# Novel Decomposition Technique on Rational-Based Neuro-Transfer Function for Modeling of Microwave Components

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## Abstract

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## 1. Introduction

## 2. The High-Sensitivity Issue in the Existing Rational-Based Neuro-TF Method

## 3. Proposed Decomposition Technique for Development of Rational-Based Neuro-TF Model

#### 3.1. Concept of the Decomposition Technique for Rational-Based Neuro-TF Model

#### 3.2. Proposed Decomposition Technique for Parameter Extraction and Model Development

- Step 1:
- Step 2:
- Step 3:
- Initialize the number M of the sub-neuro-TF models to two.
- Step 4:
- Step 5:
- Obtain the pole and residue data ${\widehat{\mathit{p}}}_{i}^{\left(k\right)}$ and ${\widehat{\mathit{r}}}_{i}^{\left(k\right)}$ by (30)–(31) for the ith sub-neuro-TF model. Convert data ${\widehat{\mathit{p}}}_{i}^{\left(k\right)}$ and ${\widehat{\mathit{r}}}_{i}^{\left(k\right)}$ into data ${\widehat{\mathit{a}}}_{i}^{\left(k\right)}$ and ${\widehat{\mathit{b}}}_{i}^{\left(k\right)}$. Obtain the training data $({\mathit{x}}_{k},{\widehat{\mathit{a}}}^{\left(k\right)})$ and $({\mathit{x}}_{k},{\widehat{\mathit{b}}}^{\left(k\right)})$ by (32)–(33) for the two neural networks $\widehat{\mathit{a}}(\mathit{x},{\mathit{w}}_{a})$ and $\widehat{\mathit{b}}(\mathit{x},{\mathit{w}}_{b})$.
- Step 6:
- Perform the preliminary training of the two neural networks and refinement training of the overall model by (34).
- Step 7:
- Use the training data (${\mathit{x}}_{k}$, ${\mathit{d}}_{k}$) to verify the trained overall model. If the training error ${E}_{Tr}$ is lower than a user-defined threshold ${E}_{t}$, go to Step 8. Otherwise, increase the number of hidden neurons and go to Step 6.
- Step 8:
- Use the testing data to verify the overall model. If the testing error ${E}_{Ts}$ is lower than the user-defined threshold ${E}_{t}$, go to Step 9. Otherwise, increase the number M of sub-neuro-TF models by one (i.e., $M=M+1$) and go to Step 4.
- Step 9:
- Stop the modeling process.

## 4. Application Examples

#### 4.1. Three-Order Waveguide Filter Modeling

#### 4.2. Four-Order Bandpass Filter Modeling

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Structure for the rational-based neuro-TF model incorporating the proposed decomposition technique.

**Figure 2.**The flow diagram of the overall model development process using the proposed decomposition technique.

**Figure 4.**Structure for the proposed rational-based neuro-TF model with $M=3$ of the three-order waveguide filter example.

**Figure 5.**Comparison of magnitude in decibels of ${S}_{11}$ different modeling approaches and EM data: (

**a**) test sample $\mathit{x}=[$14.37 14.57 9.25 6.17${]}^{T}$ (mm), (

**b**) test sample $\mathit{x}=[$14.17 15.20 8.75 6.25${]}^{T}$ (mm), and (

**c**) test sample $\mathit{x}=[$14.56 15.20 8.63 6.17${]}^{T}$ (mm) for the three-order waveguide filter example.

**Figure 6.**Comparison of $\mathrm{Re}\left({S}_{11}\right)$ and $\mathrm{Im}\left({S}_{11}\right)$ for different modeling approaches and EM data at test sample $\mathit{x}=[$14.56 15.20 8.63 6.17${]}^{T}$ (mm) for the three-order waveguide filter example. (

**a**) $\mathrm{Re}\left({S}_{11}\right)$ and (

**b**) $\mathrm{Im}\left({S}_{11}\right)$.

**Figure 8.**Structure for the proposed rational-based neuro-TF model with $M=3$ of the four-order bandpass filter example.

**Figure 9.**Comparison of magnitude in decibels of ${S}_{11}$ for different modeling approaches and EM data: (

**a**) test sample $\mathit{x}=[$3.52 4.46 3.96 3.28 3.07${]}^{T}$ (mm), (

**b**) test sample, $\mathit{x}=[$3.60 4.22 4.20 3.40 3.10${]}^{T}$ (mm), and (

**c**) test sample $\mathit{x}=[$3.36 4.38 4.00 3.36 3.02${]}^{T}$ (mm) for the four-order bandpass filter example.

**Figure 10.**Comparison of $\mathrm{Re}\left({S}_{11}\right)$ and $\mathrm{Im}\left({S}_{11}\right)$ for different modeling approaches and EM data at test sample $\mathit{x}=[$3.36 4.38 4.00 3.36 3.02${]}^{T}$ (mm) for the four-order bandpass filter example. (

**a**) $\mathrm{Re}\left({S}_{11}\right)$ and (

**b**) $\mathrm{Im}\left({S}_{11}\right)$.

**Table 1.**Comparison of the sensitivity of rational-based neuro-transfer function (neuro-TF) model response with respect to the coefficients with/without decomposition.

Coeff. | Transfer Function | Sensitivity of Transfer Function Response w.r.t the Coeff. | |
---|---|---|---|

Existing Rational-Based Neuro-TF model (without Decom-position) | ${a}_{j}$ | $y=\frac{{\displaystyle \sum _{j=1}^{N}{a}_{j}{s}^{j-1}}}{{\displaystyle 1+\sum _{j=1}^{N}{b}_{j}{s}^{j}}}$ | $\frac{{\displaystyle \partial y}}{{\displaystyle \partial {a}_{j}}}=\frac{{\displaystyle {s}^{j-1}}}{{\displaystyle 1+\sum _{j=1}^{N}{b}_{j}{s}^{j}}}=\frac{{\displaystyle {s}^{j-1}}}{{\displaystyle \prod _{j=1}^{{N}_{eff}}\left(s-{p}_{j}\right)\xb7\prod _{j=1}^{{N}_{eff}}\left(s-{p}_{j}^{*}\right)}}$ |

${b}_{j}$ | $\frac{{\displaystyle \partial y}}{{\displaystyle \partial {b}_{j}}}=\frac{{\displaystyle -{s}^{j}\sum _{j=1}^{N}{a}_{j}{s}^{j-1}}}{{\left({\displaystyle 1+\sum _{j=1}^{N}{b}_{j}{s}^{j}}\right)}^{2}}=\frac{{\displaystyle -y\xb7{s}^{j}}}{{\displaystyle \prod _{j=1}^{{N}_{eff}}\left(s-{p}_{j}\right)\xb7\prod _{j=1}^{{N}_{eff}}\left(s-{p}_{j}^{*}\right)}}$ | ||

Proposed Rational-Based Neuro-TF model (with Decom-position) | ${\widehat{a}}_{ij}$ | $y=\sum _{i=1}^{M}{y}_{i}=\sum _{i=1}^{M}\frac{{\displaystyle \sum _{j=1}^{2{N}_{i}}{\widehat{a}}_{ij}{s}^{j-1}}}{{\displaystyle 1+\sum _{j=1}^{2{N}_{i}}{\widehat{b}}_{ij}{s}^{j}}}$ | $\frac{{\displaystyle \partial y}}{{\displaystyle \partial {\widehat{a}}_{ij}}}=\frac{{\displaystyle {s}^{j-1}}}{{\displaystyle 1+\sum _{j=1}^{2{N}_{i}}{\widehat{b}}_{ij}{s}^{j}}}=\frac{{\displaystyle {s}^{j-1}}}{{\displaystyle \prod _{j=1}^{{N}_{i}}\left(s-{p}_{n(i,j)}\right)\xb7\prod _{j=1}^{{N}_{i}}\left(s-{p}_{n(i,j)}^{*}\right)}}$ |

${\widehat{b}}_{ij}$ | $\frac{{\displaystyle \partial y}}{{\displaystyle \partial {\widehat{b}}_{ij}}}=\frac{{\displaystyle -{s}^{j}\sum _{j=1}^{2{N}_{i}}{\widehat{a}}_{ij}{s}^{j-1}}}{{\left({\displaystyle 1+\sum _{j=1}^{2{N}_{i}}{\widehat{b}}_{ij}{s}^{j}}\right)}^{2}}=\frac{{\displaystyle -{y}_{i}\xb7{s}^{j}}}{{\displaystyle \prod _{j=1}^{{N}_{i}}\left(s-{p}_{n(i,j)}\right)\xb7\prod _{j=1}^{{N}_{i}}\left(s-{p}_{n(i,j)}^{*}\right)}}$ |

Geometrical Parameters (mm) | Training Samples (49 Samples) | Test Samples (49 Samples) | |||||
---|---|---|---|---|---|---|---|

Min | Max | Steps | Min | Max | Steps | ||

Case 1 (Narrower Range) | ${L}_{1}$ | 13.84 | 14.12 | 0.05 | 13.86 | 14.10 | 0.04 |

${L}_{2}$ | 15.05 | 15.35 | 0.05 | 15.07 | 15.33 | 0.04 | |

${W}_{1}$ | 8.91 | 9.09 | 0.03 | 8.93 | 9.08 | 0.03 | |

${W}_{2}$ | 5.94 | 6.06 | 0.02 | 5.95 | 6.05 | 0.02 | |

Case 2 (Increased Range) | ${L}_{1}$ | 13.70 | 14.26 | 0.09 | 13.75 | 14.21 | 0.08 |

${L}_{2}$ | 14.90 | 15.50 | 0.10 | 14.95 | 15.45 | 0.08 | |

${W}_{1}$ | 8.82 | 9.18 | 0.06 | 8.85 | 9.15 | 0.05 | |

${W}_{2}$ | 5.88 | 6.12 | 0.04 | 5.90 | 6.10 | 0.03 | |

Case 3 (Wider Range) | ${L}_{1}$ | 13.28 | 14.68 | 0.23 | 13.40 | 14.56 | 0.19 |

${L}_{2}$ | 14.44 | 15.96 | 0.25 | 14.57 | 15.83 | 0.21 | |

${W}_{1}$ | 8.55 | 9.45 | 0.15 | 8.63 | 9.38 | 0.13 | |

${W}_{2}$ | 5.70 | 6.30 | 0.10 | 5.75 | 6.25 | 0.08 |

**Table 3.**Comparisons of different rational-based neuro-TF modeling approaches for the three-order waveguide filter example.

Modeling Methods | No. of Sub-Models | ${\mathit{N}}_{\mathbf{eff}}$ or ${\mathit{N}}_{\mathit{i}}$ | No. of Hidden Neurons | Average Training Error | Average Testing Error | ||
---|---|---|---|---|---|---|---|

Case 1 (Narrower Range) | Existing Rational Neuro-TF Method | 1 | 5 | NN for Numerator | 10 | 0.630 % | 0.718 % |

NN for Denominator | 10 | ||||||

Proposed Rational Neuro-TF Method | 2 | 3 2 | NN for Numerator | 10 | 0.239 % | 0.267 % | |

NN for Denominator | 10 | ||||||

Case 2 (Increased Range) | Existing Rational Neuro-TF Method | 1 | 5 | NN for Numerator | 10 | 1.953% | 2.017% |

NN for Denominator | 10 | ||||||

Proposed Rational Neuro-TF Method | 2 | 3 2 | NN for Numerator | 10 | 0.467% | 0.496% | |

NN for Denominator | 10 | ||||||

Case 3 (Wider Range) | Existing Rational Neuro-TF Method | 1 | 5 | NN for Numerator | 10 | 5.073% | 6.604% |

NN for Denominator | 10 | ||||||

1 | 5 | NN for Numerator | 40 | 3.222% | 50.69% | ||

NN for Denominator | 40 | ||||||

Proposed Rational Neuro-TF Method | 2 | 3 2 | NN for Numerator | 10 | 1.490% | 1.840% | |

NN for Denominator | 10 | ||||||

3 | 2 2 1 | NN for Numerator | 10 | 0.746% | 0.962% | ||

NN for Denominator | 10 |

Geometrical Parameters (mm) | Training Samples (81 Samples) | Test Samples (64 Samples) | |||||
---|---|---|---|---|---|---|---|

Min | Max | Steps | Min | Max | Steps | ||

Case 1 (Narrower Range) | ${h}_{1}$ | 3.4 | 3.56 | 0.02 | 3.41 | 3.55 | 0.02 |

${h}_{2}$ | 4.3 | 4.46 | 0.02 | 4.31 | 4.45 | 0.02 | |

${h}_{3}$ | 4.0 | 4.16 | 0.02 | 4.01 | 4.15 | 0.02 | |

${h}_{c1}$ | 3.2 | 3.36 | 0.02 | 3.21 | 3.35 | 0.02 | |

${h}_{c2}$ | 2.9 | 3.06 | 0.02 | 2.91 | 3.05 | 0.02 | |

Case 2 (Wider Range) | ${h}_{1}$ | 3.3 | 3.62 | 0.04 | 3.32 | 3.6 | 0.04 |

${h}_{2}$ | 4.2 | 4.52 | 0.04 | 4.22 | 4.5 | 0.04 | |

${h}_{3}$ | 3.9 | 4.22 | 0.04 | 3.92 | 4.2 | 0.04 | |

${h}_{c1}$ | 3.1 | 3.42 | 0.04 | 3.12 | 3.4 | 0.04 | |

${h}_{c2}$ | 2.8 | 3.12 | 0.04 | 2.82 | 3.1 | 0.04 |

**Table 5.**Comparisons of different rational-based neuro-TF modeling approaches for the four-order bandpass filter example.

Modeling Methods | No. of Sub-Models | ${\mathit{N}}_{\mathbf{eff}}$ or ${\mathit{N}}_{\mathit{i}}$ | No. of Hidden Neurons | Average Training Error | Average Testing Error | ||
---|---|---|---|---|---|---|---|

Case 1 (Narrower Range) | Existing Rational Neuro-TF Method | 1 | 6 | NN for Numerator | 10 | 4.448% | 4.674% |

NN for Denominator | 10 | ||||||

1 | 6 | NN for Numerator | 40 | 2.562% | 9.121% | ||

NN for Denominator | 40 | ||||||

Proposed Rational Neuro-TF Method | 2 | 3 3 | NN for Numerator | 10 | 1.487% | 1.672% | |

NN for Denominator | 10 | ||||||

3 | 2 2 2 | NN for Numerator | 10 | 0.876% | 1.015% | ||

NN for Denominator | 10 | ||||||

Case 2 (Wider Range) | Existing Rational Neuro-TF Method | 1 | 6 | NN for Numerator | 10 | 6.809% | 8.382% |

NN for Denominator | 10 | ||||||

1 | 6 | NN for Numerator | 40 | 4.791% | 32.23% | ||

NN for Denominator | 40 | ||||||

Proposed Rational Neuro-TF Method | 2 | 3 3 | NN for Numerator | 10 | 2.252% | 4.397% | |

NN for Denominator | 10 | ||||||

2 | 3 3 | NN for Numerator | 40 | 1.877% | 16.21% | ||

NN for Denominator | 40 | ||||||

3 | 2 2 2 | NN for Numerator | 10 | 1.624% | 1.982% | ||

NN for Denominator | 10 |

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## Share and Cite

**MDPI and ACS Style**

Zhao, Z.; Feng, F.; Zhang, J.; Zhang, W.; Jin, J.; Ma, J.; Zhang, Q.-J.
Novel Decomposition Technique on Rational-Based Neuro-Transfer Function for Modeling of Microwave Components. *Micromachines* **2020**, *11*, 696.
https://doi.org/10.3390/mi11070696

**AMA Style**

Zhao Z, Feng F, Zhang J, Zhang W, Jin J, Ma J, Zhang Q-J.
Novel Decomposition Technique on Rational-Based Neuro-Transfer Function for Modeling of Microwave Components. *Micromachines*. 2020; 11(7):696.
https://doi.org/10.3390/mi11070696

**Chicago/Turabian Style**

Zhao, Zhihao, Feng Feng, Jianan Zhang, Wei Zhang, Jing Jin, Jianguo Ma, and Qi-Jun Zhang.
2020. "Novel Decomposition Technique on Rational-Based Neuro-Transfer Function for Modeling of Microwave Components" *Micromachines* 11, no. 7: 696.
https://doi.org/10.3390/mi11070696