To extend the GMRT, it is proposed to add 15 antennas for cost-effectiveness and to realize the scientific goals. In order to add the antennas to the current array, the position of each antenna using rectangular coordinates (x, y, z) is calculated using:

X = R × sin θ × cos φ

Y = R × sin θ × sin φ (1)

Z = R × cos θ

where R, θ, and φ are the latitude, longitude, and radial distance R (equal to about 6378.1 km, the value of earth’s radius), respectively. The extended configuration is placed at 110 km along the arms of the GMRT. The declination of the radio frequency source of interest for all configurations is the same (and is equal to 45˚). Spiral configurations follow the theory of a logarithmic spiral [1] .

Results in [1] show that the spiral antenna array configuration provides low SLL and good u-v coverages and therefore achieves high image resolution. Therefore, the following equations are used to calculate the gridded cells in u-v domain:

N grid = ( A t A ) × ( nosmp ) (2)

Δ u = u max − u min N grid (3)

Δ v = v max − v min N grid (4)

where nosmp, A t , A , and N grid are the number of samples in the snapshot or hour tracking observation, the total desired area, the covered area by the current configuration, and the number of gridded cells, respectively.

And Δ u , Δ v , u max , u min , v max , v min and N grid are the dimension of the cell in the u direction, the dimension of the cell in the v direction, the maximum value of u in the spatial frequency domain, the minimum value of u in the spatial frequency domain, the maximum value of v in the spatial frequency domain, the minimum value of v in the spatial frequency domain and the number of gridded cells as defined in Equation (2), respectively [1] .

Also, results in [2] show the good performance of the GA configuration in terms of SLL and sampling. The algorithm provides an optimum solution for both a snapshot observation and a 6-hour tracking observation with minimum values of overlapping. This happens due to the discrete grid locations of the plane as defined in:

N grid = nearest ( n × ( n − 1 ) × A total / A first ) (5)

where n is the number of antennas, A_total is the total area, A_first is the first calculated area of tracking observation from the first random population, and nearest rounds N_grid to the nearest integer. This selection helps distribute the samples in the u-v plane with less overlapping in both snapshot and hour tracking simultaneously [1] .

The new algorithm (DA), explained in [3] , shows good results in optimizing an interferometric array. Therefore, these three methods are used in this section to investigate an optimum configuration, in order to extend an interferometric array.

To evaluate the position of each antenna using GA and DA methods, the fitness function elaborated in [2] is used:

fitness ( k ) = − o l s a m ( k ) max ( o l ) − D ( k ) R − ( A tofsam − A sam ( k ) ) A tofsam (6)

where: fitness (k) is the fitness value of k^th baseline in n^th generation. ol_sam(k) indicates the number of overlapped samples generated by k^th baseline in n^th generation. Max(ol) provides the maximum value of overlapping that is in n^th generation. D(k) is the sample distance from the grid center. R is the gridded cell radius. A_sam(k) is the calculated area generated by k^th baseline’s samples in n^th generation. A_tofsam is the total area generated by n^th generation [2] .

Finally, the mean SLL for all the aforementioned methods are calculated using the following equation [1] :

mean S L L = − mean ( first S L L + second S L L + third S L L ) (7)

Malaysia has started to fund major research astronomy projects recently with two telescopes. In this work, we apply the proposed theory of localization using an array of antennas for astronomy applications to suppress the SLLs and/or increase the number of samples in the u-v plane coverage for the future array configuration in Malaysia.

We expect that the proposed methods can optimize the number of data samples and minimize the side lobe levels in the angular domain to enhance the image quality as much as possible. The methods discussed in this study are applied to 10 antennas which will be placed at the coordinates of the antennas in Malaysia.

The materials and methods used in this section are taken from [2] [3] . Here, the GA and DA algorithms are used to investigate the optimum solutions for ten antennas in Malaysia.

To expand the existing array, 15 antennas are added for the following configurations: 1) extended Y-shaped; 2) spiral; 3) 25^th generation using GA; 4) 150^th generation using GA; and 5) DA. Figure 1 shows the different configurations. The u-v plane coverage achieved from these different configurations are shown in Figure 2 and Figure 3 for the snapshot and 6-hour tracking observations, respectively. The results are summarized in Figures 1-3, and Table 1 & Table 2.

In the first step, extension of the Y-shaped array is investigated. In this configuration, 15 antennas are located along the three arms of the GMRT. Five antennas are added in each arm and broadened around to 110 km. This array gets a Y-shape to investigate the effect of extending the arms in the original Y-shape. This new arrangement, its snapshot observation, and 6-hour image synthesis are shown in Figure 1(a), Figure 2(a), and Figure 3(a), respectively.

Notes: 6-hour is used in hour tracking synthesize.

Notes: 6-hour is used in hour tracking synthesize.

Table 1 shows the number of overlapped samples using snapshot as well as for a 6-hour tracking observation. This parameter is valued at 89.19% using the snapshot and 88.89% for a 6-hour tracking observation. The unsampled cells ratio achieved with this configuration (expanded GMRT) are 78.67% using the snapshot observation and 75.19% for a 6-hour tracking observation.

As shown in Figure 2(a), for the u-v plane coverage using the snapshot observation, the linear ridges are not smooth, and this configuration provides a poor sensitivity and consequently a poor signal to noise ratio [1] . Therefore, the linear ridges need to be smoothened. In order to perform this requirement, the arms are curved in the next arrangement as is shown in Figure 2(b).

Arms in the spiral configuration cover the angular position of 60 to 96, 180 to 216 and 300 to 336 degrees. The angular positions of five added antennas in each arm are 62 to 70, 182 to 190 and 302 to 310 degrees. The overlapped sample ratio is equal to 86.77% for the spiral using the snapshot observation. Similarly, it is equal to 85.51% for a 6-hour tracking observation. The unsampled cells ratio from the u-v coverage indicate the percentage of the cells without any sample. This ratio is equal to 72.56% using the snapshot (see Figure 2(b)) and 67.59% for a 6-hour tracking observation (see Figure 3(b) and Table 1). This means the spiral configuration is able to sample the Fourier space of the image better than the extended Y-shape. In comparison to the extended GMRT, the lower ratios of overlapped samples and unsampled cells resulting from the spiral suggest that this configuration provides more information about the source due to a greater number of samples in the u-v plane. Therefore, the configuration of the spiral provides a better u-v plane coverage in both types of observations in comparison to the extended GMRT’s as shown in Figure 2(b) and Figure 3(b).

In the next step, the GA is applied to this extended interferometric array, to investigate the effect of the algorithm.

From the results achieved using the previous configurations (see Figure 1(a) and Figure 1(b)), the data in the spatial frequency domain needs to be spread out to get less overlapping. Therefore, the GA is applied to work on sampled data using the snapshot to provide the desired resolution.

For the GA, the optimum ratio values of mutation and crossover (25% mutation ratio and 25% crossover ratio) are used in the proposed algorithm. The number of antennas (chromosomes) is set to 45 and for each the diameter is equal to 45 m to assess the efficiency of the algorithm. To distribute the sampled data in u-v plane coverage more smoothly, the algorithm works on 150 generations to optimize the image resolution.

The position of the antennas (chromosomes) in the array for the 25^th, and the 150^th generations using the GA are shown in Figure 1(c) and Figure 1(d), respectively. The snapshot in the u-v plane coverage for the 25^th, and the 150^th generations that are shown in Figure 2(c) and Figure 2(d). These illustrate how the sampled data distribution are from the results using the extended GMRT. As can be observed qualitatively, the data in the 150^th generation improves the distribution of the data somewhat more evenly. The calculated ratios shown in Table 1 summarize the ability of the GA (when using snapshot) in distributing the samples and obtaining more samples rather than the extended GMRT.

The calculated ratios in Table 1 show that the GA can sample data with less overlapped data using the snapshot observation. Specifically, for the 150^th generation the ratio is equal to (85.05%), in comparison to 89.19% for the extended GMRT. This indicates that as the algorithm works with more generations, it distributes sampled data in the u-v plane coverage more efficiently than extended Y-shaped.

The calculated ratio of overlapped samples for a 6-hour tracking is shown in Table 1. From the results, this value is improved from 88.89% to 86.58%, and then to 84.76% using the extended GMRT, the 25^th, and the 150^th generation configurations, respectively. It means the algorithm improves the ratio of overlapped samples since it works with a greater number of generations.

When the number of generations goes up, the number of unsampled cells is lower; specifically, this percentage is 78.67% using the extended GMRT configuration and becomes 76.34%, and 74.56% values using the 25^th, and 150^th generations using the snapshot, respectively. This ratio is equal to 75.19% using the extended GMRT observation and is equal to 70.74% and 66.65% values using the 25^th, and the 150^th generations for 6-hour tracking observations, respectively.

In the last step, the DA is applied to this extended interferometric array, to investigate the effect of the algorithm. The algorithm provides an optimum solution for both the snapshot and for the hour-tracking observations with minimum ratios of overlapping.

The position of the antennas, u-v coverages at snapshot and 6-hour tracking observations in the array using DA are shown in Figure 1(e), Figure 2(e), and Figure 3(e), respectively. The calculated parameters shown in Table 1 express the ability of the DA in distributing the samples and obtaining more samples rather than the extended GMRT’s at the snapshot.

The results calculated in Table 1 show that the DA can achieve the sampled data with less overlapped data at snapshot observation among all discussed configurations in this study (79.45%). The calculated overlapped samples ratio for a 6-hour tracking is shown in Table 1. From the results, this value is equal to 83.87%. It means the algorithm improves the overlapped number of samples more efficiently than GA. When the number of generations goes up, the unsampled cells get reduced; specifically, this percentage is 72% at snapshot and 63.76% for a 6-hour tracking observation.

The calculated first SLL, mean of the first three SLL (mean SLL), and the worst SLL defined as the peak SLL are shown in Table 2. The calculated SLLs show that the spiral geometry has lower side lobes (SLL = −11.25 dB and mean SLL = −12.46 dB, and peak SLL = −10.86 dB) than the extended GMRT (SLL = −11.18 dB and mean SLL = −11.97 dB, and peak SLL = −10.59 dB) and the linear ridges using the snapshot is also smoother than for the extended GMRT (see Figure 2(b)). As such, the spiral configuration smoothens the linear ridges.

The first and the mean values of the first three SLLs of 25^th and 150^th generation using GA are also calculated Table 2. The first SLL is −11.18 dB in extended GMRT and −11.88 dB, and −12.04 dB for the 25^th, and the 150^th generation configurations, respectively. The calculated mean value of the first SLL shown in Table 2 depicts that the configurations have the mean SLL of −13.85 dB and −13.88 dB at the 25^th, and the 150^th generation, respectively (this ratio is valued at −11.97 dB using the extended GMRT). The algorithm is also able to decrease the level of the peak SLL as the number of generations goes up (this ratio is −12.55 dB and −13.75 dB using 25^th and 150^th generation configurations, respectively). The calculated first SLL, mean SLL, and peak SLL values of DA configuration are −19.3 dB, −16.86 dB, and −14.8 dB, respectively. These values show the better efficiency of the DA in comparison to the GA.

The evolution of the average fitness in each generation for the u-v plane coverage using the snapshot observation and SLL reduction are shown in Figure 4. As shown in Figure 4(a), the optimum solution occurred at the 47^th generation for the first 50 generations, and the optimum value in the range of 82^th to 150^th

generation remains constant. The fitness value for the first 50 generations in Figure 4(b) obtains the optimum solution at 23^rd generation. Since the algorithm seeks the solutions randomly, it provides different solutions each time. The aim of this study is to demonstrate optimum solutions for an extended interferometric array. It is not written for any specific constraints for astronomy applications; however, it can work with different constraints.

From the results obtained from the different configurations in this study, a spiral shape suggests good results in the angular domain as well as in the u-v domain and the GA and DA provide improved SLLs, and u-v plane coverages.

The simulated source declination is 45 degrees. The duration for the fully synthesized observation is a 6-hour tracking with 10 minutes time interval between each two samples.

As a first step, the GA is applied to an interferometric array, to investigate the effect of the algorithm.

The algorithm provides an optimum solution for both the snapshot and the hour-tracking observations with minimum values of overlapping (this happens due to the grounding of the plane as defined in (2) and (3)).

The optimum ratio values of mutation and crossover (25% mutation ratio and 25% crossover ratio) are used in the algorithm. The number of antennas (chromosomes) is fixed to 10. To distribute the sampled data in u-v plane coverage more smoothly, the u-v plane is gridded with the dimension of D_u × D_v as defined in (3) and the algorithm works on 150 generations to optimize the resolution of the image.

The positions of the antennas (chromosomes) in the array of the 25^th and 150^th generations using the GA are shown in Figure 5(a) and Figure 5(b), respectively. Using the snapshot, the u-v plane coverages for the 25^th, and 150^th generations in Figure 6(a) and Figure 6(b) and 6-hour tracking in Figure 7(a) and Figure 7(b) show how the sampled data is distributed using 25^th generation gradually to 150^th generation.

The calculated parameters shown in Table 3 express the ability of the GA to distribute the samples and obtaining more samples as the number of generations goes up at the snapshot. The calculated results summarized in Table 3 show that the GA achieves the sampled data with less overlapped data at snapshot observation from the 25^th generation (12%) to the 150^th generation (2%). This indicates

Notes: 6-hour is used in hour tracking synthesize.

that as the algorithm works with more generations, it can distribute sampled data in the u-v plane coverage more efficiently.

The calculated overlapped samples ratio for a 6-hour tracking is shown in Table 3. From the results, this value is varied from 13.34% to 2.23% using the 25^th and the 150^th generation configurations, respectively. It means the algorithm improves the overlapped samples as it works with more numbers of generations.

Since the number of generations goes up, the number of unsampled cells are decreased; specifically, the ratio is 94.89% using the 25^th generation observation and becomes 94.00% using the 150^th generations at the snapshot. This ratio is equal to 93.23% using the 25^th generation observation and becomes 92.49% using the 150^th generation for a 6-hour tracking observation.

Finally, the DA is applied to the extended interferometric array, to investigate the effect of the algorithm. The algorithm provides an optimum solution for both the snapshot and the hour-tracking observations with minimum values of overlapping (this happens due to the gridding of the plane as defined in (2) and (3)).

The position of the antennas, the u-v coverage for the snapshot observation and for a 6-hour tracking observation in the array using the DA are shown in Figure 6(c), Figure 7(c), and Figure 8(c), respectively. The calculated ratios shown in Table 3 express the ability of the DA in distributing the samples and obtaining more samples rather than the GA for the snapshot.

The calculated results that are summarized in Table 3 show that the DA can achieve the sampled data with the same efficiency as the GA for the snapshot observation. The calculated overlapped samples ratio for a 6-hour tracking is shown in Table 3. From the results, this ratio is valued at 22.28%. It means the algorithm improves the overlapped samples more efficiently than GA for a 6-hour tracking observation.

Since the number of generations goes up, the number of unsampled cells is reduced; specifically, this percentage is 91.56% at snapshot and 92.16% for a 6-hour tracking observation, which show the same efficiency of the GA.

The calculated first SLL, mean values of the first three SLLs (mean SLL), and the worst SLL defined as the peak SLL are shown in Table 4. The values of the

Notes: 6-hour is used in hour tracking synthesize.

first SLL, mean SLL, and peak SLL are −6.99 dB, −6.75 dB, and −3.03 using the 25^th generation, −5.23 dB, −8.45 dB, and −5.2 dB using the 150^th generation, and −10 dB, −9.86 dB, and −5.88 dB for the DA array, respectively. It shows that even though the GA decreases the SLL, the DA gains reasonable SLL and optimum ratios in the spatial frequency domain, with the same population.

The evolution of the average fitness in each generation for the u-v plane coverage at the snapshot and SLL reduction is shown in Figure 8. As shown in Figure 8(a), for the first 50 generations, the optimum solution is reached at the 38^th generation. In contrast, for a total of 150 generations, a better solution is obtained at the 112^th generation. Similarly, the fitness value for the first 50 generations in Figure 8(b) obtains the optimum solution at 43^th generation. Since, the algorithm seeks for the solutions randomly; it provides different solutions for each generation.