## 3 The Architecture of the Proposed ANN Model  
A radial basis function (RBF) is used to generate an artificial neural network (ANN). The RBF ANN is a type of artificial neural network that has become increasingly popular  
Figure 8: The phase characteristic of the proposed \(\lambda/4\) and \(3\lambda/4\) filters. Figure 7: Proposed equivalent filter: (**a**) layout, and (**b**) frequency response. in recent years due to its notable capabilities in functional approximation and pattern recognition. The RBF architecture consists of three layers: input, hidden and an output layer. The hidden layer uses a radial basis function as its activation function, which allows the RBF neural network to perform non-linear transformations of the input data. The output layer is typically a linear function of the weighted inputs from the hidden layer. Figure 9 shows the proposed RBF structure for the defined artificial network and its neuron model. According to this figure, the input parameters are connected to output nodes by a single hidden layer. The filter dimensions are considered to be the input parameters for the proposed RBF model, which are W\({}_{1}\), W\({}_{2}\), W\({}_{3}\), L\({}_{1}\), L\({}_{2}\), L\({}_{3}\), S\({}_{1}\), and S\({}_{2}\), while the output parameters of the RBF model are cutoff frequency \(\left\{f_{\rm c}\right\}\), stopband bandwidth (STBW), and insertion loss (IL) parameters. The maximum number of 70 neurons are considered in the hidden layer of the proposed ANN model. Additionally, a gradient descent-based training algorithm of Bayesian regularization backpropagation algorithm is used for the training process. The Bayesian regularization algorithm combines the advantages of regularized least-squares and backpropagation methods to train the RBF network efficiently. Applied regularization techniques can prevent the over fitting problem from occurring in the network. In addition, regularization helps to control the complexity of the RBF model and improve its generalization performance by adding regularization terms to the objective function during training. The mean-squared-error (MSE) value is used to quantify the average squared difference between the network predicted outputs and the target outputs during the training process. Lower MSE values indicate a better training performance and the training goal is considered as zero in the proposed model, which is the desired value for the performance metric of MSE. The Gaussian activation function is utilized for the hidden layer in the proposed model. The spread (width) of the radial basis functions in the hidden layer is considered to be equal to 1, which determines the influence of each radial basis function on the network output. The performance step size and the transfer function step size, which are considered during performance gradient calculation, are equal to 0.001 and 0.01, respectively. The 'newrb' function in the Neural Network Toolbox in MATLAB software is used to design the presented RBF ANN model. Figure 9: (a) Proposed structure of the RBF-ANN model including a single hidden layer. (b) Neuron model of the RBF network. The analysis of the presented RBF network can be written as shown in Equations (1)-(3) [43; 44]. The Gaussian function, which is used as the radial basis function in the RBF network, is written in Equation (1). \[\varphi(x,c,\sigma)=e^{(-\frac{|x-c|^{2}}{2\sigma^{2}})}, \tag{1}\]  
where '\(x\)' is the input vector, '\(c\)' is the center of radial basis function, and '\(\sigma\)' is the width of the radial basis function. By applying the radial basis function to the input pattern and the center of the neuron, the activation of a neuron in the RBF can be computed, which is written in Equation (2). \[a_{i}=\varphi(x,c_{i},\sigma_{i}) \tag{2}\]  
In this equation, '\(x\)' is the input vector. Additionally, '\(c_{i}\)' and '\(\sigma_{i}\)' are the center and width of the \(i\)th neuron.