Enhancing Beamforming Efficiency: Utilizing Taguchi Optimization and Neural Network Acceleration

1. Introduction

Table 1. Comparison of Antenna Array Radiation Pattern Synthesis Methods [24,25,26,27].

2. Phased Antenna Arrays

3. Taguchi Method

3.1. Amplitude Synthesis of Antenna Array Radiation Pattern under Constraints

In this type of synthesis, constraints are imposed on the levels of sidelobes and the locations of nulls in the radiation pattern, which the beam towards interfering signals should exhibit. In this initial synthesis type, the Taguchi optimization procedure will be applied to three linear antenna arrays, each with a different number of sources (10, 16, and 24), in order to minimize sidelobes [46,49].

3.1.1. 10-Element Antenna Array

The array factor is described by the following equation.

$$AF\left(\theta \right)=2\sum _{n=1}^{5}a\left(n\right)e^{j\varphi \left(n\right)}cos\left[\beta d\left(n\right)sin\left(\theta \right)\right]$$

(10)

To minimize sidelobe levels, the fitness function is chosen based on the optimization objective.

$$fitness=min\left(max\left\{20log\left|AF\left(\theta \right)\right|\right\}\right)$$

(11)

subject to constraints $\theta \in \left[0^{\circ },{76}^{\circ }\right],\left[{104}^{\circ },{180}^{\circ }\right]$

To simplify the task and enhance understanding of this application, let’s elaborate on the Taguchi optimization procedure:

First Step: Initialization Problem

The first step involves selecting an appropriate experimental design and a well-structured fitness function. There are 5 parameters that need optimization. Therefore, the chosen orthogonal experimental design should have 5 columns ($k=5$) to represent these parameters. To capture nonlinear effects, three levels ($s=3$) for each parameter are deemed sufficient. Typically, an orthogonal experimental design with a strength of 2 ($t=2$) is effective for most problems. In summary, an orthogonal experimental design with 5 columns, 3 levels, and a strength of 2 is required.

Second Step: Designating Input Parameters

After accessing the online database OA , an orthogonal array (OA) (27, 5, 3, 2) is available, as indicated in Table 2. The fitness function is chosen based on the optimization objective. In this optimization example, the function is selected to achieve a low sidelobe level. The input parameters need to be selected for conducting experiments. When the orthogonal experimental design is chosen, numerical values corresponding to the three levels of each input parameter must be determined in the first iteration. The value for level 2 is selected at the midpoint of the optimization range. The values for levels 1 and 3 are calculated using the following equations.

$$D{N}_1=\frac{A_{max}+A_{min}}{S+1}$$

(12)

$$A{\left(n\right)}_2^1=\frac{A_{max}+A_{min}}{2}$$

(13)

$$A{\left(n\right)}_1^1=A{\left(n\right)}_1^2-D{N}_1$$

(14)

$$A{\left(n\right)}_3^1=A{\left(n\right)}_1^2+D{N}_1$$

(15)

Numerical Application:

$$D{N}_1=\frac{1+0}{3+1}=0.25$$

(16)

$$A{\left(n\right)}_2^1=\frac{1+0}{2}=0.5$$

(17)

$$A{\left(n\right)}_1^1=0.5-0.25=0.25$$

(18)

$$A{\left(n\right)}_3^1=0.5+0.25=0.75$$

(19)

Table 2. Taguchi Experimental Design (N=27, K=5, S=3, t=2)

Table 2. Taguchi Experimental Design (N=27, K=5, S=3, t=2)

Experiment	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{a}}_5$
1	1	1	1	1	1
2	2	1	2	2	2
3	3	1	3	3	3
4	1	2	1	2	2
5	2	2	2	3	3
6	3	2	3	1	1
7	1	3	1	3	3
8	2	3	2	1	1
9	3	3	3	2	2
10	1	1	2	1	2
11	2	1	3	2	3
12	3	1	1	3	1
13	1	2	2	2	3
14	2	2	3	3	1
15	3	2	1	1	2
16	1	3	2	3	1
17	2	3	3	1	2
18	3	3	1	2	3
19	1	1	3	1	3
20	2	1	1	2	1
21	3	1	2	3	2
22	1	2	3	2	1
23	2	2	1	3	2
24	3	2	2	1	3
25	1	3	3	3	2
26	2	3	1	1	3
27	3	3	2	2	1

Using these equations, Table 2 can be converted into numerical values as shown in Table 3. $a{\left(n\right)}_1^2=0.5;a{\left(n\right)}_1^3=0.75;a{\left(n\right)}_1^1=0.25$

Table 3. The numerical values of the levels in the first iteration.

Table 3. The numerical values of the levels in the first iteration.

Experiment	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{a}}_5$
1	0.25	0.25	0.25	0.25	0.25
2	0.5	0.25	0.5	0.5	0.5
3	0.75	0.25	0.75	0.75	0.75
4	0.25	0.5	0.25	0.5	0.5
5	0.5	0.5	0.5	0.75	0.75
6	0.75	0.5	0.75	0.25	0.25
7	0.25	0.75	0.25	0.75	0.75
8	0.5	0.75	0.5	0.25	0.25
9	0.75	0.75	0.75	0.5	0.5
10	0.25	0.25	0.5	0.25	0.5
11	0.5	0.25	0.75	0.5	0.75
12	0.75	0.25	0.25	0.75	0.25
13	0.25	0.5	0.5	0.5	0.75
14	0.5	0.5	0.75	0.75	0.25
15	0.75	0.5	0.25	0.25	0.5
16	0.25	0.75	0.5	0.75	0.25
17	0.5	0.75	0.75	0.25	0.5
18	0.75	0.75	0.25	0.5	0.75
19	0.25	0.25	0.75	0.25	0.75
20	0.5	0.25	0.25	0.5	0.25
21	0.75	0.25	0.5	0.75	0.5
22	0.25	0.5	0.75	0.5	0.25
23	0.5	0.5	0.25	0.75	0.5
24	0.75	0.5	0.5	0.25	0.75
25	0.25	0.75	0.75	0.75	0.5
26	0.5	0.75	0.25	0.25	0.75
27	0.75	0.75	0.5	0.5	0.25

Third Step: Conducting Experiments and Building a Response Table

After determining the input parameters, the fitness function for each experiment can be calculated. For instance, the fitness value for experiment 1 (i.e., the first row of Table 3) is computed using Equation (14), resulting in 12.97. Then, the fitness value in the Taguchi method is converted to signal-to-noise ratio (S/N), denoted as ( $\eta$), using formula (14). The corresponding fitness values and the S/N ratios are listed in Table 4. These results are then used to construct a Response Table 4 for the first iteration by averaging the $S/N$ ratios for each parameter." The corresponding signal-to-noise ratio values are listed in Table 5. These results are then used to construct a Response Table (Table 5) by averaging the signal-to-noise ratios ( $S/N$) for each parameter using the following equation (3.9)- [46,49].

$$\overline{\eta }(m,n)=\frac{1}{N}{\int }_{i,OA(i,n)=m}^{ }\eta \left(i\right)$$

(20)

• n: Number of parameters
• m: Number of levels (1, 2, 3)
• N: Number of level combinations
• i: i-th iteration

For example, the average of the ratios (S/N) for $A\left(4\right){\mid }_1^2$ and $A\left(5\right){\mid }_1^3$ is.

$$\overline{\eta }(2,4)=\frac{1}{9}{\int }_{i,OA(1,4)=2}^{ }\eta \left(1\right)=-20.03dB$$

(21)

$$\overline{\eta }(3,5)=\frac{1}{9}{\int }_{i,OA(1,5)=3}^{ }\eta \left(1\right)=-19.24dB$$

(22)

Table 4. Experiments and Fitness Values.

Table 4. Experiments and Fitness Values.

Experiment	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{a}}_5$	Fitness R	(S/N)
1	0.25	0.25	0.25	0.25	0.25	12.97	-22.26
2	0.5	0.25	0.5	0.5	0.5	11.19	-20.98
3	0.75	0.25	0.75	0.75	0.75	10.56	-20.47
4	0.25	0.5	0.25	0.5	0.5	9.91	-19.92
5	0.5	0.5	0.5	0.75	0.75	9.70	-19.73
6	0.75	0.5	0.75	0.25	0.25	13.86	-22.83
7	0.25	0.75	0.25	0.75	0.75	8.67	-18.76
8	0.5	0.75	0.5	0.25	0.25	15.53	-23.82
9	0.75	0.75	0.75	0.5	0.5	16.81	-24.51
10	0.25	0.25	0.5	0.25	0.5	9.32	-19.39
11	0.5	0.25	0.75	0.5	0.75	9.31	-19.38
12	0.75	0.25	0.25	0.75	0.25	7.61	-17.63
13	0.25	0.5	0.5	0.5	0.75	8.28	-18.36
14	0.5	0.5	0.75	0.75	0.25	9.88	-19.90
15	0.75	0.5	0.25	0.25	0.5	10.99	-20.82
16	0.25	0.75	0.5	0.75	0.25	9.03	-19.12
17	0.5	0.75	0.75	0.25	0.5	13.93	-22.88
18	0.75	0.75	0.25	0.5	0.75	11.27	-21.04
19	0.25	0.25	0.75	0.25	0.75	6.84	-16.70
20	0.5	0.25	0.25	0.5	0.25	10.13	-20.11
21	0.75	0.25	0.5	0.75	0.5	9.70	-19.73
22	0.25	0.5	0.75	0.5	0.25	8.26	-18.34
23	0.5	0.5	0.25	0.75	0.5	10.97	-20.81
24	0.75	0.5	0.5	0.25	0.75	10.95	-20.78
25	0.25	0.75	0.75	0.75	0.5	8.28	-18.36
26	0.5	0.75	0.25	0.25	0.75	7.90	-17.96
27	0.75	0.75	0.5	0.5	0.25	21.51	-26.65

To identify the optimal level value for each parameter, it is necessary to find the highest signal-to-noise (S/N) ratio within each column of Table 4. This is highlighted in Table 5. Once the optimal levels are identified, a confirmation test is conducted using the corresponding numerical values of these optimal levels in the response table, Table 6 [46,49,84].

Table 5. Response table (in decibels) after the first iteration-Element Levels (dB).

Table 5. Response table (in decibels) after the first iteration-Element Levels (dB).

Elements	1	2	3	4	5
Level 1	-19.02	-19.63	-19.92	-20.83	-21.18
Level 2	-20.62	-20.17	-20.95	-21.03	-20.82
Level 3	-21.61	-21.46	-20.38	-19.39	-19.24

If the results from the current iteration do not meet the termination criteria, which are examined using Equation (23), and after determining the optimal level values from the current iteration that are used as central values for the next iteration (24) in our case, the central values for the next iteration are $a{\left(1\right)}_1^1$;$a{\left(2\right)}_1^1$;$a{\left(3\right)}_1^1$;$a{\left(4\right)}_1^1$;$a{\left(5\right)}_1^1$; the optimization process is repeated in the subsequent iteration, reducing the optimization interval using Equation (25).

$$\frac{D{N}_{i+1}D{N}_1}{<}0.01$$

(23)

$$a_n{\mid }_{i+1}^2=a_n{\mid }_i^{opt}$$

(24)

$$D{N}_{i+1}=RR\left(i\right)\times D{N}_1=r{r}^i\times D{N}_1$$

(25)

In this optimization example, the reduced function ($rr$) is set to $0.9$.

Table 6. Optimized values of $a_i$ and maximum SLL for networks

Table 6. Optimized values of $a_i$ and maximum SLL for networks

Optimized values of ${\mathit{a}}_{\mathit{i}}$	Maximum SLL
1.0000	-13.1526
0.8413	-22.8982
0.9322
0.7675
0.6049
0.5715
0.4746
0.4877

This first type of synthesis (amplitude synthesis under constraint) aims to minimize the secondary lobes as much as possible, using our optimization technique. In Figure 3(a), the obtained results demonstrate successful minimization of the secondary lobe level (SLL or Side-Lobe-Level) down to $-25.2722$ dB, where the gain is approximately $12.3$ dB higher compared to the secondary lobe level of the uniform linear antenna array (SLL = $-12.9651$ dB). The optimization objective is achieved after 80 iterations as indicated in Figure 3(b). The excitation weights of this antenna array optimized by the Taguchi method are mentioned in Table 7.

Table 7. Table of elements and weights.

Table 7. Table of elements and weights.

Elements	1	2	3	4	5
Weights	1.0000	0.8984	0.7187	0.5015	0.3857

By comparing the results obtained with those of PSO (Particle Swarm Optimization) (see Figure 4), we observe:

• A gain of 0.6 dB in terms of secondary lobes minimization.
• A convergence speed of 80 iterations for the Taguchi method.
• The real-time required for our digital optimization tool is approximately 10 seconds.

With this method, we aim to minimize the error criterion; this amounts to minimizing the difference between the desired pattern (template) and the pattern obtained through synthesis.

3.1.2. 16-Element Antenna Array

In this example, the array factor is described by the following Equation (26).

$$AF\left(\theta \right)=2\sum _{n=1}^{8}a\left(n\right)cos\left[\beta d\left(n\right)cos\theta \right]$$

(26)

The fitness function is chosen based on the optimization objective:

$$fitness=min\left(max\left\{20log\left|AF\left(\theta \right)\right|\right\}\right)$$

(27)

Under constraint $\theta \in \left[0^{\circ },{80}^{\circ }\right],\left[{100}^{\circ },{180}^{\circ }\right]$

To use the Taguchi method, the following steps [3.3]-[3.1] must be followed:

• Step 1: Determine the number of parameters ($k=8$).
• Step 2: Determine the number of levels ($s=3$).
• Step 3: Determine the strength ($t=2$).
• Step 4: Determine the OA experimental design ($27,8,3,2$).
• Step 5: Determine the reduced function ($rr=0.9$).
• Step 6: Determine the convergence value $=0.01$.

Figure 5 (a) shows that the level of side lobes (SLL or Side-Lobe-Level) is minimized down to (SLL=-31.3151 dB). The gain is approximately 18.17 dB compared to the level of side lobes of the uniform array (SLL=-13.148 dB).

• As shown in Figure 5(b), the optimization objective is achieved after 66 iterations.
• The excitation weights of this optimized antenna array using the Taguchi method are indicated in Table 8.
• The results obtained compared to those of PSO (Figure 5) show that:

  -

  A gain of 0.8 dB in terms of minimizing the side lobes.

  -

  A convergence speed of 66 iterations for the Taguchi method.

  -

  9 seconds as real-time required for our digital optimization tool.

Table 8. Table of elements and weights (Optimized weights by Taguchi method for $a\left(n\right)$ (16 elements)).

Table 8. Table of elements and weights (Optimized weights by Taguchi method for $a\left(n\right)$ (16 elements)).

Elements	1	2	3	4	5	6	7	8
Weights	1.0000	0.9500	0.8575	0.7317	0.5861	0.4381	0.2988	0.2552

3.1.3. 24-Element Antenna Array

In this example, the network factor is described by the following equation

$$AF\left(\theta \right)=2\sum _{n=1}^{12}a\left(n\right)cos\left[\beta d\left(n\right)cos\theta \right]$$

(28)

The fitness function is chosen based on the optimization objective:

$$fitness=min\left(max\left\{20log\left|AF\left(\theta \right)\right|\right\}\right)$$

(29)

Under constraint $\theta \in \left[0^{\circ },{82}^{\circ }\right],\left[{102}^{\circ },{180}^{\circ }\right]$

To use the Taguchi method, you need to go through the following steps:

• Step 1: Determine the number of parameters ($k=12$).
• Step 2: Determine the number of levels ($s=3$)
• Step 3: Determine the strength ($t=2$).
• Step 4: Determine the OA experimental design ($27,12,3,2$).
• Step 5: Determine the reduced function ($rr=0.8$).
• Step 6: Determine the convergence value $=0.001$.

By following the same procedure as for examples 1 and 2, the results obtained in this example can be summarized as follows:

• The optimized maximum SLL by the Taguchi method = -39.2263 dB (Figure 6 (a)).
• The gain = 25.2718 dB (since the level of side lobes of the uniform array is SLL = -13.9545).
• The number of iterations = 73 (Figure 6 (b)).
• The excitation weights of the optimized antenna array by the Taguchi method are listed in Table 9.
• The comparative study between the results obtained by the Taguchi method and those by the PSO method indicates a considerable gain of approximately 3.7 dB [46].

Table 9. Optimized weights by Taguchi method for an antenna array with 24 elements.

Table 9. Optimized weights by Taguchi method for an antenna array with 24 elements.

Elements	Weights
1	1.0000	7	0.5292
2	0.9717	8	0.4203
3	0.9171	9	0.3182
4	0.8399	10	0.2275
5	0.7454	11	0.1512
6	0.6397	12	0.1262

In summary, the results of the three examples lead us to conclude that there is a proportionality between the number of elements on one hand and the convergence speed on the other hand, such that the higher the number of elements, the faster the convergence speed will be.

4. Neural Networks for Synthesis and Optimization of Antenna Arrays

4.1. Taguchi-Neural Network Architectures

When integrating neural networks with the Taguchi method for antenna array optimization, a comprehensive understanding of the underlying mathematical framework is crucial [64,68,75]. The Taguchi method typically employs a loss function, often derived from the signal-to-noise ratio (SNR), aimed at optimizing antenna array performance. One common formulation is the smaller-the-better loss function:

$$L=-10{log}_{10}\left(\frac{1}{n}\sum _{i=1}^n{\left(\frac{1}{{\varphi }_i}\right)}^2\right)$$

(34)

Here, ${\varphi }_i$ represents the observed performance from individual experiments. This loss function encapsulates the overall quality of the antenna array, with lower values indicating better performance.

Neural networks are then utilized to predict the output performance $\widehat{\varphi }$ based on input parameters X and network parameters $\theta$. Mathematically, this relationship is expressed as $\widehat{\varphi }=f(X;\theta )$, where the function f encapsulates the mapping learned by the neural network.

The optimization objective is to minimize the loss function $L\left(X\right)$, subject to any parameter constraints. During the training phase of the neural network, optimization algorithms such as gradient descent are employed to update the network parameters $\theta$ iteratively. Specifically, the parameters are updated according to:

$$\theta :=\theta -\alpha \nabla J\left(\theta \right)$$

(35)

Where $\alpha$ represents the learning rate, and $\nabla J\left(\theta \right)$ denotes the gradient of the loss function with respect to the network parameters.

These equations form the backbone of the mathematical framework when combining neural networks with the Taguchi method for antenna array optimization. By leveraging neural networks, engineers can effectively design and optimize antenna arrays, leading to enhanced performance and efficiency in various applications [70,83,85].

Table 10. Comparison of Training Algorithms for Neural Networks with Performance [74,81]

Table 10. Comparison of Training Algorithms for Neural Networks with Performance [74,81]

Abbreviation	Algorithm	Performance
LM	Levenberg-Marquardt	High convergence rate
BFG	BFGS Quasi-Newton	Fast convergence
RP	Resilient Backpropagation	Robust to noise
BR	Bayesian Regularization	Effective for small data
SCG	Scaled Conjugate Gradient	Memory efficient
CGB	Conjugate Gradient with Powell/Beale Restarts	Balanced performance
CGF	Fletcher-Powell Conjugate Gradient	Stable convergence
CGP	Polak-Ribiére Conjugate Gradient	Good for sparse data
OSS	One-Step Secant	Fast convergence
GDX	Variable Learning Rate Backpropagation	Adaptive learning rate
GD	Basic Gradient Descent	Simple, easy to implement
GDM	Gradient Descent with Momentum	Accelerated convergence

We employed the Backpropagation algorithm, a widely-used method in neural network training, to train our model. The neural network architecture consisted of three layers: an input layer with 15 neurons, a hidden layer with 50 neurons, and an output layer with 10 neurons (see Figure 11) [75,76].

To measure the performance of the network during training, we utilized the Mean Squared Error (MSE) as the performance criterion. Our goal was to minimize the MSE to below 0.0001, indicating a high level of accuracy in the model’s predictions (see Figure 12).

Figure 12. Taguchi- Neural network Training (Epoch 290,Performance goal met) .

Figure 12. Taguchi- Neural network Training (Epoch 290,Performance goal met) .

Figure 13. Taguchi- Neural network Mean Squared Error(mse) .

Figure 13. Taguchi- Neural network Mean Squared Error(mse) .

Figure 14. Taguchi- Neural network Training; $R=0.99933$ .

Figure 14. Taguchi- Neural network Training; $R=0.99933$ .

Figure 15. SOM Topology and Neighbor Connections.

Figure 15. SOM Topology and Neighbor Connections.

During the training process, we set parameters to monitor the progress and control the learning dynamics. We configured the network to display progress updates every 50 training epochs, allowing us to track the improvement of the model over time. Additionally, we set the learning rate to 0.03 to regulate the size of weight updates during training.

To ensure effective convergence of the model, we specified a maximum number of training epochs at 30000. This parameter defines the number of iterations the training algorithm goes through to optimize the network parameters. Furthermore, we employed a momentum coefficient of 0.9, which helps accelerate convergence by considering past weight updates during parameter optimization [75,77].

With a 15-neuron input layer and a 10-neuron output layer in a Self-Organizing Map (SOM), each neuron in the output layer represents a cluster or grouping of similar data from the input layer. The SOM reduces the dimensionality of the input space into a 10-dimensional output space, preserving spatial relationships between similar data points. Each output neuron acts as a centroid or representative of a cluster of similar data, grouping input data points that are closest in terms of distance. This interpolation of data into the output space allows for information aggregation and helps mitigate overfitting issues. With only 10 neurons in the output layer, visualization and interpretation of the resulting map are facilitated, allowing for an understanding of how different clusters are spatially organized. In summary, this configuration provides a compact representation of input data, easing visualization, interpretation, and data analysis while preserving relationships between similar data points [78,79,81,88,89].

Figure 16. SOM Neighbor Weight Distances and Samples Hits.

Figure 16. SOM Neighbor Weight Distances and Samples Hits.

SOM Neighbor Weight Distances and Sample Hits are crucial metrics for comprehending Self-Organizing Maps (SOMs). Neighbor Weight Distances measure the similarity or dissimilarity between adjacent neurons’ weights. Smaller distances indicate a smooth transition in representing input space, suggesting similar responses to similar input patterns among neighboring neurons [83,84,85,86]. Conversely, larger distances imply a more abrupt transition, indicating differing responses to similar inputs. Sample Hits track how often each neuron is selected as the best matching unit (BMU) for input samples during training. Neurons with higher hits represent densely populated regions of the input space, while lower-hit neurons represent sparser regions [88,89]. Analyzing Neighbor Weight Distances helps evaluate the map’s ability to capture the underlying structure of input data. Sample Hits reveal data distribution and density, aiding in identifying clusters or patterns. Together, these metrics provide insight into how the SOM organizes and represents input data, facilitating interpretation and understanding of its performance [65,76,81,84].

Figure 17. SOM input Planes.

Figure 17. SOM input Planes.

5. Conclusions

Abbreviations

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