Intelligent Design Prediction of a Circular Polarized Antenna for CubeSat Application Using Machine Learning Algorithms

Abstract

This paper presents an intelligent design method for a corner-truncated microstrip patch antenna (CTMPA) operating at 32 GHz using various well-known machine learning (ML) techniques. Our objectives are to obtain a gain of >5 dBic across a 10% bandwidth, an axial ratio (AR) of <3 dB, and a return loss of <−10 dB. First, a dataset of 715 full-wave simulated samples is analyzed with four distinct antenna characteristics (viz. features), along with the related computed $|S_{11}|$, gain, and AR. Using mean absolute error (MAE), mean square error (MSE), root mean square error (RMSE), and R${}^2$ score, 12 ML regression models were examined to compare the training data with the new predicted values. Next, the model that best satisfies our objectives was chosen. Results showed that the artificial neural network (ANN) followed by k-nearest neighbor (KNN) regression produced the lowest error compared to all tested ML models. The design parameters that achieved our intended objectives were computed using the predicted results. The predicted design was validated using a full-wave simulation and a prototype measurement.

1. Introduction

Millimeter-wave (mmWave) bands are necessary for future satellite and deep space communications because of their larger bandwidth, improved signal penetration, smaller antenna size, lower levels of interference, and higher data rates [1]. Specifically, the International Telecommunication Union (ITU) has allocated a part of the Ka-band that extends from 31.8 GHz to 32.3 GHz for CubeSat deep space downlink communication [2]. In addition to the power and size constraints, CubeSat antennas are expected to have full polarization diversity (viz. left-hand and right-hand circular) with high gain to compensate for the multipath fading, boost capacity, improve signal strength, and increase the signal-to-noise ratio (SNR). Adding additional design parameters and degrees of freedom can only satisfy such criteria, implying more time-consuming and computationally exhaustive full-wave simulations [3].

Conversely, antenna engineers have recently developed novel design strategies utilizing machine learning (ML) to reduce computational time by several orders of magnitude [4,5,6]. ML models were first applied in the early 2000s to predict the direction of arrival (DoA) in smart antenna arrays, such as support vector regression (SVR) [7,8,9]. Notably, it has been less than a decade since engineers began utilizing ML models for antenna design and wireless communication purposes [10]. Among the various ML-based models, notable examples include artificial neural network (ANN), Gaussian process regression (GPR), SVR, and k-nearest neighbor (KNN), among others [11]. ML models can predict outcomes for new inputs by leveraging probabilistic knowledge acquired from provided datasets. This knowledge encompasses non-linear characteristics within the data [12]. In [13], a modified KNN model was introduced to optimize the dipole antenna that reduced traditional simulation time significantly. Recently, support vector machines (SVMs) were used to categorize antennas according to their intended usage. Subsequently, ensemble learning models have been employed to forecast antenna performance, particularly for circular patch and horn antennas [14]. Polarization considerations often need more attention when applying ML algorithms. While the design process for linear-polarized antennas is straightforward, it becomes more intricate for corner-truncated microstrip patch (CTMPA) antennas, requiring time-consuming full-wave optimization. ML can reduce the optimization process by several orders of magnitude for the antenna design optimization. The utilization of ML algorithms for predicting CTMPA designs is an area that continues to be studied.

In this paper, we present an intelligent and efficient design method to optimize a CTMPA operating at 32 GHz using ML algorithms (see Figure 1). A dataset of features (viz. antenna design parameters) and targets ($|S_{11}|$, gain, and axial ratio (AR)) was initially produced using full-wave simulation. A total of 715 full-wave simulated datasets were created using arbitrary values of four different antenna features. The datasets were then trained using 12 different ML regression models for target prediction. Next, the performance of each model was analyzed using mean absolute error (MAE), mean square error (MSE), root mean square error (RMSE), and R${}^2$ score to find the best model. The ANN regression produced the lowest error among the regression models tested. Using the outcome of this model, we predicted the design parameters that achieve our desired goals. Indeed, the predicted results help us find the antenna size (L and W), corner length ($cl$), and feeding position (d). The predicted design (shown in Figure 1) was finally validated using full-wave simulation and fabrication, indicating that the antenna is suitable for the allocated deep space frequency band (viz. 31.8–32.3 GHz).

2. Antenna Design Using Machine Learning Approach

Figure 1 and Figure 2 show the top view and side view of the patch antenna with its different features ($cl$, d, L, and W), respectively. The substrate is RT/Duroid 5880 (${ϵ}_r$ = 2.2 and tan$\delta$ = 0.009), with a thickness of 0.79 mm. The antenna shows two possible orthogonal feeds: Position 1 and Position 2. Each position is used to realize either left-hand circular polarization (LHCP) or right-hand circular polarization (RHCP). The antenna is not aimed at serving the purpose of dual polarization (LHCP and RHCP). Hence, LHCP and RHCP will not be obtained simultaneously. The antenna is a square corner-truncated patch. We anticipate that both LHCP and RHCP can be obtained for the same L, W, and cl values by feeding the antenna at a distance ‘d’ away from the edge of the antenna at Position 1 and Position 2 in an alternating manner. Therefore, considering the same value of ‘d,’ we can use ML algorithms to predict the LHCP and RHCP antenna design parameters.

2.1. Data Generation and Model Building

The complete framework to find an effective ML model is shown in Figure 3. The full-wave simulation was used to generate 715 arbitrary datasets of four features ($cl$, d, L, and W) and three corresponding targets ($|S_{11}|$, gain, and AR) at 32 GHz. To generate the dataset, we conducted a parametric analysis using the High-Frequency Structure Simulator (HFSS). The initial values of cl, dl, L, and W ranged from 0.1 mm, 0.2 mm, 2 mm, and 2 mm, respectively, with increments of 0.1 mm. The final values reached 1.8 mm, 0.8 mm, 4 mm, and 4 mm, respectively. Since we employed a square-shaped microstrip patch antenna, the values of L and W were kept similar. The data distribution is visually represented in the histogram plot shown in Figure 4. The datasets are then trained using 12 different ML regression models based on linear regression (LR), ridge regression (RR), lasso regression (LAR), elastic net regression (ENR), decision tree regression (DTR), bagging regression (BGR), random forest regression (RFR), gradient boosting regression (GBR), polynomial regression (PR), SVR, KNN, and ANN.

2.1.1. Linear Regression

The LR is one of the most straightforward techniques in statistics and ML. LR results in a linearly separable relation between the dependent (targets) and independent (features) variables from the observed antenna dataset. This approach minimizes the cost function during the fitting phase, resulting in quick calculations for predicting antenna parameters [14].

2.1.2. Ridge Regression

The RR uses a regularization method (L2 regularization) to estimate the magnitude of the coefficient of multiple regression models when independent antenna variables (features) are highly correlated. This technique lowers the standard errors by introducing some bias into the usual linear regression for antenna parameter prediction [15].

2.1.3. Lasso Regression

The LAR uses a regularization (L1 regularization) that employs the shrinkage method to optimize the cost function in the event that there are many antenna features [16].

2.1.4. Elastic Net Regression

The L1 and L2 penalties of the lasso and ridge methods are linearly combined in the regularized regression technique, known as the elastic net [17].

2.1.5. Decision Tree Regression

The DTR is a widely used method for its practicality and efficiency. A decision tree uses a tree-like structure to generate regression models. It incrementally develops an associated decision tree while segmenting the antenna dataset into smaller sections [18].

2.1.6. Bagging Regression

The BGR is an ensemble meta-estimator that fits base regressors one at a time to random subsets of the original dataset. Then, it combines each prediction into a single final prediction. When employed with decision trees, BGR dramatically increases the stability of models by enhancing accuracy and lowering variance, which removes the over-fitting concern of antenna parameters [19].

2.1.7. Random Forest Regression

The RFR is a supervised learning technique that leverages the ensemble learning approach for regression. The ensemble learning method combines predictions from various ML algorithms to provide predictions of different antenna parameters [20].

2.1.8. Gradient Boosting Regression

The GBR is a widely adopted ML technique for predictive modeling, particularly in regression problems. It is a family of ensemble methods that combines many models trained to make antenna parameter predictions. The operational mechanism of GBR involves the iterative addition of weak decision trees to the ensemble, each trained to address the errors generated by the previous model in the sequence [21].

2.1.9. Support Vector Regression

The SVR is a popular ML method representing a variation of the SVM approach. The SVR technique can be employed to address both classification and regression problems. Regarding antenna parameter prediction, SVR endeavors to identify a function that minimizes margin violations while simultaneously fitting the training data [22].

2.1.10. Polynomial Regression

The connection between independent variables (antenna characteristics) and dependent variables (antenna objectives) can be represented as an nth-degree polynomial. During the training of regression models, the fitting curve can be identified. PR is particularly well-suited for antennas that exhibit highly nonlinear behavior [23]. In our case, we employ an 8th-order PR.

2.1.11. K-Nearest Neighbor Regression

The KNN is a straightforward ML algorithm applied for classification and regression. KNN saves all of the available data, and new input is categorized based on how similar it is to the already recorded data. Typically, a distance metric, such as the Manhattan distance or Euclidean distance, is used to construct the similarity measure. A hyperparameter that needs to be set beforehand is the value of k. A proper way to find the value of k using the elbow method is shown in Figure 5. It is a representation of k vs. error (RMSE). The value of RMSE is lowest for $k=5$, and that was determined using grid search cross-validation (GridsearchCV). KNN provides the new data point of the antenna during regression, which shows the average value of the k nearest neighbors [13].

2.1.12. Artificial Neural Network Regression

ANN regression proves highly effective when dealing with highly non-linear datasets. Including activation functions within each hidden layer of an ANN facilitates its capacity to comprehend intricate non-linear associations between input features and the target variable. This capability enables ANNs to make highly accurate predictions, particularly when complex, non-linear relationships exist in the data [4]. Our study employs a network architecture comprising five hidden layers to explore and leverage these capabilities. We initially started with fewer hidden layers for ANN regression. However, we discovered highly non-linear connections between the input features and the target variable. As a result, the model’s capacity to approximate those relationships was improved by adding more hidden layers (five). Figure 6 shows the epochs vs. loss for ANN regression.

The precision of all the above-mentioned ML algorithms depends on the antenna dataset. Therefore, finding an efficient ML model to predict antenna parameters is necessary. The complete framework to find an effective ML model is shown in Figure 3. The datasets are split into two categories, where 70% of the data is used to train the model, and 30% is used for testing. We used Scikit-learn extensively for different regressions, error metrics, hyperparameter tuning, and cross-validation throughout our study. The model validation using different error metrics is described in the following section.

2.2. Model Validation and Antenna Prediction

The performance of each model is analyzed by computing and comparing the MAE, MSE, RMSE, and R${}^2$ score to find the best model using the Python programming language on Jupyter Notebook. The lowest MAE, MSE, and RMSE and the highest R${}^2$ score are desirable for accurate prediction. Notably, the ANN model exhibits the lowest MAE, MSE, and RMSE and the highest R${}^2$ score.

Table 1. Error metrics showing MAE, MSE, RMSE, and R${}^2$ score for different algorithms.

Algorithm	MAE	MSE	RMSE	R${}^2$
LR	1.84	9.67	3.11	0.60
RR	1.84	9.67	3.11	0.60
LAR	1.84	9.66	3.10	0.60
ENR	1.847	9.63	3.10	0.61
SVR	1.26	9.58	3.25	0.67
DTR	0.34	3.52	1.87	0.85
BGR	0.22	1.55	1.24	0.95
RFR	0.20	1.54	1.24	0.93
GBR	0.20	1.54	1.24	0.93
PR (8th)	0.65	1.37	1.17	0.88
KNN	0.11	0.91	0.95	0.96
ANN	0.46	0.33	0.57	0.97

shows the error metrics for all regression models. The second model, which shows better results, is KNN. Figure 7 shows the bar chart indicating MAE, MSE, RMSE, and R${}^2$ scores for all regression models used in this paper. Figure 8, Figure 9, Figure 10 and Figure 11 show the residual $|S_{11}|$ results for LR, DTR, KNN, and ANN regression, respectively. Figure 12, Figure 13, Figure 14 and Figure 15 show the actual and predicted $|S_{11}|$ results for LR, DTR, KNN, and ANN regression, respectively.

The ANN regression shows a more accurate prediction compared to LR and DTR. For instance, in the LR model, the large number of residuals is responsible for the highest MAE, MSE, and RMSE and the lowest R${}^2$ scores, as shown in Table 1. We note that the KNN can be a secondary option, as it shows results close to ANN. However, PR, BGR, GBR, RFR, and DTR models can also be used as secondary selections with comparatively lower MAE, MSE, and RMSE, among other models (LR, RR, LAR, ENR, and SVR).

Table 2. Predicted $|S_{11}|$, gain, and AR for $cl$ = 0.88 mm, d = 0.58 mm, L, and W = 2.73 mm using different algorithms.

Algorithm	$|{\mathit{S}}_{11}|$ (dB)	Gain (dBic)	AR (dB)
LR	−17.43	6.10	5.84
RR	−17.43	6.10	5.84
LAR	−17.42	6.10	5.86
ENR	−17.41	6.10	5.86
SVR	−15.87	7.42	5.63
DTR	−14.43	7.61	1.16
BGR	−14.32	7.67	1.21
RFR	−14.40	7.69	1.16
GBR	−13.24	7.82	1.65
PR (8th)	−13.48	7.8	1.57
KNN	−13.45	7.64	0.82
ANN	−13.70	7.63	0.71
Full-wave Simulation	−13.68	7.6	0.53

shows the predicted $|S_{11}|$, realized gain, and AR for an arbitrary antenna ($cl$ = 0.88 mm, d = 0.58 mm, L, and W = 2.73 mm) at 32 GHz, with values of −13.70 dB, 7.63 dBic, and 0.71 dB, respectively. The features or design variables are chosen arbitrarily, as our model can predict design targets of a circular polarized (CP) antenna for any given dataset.

Designing a circular polarized CTMPA is more complex than designing a linearly polarized square patch antenna, as the design parameters ($|S_{11}|$, gain, and AR) can vary significantly with changes in feeding point (d), corner length ($cl$), and antenna dimensions (L and W). The conventional full-wave simulation is used to optimize the CP antenna’s gain, return loss, and AR. However, this is time-consuming and often exhausts resources. Our approach offers a way to predict the design parameters of CTMPA before conducting actual simulations, making it much faster and more efficient. To the best of our knowledge, we are the first to introduce different ML algorithms for predicting the design parameters of a CTMPA antenna.

2.3. Antenna Design Validation Using Full-Wave Simulation

We validated the predicted antenna ($cl$ = 0.88 mm, d = 0.58 mm, L, and W = 2.73 mm) using full-wave simulation (High-Frequency Structure Simulator (HFSS)). The target prediction using all ML algorithms and true values (full-wave simulation) is shown in Table 2. As expected, the simulated $|S_{11}|$, realized gain, and AR are almost identical to the ANN prediction at 32 GHz, with values of −13.68 dB, 7.6 dBic, and 0.53 dB, respectively, as shown in Figure 16. Our objectives are to obtain a gain of >5 dBic across a 10% bandwidth, an axial ratio (AR) of <3 dB, and a return loss of <−10 dB. Meanwhile, we obtained a gain of >6.4 dBic across a 14% bandwidth (29.5 GHz–34 GHz) and an axial ratio (AR) of <3 dB (31 GHz–33 GHz).

We validated the LHCP and RHCP using the same antenna parameters by switching the orthogonal feeding position from Position 1 to Position 2, as illustrated in Figure 1. Our presented antenna is a square patch with corner perturbation, providing LHCP and RHCP alternatively using two orthogonal feedings. Our ML model accurately predicts the design of both individual LHCP and RHCP antennas. Figure 17 shows the realized gain and AR at 32 GHz for both LHCP and RHCP when the feeding position is in Position 1 and Position 2, respectively. The broadside realized gain is 7.6 dBic at 32 GHz in the elevation plane for both polarizations. Additionally, the simulated AR at the broadside is 0.53 dB. As such, the simulated result greatly agrees with the prediction result. This verification demonstrates that the ANN model outperforms all other ML regression models in terms of prediction.

ANN regression can be used where a non-linear relationship among the data frame is dominant. Due to the presence of an activation function in each hidden layer, ANN can learn the non-linear complex relationship between the features and target and can predict results accurately. In contrast, KNN stores all available data and categorizes new input based on its similarity to the recorded data using a distance metric, such as the Manhattan or Euclidean distance. However, in other models, all the data are not saved. For instance, linear regression predicts a linear relationship between dependent and independent variables and identifies the optimal fitting line/curve for efficient prediction. In such cases, the best result is obtained when the variables have a linear relationship with each other. However, our dataset does not exhibit a linear relationship among all variables. Instead, datasets are random. Therefore, ANN and KNN would be ideal models for our prediction. Additionally, bagging regression, random forest regression, gradient boosting regression, and decision tree regression can be used as secondary options for prediction, as these ML techniques use small, tree-like structures for training and prediction. Figure 18 shows the antenna’s efficiency for both polarizations, and 94% efficiency was achieved at 32 GHz.

2.4. Antenna Prediction Time

The simulation was done on a 2.9 GHz processor with 80 GB of RAM installed. The maximum time to complete the sweep in a single frequency (32 GHz) is about 63,000 ms. In contrast, the target prediction time for the ML algorithm is only a few milliseconds using the same operating system, as shown in

Table 3. Time comparison for the target ($|S_{11}|$, gain, and AR) prediction of any random features (viz. L, W, $cl$ and d) at 32 GHz using different methods.

Method	Time (ms)
LR	3.67
RR	4.78
LAR	4.95
ENR	3.98
SVR	14.9
DTR	4.98
BGR	41.7
RFR	36.9
GBR	53.8
PR (8th order)	7.6
KNN	5.96
ANN	4.87
Full-wave Simulation	63,000

. Indeed, the prediction time decreases several orders of magnitude (∼10${}^4$) using the ML approach compared to conventional full-wave simulation.

3. Fabrication and Measurement

We fabricated a prototype for the LHCP to validate the concept. A standard milling machine was used for the fabrication. A 2.4 mm connector was used to feed the antenna for measurement. Figure 19 shows the simulated and measured $|S_{11}|$ of the LHCP antenna. The predicted $|S_{11}|$ is −13.7 dB, the simulated $|S_{11}|$ is −13.68 dB, and the measured $|S_{11}|$ is −12.75 dB at 32 GHz. Therefore, the measured data agree well with the simulated and predicted values. Figure 20 shows the pattern measurement setup at the Ka-band (26.5 GHz–40 GHz). Figure 21 shows the measured realized gain of the LHCP antenna for both the E plane and the H plane at 32 GHz. The maximum gain at 32 GHz is 5.01 dBic, and the 3 dB beam width is ${68}^0$, whereas the simulated beam width is ${58}^0$. Figure 22 shows the simulated and measured realized gain and AR of the LHCP antenna. The predicted, simulated, and measured realized gains are 7.63 dBic, 7.6 dBic, and 5.01 dBic, respectively, at 32 GHz. The predicted, simulated, and measured AR are 0.71 dB, 0.53 dB, and 2.3 dB at 32 GHz. The slight discrepancy between the simulation and fabrication outcomes is due to fabrication inaccuracy and soldering of the connector. During the soldering process, we encountered a challenge where the antenna’s ground plane did not securely attach to the connector base. To address this issue within the constraints of our fabrication process, we applied additional soldering resin to establish a connection. We believe this irregularity in the ground plane’s current distribution contributed to discrepancies between the simulated and measured data. Under ideal conditions, we would expect these discrepancies to be significantly reduced. However, the result is acceptable for validating the proof of concept. The properties of the different ML algorithms for designing the antenna are summarized in

Table 4. Our work with reported work in the literature.

Reference	Algorithms	Antenna Type	Polarization	Predicted Objective
[4]	Lasso, KNN, and ANN	T-monopole	N/A	$|S_{11}|$
[13]	KNN	Dipole	N/A	$|S_{11}|$
[14]	SVM	Patch and Horn	N/A	$|S_{11}|$
[24]	LR	Patch	N/A	$|S_{11}|$
[25]	7 Models	Patch	N/A	$|S_{11}|$
Our Work	12 Models	CTMPA	Circular	$|S_{11}|$, AR, and gain

. Our study introduces an approach that considers multiple objectives, including $|S_{11}|$, AR, and gain, as well as antenna polarization. This contrasts with most other research efforts, where the primary focus is estimating $|S_{11}|$.

4. Conclusions

This paper presents an efficient and intelligent ML approach to predict the $|S_{11}|$, gain, and AR of a circularly polarized microstrip antenna. The goal is to achieve full polarization diversity with a wide 3 dB axial ratio (30.9 GHz–32.8 GHz) and a stable gain of >5 dBic. Twelve different ML regression models were evaluated using four error metrics (MAE, MSE, RMSE, and R${}^2$ score). Among them, the ANN model performed best. A model was then predicted and validated using full-wave simulation. Also, a prototype was fabricated and measured. The predicted, simulated, and measured results agreed well, indicating that this approach can save significant computational resources and time in designing an efficient circular polarized antenna for CubeSat downlink applications.