Design geometry and analysis

The planned UWB BPF with notch characteristics and structural design progression are elaborated on in this section. Figure 1 describes the schematic layout of the designed structure with optimum dimensions. It is built on a substrate called RT/Duroid 5880 with a dielectric constant of 2.2, a tanδ value of 0.0009, and a thickness of 0.787 mm. A CSRR is engraved on the substrate’s underside. Two parallel open stub microstrip lines with dimensions of 4.5 × 0.25 mm are coupled to the CRR on the upper part of the substrate. These microstrip lines are open-circuited and joined with 50-Ω source and load lines. EM simulations are carried out with Ansys High-Frequency Structure Simulator (HFSS) to examine the performance of the suggested design. The fabricated prototype of the designed UWB filter with a notch has been illustrated in Fig. 2

Figure 1. Schematic of the designed UWB BPF along notch characteristics. (All structural metrics are displayed in mm: l ₁ = 4.5, l ₂ = 7.4, w ₁ = 0.25, w ₂ = 3.5, d = 2.0, s = 0.3, and g = 0.2).

Figure 2. Fabricated prototype of the proposed filter (a) top view, (b) bottom view.

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The corresponding lumped model of the circuit of the developed UWB BPF with a notch is depicted in Fig. 3 to allow for an examination of its characteristics. Lumped inductance L ₁ was generated because of the RR and coupling capacitance C ₁, mainly due to the gap (g) between the two RRs. Because of the open stub, the capacitance C ₃ and inductance L ₂ appear. The substrate generates the capacitance C ₂ between the top and bottom copper structure. The CSRR at the bottom of the structure generates the shunt lumped component inductance L_s and L_{s 1}, along with capacitance C _S. The circuit model has been designed on the Advanced Design System (ADS) circuit simulator, shown in Fig. 4. It consists of input and output feed as Term G1 and Term G2, and the frequency span for simulation is 0.01–13 GHz with step size of 0.01 GHz. The value of lumped elements used in equivalent circuits is represented by the variables (VAR). The following are the extracted lumped values of the intended UWB filter equivalent circuit using the ADS circuit simulator: L ₁ = 0.727 nH, L ₂ = 3.1 nH, L_s = 1.51 nH, L_{s 1} = 4.74 nH, C ₁ = 10.0 pF, C ₂ = 5.0 pF, and C ₃ = 0.07 pF. The scattering parameters of the designed circuit model on the ADS circuit simulator are illustrated in Fig. 5, which is in close agreement with EM simulated results.

Figure 3. Constructed circuit model of the designed filter.

Figure 4. Constructed circuit model of designed filter on ADS circuit simulator.

Figure 5. ADS circuit simulated scattering parameters of proposed UWB BPF with a notch.

The synthesis of the loaded elements has been computed using the formulas for dispersed elements as reference [Reference Bahl14], based on lumped element values of the equivalent circuit model based on the used substrate and its characteristics.

(1-a)\begin{equation}{C_1}\left(\,{pF} \right) = {\varepsilon _0}{\varepsilon _r}\left( {\frac{{{l_2} \times {w_2}}}{g}} \right)\end{equation}

(1-b)\begin{equation}{C_2}\left(\,{pF} \right) = {\varepsilon _0}{\varepsilon _r}\left( {\frac{{{A_{metal}}}}{{2{h_{sub}}}}} \right)\end{equation}

(1-c)\begin{equation}{L_1} \approx \frac{{{Z_{oc}}}}{{2\pi f}}\cot \left( {{\beta _{oc}}{l_2}} \right)\end{equation}

(1-d)\begin{equation}{L_2} \approx \frac{{{Z_{oc}}}}{{2\pi f}}\cot \left( {{\beta _{oc}}{l_1}} \right)\end{equation}

(1-e)\begin{equation}{C_3} \approx \frac{{{Z_{oc}}}}{{2\pi f}}\tan \left( {{\beta _{oc}}{l_1}} \right)\end{equation}

(1-f)\begin{equation}{L_s} = 400\pi r\left( {\ln \frac{{8s}}{{t + p}} - 0.5} \right)nH\end{equation}

(1-g)\begin{equation}{C_s} = 8.85\left( {\frac{{pt}}{d} + \frac{{2\pi t}}{{\ln \left[ {\frac{{2.4t}}{p}} \right]}} + \frac{{2t}}{\pi }\ln \frac{{4s}}{d}} \right)pF\end{equation}

where ε_r is the dielectric constant of the substrate,

h _sub is substrate thickness.

t is the thickness of printed copper,

A _metal is the metallic area of the filter structure, and

g ₀ is the gap between rectangular resonators

Z _oc is the characteristic impedance of the open-circuited strip,

l ₁ is the length of the open stub.

β _oc is the propagation constant, and

l ₂ represents the length of the rectangular resonator,

f denotes the center frequency of the UWB response.

s represents the gap of CSRR,

d is the gap between the ring and

p is the perimeter of the rectangular ring.

The structural layout of the proposed filter provides UWB response with a notch band, having TZ at the lower and upper sides of the passband. These TZ can be controlled independently by varying the structural parameters of the designed structure, as shown in Fig. 6. From Fig. 6(a), it can be predicted that the lower TZ can be controlled without varying the upper TZ. While the CSRR’s gap increases, the TZ frequency shifts toward a lower value without affecting the upper TZ. Similarly, in Fig. 6(b), the upper TZ can be controlled without affecting the lower TZ. While increasing the width “w ₁” of the open stub, the Upper TZ shifted toward a lower value without any change in other properties. The simulated and measured scattering parameters of the designed filter are illustrated in Fig. 7. The simulated and measured values indicate some change, but overall, there is good agreement between the findings. Connector losses, soldering process flaws, variations in the substrate thickness, or dielectric constant could cause the discrepancy between the measured and simulated findings. The filter’s passband ranges from 3.0–10.7 GHz, and its band rejection ranges from 6.25–6.75 GHz. The input reflection coefficient (IRC) [Reference Bird15] in the filter’s passband is more than 15.0 dB, and the insertion loss is only about 0.1 dB

Figure 6. Variation of transmission zeros with respect to: (a) Gap “s” of CSRR, (b) Width “w₁” of the open stub.

Figure 7. Scattering parameters of designed UWB BPF along notch characteristics.

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The variation of center frequency and bandwidth with respect to the length of the CSRR is illustrated in Fig. 8. Here, the 3 dB bandwidth has been calculated by taking the difference of higher 3 dB cutoff frequency (f ₂) and lower 3 dB cutoff frequency (f ₁) of the proposed UWB filter. At an optimized value of length of CSRR “l_g” (6.5 mm), the higher and lower 3 dB cutoff frequencies are 10.7 and 3.0 GHz, respectively. So, the bandwidth is 7.7 GHz by taking the difference of both the cutoff frequency. The center frequency has calculated by taking the average of both cutoff frequency, its value is 6.85 GHz. While increasing the length of CSRR “l_g,” the 3 dB bandwidth increases and its centre frequency gradually decreases.

Figure 8. Variation of bandwidth and centre frequency with respect to length “l_g” of CSRR.

Optimization of the UWB filter through machine learning

The machine learning (ML) flow chart of the proposed UWB filters is displayed in Fig. 9 [Reference Bird15–Reference Cao18]. The HFSS software designs the proposed UWB filter. An enormous amount of data is needed to use the ML algorithms in the proposed filter. The preparation of the dataset involves changing the UWB filter’s multiple parameters. A total of 322,677 datasets were generated, as given in Table 1.

Figure 9. Block diagram of machine learning in the proposed UWB filter.

Table 1. Dataset

The six variables d, g, W ₁, W_{g 1}, W_{s 1}, and frequency generate the dataset. The S parameters depend on the remaining parameters. The training and testing datasets are created from the entire dataset. The ML algorithms are trained on 80% of the dataset, with the remaining 20% being utilized for testing the models that have been trained. Five ML methods are employed in this work: artificial neural network (ANN), K-nearest neighbour (KNN), decision tree (DT), RF, and extreme gradient boosting (XGB) [Reference Goudos, Diamantoulakis, Matin, Sarigiannidis, Wan and Karagiannidis19–Reference Rai, Ranjan and Chowdhury26, Reference Sharma, Zhang and Xin29]. The mean square error (MSE), mean absolute error (MAE), and R ² score for each ML method are given in Table 2. Their R ² score shows the accuracy of our ML algorithms. The accuracy, testing time, and training time of several ML methods are shown in Figs. 10–12, respectively. Compared to other ML algorithms, RF ML methods had the best accuracy, as shown in Table 2 and Fig. 13. The S parameters of the proposed UWB filter are accurately predicted by the RF ML algorithm, as shown in Fig. 14. Equations (2) and (3) provide the mathematical expression for the ${R^2}$score and MSE, respectively.

Figure 10. Actual and predicted values of S-parameters in various ML algorithms.

Figure 11. Testing time of ML algorithms.

Figure 12. Training time of ML algorithms.

Figure 13. Accuracy of ML algorithms.

Figure 14. Comparison of HFSS and RF ML algorithm.

(2)\begin{equation}{R^2} = 1 - \frac{{Sum\;of\;squares\;of\;residuals}}{{Total\;sum\;of\;squares}}\end{equation}

(3)\begin{equation}MSE = \frac{1}{n}\mathop \sum \limits_{i = 1}^n {\left( {{Y_i} - {Y_j}} \right)^2}\end{equation}

Where Y_i has observed values,

n is the number of data points, and

Y_j is predicted values.

Table 2. MAE, MSE, and R ² score of various ML algorithms

Table 3 compares the proposed UWB BPF with other comparable studies described in the literature. It includes electrical size, passband bandwidth, notch frequency band, IRC, insertion loss, number of TZs, used substrate, and techniques. The comparison table reveals that the suggested filter performs better than the previously published studies. The size is represented in λ_g, representing guided wavelength at a lower cutoff frequency. The presented UWB BPF has an electrical size of 0.15 × 0.10 λ_g.

Table 3. Comparison of designed structure with similar work reported in the literature

Note: IRC = input reflection coefficient, TZ = transmission zeros.