Deep-learning-based neural network for design of a dual-band coupled-line trans-directional coupler

A deep-learning-based model to automate the design of a dual-band coupled-line trans-directional (CL-TRD) coupler can greatly improve upon the current techniques that rely on complicated analysis. In this paper, we propose a convolutional neural network (CNN), which is a type of deep learning, which can rapidly output the parameters of dual-band directional couplers corresponding to theoretical (ideal) specifications of the electrical parameters through an inverse model. The neural network training data are generated using the HFSS electromagnetic simulation tool by varying the geometrical design parameters of the coupler. To validate the robustness of the CNN inverse model, it is applied on a 3-dB dual-band CL-TRD coupler operating at 1.2/4 GHz, and compared with a shallow neural network, namely, a radial basis function neural network (RBFNN). The coupler parameters designed by both neural networks are verified by HFSS. The results reveal that the CNN-simulated S-parameters and output port phase difference are in good agreement with the ideal values compared with those of the RBFNN, with greater accuracy and speed. The designed coupler was fabricated and measured for verification.


Introduction
Directional couplers play a vital role in microwave systems such as radar systems, balanced mixers, amplifiers, and beamforming networks [1,2].A closed analytical solution of different couplers can be developed for ideal transmission line sections.However, for practical implementation using real transmission line sections such as microstrip lines, a tuning process is required to include different parameters such as discontinuities and coupling between line sections.Thus, the most common approach to design directional couplers in practice relies heavily on trial and error.Therefore, it is necessary to repeatedly run electromagnetic (EM) simulators, such as the high-frequency structure simulator (HFSS), to calculate electrical properties from physical parameters and update these physical parameters continuously for many iterations until all electrical specifications are met.This approach is quite time-consuming.Moreover, for the case of dual-band operation, the tuning process of the coupler becomes more complicated.
On the other hand, the design parameters of the coupler can be extracted using inverse modeling based on the required electrical parameters.The neural network (NN) is considered a good candidate for solving inverse problems [3,4].NNs have recently been utilized as a robust inverse model in designing microwave components [5,6].The key advantage of inverse modeling is that it does not require iterative processes to obtain the design parameters.Hence, it is much faster than EM-simulation-based methods.Moreover, the NN can be used for repetitive designs once it is well trained.Recently, in [7] and [8], NN inverse modeling was applied to the design of directional couplers.However, these approaches used conventional (shallow) NN techniques 1 3 which are not suitable for high-dimensional problems with many input variables because the model training in this case becomes too complicated.
On the other hand, deep-learning-based approaches [9] can deal with such high-dimensional inverse problems [10,11].A convolutional neural network (CNN) is a deep learning structure.It has the advantage of fast training, as it has a small number of parameters due to its parallel structure and shared weights.Owing to its ability to extract the hidden features in the input data, the CNN can deal with any degree of correlation among data [12].It has recently been applied to extract the coupling matrix of microwave filters [13].In this paper, a CNN is applied as an inverse model to extract the design variables of a dual-band coupled-line trans-directional (CL-TRD) coupler using its required magnitude of S-parameters and phase difference (PD) between its output ports at the two required bands.
The proposed dual-band coupler consists of symmetric coupled line sections loaded by shunt capacitors.The dimensions of these coupled line sections, the values of the loading shunt capacitors, and the spacing distances between the capacitors represent the output variables of the CNN.The required magnitudes of the S-parameters and the corresponding PD represent the inputs to the CNN.The CNN is compared with a radial basis function neural network (RBFNN) [13,14].The RBFNN model is a standard three-layer feedforward NN, which consists of an input layer, an output layer, and a hidden layer.Thus, it is a shallow (non-deep) NN, since it has only one hidden layer.The CNN is found to be faster than the RBFNN, with much greater accuracy in designing the CL-TRD coupler.
This paper is organized as follows.Section 2 presents the formulation of the proposed problem.Section 3 presents the RBFNN used to model this problem, and Sect. 4 presents the CNN used to model the problem.To verify the performance of the CNN-based inverse model, the simulation results are presented in Sect.5, including a comparison between the results of the two models and the results of the full-wave analysis of the proposed problem.Section 6 presents a 3-dB fabricated prototype of the coupler and the measured results, followed by concluding remarks in Sect.7.

Formulation of the problem
We propose the design of a 3-dB dual-band CL-TRD coupler [2] as shown in Fig. 1 using NN inverse modeling.The coupler is designed to operate in two center frequencies, 1.2 GHz and 4 GHz.The required coupling coefficient in the two bands is 3 dB.The phase difference between the two output ports is + 90 • for the lower band and -90 • for the upper band.The proposed coupler includes eight geometrical design dimensions: l 1 , l 2 , s 1 , s 2 , w 1 and w 2 , as shown in Fig. 1.In addition, the coupled line sections are loaded by lumped capacitors c 1 and c 2 for lines l 1 and l 2 , respectively.Three loading capacitors are used per line as shown in Fig. 1.The spacing distance between these capacitors is x 1 and x 2 , respectively, which represent two additional geometrical parameters.The input port is port 1.For ideal operation, the input power is coupled to port 3, while the through signal is at port 4. In this case, port 2 is isolated.The four ports of the coupler are designed to have a characteristic impedance Z 0 = 50Ω .The electrical parameters of the coupler are the magnitudes of the S-parameters S 11 , S 12 , S 13 , and S 14 and the phase difference (PD) between output ports 3 and 4. The proposed coupler is designed based on a Rogers RO4003 substrate, with r = 3.38 , loss tangent of 0.0027 and substrate thickness of 1.524 mm.Different simulations with random input parameters are obtained using full-wave analysis.These simulations are used to train and validate the proposed NN.
Figure 2 shows the NN inverse model for designing the coupler variables given its electrical parameters.1.
The training and validation patterns of this problem are generated by the HFSS EM simulator.These patterns are obtained using random values within the mentioned intervals for the corresponding ten parameters of the coupler.A total of 1535 data sets are generated, of which 90% are randomly chosen for training and the remaining 10% for validation.For each sample of generated data sets, the data of coupler variables and the corresponding S-parameters plus PD are swapped for training the NN inverse model.The trained NN is tested by the ideal set of S-parameters and PD (which is never used in the training); then the output variables are simulated, and the simulated S-parameters and PD are compared to the ideal ones.

RBFNN model
The RBFNN is a special three-layer feed-forward network which consists of an input layer, an output layer, and a single hidden layer as shown in Fig. 3.In the hidden layer, the nonlinear functions are usually considered to be Gaussian functions of appropriately chosen means and variance.The weights from the hidden to the output layer are determined by considering a supervised learning procedure.Assume that input, hidden, and output layers have J, K, and P nodes, respectively.The network output vector is given by where z = [z 1 z 2 … z J ] is the input vector, c k and σ k are the center vector and the standard deviation (spread parameter) of the kth Gaussian function, respectively, and {ω p,k ; p = 1,…, P; k = 1,…, K} is the weight from the kth hidden node to pth output node.
In the present case, the RBFNN consists of three layers: input, hidden, and output layer.The input layer consists of 1105 neurons, which is the same number as the input data.The hidden layer has 350 neurons, while the output layer consists of 10 neurons, which correspond to the required outputs.

CNN model
CNNs have one or more convolutional layers to extract the discriminative features from the input data via feature maps.After all the convolutional layers, these learned features are then aggregated to the vectors by the fully connected layers for the regression task.The proposed CNN architecture is shown in Fig. 4.
The feature map of the convolutional layer is defined as (1)

ReLU
Fig. 4 The proposed CNN structure where x l j is the jth feature map at the lth layer, f Conv () is the activation function of the convolutional layer, M is the number of feature maps at layer l -1, x l−1 i is the ith feature map at the (l -1)th layer, the symbolε * ε is the operation of convolution, k l ij is the kernel between the ith feature map at the (l -1)th layer and the jth feature map at the lth layer, and b l j is the bias for the jth feature map at the lth layer.The fully connected layer is defined as where y i is the value of the ith output from the fully con- nected layer, f FC () is the activation function of the fully con- nected layer, N is the number of feature maps before the fully connected layer, M is the number of neurons of each feature map, w is the weight, and m is the number of outputs from the fully connected layer.
In the present case, there are three convolutional (Conv) layers.As in the case of RBFNN, the input layer consists of 1105 neurons, which is the same number as the input data, and the first Conv layer includes N = eight feature maps (FMs).The size of all FMs in the present analysis is 2 × 2, which corresponds to M = 4 in (2).The second and third Conv layers include N = 16 and N = 32 FMs, respectively.All Conv layers have a stride of 1 and the "same" padding.Then, there is a single fully connected (FC) layer with 150 nodes.Finally, there is an output layer with 10 nodes.A dropout with a rate of 50% is utilized to reduce overfitting.The activation functions of Conv layers are rectified linear unit (ReLU).The FC layer uses exponential linear unit (ELU) as an activation function.Because of the regression problem, a linear activation is used at the output layer.

Simulation results
To verify the performance of the CNN-based inverse model, it is applied for the design of the dual-band CL-TRD coupler shown in Fig. 1.The Adam (adaptive momentum) optimization algorithm [15] is used to update the network weights, (2) and the loss function used for this network is the mean squared error.The value of the learning rate is 0.001.The batch size is 4 and the number of epochs is 10.To further verify the performance of the CNN, it is compared to that of the RBFNN.The CNN and RBFNN are implemented in MATLAB R2020b using the Deep Learning Toolbox.Both NNs have the same number of inputs (1105) and outputs (10) as well as the same size of training and validation data sets (1381 and 154, respectively).Both the trained networks are tested by the ideal S-parameters and PD (which never appear during the training), and then the extracted variables from both networks are simulated in HFSS to compare the simulation results with the ideal ones.
Table 1 shows the designed coupler variables (the eight dimensions and the two values of the loading capacitors) using both RBFNN and CNN, and also shows the optimized variables obtained using the optimization tool included in the HFSS simulator.It can be noted that the variable values of the CNN are much closer to the ideal case than those of the RBFNN.Although the shallow RBFNN has 350 hidden neurons, it cannot represent this high-dimensional input-output relationship effectively.From this result, it can be concluded that the proposed CNN modeling technique is more appropriate for this high-dimensional modeling problem.
Table 2 shows the training and execution (testing) times and the percentage root mean square error (RMSE) between the optimized variables of [2] and those extracted by both NNs.It can be seen that the CNN inverse modeling for the design of the dual-band coupler yields extremely high accuracy, and less training and execution time than the RBFNN modeling.The smaller error of the CNN and its higher speed indicate that it is a good candidate for modeling high-dimensional problems like dual-band CL-TRD coupler.
Figure 5 shows the simulated frequency responses of the designed couplers based on CNN and RBFNN corresponding to the final coupler variables in Table 1 compared to the ideal responses.Good agreement is observed between the design based on CNN and the ideal responses.This is not the case for the design based on RBFNN.It can also be noted that the CNN has the capability to extract the hidden features in the input data, S-parameters and PD, automatically.On the other hand, because the RBFNN is shallow, it cannot

Experimental results
To validate the proposed CNN-based inverse model, the designed coupler was fabricated and measured.The fabricated prototype is shown in Fig. 6a.The overall size of the coupler is 90 mm × 15 mm.The fabricated coupler is measured using the Rhode and Schwarz ZVA67 vector network analyser, as shown in Fig. 6b.A comparison between the measured S-parameters and the corresponding simulated

Conclusion
A CNN-based inverse model is proposed to design a dualband CL-TRD coupler based on the required S-parameters and phase difference between its output ports.The results show that the proposed CNN method can be reliably used to perform the design of microwave directional couplers.Compared to a shallow neural network like RBFNN, the deeplearning CNN is much more accurate and faster in extracting the coupler design variables to obtain the required performance.The designed coupler is fabricated and measured for verification.The results show good agreement between simulation and measurement.
Fig.1.In addition, the coupled line sections are loaded by lumped capacitors c 1 and c 2 for lines l 1 and l 2 , respectively.Three loading capacitors are used per line as shown in Fig.1.The spacing distance between these capacitors is x 1 and x 2 , respectively, which represent two additional geometrical parameters.The input port is port 1.For ideal operation, the input power is coupled to port 3, while the through signal is at port 4. In this case, port 2 is isolated.The four ports of the coupler are designed to have a characteristic impedance Z 0 = 50Ω .The electrical parameters of the coupler are the magnitudes of the S-parameters S 11 , S 12 , S 13 , and S 14 and the phase difference (PD) between output ports 3 and 4. The proposed coupler is designed based on a Rogers RO4003 substrate, with r = 3.38 , loss tangent of 0.0027 and substrate thickness of 1.524 mm.Different simulations with random input parameters are obtained using full-wave analysis.These simulations are used to train and validate the proposed NN.Figure2shows the NN inverse model for designing the coupler variables given its electrical parameters.The input to the NN is the vectors |S 11 |, |S 12 |, |S 13 |, and |S 14 |, representing the scalar magnitudes of the four S-parameters, besides the vector PD (in degrees).Each vector of the five input vectors is simulated at 221 frequency points in a range from 0.4 GHz to 4.8 GHz, which corresponds to a frequency step of 20 MHz.Therefore, the total number of inputs is 1105.The output of the NN inverse model is a vector of eight dimensions of the coupler l 1 , l 2 , s 1 , s 2 , w 1 , w 2 , x 1 and x 2 and the two values of the capacitors c 1 and c 2 , as shown in

Fig. 5 Fig. 6 a
Fig. 5 Comparison of the simulated a return loss, b isolation, c coupling, d through and e phase difference results and the simulated responses of RBFNN and CNN

Fig. 7
Fig. 7 Simulated and measured a return loss, b isolation, c coupling, d through and e phase difference results of the fabricated coupler

Table 1
The coupler design parameters for the proposed dual band

Table 2
Percentage RMSE and computation time of RBFNN and CNN strengthen the network training process by reconstructing the input S-parameters and PD.