WCA-Based Low-PSLL and Wide-Nulling Beampattern Synthesis for Radar Applications

Abstract

There are many unavoidable array errors in practical scenarios, which would drastically increase the sidelobe level (SLL) and distort the performance of radar systems accordingly. In this paper, an effective beampattern synthesis approach is proposed to realize a low peak sidelobe level (PSLL) and wide-nulling in the presence of array errors, thus improving the consequent performance of the radar. In particular, the covariance matrix of the sidelobe region (CMSR) is incorporated into the optimization. Considering the randomness of array errors, the statistical mean method is adopted to obtain a more accurate calculation of the CMSR in the presence of array errors based on a Monte Carlo trial. To efficiently and effectively solve the optimization problem, an advanced metaheuristic algorithm, i.e., the water cycle algorithm (WCA), is adopted when seeking the corresponding optimal weight vectors. Numerical results are provided and discussed to demonstrate the effectiveness of the proposed approach including the results based on a wideband linearly spaced magneto-electric (ME) dipole array.

1. Introduction

With the increasing complexity of the electromagnetic environment, radar systems are facing many great challenges, especially the demand of anti-jamming capability when monitoring or detecting targets [1,2,3,4]. At present, array antennas are extensively utilized in radar systems since they have the ability to provide high gain beam, multi-beam and beam scanning [5,6,7,8]. In practical scenarios, it is likely for a jammer to generate interference through sidelobe regions of an array antenna with high possibility, so it is important for the array antenna to maintain the sidelobes at a specified low level for radar applications [9]. It is also noticed that the non-Gaussian noise has an important influence on the target detection performance of a radar system [10,11]. The peak sidelobe level (PSLL) is one of the critical systematic parameters of an array antenna in the task of reducing background noise and interference for radar systems [12]. A low PSLL can prevent interference energy from entering the radar receiver through the sidelobe, thus effectively improving the anti-jamming performance of radar systems. Moreover, the generation of null(s) in the direction of interference sources can further enhance the anti-jamming capability of radar systems [13]. However, the deviation of the antenna position or the rapid movement of the interference source would cause the null(s) of the antenna beampattern to become unaligned with the interference source. Under this circumstance, the wide-nulling technique has emerged as a way to increase the output signal-to-interference-plus-noise ratio (SINR) of radar systems by widening the null region of the beampattern around the interference direction [14]. Therefore, either the low PSLL or wide-nulling beampattern synthesis of the array antenna can play an important role in improving the performance of radar systems.

Firstly, many typical and traditional amplitude tapering methods have been proposed, such as Chebyshev distribution [15,16], Taylor distribution [17], Gaussian distribution [18], Fourier transform [19] and hybrid distributions [20,21,22], to achieve a low sidelobe level (SLL). Secondly, with the development of metaheuristic optimization algorithms, more and more scholars have tried to utilize these algorithms to achieve a low SLL and wide null(s) by adjusting the amplitudes, phases and positions of the array elements. The genetic algorithm (GA) was introduced to obtain a low SLL beampattern synthesis and array thinning in [23,24,25,26]. Meanwhile, it was also applied to achieve multi-nulls by only adjusting the amplitudes and phases of a relatively small number of elements at the edge of the array [27]. Compared with the GA, the particle swarm optimization (PSO) algorithm is much easier to understand and implement, and requires a minor mathematical preprocessing [28]. The PSO algorithm was used to achieve a low SLL and null control by optimizing the positions of the array elements in [29,30,31]. The hybridization of the PSO and ant lion optimization was employed to achieve sidelobe suppression by optimizing the amplitudes and phases of excitations, and element spacing in [32]. Moreover, the differential evolution (DE) algorithm was utilized to realize a low SLL by properly optimizing the amplitudes, phases and positions of the array elements in [33,34,35]. A quantized DE algorithm and bee algorithm were exploited for interference suppression, which can form single, multiple and wide nulls in prescribed directions by controlling the rotation phases and amplitudes of each element, in [36,37]. An analytical approach was proposed for the design of a uniformly spaced array to achieve an as low as possible SLL by only adjusting the mainlobe region in [38]. A flexible covariance matrix tapering (CMT)-based null-widening method was proposed in [39], which could produce wide nulls with a different desired width and asymmetry. A binary bat algorithm (BBA)-based amplitude control beamformer was proposed to create a pattern nulling for the uniform linear array to prevent interference in [40], which has superiority in the convergence rate compared to binary PSO. Moreover, a BBA-based phase control approach for half-wave dipole uniform linear array (DULA) pattern nulling to suppress unknown interference was presented in [41]. In practical applications, there are many inevitable array errors, such as the mutual coupling effects among elements, amplitude errors, phase errors and element position errors, which exist either inherently or are induced by the limited processing techniques and installing accuracy. These errors would drastically influence the beampattern shape, beam steering and SLL of the array antenna, and further degrade the radar performance of sensitivity and ambiguity suppression [42]. Nevertheless, the array errors are not taken into consideration in most of the above reported investigations.

To improve the reliability of radar systems, it is of great importance to take these errors into consideration when implementing related researches. In [43], an analytical approach was presented to investigate the influence of phase shifter errors on the beam collection efficiency of a planar array. Amplitude errors are taken into consideration in [44] and [45]. In [46], an approach based on statistical analysis was proposed for the analysis of the influence of array antenna position errors on the beam collection efficiency. In [47], the improved DE was adopted to achieve array synthesis for optimal microwave power transmission in the presence of excitation errors. A conical method for the representation of position errors is proposed in [48], which considers both the relevance and the randomness of adjacent planar array elements. In [49], an approach was proposed to design sum and difference patterns with common nulls and low SLLs simultaneously in the presence of array errors for radar applications. After a thorough literature review, it was found that most of the current investigations only consider one or two kinds of factors when implementing relative researches, and quite limited investigations have been conducted on either the low PSLL or wide-nulling beampattern synthesis for radar applications in the presence of unavoidable array errors in the state-of-the-art methods. Inspired by the observation of the water cycle and how rivers and streams flow downhill in nature, Eskandar et al. [50] proposed a new metaheuristic algorithm, i.e., WCA, in 2012. This algorithm has the advantages of a fast optimization rate and high iterative accuracy. This simple and robust algorithm has good potential for finding all global optimal solutions of multimodal functions and benchmark functions [51,52]. At present, it has only been introduced in array beampattern synthesis [53] and antenna design [54]. Aiming at improving the performance of radar systems in monitoring or detecting targets, an effective low PSLL and wide-nulling beampattern synthesis approach is proposed based on the advanced WCA in the presence of array errors. The main contributions of this work are summarized as follows.

(1)

A more general array error model is established by summarizing all of the above mentioned factors on the array beamforming as the amplitude and phase response errors satisfying the Gaussian random distribution and zero mean uniformly random distribution, respectively.

(2)

The WCA-based low PSLL and wide-nulling beampattern synthesis approach is proposed in the presence of array errors with proper constraints imposed on the mainbeam width and the PSLL. In particular, the covariance matrix of the sidelobe region (CMSR) is incorporated into the optimization problem based on the above error model.

(3)

Theoretical analysis based on the signal processing technique and electromagnetic simulation based on the Ansoft HFSS workbench are combined to verify the effectiveness of the proposed low PSLL and wide-nulling beampattern synthesis in the presence of errors.

The rest of the paper is arranged as follows. In Section 2, the basic signal models are established, and the consequent problems are introduced. The proposed WCA-based beampattern synthesis approach is presented in Section 3 to separately achieve low PSLL and wide null(s) in the presence of array errors. Theoretical results using the MATLAB workbench and electromagnetic simulation results using the Ansoft HFSS workbench are provided and analyzed in Section 4 to verify the effectiveness of the proposed approach. Finally, Section 5 summarizes the work and draws the conclusion of this paper.

2. Signal Models and Problem Formulation

Take a uniformly spaced linear array composed of M elements into consideration, the inter-element spacing of which is denoted by d, as depicted in Figure 1. Set the first element as the reference point, then the steering vector of the array can be expressed as [55]

$$\mathbf{a}(\theta )=[\exp \left\{j\overrightarrow{k}\cdot {\overrightarrow{x}}_1\right\},\cdots ,\exp \left\{j\overrightarrow{k}\cdot {\overrightarrow{x}}_m\right\},\cdots ,\exp \left\{j\overrightarrow{k}\cdot {\overrightarrow{x}}_M\right\}{]}^{\mathbf{T}}$$

(1)

where $\mathbf{a}(\theta )\in C^{M\times 1}$, $\overrightarrow{k}=\frac{2\pi }{\lambda }{[\sin (\theta ),\cos (\theta )]}^{\mathbf{T}}$ stands for the wave vector with $\lambda$ being the operational wavelength, the superscript ${}^{\mathbf{T}}$ denotes the transpose operator and $\theta$ is defined as the angle deviating from the positive y-axis. ${\overrightarrow{x}}_m={\left[\left(m-1\right)d,0\right]}^{\mathrm{T}}$, $m=1,\cdots ,M$ is the position vector of the mth element.

In practical scenarios, it is known that many unavoidable errors exist due to many factors, such as the exciting errors, position errors, failure element(s) and the mutual coupling effects among elements. Therefore, the actual steering vector cannot be exactly expressed by Equation (1). In this case, ${\mathbf{a}}_e\left(\theta \right)$ is utilized in lieu of $\mathbf{a}\left(\theta \right)$ when formulating the actual power pattern of the array, i.e.,

$${\mathbf{P}}_{act}(\theta )={\left|{\mathbf{w}}^{\mathbf{H}}{\mathbf{a}}_e(\theta )\right|}^2={\mathbf{w}}^{\mathrm{H}}{\mathbf{R}}_e(\theta )\mathbf{w}$$

(2)

where the superscript ${}^{\mathbf{H}}$ denotes the conjugate transpose operator, $\mathbf{w}\in {\mathbf{C}}^{M\times 1}$ represents the weight vector of the array and ${\mathbf{R}}_e(\theta )={\mathbf{a}}_e(\theta ){\mathbf{a}}_e{}^{\mathrm{H}}(\theta )\in {\mathbf{C}}^{M\times M}$ is the actual covariance matrix of the array in the presence of array errors. The aforementioned errors will drastically distort the array beampattern and consequently weaken the performance of radar systems. Usually, the above mentioned factors are merged together, and it is quite difficult to individually distinguish their influence on the array beamforming. In this case, it is reasonable to summarize the effects of all these factors on the array beamforming as the amplitude and phase response errors. Therefore, the actual steering vector can be constructed as

$${\mathbf{a}}_e(\theta )=e\odot \mathbf{a}(\theta )$$

(3)

where $e={[e_1,\cdots ,e_m,\cdots ,e_M]}^{\mathrm{T}}, m=1,\cdots ,M\in C^{M\times 1}$ is the array error vector, and $\odot$ stands for the Schur–Hadamard product. Specifically

$$e_m=(1+{\xi }_m)\exp \left\{j{\zeta }_m\right\}$$

(4)

where the parameters ${\xi }_m$ and ${\zeta }_m$ represent the amplitude and phase response errors, respectively, and satisfy the Gaussian random distribution and zero mean uniformly random distribution, respectively. Herein, it should be highlighted that the maximum amplitude and phase response errors can be preliminarily estimated by measurement.

As is shown in Equation (2), ${\mathbf{R}}_e(\theta )$ is important in forming the array pattern. To analyze the properties of ${\mathbf{R}}_e(\theta )$, we first examine the characteristic of $e$. Since Equation (4) can be equivalently written as $e_m=\left(1+{\xi }_m\exp \left\{j{\zeta }_m\right\}\right)+\left(\exp \left\{j{\zeta }_m\right\}-1\right)$, the expectation value of $e_m$ can be expressed as

$$E\left\{e_m\right\}=1+E\left\{{\xi }_m\right\}\exp \left\{jE\left\{{\zeta }_m\right\}\right\}+\left(\exp \left\{jE\left\{{\zeta }_m\right\}\right\}-1\right)=1$$

(5)

where $E\left\{·\right\}$ denotes the expectation operator. Therefore, the expectation of the actual covariance matrix can be represented as

$$E\left\{{\mathbf{R}}_e(\theta )\right\}=E\left\{\mathbf{R}(\theta )\odot \mathbf{E}\right\}=\mathbf{R}(\theta )$$

(6)

where $\mathbf{E}=e{e}^{\mathbf{H}}\in C^{M\times 1}$ is the error matrix. From Equation (6), it is seen that the expectation of ${\mathbf{R}}_e(\theta )$ in the presence of array errors converges to $\mathbf{R}(\theta )$. Namely, the existence of errors introduces a disturbance on the covariance matrix $\mathbf{R}(\theta )$.

Generally, the CMSR is commonly utilized when implementing the beampattern synthesis. To accurately calculate the CMSR in the presence of array errors, the statistical mean method is exploited. Specifically, $L$ Monte Carlo trials are executed, and then the mean of the CMSR in the existence of array errors can be written as

$${\hat{\mathbf{R}}}_e(\theta )=\frac{1}{L}{\displaystyle \sum _{l=1}^L{\mathbf{R}}_e^l(\theta )}, \theta \in \left[\left.-\frac{\pi }{2},{\theta }_1\right)\right.\cup \left({\theta }_2,\frac{\pi }{2}\right]$$

(7)

where ${\mathbf{R}}_e^l(\theta )$ is the CMSR obtained in the lth Monte Carlo trial with array errors taken into consideration, ${\theta }_1$ and ${\theta }_2$ denote the two boundaries of array sidelobe region, and ${\theta }_1<{\theta }_2$. Then the estimated ${\hat{\mathbf{R}}}_e(\theta )$ will be incorporated into the WCA-based low PSLL and wide-nulling beampattern synthesis presented in Section 3.

At the end of this section, an illustration is provided to show the possible negative influence that the array errors would have on the array beampattern. As depicted in Figure 2, four normalized patterns are presented in four different situations. Specifically, pattern 1 and pattern 2 are generated with ${\mathbf{w}}_0$, respectively, in an ideal situation and in the presence of 10% array errors, while pattern 3 and pattern 4 are formed with ${\mathbf{w}}_{null}$, respectively, in an ideal situation and in the presence of 10% array errors, where ${\mathbf{w}}_0\in R^{8\times 1}$ is an all one vector and ${\mathbf{w}}_{null}\in R^{8\times 1}$ denotes the wide-nulling weight vector where the wide null is located at the angle region from 30° to 40° with a null level (NL) of −35 dB. Comparing pattern 2 with pattern 1, it is observed that the PSLL is increased from −12.8 dB to −8.8 dB due to the existence of array errors. Comparing pattern 4 with pattern 3, it is seen that not only is the wide null distorted, but also the PSLL is increased (from −20 dB to −16.3 dB) when the array errors exist.

3. The WCA-Based Low PSLL and Wide-Nulling Beampattern Synthesis Approach in the Presence of Array Errors

As aforementioned, it is of significant importance to improve the PSLL performance of radar systems, especially in the presence of array errors. In this section, we aim to find two optimal complex-valued weight vectors based on WCA separately for a low PSLL and wide-nulling array beamforming in the presence of array errors. It is known that the gain of the array antenna is significant for radar systems since a higher gain tends to mean a good output SINR. Nevertheless, the value of gain would be greatly reduced when the mainlobe width is widened for the sake of a low PSLL. Meanwhile, the mainlobe distortion would appear if the mainlobe width is set too wide. Therefore, after careful consideration, we set the constraint imposed on the mainlobe width so that it is only slightly widened when constructing the two optimization problems. In this way, the first obtained weight vector can enable the array to generate a beampattern with a slightly widened mainlobe and a PSLL that is as low as possible, while the second one allows the array to form wide nulls(s) at the prescribed region(s), a PSLL that is as low as possible and a slightly widened mainlobe, simultaneously, in the presence of array errors. In the following, a brief introduction of WCA is firstly provided in Section 3.1, the fitness functions for a low PSLL and wide-nulling beampattern synthesis in the presence of array errors are separately presented in Section 3.2 and Section 3.3 followed by the optimization procedures based on WCA in Section 3.4.

3.1. Brief Introduction of WCA

Before constructing the corresponding optimization models, a brief introduction of the WCA is provided in this subsection. This algorithm combines the procedure of seeking the optimal solution with the process of the water cycle in nature, takes the fitness value calculated by the objective function as the guide to the river confluence, takes the evaporation as the assistant to get out of the local optimal solution and finally finds the optimal solution [50]. Similar to other metaheuristic algorithms, the WCA initially generates a group of raindrops in the solution hyperspace and divides these raindrops into the sea, rivers and streams according to their fitness values. Specifically, the raindrop, the fitness value of which is minimum, is initially set as the optimal one, i.e., the sea. A number of raindrops, whose fitness values are relatively small, are regarded as rivers, and the rest of the raindrops are streams. After the division, the WCA enters into the iterative optimization process and updates their positions based on the principle that streams flow to the rivers and the sea, and rivers flow to the sea until the end of the algorithm. During this procedure, evaporation and rainfall happen when the designated constraint is satisfied.

3.2. Fitness Function Construction of the Low PSLL Optimization in the Presence of Array Errors

For the low PSLL optimization problem, the goal is to find the optimal weight vector ${\mathbf{w}}_{PSLL}$, which can minimize the maximum energy transmitted from the sidelobe region subject to a distortionless response for a specified angle-of-interest in the presence of array errors. Note that the calculated CMSR, taking the errors into consideration, i.e., ${\hat{\mathbf{R}}}_e(\theta )$, is incorporated into the optimization problem. Therefore, the objective function and constraints of the low PSLL optimization problem in the presence of array errors can be established as

$$\underset{{\mathbf{w}}_{PSLL}}{\min } {\mathbf{w}}_{PSLL}{}^{\mathrm{H}}{\hat{\mathbf{R}}}_e(\theta ){\mathbf{w}}_{PSLL}$$

(8)

subject to

$$P_{low-PSLL}(\theta ){|}_{\theta ={\theta }_0}=1$$

(9a)

and

$$P_{low-PSLL}(\theta ){|}_{\theta ={\theta }_1}=P_{low-PSLL}(\theta ){|}_{\theta ={\theta }_2}=0$$

(9b)

where $P_{low-PSLL}$ denotes the generated beampattern with the potential solution in the current iteration, and ${\theta }_1<{\theta }_0<{\theta }_2$. The mainlobe width is ${\theta }_2-{\theta }_1$, which is slightly larger than its counterpart before optimization. Specifically, Equation (9a) is exploited to ensure that the generated pattern can be steered to the desired angle of ${\theta }_0$, while Equation (9b) is utilized to guarantee that the gain loss is not too much.

According to the above analysis and description, the fitness function for low PSLL optimization can be constructed as

$$f_1({\mathbf{w}}_{PSLL})=\underset{{\mathbf{w}}_{PSLL}}{\min }\underset{\theta \in [-\pi /2,{\theta }_1)\cup ({\theta }_2,\pi /2]}{\max }{\mathbf{w}}_{PSLL}{}^{\mathrm{H}}{\hat{\mathbf{R}}}_e(\theta ){\mathbf{w}}_{PSLL}$$

(10)

which decreases as the optimization proceeds. Note that the minimum criterion of the PSLL is adopted. In fact, when the PSLL is reduced, the levels of the remaining sidelobes can also be well controlled.

3.3. Fitness Function Construction of Wide-Nulling Optimization in the Presence of Array Errors

For the wide-nulling optimization problem, the goal is to find the optimal weight vector ${\mathbf{w}}_{null}$, which makes the generated beampattern $P_{g-null}\left(\theta \right)$ best approximate the desired pattern $P_{d-null}\left(\theta \right)$ in some sense, where $P_{d-null}\left(\theta \right)$ exhibits wide null(s) in designated positions. The weighted integral error function, which is defined as the integral of the error between the current generated pattern and the desired one in angle space, can be constructed as

$$J\left({\mathbf{w}}_{null}\right)={\displaystyle \int W\left(\theta \right)\left|P_{g-null}\left(\theta \right)-P_{d-null}\left(\theta \right)\right|}d\theta$$

(11)

where $W\left(\theta \right)\ge 0$ is the weight function. The characteristics of the desired pattern are described as follows

$$P_{d-null}\left(\theta \right){|}_{\theta ={\theta }_0}=1$$

(12a)

$$P_{d-null}\left(\theta \right){|}_{\theta ={\theta }_1}=P_{d-null}\left(\theta \right){|}_{\theta ={\theta }_2}=0$$

(12b)

$$P_{d-null}\left(\theta \right)\left|{}_{\theta \in \left[{\theta }_3,{\theta }_4\right]}\right.=\eta$$

(12c)

$$P_{d-null}\left(\theta \right)\left|{}_{\theta \in {\psi }_s}\right.\le \gamma$$

(12d)

where $\eta$ and $\gamma$, respectively, denote the desired NL and the PSLL, $\left[{\theta }_3,{\theta }_4\right]$ and ${\psi }_s$ are, respectively, the wide null region and sidelobe region, and ${\theta }_3<{\theta }_4$. Note that only one wide null region is presented here as an illustration. Specifically, Equation (12a) is also utilized to ensure that the maximum can be generated in the angle of ${\theta }_0$, and Equation (12b) denotes the constraint imposed on the mainlobe width. Equation (12c) is exploited to form a wide null with given NL in the designated position, while Equation (12d) presents the constraint imposed on the acceptable PSLL. Note that the estimated CMSR ${\hat{\mathbf{R}}}_e(\theta )$ in the presence of array errors is utilized in the generation of $P_{g-null}\left(\theta \right)$. Therefore, the objective function can be constructed as

$$\underset{{\mathbf{w}}_{null}}{\min } J\left({\mathbf{w}}_{null}\right)$$

(13)

as ${\mathbf{w}}_{null}$ is updated iteratively. After the adoption of discretization, Equation (11) can be represented as

$${\displaystyle \sum _{i=1}^{I}W\left({\theta }_i\right)\left|P_{g-null}\left({\theta }_i\right)-P_{d-null}\left({\theta }_i\right)\right|}$$

(14)

where ${\theta }_i\in \left[-\pi /2,{\theta }_2\right)\cup \left({\theta }_2,\pi /2\right]$ and I denotes the number of discrete points. Hence, the fitness function for wide-nulling beampattern synthesis can be constructed as

$$f_2\left({\mathbf{w}}_{null}\right)=\underset{{\mathbf{w}}_{null}}{\min } {\displaystyle \sum _{i=1}^{I}W\left({\theta }_i\right)\left|P_{g-null}\left({\theta }_i\right)-P_{d-null}\left({\theta }_i\right)\right|}$$

(15)

which is constantly decreased as ${\mathbf{w}}_{null}$ is updated iteratively.

3.4. Low PSLL and Wide-Nulling Optimization Approach Based on WCA

In this subsection, we aimed to find the optimal weight vector for the low PSLL and wide-nulling optimization based on WCA. For the sake of simplicity, $w$ is adopted to represent ${\mathbf{w}}_{PSLL}$ in Section 3.2 and ${\mathbf{w}}_{null}$ in Section 3.3 during the optimization procedures. Initially, K raindrops are generated in the solution space, each of which is a 2M-dimensional vector including the real and imaginary parts of the weight vector, i.e.,

$$Total Population={\left[w_1,\cdots ,w_k,\cdots ,w_K\right]}^{\mathbf{T}}$$

(16)

where $w_k\in {ℝ}^{2M\times 1}, k=1,\cdots ,K$, which is generated according to

$$w_{km}=F_{\min }+{\alpha }_{km}\left(F_{\max }-F_{\min }\right), m=1,\cdots ,2M$$

(17)

where $w_{km}$ is the mth entry of $w_k$, and ${\alpha }_{km}$ is a random number between 0 and 1 satisfying uniform distribution, and $F_{\max }$ and $F_{\min }$ denote the upper and lower bounds of the problem, respectively. Then, evaluate these K raindrops according to Equation (10) for the low PSLL optimization problem or Equation (15) for the wide-nulling optimization problem, reorder these obtained fitness values from small to large and store them in a column vector

$$f={\left[f_1,\cdots ,f_i,\cdots f_K\right]}^{\mathbf{T}}$$

(18)

For the first step, the raindrop, whose fitness value is $f_1$, is set as the sea and denoted as $w_{sea}$. In the sequel, $K_r$ second best individuals are chosen and set as rivers. The rest of the raindrops are considered as the streams, whose population is $K_s$ and $K_s=\left(K-K_r-1\right)$.

As descripted in Section 3.1, each stream flows to a river or the sea. Depending on the flow intensity, the sea and each river absorb the streams. Therefore, the number of streams flowing to a river and the sea varies. The specific calculation formulas are

$$C_i=f_i-f_{K_r+2}, i=1,\cdots ,K_r+1$$

(19)

and

$$N_i=round\left\{\left|\frac{C_i}{{\displaystyle \sum _{i=1}^{K_r+1}{C}_i}}\right|\times K_s\right\}, i=1,\cdots ,K_r+1$$

(20)

where $round\left\{·\right\}$ denotes the rounding operator and $N_i$ is the number of streams flowing to the specific rivers or the sea. It should be noted that $C_i<0$, which means the larger $C_i$ is, the closer its corresponding weight vector is to the optimal solution, and the more rivers or streams the corresponding weight vector needs to connect. Note that, in rare cases, ${\displaystyle \sum _i{N}_i}\ne K_s$ due to the rounding operation. By randomly adding or subtracting 1 from the number of streams that connect to the rivers or the sea, ${\displaystyle \sum _i{N}_i}=K_s$ can be guaranteed.

In the following, before updating the positions of streams and rivers, the connecting line along a stream and the river the stream flowing to is defined as x

$$x\in \left(0, \beta d_{s-r}\right), \beta \in \left(1,2\right]$$

(21)

where $d_{s-r}$ denotes the distance between the current stream and the river it is flowing to, and $\beta \in \left(1,2\right]$ is utilized to ensure that the location of the new stream can be anywhere between the current stream and the river. This concept can also be used in rivers flowing to the sea. The location of streams and rivers is updated as follows

$$w_{\mathrm{stream}}(t+1)=w_{\mathrm{stream}}(t)+rand\times \beta \times \left(w_{\mathrm{river}}(t)-w_{\mathrm{stream}}(t)\right)$$

(22a)

$$w_{\mathrm{stream}}(t+1)=w_{\mathrm{stream}}(t)+rand\times \beta \times \left(w_{\mathrm{sea}}(t)-w_{\mathrm{stream}}(t)\right)$$

(22b)

and

$$w_{\mathrm{river}}(t+1)=w_{\mathrm{river}}(t)+rand\times \beta \times \left(w_{\mathrm{sea}}(t)-w_{\mathrm{river}}(t)\right)$$

(22c)

Generally, $\beta =2$ is set. Equations (22a) and (22b) respectively provide the position renewal process of streams flowing into rivers and the sea, while Equation (22c) represents the corresponding process of rivers flowing into the sea. If the fitness value of the updated stream (river) is smaller than that of the river (sea), exchange their positions. Figure 3 provides the schematic view of the WCA.

With the progress of the WCA, rivers and streams gradually approach the sea. At that moment, to enhance the search ability of the WCA and prevent it from converging prematurely into local optimal solutions, evaporation and rainfall operations are performed. The evaporation conditions for rivers and streams are

$$‖w_{\mathrm{sea}}-w_{\mathrm{river}}^i‖<{\tau }_1, i=1,\cdots ,K_r$$

(23a)

and

$$‖w_{\mathrm{sea}}-w_{\mathrm{stream}}^j‖<{\tau }_2, j=1,\cdots ,K_{s-s},$$

(23b)

respectively, where ${\tau }_1$ and ${\tau }_2$ are quite small (usually close to 0), and $K_{s-s}$ denotes the number of streams flowing into the sea. ${\tau }_1$ and ${\tau }_2$ control the search intensity near the sea, which is adaptively decreased as

$${\tau }_1^{i+1}={\tau }_1^i-\frac{{\tau }_1^i}{N_{iter}}, i=1,\cdots ,K_r$$

(24a)

and

$${\tau }_1^{i+1}={\tau }_1^i-\frac{{\tau }_1^i}{N_{iter}}, j=1,\cdots ,K_{s-s}$$

(24b)

where $N_{iter}$ denotes the maximum number of iterations.

When Equation (23a) is satisfied, the streams will be regenerated in the problem space, i.e.,

$$w_{\mathrm{Stream}m}^{\mathrm{New}}\left(t+1\right)=F_{\min }+{\alpha }_m\left(F_{\max }-F_{\min }\right), m=1,\cdots ,2M$$

(25)

where $w_{\mathrm{Stream}m}^{\mathrm{New}}\left(t+1\right)$ is the mth element of $w_{\mathrm{Stream}}^{\mathrm{New}}\left(t+1\right)$. When Equation (23b) is satisfied, the streams will be regenerated near the sea, i.e.,

$$w_{\mathrm{Stream}}^{\mathrm{New}}\left(t+1\right)=w_{\mathrm{Sea}}\left(t\right)+\sqrt{\sigma }\times r$$

(26)

where $\sigma$ is a coefficient showing the range of the searching region around the sea, and $r\in {ℝ}^{2M\times 1}$ denotes a normally distributed random vector. In a mathematical view, $\sigma$ defines the variance, i.e., the generated streams are around the sea with variance $\sigma$, which is usually set as 0.1. To make the optimization procedures clearer, Figure 4 provides the flowchart of the proposed low PSLL and wide-nulling beampattern synthesis in the presence of array errors based on the WCA.

From the above descriptions, it is known that ${\hat{\mathbf{R}}}_e(\theta )$ is estimated first before the optimization. The larger L is, the closer ${\hat{\mathbf{R}}}_e(\theta )$ is to the real one. During the iteration, a set of second best selected solutions, i.e., rivers, act as guidance to lead other individuals towards better positions (as depicted in Figure 3) and prevent the search in inappropriate regions in near-optimum solutions (indicated by Equation (22)) simultaneously. Moreover, the evaporation and raining implementations are utilized in the proposed approach to prevent the algorithm being trapped in local solutions. In this way, the convergence robustness of the proposed approach can be guaranteed.

When the optimization stops, an optimal weight vector ${\mathbf{w}}_{opt}={\mathbf{w}}_{sea}$ is obtained. Hence, the resultant low PSLL/wide-nulling beampattern in the presence of array errors can be represented as

$${\hat{P}}_g(\theta )={\mathbf{w}}_{opt}^{\mathrm{H}}{\mathrm{R}}_e(\theta ){\mathbf{w}}_{opt}$$

(27)

Note that ${\mathbf{w}}_{opt}$ is robust against array errors in array beamforming since it is obtained with ${\hat{\mathrm{R}}}_e(\theta )$.

4. Numerical Results and Discussions

In this section, experiments are conducted to validate the effectiveness of the proposed approach. Specifically, the comparisons among PSO, GA and WCA are firstly conducted in Section 4.1 based on different test functions in terms of efficiency and accuracy. Numerical results obtained by low PSLL synthesis are provided in Section 4.2, while those generated by wide-nulling synthesis are presented in Section 4.3. Both of the theoretical analyses using THE MATLAB workbench and electromagnetic simulation using THE Ansoft HFSS workbench are utilized to test the effectiveness of the proposed approach. In particular, a wideband magneto-electric (ME) dipole antenna is utilized to form a uniformly spaced linear array.

4.1. Comparison with PSO and GA

In this part, the search ability and stability of the PSO, GA and WCA are compared. Two classical benchmark functions, i.e., the Sphere function and the Rastrigin function, are selected to compare these three algorithms from the point of the available best fitness value and the convergence rate. The sphere function is a spherical unimodal function, which has difficulty finding the global optimal solution. The Rastrigin function image is similar to a single peak Sphere function, and its optimal solution has multiple local peaks. Figure 5 provides the images of these two benchmark functions when n = 2 as an illustration. The basic settings of these two functions are presented in

Table 1. Benchmark function settings.

Function Name	Benchmark Function	Number of Iterations	Range of Variable
Sphere	$F_1={\displaystyle \sum _{i=1}^n{x}_i^2}$	500	$x_i\in [-10,10]$
Rastrigin	$F_2={\displaystyle \sum _{i=1}^n(x_i^2}-10\cos 2\pi x_i+10)$	1000	$x_i\in [-10,10]$

. Note that the optimal values of the two functions are 0.

Simulation experiments using the PSO, GA and WCA are conducted to obtain the optimal solution of the above two functions. Upper and lower bounds of the problem are [−10, 10], and the dimension of the search space n = 2, 5, 10 is chosen. Other parameter settings of each algorithm are as follows: (1) the number of chromosomes and iterations of the GA are set to 200 and 1000, respectively, and the crossover probability and mutation probability are set to 0.8 and 0.1, respectively [25]; (2) the number of particles and iterations of the PSO are set to 200 and 1000, respectively, w = 0.8 and $c_1=c_2=1.5$ [28]; (3) the number of raindrops and iterations of the WCA are set to 200 and 1000, respectively, $K_r$ = 3, $\tau$ = 1 × 10⁻¹⁶ and $\beta$ = 2 [50]. The results of the simulation experiments are shown in Figure 6,

Table 2. The optimal fitness value for Sphere function optimization.

Sphere	n = 2	n = 5	n = 10
PSO	1.940 × 10−43	2.618 × 10−36	1.010 × 10−25
GA	4.517 × 10−6	5.987 × 10−6	6.450 × 10−5
WCA	0	0	0

and

Table 3. The optimal fitness value for Rastrigin function optimization.

Rastrigin	n = 2	n = 5	n = 10
PSO	0	1.99	8.955
GA	4.176 × 10−5	0.001	0.019
WCA	0	0	0

. From Figure 6, it is clearly seen that the WCA has the fastest convergence rate with a higher precision when compared with the PSO and GA.

4.2. Low PSLL Beampattern Based on WCA

In this subsection, experiments are conducted to validate the effectiveness of the proposed low PSLL beampattern synthesis based on WCA. Specifically, the theoretical results using the MATLAB workbench are firstly provided in part 4.2.1 with the existing 10% amplitude and phase response errors. Then the electromagnetic results using the Ansoft HFSS workbench are presented in part 4.2.2 to further test the effectiveness of the proposed approach. Note that a 24-mm-spaced linear ME dipole array antenna, working at frequency from 6.67 GHz to 10.8 GHz in terms of active VSWR < 2.5, is designed since a wideband array antenna is commonly utilized in radar applications. The corresponding simulation parameters are provided in

Table 4. Simulated parameters.

Parameters	Values	Parameters	Values	Parameters	Values
$M$	8	$\beta$	2	$N_{iter}$	200
d	24 mm	$K_r$	10	$\tau$	1 × 10−16
$K$	50	$\sigma$	0.1

.

4.2.1. Simulation Result on MATLAB Workbench

In this part, the beampatterns at four frequencies, i.e., 7 GHz, 8 GHz, 9 GHz and 10 GHz, in the presence of 10% array errors, are taken into consideration. It is seen from Table 4 that the inter-element spacing is fixed, i.e., 24 mm, through the experiments. Therefore, the electric sizes of the inter-element spacing are 0.56, 0.64, 0.72 and 0.80 at these four frequencies, respectively. The corresponding amplitude and phase parts of the optimal weight vectors at these four frequencies are provided in

Table 5. Optimal weight vectors for Low PSLL optimization at four frequencies.

7 GHz		8 GHz		9 GHz		10 GHz
Amp	Pha (°)	Amp	Pha (°)	Amp	Pha (°)	Amp	Pha (°)
0.231	15.8	0.375	0.0	0.351	5.3	0.280	0.0
0.515	0.9	0.593	8.4	0.606	1.8	0.506	4.6
0.756	5.7	0.844	8.2	0.865	0.6	0.800	1.8
0.993	3.0	0.999	11.0	0.986	0.1	0.991	5.4
1.000	2.5	1.000	11.0	1.000	1.5	1.000	6.0
0.887	5.7	0.842	9.4	0.810	0.4	0.857	3.2
0.689	0.0	0.598	10.5	0.519	0.0	0.579	7.0
0.315	6.2	0.365	6.8	0.297	4.2	0.325	9.1

.

Figure 7 presents the convergence curves at 7 GHz, 8 GHz, 9 GHz and 10 GHz, in the presence of 10% array errors. From this figure, it is seen that the fitness value decreases rapidly in the first 12 iterations, gently from 12 to 20 iterations and remains almost unchanged after 20 iterations at 7 GHz. It is almost the same situation for 8 GHz, 9 GHz and 10 GHz. Namely, the proposed WCA-based low PSLL beampattern synthesis approach converges within 20 iterations.

Figure 8 provide the corresponding normalized beampatterns at these four frequencies before and after applying the proposed low-PSLL beampattern synthesis approach. In particular, the patterns generated with $w_0$ in an ideal situation and in the presence of 10% errors are also provided as a comparison. It is observed that the mainbeam widths are, respectively, 26°, 22.5°, 20° and 18° at these four frequencies. Therefore, the constraints imposed on the mainbeam width at these four frequencies are set as 32°, 28°, 26° and 24°, respectively, each of which is slightly larger than its counterpart before the synthesis. As has been stated in Section 3, to improve the robustness of the low PSLL synthesis approach against the random characteristic of array errors, the statistical mean method is adopted in calculating the CMSR. Therefore, three sets of errors are adopted to verify the robustness of the obtained weight vectors, each of which satisfies the properties presented in Equation (4). As depicted in Figure 8, it is seen that the 8-element array exhibits a PSLL around −13.0 dB, i.e., −12.82 dB at 7 GHz, −12.81 dB at 8 GHz, −12.80 dB at 9 GHz and −12.8 dB at 10 GHz. The PSLLs are increased to −9.8 dB, −8.5 dB, −10.4 dB and −11.3 dB at these four frequencies with existing 10% array errors. After applying the obtained optimal weight vectors presented in Table 5, the PSLL is reduced to below −21.0 dB at 7 GHz and 8 GHz, and below −20 dB at 9 GHz and 10 GHz for all of the three cases, which demonstrates the robustness of the proposed approach. The detailed PSLLs are provided in

Table 6. The PSLLs after applying the proposed low PSLL beampattern synthesis approach at 7 GHz, 8 GHz, 9 GHz and 10 GHz.

7 GHz (dB)			8 GHz (dB)			9 GHz (dB)			10 GHz (dB)
Case 1	Case 2	Case 3	Case 1	Case 2	Case 3	Case 1	Case 2	Case 3	Case 1	Case 2	Case 3
−22.0	−21.1	−21.2	−21.3	−21.0	−22.5	−20.5	−20.4	−20.8	−20.3	−20.2	−21.4

at these four frequencies.

4.2.2. Verification on HFSS Workbench

In practical scenarios, mutual coupling effects among elements are unavoidable, especially those between adjacent elements, which can cause large array errors and would distort the performance of the array accordingly. Therefore, the effectiveness of the proposed WCA-based low PSLL approach is further verified on the Ansoft HFSS workbench in this part. Proposed by Prof. Luk in 2006 [56], the ME dipole antenna possesses many excellent properties, such as wideband, a low cross-polarization level and a stable radiation pattern across the operating frequency band, which are desirable for radar applications. Therefore, the ME dipole antenna is chosen as the array element, and is designed with the size of 18 mm × 18 mm × 7.5 mm, the operating frequency of which is from 6.77 GHz to 10.91 GHz in terms of VSWR < 2. Then it is utilized to form a 24-mm-spaced 8-element linear array, as depicted in Figure 9. In addition, a box-shaped structure is utilized on the periphery of the array to reduce the back radiation.

Figure 10 presents the simulated parameters of the 8-element linear array. It is seen from Figure 10a that the overlapped operating frequency of the array is from 6.67 GHz to 10.8 GHz in terms of an active VSWR ≤ 2.5. Meanwhile, the peak realized gain of 18.4 dBi can be achieved across the operating frequency band. Figure 10b provides the S parameters between the 5th element and the other elements of the array. It is seen from this figure that the strongest mutual coupling effects appears between the 4th element (or the 6th element) and the 5th element, which can be as high as −14 dB.

Figure 11 provides the co-polarization and cross-polarization patterns of the 8-element array antenna at xoz-plane before and after applying the optimal weight vectors listed in Table 5. In particular, the corresponding patterns generated with $w_0$ are provided as a comparison. After a careful observation, several remarks can be drawn in the following. (1) The PSLLs of the array are all around −13.0 dB at all of the four frequencies before applying the obtained optimal weight vectors. (2) The PSLLs of the array are separately reduced by 6.6 dB, 4.4 dB, 7.2 dB and 5.0 dB at these four frequencies after applying the obtained optimal weight vectors. (3) The peak cross-polarization levels (PCPLs) of the array are also slightly reduced after applying the optimal weight vectors. To summarize, the beampattern of the 8-element linear array exhibits good performance after applying the obtained weight vectors, which in turn demonstrates the effectiveness of the proposed WCA-based low PSLL synthesis approach in the presence of strong mutual coupling effects.

Table 7. The PSLLs and PCPLs of the 8-element linear array before and after applying the proposed low PSLL beampattern synthesis approach at 7 GHz, 8 GHz, 9 GHz and 10 GHz.

	7 GHz (dB)	8 GHz (dB)	9 GHz (dB)	10 GHz (dB)
PSLL (before)	−13.5	−13.3	−13.2	−13.0
PSLL (after)	−20.1	−17.7	−20.4	18.0
PCPL (before)	35.4	28.0	22.9	19.0
PCPL (after)	37.6	28.5	23.0	19.2

provides the PSLLs and PCPLs of the 8-element linear array before and after applying the proposed low PSLL beampattern synthesis approach at 7 GHz, 8 GHz, 9 GHz and 10 GHz.

4.3. Wide-Nulling Beampattern Based on WCA

In this subsection, numerical results are provided to verify the effectiveness of the proposed wide-nulling beampattern synthesis based on WCA. Specifically, (1) a wide null at the angle region from 50° to 60 ° is of interest at 7 GHz; (2) two wide nulls at the angle region from −60° to −40° and 40° to 60° are considered at 8 GHz; (3) two wider nulls from −60° to −30° and 30° to 60° are taken into consideration at 9 GHz; and (4) two narrower nulls from −65° to −55° and 55° to 65° are taken into consideration at 10 GHz. The specific constraints of NL are provided in

Table 8. The parameters’ settings of wide-nulling at 7 GHz, 8 GHz, 9 GHz and 10 GHz, respectively.

	7 GHz	8 GHz	9 GHz	10 GHz
Null Region	50°~60°	−60°~−40° and 40°~60°	−60°~−30° and 40°~60°	−65°~−55° and 55°~65°
NL	−50 dB	−35 dB	−30 dB	−40 dB
PSLL	−20 dB	−20 dB	−20 dB	−20 dB

. Similarly, both of the theoretical results using the MATLAB workbench and the electromagnetic results using the Ansoft HFSS workbench are provided, respectively, in Section 4.3.1 and Section 4.3.2. The corresponding simulation parameters are the same as those in Section 4.2.

4.3.1. Simulation Result on MATLAB Workbench

In this part, the simulation results obtained on the MATLAB workbench are provided to verify the effectiveness of the proposed wide-nulling beampattern synthesis based on the WCA. In addition, the beampatterns generated at four frequencies, i.e., 7 GHz, 8 GHz, 9 GHz and 10 GHz are taken into consideration in the existence of 10% array errors. The electric sizes of the inter-element spacing and the constraints imposed on the mainbeam width are inconsistent with those in Section 4.2. The corresponding amplitude and phase parts of the obtained optimal weight vectors at these four frequencies are provided in

Table 9. Optimal weight vectors for wide-nulling optimization at four frequencies.

7 GHz		8 GHz		9 GHz		10 GHz
Amp	Pha (°)	Amp	Pha (°)	Amp	Pha (°)	Amp	Pha (°)
0.311	20.5	0.315	6.1	0.329	2.2	0.199	4.5
0.847	5.7	0.775	3.2	0.733	1.3	0.377	0.7
0.983	0.2	0.825	0.0	0.997	0.0	0.673	1.6
1.000	0.0	1.000	0.0	1.000	0.0	0.912	0.1
0.975	0.9	0.999	0.0	1.000	0.0	0.999	0.0
0.947	5.2	0.849	0.0	1.000	0.0	1.000	0.0
0.801	3.0	0.864	3.4	0.809	5.6	0.608	0.0
0.399	24.4	0.363	5.8	0.345	14.9	0.370	12.87

.

The corresponding convergence curves at 7 GHz, 8 GHz, 9 GHz and 10 GHz, in the presence of 10% array errors, are provided in Figure 12. It is observed from this figure that the four convergence curves with 7 GHz, 8 GHz, 9 GHz and 10 GHz remain almost unchanged after 20 iterations. It is worth mentioning that the excellent result confirms the fast convergence rate of the proposed wide-nulling beampattern synthesis approach based on the WCA.

The normalized beampatterns after employing the proposed wide-nulling beampattern synthesis approach at 7 GHz, 8 GHz, 9 GHz and 10 GHz are presented in Figure 13. Note that the desired beampatterns are also provided as a comparison. Meanwhile, the constraints on the mainbeam widths are also set to 32°, 28°, 26° and 24° at these four frequencies, respectively. At each frequency, three sets of errors satisfying the properties in Equation (4) are also adopted to show the robustness of the obtained weight vectors against errors. It is seen from Figure 13a that a wide null is generated at the desired angle region from 50° to 60° with an NL around −50 dB and a PSLL around −16 dB for all of the three cases. From Figure 13b, it can be observed that two wider nulls at the desired angle regions from −60° to −40° and from 40° to 60° emerge with an NL around −35 dB and a PSLL around −14 dB. In Figure 13c, it is seen that two wider nulls at the desired angle regions from −60° to −30° and from 30° to 60° are generated with an NL around −30 dB and a PSLL around −15 dB. As depicted in Figure 13d, it is observed that two narrower nulls at the desired angle regions from −65° to −55° and from 55° to 65° emerge with an NL around −40 dB and a PSLL around −20 dB. Although there are some slight departures from the settings of the wide null(s), they are basically within an acceptable range. Therefore, the experiment results demonstrate the robustness of the proposed wide-nulling beampattern synthesis approach.

4.3.2. Verification on HFSS Workbench

In this part, the obtained optimal weight vectors in Table 9 are applied in the 8-element linear ME dipole array antenna depicted in Figure 9 to demonstrate the effectiveness of the proposed wide-nulling beampattern synthesis in the presence of strong mutual coupling effects. Figure 14 presents the co-polarization and cross-polarization patterns of the 8-element array antenna at the xoz-plane before and after applying the proposed approach. Similarly, the corresponding patterns generated with $w_0$ are also provided for a more intuitive comparison. After careful observation of the results, several remarks are drawn as follows. (1) All of the obtained beampatterns at these four different frequencies form satisfactory wide null(s) at the corresponding desired angle region with acceptable NLs. (2) The PSLLs are reduced by 2.4 dB, 3.0 dB, 2.1 dB and 4.6 dB, respectively, at these four frequencies after applying the proposed wide-nulling beampattern synthesis approach. (3) The PCPLs of the array are also slightly reduced after applying the proposed approach. To summarize, the beampatterns of the 8-element ME dipole linear array show excellent performance after applying the obtained weight vectors, which in turn demonstrate the effectiveness of the proposed WCA-based wide-nulling synthesis approach in the presence of mutual coupling effects.

Table 10. The PSLLs and PCPLs of the 8-element linear array before and after applying the proposed wide-nulling approach at 7 GHz, 8 GHz, 9 GHz, and 10 GHz.

	7 GHz (dB)	8 GHz (dB)	9 GHz (dB)	10 GHz (dB)
PSLL (before)	−13.5	−13.3	−13.2	−13.0
PSLL (after)	−15.9	−16.3	−15.3	−17.6
PCPL (before)	35.4	28.0	22.9	18.9
PCPL (after)	35.5	28.5	23.8	19.0

provides the PSLLs and PCPLs of the 8-element linear array before and after applying the proposed wide-nulling approach at these four frequencies.

5. Conclusions

To improve the anti-jamming performance of radar systems, an effective low PSLL and wide-nulling beampattern synthesis approach is proposed in this paper based on the WCA in the presence of array errors. A more general error model is established and then adopted in calculating the CMSR, which is incorporated into the optimization problem. Numerical results including those based on a linearly spaced ME dipole array antenna demonstrate the effectiveness of the proposed approach.