Dual Band Antenna Design and Prediction of Resonance Frequency Using Machine Learning Approaches

: An inset fed-microstrip patch antenna (MPA) with a partial ground structure is constructed and evaluated in this paper. This article covers how to evaluate the performance of the designed antenna by using a combination of simulation, measurement, creation of the RLC equivalent circuit model, and the implementation of machine learning approaches. The MPA’s measured frequency range is 7.9–14.6 GHz, while its simulated frequency range is 8.35–14.25 GHz in CST microwave studio (CST MWS) 2018. The measured and simulated bandwidths are 6.7 GHz and 5.9 GHz, respectively. The antenna substrate is composed of FR-4 Epoxy, which has a dielectric constant of 4.4 and a loss tangent of 0.02. The equivalent model of the proposed MPA is developed by using an advanced design system (ADS) to compare the resonance frequencies obtained by using CST. In addition, the measured return loss of the prototype is compared with the simulated return loss observed by using CST and ADS. At the end, 86 data samples are gathered through the simulation by using CST MWS, and seven machine learning (ML) approaches, such as convolutional neural network (CNN), linear regression (LR), random forest regression (RFR), decision tree regression (DTR), lasso regression, ridge regression, and extreme gradient boosting (XGB) regression, are applied to estimate the resonant frequency of the patch antenna. The performance of the seven ML models is evaluated based on mean square error (MSE), mean absolute error (MAE), root mean square error (RMSE), and variance score. Among the seven ML models, the prediction result of DTR (MSE = 0.71%, MAE = 5.63%, RMSE = 8.42%, and var score = 99.68%) is superior to other ML models. In conclusion, the proposed antenna is a strong contender for operating at the entire X-band and lower portion of the Ku-band frequencies, as evidenced by the simulation results through CST and ADS, it measured and predicted results using machine learning approaches.

error (MAE), root MSE (RMSE), and variance score are observed to validate the prediction of the f r using the mentioned models [32].
Recently, a new technique has been explored by using multiple regression algorithms along with convolutional neural networks (CNN) to predict the f r by using CST electromagnetic (EM) simulation tools. Furthermore, the simulated f r using CST and ADS and measured f r is compared with the predicted f r using the proposed ML algorithms.
The main contributions of this research work are summarized as follows: i Simulate and analyze the performance of a microstrip patch antenna using CST EM simulation tools. ii Validate the CST simulation results using ADS simulation software, and the simulated S 11 is compared with the measured S 11. iii The resonance frequency ( f r ) is predicted using six ML regression algorithms and CNN. A comparative study of the different models based on the different predicted results is incorporated.
The remaining sections of this study are organized as follows. Section 2 explains several generic design formulas and the geometrical structure of the proposed antenna and the required performance analysis of the various machine learning algorithms. The results of the necessary simulations and measurements are presented in Section 3 and, consequently, the concluding remarks are highlighted in Section 4.

Design Methodology
In microstrip patch antennas, the size and shape of the patch determine the antenna's resonance frequency, bandwidth, return loss, gain, and radiation pattern. Radiation efficiency and resonance frequency are two of the many variables whose values are directly determined by the patch's width and length. The feasible and efficient dimensions (length and width) of a rectangular microstrip antenna can be calculated using the formulae below. Based on the transmission line model, the dimensions of a radiation patch can be calculated using the following equations [33,34]: where and feedline W 0 is evaluated from the input impedance Z in : Here, c is the speed at which light travels through a vacuum, ( f r ) is the resonance frequency, and ε r is the relative permittivity (also called the dielectric constant) of the dielectric material. In real life, the fields do not just stay on the patch. The fringing field is a field component that goes beyond the patch's physical limits (L × W). The effective patch width could be calculated using the effective dielectric constant (ε re f f ) for the width of the patch (W) while considering the influence of the fringing field effect. The effective dielectric constant (ε re f f ) is determined by the following equation: Here, h is the thickness of dielectric substrate. To account for the influence of the fringing field given a patch length L, both ends of the line must have a length added to them.
∆L e f f The antenna has a total footprint of 28.2 × 23.8 × 1.6 mm 3 with a patch area of 28.2 × 23.8 mm 2 . Annealed copper is used as a patch and ground plane material and is matched to the real-world antenna construction material.
FR4 is, rightly, the most commonly used material in PCB construction. Boards from FR4 are robust, water resistant, and provide sound insulation between copper layers that minimize interference and support good signal integrity. This research investigated the feasibility of using the FR-4 substrate for microstrip antennas throughout a wide frequency range (8)(9)(10)(11)(12). The purpose of this investigation was to examine the FR-4 substrate as a potential option for designing an X-band microstrip antenna, aiming to achieve a high degree of agreement between simulated and measured results. Due to its inexpensive cost and widespread availability, FR-4 was selected for this research because it can be utilized for prototyping microstrip antennas. A microstrip feed line with a 50 Ω impedance was used to feed the input signal to the antenna. CST carries out the preliminary design and performance optimization. The dimensional parameters of the proposed MPA are presented in Table 1. The proposed MPA utilizes FR-4 Epoxy dielectric material, which has a relative permittivity of ε r = 4.4 and loss tangent of 0.007. The antenna features a rectangle-shaped patch with two inset slot cuts and a partial ground plane with a smaller area than the substrate. The parametric analysis in CST derives the optimal dimensions of insets and ground planes. Optimized dimensions of inset slots in combination with a partial ground plane give an excellent wideband response. Figure 1a,b depict a simulated 3D view; the partial ground plane of the proposed MPA and Figure 2a shows the dimensions of substrate, patch, and feedline of the proposed X-band MPA and Figure 2b depicts the fabricated prototype view of the proposed wideband MPA that has the capability to satisfy X-band applications requirement.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 18 fabricated prototype view of the proposed wideband MPA that has the capability to satisfy X-band applications requirement.

Result Analysis of the Proposed MPA
The simulated and measured results of the proposed MPA are discussed in this section. The simulated S11 using CST is also compared with the results obtained from the advanced design system (ADS). Finally, different machine learning algorithms are discussed briefly to predict the proposed antenna's resonance frequency.

Simulated and Measured Results
The simulated and measured S11 plot is shown in Figure 3. The simulated and measured S11 plot ensures that the proposed MPA will provide satisfactory performance across the whole X-band. Moreover, the S11 graph shows that the antenna will be suitable for the entire X-band and a portion of the lower part of the Ku-band. The MPA provides two resonance frequencies of 9 GHz and 13 GHz with return loss magnitudes of S11 −35 dB and −18 dB, respectively. The MPA offers a −10 dB impedance bandwidth of 5.8 GHz, ranging from 8.35 GHz to 14.15 GHz. However, the measured return loss graph is slightly different from the simulated return loss graph. It may have occurred because the antenna is excited using a waveguide port during simulation, but practically the antenna is excited using the SMA connector. The connector loss influences the response of the antenna. In  fabricated prototype view of the proposed wideband MPA that has the capability to satisfy X-band applications requirement.

Result Analysis of the Proposed MPA
The simulated and measured results of the proposed MPA are discussed in this section. The simulated S11 using CST is also compared with the results obtained from the advanced design system (ADS). Finally, different machine learning algorithms are discussed briefly to predict the proposed antenna's resonance frequency.

Simulated and Measured Results
The simulated and measured S11 plot is shown in Figure 3. The simulated and measured S11 plot ensures that the proposed MPA will provide satisfactory performance across the whole X-band. Moreover, the S11 graph shows that the antenna will be suitable for the entire X-band and a portion of the lower part of the Ku-band. The MPA provides two resonance frequencies of 9 GHz and 13 GHz with return loss magnitudes of S11 −35 dB and −18 dB, respectively. The MPA offers a −10 dB impedance bandwidth of 5.8 GHz, ranging from 8.35 GHz to 14.15 GHz. However, the measured return loss graph is slightly different from the simulated return loss graph. It may have occurred because the antenna is excited using a waveguide port during simulation, but practically the antenna is excited using the SMA connector. The connector loss influences the response of the antenna. In

Result Analysis of the Proposed MPA
The simulated and measured results of the proposed MPA are discussed in this section. The simulated S 11 using CST is also compared with the results obtained from the advanced design system (ADS). Finally, different machine learning algorithms are discussed briefly to predict the proposed antenna's resonance frequency.

Simulated and Measured Results
The simulated and measured S 11 plot is shown in Figure 3. The simulated and measured S 11 plot ensures that the proposed MPA will provide satisfactory performance across the whole X-band. Moreover, the S 11 graph shows that the antenna will be suitable for the entire X-band and a portion of the lower part of the Ku-band. The MPA provides two resonance frequencies of 9 GHz and 13 GHz with return loss magnitudes of S 11 −35 dB and −18 dB, respectively. The MPA offers a −10 dB impedance bandwidth of 5.8 GHz, ranging from 8.35 GHz to 14.15 GHz. However, the measured return loss graph is slightly different from the simulated return loss graph. It may have occurred because the antenna is excited using a waveguide port during simulation, but practically the antenna is excited using the SMA connector. The connector loss influences the response of the antenna. In addition, the near-field scattering objects, the losses due to the feed connector, and the coaxial cable also affect the response of the antenna performance. Figure 4 shows that the VSWR is less than 1.5 at both resonance frequencies, ensuring good impedance matching characteristics.
the coaxial cable also affect the response of the antenna performance. Figure 4 shows that the VSWR is less than 1.5 at both resonance frequencies, ensuring good impedance matching characteristics.
The surface current distribution of the designed MPA indicates that the antenna has the most current in the middle of its length and the least current near its edges, as illustrated in Figure 5. To validate the simulated and measured S11, the equivalent RLC model of the proposed antenna is designed using Agilent ADS software and the resonance frequencies obtained from the RLC equivalent model of the proposed MPA using ADS Agilent software are almost equal to the simulated (using CST) and measured resonance frequencies. In addition, the S11 using ADS is −34 dB and −35 dB at both resonance frequencies.   that the VSWR is less than 1.5 at both resonance frequencies, ensuring good impedance matching characteristics.
The surface current distribution of the designed MPA indicates that the antenna has the most current in the middle of its length and the least current near its edges, as illustrated in Figure 5. To validate the simulated and measured S11, the equivalent RLC model of the proposed antenna is designed using Agilent ADS software and the resonance frequencies obtained from the RLC equivalent model of the proposed MPA using ADS Agilent software are almost equal to the simulated (using CST) and measured resonance frequencies. In addition, the S11 using ADS is −34 dB and −35 dB at both resonance frequencies.   The surface current distribution of the designed MPA indicates that the antenna has the most current in the middle of its length and the least current near its edges, as illustrated in Figure 5. To validate the simulated and measured S 11 , the equivalent RLC model of the proposed antenna is designed using Agilent ADS software and the resonance frequencies obtained from the RLC equivalent model of the proposed MPA using ADS Agilent software are almost equal to the simulated (using CST) and measured resonance frequencies. In addition, the S 11 using ADS is −34 dB and −35 dB at both resonance frequencies.
The radiation pattern of the proposed antenna is depicted in Figure 6, which shows the main lobe direction, main lobe magnitude, side lobe level (SLL), and a 3-dB beam width. At the two distinct resonance frequencies, the 3-dB beam width is 148.2 degrees for 9 GHz and 71 degrees for 13 GHz. The SLL at a resonance frequency of 13 GHz is −6.3 dB and at a frequency of 9 GHz it is −10 dB. Gain is a measurement of the energy delivered to the main beam. The gain vs. frequency curve of the proposed antenna is presented in Figure 7. From the figure, the gain of the microstrip patch antenna varies from 2.2 dB to 6.25 dB in the entire simulated frequency range. The designed antenna gained 4.0614 dB at 9 GHz and 3.4589 dB at 13 GHz. The radiation pattern of the proposed antenna is depicted in Figure 6, which shows the main lobe direction, main lobe magnitude, side lobe level (SLL), and a 3-dB beam width. At the two distinct resonance frequencies, the 3-dB beam width is 148.2 degrees for 9 GHz and 71 degrees for 13 GHz. The SLL at a resonance frequency of 13 GHz is −6.3 dB and at a frequency of 9 GHz it is −10 dB. Gain is a measurement of the energy delivered to the main beam. The gain vs. frequency curve of the proposed antenna is presented in Figure 7. From the figure, the gain of the microstrip patch antenna varies from 2.2 dB to 6.25 dB in the entire simulated frequency range. The designed antenna gained 4.0614 dB at 9 GHz and 3.4589 dB at 13 GHz.  The radiation pattern of the proposed antenna is depicted in Figure 6, which shows the main lobe direction, main lobe magnitude, side lobe level (SLL), and a 3-dB beam width. At the two distinct resonance frequencies, the 3-dB beam width is 148.2 degrees for 9 GHz and 71 degrees for 13 GHz. The SLL at a resonance frequency of 13 GHz is −6.3 dB and at a frequency of 9 GHz it is −10 dB. Gain is a measurement of the energy delivered to the main beam. The gain vs. frequency curve of the proposed antenna is presented in Figure 7. From the figure, the gain of the microstrip patch antenna varies from 2.2 dB to 6.25 dB in the entire simulated frequency range. The designed antenna gained 4.0614 dB at 9 GHz and 3.4589 dB at 13 GHz.

RLC Lumped Element Extraction and Equivalent Circuit of the Proposed MPA Using ADS
The equivalent RLC model of the proposed MPA using Agilent ADS software is presented in Figure 8. The values of R, L, and C have been chosen so that the equivalent impedance of the designed antenna is reasonably matched with the characteristic impedance (50 Ω) of the transmission line. An optimal impedance match between the antenna

RLC Lumped Element Extraction and Equivalent Circuit of the Proposed MPA Using ADS
The equivalent RLC model of the proposed MPA using Agilent ADS software is presented in Figure 8. The values of R, L, and C have been chosen so that the equivalent impedance of the designed antenna is reasonably matched with the characteristic impedance (50 Ω) of the transmission line. An optimal impedance match between the antenna and the transmission line is required to transfer maximum power from the input port of the antenna to its antenna structure [35]. For the resonance frequencies of 9 GHz and approximately 13 GHz, resistance R1 = 52.56 Ω, inductance L1 = 49.8 pH, and capacitance C1 = 6.369 pF are assigned to the parallel RLC circuit, along with resistance R2 = 51.88 Ω, inductance L2 = 135.8 pH, and capacitance C2 = 1.10 pF as presented in Figure 8. As the antenna input port, the input terminal source in ADS is configured as a terminal block (Term is G) with a 50 Ω characteristic impedance and acts as an input port for the antenna. The values of R, L, and C are considered in such a manner that the equivalent circuit impedance is equal to the antenna input port impedance. The impedance matching performance is observed in ADS using the S-parameter block. In the S-parameter block, the intended frequency range is swept from 7 GHz to 17 GHz in 10 KHz increments. The plot of the return loss response (dB (S (1,1)) is chosen for plotting the resonant circuit's output result. The resonant circuit yields a resonant frequency of 9.0 GHz with a return loss of −33 dB, and at 13 GHz with a return loss of -35 dB, as shown in Figure 9.

RLC Lumped Element Extraction and Equivalent Circuit of the Proposed MPA Using ADS
The equivalent RLC model of the proposed MPA using Agilent ADS software is presented in Figure 8. The values of R, L, and C have been chosen so that the equivalent impedance of the designed antenna is reasonably matched with the characteristic impedance (50 Ω) of the transmission line. An optimal impedance match between the antenna and the transmission line is required to transfer maximum power from the input port of the antenna to its antenna structure [35]. For the resonance frequencies of 9 GHz and approximately 13 GHz, resistance 1 = 52.56 Ω, inductance 1 = 49.8 pH, and capacitance 1 = 6.369 pF are assigned to the parallel RLC circuit, along with resistance 2 = 51.88 Ω, inductance 2 = 135.8 pH, and capacitance 2 = 1.10 pF as presented in Figure   8. As the antenna input port, the input terminal source in ADS is configured as a terminal block (Term is G) with a 50 Ω characteristic impedance and acts as an input port for the antenna. The values of R, L, and C are considered in such a manner that the equivalent circuit impedance is equal to the antenna input port impedance. The impedance matching performance is observed in ADS using the S-parameter block. In the S-parameter block, the intended frequency range is swept from 7 GHz to 17 GHz in 10 KHz increments. The plot of the return loss response (dB (S (1,1)) is chosen for plotting the resonant circuit's output result. The resonant circuit yields a resonant frequency of 9.0 GHz with a return loss of −33 dB, and at 13 GHz with a return loss of -35 dB, as shown in Figure 9.

Machine Learning-Based Resonance Frequency Prediction
To achieve simplicity in the design of the antenna and predict the antenna performances, different ML and/or DL models are introduced. Using those models, the design

Machine Learning-Based Resonance Frequency Prediction
To achieve simplicity in the design of the antenna and predict the antenna performances, different ML and/or DL models are introduced. Using those models, the design and optimization of the antennas are much easier than with the traditional EM simulation tools. In this study, we are investigating and showing a wide variety of regression methods along with convolutional neural networks (CNN). All these algorithms are used to predict the resonance frequency of the proposed MPA. A brief description of the different models that are used for the prediction of resonance frequency is provided in the following subsection.

Brief Description of the Learning Models
Machine Learning Models: Machine learning creates algorithms from data and then uses those algorithms to generate predictions based on other data. Multiple methods, including regression, classification, and deep learning, see heavy rotation in the machine learning toolkit (Neural Network). To solve this problem, we are investigating and showing various regression methods. It is common practice to employ regression when making predictions and forecasts.

Convolutional Neural Network (CNN):
CNN has been proposed for use on one-dimensional data in addition to its widespread application in image recognition and text analysis. In convolutional neural networks, there are three types of layers: input, hidden, and output. Both the input and output layers are activated linearly. However, since neural networks can have multiple hidden layers, their activation functions are often nonlinear. Hidden layers in convolutional neural networks (CNNs) convolve the input and send the result to the next layer [36,37]. Table 2 shows the hyperparameter settings of the CNN model. The CNN architecture is shown in Figure 10.

Linear Regression:
Linear regression is a well-known approach in the fields of statistics and machine learning. This is a mathematical technique used in forecasting or predicting results. A linear regression model aims to establish a correlation between a set of characteristics and a continuous dependent variable [38].

Random Forest Regression:
Classification and regression with random forests involve constructing a set of tree predictors, each of which is built with a random vector that is chosen independently of the input vector. Regression with tree predictor uses numerical values in place of class labels. Random forest regression constructs a tree when using variables at each node [39].

Decision Tree Regression:
Decision tree regression considers an object's qualities and trains a model to predict future data and give useful continuous output. It is a tree structure for quantitatively forecasting the results of the dependent variable [40]. Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 18 Figure 10. CNN architecture.

Linear Regression:
Linear regression is a well-known approach in the fields of statistics and machine learning. This is a mathematical technique used in forecasting or predicting results. A linear regression model aims to establish a correlation between a set of characteristics and a continuous dependent variable [38].

Random Forest Regression:
Classification and regression with random forests involve constructing a set of tree predictors, each of which is built with a random vector that is chosen independently of the input vector. Regression with tree predictor uses numerical values in place of class labels. Random forest regression constructs a tree when using variables at each node [39].

Lasso Regression:
Lasso regression is one type of linear regression that employs the shrinkage method. Researchers often turn to lasso regressions for modeling scenarios with many features [41] because of its efficiency in executing feature selection.

Ridge Regression:
A valuable method for analyzing multiple regression on data that exhibits multicollinearity is known as ridge regression. It is a regularization model where an extra variable (tune parameter) is added and optimized to address numerous variables in linear regression [42].

XGB Regression:
When it comes to fixing regression or classification issues, the most efficient method is extreme gradient boosting (XGBoost). This approach uses a gradient boosting framework to make use of decision trees. It provides features that significantly affect the model's efficacy [43].
The data generation flow diagram is shown in Figure 11. There are six regression models, linear regression, random forest regression, decision tree regression (DTR), lasso regression, ridge regression, XGB regression, and CNN, used to predict the resonance frequency. The comparisons between the simulated and the predicted resonance frequencies of the 18 test data samples using DTR and CNN are shown in Tables 3 and 4

Ridge Regression:
A valuable method for analyzing multiple regression on data that exhibits multicollinearity is known as ridge regression. It is a regularization model where an extra variable (tune parameter) is added and optimized to address numerous variables in linear regression [42].

XGB Regression:
When it comes to fixing regression or classification issues, the most efficient method is extreme gradient boosting (XGBoost). This approach uses a gradient boosting framework to make use of decision trees. It provides features that significantly affect the model's efficacy [43].
The data generation flow diagram is shown in Figure 11. There are six regression models, linear regression, random forest regression, decision tree regression (DTR), lasso regression, ridge regression, XGB regression, and CNN, used to predict the resonance frequency. The comparisons between the simulated and the predicted resonance frequencies of the 18 test data samples using DTR and CNN are shown in Table 3 and Table 4, respectively

Performance Evaluation of the ML Models
Different performance metrics of the ML models are investigated to validate the prediction of the resonance frequency with respect to the simulated and the measured resonance frequency. In this research work, the mean absolute error (MAE), the mean squared error (MSE), the root mean squared error (RMSE), and the variance score are observed as

Performance Evaluation of the ML Models
Different performance metrics of the ML models are investigated to validate the prediction of the resonance frequency with respect to the simulated and the measured resonance frequency. In this research work, the mean absolute error (MAE), the mean squared error (MSE), the root mean squared error (RMSE), and the variance score are observed as performance indicators of the ML regression models. Predicting errors are computed, and prediction models are assessed using these metrics. The mean absolute error, mean squared error, and root-mean-squared error are well-known scale-dependent measurements based on absolute and squared values [44,45]. The MAE considers the average amount of error over a group of projections but ignores the direction of the mistake, giving less weight to extreme predictions [46]. In [47], the MAE is mathematically formulated as: where n = number of errors and |Pi − Oi| = Absolute error. The MSE is a common statistic used in estimation. The root-mean-squared error (RMSE) is used in place of MSE by taking the square root. The root-mean-squared error (RMSE) quantifies how far estimates deviate from reality. The definition of RMSE is [48]: Another performance indicator is the explained variance score that reflects the error scatter in a data collection. It can be defined as [49]: Figure 12 shows that a total of 86 samples were used to predict the resonance frequency of the designed MPA. Among the 86 samples, 68 samples were used to train the models and 18 samples were used to test the models' performance. The number of training samples and the number of sampling data determines the accuracy of the prediction model. The 86 data samples were generated using CST simulation software by varying the different dimensional parameters of the proposed MPA. Figure 12 shows that a total of 86 samples were used to predict the resonance frequency of the designed MPA. Among the 86 samples, 68 samples were used to train the models and 18 samples were used to test the models' performance. The number of training samples and the number of sampling data determines the accuracy of the prediction model. The 86 data samples were generated using CST simulation software by varying the different dimensional parameters of the proposed MPA.  Figure 13 shows a heat map plot (correlation matrix) which represents each of the ten attributes of antenna design parameters named Lp, Ls, Lg, Wp, h, Ws, Mtp, Mt, S11, and frequency on the y-and x-axis of the plot. These values represent the input dimensions of the proposed antenna structure, as shown in Table 1 and Figure 2 in the previous section. The maximum correlation occurs at the center of the plot with a value of 1.00. As the point moves away from the maximum line (1.00), the correlation decreases accordingly. In this mapping plot, the correlation value ranges from 0.03 to 0.72.   Figure 13 shows a heat map plot (correlation matrix) which represents each of the ten attributes of antenna design parameters named Lp, Ls, Lg, Wp, h, Ws, Mtp, Mt, S11, and frequency on the y-and x-axis of the plot. These values represent the input dimensions of the proposed antenna structure, as shown in Table 1 and Figure 2 in the previous section. The maximum correlation occurs at the center of the plot with a value of 1.00. As the point moves away from the maximum line (1.00), the correlation decreases accordingly. In this mapping plot, the correlation value ranges from 0.03 to 0.72.    Figure 14 shows the simulated vs. predicted frequency of test data samples using DTR and CNN, respectively, for 18 test samples. In the analysis, the frequency tuning ranges from 7 GHz to 12 GHz. It is shown in Figure 14a that there is a small deviation (close to 0) between the actual and predicted frequencies for DTR. Among the 18 test samples, 3 test samples were accurately predicted, with zero percentage of error. In addition, the percentage of error in most cases is less than one. As a result, the predicted result almost perfectly matches the simulated result, as shown in Figure 14a. Somehow, there are deviations for CNN with magnitudes ranging from 0.95446% to 20.65657% as depicted in Figure 14b. It is found that the percentage of predicted errors in CNN is slightly higher than in DTR. Hence, DTR is selected for better prediction performance as compared to CNN.

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(close to 0) between the actual and predicted frequencies for DTR. Among the 18 test samples, 3 test samples were accurately predicted, with zero percentage of error. In addition, the percentage of error in most cases is less than one. As a result, the predicted result almost perfectly matches the simulated result, as shown in Figure 14a. Somehow, there are deviations for CNN with magnitudes ranging from 0.95446% to 20.65657% as depicted in Figure 14b. It is found that the percentage of predicted errors in CNN is slightly higher than in DTR. Hence, DTR is selected for better prediction performance as compared to CNN. The performance metrics of ML for CNN, linear regression, random forest regression, decision tree regression (DTR), lasso regression, ridge regression, and XGB regression algorithms are tabulated in Table 5. The magnitude values of MAE, MSE, RMSE, and the variant score are evaluated to determine the error performance of each algorithm. It is shown that the DTR model generates a small percentage error for MAE, MSE, RMSE, and the variant score magnitude with values of 5.63%, 0.71%, 8.42%, and 99.68%, respectively. The DTR method has performed better than the rest of the regression models and yields the highest quality results in all four scenarios. It is also shown that the XGB model is considered as a secondary selection, where it generates values of 7.03%, 1.06%, 10.27%, and 99.54%, respectively. It depicts the XGB model, performing close to DTR. Somehow, the performance of the CNN and Lasso models produced lower percentage magnitudes of 76.41%, 92.49%, 96.17%, and 57.62% and 55.01%, 63.25%, 79.53%, and 70.97%, respectively. Hence, they are not good for method selection for proposed research work. The performance comparison for all algorithms viewed in the bar chart is illustrated in Figure  15.  The performance metrics of ML for CNN, linear regression, random forest regression, decision tree regression (DTR), lasso regression, ridge regression, and XGB regression algorithms are tabulated in Table 5. The magnitude values of MAE, MSE, RMSE, and the variant score are evaluated to determine the error performance of each algorithm. It is shown that the DTR model generates a small percentage error for MAE, MSE, RMSE, and the variant score magnitude with values of 5.63%, 0.71%, 8.42%, and 99.68%, respectively. The DTR method has performed better than the rest of the regression models and yields the highest quality results in all four scenarios. It is also shown that the XGB model is considered as a secondary selection, where it generates values of 7.03%, 1.06%, 10.27%, and 99.54%, respectively. It depicts the XGB model, performing close to DTR. Somehow, the performance of the CNN and Lasso models produced lower percentage magnitudes of 76.41%, 92.49%, 96.17%, and 57.62% and 55.01%, 63.25%, 79.53%, and 70.97%, respectively. Hence, they are not good for method selection for proposed research work. The performance comparison for all algorithms viewed in the bar chart is illustrated in Figure 15.

Conclusions
This article discusses the integration of simulation, measurement, development of the RLC equivalent circuit model, and applying machine learning approaches to evaluate the performance of the proposed antenna. In terms of frequency, the designed antenna supports the whole X-band as well as some portions of the Ku-band. The prototype was built and analyzed to confirm the intended performance. In addition, the RLC equivalent model of the proposed MPA designed with the ADS Agilent software yields resonance frequencies nearly identical to those generated by simulation (with CST) and measurement. Furthermore, six machine learning and one deep learning (CNN) algorithm have been developed to determine the resonant frequency of the MPA. When the predicted and simulated resonant frequencies are compared, it is observed that they are almost identical. Different performance metrics, such as MAE, MSE, RMSE, and variance scores, are calculated to validate the prediction using the learning algorithms. These metrics are obtained through the process of computing the results of the prediction. The predicted results show that the error performances of the decision tree regression model are comparatively better than other models. The MAE, MSE, RMSE, and variance scores (in percentage) of the DTR model are 5.63, 0.71, 8.42, and 99.68, respectively. The XGB model (MAE = 7.03%, MSE = 1.06%, RMSE = 10.27%, and var score = 99.54%) performs better than the other learning models that were introduced in this study, except DTR. The performance of the deep learning model (CNN) is slightly lower than the presented regression models, which may have occurred due to the inadequate number of data samples for the CNN model. Despite the fact that the proposed MPA has two resonant frequencies, we have only predicted one (9 GHz) using ML models. In addition, the designed MPA has a lower gain of 4.06 dB at 9 GHz and 3.46 dB at 13 GHz. The measured resonance frequency range (7.90 GHz to 14.6 GHz) does not quite correspond to the predicted resonance frequency range (8.35 GHz to 14.25 GHz). In the future, we will generate an adequate number of data samples to achieve better results using DL models, such as CNN, and predict the multiple frequencies for a multiband antenna. Furthermore, we will develop the ML models to predict the return loss, gain, length, and width of the proposed antenna. Moreover, we will ensure better impedance matching between the proposed MPA and the SMA connector so that the simulated and measured frequencies are completely matched. Finally, it can be concluded that the simulated, measured, and predicted results ensure the reliability of the proposed antenna in the whole X-band and part of Ku-band applications.

Conclusions
This article discusses the integration of simulation, measurement, development of the RLC equivalent circuit model, and applying machine learning approaches to evaluate the performance of the proposed antenna. In terms of frequency, the designed antenna supports the whole X-band as well as some portions of the Ku-band. The prototype was built and analyzed to confirm the intended performance. In addition, the RLC equivalent model of the proposed MPA designed with the ADS Agilent software yields resonance frequencies nearly identical to those generated by simulation (with CST) and measurement. Furthermore, six machine learning and one deep learning (CNN) algorithm have been developed to determine the resonant frequency of the MPA. When the predicted and simulated resonant frequencies are compared, it is observed that they are almost identical. Different performance metrics, such as MAE, MSE, RMSE, and variance scores, are calculated to validate the prediction using the learning algorithms. These metrics are obtained through the process of computing the results of the prediction. The predicted results show that the error performances of the decision tree regression model are comparatively better than other models. The MAE, MSE, RMSE, and variance scores (in percentage) of the DTR model are 5.63, 0.71, 8.42, and 99.68, respectively. The XGB model (MAE = 7.03%, MSE = 1.06%, RMSE = 10.27%, and var score = 99.54%) performs better than the other learning models that were introduced in this study, except DTR. The performance of the deep learning model (CNN) is slightly lower than the presented regression models, which may have occurred due to the inadequate number of data samples for the CNN model. Despite the fact that the proposed MPA has two resonant frequencies, we have only predicted one (9 GHz) using ML models. In addition, the designed MPA has a lower gain of 4.06 dB at 9 GHz and 3.46 dB at 13 GHz. The measured resonance frequency range (7.90 GHz to 14.6 GHz) does not quite correspond to the predicted resonance frequency range (8.35 GHz to 14.25 GHz). In the future, we will generate an adequate number of data samples to achieve better results using DL models, such as CNN, and predict the multiple frequencies for a multiband antenna. Furthermore, we will develop the ML models to predict the return loss, gain, length, and width of the proposed antenna. Moreover, we will ensure better impedance matching between the proposed MPA and the SMA connector so that the simulated and measured frequencies are completely matched. Finally, it can be concluded that the simulated, measured, and predicted results ensure the reliability of the proposed antenna in the whole X-band and part of Ku-band applications.