A Multiobjective Array Beamforming Method for Arrays of Flexible Shape
Abstract
:1. Introduction
2. Array Beamforming
3. Beamforming Algorithms
3.1. Problem Description
3.2. Classical PSO
3.2.1. Particle Swarm Optimization
3.2.2. Parameters in PSO
- Inertia weight, w, which is linearly decreasing with the iteration number, is defined as,
- Two random parameters, and are uniformly distributed values in the interval .
- Swarm size, is also named population size or the number of particles, which is usually set between 70 and 100 to be safe. For higher-dimensional problems, PSO often demonstrates a better performance with a larger swarm size [36].
3.2.3. Position and Velocity Initialization
3.2.4. Position and Velocity Bounds
3.3. Modified PSO Algorithm
3.3.1. Establishing Initialization Strategy
- First, work out the weights following a Taylor distribution, the entries required comprise the total number of array elements and the desired peak sidelobe level;
- Obtain the initial particle positions by adding the Taylor weights with random values, and the calculation is described in detail in the following section;
- Assemble the particle swarm using the initial particle positions, ready to carry out the optimizing iterations.
3.3.2. Procedure for the Modified PSO
Algorithm 1: Modified PSO |
Input: array elements N, , , fitness function , position upper bound , position lower bound , Output: Optimized
|
3.4. Fitness Function Definition
4. Numerical Analysis
4.1. One Main Lobe
4.1.1. Single Null
4.1.2. Multiple Nulls
4.1.3. Broad Nulls
4.2. Multiple Beams
4.2.1. Two Main Lobes
4.2.2. Thirteen Main Lobes
4.3. Modified PSO versus Classic PSO
4.4. Beamwidth
5. Results and Discussion
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Scenario | Array Elements | Null Number or Type | Main Lobe Number | DNDL | DSLL |
---|---|---|---|---|---|
1 | 16 | 1 | 1 | −70 dB | −20 dB |
2 | 16 | 8 | 1 | −60 dB | −20 dB |
3 | 16 | range | 1 | −50 dB | −20 dB |
4 | 16 | 3 | 2 | −60 dB | −20 dB |
5 | 100 | 12 | 13 | −50 dB | −20 dB |
Angle (Degree) | Depth (dB) | Angle (Degree) | Depth (dB) |
---|---|---|---|
25 | −63.24 | 110 | −60.03 |
40 | −60.26 | 125 | −61.42 |
55 | −60.91 | 140 | −60.43 |
70 | −60.05 | 155 | −61.01 |
Angle (Degree) | Depth (dB) | Angle (Degree) | Depth (dB) | Angle (Degree) | Depth (dB) |
---|---|---|---|---|---|
23 | −50.09 | 72 | −50.49 | 120 | −50.35 |
36 | −49.98 | 84 | −50.04 | 132 | −50.94 |
48 | −58.07 | 96 | −53.02 | 144 | −50.00 |
60 | −55.53 | 108 | −52.61 | 157 | −49.99 |
Scenario | Average Fitness Value | Average Execution Time (s) | |||
---|---|---|---|---|---|
Classic PSO | Modified PSO | Classic PSO | Modified PSO | ||
1 | 100 | 2.089 | 0 | 67.97 | 3.97 |
2 | 200 | 2.729 | 0.093 | 129.83 | 85.28 |
3 | 100 | 3.239 | 0.027 | 64.97 | 39.12 |
4 | 100 | 2.009 | 0.016 | 46.82 | 33.89 |
Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 | |||||
---|---|---|---|---|---|---|---|---|
Modified PSO | Classic PSO | Modified PSO | Classic PSO | Modified PSO | Classic PSO | Modified PSO | Classic PSO | |
E1 | 0.64 + j0.60 | 0.57 + j0.68 | 0.26 + j0.37 | 0.16 + j0.01 | 0.29 + j0.42 | 0.35 + j0.35 | 0.26 + j0.15 | 0.39 + j0.98 |
E2 | 0.41 + j0.23 | 0.35 + j0.59 | 0.31 + j0.26 | 0.25 + j0.19 | 0.43 + j0.01 | 0.27 + j0.45 | 0.12 + j0.32 | 0.63 + j0.51 |
E3 | 0.61 + j0.36 | 0.48 + j0.39 | 0.38 + j0.31 | 0.43 + j0.47 | 0.43 + j0.46 | 0.76 + j0.36 | 0.11 + j0.12 | 0.32 + j0.96 |
E4 | 0.69 + j0.58 | 0.61 + j0.49 | 0.59 + j0.63 | 0.53 + j0.49 | 0.45 + j0.54 | 0.27 + j0.51 | 0.35 + j0.49 | 0.79 + j0.96 |
E5 | 0.61 + j0.59 | 0.59 + j0.63 | 0.63 + j0.62 | 0.38 + j0.51 | 0.80 + j0.77 | 0.75 + j0.58 | 0.48 + j0.70 | 0.92 + j0.79 |
E6 | 0.79 + j0.73 | 0.36 + j0.92 | 0.78 + j0.77 | 0.66 + j0.74 | 0.91 + j0.84 | 0.84 + j0.70 | 0.82 + j0.80 | 1.00 + j0.77 |
E7 | 0.79 + j0.82 | 0.54 + j0.64 | 0.83 + j0.88 | 0.80 + j0.79 | 0.87 + j0.72 | 0.78 + j0.72 | 0.90 + j0.66 | 1.00 + j0.51 |
E8 | 0.74 + j0.87 | 0.45 + j0.63 | 0.70 + j0.73 | 0.60 + j0.68 | 0.96 + j0.86 | 0.72 + j0.72 | 0.75 + j0.87 | 0.74 + j1.00 |
E9 | 0.77 + j0.84 | 0.68 + j0.89 | 0.71 + j0.75 | 0.53 + j0.65 | 0.86 + j0.81 | 0.61 + j0.87 | 0.84 + j0.89 | 0.90 + j0.83 |
E10 | 0.84 + j0.80 | 0.75 + j0.63 | 0.84 + j0.89 | 0.67 + j0.64 | 0.86 + j0.80 | 0.72 + j0.55 | 0.76 + j0.74 | 0.79 + j0.00 |
E11 | 0.76 + j0.73 | 0.37 + j0.86 | 0.77 + j0.75 | 0.73 + j0.67 | 0.84 + j0.87 | 0.62 + j0.46 | 0.45 + j0.86 | 0.80 + j0.91 |
E12 | 0.69 + j0.71 | 0.67 + j0.73 | 0.60 + j0.59 | 0.72 + j0.68 | 0.71 + j0.73 | 0.58 + j0.52 | 0.65 + j0.68 | 0.77 + j1.00 |
E13 | 0.55 + j0.47 | 0.49 + j0.36 | 0.59 + j0.69 | 0.58 + j0.48 | 0.43 + j0.72 | 0.68 + j0.43 | 0.71 + j0.37 | 0.58 + j0.29 |
E14 | 0.57 + j0.43 | 0.39 + j0.56 | 0.40 + j0.42 | 0.33 + j0.29 | 0.49 + j0.63 | 0.36 + j0.41 | 0.58 + j0.01 | 0.75 + j0.00 |
E15 | 0.39 + j0.40 | 0.12 + j0.27 | 0.30 + j0.33 | 0.41 + j0.37 | 0.04 + j0.49 | 0.48 + j0.41 | 0.26 + j0.37 | 0.51 + j0.76 |
E16 | 0.87 + j0.63 | 0.56 + j0.54 | 0.24 + j0.34 | 0.40 + j0.18 | 0.35 + j0.35 | 0.21 + j0.00 | 0.21 + j0.21 | 0.31 + j0.05 |
Element Number | FNBW (Degree) | HPBW (Degree) | Target SLL (dB) | Target Null Angle (Degree) | Obtained SLL (dB) | Obtained NDL (dB) |
---|---|---|---|---|---|---|
16 | 15.1 | 6.4 | −15 | 135 | −15.02 | −64.25 |
16 | 15.8 | 6.6 | −15 | 85 | −15.21 | −68.45 |
16 | 15.5 | 6.8 | −15 | 20 | −15.67 | −67.02 |
16 | 18.1 | 7.2 | −20 | 135 | −20.03 | −68.66 |
16 | 17.3 | 7.1 | −20 | 85 | −20.07 | −72.50 |
16 | 17.5 | 7.1 | −20 | 20 | −21.00 | −65.05 |
16 | 19.9 | 7.7 | −25 | 135 | −25.19 | −64.25 |
16 | 19.6 | 7.6 | −25 | 85 | −25.02 | −62.37 |
16 | 20.2 | 7.7 | −25 | 20 | −25.21 | −60.01 |
E1 | E2 | E3 | E4 | E5 | E6 | E7 | E8 | ||
---|---|---|---|---|---|---|---|---|---|
Analytic | Amplitude | 0.76 | 0.85 | 1.11 | 1.27 | 1.27 | 1.11 | 0.85 | 0.76 |
Determination | Phase (rad) | 8.94 | 6.46 | 3.89 | 1.25 | −1.47 | −4.26 | −7.12 | −10.03 |
Modified PSO | Amplitude | 0.63 | 0.69 | 0.75 | 0.75 | 0.68 | 0.60 | 0.42 | 0.42 |
Phase (rad) | 8.98 | 6.46 | 4.02 | 1.34 | −1.36 | −3.81 | −6.80 | −9.87 |
Ref. | Algorithm | Design | Function | Performance |
---|---|---|---|---|
[25] | GWO | Uniform linear array. | Optimal antenna positions and amplitudes for achieving minimum SLL along with null placement and suppression of the first side lobe. | Lower SLL was obtained; however, GWO is more computationally time-consuming than PSO. |
[26] | WOA | Linear aperiodic array. | WOA was applied to the synthesis of uniformly excited broadside linear aperiodic arrays with sidelobe suppression and null steering under constraints on beamwidth requirements. | The average fitness result of WOA is slightly higher than that of CLPSO [42]. WOA has fewer iterations than CLPASO [42] and higher iterations than IWO-WDO [15]. |
[11] | Mayfly Algorithm (MA) | Uniform and sparse linear array. | MA was applied for sidelobe suppression and null placement in the following two ways: by optimizing amplitudes while maintaining uniform spacing and by optimizing the antenna positions while assuming a uniform amplitude excitation. | MA is able to obtain a considerable improvement in peak SLL suppression and null control. However, MA requires a longer computation time and more parameters than PSO. |
[15] | Hybrid invasive weed optimization and wind-driven optimization (IWO/WDO) | Linear sparse array. | The proposed algorithm is implemented to synthesize the uniformly excited linear sparse array pattern having an SLL and null control with a constraint on beamwidth by optimizing the element position only. | The Hybrid algorithm has improved performance in terms of null control, beamwidth control, and the rate of convergence. The proposed algorithm requires fewer iterations for convergence compared to PSO. |
[43] | Cuckoo Search (CS) | Large scale concentric circular antenna array (CCAA). | A hybrid approach to suppress the SLL of large-scale CCAA is proposed, which includes an improved discrete cuckoo search algorithm (IDCSA) for thinning the CCAA and a cuckoo search-invasive weed algorithm (CSIWA) for further optimizing the thinned CCAA. | The proposed IDCSA and CSIWA can obtain a lower maximum SLL but do not have advantages in terms of processing time. |
This work | Modified PSO | Uniform linear array. | A modified PSO algorithm is proposed for optimizing multiple objectives in radiation pattern, including SLL, null, and main lobe direction, by optimizing the excitation phase and amplitude of the elements. | The proposed algorithm has shown good performance compared to classical PSO in multiple design scenarios, and its effectiveness was verified with a curved array. |
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Meng, C.; Zhang, Y.; Temiz, M.; El-Makadema, A. A Multiobjective Array Beamforming Method for Arrays of Flexible Shape. Electronics 2024, 13, 752. https://doi.org/10.3390/electronics13040752
Meng C, Zhang Y, Temiz M, El-Makadema A. A Multiobjective Array Beamforming Method for Arrays of Flexible Shape. Electronics. 2024; 13(4):752. https://doi.org/10.3390/electronics13040752
Chicago/Turabian StyleMeng, Chenfeng, Yongwei Zhang, Murat Temiz, and Ahmed El-Makadema. 2024. "A Multiobjective Array Beamforming Method for Arrays of Flexible Shape" Electronics 13, no. 4: 752. https://doi.org/10.3390/electronics13040752