Planar Array Failed Element(s) Radiation Pattern Correction: A Comparison

: Phased arrays are widely used in different ﬁelds, such as broadcasting, radar, optics, and space communications. The principle of phased arrays is to generate a directed signal from a large number of antennas to be steered at any desired angle. This, however, increases the probability of defective elements in an array. Faulty elements in an array cause asymmetry and result in increased sidelobe levels which rigorously distort the radiation pattern. Increased sidelobe radiation wastes energy and can cause interference by radiating and receiving signals in unintended directions. Therefore, it is necessary to ﬁnd a method that can provide accuracy in the radiation pattern transmitted or received in the presence of failed element(s) in an array. This paper compares the few available optimization methods, namely, simulated annealing (SA), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Pattern Search (PS) methods. For each method, various types of failures were examined, and the most suitable techniques to recover the far-ﬁeld radiation are recommended. The optimization is then carried out by selecting the optimal weights of the remaining working elements in the planar array. The optimized radiation pattern’s efﬁciency was evaluated by comparing the Signal to Noise Ratio (SNR) value of the optimized radiation with reference and failed radiation patterns. The PSO method showed a better performance compared to all the other methods in reducing the failed radiation pattern’s SNR value. In various types of failure tests, this method reduced the failed radiation pattern’s SNR from 1 to 10 dB. This method also successfully produced a radiation pattern that closely matches the reference pattern before any failed element(s) are presented in the array. The life cycle of a planar array system with faulty elements can be increased by optimizing the remaining active elements in the array with the PSO method. It also reduces the cost of restoring and replacing the failed elements in an array regularly. This approach also prevents near-ﬁeld measurement that requires complicated processes using costly equipment.


Introduction
Phased array antennas are used to form directed signals that can be steered at the desired angle. Some specific applications of phased arrays include broadcasting, radar, optics, and space communication. The current trend in the telecommunication industry with Massive-Input-Massive-Output (MIMO) uses a large number of multiple antennas at the transmitter and receiver ends. Its base station is equipped with tens to hundreds of antennas in a single antenna array. The usage of phased arrays in space communication is also increasing [1].
This large array distribution in a system increases the probability of more antenna system elements failing, and the failures are unavoidable [2]. The failed elements cause the array to lose its symmetry, resulting in a distorted radiation pattern with increased sidelobe levels, weakening null depth and null shift [1]. This causes transmitters to waste energy and cause interference to other equipment by radiating signals in unwanted directions. While in receivers, the noise level is increased by receiving interfering signals. This will reduce the efficiency of the transmitter and the receiver.
Research on failure elements in arrays has existed since the end of the 20th century and the beginning of the 21st century because of its continuous usage in various fields. Now it is regaining its attention due to its mass usages in wireless and communication systems and other fields. The need for failure correction techniques is still necessary.
Replacing the failed element with a new element is not a possible solution for applications such as space platforms, where it is complex, time-consuming, and expensive. So, optimization-based far-field radiation pattern correction methods with remaining active elements in the array are being researched [3][4][5][6].
Recent research on correcting failed radiation patterns are with grey wolf optimizer hybridized with an interior point algorithm [7], Improved Particle Swarm Optimization with Wavelet Mutation [8], Particle Swarm Optimization (PSO) and Bacteria Foraging Optimization (BFO) [5], Whale Optimization Algorithm (WOA) and Chaotic Whale Optimization Algorithm (CWOA) [9], and Backtracking Search Optimization [10]. However, this research focuses on the linear array, which consists of a smaller number of elements in an array. Failed pattern correction with the planar array is limited. Some available research uses the Simulated Annealing Technique [11], Artificial Neural Network with Hopfield network [12], and Particle Swarm Optimization [13]. In study [11], the number of faulty elements in an array is very little, while the current communication systems have more probability of element failure in an array. Other than that, reference [13] has investigated a 5 × 5 planar array with 2 to 20 element failures in the array.
Most of the papers in the literature used a meta-heuristic search for restoring the radiation pattern. However, each method has its disadvantages, such as solutions converge prematurely, easily fall into a local optimum, and adequate parameter tuning [14]. An optimization algorithm developed varies case by case. Similarly, optimization methods developed for linear arrays and smaller array sizes will not be suitable for a large planar array. This paper compares four different optimization methods; simulated annealing (SA), genetic algorithm (GA), particle swarm optimization (PSO), and pattern search (PS) method, and proposes a method that is suitable for large planar array radiation pattern correction at the end of the paper. A failed radiation pattern is corrected by optimizing the weights of the remaining working elements in the array. The method will improve the overall efficiency of any system with a planar array and reduce the cost of repairing and replacing the failed element(s) in an array. This method also avoids near fields measurement, which involves complex processes and uses expensive equipment.

Design Methodology
The design scheme presented in this paper is divided into two parts; the first part is developing a planar array model with an 8 × 8 element array. Secondly, four different optimization processes were applied to the planar array model, described in Section 2.2. A flowchart of the whole process is shown in Figure 1.
This paper illustrates a suitable optimization method for failed planar array radiation patterns where the optimization is done in MATLAB through simulation. All the simulation was done with a 64-bit operating system that contains an Intel(R) Core (TM) i7-4790 CPU processor and 16 GB installed memory. This simulation is for research purposes only. To be implemented in real-time, the optimization can be programmed in C language and processed in real-time on Digital Signal Processor (DSP) embedded processor platform, which will make the process faster. This simulation has the limitation of computer capability and software limitations. This paper illustrates a suitable optimization method for failed planar array radiation patterns where the optimization is done in MATLAB through simulation. All the simulation was done with a 64-bit operating system that contains an Intel(R) Core (TM) i7-4790 CPU processor and 16 GB installed memory. This simulation is for research purposes only. To be implemented in real-time, the optimization can be programmed in C language and processed in real-time on Digital Signal Processor (DSP) embedded processor platform, which will make the process faster. This simulation has the limitation of computer capability and software limitations.

Array Beam Pattern Generation
Beam pattern is also known as spatial signal processing. It transmits and receives sound waves in the desired direction by phase shift and time delay. It is well known for its advantage of high gain, focused directivity, reduced interference, and improved device ability, producing better signal quality than omnidirectional signals. The phased array beamforming method [15] is used to generate a beam pattern with × = (8 × 8) elements, an ultrasonic frequency of = 100 kHz, and the speed of sound in air at 20 ° is = 344 m/s, the inter-element distance avoiding grating is:

Array Beam Pattern Generation
Beam pattern is also known as spatial signal processing. It transmits and receives sound waves in the desired direction by phase shift and time delay. It is well known for its advantage of high gain, focused directivity, reduced interference, and improved device ability, producing better signal quality than omnidirectional signals. The phased array beamforming method [15] is used to generate a beam pattern with M × N = (8 × 8) elements, an ultrasonic frequency of f c = 100 kHz, and the speed of sound in air at 20 • C is c = 344 m/s, the inter-element distance d avoiding grating is: The array factor beam pattern was normalized with equal amplitudes of w(m) = 1. Planar array beam pattern G p is formed by the superposition of two-beam patterns, S x on the x-axis and S y on the y-axis. The beamforming output for the planar array pattern is given in (3). where and S y (θ, ϕ) = N ∑ n=1 w y (n)e i(2n−N−1) * arg y (6) θ and ϕ = −90:1:90, are the scanning angle of x and y dimensions; γ is steering angle; w x (m) = w y (n) = w(m) = 1, uniform weight on each element; d is the inter-element distance between array elements; M × N, Array elements, and m, n is the index. Figure 2 shows M × N elements in an array with uniform inter-element distance [15].

Figure 2.
× elements in an array with uniform inter-element distance. Figure 2. M × N elements in an array with uniform inter-element distance.

Failed Beam Pattern Generation
Failed array weights, w f x (q) and w f y (q), are used to simulate failed beam patterns. Failed element weights in w f x (q) and w f y (q) were assigned as zero instead of one in w x (m) and w y (n). For this purpose, elements from Rows 4 and 7 (24 elements) in the array are failed in an 8 × 8 element array. The beam pattern formulation of failed elements is as (8). is as (8). Figure 3 shows the planar array with Row 4 and Row 7 failed elements. The failed elements in the figure below are not shaded, while the functional elements are shaded. Figure 3. Failed elements in a planar array with non-uniform inter-element distance.

Performance Measure of the Radiation Patterns
The gain performance of the radiation patterns is measured with SNR, . It is set to be the ratio of the main beam above the −3 dB, 3 to the side lobes as (9) of each radiation pattern.

Optimization Algorithms
Lately, metaheuristic algorithms have been used in numerous fields, such as engineering, economics, politics, and management, to solve complex real-life problems [16]. A proper balance between intensification and diversification elements is required to solve real-life problems effectively [17]. A carefully chosen optimization algorithm for a specific cause can effectively solve many nonlinear optimization problems [18].
In this paper, four different optimization methods, SA [19], GA [20], PSO [21], and PS method [22], were used to optimize the remaining working element in the planar array. An optimization method that produces SNR similar to the initial radiation pattern was proposed at the end of the paper for a planar array failed radiation pattern correction.
To optimize the weights, an objective function is created. In the objective function, SNR is calculated with (8) for a reference radiation pattern , and a current optimized pattern . The optimization will compare both SNR values and minimize the objective function. The objective function is defined as (10):

Performance Measure of the Radiation Patterns
The gain performance of the radiation patterns is measured with SNR, η. It is set to be the ratio of the main beam above the −3 dB, G BW 3db to the side lobes as (9) of each radiation pattern.

Optimization Algorithms
Lately, metaheuristic algorithms have been used in numerous fields, such as engineering, economics, politics, and management, to solve complex real-life problems [16]. A proper balance between intensification and diversification elements is required to solve real-life problems effectively [17]. A carefully chosen optimization algorithm for a specific cause can effectively solve many nonlinear optimization problems [18].
In this paper, four different optimization methods, SA [19], GA [20], PSO [21], and PS method [22], were used to optimize the remaining working element in the planar array. An optimization method that produces SNR similar to the initial radiation pattern was proposed at the end of the paper for a planar array failed radiation pattern correction.
To optimize the weights, an objective function is created. In the objective function, SNR is calculated with (8) for a reference radiation pattern η G p , and a current optimized pattern η G ponew . The optimization will compare both SNR values and minimize the objective function. The objective function is defined as (10): The planar array weights, w f x and w f y , are vector quantities that require optimization techniques. Minimization in the objective function will return optimized weights of remaining working elements in the array that can generate an optimized beam pattern of the array in the presence of failure elements.

Planar Array Modeling Results
The methodology developed was to examine a planar array with 8 × 8 and 6 × 6 elements with λ/2 interelement distance. Equation (3) was used to generate a beam pattern of reference radiation pattern without any failed elements in an array. Figure 4 shows an example of a reference radiation pattern for 8 × 8 elements.

Planar Array Modeling Results
The methodology developed was to examine a planar array with 8 × 8 and 6 × 6 elements with 2 ⁄ interelement distance. Equation (3) was used to generate a beam pattern of reference radiation pattern without any failed elements in an array. Figure 4 shows an example of a reference radiation pattern for 8 × 8 elements. Next, the failed radiation beam pattern is generated with Equation (8) by designating the failed element's weight as zero. It is assumed that the failed element(s) do not contribute to the radiation pattern. In this analysis, the probability of faulty elements is fixed to a maximum of 25 elements (varies on case basis) in an 8 × 8 array and 12 elements for a 6 × 6 array. Figure 5a shows 25 elements failed in a random location radiation pattern and its failed radiation pattern in Figure 5b. Next, the failed radiation beam pattern is generated with Equation (8) by designating the failed element's weight as zero. It is assumed that the failed element(s) do not contribute to the radiation pattern. In this analysis, the probability of faulty elements is fixed to a maximum of 25 elements (varies on case basis) in an 8 × 8 array and 12 elements for a 6 × 6 array. Figure 5a shows 25 elements failed in a random location radiation pattern and its failed radiation pattern in Figure 5b

Optimization Algorithm Results
Then, the failed radiation pattern was restored by optimizing the remaining active elements of the array. The optimization methods described in Section 2.2 were applied to the failed radiation pattern following the optimization flow. New weights for the remaining working element were assigned to calculate SNR for the current radiation pattern. The objective function is evaluated as in (10). The objective function minimizes the error between the current radiation pattern and the reference radiation pattern without any failed

Optimization Algorithm Results
Then, the failed radiation pattern was restored by optimizing the remaining active elements of the array. The optimization methods described in Section 2.2 were applied to the failed radiation pattern following the optimization flow. New weights for the remaining working element were assigned to calculate SNR for the current radiation pattern. The objective function is evaluated as in (10). The objective function minimizes the error between the current radiation pattern and the reference radiation pattern without any failed

Optimization Algorithm Results
Then, the failed radiation pattern was restored by optimizing the remaining active elements of the array. The optimization methods described in Section 2.2 were applied to the failed radiation pattern following the optimization flow. New weights for the remaining working element were assigned to calculate SNR for the current radiation pattern. The objective function is evaluated as in (10). The objective function minimizes the error between the current radiation pattern and the reference radiation pattern without any failed elements in the array. The iteration continues until the optimization meets the exit condition where the optimization stops.
An example of results from applying the PS optimization method to 25 failed element radiation patterns is shown in Figure 6 and the optimized weights are shown in Table 1. elements in the array. The iteration continues until the optimization meets the exit condition where the optimization stops. An example of results from applying the PS optimization method to 25 failed element radiation patterns is shown in Figure 6 and the optimized weights are shown in Table 1.  Next, the optimization methods were implemented for eight different types of failure as listed in Table 2 to analyze the efficiency of the methods towards different types of failure in 8 × 8 and 6 × 6 planar arrays, respectively.  Next, the optimization methods were implemented for eight different types of failure as listed in Table 2 to analyze the efficiency of the methods towards different types of failure in 8 × 8 and 6 × 6 planar arrays, respectively. The results of the 8 × 8 planar array in Rows 4 and 7 that failed for all four optimization methods are shown. Figure 7a-c represents the initial, failed, and optimized radiation patterns of the SA method. The corresponding optimized weights are shown in Table 3. In this case, the optimized radiation pattern has a higher SNR value compared to failed radiation pattern. This method could not meet the objective where optimized weights in a planar array should generate an optimized radiation pattern that matches the reference radiation pattern with a lower SNR value than the failed radiation pattern SNR.  The results of the 8 × 8 planar array in Rows 4 and 7 that failed for all four optimization methods are shown. Figure 7a-c represents the initial, failed, and optimized radiation patterns of the SA method. The corresponding optimized weights are shown in Table 3. In this case, the optimized radiation pattern has a higher SNR value compared to failed radiation pattern. This method could not meet the objective where optimized weights in a planar array should generate an optimized radiation pattern that matches the reference radiation pattern with a lower SNR value than the failed radiation pattern SNR.
(a)   Results of the GA method with a similar type of failure are shown in Figure 8a-c. The optimized weights for this Type 1 failure are shown in Table 4. A lower SNR value was observed at the optimized radiation pattern, but the main lobe level is compromised. For failure Types 2, 3, 6, and 8, this method performed less accurately, as described in Table 2.  Results of the GA method with a similar type of failure are shown in Figure 8a-c. The optimized weights for this Type 1 failure are shown in Table 4. A lower SNR value was observed at the optimized radiation pattern, but the main lobe level is compromised. For failure Types 2, 3, 6, and 8, this method performed less accurately, as described in Table  2.
(a)   The PSO method results for the same type of failure are shown in Figure 9a-c. The optimized weights using this method are shown in Table 5. Here, the method has matched the SNR value of the optimized radiation pattern with the reference radiation pattern. The main lobe level is also maintained. From an SNR value of 19 in the failed radiation pattern, the method has successfully reduced the value to 15 in the optimized radiation pattern.  The PSO method results for the same type of failure are shown in Figure 9a-c. The optimized weights using this method are shown in Table 5. Here, the method has matched the SNR value of the optimized radiation pattern with the reference radiation pattern. The main lobe level is also maintained. From an SNR value of 19 in the failed radiation pattern, the method has successfully reduced the value to 15 in the optimized radiation pattern.
(a)   Next, the results of a similar type of failure with the PS method are shown in Figure 10a-c. The optimized weights for this Type 1 failure are shown in Table 6. This method has also reduced the failed radiation pattern SNR value after optimization without compromising the main lobe level. The failed SNR value of 19 was reduced to an SNR value of 17 for the optimized radiation pattern.   Figure  10a-c. The optimized weights for this Type 1 failure are shown in Table 6. This method has also reduced the failed radiation pattern SNR value after optimization without compromising the main lobe level. The failed SNR value of 19 was reduced to an SNR value of 17 for the optimized radiation pattern.
(a)    Table 7 shows a compilation of reference, failed, and optimized SNR values for eight types of failures with two different array sizes, namely 8 × 8 and 6 ×6 planar arrays for all four optimization methods. explorations, PSO searches for the best solution space while GA searches via crossover [23]. This may lead to PSO providing the best solution for a problem. On the other hand, pattern search uses a directed move in a search space, while GA uses a sample search space and many generations. PS finds the best performance for a comparatively smaller search space and less computing time than GA [24]. In terms of time for the optimizations to converge, SA took 1439.593 s, PS took 1520.216 s, GA took 8120.882 s, and PSO took 9079.376 s on average. SA and GA took shorter time periods to converge, but the methods failed to meet the objective. On the other hand, the methods that provide expected results, PS and PSO, have a huge difference among them in terms of the time to converge. For real-time implementation, the optimization can be programmed in the 'C' language and processed in real-time on a Digital Signal Processor (DSP) embedded processor platform. This will make the process faster. The PSO method is suggested for better accuracy in matching the SNR value of the reference radiation pattern.
This summarizes that the PSO method provides a better solution for failed planar array radiation pattern correction with the highest SNR reduction with a higher time to converge. It is also shown that the weights of the remaining working elements in a planar array can be optimized to produce a radiation pattern with a reduced SNR value to enhance the efficiency of a system with the presence of faulty elements in an array without compromising main lobe levels.

Conclusions
The objective of this research was to investigate the performance of different optimization methods for selecting the most suitable techniques for recovering far-field radiation. The findings indicate that the SA and GA methods are not suitable for failed planar array radiation pattern correction. The PSO method provided the highest SNR reduction (1 to 10 dB) among all the methods. The PS method also provided an ideal solution for this case where it managed to reduce 2 to 5 dB of failed SNR values. The PSO and PS methods took 9079.376 s and 1520.216 s, respectively, to converge. Next, users can compare cost and time factors upon applying these optimization methods in real life to correct failed planar array correction.
Both PSO and PS methods successfully find optimum weights of the remaining working element to provide a beam pattern similar to the beam pattern without any failure in the array. The optimized weights with failed elements can successfully provide a beam pattern that gives reduced SNR compared to failed patterns that closely match the reference radiation pattern.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.