BA-Based Low-PSLL Beampattern Synthesis in the Presence of Array Errors

In practical scenarios, there are always array errors, which would increase the sidelobe level (SLL) of the array and distort the performance of the electronic devices consequently. Nevertheless, most of the reported works do not take these unavoidable errors into consideration when implementing array beampattern synthesis. To remedy this problem, a low sidelobe beampattern synthesis approach is proposed in the presence of array errors based on the bat algorithm (BA). In particular, the covariance matrix of the sidelobe region in the presence of array errors is incorporated into the optimization problem. Generally, there are many factors that would contribute to array errors in practical scenarios. Therefore, to reduce the influence of the uncertainty characteristic of the errors on the low-SLL beampattern synthesis, the statistical mean method is utilized to obtain a robust calculation of the covariance matrix in the presence of array errors. Theoretical analysis using signal processing technique and electromagnetic simulation using Ansoft HFSS workbench are combined to testify the effectiveness of the proposed approach. Numerical results show that peak SLLs (PSLLs) of around -22dB and -20dB can be respectively achieved in the above two simulation situations with the obtained weight vectors.


I. INTRODUCTION
Low-sidelobe beampattern synthesis is one of the most important issues in array antenna synthesis since it can reduce the energy leakage in the region out-of-interest, thus improving the array energy efficiency and reducing the influence of the interferences and/or clutter on the array performance in sidelobe-region simultaneously [1]- [2]. Typical amplitude tapering methods were firstly proposed to reduce the sidelobe level (SLL) of the array, such as Chebyshev distribution [3], Taylor distribution [4], Gaussian distribution [5], and hybrid distribution [6]. In general, not only the array amplitudes, but also the phases and positions of the elements in an array can be properly adjusted to achieve low SLL [7]. A series of global optimization methods, such as particle swarm optimization (PSO) algorithm [8], genetic algorithm (GA) [9], hybrid algorithm [10], and et. al., have been proposed to obtain complex exciting weight values of the elements for low SLL array design. A pencil beam in the broadside direction is produced with low SLL by iteratively modifying the positions of the elements [11].
In practical scenarios, array errors always exist unavoidably due to many factors, such as element position errors, mutual coupling effects among elements, amplitude and phase errors of exciting currents, which would increase the SLL and distort the performance of the electronic devices consequently. At present, some types of array errors have been taken into consideration when implementing relative research. In [12], an interval analysis method accompanied by convex programming was proposed to design robust beamformer weights for array antennas with amplitude errors. In [13], a novel matrix-based interval arithmetic method was proposed for linear antenna array pattern tolerance analysis with excitation amplitude errors. The iterative Fourier transform technique was utilized to mitigate the degradation of sidelobe performance of array antenna caused by quantization of its taper across the array in [14]. In [15], the differential evolution (DE) algorithm was adopted to synthesize array antenna for microwave power transmission with excitation errors taken into consideration. In [16], a conical method was proposed for the representation of position errors by considering both the relevance and the randomness of the adjacent planar array elements. In [17], Schmid presented the worst-case boundaries and a statistical analysis of the beampattern deviation for linear, angleindependent calibration error and mutual coupling models. However, it is found that most of the state-of-the-art works only consider one or two factors that contribute(s) to array errors after a thorough literature review.
Inspired by the echolocation behavior of bats, Dr. Yang proposed a new metaheuristic algorithm [18], i.e., the bat algorithm (BA), in 2010. It has been demonstrated that this algorithm is a powerful optimization algorithm, which is superior to the famous standard PSO and GA in terms of accuracy and efficiency [18]- [19]. Since then, the BA has been applied in some engineering applications including the array beamforming [19]- [22] and antenna design [23]. Nevertheless, the applications of this algorithm in antenna field is quite limited. Under this circumstance, we aim to propose an efficient low-PSLL beampattern synthesis approach in the presence of array errors based on this algorithm to utilize its excellent performance. A more general array error model established in [24] is adopted during the low-PSLL optimization procedure, where the above mentioned errors are boiled down to the amplitude and phase response errors. Meanwhile, the covariance matrix of the sidelobe region is incorporated into the optimization problem. To reduce the influence of the uncertainty characteristic of the errors, the statistical mean method is utilized to obtain a more robust calculation of the covariance matrix in the presence of array errors. Both of the theoretical analysis using signal processing technique and electromagnetic simulation using Ansoft HFSS workbench are adopted to testify the effectiveness of the proposed approach.
The remainder of this paper is organized as follows. Section II firstly formulates the signal model of the problem. On this basis, Section III derives the proposed low-PSLL beampatten synthesis approach in the presence of array errors based on BA. Numerical results are presented in Section IV to demonstrate the effectiveness of the proposed approach, and the conclusion is drawn in Section V.

II. PROBLEM FORMULATION
As depicted in Fig. 1., consider a uniform linear array (ULA) consisting of M elements, the inter-element spacing of which is d. Taking the first element of the ULA as the reference point, the steering vector of the array can be written as [25]   is the position vector of the mth element. Then the beampattern of the array can be expressed as In the following, a more general array error model is established. It is known that the amplitude and phase responses are two key parameters that play quite important roles in array beamforming [26]. When these two factors are determined, the beampattern of the array is determined. In practical scenario, there exist many kinds of array errors and these errors always occur simultaneously. Thus, it is unreal to separate a specific kind of array error and explore its effect on the array beampattern. All these array errors have influences on the amplitude and phase responses, which causes the array beampattern error. Therefore, it is reasonable that the influence of these errors on the array beampattern are boiled down to the amplitude and phase response errors [24]. In specific, the array error vector can be expressed as These errors would drastically increase the SLL of the array and distort the performance of the array correspondingly. As an illustration, Fig. 2 presents four normalized array patterns, i.e., pattern 1: the ideal one generated with uniform weight vector; pattern 2: the pattern generated with uniform weight vector in the presence of array errors; pattern 3: the ideal one generated with -20dB Chebyshev window; and pattern 4: the pattern generated with -20dB Chebyshev window in the presence of array errors. From Fig. 2, it can be seen that the PSLLs of the array are increased by 4.2dB (from -13.2dB to -9.0dB) and 6.5dB (from -20.0dB to -13.5dB ) when utilizing the uniform weight vector and -20 Chebyshev tapering vector, respectively, in the presence of 10% errors. Note that the specific increased value is not a fixed number since the errors cannot be determined. Herein, Fig. 2 is utilized to demonstrate that large SLL increment would appear in case of array errors.

III. THE PROPOSED LOW-SIDELOBE BEAMPATTERN SYNTHESIS APPROACH IN THE PRESENCE OF ARRAY ERRORS
The aim of this section is to find the optimal complexvalued weight vector w which can minimize the maximum power from the sidelobe region subject to a distortionless constraint for a specified steering vector of interest in the presence of array errors. In particular, the covariance matrix of the sidelobe region is incorporated into the optimization problem. As has been pointed out in Section II, the induced amplitude and phase response errors of the array comply with random distribution. Therefore, to reduce the influence of the uncertainty characteristic of the errors on the low-SLL beampattern synthesis, the statistical mean method is utilized to obtain a more robust calculation of the covariance matrix of the sidelobe region in the presence of array errors. Executing L Monte Carlo trials, the mean value of the covariance matrix in the presence of array errors can be expressed as subject to and     where   g P  denotes the generated pattern in the presence of array errors. Specifically, Eq. (7a) is utilized to guarantee that the mainbeam of the generated beampattern is steered to the desired direction 0  , and Eq. (7b) provides the constraints imposed on the mainbeam width which is 2 1 - In this way, the fitness function can be represented as which is decreased as the weight vector w , i.e., the position of bat in BA, updated iteratively. It is noted that the PSLL and integrated SLL are the two main metrics for evaluating the array beampattern. Actually, if the PSLL of the beampattern is well controlled, the integrated SLL can also be controlled. Herein, we only consider the PSLL and do not incorporate the integrated SLL.
In the sequel, we define the rule how the position i w and and where t g w represents the global best position of all the bats at time step t iteration, min F , max F and  are system parameters that controls the algorithm performance. It is seen from Eq. (9) that, the velocity is utilized to update the position of the bat. As the iteration proceeds, In nature, bats utilize the ultrasonic wave to find preys. When a bat has found its prey, the loudness of the ultrasonic wave decreases and the rate of pulse emission increases. Therefore, two more judgement parameters are introduced in BA, i.e., the rate i r and the loudness i A . It should be highlight that these two parameters will be updated only if the new solutions are improved, which means that these bats are moving towards the optimal solution. In specific, their loudness and emission rates will be updated in the following way    is predesigned. Theoretically, it can be set as any value, which is not less than the counterpart of the array generated with uniform weight vector in ideal scenario. After calculating the positions of all bats, find the minimum one among all of the maximum values. Then, set the position of the bat, which can generate the minimum value, as the best solution g w , and record the minimum value as ( ) g f w in this current iteration. From this algorithm, it is seen that an optimal weight vector opt w is obtained after the optimization.
Algorithm1: Pseudo code of the BA-based low-SLL optimization algorithm

IV. NUMERICAL RESULTS
In this section, numerical results including the results generated with signal processing method and those obtained using Ansoft HFSS workbench are provided to investigate the effectiveness of the proposed approach in the presence of array errors. Table I Fig. 3(a), from which it can be seen that the fitness values of the proposed optimization approach drops faster with the increment of  . Finally, we set 0.01   for a moderate convergence of the algorithm. In Fig. 3(b), we explore the convergence of the proposed algorithm as well as the PSO and DE algorithms. It is seen that the BA-based algorithm has higher efficiency compared with the state-of-the-art heuristic algorithms. The reason is that BA optimization adjust its parameters within the algorithms itself, which accelerates the convergence of the algorithm. As described in Section III, the statistical mean method is adopted when calculating the covariance matrix of the sidelobe region. In this Section, two situations, i.e., the beam steered to 0° and 30°, are considered. The accordingly amplitude and phase parts of the generated optimal weight vectors in these two situations are provided in Tables II-III. To demonstrate the robustness of the obtained opt w in low-SLL beampattern synthesis, we adopt four sets of randomly generated errors cases, i.e., Case 1, Case 2, Case 3 and Case 4, respectively under the array error of 10% and 15%. The amplitude and phase parts of these four sets of errors are provided in Fig. 4 In practice, the array error can be smaller than that considered in this example. It is possible to adjust the array error based on the experimental results. The proposed algorithm is feasible for a relatively wide range of the array error. Fig . 5 provides the corresponding normalized array patterns in these four cases after applying the obtained weight vectors opt w when the beam is steered to 0° and 30° at the array error of 10% and 15%, respectively. As can be seen from Figs. 5(a) and (b), the generated PSLLs are all around -23.0dB and -21.5dB, respectively, for 0 =0   and 0 =30   when exists 10% array error. As the errors in these four cases are different from each other, it can be concluded that the obtained weight vector is robust against the array error, which in turn demonstrates the effectiveness of the proposed approach. Besides, it is seen that the available PSLL is increased with respect to the increment of steered angle. Table III presents the specific PSLLs of the generated patterns in these four cases after applying the proposed approach. As comparison, the beampatterns are plotted for the 15% array error in Figs. 5(c) and (d). It is seen that the PSLLs of the beampattern are slightly higher. In the simulation example, it becomes -19.0dB for steered angle 0° while -17.4dB for steered angle 30°. Nevertheless, the proposed algorithm is still effective for this increased array error situation.   Mutual coupling effect among elements is unavoidable in practical scenario, which is a common factor that can cause large array errors. The existence of mutual coupling effect would increase the SLL of the array and distort the performance of the array accordingly. To further check the effectiveness of the proposed approach, electromagnetic simulation results using Ansoft HFSS workbench are provided. Specifically, a 16-element microstrip array antenna is designed, the mutual coupling of which is around -13.5dB among two adjacent elements. Figs. 6 (b)-(c) present the patterns after applying the obtained weights. The original patterns and the patterns applied with -20dB Chebywin weights are also provided as comparison. In particular, the original patterns are provided to show the original PSLLs of the array before implementing the low-PSLL beampattern synthesis. The patterns applied with -20dB Chebywin weights are provided to show the obtained PSLLs of amplitude tapering method when the mutual coupling exists. It is known that the beampattern generated with Chebyshev tapering weight vector has equal sidelobe levels without taking the element pattern into consideration. The resultant beampattern of the array is the product of the array pattern and the element pattern. The element pattern exhibit the property that its sidelobes level decrease as the sidelobe grow farther away from the mainlobe. Namely, the resultant beampattern can be viewed as the weighted array pattern, where the element pattern acts as the weight. Therefore, the black curve sidelobes level decrease as the sidelobe grow farther away from the mainlobe. It is seen from Figs. 6(b) and (c) that the PSLLs of -16.1dB and -15.0dB are achieved, respectively, for 0 =0   and 0 =30   after applying the -20dB Chebywin weights, while the PSLLs of -20.8dB and -18.5dB are obtained with the proposed approach. Besides, we also testify other heuristic optimization algorithms in this experiment, including PSO and DE algorithms. It is seen that the proposed BA-based algorithm performs slightly better than the PSO and DE algorithms. Namely, the proposed approach is still effective when strong mutual coupling effects exist. The specific PSLLs of the patterns in Figs. 6(b)-(c) are provided in Table V. From Fig. 6, it is seen that with the increment of 0  , the PSLLs are increased in both situations. It is also observed that we can draw the same conclusion with beapatterns pointing towards 0° and 30°. Besides, we find that the proposed approach is feasible for the beampatterns with beam steering direction within -30°3 0°. At present, the RF circuit design for the array still needs more efforts at present. Therefore, we plan to build the linear array as well as the corresponding RF circuit to further test the performance of the array in the near future.

V. CONCLUSION
In this paper, an effective beampattern synthesis approach is proposed to achieve low PSLL in the presence of errors based on the advanced BA. In specific, the covariance matrix in the presence of array errors is incorporated into the optimization problem. Meanwhile, the statistical mean method is utilized to obtain a robust calculation of the covariance matrix. Numerical results show that the proposed approach is superior over PSO and DE algorithms in terms of convergence rate and the performance of the obtained array pattern.