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2757–2760, Aug. 2008. [21] X.-K. Wang, Y.-C. Jiao, Y, Liu, and Y.-Y. Tan, “Synthesis of large planar thinned arrays using IWO-IFT algorithm,” Prog. Electormagn. Res., vol. 136, pp. 29–42, 2013. [22] X.-K. Wang, Y.-C. Jiao, and Y.-Y. Tan, “Synthesis of large thinned planar arrays using a modified iterative Fourier technique,” IEEE Trans. Antennas Propag., vol. 62, no. 4, pp. 1564–1571, Apr. 2014.

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[23] K. Yang, Z. Zhao, and Q. H. Liu, “Fast pencil beam pattern synthesis of large unequally spaced antenna arrays,” IEEE Trans. Antennas Propag., vol. 61, no. 2, pp. 627–634, Feb. 2013. [24] D. F. Kelley and W. L. Stutzman, “Array antenna pattern modeling methods that include mutual coupling effects,” IEEE Trans. Antennas Propag., vol. 41, no. 12, pp. 1625–1632, Dec. 1993. [25] P. You, Y. Liu, X. Huang, L. Zhang, and Q. H. Liu, “Efficient phase-only linear array synthesis including coupling effect by GA-FFT based on least-square active element pattern expansion method,” Electron. Lett., vol. 51, no. 10, pp. 791–792, 2015. [26] X. Huang, Y. Liu, P. You, M. Zhang, and Q. H. Liu, “Fast linear array synthesis including coupling effects utilizing iterative FFT via least-squares active element pattern expansion,” IEEE Antennas Wireless Propag. Lett., vol. 16, pp. 804–807, 2016. [27] Ansoft High Frequency Structural Simulator (HFSS), Ver. 13, Ansoft Corp., Pittsburgh, PA, USA, 2010. Yanhui Liu (M’15) received the B.S. and Ph.D. degrees in electrical engineering from the University of Electronic Science and Technology of China (UESTC), Sichuan, China, in 2004 and 2009, respectively. From 2007 to 2009, he was a Visiting Scholar with the Department of Electrical Engineering, Duke University, Durham, NC, USA. Since 2011, he has been with Xiamen University, Xiamen, China, where he is currently a Full Professor with the Department of Electronic Science. He has authored or co-authored over 90 peer-reviewed journal and conference papers. He holds several granted Chinese patents. His current research interests include antenna array design, array signal processing, and microwave imaging methods. Dr. Liu received the UESTC Outstanding Graduate Award in 2004 and the Excellent Doctoral Dissertation Award of Sichuan Province of China in 2012. He is serving as a Reviewer for several international journals, including IEEE T RANSACTIONS ON A NTENNAS AND P ROPAGATION, the IEEE T RANSAC TIONS ON G EOSCIENCES AND R EMOTE S ENSING , the IEEE A NTENNAS AND W IRELESS P ROPAGATION L ETTERS , the IEEE M ICROWAVE AND W IRELESS C OMPONENTS L ETTERS , IET Microwave, Antennas and Propagation, and Digital Signal Processing. Xin Huang received the B.S. degree in electronic information science and technology from Xiamen University, Xiamen, China, in 2014, where he is currently pursuing the M.S. degree with the Institute of Electromagnetics and Acoustics.

Kai Da Xu (S’13–M’15) received the B.S. and Ph.D. degrees in electromagnetic field and microwave technology from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 2009 and 2015, respectively. From 2012 to 2014, he was a Visiting Researcher with the Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA, under the financial support from the China Scholarship Council. He is currently an Assistant Professor with the Institute of Electromagnetics and Acoustics, and the Department of Electronic Science, Xiamen University, Xiamen, China. He has authored or co-authored over 60 papers in peer-reviewed journals and conference proceedings. His current research interests include RF/microwave and mm-wave circuits, antenna arrays, and nanoscale memristors. Dr. Xu received the UESTC Outstanding Graduate Awards in 2009 and 2015, respectively. He was a recipient of the National Graduate Student Scholarship in 2012, 2013, and 2014 from the Ministry of Education, China. He is serving as a Reviewer for several IEEE and IET journals, including the IEEE T RANSACTIONS ON M ICROWAVE T HEORY AND T ECHNIQUES , the IEEE T RANSACTIONS ON E LECTRON D EVICES , the IEEE T RANSACTIONS ON C OMPUTER -A IDED D ESIGN OF I NTEGRATED C IRCUITS AND S YSTEMS , the IEEE T RANSACTIONS ON A PPLIED S UPERCONDUCTIVITY, the IEEE M ICROWAVE AND W IRELESS C OMPONENTS L ETTERS , IET Microwaves Antennas & Propagation, and Electronics Letters.

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Zhengyong Song received the Ph.D. degree in physics from Fudan University, Shanghai, China, in 2013. From 2013 to 2014, he was a Post-Doctoral Research Fellow with the Department of Physics and Applied Physics, Nanyang Technological University, Singapore. He is currently an Assistant Professor with the Department of Electronic Science, Xiamen University, Xiamen, China. His current research interests include electromagnetic field and wave, photonics, physics, metamaterials, and plasmonics. Shiwen Yang (M’00–SM’04) was born in Sichuan, China, in 1967. He received the B.S. degree in electronic science and technology from East China Normal University, Shanghai, China, and the M.S. degree in electromagnetics and microwave technology and the Ph.D. degree in physical electronics from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1989, 1992, and 1998, respectively. From 1994 to 1998, he was a Lecturer with the Institute of High Energy Electronics, UESTC. From 1998 to 2001, he was a Research Fellow with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. From 2002 to 2005, he was a Research Scientist with Temasek Laboratories, National University of Singapore, Singapore. Since 2005, he has been a Full Professor with the School of Electronic Engineering, UESTC. He has been a Chang-Jiang Professor nominated by the Ministry of Education of China since 2015. He has authored or co-authored over 200 technical papers. His current research interests include antennas, antennas arrays, optimization techniques, and computational electromagnetics. Dr. Yang was a recipient of the Foundation for China Distinguished Young Investigators presented by the National Science Foundation of China in 2011. He is the Chair of the IEEE /AP-S/EMC-S Joint Chengdu Chapter, and served as an Editorial Board Member for IJAP and the Chinese Journal of Electronics.

Qing Huo Liu (S’88–M’89–SM’94–F’05) received the B.S. and M.S. degrees in physics from Xiamen University, Xiamen, China, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana–Champaign, Champaign, IL, USA. He was with the Electromagnetics Laboratory, University of Illinois at Urbana-Champaign, as a Research Assistant from 1986 to 1988, and as a PostDoctoral Research Associate from 1989 to 1990. He was a Research Scientist and a Program Leader with Schlumberger-Doll Research, Ridgefield, CT, USA, from 1990 to 1995. From 1996 to 1999, he was an Associate Professor with New Mexico State University, Las Cruces, NM, USA. Since 1999, he has been with Duke University, Durham, NC, USA, where he is currently a Professor of Electrical and Computer Engineering. He has authored over 400 papers in refereed journals and 500 papers in conference proceedings. His current research interests include computational electromagnetics and acoustics, inverse problems, and their application in nanophotonics, geophysics, biomedical imaging, and electronic packaging. Dr. Liu is a fellow of the Acoustical Society of America, the Electromagnetics Academy, and the Optical Society of America. He received the 1996 Presidential Early Career Award for Scientists and Engineers from the White House, the 1996 Early Career Research Award from the Environmental Protection Agency, and the 1997 CAREER Award from the National Science Foundation. He served as an IEEE Antennas and Propagation Society Distinguished Lecturer from 2014 to 2016. He received the ACES Technical Achievement Award in 2017. He currently serves as the founding Editorin-Chief of the new IEEE J OURNAL ON M ULTISCALE AND M ULTIPHYSICS C OMPUTATIONAL T ECHNIQUES , the Deputy Editor-in-Chief of Progress in Electromagnetics Research, an Associate Editor of the IEEE T RANSACTIONS ON G EOSCIENCE AND R EMOTE S ENSING , and an Editor of the Journal of Computational Acoustics.