# Statistical analysis and modeling of soft or flexible antennas in fluctuating conditions

Jinxin Du

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## To cite this version:

Jinxin Du. Statistical analysis and modeling of soft or flexible antennas in fluctuating conditions. Electromagnetism. Télécom ParisTech, 2018. English. NNT: 2018ENST0036. tel-04501004

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Submitted on 12 Mar 2024

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ParisTech

INSTITUT DES SCIENCES ET TECHNOLOGIES
PARIS INSTITUTE OF TECHNOLOGY

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2018-ENST-0036

EDITE ED 130

Doctorat ParisTech

THÈSE

pour obtenir le grade de docteur délivré par

Télécom ParisTech

Spécialité "Communications & Electronique"

présentée et soutenue publiquement par

Jinxin DU

Le 03/07/2018

Statistical analysis and modeling of soft or flexible antennas
in fluctuating conditions

Directeur de thèse : **Christophe ROBLIN**

Jury

M. Ala SHARAIHA, Professeur, IETR - Université de Rennes 1

Président du jury

M. Philippe BESNIER, Directeur de Recherche CNRS, IETR

Rapporteur

M. Hendrik ROGIER, Professeur, Université de Ghent

Rapporteur

Mme Shermila MOSTARSHEDI, Maître de conférence, Université Paris-Est Marne-La-Vallée

Examinatrice

Mme Odile PICON, Professeur émérite, Université Paris-Est Marne-La-Vallée

Examinatrice

M. Joe WIART, Titulaire de la chaire C2M, Télécom ParisTech

Examinateur

M. Christophe ROBLIN, Maître de conférence, Télécom ParisTech

Directeur de thèse

T
H
È
S
E

Télécom ParisTech

école de l'Institut Mines Télécom - membre de ParisTech

46, rue Barrault - 75634 Paris Cedex 13 - Tél. +33 (0)1 45 81 77 77 - www.telecom-paristech.fr
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Statistical Analysis and Modeling of Soft or Flexible
Antennas in Fluctuating Conditions

Jinxin DU

Supervisor: Prof. Christophe ROBLIN

Département Communications et Electronique (COMELEC)
Télécom ParisTech (Institut Mines-Télécom and LTCI), Université Paris-Saclay
46 rue Barrault, Paris CEDEX 13, 75634, France
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Acknowledgement

I wish to begin this dissertation by expressing my sincere thanks to people who have
helped me a lot during the past three and a half years. Without their help, I cannot fulfill my
thesis in such an enthusiastic and pleasant way. This thesis was conducted in the RFM
(Radio-Fréquences et Micro-ondes) group of the COMELEC (Communications et
Electronique) department at Télécom ParisTech.

First of all, I would like to express my greatest gratitude to my supervisor, Prof. Christophe Roblin, who offered me this interesting subject, accompanied me and supported me throughout this important period of my life. He is always available with great patience for discussions whenever I encountered any kind of question or confusion. His encouragement and confidence motivate me to keep moving forward. His rigorous research attitude, extensive expertise and good personal character have impressed me deeply and will also have a far-reaching impact on my future career.

I would also like to thank Prof. Alain Sibille, Prof. Joe Wiart, and Prof. David Lautru who shared their valuable time, experiences and vast knowledge in many fruitful discussions. My appreciation also goes to M. Karim Ben Kalaia and M. Antoine Khy, who helped me a lot in prototyping and experiments. I am especially grateful to Prof. Bruno Sudret' team who offered me the opportunity to use their mathematical toolbox UQLab for academic purpose, which saves me a lot of time for the implementation of numerical algorithms.

I would like to thank the reviewers Prof. Hendrik Rogier and Prof. Philippe Besnier for their extremely admirable work in reviewing my manuscript and writing very relevant evaluation reports. I would also like to thank the examiners Prof. Ala Sharaiha, Prof. Joe Wiart, Prof. Shermila Mostarshedi, and Prof. Odile Picon for their constructive remarks and suggestions on my manuscript, as well as for their valuable discussions in numerous occasions during my thesis.

I am also very grateful to my colleagues in RFM group, M. Xavier Begaud, M. Jean-
Christophe Cousin, Mme. Anne-Claire Lepage, M. Bernard Huyart, Mme. Florence Besnard,
Mme. Chantal Cadiat and Mme. Yvonne Bansimba for their kind help in a number of ways. I
will forever be thankful to my friends Heming Huang, Xin You, Cheng Wang, Wei Wei,
Longguang Li, Mengdi Song, Jianan Duan, Jose Enriquez Gonzalez, Chetan Joshi, Taguhi
Chalumyan, and Wasyhun Gemechu for their warm company and encouragements during my
PhD study.

Finally, I would like to express my deepest gratitude to my family who always
accompanies and supports me with unconditional love and care.
Table of contents

<table><tr><td>List of Figures</td><td>VI</td></tr><tr><td>List of Tables</td><td>IX</td></tr><tr><td>List of Acronyms</td><td>X</td></tr><tr><td>1. Introduction</td><td>1</td></tr><tr><td>1.1 Motivations</td><td>1</td></tr><tr><td>1.2 Thesis contributions</td><td>3</td></tr><tr><td>1.3 Organization of the thesis</td><td>3</td></tr><tr><td>2. State of the art</td><td>5</td></tr><tr><td>2.1 Soft or flexible antennas</td><td>5</td></tr><tr><td>2.1.1 Mainstream design technologies</td><td>5</td></tr><tr><td>2.1.2 Advantages and application prospects</td><td>6</td></tr><tr><td>2.1.3 Stochastic characteristics and arising concerns</td><td>7</td></tr><tr><td>2.1.4 Increasing demand for efficient modeling methods</td><td>9</td></tr><tr><td>2.2 Literature review for antenna modeling</td><td>10</td></tr><tr><td>2.2.1 FoMs modeling</td><td>10</td></tr><tr><td>2.2.2 Trend of complete FF modeling</td><td>12</td></tr><tr><td>2.3 Aims to be achieved by our work</td><td>12</td></tr><tr><td>3. Methodologies for modeling the ATF of variable antennas</td><td>14</td></tr><tr><td>3.1 Introduction</td><td>14</td></tr><tr><td>3.2 Field parsimony</td><td>16</td></tr><tr><td>3.2.1 Methods for parsimoniously representing scalar spherical functions</td><td>17</td></tr><tr><td>3.2.1.1 Double Fourier Series (DFS) expansion</td><td>17</td></tr><tr><td>3.2.1.2 Scalar Spherical Harmonics (SSH) expansion</td><td>17</td></tr><tr><td>3.2.2 Vector Spherical Harmonics (VSH) expansion</td><td>20</td></tr><tr><td>3.2.2.1 Principle</td><td>20</td></tr><tr><td>3.2.2.2 Test of the performance of VSH</td><td>21</td></tr><tr><td>3.3 Deterministic modeling of ATF: PS</td><td>23</td></tr><tr><td>3.3.1 Modeling procedures</td><td>24</td></tr><tr><td>3.3.2 Application to a canonical dipole antenna - with one input variable</td><td>25</td></tr><tr><td>3.3.2.1 Design of a deformable dipole</td><td>25</td></tr><tr><td>3.3.2.2 Field parsimony with VSH expansion</td><td>27</td></tr><tr><td>3.3.2.3 Surrogate model extraction</td><td>28</td></tr></table>
<table><tr><td>3.3.2.4</td><td>Model accuracy assessment</td><td>28</td></tr><tr><td>3.3.2.5</td><td>Verification based on interpolation and extrapolation</td><td>29</td></tr><tr><td>3.3.2.6</td><td>Modeling results for severer deformations</td><td>30</td></tr><tr><td>3.3.3</td><td>Application to a canonical dipole antenna - with two input variables</td><td>32</td></tr><tr><td>3.3.4</td><td>Discussion</td><td>33</td></tr><tr><td>3.4</td><td>Statistical modeling of the ATF's: PCE</td><td>34</td></tr><tr><td>3.4.1</td><td>Introduction</td><td>34</td></tr><tr><td>3.4.2</td><td>Principle of the Polynomial Chaos Expansion (PCE)</td><td>35</td></tr><tr><td>3.4.2.1</td><td>Formulation</td><td>35</td></tr><tr><td>3.4.2.2</td><td>Construction of the orthonormal polynomial basis</td><td>36</td></tr><tr><td>3.4.2.3</td><td>Computation of the PCE coefficients</td><td>37</td></tr><tr><td>3.4.2.4</td><td>Error indicator for assessing the model accuracy</td><td>45</td></tr><tr><td>3.4.2.5</td><td>Post-processing with PCE model</td><td>47</td></tr><tr><td>3.4.2.6</td><td>Experimental design (ED)</td><td>47</td></tr><tr><td>3.4.3</td><td>Methodology based on VSH expansion and PCE</td><td>49</td></tr><tr><td>3.4.3.1</td><td>Iterative procedure</td><td>50</td></tr><tr><td>3.4.3.2</td><td>Validation on a canonical dipole antenna</td><td>51</td></tr><tr><td>3.5</td><td>Improvements of the statistical method</td><td>54</td></tr><tr><td>3.5.1</td><td>Frequency behavior</td><td>54</td></tr><tr><td>3.5.2</td><td>Enhanced parsimony</td><td>55</td></tr><tr><td>3.5.2.1</td><td>Sparse VSH expansion</td><td>56</td></tr><tr><td>3.5.2.2</td><td>Correlation analysis on the selected dominant VSH modes</td><td>57</td></tr><tr><td>3.5.3</td><td>Appropriate metrics</td><td>58</td></tr><tr><td>3.5.4</td><td>Iterative modeling procedure</td><td>59</td></tr><tr><td>3.5.5</td><td>Application to a realistic deformable textile patch antenna</td><td>60</td></tr><tr><td>3.5.5.1</td><td>Design of the textile patch antenna</td><td>61</td></tr><tr><td>3.5.5.2</td><td>Joint distribution of the random input parameters</td><td>61</td></tr><tr><td>3.5.5.3</td><td>Pre-evaluation of the ATF's</td><td>70</td></tr><tr><td>3.5.5.4</td><td>Prototyping and control measurements</td><td>71</td></tr><tr><td>3.5.5.5</td><td>Preliminary study: sensitivity analysis</td><td>72</td></tr><tr><td>3.5.5.6</td><td>Modeling procedure and results</td><td>73</td></tr><tr><td>3.5.5.7</td><td>Validation of the surrogate model</td><td>75</td></tr><tr><td>3.5.5.8</td><td>Application examples of the surrogate model</td><td>77</td></tr><tr><td>3.5.5.9</td><td>Discussion about the model accuracy</td><td>78</td></tr><tr><td>3.5.5.10</td><td>Discussion about the computational cost</td><td>79</td></tr><tr><td>3.6</td><td>Conclusion</td><td>80</td></tr></table>
4. Antenna surrogate models in radio links analysis .................................................. 82

4.1 Introduction.................................................................................................................. 82

4.2 Effective performance of terminal antenna.................................................................. 83

4.3 Local propagation environments ............................................................................ 84

4.4 Effective performance and discussion .................................................................... 86

4.4.1 Relative Effective Gain (REG)........................................................................ 86

4.4.2 Relative Average EG (RAEG) in sub-bands .................................................. 89

4.4.3 RAEG for restricted azimuthal DoAs.................................................................. 91

4.5 Conclusion.................................................................................................................... 93

5. Statistical modeling of the reflection coefficient of variable antennas ...................... 94

5.1 Introduction.................................................................................................................. 94

5.2 Identification.................................................................................................................. 95

5.2.1 Technique 1: Polynomial fitting........................................................................ 95

5.2.2 Technique 2: Levy's complex curve fitting with rational functions............ 96

5.3 Surrogate model extraction.................................................................................... 96

5.4 Applications.................................................................................................................. 97

5.4.1 Application to a canonical dipole ........................................................................ 97

5.4.2 Application to a textile patch............................................................................ 99

5.5 Conclusion.................................................................................................................... 102

6. Conclusions and Perspectives .................................................................................. 104

6.1 Conclusions.................................................................................................................. 104

6.2 Perspectives .................................................................................................................. 105

References .......................................................................................................................... 107

Publications.......................................................................................................................... 119

Annex................................................................................................................................. 120

I. Introduction.................................................................................................................. 120

II. CPW-Fed IIFA on FR4 Substrate............................................................................ 122

a. Design of the antenna.................................................................................................. 122

b. Simulation and measurement results .................................................................... 122

III. CPW-Fed IIFA on FR4 Substrate............................................................................ 125

a. Design of the antenna.................................................................................................. 125

b. Simulation results........................................................................................................ 126

c. Test on a body phantom .......................................................................................... 127

IV. Conclusion.................................................................................................................... 128
TABLE OF CONTENTS

V

<table><tr><td>Reference</td><td>128</td></tr><tr><td>Résumé français</td><td>131</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>Figure 3.17: RRMSE of reconstructed fields with different approaches for 200 random values of the parameter κ ∈ [0, 1/10] mm<sup>-1</sup></td>
      <td>32</td>
    </tr>
    <tr>
      <td>Figure 3.18: Illustration of the joint bending & torsion phenomenon of a dipole antenna</td>
      <td>32</td>
    </tr>
    <tr>
      <td>Figure 3.19: RRMSE of reconstructed fields with different approaches for bivariate case</td>
      <td>33</td>
    </tr>
    <tr>
      <td>Figure 3.20: The LARS algorithm in the case of M = 2 covariates</td>
      <td>44</td>
    </tr>
    <tr>
      <td>Figure 3.21: An example showing the overfitting problem when approximating an empirical sample with high order polynomials</td>
      <td>46</td>
    </tr>
    <tr>
      <td>Figure 3.22: Illustration of a dipole undergoing different deformations: (a) original dipole; (b) stretched dipole; (c) curved dipole; (d) curved and twisted dipole</td>
      <td>51</td>
    </tr>
    <tr>
      <td>Figure 3.23: Joint effect of stretching, bending, and torsion (ξ, κ, α) on the dipole: (a) reflection coefficient |S<sub>11</sub>|; (b) radiation patterns (in E/H plane) at f = 2.3 GHz</td>
      <td>52</td>
    </tr>
    <tr>
      <td>Figure 3.24: e<sub>LOO</sub> of the PCE models derived from the iteratively enriched EDs</td>
      <td>52</td>
    </tr>
    <tr>
      <td>Figure 3.25: (a) Radiation patterns (in E/H plane) of the initial, reconstructed and generated fields for the most deformed case at f = 2.3 GHz; (b) RRMSE of 500 generated dipoles for random values of (ξ, κ, α)</td>
      <td>53</td>
    </tr>
    <tr>
      <td>Figure 3.26: Wearable textile patch antenna: (a) un-deformed plane patch; (b) patch undergoing the joint effect of crumpling, bending and thickness variation</td>
      <td>60</td>
    </tr>
    <tr>
      <td>Figure 3.27: Experimental setup (a) and simulation design (b) for bending analysis</td>
      <td>63</td>
    </tr>
    <tr>
      <td>Figure 3.28: Center frequency shift under bending (R<sub>b</sub> = 31.75 mm) for increasing substrate compression Δh/h<sub>0</sub> (by simulation)</td>
      <td>65</td>
    </tr>
    <tr>
      <td>Figure 3.29: Statistic of the medium human arm radius, derived from the database 2015-2016 provided by the US National Center for Health Statistics</td>
      <td>69</td>
    </tr>
    <tr>
      <td>Figure 3.30: Experimental setup for measuring the ATFs</td>
      <td>71</td>
    </tr>
    <tr>
      <td>Figure 3.31: Three representative patches for which: the prototypes, the simulated/measured reflection coefficient |S<sub>11</sub>|, and the simulated/measured G<sub>r</sub> at 2.45 GHz in principal cut-planes for the co-polarization and cross-polarization components are presented in columns</td>
      <td>72</td>
    </tr>
    <tr>
      <td>Figure 3.32: S<sub>WT</sub> of the 6 random input parameters</td>
      <td>73</td>
    </tr>
    <tr>
      <td>Figure 3.33: ATF reconstruction errors for different power threshold E<sub>c_th</sub></td>
      <td>74</td>
    </tr>
    <tr>
      <td>Figure 3.34: Prediction error e<sub>WTLOO</sub> of surrogate models iteratively constructed by enriching the ED</td>
      <td>75</td>
    </tr>
    <tr>
      <td>Figure 3.35: Comparison between the errors of two successive surrogate models for different EDs</td>
      <td>75</td>
    </tr>
    <tr>
      <td>Figure 3.36: Error of the model-generated ATFs w.r.t. the reference MC sample</td>
      <td>76</td>
    </tr>
    <tr>
      <td>Figure 3.37: EM-simulated (left column) and model-generated (right column) G<sub>r</sub> patterns for a representative average-error configuration</td>
      <td>77</td>
    </tr>
    <tr>
      <td>Figure 3.38: EM-simulated (left column) and model-generated (right column) G<sub>r</sub> patterns for a representative maximal-error configuration</td>
      <td>77</td>
    </tr>
    <tr>
      <td>Figure 3.39: Comparison between the CDFs of: (a) -10 dB beamwidth, (b) maximal G<sub>r0</sub> computed from the model and EM-simulations from the MC sample</td>
      <td>78</td>
    </tr>
  </tbody>
</table>
Figure 3.40: Prediction error $e_{WTLOO}$ of surrogate models iteratively constructed by enriching the ED till large cardinal. .................................................................................... 79

Figure 3.41: Comparison between the CDFs of the model-generated and the EM-simulated dominant VSH coefficients from the MC sample. ................................................................ 79

Figure 4.1: Illustration of the radio link of a wireless communication system. .................................... 83

Figure 4.2: CDFs of the relative effective gain of the textile patch antenna in different propagation environments: color thin lines – partial CDFs; red thick line – global CDF. .................................................................................................................... 87

Figure 4.3: Allowable sub-bands for WiFi standard in the 2.4 – 2.5 GHz range. ................................. 89

Figure 4.4: CDFs of the relative average effective gain of the textile patch antenna in different propagation environments: color thin lines – partial CDFs; red thick line – global CDF. .................................................................................................................... 90

Figure 4.5: CDFs of the relative average effective gain of the textile patch antenna ($\mu_{\phi} \sim U([-90^{\circ}, 90^{\circ}])$) in different propagation environments: color thin lines – partial CDFs; red thick line – global CDF. .................................................................................... 92

Figure 5.1: Dipole – Approach I: (a) $e_{LOO}$ of the PCE models of characteristic parameters for different ED; (b) err of reconstruction in case of $N_{ED} = 260$. .................................... 98

Figure 5.2: Dipole – Approach I: comparison of the frequency responses of the initial, the identified (with polynomial fitting), and the final surrogate model for: (a) the worst case (i.e. with maximal err); (b) the median case (i.e. with median err). ........................ 98

Figure 5.3: Dipole – Approach II: (a) $e_{LOO}$ of the PCE models of characteristic parameters for different ED; (b) err of reconstruction in case of $N_{ED} = 120$. .................................... 99

Figure 5.4: Dipole – Approach II: comparison of the frequency responses of the initial, the identified (with Levy's), and the final surrogate model for: (a) the worst case (i.e. with maximal err); (b) the median case (i.e. with median err). .................................... 99

Figure 5.5: Textile patch undergoing random variations in length, bending radius, substrate height and permittivity. .................................................................................... 100

Figure 5.6: Patch – Approach I: (a) $e_{LOO}$ of the PCE models of characteristic parameters for different ED; (b) err of reconstruction in case of $N_{ED} = 200$. .................................... 100

Figure 5.7: Patch – Approach II: (a) $e_{LOO}$ of the PCE models of characteristic parameters for different ED; (b) err of reconstruction in case of $N_{ED} = 280$. .................................... 100

Figure 5.8: Patch – Approach I: comparison of the frequency responses of the initial, the identified (with polynomial fitting), and the final surrogate model for: (a) the worst case (i.e. with maximal err); (b) the median case (i.e. with median err). ........................ 101

Figure 5.9: Patch – Approach II: comparison of the frequency responses of the initial, the identified (with Levy's), and the final surrogate model for: (a) the worst case (i.e. with maximal err); (b) the median case (i.e. with median err). .................................... 101
List of Tables

<table><tr><td>Table I: Some examples of planar soft/flexible antennas [24]-[27], [4]</td><td>6</td></tr><tr><td>Table II: Parametric study of a deformable dipole: input parameters, modeling parameters and results</td><td>33</td></tr><tr><td>Table III: Random input parameters of the textile patch</td><td>61</td></tr><tr><td>Table IV: K-S test on the 27 dominant VSH coefficients</td><td>79</td></tr><tr><td>Table V: Characteristics of 4 WINNER2/+ propagation environments</td><td>86</td></tr><tr><td>Table VI: Statistics of the relative effective gains</td><td>87</td></tr><tr><td>Table VII: Statistics of the relative average effective gains</td><td>90</td></tr><tr><td>Table VIII: Statistics of the relative average effective gains (μ<sub>φ</sub> ~ U([-90°, 90°]))</td><td>92</td></tr><tr><td>Table IX: Design parameters and variability range for the textile patch</td><td>100</td></tr></table>
List of Acronyms

<table><tr><td>AMPS</td><td>Advanced Mobile Phone System</td></tr><tr><td>ANN</td><td>Artificial Neural Networks</td></tr><tr><td>ANOVA</td><td>Analysis of Variance</td></tr><tr><td>ATF</td><td>Antenna Transfer Function</td></tr><tr><td>AUT</td><td>Antenna Under Test</td></tr><tr><td>CAD</td><td>Computer Aided Design</td></tr><tr><td>CDF</td><td>Cumulative Density Function</td></tr><tr><td>CP</td><td>Co-Polarization</td></tr><tr><td>DFS</td><td>Double Fourier Series</td></tr><tr><td>DoA</td><td>Directions of Arrival</td></tr><tr><td>ED</td><td>Experimental Design</td></tr><tr><td>EG</td><td>Effective Gain</td></tr><tr><td>EM</td><td>Electromagnetic</td></tr><tr><td>FDTD</td><td>Finite-Difference Time-Domain</td></tr><tr><td>FEM</td><td>Finite Element Method</td></tr><tr><td>FF</td><td>Far Field</td></tr><tr><td>FoM</td><td>Figures of Merit</td></tr><tr><td>GNSS</td><td>Global Navigation Satellite Systems</td></tr><tr><td>gPCE</td><td>generalized Polynomial Chaos Expansion</td></tr><tr><td>GPR</td><td>Gaussian Process Regression</td></tr><tr><td>GPS</td><td>Global Positioning System</td></tr><tr><td>GQ</td><td>Gaussian Quadrature</td></tr><tr><td>HLARS</td><td>Hybrid Least Angle Regression Sampling</td></tr><tr><td>IFA</td><td>Inverted F-shape Antenna</td></tr><tr><td>IIFA</td><td>Integrated IFA</td></tr><tr><td>IoT</td><td>Internet of Things</td></tr><tr><td>ISM</td><td>Industrial Scientific Medical</td></tr><tr><td>LARS</td><td>Least Angle Regression Sampling</td></tr><tr><td>LCP</td><td>Liquid-Crystal Polymer</td></tr><tr><td>LHS</td><td>Latin Hypercube Sampling</td></tr><tr><td>LOOE</td><td>Leave-One-Out Error</td></tr><tr><td>LOS</td><td>Line-Of-Sight</td></tr><tr><td>LTI</td><td>Linear Time Invariant</td></tr><tr><td>MC</td><td>Monte-Carlo</td></tr><tr><td>MCS</td><td>Monte Carlo Sampling</td></tr><tr><td>MEG</td><td>Mean Effective Gain</td></tr><tr><td>MIMO</td><td>Multiple-Input and Multiple Output</td></tr><tr><td>NB</td><td>Narrow-Band</td></tr><tr><td>NLOS</td><td>Non-Line-Of-Sight</td></tr><tr><td>OLS</td><td>Ordinary Least Square</td></tr><tr><td>PCE</td><td>Polynomial Chaos Expansion</td></tr><tr><td>PDF</td><td>Probability Density Function</td></tr><tr><td>PDMS</td><td>Polydimethylsiloxane</td></tr><tr><td>PEC</td><td>Perfect Electrical Conductor</td></tr></table>
<table><tr><td>PET</td><td>Polyethylene Terephthalate</td></tr><tr><td>PS</td><td>Parametric Study</td></tr><tr><td>PTFE</td><td>Polytetrafluoroethene</td></tr><tr><td>QoS</td><td>Quality of Service</td></tr><tr><td>RAEG</td><td>Relative Average Effective Gain</td></tr><tr><td>REG</td><td>Relative Effective Gain</td></tr><tr><td>RMSE</td><td>Root Mean Square Error</td></tr><tr><td>RRMSE</td><td>Relative Root Mean Square Error</td></tr><tr><td>SCM</td><td>Stochastic Collocation Method</td></tr><tr><td>SEM</td><td>Singularity Expansion Method</td></tr><tr><td>SSH</td><td>Scalar Spherical Harmonics</td></tr><tr><td>ST</td><td>Stochastic Testing</td></tr><tr><td>SVR</td><td>Support Vector Regression</td></tr><tr><td>TE</td><td>Transverse Electric</td></tr><tr><td>TF</td><td>Transfer Function</td></tr><tr><td>TM</td><td>Transverse Magnetic</td></tr><tr><td>TSI</td><td>Total Sobol's Indices</td></tr><tr><td>UE</td><td>User End</td></tr><tr><td>UWB</td><td>Ultra-Wideband</td></tr><tr><td>VNA</td><td>Vector Network Analyzer</td></tr><tr><td>VSH</td><td>Vector Spherical Harmonics</td></tr><tr><td>WBAN</td><td>Wireless Body Area Network</td></tr><tr><td>WLAN</td><td>Wireless Local Area Network</td></tr><tr><td>XP</td><td>Cross-Polarization</td></tr><tr><td>XPR</td><td>Cross-Polarization power Ratio</td></tr></table>
<figure><img src="image_8.png" /></figure>
Chapter 1

# 1. Introduction

## 1.1 Motivations

Since 1980s when the first generation (1G) of mobile communication standard (i.e. Advanced Mobile Phone System, AMPS, developed by Bell Labs) came out, wireless communication technologies have achieved leaping growth and have brought great convenience into people's life. Today, people are highly dependent on all kinds of wireless services such as the cellular networks (e.g. 2G, 3G, 4G), the GNSS (Global Navigation Satellite Systems, e.g. the Global Positioning System or GPS), the Wireless Local Area Networks (WLANs, e.g. Wi-Fi), and the Wireless Body Area Networks (WBANs, e.g. Wi-Fi, Bluetooth, ZigBee). The pace of revolution has never stopped and is even much faster than ever before. We are now almost ready to welcome the 5G technology thanks to which not only people but also many “things” (e.g. machines, vehicles, sensors...) will be connected, forming an immense Internet of Things (IoT). The IoT is supposed to contribute to an “intelligent” world (e.g. smart grid, intelligent transportation, real-time health care, intelligent home, office, factory, etc.) and to improve the quality of people’s life to a large extent. With the pervasive expansion of wireless technologies, the requirement for wireless electronic devices will certainly experience explosive growth in coming years. Figure 1.1 shows the prediction of IoT devices growth in the next 10 years [1].

<figure><img src="image_9.png" /><figcaption>Figure 1.1: IoT electronic devices growth forecasted by IHS (Information Handling Services) 2016.</figcaption></figure>
Being the unique interface to communicate with the outside, antenna is extremely vital component for all wireless electronic devices. Successes in wireless communication industry would have not been achieved without the continuing advances in antenna technology. In recent years, antennas have the trend to be highly integrated, miniaturized, and arrayed in order to meet with the challenging demand of massive deployment, short R&D cycle, low cost and high performance. Moreover, some other advantageous properties are also indispensable in specific application scenarios. In particular, in the context of WBANs where antennas should be integrated into garments or pasted directly onto the human body, softness or flexibility permitting them to conform to various body shapes and sizes would be really useful.

However, softness or flexibility implies geometric deformation (e.g. bending, torsion, crumpling...) which can drastically perturb the behavior (e.g. radiation pattern, impedance...) of antennas. The perturbed antennas are no longer *fixed* or *static*, but *varying* or *dynamic* seeing that their behavior is varying randomly during the operation. In addition to self-deformations, external randomness from the close (possibly fluctuating) environment can also disturb antennas. For example, variation of the temperature and humidity may cause changes of antennas' material properties such as the permittivity and conductivity; nearby surroundings, notably the human body, could perturb antennas' nearfield zone resulting in strong absorption, reflection, scattering, or the joint effect of them. This effect becomes remarkable as antennas become miniaturized.

Nowadays, with the widespread use of *flexible electronics* (such as textile electronics) and the increasing complexity of operating environments, the presence of randomness and uncertainties on the antennas is more realistic than ever before in almost all application contexts. This brings severe challenges in antenna design as well as radio links analysis. How to characterize antennas' *in situ* performance becomes an inescapable concern and is attracting increasing attention. Moreover, in order to guarantee the *in situ* functionality of antennas, the traditional solution is to add an ad hoc gain margin to roughly account for those random effects that can be encountered in real world. However, this usually induces high cost and waste of precious space and/or power resources. To avoid excessive margin values, quantitatively and precisely characterizing the behavior of perturbed antennas is indispensable and is attracting increasing attention of antenna engineers.

Conventionally, the characterization of a classical deterministic antenna (e.g. rigid or large sized, or clearly isolated, etc.) can be carried out by means of measurements or full-wave simulations. Yet, this approach becomes inappropriate for antenna problems under uncertain conditions where a large number of random sources intervene. Indeed, the combinatory of situations can be so explosive that executing measurements or simulations
one by one would be too time-consuming and even physically impossible for us at this time.
Under such circumstances, more efficient characterization approaches are urgently required.

This thesis is dedicated to investigate efficient methodologies for quantitatively analyzing and modeling the random effects on the “overall” behavior, i.e. the *complete* radiated Far Field (FF) as well as the reflection coefficient, of soft or flexible antennas in fluctuating conditions. Notably, statistical methodologies which provide good modeling accuracy while minimizing the modeling cost are presented. The principle is to try to carry out only very limited number of pre-evaluations (by measurement or simulation), and make full use of them to extract efficient surrogate models for the prediction in other possible situations. The applicative perspectives of such surrogate models are to substitute real antennas or cumbersome full-wave simulations in various types of analyses, notably when the *complete* knowledge (including polarization, amplitude and phase of the field) of the antenna under study is mandatory. Such application contexts comprise for example, joint antenna/directional channel modeling, antenna plugged-in asymptotic simulators (e.g. ray tracing), radio links analysis, MIMO (Multiple-Input and Multiple Output) systems characterization, and beamforming designs.

**1.2 Thesis contributions**

This thesis is focused on the investigation of efficient methodologies for quantitatively characterizing the “*overall*” behavior of soft/flexible or disturbed antennas that are subject to random uncertainties. The main topics covered are:

* Parsimonious description of the antenna's *complete* radiated Far Field (FF) or its equivalent, the Antenna Transfer Function (ATF).

* Deterministic methodology for quantitatively modeling the ATF.

* Statistical methodology for quantitatively modeling the ATF in multivariate cases.

* Statistical methodology for quantitatively modeling the antenna's reflection coefficient $S_{11}(f)$ in multivariate cases.

* Investigation of the usage of antenna surrogate models for radio links analysis, where antenna models are implemented in typical stochastic propagation environments.

**1.3 Organization of the thesis**

The organization of this thesis is as follows:

*Chapter 2* introduces the state of the art of soft or flexible antennas. In the first part, the design technologies, the advantages and broad application prospects, as well as the arising concerns of soft or flexible antennas are briefly introduced. In the second part, the state of the
art on antenna modeling is reviewed, and the aims that we want to achieve in this thesis are outlined.

*Chapter 3 proposes methodologies for modeling the antenna's complete radiated FF or ATF. First, the field parsimony technique based on the Vector Spherical Harmonics (VSH) expansion is introduced. Then, deterministic as well as stochastic modeling methodologies are proposed. All methodologies have a Two-Step process, that is, field parsimony followed by surrogate model extraction. The deterministic methodology is a combination of the VSH expansion and the conventional parametric study; and the statistical methodology is a combination of the VSH expansion and the advanced Polynomial Chaos Expansion (PCE) method.*

*Chapter 4 investigates the utility of accurate antenna surrogate models for radio links analysis, where the antenna's effective performance is quantified by taking into account both the inherent uncertainties of the antenna itself and those from the stochastic local propagation environments.*

*Chapter 5 presents a statistical methodology for modeling the complex reflection coefficient $S_{11}(f)$ of narrow-band (NB) variable antennas. This approach is split into two steps: identification, and model extraction using the PCE method.*

*Chapter 6 summarizes the work realized during this thesis and presents some perspectives for future researches.*
Chapter 2

**2. State of the art**

This chapter presents the research context of this thesis. Firstly, the soft or flexible
antennas technology and their broad application prospects are briefly introduced. The arising
concerns related to the stochastic characteristics of soft/flexible antennas are also explained.
Then, the increasing demand for efficient modeling methods is discussed. Finally, the state of
the art on antenna modeling is reviewed, and the aims that we want to achieve in this thesis
are outlined.

**2.1 Soft or flexible antennas**

**2.1.1 Mainstream design technologies**

"Soft/flexible antennas" is a relative concept with respect to classical non-deformable rigid antennas. Generally, this term refers to a broad category of antennas made of soft or flexible materials whose geometric shape can be more or less changed for purpose of gaining some flexibility during usage. Such kind of flexibility is in high demand not only in a priori unknown application contexts (e.g. uncertainty due to the morphological distribution of the user population), but also for "dynamic" situations where real-time flexibility is mandatory (e.g. time-varying geometric deformations due to movement).

Typical representatives of soft/flexible antennas are those planar shape antennas designed for wearable communications. These antennas are composed of thin soft/flexible conductive layers as their ground plane and radiating part, and dielectric layers as their substrate part. Popularly adopted conductive materials include copper foil, conductive fibers (or *E*-fibers, e.g. copper-, silver- or nickel-plated), conductive inks, and graphene, etc. [2]-[7]. As for the dielectric material, textiles such as denim, cotton, yarn, fleece, etc. are the most common used because of their low-cost, drapability and biocompatibility [8]. In addition, paper, rubber, polymers (e.g. LCPs (liquid-crystal polymers), PDMS (polydimethylsiloxane), PET (polyethylene terephthalate) and PTFE (polytetrafluoroethene), etc.) are also good candidates for dielectric materials [9]-[13]. These thin layers are integrated to realize the specified structure using proper assemblage technologies such as adhesive laminating, sewing, embroidery, screen-printing, ink-jet, and evaporation deposition [2], [14], [15]. In recent years, studies addressing such antennas are more than abundant. Some examples are given in Table I.
Other types of soft/flexible antennas include those classic bendable or foldable wire antennas, as well as those recently emerging micro-fluidic antennas which encapsulate conductive liquids in flexible plastic structures [16]-[19]. Tunable antennas (or antenna arrays) whose “electrical shape” is reconfigurable may also be considered as somewhat “soft/flexible” [20]-[23].

<table><caption>Table I: Some examples of planar soft/flexible antennas [24]-[27], [4].</caption><thead><tr><th>Photo</th><th>Conducting part</th><th>Dielectric part</th><th>Assemblage</th></tr></thead><tbody><tr><td></td><td>copper tape</td><td>denim</td><td>adhesive</td></tr><tr><td></td><td>single-walled carbon nanotube composite</td><td>paper</td><td>inkjet printing</td></tr><tr><td></td><td>silver loaded epoxy</td><td>PDMS</td><td>inkjet printing</td></tr><tr><td></td><td>copper and nickel coated fibers</td><td>felt</td><td>adhesive</td></tr><tr><td></td><td>E-fibers</td><td>polymer composites</td><td>embroidery</td></tr></tbody></table>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

### 2.1.2 Advantages and application prospects

Compared with traditional non-deformable rigid antennas, soft/flexible antennas have some unparalleled advantages namely:

* Deformability/conformability (e.g. stretchability, flexibility, compressibility, drapability) or re-configurability.

* Lightness.

* Low-profile (non-protruding, unobtrusiveness).

* Washability and long-lasting.

* Low-cost.

All these benefits greatly improve the user experience of electronic devices in terms of comfort, beauty and economy, hence making them ideal candidates notably for the explosively emerging *on-body/off-body* WBAN applications. Figure 2.1 illustrates some existing or developing applications of soft/flexible antennas in the framework of WBANs [28]-[31]. It can be seen that many human activity domains are covered: health care – monitoring of the physiological signals such as heart rate, breathing rate, brain wave, blood
pressure, and body temperature, etc. of patients; sportive activities – monitoring of those
fitness parameters of the athletes; security/emergency – positioning, and monitoring of the
vital signals of fire fighters; and entertainment – remote controlling in virtual reality and
computing games.

<figure><img src="image_15.png" /><figcaption>Figure 2.1: Some existing application examples of wearable antennas: (a) health monitoring; (b) smart T-shirt for athletes; (c) smart garment for fire fighters; (d) smart gloves for remote controlling.</figcaption></figure>

**2.1.3 Stochastic characteristics and arising concerns**

However, just as everything has its two sides, despite the advantages mentioned above,
the softness/flexibility is accompanied by negative effects that would not occur (at least not
significantly) for non-deformable rigid antennas. In fact, the performance of the latter is
conventionally considered to be totally determined by the directivity and the radiation
patterns of the transmitting & receiving antennas once they are aligned. By contrast, under the
condition of softness/flexibility, the electromagnetic (EM) characteristics of antennas are no
longer fixed, but are time-varying, often in a stochastic manner, with regard to the
uncertainties coming from some intrinsic and/or external sources. Typically, such random
uncertainties can take place in both the fabrication and the usage phase. In the former phase,
random uncertainties are introduced due to:

- The imprecision of the characterization of material properties. Note that for most of the soft/flexible materials, their EM properties are not as stable as those of the materials used for non-deformable rigid antennas due to the inhomogeneity and variability, hence they can only be quantified statistically and globally, inevitably introducing some uncertainties.
- The fabrication tolerances. On the one hand, the geometrical precision that can be achieved by existing technologies for soft/flexible materials is far less than that for rigid materials; on the other hand, the fabrication procedure particularly the assemblage step would exert parasite stresses (e.g. compression, creasing, stretching, etc.) inside the material, causing random variability in the material properties as well as on antennas' geometric shape.

And during the usage phase, random uncertainties are caused by:

- The global geometric deformations (e.g. bending, torsion, crumpling, thinning, elongation, etc.) which are caused by external forces and which induce variations in the current distribution and possibly of the material properties of the considered antenna.

- The morphological distribution of the user population, in terms of age (baby, child, teen, adult, senior), gender (male/female), posture (sitting, standing, walking...), device placement (arm, wrist, chest, shoulders, back...), etc.

- The fluctuating environment, whose influence is twofold: first, the temperature and humidity (from moisture, rain, etc.) may cause significant variations in material properties (e.g. conductivity and permittivity); second, the proximity effect of immediate surroundings notably the human body could drastically perturb the reactive near-field of the considered antenna, causing strong effect of absorption, blockage, and reflection, etc. [32], [33]. In such situation, the immediate environment cannot be discriminated from the antenna itself. In other words, the antenna plus its immediate surroundings turn into a new “effective antenna”. Therefore, all randomness due to the interaction between the antenna and its neighboring surroundings must be considered as inherent stochastic characteristics of the new “effective antenna”.

- The operational frequencies. Apparently, when an antenna operates over different frequency bands its behavior can be quite different. This effect can be considered as a stochastic influence seeing that nowadays many electronic devices support multi-bands (even the so-called “full bands”) allowing to switch between frequencies (or even to use carrier aggregation of the signal over non-contiguous spectrum allocations, e.g. in LTE Advanced) according to the supervised real-time channel quality, while the sub-band selection is by essence random.

Additionally, it is worth pointing out that the stochastic properties of the local propagation environment of a considered antenna, that's to say, the spatial and polar distribution of the multipath incoming waves, also have strong influence on the effective performance of the antenna. However, this aspect of uncertainty can be discriminated from those inherent antenna uncertainties, and can be characterized separately by stochastic
channel models (cf. Chapter 4 for more thorough discussion). Therefore, this aspect of
uncertainty will not be included in our investigations. Henceforward, for sake of clarity, we
call the antennas whose behavior/characteristics vary randomly during their operation due to
their inherent uncertainties as “*Variable antennas*”. This would include those soft/flexible
antennas and other types of deformable antennas, as well as those proximity-
interacted/disturbed antennas.

Those aforementioned inherent uncertainties can result in drastic changes in the behavior
of a given variable antenna, for example:

* Impedance mismatch or detuning.

* Radiation efficiency degradation (due to absorption).

* Re-direction or distortion of the radiation pattern (polarization, amplitude and phase).

Clearly, to ensure the reliable operation of a variable antenna, we have to quantitatively
characterize those random effects.

2.1.4 Increasing demand for efficient modeling methods

Conventionally, for designing or characterizing a “classical” deterministic antenna (such
as a non-deformable rigid, or large-sized, or clearly isolated one), a canonical process
combining full-wave simulations with prototyping & measurements is often adopted.
Concretely, preliminary analyses based on full-wave simulations (using some commercially
available simulators e.g. HFSS, CST, etc. or self-implemented numeric algorithms e.g. FEM
(Finite Element Method), FDTD (Finite-Difference Time-Domain), etc.) are firstly carried out
to visualize the behavior of the antenna under study and eventually, a parametric analysis can
be applied for a rough search of the optimal parameters. Then, some prototypes are fabricated,
and their typical characteristics (e.g. radiation pattern, reflection coefficient S11) are measured
to bring to light the realistic performance of the antenna. After that, modifications on design
parameters can be made if necessary, and eventually a new round of test could be launched.

However, this process becomes completely incapable for variable antenna problems since
the combinatory of random-uncertainties-involved situations can be so explosive that the
characterization cost could immediately surpass one’s affordability. Therefore, more
appropriate modeling methods which allow to efficiently characterizing those random effects
with good precision are of paramount importance, especially in the application contexts of:

- Predicting the real *in situ* performance of variable antennas.

- Acquiring better precision in the assessment of end-to-end radio links performance.

- Saving the cost of maintenance.

- Accelerating the design cycle of an application-dependent variable antenna.
## 2.2 Literature review for antenna modeling

Antenna modeling aims to efficiently characterize the impact of those random uncertainties on the antenna's behavior. The modeling of variable antennas dates as early as the pioneering work of Shifrin in late sixties [34]. More attention has been paid to these issues notably during the last decade (see e.g. [35]). Most of the existing researches had focused on the modeling of some primary relevant characteristics of antennas, for example, the total efficiency, reflection coefficient/impedance, resonant frequency, beamwidth, axe ratio, maximal realized gain, etc. These scalar quantities, also referred as Figures of Merit (FoMs) of interest, describe either partially or averagely the behavior of a given antenna, and a good knowledge on them is often sufficient for most of classical application contexts. Recently, with the increasing demand for *complete* antenna models (i.e. containing the precise information of polarization, amplitude and phase of the field) in e.g. joint directional channel/antenna, MIMO (Multiple-Input and Multiple Output) system or beamforming analysis, an *overall* modeling that must comprise both the *complete* radiated Far Field (FF) and the reflection coefficient (which describe respectively the full transmission and reflection properties of the considered antenna, see Figure 2.2) becomes a highly relevant and timely topic.

<figure><img src="image_16.png" /><figcaption>Figure 2.2: Schematic of a transmitting antenna, fed by a source of internal impedance $Z_g$, via a transmission line of characteristic impedance $Z_c$. $a_1$, $b_1$ are respectively the incident and reflected partial wave. $P_{inc}$, $P_r$, and $P_a$ are respectively the incident, reflected, and received power by the antenna. $P_{loss}$ is the conductive & dielectric power loss of the antenna, and $P_{rad}$ is the radiated power by the antenna. $k$ is the vector wave number. $\hat{\mathbf{r}} = (\theta, \varphi)$ is the unit radial vector. $\mathbf{E}_{\text{rad}}(\hat{\mathbf{r}})$ is the electric FF.</figcaption></figure>

### 2.2.1 FoMs modeling

The Monte-Carlo (MC) method, due to its simplicity, was first used, e.g. in [36], to assess the statistics of some FoMs of interest for the considered problems. However, the computational cost of MC is high, even prohibitive for some EM problems as it requires a
large number of realizations due to its slow convergence ($O(1/\sqrt{N})$, $N$ being the number of realizations). Moreover, MC is not effective for sensitivity analysis, i.e. for quantifying the significance of each random effect.

Quite recently, more efficient statistical approaches have been proposed, allowing to quantitatively account for those random effects.

For instance, in [37] the analysis of variance (ANOVA) method combined with a regression technique is used to characterize the individual and joint effect of bending, shearing and stretching on the bandwidth of a textile antenna.

In [38], the randomness in antenna design parameters was addressed by means of small signal analysis, which expands the FoMs of interest as a Taylor series of the design parameters. This approach is mainly suited for relatively small random variations of the input parameters.

Support Vector Regression (SVR) had also been used for antenna modeling. FoMs that have been characterized in this manner include e.g. the impedance (at the resonant frequency) of a microstrip patch antenna [39], and the axial ratio of a circularly polarized antenna against variable design parameters [40]. In general, the SVR has advantage of mapping to convex quadratic optimization problems hence it does not suffer from local minima, but it might not be very efficient for highly non-linear multiple input problems. An improved approach combining the Gaussian process regression with the standard SVR was proposed in [41] and had been used to model the reflection coefficient of a square slot antenna against the frequency as well as some tunable geometry variables. This approach is more robust than the classic SVR against strong non-linearity.

Artificial Neural Networks (ANNs) also gains great consideration in antenna design and modeling problems [42]. For example, in [43] and [44] the resonant frequency, in [45] and [46] the input impedance, and in [47] the radiation pattern as well as the reflection coefficient of the considered antennas against variable design parameters were successfully modeled with ANNs. However, ANNs is a rather complicated method that is not straightforward to implement due to the lack of effective methods for determining the number of hidden layers and nodes. In addition, the “overfitting” problem – a model having too high degree of freedom to closely fit a particular set of observations may lose its ability for predicting future observations – is another drawback that should be paid careful attention when applying the ANNs.

Most recently, Stochastic Collocation Methods (SCM) based on the generalized Polynomial Chaos Expansion (gPCE) theory [48] have been investigated to characterize random effects on several classical FoMs, namely the resonant frequency, the input impedance, axial ratio and reflection coefficient (at a fixed operating frequency) of textile
antennas undergoing random uncertainties [49]-[51], where SCM shows its effectiveness in terms of good modeling accuracy and acceptable implementation complexity.

### 2.2.2 Trend of complete FF modeling

A complete knowledge of antennas’ radiated FF rather than the FoMs, is mandatory for analyzing in a reliable way radio links in realistic complicated conditions. Moreover, once the FF is fully characterized, classical FoMs can be deduced from it straightforwardly.

Pioneering attempts of FF modeling were carried out in [52]-[54], where a canonical MC method combining the Vector Spherical Harmonics (VSH) expansion and the Singularity Expansion Methods (SEM) [55]-[57] was used to model the FF radiated by Ultra-Wideband (UWB) antennas whose design parameters are randomly varying.

Most recently, an attempt of quantitatively modeling the complete radiated field (in the Near-Field Fresnel region) of a textile antenna subject to uncertainties was presented in [58]. A “macromodel” was constructed by applying the gPCE to a VSH expansion of the radiated field. This work is an excellent example for demonstrating the interest and high efficiency of resorting to complete antenna models in radio links analysis, and the assessment of various performance indicators, such as the power transfer efficiency of the considered wireless power transfer system in this work.

Nevertheless, except these primary attempts mentioned above, the complete FF modeling of variable antennas has not yet been fully investigated to date.

## 2.3 Aims to be achieved by our work

In this thesis, we investigate efficient methodologies for quantitatively modeling the overall behavior of variable antennas. This includes the modeling of the antenna’s complete radiated FF as well as the reflection coefficient $S_{11}(f)$. Particularly, we use the Antenna Transfer Function (ATF) [59], defined in (2.1), as the equivalent representation for the FF. The benefit is that ATF is independent of the input power, hence is a representation of the intrinsic properties of the considered antenna.

$$ \mathcal{H}(f, \hat{\mathbf{r}}) \triangleq \frac{re^{jkr}}{a_1(f)\sqrt{\eta_0}} \mathbf{E}^{\infty}(f, \mathbf{r}) = \mathcal{H}_{\theta}(f, \hat{\mathbf{r}})\hat{\theta} + \mathcal{H}_{\varphi}(f, \hat{\mathbf{r}})\hat{\varphi} \quad (2.1) $$

where $a_1$ is the incident wave at antenna port, $\eta_0$ the free space impedance, $\hat{\mathbf{r}} = (\theta, \varphi)$ the unit radial vector, $k$ the scalar wave number, $j = \sqrt{-1}$ and $\mathbf{E}^{\infty}$ the electric FF (see Figure 2.2).

Compared to existing works, the novel features or improvements that we are aiming to achieve in our proposed modeling methodology are:
1) Wide range of applications: suitable for various types of random uncertainties, particularly including the frequency behavior; and for relatively high dimensionality (i.e. large number of input parameters) and large range of variability.

2) Efficiency: minimized computational cost for constructing antenna models.

3) Versatility: scalable/adjustable according to required modeling accuracy.

And the antenna models constructed in such way should meet the properties of:

1) Explicitness: quantitative generation of the corresponding ATF & $S_{11}(f)$ for any new set of input parameters values.

2) Compactness: having a compact/parsimonious formulation.

3) Accuracy: meeting the application-dependent requirement on accuracy.

4) Efficiency: much faster than numerical full-wave simulators for ATF & $S_{11}(f)$ generation.

And they can be then incorporated into various types of higher level analysis, such as the joint antenna/channel modeling, antenna plugged-in asymptotic simulators (e.g. ray tracing), or used for a quick statistic study of some secondary observables, Figure 2.3.

<figure><img src="image_17.png" /><figcaption>Figure 2.3: Antenna surrogate models for higher lever analysis.</figcaption></figure>
Chapter 3

# 3. Methodologies for modeling the ATF of variable antennas

## 3.1 Introduction

As discussed in the previous chapter, quantitatively characterizing the ATF is essential for knowing the *overall* behavior of antennas which are undergoing random uncertainties due to intrinsic deformations or external disturbances from the close (possibly fluctuating) environment.

Consider an antenna which is subject to the joint effects of *M* random uncertainties (e.g. those listed in § 2.1.3), denoted as a random vector *X* = (*X*₁, *X*₂, ..., *X*ₘ)ᵀ. Then the variable antenna can be taken as a “black box” with *X* being its input and the *X*-dependent ATF *H*ₓ(*f*, *r̂*) being its output. Our objective is to represent the “black box” with explicit mathematical functions which permit, for any given values of *X*, to generate the estimate of the corresponding output *Ĥ*(*X*, *f*, *r̂*). In addition, the difference between the estimate *Ĥ*(*X*, *f*, *r̂*) and the real observation *H*ₓ(*f*, *r̂*) must be minimized to a satisfactory level. For the purpose of convenience, we refer henceforth the aimed “explicit mathematical functions” or directly the estimate *Ĥ*(*X*, *f*, *r̂*) as “Surrogate model” in the sense that it can serve as eminently suitable substitute for the real antenna (or cumbersome full-wave simulated antenna) in various types of analyses.

Constructing such surrogate models is not easy as long as we notice the cumbersome expression of the ATF, not to mention the complexity introduced by the vast sources of randomness. Indeed, from (2.1) we can see the ATF is not a simple scalar-valued parameter as any of the aforementioned FoMs, but a frequency-dependent, complex- and vector-valued function defined on the surface of a sphere. In other words, instead of having only one single observable (e.g. FoM) to monitor when the random uncertainties intervene, we are now having to oversee an infinity of observables – both the amplitude and the phase of both polarimetric components on the whole spherical surface will be impacted – an extremely complicated multi-input/infinite-output problem which is intractable by using conventional identification methods.
To deal with this daunting challenge we propose in this thesis a “Two-Step” modeling approach: “field parsimony” in the first step, followed by “surrogate model extraction” in the second step. The aim of the first step is to substitute the cumbersome ATF by a parsimonious/compact representation, for example, in the form of a limited number of discrete characteristic parameters. Thereby, the initial intractable multi-input/infinite-output problem can be downgraded to a multi-input/multi-output one, leading to the possibility for a quantitative characterization. In the second step, the relation between the random input and the parsimonious ATF will be interpreted via dedicated deterministic or statistical approaches, from where surrogate models will be derived. The principle of the “Two-Step” approach is illustrated in Figure 3.1, where the field parsimony operation is denoted $\mathcal{F}_p$.

<figure><img src="image_18.png" /><figcaption>Figure 3.1: Principle of the “Two-Step” approach for modeling the ATF of variable antennas.</figcaption></figure>
The organization of this chapter is that: first, the method for realizing the “*field parsimony*” will be presented; then, dedicated methods, both deterministic and statistical ones, for extracting antenna surrogate models will be introduced, and complete modeling procedure will be synthesized; for each proposed methodology, application examples will be given to justify its effectiveness as well as to bring to light its limitations.

## 3.2 Field parsimony

When the question of “*field parsimony*” is inevitably raised, the first solution we may think of intuitively is to discretize the ATF on the sphere, i.e. substitute the continuous function by a finite set of (spatial) samples considered as output observables. However, this proves not to be a good idea for two reasons. First, to guarantee perfect reconstruction of the original field the Shannon’s sampling theorem must be respected. That’s to say, the sampling rate should be at least twice higher than the highest spatial variation frequency contained in the field in both elevation and azimuth, i.e. for all look directions. Therefore, a good knowledge of the spectral distribution of the field under study is required prior to the sampling procedure. However, it is not easy or even impossible to have such kind of prerequisite for an antenna (except for those very simple types such as dipoles and monopoles), not to mention that when it is undergoing random uncertainties. Second, a sampling strategy suitable for one antenna configuration (corresponding to one state of deformation or environment) may not be suitable for other configurations (corresponding to other states of deformation or environment). In other words, the observables chosen are different from one configuration to another, leading in total to a prohibitive number of observables to be monitored, which is certainly not conducive to the aim of parsimony. Briefly, there is no universal principle to follow to obtain a satisfactory discretization solution suitable for all antenna configurations. In practice, we usually adopt a uniform and oversampling strategy for the purpose of conserving the information completeness as well as facilitating the data collection process. Therefore, more sophisticated methods should be investigated in order to efficiently represent the continuous ATF, or the oversampled sequence of the latter.

In the first part of § 3.2, two methods for parsimoniously representing those scalar functions defined on the surface of a sphere will be reviewed. These methods are latter applied separately to the two polarization components $H_{\theta}$ and $H_{\phi}$ (which are complex-valued scalar functions) of the ATF, and the limitations of these scalar methods will be discussed. In the second part, a parsimonious method of representation which is directly applicable onto those vector-valued functions, e.g. the ATFs, will be presented. The advantage of the latter will be demonstrated by comparing to the scalar methods.
### 3.2.1 Methods for parsimoniously representing scalar spherical functions

#### 3.2.1.1 Double Fourier Series (DFS) expansion

Double Fourier Series (DFS) expansion is a simple and fast method for spectral analysis of scalar functions defined on the surface of a sphere. It is well-known that classical Fourier series permit to decompose a scalar function defined on the circle (or more generally, a periodic function) into a weighted sum of trigonometric (sine and cosine) functions. Analogically, DFS allows to expanding a scalar function $f(\theta, \varphi)$ defined on the surface of a sphere as a weighted sum of products of exponentials of separable spherical angular coordinates $(\theta, \varphi)$:

$$f(\theta, \varphi) = \sum_{n=-\infty}^{\infty} \sum_{m=-\infty}^{\infty} a_{nm} e^{j n \theta} e^{j m \varphi} \qquad (3.1)$$

On the viewpoint of duality, the Fourier coefficients {$a_{nm}$} are the equivalent (in the dual space) of the continuous function $f(\theta, \varphi)$ (in its original space), hence the latter can be exactly recovered from the former. In practice, the expansion is always truncated according to the required tradeoff between accuracy and sparsity. In this way, a parsimonious while complete representation – in the form of a limited number of Fourier coefficients {$a_{nm}$} – of the continuous function $f(\theta, \varphi)$ is obtained.

The DFS expansion was originally used for solving partial differential equations (elliptic or hyperbolic) which are defined on the surface of a sphere, such as the vorticity and shallow-water equations in the fluid mechanics domain [60]-[63]. Recently, newly emerging applications include the computation of gravitational fields (e.g. near black holes) [64] and some other space-time spectral analyses [65]. In the antenna-related domain, the DFS expansion could be useful for investigating some scalar-valued, spatially distributed performance indicators. For example, in [66] the DFS expansion was used to get a parsimonious representation for the radiation gain function $G(\theta, \varphi)$ before carrying out the statistical analysis of some uncertainty effects on $G(\theta, \varphi)$.

#### 3.2.1.2 Scalar Spherical Harmonics (SSH) expansion

The Scalar Spherical Harmonics (SSH) expansion is the appropriate Fourier transform for scalar functions defined on the surface of a sphere. The basis functions adopted by the SSH are Laplace's scalar spherical harmonics, denoted $Y_{nm}(\theta, \varphi)$ as in (3.2), instead of simple sine/cosine waves by the DFS:

$$Y_{nm}(\theta, \varphi) = P_n^m(\cos\theta) e^{-j m\varphi} \qquad (3.2)$$

where $P_n^m$ is the associated Legendre function (1st kind) of degree $n$ and order $m$ [67].
Usually, the spherical harmonics are normalized as (3.3) with regard to the inner product (3.4), forming hence a complete orthonormal basis in the Hilbert space of square-integrable scalar-valued functions defined on the surface of a sphere. In this way, any function $f(\theta, \varphi)$ belonging to this Hilbert space can be expanded as a linear combination of “spherical modes” as (3.5), where the modal coefficients $f_{nm}$ can be obtained by projection
$$f_{nm} = \langle f(\theta, \varphi), \hat{Y}_{nm}(\theta, \varphi) \rangle.$$

$$\hat{Y}_{nm}(\theta, \varphi) = \sqrt{\frac{2n+1}{4\pi} \frac{(n-m)!}{(n+m)!}} Y_{nm}(\theta, \varphi) \quad (3.3)$$

$$\langle f, g \rangle = \int_{0}^{\pi} \int_{0}^{2\pi} f g^* \sin\theta d\theta d\varphi \quad (3.4)$$

$$f(\theta, \varphi) = \sum_{n=0}^{N} \sum_{m=-n}^{n} f_{nm} \hat{Y}_{nm}(\theta, \varphi) \quad (3.5)$$

The SSH expansion method has gained much more popularity than the DFS expansion method in physics since mathematically, the former offers more proper spectral representation for spherical functions. In the antenna-related domain, the SSH expansion method is also quite useful for spectral analysis of the radiation gain function $G(\theta, \varphi)$ or the realized gain function $G_r(\theta, \varphi)$.

However, both DFS and SSH expansion methods are unsuitable for representing vector-valued functions defined on a spherical surface, e.g. the ATF $\mathcal{H}(\hat{\mathbf{r}})$ encountered in our problem. This is because the two polarization components ($\mathcal{H}_{\theta}$ and $\mathcal{H}_{\varphi}$) of the ATF cannot be decomposed separately by using DFS or SSH method due to the uniqueness of vector-valued functions such as $\mathcal{H}$ at the two poles ($\theta = 0$ and $\theta = \pi$). In fact, given that the zenith point ($\theta = 0, \varphi$) is associated with an infinity of $\varphi$-dependent local coordinates system ($\hat{\theta}_{(0,\varphi)}, \hat{\varphi}_{(0,\varphi)}$), the unique vector $\mathcal{H}(\theta=0,\varphi)$ could have an infinity of possibilities of decomposition in $\mathcal{H}_{\theta}$ and $\mathcal{H}_{\varphi}$ depending on the chosen local coordinates system:

$$\mathcal{H}(\theta=0, \varphi) = \mathcal{H}_{\theta}(0, \varphi) \hat{\theta}_{(0, \varphi)} + \mathcal{H}_{\varphi}(0, \varphi) \hat{\varphi}_{(0, \varphi)} \quad (3.6)$$

As a result, an infinity of values of $\mathcal{H}_{\theta}$ and $\mathcal{H}_{\varphi}$ are associated to this single point, which is evidently contradictory to a properly defined function. The same indetermination exists also at the nadir point ($\theta = \pi, \varphi$). As a result, because of the mathematical difficulty, significant errors can occur when the DFS or the SSH method is applied independently onto the two polarization components. Besides, the same problem occurs also if the partial (or “polarimetric”) gains $G_{\theta,\varphi}$ or $G_{r\theta,\varphi}$ instead of the total gains $G$ or $G_r$ are considered. Note that
the total gains $G$ and $G_r$ are proportional to $||\mathcal{H}||^2$, thence having unique values at the poles and are regularly defined on the whole sphere; by contrast, the polarization components which are proportional to $|\mathcal{H}_{\theta}|^2$ or $|\mathcal{H}_{\varphi}|^2$ are not univocally defined at the poles.

<figure><img src="image_19.png" /><figcaption>Figure 3.2: Patch antenna used for field parsimony test.</figcaption></figure>

For the purpose of illustration, we applied the SSH expansion method to decompose the total realized gain function $G_r(\theta, \varphi)$ as well as its $\varphi$-component $G_{r\varphi}(\theta, \varphi)$ (of which the polar values are chosen at $(\theta=0/\pi, \varphi=0)$) of a classical patch antenna operating at 2.45 GHz (Figure 3.2). Results are shown in Figure 3.3 and 3.4. As expected, it can be seen that the SSH permits to represent the total $G_r$ with excellent precision of reconstruction (the maximal difference of $G_r$ in the -8 dB main radiation direction is only 0.07 dB), but is not able to recover the $\varphi$-component $G_{r\varphi}$ accurately especially in the vicinity of the two poles, even with a very high expansion degree $N_{SSH} = 30$. The failure of the SSH method for representing the polarimetric components of $G$ (or $G_r$) has also been pointed out in other people's work, e.g. in [68].

<figure><img src="image_20.png" /><figcaption>Figure 3.3: SSH expansion applied on the total realized gain $G_r(\theta, \varphi)$ of a patch antenna: (a) initial $G_r$; (b) reconstructed $G_r$ with the SSH modes ($N_{SSH} = 5$).</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>Figure 3.4: SSH expansion applied on the polarimetric realized gain $G_{r\varphi}$ ($\theta, \varphi$) of a patch antenna: (a) initial $G_{r\varphi}$; (b)-(e) reconstructed $G_{r\varphi}$ with the SSH modes ($N_{SSH} = 5, 10, 20$ and $30$ respectively).</figcaption></figure>

To summarize, although the DFS and the SSH expansion methods are efficient for parsimoniously representing the total gains $G$ or $G_r$, they are insufficient for decomposing neither the polarimetric components nor the vector-valued ATF. We present in the following part a parsimony method which is directly applicable to vector-valued functions named Vector Spherical Harmonics (VSH) expansion method.

### 3.2.2 Vector Spherical Harmonics (VSH) expansion

#### 3.2.2.1 Principle

The Vector Spherical Harmonics (VSH) expansion is the generalization of the SSH expansion for the Hilbert space consisting of square-integrable vector-valued functions defined in the 3D spherical coordinates system ($r, \theta, \varphi$). Hansen explained thoroughly in [70] the principle of using the VSH expansion as an efficient approach to represent the radiation source and to enable the near-field/far-field transformations of the EM waves in the context of antenna applications. In the framework of our problem we are interested in the far-field expression of the *E*-field, or the ATF $\mathcal{H}(f, \hat{r})$, so we simplify the formulation proposed in [70] by excluding the *r*-dependence. Thereby, we are in the framework of vector-valued functions defined on the surface of a sphere, for which the appropriate orthonormal basis are the simplified VSHs [71] as:
$$ 
\left\{ 
\begin{aligned}
\hat{\psi}_{nm}^{TM}(\hat{\mathbf{r}}) &= v_{nm} \left[ \partial_{\theta} Y_{nm} \hat{\theta} + \frac{1}{\sin\theta} \partial_{\varphi} Y_{nm} \hat{\varphi} \right] \\
\hat{\psi}_{nm}^{TE}(\hat{\mathbf{r}}) &= v_{nm} \left[ \frac{1}{\sin\theta} \partial_{\varphi} Y_{nm} \hat{\theta} - \partial_{\theta} Y_{nm} \hat{\varphi} \right]
\end{aligned}
\right. 
\qquad (3.7) 
$$

where TM and TE represent respectively the Transverse Magnetic and the Transverse Electric modes; $Y_{nm}$ are the SSHs (3.2); $v_{nm}$ are the normalization coefficients:

$$ v_{nm} = \left[ \frac{2n + 1}{4\pi n(n + 1)} \frac{(n - m)!}{(n + m)!} \right]^{1/2} $$

Then, the ATF $\mathcal{H}(f, \hat{\mathbf{r}})$ which is well-defined, regular and continuous on the whole spherical surface (including the two poles), can be expanded into a series of VSH modes as stated in (3.8).

$$ \mathcal{H}(f, \hat{\mathbf{r}}) = \sum_{n=1}^{+\infty} \sum_{m=-n}^{n} \sum_{u=1}^{2} H_{nm}^{u}(f) \hat{\psi}_{nm}^{u}(\hat{\mathbf{r}}) \qquad (3.8) $$

where

* $u=1, 2$ represents the TM or TE mode.

* $n, m$ represent respectively the degree and the order of expansion. It is demonstrated that the power of the FF is mainly retained in VSH modes of lower degrees, and vanishes rapidly as $n$ increases. The maximal degree (denoted as $N_{VSH}$) beyond which the power contribution can be neglected is related to the radius ($r_0$) of the minimal sphere circumscribing the antenna by an empirical rule: $N_{VSH} = kr_0 + O(\sqrt[3]{kr_0})$ [72]. Therefore, in general the VSH expansion is truncated at a proper $N_{VSH}$ according to the required precision.

* $H_{nm}^u(f)$ are the complex-valued VSH coefficients which can be obtained by projection: $H_{nm}^u(f) = \langle \mathcal{H}(f, \hat{\mathbf{r}}), \hat{\psi}_{nm}^u(\hat{\mathbf{r}}) \rangle$.

### 3.2.2.2 Test of the performance of VSH

To illustrate the superior advantage of the VSH method over the SSH method for sparsely representing vector-valued ATFs, we carried out a test on the same patch antenna used in § 3.2.1 as follows:

1) We applied the VSH expansion on the initial ATF $\mathcal{H}^{init}$ at a truncated degree of $N_{VSH} = 5$.

2) We reconstructed the ATF from the VSH coefficients obtained in 1) by reversing the expansion procedure. Denote the reconstructed ATF as $\mathcal{H}^{rcs}$.
3) We calculated the total realized gain $G_r$ as well as the φ-component $G_{r\phi}$ (of which the polar values are chosen at ($\theta = 0/\pi$, $\phi = 0$)) respectively from $\mathcal{H}^{\text{init}}$ and $\mathcal{H}^{\text{rcs}}$; the comparison results are shown in Figure 3.5 and 3.6.

We see that both $G_r$ and $G_{r\phi}$ are accurately recovered from the VSH parsimony operation (the maximal difference in the -8 dB main radiation direction is 0.04 and 0.08 dB respectively). Particularly, the good recovery of the polarimetric components of $G_r$ confirms the good recovery of the ATF components since we have $G_{r\phi,\theta} = |\mathcal{H}_{\phi,\theta}|^2$ [59]. Comparing to the unsatisfactory results when using the SSH expansion method (Figure 3.4 (b)-(e)), we see immediately the benefits of the VSH expansion method. In the subsequent modeling, we will adopt the VSH expansion method to achieve the purpose of “field parsimony”.

<figure><img src="image_22.png" /><figcaption>Figure 3.5: VSH expansion applied on the ATF of a patch antenna: (a) initial $G_r$; (b) reconstructed $G_r$ with the VSH modes ($N_{\text{VSH}} = 5$).</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Figure 3.6: VSH expansion applied on the ATF of a patch antenna: (a) initial $G_{r\phi}$; (b) reconstructed $G_{r\phi}$ with the VSH modes ($N_{\text{VSH}} = 5$).</figcaption></figure>
## 3.3 Deterministic modeling of ATF: PS

Once the cumbersome ATF is transformed into a parsimonious form, we can perform the second step of the “Two-Step” methodology, that’s to say, extract the surrogate model in order to explicitly interpret the relation between the random input variables and the parsimonious ATF (i.e. the selected VSH coefficients $H_{nm}^u(f)$) as illustrated in Figure 3.7.

<figure><img src="image_24.png" /><figcaption>Figure 3.7: Surrogate model relating the random input variables and the parsimonious ATF.</figcaption></figure>

We investigate first a deterministic modeling method named Parametric Study (PS). PS is an intuitive and canonical approach which provides non-intrusive, direct, and fast insights into the input-output relation of an unknown system. The underlying idea is simple – make varying only one parameter at a time while keeping the other parameters fixed, and observe how the FoMs of interest are evolving due to the single parameter effect. Because of its simplicity, PS has been widely used in many domains of applied physics and has been implemented in various types of commercially available Computer Aided Design (CAD) tools. In the domain of antenna design, PS can be used to analyze the sensitivity of the antenna on certain design parameters, and to hunt for the optimal parameter value. Some application examples can be found in [73]-[77]. Here, we adopt the PS to extract surrogate models to quantify the impact of some input variable $x$ on the parsimonious ATF.

In order to evaluate the accuracy of the derived surrogate model, we define a performance indicator named the Relative Root Mean Square Error (RRMSE) of the realized gain $G_r$ as:

$$RRMSE[G_r(f)] = \frac{RMSE[G_r(f)]}{\langle G_r^{init}(f, \hat{\mathbf{r}}) \rangle_{(\theta, \phi)}} \quad (3.9)$$

with RMSE being the Root Mean Square Error of $G_r$:

$$RMSE[G_r(f)] = \sqrt{\frac{1}{4\pi} \iint_{\Omega} [G_r^{init}(f, \hat{\mathbf{r}}) - G_r^{rcs}(f, \hat{\mathbf{r}})]^2 d\Omega} \quad (3.10)$$

where $G_r^{init}$ and $G_r^{rcs}$ are respectively the realized gain of the initial field and that of the reconstructed field from the surrogate model; and $\langle G_r^{init}(f, \hat{\mathbf{r}}) \rangle_{(\theta, \phi)}$ is the realized gain of the initial field averaged over the whole sphere ($\Omega=4\pi \text{ str}$) and is in fact the total radiation efficiency $\eta$ of the antenna:
$$ \eta(f) = \langle G_r^{\text{init}}(f, \hat{\mathbf{r}}) \rangle_{(\theta, \varphi)} = \frac{1}{4\pi} \iint_{\Omega} G_r^{\text{init}}(f, \hat{\mathbf{r}}) d\Omega $$

$d\Omega = \sin\theta d\theta d\varphi$ being the infinitesimal solid angle, and reminding $G_r(f, \hat{\mathbf{r}}) = \|\mathcal{H}(f, \hat{\mathbf{r}})\|^2$ [59].

Note that till now, the general expression of all physical quantities are frequency dependent, which means the antenna's frequency behavior has not been revealed yet in the surrogate model. However, from the open literatures we are aware that the frequency behavior is not an easy task to be addressed. To mitigate, some approaches were restricted to a single fixed frequency, e.g. in [49]-[51], [58], whereas other authors have chosen to investigate the FoMs of interest averaged over a range of frequencies, e.g. in [66], [68]. Here in our first attempt, we choose to temporarily bypass this thorny issue by simply fixing the frequency at a nominal value, e.g. the antenna's resonant frequency. An advanced modeling technique permitting to include the frequency behavior in the surrogate model will be presented in § 3.5.

### 3.3.1 Modeling procedures

An iterative modeling procedure combining the VSH expansion and the PS to extract surrogate model of an antenna undergoing a variable input parameter $x$ is proposed hereby (Figure 3.8):

1. Generate a set of sampling points $\{x_k\}$ which cover the support of definition of the input parameter $x$. This set of points is also called the Experimental Design (ED).

2. For each value $x_k$ reconfigure the antenna accordingly, carry out the EM simulation or measurement, and compute the ATF $\mathcal{H}_k(\hat{\mathbf{r}})$.

3. Apply the VSH expansion onto each ATF to get the VSH coefficients $\{H_{nm}^u, n \le N_{\text{VSH}}\}_k$ up to the degree of truncation $N_{\text{VSH}}$ depending on the required precision.

4. Reconstruct the ATFs from the VSH coefficients, and assess the RRMSE to verify the relevance of the choice of $N_{\text{VSH}}$. If the RRMSE is satisfactory then select the most energetically dominant VSHs; otherwise, restart from the Step 3 with a higher $N_{\text{VSH}}$.

5. Fit the evolution curve of those dominant VSH coefficients in terms of $x$ with polynomial functions: $\{ H(x) = \sum_{k=0}^{P} \beta_k(f) P_k(x) \}_{nm}^u$.

6. Reconstruct the ATFs from the whole set of polynomial coefficients $\{\beta_0, \beta_1, \beta_2, \dots\}_{nm}^u$, and assess the reconstruction error in terms of the RRMSE. If the model accuracy is satisfactory then stop; otherwise, return to Step 5 and increase the polynomial degree.

Eventually, a deterministic model relating quantitatively and explicitly the input parameter $x$ and the output parameters $H_{nm}^u(x)$ via the whole set of polynomial coefficients
$\{\beta_0, \beta_1, \beta_2, \dots\}_{nm}^u$ is established, allowing the ATF prediction for any value of $x$, thanks to the inverse VSH transform.

<figure><img src="image_25.png" /><figcaption>Figure 3.8: Algorithm flowchart of the parametric study procedure.</figcaption></figure>

### 3.3.2 Application to a canonical dipole antenna - with one input variable

We are going to quantitatively valid the proposed PS approach in a rather academic framework. The procedure is applied to the half-wave dipole which is a simple and very well-known canonical antenna (so an ideal “test antenna”) to verify its relevance, and assess its accuracy and robustness [69].

#### 3.3.2.1 Design of a deformable dipole

A dipole which is made of perfect electrical conductor (PEC) material, with length of $l_{dip}$ = 58.14 mm and section radius of 1 mm is designed (Figure 3.9 (a)). In its straight state, the
dipole is well matched (i.e. $|S_{11}| < -10$ dB) over a frequency band of [2.1, 2.4] GHz. This dipole is assumed to be deformable and may undergo bending (Figure 3.9 (b)), torsion, or any combination of the two. In this example, only the bending effect is studied, i.e. the bending curvature $\kappa$ (or the bending radius $R_b = 1/\kappa$) is considered as the only input variable. The frequency behavior will not be included and $f$ is fixed at 2.3 GHz.

Electromagnetic simulations are undertaken in CST® MWS (with the Frequency domain solver and Tetrahedral adaptive mesh type) to obtain radiation fields which are sampled every 5° in both elevation ($\theta \in [0, 180°]$) and azimuth ($\phi \in [0, 360°]$).

<figure><img src="image_26.png" /><figcaption>Figure 3.9: Illustration of the bending phenomenon of a dipole antenna.</figcaption></figure>

Figure 3.10 shows the bending effect on the reflection coefficient $S_{11}$. The black line is the $|S_{11}|$ in straight state; the blue, red and purple lines correspond respectively to $R_b = 60, 30$ and 10 mm. Comparing to the straight state, the first two bending states present an upward shift of the resonant frequency, while the last one presents also a strong impedance mismatch. We are going to construct a surrogate model to quantitatively model the bending effect on the ATF of the dipole.

In the first stage, we choose to limit the bending effect in relatively moderate level. The sampling points of $\kappa$ are taken such that the corresponding bending radius $R_b$ takes values in the set $\{\infty\} \cup \{500:-50:200\} \cup \{150:-10:60\}$ (in mm). Note that $R_b = \infty$ (i.e. $\kappa = 0$) corresponds to the straight state.

<figure><img src="image_27.png" /><figcaption>Figure 3.10: Reflection coefficient $|S_{11}|$ of the dipole in different bending states ($R_b$ in mm).</figcaption></figure>
### 3.3.2.2 Field parsimony with VSH expansion

The ATFs are computed from the FFs according to (2.1) and are expanded using the VSH method for degrees $n$ up to $N_{VSH} = 3$ and thus in total 30 VSH modes ($1 \le n \le N_{VSH}$, $-n \le m \le n$ and $u = 1, 2$) are obtained. Then we notice that among these 30 VSH modes only 6, i.e. $(u, n, m) \in \{(1,1, \pm 1), (2,1,0), (2,2, \pm 1), (2,3,0)\}$ are dominant in terms of power, containing more than 99.9 % of the total power.

After that, the ATFs are reconstructed separately from the 30 VSH modes and from the 6 dominant modes. The bent dipole with $R_b = 60$ mm (which is the largest deformation in the current case) is taken for the illustration of results in the following context. The maximal difference in $G_r$ between the reconstructed field from those 30 modes and the initial field is only about 0.03 dB, which justifies a very good accuracy with truncation to degree $N_{VSH} = 3$ in the expansion. Furthermore, the deviation in $G_r$ of the reconstructed field from the 6 dominant modes is less than 0.1 dB compared to the initial field, which indicates that the approximation with only 6 dominant VSH modes is still very good. The RRMSE of the reconstructed fields are also computed, as shown in Figure 3.11. It is noted that the RRMSE of the reconstruction from 30 modes (the red dash line) is less than 0.2 %. This error is mainly due to the truncation of expansion ($N_{max} = 3$), a small part being related to the angular sampling inducing some approximations in the numerical quadrature performed in the VSH transform. The RRMSE of the reconstruction from 6 dominant modes (the blue dash line) is less than 0.4 %, confirming again the conservation of a high accuracy using only 6 dominant VSH modes instead of 30 ones.

<figure><img src="image_28.png" /><figcaption>Figure 3.11: RRMSE of the reconstructed fields with different approaches.</figcaption></figure>

Furthermore, as the dipole presents symmetries with respect to both the equatorial and meridian planes, the VSH coefficients verify relation:

$$ \begin{cases} H_{n,-m}^{u} = (-1)^{m+u} H_{n,m}^{u} \\ H_{n,m}^{u} = 0 \quad \text{for } m+n+u \text{ even} \end{cases} \qquad (3.11) $$

which implies simple relations:
$$H_{1,-1}^{1} = H_{1,1}^{1}, \quad H_{2,-1}^{2} = -H_{2,1}^{2} \qquad (3.12)$$

As a result, it remains only 4 VSH coefficients (i.e. $H_{1,1}^1$, $H_{1,0}^2$, $H_{1,1}^2$ and $H_{3,0}^2$) to be modeled.

### 3.3.2.3 Surrogate model extraction

In this step, the evolution of these 4 VSH coefficients as function of the curvature $κ$ will be fitted with polynomials. As the VSH coefficients are complex values, their real and imaginary parts are fitted separately. This is done using the Matlab Curve Fitting Tool. Linear fits match well for all coefficients but one, which requires a quadratic fit. Finally we get a surrogate model with 17 polynomial coefficients. Figure 3.12 shows an example where the real and imaginary parts of the VSH coefficients $H_{1,1}^1$ and $H_{3,0}^2$ are fitted.

<figure><img src="image_29.png" /><figcaption>Figure 3.12: Polynomial fitting of the VSH coefficients $H_{1,1}^1$ and $H_{3,0}^2$.</figcaption></figure>

### 3.3.2.4 Model accuracy assessment

To evaluate the model's accuracy, ATFs are reconstructed using the derived surrogate model: firstly the 6 dominant VSH coefficients are computed from the polynomial coefficients and the equations (3.12); then ATFs are computed by inversing the VSH expansion procedure. The difference in $G_r$ between the reconstructed fields from the 6 dominant VSH coefficients with and without polynomial fitting is observed to be only 0.02 dB, which implies that the fitting is accurate enough. In Figure 3.11 the magenta dash line shows the RRMSE of the model-reconstructed field. It can be seen that there isn't much difference when comparing to the case where the 6 original dominant VSHs are used for reconstruction, indeed, the RRMSE remains inferior to 0.4 %, which indicates the good accuracy of the parametric modeling procedure.

The efficiency of parsimony of the surrogate model can be evaluated in terms of the data compression rate $τ$:

$$τ = 1 - N_{res} / N_{init}$$
where $N_{init}$ is the number of sampled data from the initial field, and $N_{rcs}$ is the number of parameters of the surrogate model.

In one hand, the number of real sampled data representing the radiated FF is: $N_{init} = N_K \times N_\theta \times N_\phi \times 2 \times 2$, where $N_K$, $N_\theta$ and $N_\phi$ represent the number of values of the input parameter, the number of sampling points in elevation and azimuth, and the last two factors “2” correspond to the two polarization components $H_\theta$ and $H_\phi$ which are complex quantities. This number must be divided by 4 to account for the two symmetries. In the other hand, the number of parameters of the final model $N_{rcs}$ are the 17 polynomial coefficients. As a result, for an angular step of 5° both in elevation and azimuth, the data compression rate is 99.96%.

### 3.3.2.5 Verification based on interpolation and extrapolation

It is necessary to verify if the proposed model is still valid for other range of values of the curvature $\kappa$. For this purpose, a verification based on both interpolation i.e. within the modeling interval ($\kappa \in [0, 1/60]$ mm⁻¹) and extrapolation i.e. outside the modeling interval ($\kappa \in [1/60, 1/30]$ mm⁻¹) has been carried out. More precisely, firstly 150 values of $\kappa$ are generated randomly following the uniform distribution in the total interval $[0, 1/30]$ mm⁻¹. Secondly, the dipole's FF for each value of $\kappa$ is successively obtained by new CST® simulations, expanded over 30 VSH modes and reconstructed separately with 30 original VSH modes, with 6 original dominant VSH modes and with 6 polynomial-fitted dominant VSH modes. At last, reconstructed fields are compared to the simulated ones and the accuracy is assessed by means of RRMSE as shown in Figure 3.13.

<figure><img src="image_30.png" /><figcaption>Figure 3.13: RRMSE of reconstructed fields with different approaches for 150 random values of the parameter $\kappa \in [0, 1/30]$ mm⁻¹.</figcaption></figure>
It is noted that the RRMSE of reconstructed fields remains always less than 0.4 % for all interpolation values of $κ$ and is still inferior to 1.4 % for all extrapolation values of $κ$ despite that an average variation of $G_r$ (relative to the straight dipole) of at most 6 % is caused due to the bending effect. In addition, the added error due to the polynomial fitting of the VSH coefficients (magenta curve in Figure 3.13) appears marginal compared to the error due to the modal truncation (blue curve). In other words, besides the 6 most dominant VSH modes, some less dominant ones are also needed if a better accuracy is required in the extrapolation interval. There is of course always a trade-off between the parsimony efficiency of the model and its accuracy, and the parametric model is fully adjustable according to the priority of criteria. In conclusion, the proposed model is both accurate and efficient for the whole bending range ($κ ∈ [0, 1/30] mm^{-1}$), corresponding typically to “variational” deformations.

### 3.3.2.6 Modeling results for severer deformations

The moderate range of deformation chosen previously (for which the $|S_{11}|$ is slightly affected; see Figure 3.10) was intended to illustrate the principles of parametric modeling and its effectiveness. However, to verify the robustness of the proposed algorithm more severe deformations are now considered, with curvatures up to $1/10 mm^{-1}$ (corresponding to $R_b \sim l_{dip}/6$). For the most curved dipole (with $R_b = 10$ mm) the $|S_{11}|$ is shown in Figure 3.10 (purple curve). In Figure 3.14 the brown lines show the corresponding changes of the radiation patterns. It can be seen from both figures that the effect of bending is now really significant. Actually, it corresponds to an average variation of $G_r$ (relative to the straight dipole) of about 80 %.

<figure><img src="image_31.png" /><figcaption>Figure 3.14: $G_r$ patterns of the initial field (both for the straight and curved dipole with $R_b = 10$ mm) and the reconstructed fields with different approaches (only for the curved dipole with $R_b = 10$ mm).</figcaption></figure>
<figure><img src="image_32.png" /><figcaption>Figure 3.15: Polynomial fitting of the VSH coefficient $H_{1,0}^1$ (severe bendings: $\kappa \in [0, 1/10] \text{ mm}^{-1}$).</figcaption></figure>

A new surrogate model is derived from 20 sampling points of the curvature for $\kappa \in [0, 1/10] \text{ mm}^{-1}$. The model consists of 8 dominant VSH coefficients, each of which is fitted with polynomials of order 3 (an example is shown in Figure 3.15). In Figure 3.14 the green lines show the radiation patterns for the field reconstructed with the proposed model. We can see that the deviation from of the initial field patterns (the brown lines) is very low (with a maximal difference of 0.2 dB). The *RRMSE* of fields reconstructed with the surrogate model is assessed, as shown in Figure 3.16. It is noted that the *RRMSE* remains below 4 %, whereas the average relative variation of $G_r$ is as high as 80 %, demonstrating the robustness of the model. A similar verification process based on interpolation as in the previous example is carried out for the validation of the proposed model. In order to demonstrate the predictive capabilities of the model, two hundred random dipoles are generated with the proposed model and are compared to their simulated (with CST®) counterparts. This constitutes a sort of Monte Carlo test. It is noted that the *RRMSE* of $G_r$ remains also always below 4 % for all values of $\kappa$ (Figure 3.17). Finally the data compression rate of the constructed surrogate model is evaluated at 99.92 %.

<figure><img src="image_33.png" /><figcaption>Figure 3.16: RRMSE of reconstructed fields with different approaches for the most severe bending state.</figcaption></figure>
<figure><img src="image_34.png" /><figcaption>Figure 3.17: RRMSE of reconstructed fields with different approaches for 200 random values of the parameter $κ \in [0, 1/10] \text{ mm}^{-1}$.</figcaption></figure>

### 3.3.3 Application to a canonical dipole antenna - with two input variables

In a second step, we investigate the joint effect of two input variables, namely bending and torsion (Figure 3.18), by using the PS approach. The torsion phenomenon is characterized by the torsion rate $α$, defined as:

$$α = \frac{θ}{|h|}$$

where $θ$ is the rotation angle of the dipole around the vertical axis at height $h$ along the z'Oz axis.

<figure><img src="image_35.png" /><figcaption>Figure 3.18: Illustration of the joint bending & torsion phenomenon of a dipole antenna.</figcaption></figure>

In this case, 8 dominant VSH modes are finally chosen from the total 30 ones. The surrogate model is constructed by following the procedure presented in § 3.3.1 except that bivariate polynomials are used for fitting the sampled ATFs data. The performance of the final surrogate model is evaluated in terms of the metric RRMSE, as shown in Figure 3.19. The red surface shows the RRMSE of the fields reconstructed from 30 VSH modes, serving as the reference for comparison. The bleu surface shows the RRMSE of the fields reconstructed from the 8 most dominant VSH modes, and the magenta surface shows the RRMSE of the fields predicted by the surrogate model. It can be seen that, on the one hand, the magenta
curve overlaps well with the blue one, which implies the satisfactory accuracy of the PS modeling; on the other hand, for both the magenta and blue cases, the *RRMSE* is quite low when the two input parameters are small, but increases significantly for larger values of the two input parameters, which implies the necessity of including more VSH modes in order to obtain better accuracy in the case of large variations. Other modeling results are summarized in Table II.

<figure><img src="image_36.png" /><figcaption>Figure 3.19: *RRMSE* of reconstructed fields with different approaches for bivariate case.</figcaption></figure>

<table><caption>Table II: Parametric study of a deformable dipole: input parameters, modeling parameters and results.</caption><thead><tr><th colspan="2">PS study on a deformable dipole</th><th>Case 1</th><th>Case 2</th></tr></thead><tbody><tr><td rowspan="3">Inputs</td><td>Input variable(s)</td><td>κ (mm<sup>-1</sup>)</td><td>(κ, α) (mm<sup>-1</sup> × rad·mm<sup>-1</sup>)</td></tr><tr><td>Variability range</td><td>[0, 0.1]</td><td>[0, 0.05] × [0, 0.06]</td></tr><tr><td>Maximal field variation due to deformations (RRMSE)</td><td>80%</td><td>36.50%</td></tr><tr><td rowspan="4">Modeling parameters</td><td>Cardinal of the ED</td><td>20</td><td>10 × 13 = 130</td></tr><tr><td>Number of selected dominant VSH modes</td><td>8</td><td>8</td></tr><tr><td>Maximal order for the polynomial fitting</td><td>3</td><td>3</td></tr><tr><td>Number of polynomial coefficients in surrogate model</td><td>40</td><td>144</td></tr><tr><td rowspan="2">Modeling results</td><td>Model accuracy (RRMSE)</td><td>&lt; 4 %</td><td>&lt; 5.5 %</td></tr><tr><td>Data compression rate, τ</td><td>1.33 × 10<sup>3</sup></td><td>9.62 × 10<sup>3</sup></td></tr></tbody></table>

### 3.3.4 Discussion

In this section, we combined the VSH expansion method with the PS to construct surrogate models to quantitatively characterize the ATF of antennas undergoing one or two types of deformations, corresponding to one or two input variables. The constructed surrogate
models have been tested on a canonical dipole antenna where the modeling results are relatively satisfactory in terms of accuracy, parsimony and efficiency. However, despite that PS is a fast and simple approach, it has two major methodological flaws: firstly, PS well suited for univariate or at most bivariate problems but cannot be generalized to multivariate cases especially when the number of input parameters is really large, because the number of required pre-evaluations would grow explosively; secondly, since there is no technique (such as cross-validation, early stopping, model comparison) adopted for ensuring the model’s ability of generalization, i.e. for reliably predicting new set of observations, severe “overfitting” phenomenon risks to occur during the fitting. As a result, this simple deterministic methodology is rarely applicable for realistic complex problems where generally a large number of sources of uncertainties intervene. Nevertheless, this primary methodology justifies the relevance of the “Two-Step” modeling principle of “field parsimony” followed by “surrogate model extraction”, thereby, has paved our way for further investigation of more sophisticated, efficient and robust modeling approaches, e.g. the statistical approaches.

## 3.4 Statistical modeling of the ATFs: PCE

### 3.4.1 Introduction

Conventional deterministic approaches which are suitable for designing or modeling those “classical” antennas e.g. rigid, or large-sized, or clearly isolated antennas, etc. are now inappropriate for variable antennas under uncertain conditions, especially when the number of random parameters is really large. Indeed, the combinatory of situations can be so explosive that only statistical methods can offer an efficient framework of analysis.

Some tentative researches on statistical modeling the FF or ATF are briefly reviewed in § 2.2.2. Particularly, the work of Rossi et al. [58] corroborated the feasibility for constructing quantitative antenna models by resorting to the combination of VSH expansion and gPCE methods. However, this method still has some limitations for treating much more complex problems of large dimensionality and high expansion degrees. A more detailed discussion will be given in § 3.4.2.3.

In this section, we propose a new statistical methodology for modeling the *complete* FF radiated by variable antennas. Our methodology has the similar principle of first applying the VSH expansion on the ATFs, then proceeding the PCE on the VSH coefficients. The novelty of our approach is that the unknown PCE coefficients will be solved in a more efficient way that permits to reduce drastically the pre-evaluation cost, notably in the situations where a significant number of random inputs are considered and high expansion degrees are required.
An adaptive modeling procedure is also proposed in order to control the convergence of the
surrogate model.

The organization of this section is as follows: first, the principle of the PCE is briefly introduced; second, our proposed statistical methodology for modeling the ATF is detailed; finally, the application of the proposed methodology to a deformable dipole antenna is presented and the modeling results are analyzed.

3.4.2 Principle of the Polynomial Chaos Expansion (PCE)

PCE is an efficient method for characterizing the uncertainty propagation characteristics
of an uncertain system. PCE was originally introduced by Norbert Wiener in the 1930s to
model Gaussian processes with Hermite polynomials [78]. Here, “chaos” simply refers to
“uncertainty”. According to the Cameron-Martin theorem [79], any stochastic process (not
limited to Gaussian processes) with finite second-order moment can be represented by
Hermite polynomials, but the convergence rate is optimal only for Gaussian processes. In the
1970s, Ogura Found that Charlier polynomials are more appropriate for representing Poisson
processes [80]. Then it was discovered that many of orthogonal polynomials belong to the so-
called Askey-scheme [81]. In the 2000s, Xiu and Karniadakis generalized the scope of PCE to
other types of stochastic process and proved numerically the optimal (exponential)
convergence rate when using dedicated polynomial basis for a given stochastic process,
known as the gPCE [48]. PCE has been verified to be more computational efficient than the
“brute force” MC approach in a wide range of applications including e.g. vehicle dynamics
[82]-[86], and electronics (circuits, interconnects, and multiport systems) [87]-[91]. In recent
years, PCE has been adopted in electromagnetism to assess the human exposure to the EM
field or the scattered field due to environmental uncertainties [92]-[95]. Some tentative
researches on the flexible antennas are already briefly reviewed in § 2.2.

3.4.2.1 Formulation

Consider a physical system which undergoes random uncertainties. Denoted the input-
output relation of the considered system as $Y = \mathcal{M}(X)$, where $X = (X_1, X_2, ..., X_M)$ is the
random input variables which are defined in the input space $D_X \subset \mathbb{R}^M$, and $Y$ is a scalar
output of interest of the system which is consequently also a random quantity. Suppose that
the joint distribution, i.e. the Probability Density Function (PDF) $f_X$, of the input variables is
known, and that the output $Y$ has finite second-order moment, i.e.
$$ \mathbb{E}[Y^2] = \int_{D_X} \mathcal{M}^2(\mathbf{x}) f_X(\mathbf{x}) d\mathbf{x} < \infty $$
(note that this applies to most physical systems), then $Y$
belongs to the Hilbert space $L^2(\mathbb{R}^M, \mathbb{R})$ consisting of all the $f_X$-square-integrable real-
valued M-dimensional functions, associated with the inner product
<u, v> = ∫<sub>D<sub>x</sub></sub> u(x)v(x)f<sub>x</sub>(x)dx. By appropriately defining the orthonormal multivariate
polynomial basis {Φ<sub>α</sub>(X)} with respect to the inner product:

$$
\langle \Phi_{\alpha}(\mathbf{x}), \Phi_{\beta}(\mathbf{x}) \rangle = \int_{D_{\mathbf{x}}} \Phi_{\alpha}(\mathbf{x}) \Phi_{\beta}(\mathbf{x}) f_{\mathbf{x}}(\mathbf{x}) d\mathbf{x} = \delta_{\alpha\beta} \quad (3.13)
$$

with $\delta$ being the Kronecker delta function, $Y$ can be expanded in terms of this basis as:

$$
Y = \mathcal{M}(X) \approx \sum_{\alpha \in \mathcal{A}^{M, N_{\text{PCE}}}} y_{\alpha} \Phi_{\alpha}(X) \quad (3.14)
$$

where $\boldsymbol{\alpha} = (\alpha_1, \alpha_2, \dots, \alpha_M) \in \mathbb{N}^M$ is the multi-indices that indicate the degree $\alpha_i$ of the $i$-th variable $X_i$ in the basis $\Phi_\alpha$; for computational purpose, the expansion is often truncated up to a maximal total degree, denoted as $N_{\text{PCE}}$; $\mathcal{A}^{M,N_{\text{PCE}}} = \{\boldsymbol{\alpha} \in \mathbb{N}^M \mid \|\boldsymbol{\alpha}\|_1 \le N_{\text{PCE}}\}$ is the set of multi-indices of all basis with total degree less than $N_{\text{PCE}}$ ($\|\boldsymbol{\alpha}\|_1 = \alpha_1 + \alpha_2 + \dots + \alpha_M$ being the $l_1$-norm); and $y_{\alpha} \in \mathbb{R}$ are the expansion coefficients to be determined.

It is clear that the formulation of a PCE modeling problem consists of three main steps:

1) Choose the random input variables $X$ and determine their joint PDF (i.e. the variability range and the distribution); choose the output of interest $Y$.

2) Construct the appropriate orthonormal multivariate polynomial basis {$\Phi_{\alpha}(X)$}.

3) Compute the unknown PCE coefficients $y_{\alpha}$.

The first step is obviously application-dependent. The selection of input/output depends on what uncertainty phenomenon is actually of interest. The estimation of the joint PDF of X can be made upon one's work experience, literature review and experimental measurements, and is usually a trade-off after taking comprehensive consideration of the complexity and the feasibility. Note that, "modeling" is always of the sense to interpret the complex real world in a simplified way with some unavoidable loss of accuracy. The steps 2) and 3) are detailed in the following.

**3.4.2.2 Construction of the orthonormal polynomial basis**

The direct construction of orthonormal polynomial basis according to the definition in (3.13) is difficult and even infeasible for most of arbitrary probability distributions. It can only be made for some basic well-known distributions such as uniform, Gaussian, Gamma, Beta, etc., or "quasi-arbitrary" distributions defined by a finite number of mutually dependent basic random variables, see Soize's work [96]. Therefore, for a general uncertain system having random input variables with an arbitrary joint distribution, an appropriate solution is to transform the input variables into basic variables prior to the building of polynomial basis.
*For univariate problems:* the Askey-scheme and its extension [81], [48] summarize the appropriate basis for several basic random distributions, for example, Hermite polynomials are appropriate basis for the Gaussian distribution, whereas Legendre polynomials are appropriate basis for the uniform distribution. In the situation of other types of distribution of X, if the Cumulative Density Function (CDF) $F_X(x)$ is known, then X can be transformed into a uniform variable by using the inverse CDF transform as in (3.15):

$$X = F_X^{-1}(U) \qquad (3.15)$$

where $U \sim U(0,1)$ is a uniform random variable on $[0, 1]$; or into a standard Gaussian distribution by using a conventional normal translation as in (3.16):

$$X = F_X^{-1} \circ \Phi(Y) \qquad (3.16)$$

where $Y \sim N(0,1)$ is a standard normal variable, and $\Phi$ is the CDF of $Y$. Thereby, an arbitrary distribution can be easily handled after being transformed to a more “friendly” basic distribution.

*For multivariate problems:* if the input variables are independent each other, then the basis is simply the tensor product of the univariate basis $\phi_{\alpha_i}^{(i)}(X_i)$ which are constructed beforehand based on the marginal probability density function of each variable $f_{X_i}(x_i)$:

$$\Phi_{\alpha}(X) = \prod_{i=1}^{M} \phi_{\alpha_i}^{(i)}(X_i) \qquad (3.17)$$

$$\langle \phi_{\alpha_j}^{(i)}(x_i), \phi_{\alpha_k}^{(i)}(x_k) \rangle = \int_{D_{X_i}} \phi_{\alpha_j}^{(i)}(x_i) \phi_{\alpha_k}^{(i)}(x_k) f_{X_i}(x_i) dx_i = \delta_{jk} \qquad (3.18)$$

In the context of dependent variables, some iso-probabilistic transformation techniques can be applied under appropriate conditions to transform the correlated variables from the initial physical space (X) to uncorrelated variables in the standardized space (U). For example, in the situations where the copula (a function which relates the joint distribution function $F_X(x_1, \dots, x_M)$ to the marginal distribution functions $\{F_{X_1}(x_1), \dots, F_{X_M}(x_M)\}$ [97]) is supposed to be Gaussian or more generally elliptical, the Nataf's transform can be applied [98], [99]; while in the situations where the conditional distributions are known, the Rosenblatt's [100] can be applied. Note however that in practice, for realistic problems such as those we are considering, the assessment of the empirical copula can be a challenging task, in particular if it must be performed experimentally.

### 3.4.2.3 Computation of the PCE coefficients

There exist several approaches for computing the PCE coefficients. Depending on whether the solving process relies on prerequisite knowledge on the system under study, these
approaches are classified into two categories – intrusive approaches and non-intrusive approaches. The intrusive approaches, typically the Galerkin’s scheme, require prior knowledge on the explicit equation for $M(\mathbf{x})$ to assess the PCE coefficients. Obviously, this is not practical because for the great majority of situations we cannot have such kind of explicit form for $M(\mathbf{x})$ due to its complexity. Besides, in Galerkin’s scheme all the unknown coefficients are obtained simultaneously by solving a coupled deterministic equation by modifying an existing deterministic solver [101]. In other words, a new program is required for the solution, and this is rather like developing an adjoint for a computer code. In contrast, the non-intrusive approaches consider the system under study as a black-box hence no prior knowledge is required, and the PCE coefficients are the result of the post-processing of model evaluation (over the ED) – a set of decoupled equations are solved by repeatedly calling an existing deterministic solver. It is consequently not necessary to modify underlying codes or to implement new codes. Hereafter, we will review some popular non-intrusive approaches, namely, the projection approaches which are based on the Monte-Carlo (MC) estimation or the Gaussian Quadrature (GQ) estimation, and the least square regression approaches which are based on the Ordinary Least Square (OLS) algorithm or the Least Angle Regression Sampling (LARS) algorithm.

## ♦ Projection

### ✧ Monte-Carlo (MC) estimation

Thanks to the orthonormality of the polynomial basis, the PCE coefficients $y_\alpha$ can be obtained by projecting $M(\mathbf{x})$ onto the basis $\Phi_\alpha(\mathbf{x})$:

$$ \langle M(\mathbf{x}), \Phi_{\alpha}(\mathbf{x}) \rangle = \left\langle \sum_{\beta \in \mathcal{A}^{M(\mathbf{x})}_{\text{PCE}}} y_{\beta} \Phi_{\beta}(\mathbf{x}), \Phi_{\alpha}(\mathbf{x}) \right\rangle = y_{\alpha} \quad (3.19) $$

Thereby, we only have to estimate the following integration:

$$ \langle M(\mathbf{x}), \Phi_{\alpha}(\mathbf{x}) \rangle \triangleq \int_{D_X} M(\mathbf{x}) \Phi_{\alpha}(\mathbf{x}) f_X(\mathbf{x}) d\mathbf{x} \quad (3.20) $$

This can be done via the MC method which considers the integration as the expectation of the random function $M(\mathbf{X})\Phi_\alpha(\mathbf{X}) : \mathbb{E}[M(\mathbf{X})\Phi_\alpha(\mathbf{X})] \triangleq \int_{D_X} M(\mathbf{x})\Phi_\alpha(\mathbf{x})f_X(\mathbf{x})d\mathbf{x}$. Numerically, the estimation of this expectation can be obtained by averaging $M(\mathbf{X})\Phi_\alpha(\mathbf{X})$ over a substantial number of realizations sampled according to the joint PDF of $\mathbf{X}$:

$$ y_{\alpha} = \mathbb{E}[M(\mathbf{X})\Phi_{\alpha}(\mathbf{X})] \approx \frac{1}{N} \sum_{s=1}^{N} M(\mathbf{x}^{(s)})\Phi_{\alpha}(\mathbf{x}^{(s)}) \quad (3.21) $$
Simplicity is the major advantage of the MC method. However, the great drawback of its slow convergence (O(1/√N)) limits its practical application.

## ♦ Gaussian Quadrature (GQ) estimation

A more efficient method for estimating the integration (3.20) is to resort to the GQ method (applicable because orthonormal polynomial families are involved), which approximates the integration by a weighted sum of integrand values at specified points, named Gaussian nodes, within the domain of integration [102]:

$$y_{\alpha} = \int_{D_x} \mathcal{M}(\mathbf{x}) \Phi_{\alpha}(\mathbf{x}) f_x(\mathbf{x}) d\mathbf{x} \approx \sum_{k_1=1}^{n_1} \dots \sum_{k_M=1}^{n_M} w_1^{k_1} \dots w_M^{k_M} \mathcal{M}(x_1^{k_1}, \dots, x_M^{k_M}) \Phi_{\alpha}(x_1^{k_1}, \dots, x_M^{k_M}) \quad (3.22)$$

where $x_i^{k_i}$ is the $k_i$-th Gaussian node for the $i$-th input variable $x_i$ and $w_i^{k_i}$ its corresponding weight. They are determined uniquely by the marginal probability density function $f_x(x_i)$.

It is proven theoretically that for a 1-dimensional integrand which is approximately polynomial of total degree $n$ or less, a GQ scheme with $(n+1)/2$ nodes can provide exact estimation result. Suppose that a 1-dimensional system $\mathcal{M}(x)$ is polynomial of degree $N_{PCE}$ and consider applying the PCE of degree $N_{PCE}$, then the integrand in (3.22) is polynomial of degree less or equal than $2N_{PCE}$. As a result, a GQ scheme of $(N_{PCE} + 1)$ nodes is required for exact estimation in (3.22).

In the situation of a $M$-dimensional system, the GQ scheme can be constructed as the tensor product of each 1-dimensional GQ scheme:

$$Q_M^{N_{PCE}} \triangleq Q_1^{n_1} \otimes Q_2^{n_2} \otimes \dots \otimes Q_M^{n_M} \quad (3.23)$$

with $Q_k^{n_k}$ ($k=1, \dots, M$) being the 1-dimensional GQ scheme for approximating the polynomial (of the variable $x_k$) of degree $n_k$, and $(n_1 + n_2 + \dots + n_M) = N_{PCE}$. Thereby the total number of required nodes is:

$$N_{GQ-nodes} = \prod_{i=1}^{M} (n_i + 1) \quad (3.24)$$

If we assume $n_i \equiv N_{PCE}$ constant for all input variables, then $N_{GQ-nodes} = (N_{PCE} + 1)^M$ which grows too fast (exponentially) with respect to the dimensionality $M$, causing unaffordable computational cost. This problem is known as the “curse of dimensionality”. As a result, the full tensor product grids are rarely applicable but only for low dimensions, e.g. $M \le 5$. Actually, the full tensor product grids are able to exactly approximate the integration for an integrand of degree less or equal to $M \cdot N_{PCE}$, which is far greater than $2N_{PCE}$ for $M \gg 2$, thence necessarily leading to high overhead computational cost.
In order to mitigate this “curse of dimensionality”, Smolyak first proposed sparse grids scheme [103] in which only a subset of the full tensor product grids are chosen:

$$Q_{M,N_{\text{PCE}}}^{S} = \sum_{N_{\text{PCE}}-M+1 \le n \le N_{\text{PCE}}} (-1)^{N_{\text{PCE}}-n} C_{M-1}^{N_{\text{PCE}}-n} Q_{M}^{n} \quad (3.25)$$

where $Q_{M,N_{\text{PCE}}}^{S}$ denotes the sparse grids scheme for constructing a M-dimensional PCE model with total degree of $N_{\text{PCE}}$; $Q_{M}^{n}$ denotes the full tensor product grids for constructing a M-dimensional PCE model with total degree of n; $C_{n}^{k} = n!/(k!(n-k)!)$ is the binomial coefficient. The total number of required nodes in a Smolyak’s sparse scheme is estimated by $\approx (2M)^{N_{\text{PCE}}}/N_{\text{PCE}}!$ [104], which has a polynomial (instead of exponential) dependence on the dimensionality M. The Smolyak’s sparse scheme has been used in applications e.g. [105]-[108] and have been demonstrated to be more computationally efficient than full tensor product scheme for high degrees of expansion.

Quite recently, Zhang proposed another type of sparse grids scheme called Stochastic Testing (ST) [101] where only $K = C_{M+N_{\text{PCE}}}^{N_{\text{PCE}}}$ nodes, i.e. as the same number of the unknown PCE coefficients, are selected from the full tensor product grids. The selection logic prioritizes those nodes with large weight values and try to make the following matrix as well-conditioned as possible:

$$\Phi(\mathcal{X}) \triangleq [\Phi_{\alpha_1}(\mathcal{X}), \Phi_{\alpha_2}(\mathcal{X}), \dots, \Phi_{\alpha_K}(\mathcal{X})]$$

where $\mathcal{X} = \{x^{(1)}, x^{(2)}, \dots, x^{(N_{\text{ED}})}\}^T$ is a set of points spanning the input variables space $D_{\mathcal{X}}$, conventionally referred as the Experimental Design (ED). It is easy to verify that this number of required nodes also has a polynomial dependence on M. The ST algorithm have been implanted in some stochastic collocation (SC) modeling such as in [58] where 3 random input parameters with relatively limited ranges of variability are considered. In such case a relatively low expansion degree ($N_{\text{PCE}} = 6$) is sufficient and the SC/ST was demonstrated to be a very efficient method. However, if additional random input parameters and/or larger variability ranges were to be considered, a higher expansion degree would be necessary. Under such situations, the SC approach would be less efficient than the least-squares based solutions that we will present in the next section.

To summarize, the GQ-based algorithms have the superiority over others in terms of the modeling exactness. The main drawbacks of GQ are: first, the “curse of dimensionality” limits their application to small to moderate dimensionalities, and also for relatively low polynomial expansion degrees (otherwise, a similar “curse of expansion degree” would occur). Second, the GQ scheme (i.e. nodes and weights) is so closely linked to the expansion degree $N_{\text{PCE}}$ that, if a higher $N_{\text{PCE}}$ is required, the GQ scheme
would be totally different, hence completely new pre-evaluations are required. Third, relating to the second one, the pre-evaluations should be made exactly at those pre-determined GQ nodes; however, it may happen in practice that, it is difficult (or even impossible) to build up the exact conditions corresponding to those nodes, in which situation the modeling would be completely blocked. Fourth, relating to the third one, in the situation where we already have some evaluation results at some points other than the GQ nodes, these results could be barely taken advantage of to save some computational cost. All these above-mentioned drawbacks reduce the efficiency of the GQ-based projection approaches in realistic applications.

## Least square regression

### Ordinary Least Square (OLS) regression

The equation (3.14) rewrites as the sum of a truncated series and a residual error:

$$Y = M(X) = \sum_{i=0}^{N_b-1} y_{\alpha_i} \Phi_{\alpha_i}(X) + \varepsilon_{\text{PCE}} \quad (3.26)$$

with $N_b = C_{M+N_{\text{PCE}}}^{N_{\text{PCE}}}$ being the number of polynomial basis for a PCE of degree $N_{\text{PCE}}$; and $\varepsilon_{\text{PCE}}$ is the residual error of the approximation.

The PCE coefficients can be investigated upon the knowledge of the system responses over an ED $\mathcal{X} = \{x^{(1)}, x^{(2)}, \dots, x^{(N_{\text{ED}})}\}^T$. Denote the corresponding system responses as $\mathcal{Y} = \{Y^{(1)}, Y^{(2)}, \dots, Y^{(N_{\text{ED}})}\}^T$. Thereby, we get a system of linear equations:

$$\mathcal{Y} = \Phi(\mathcal{X})\mathcal{y} + \varepsilon_{\text{PCE}} \quad (3.27)$$

where $\Phi(\mathcal{X})$ is the $N_{\text{ED}} \times N_b$ matrix of which the $i$-th column is the evaluation of the basis $\Phi_{\alpha_{i-1}}$ over the ED $\mathcal{X}$, i.e. $\Phi_{\alpha_{i-1}}(\mathcal{X})$; $\mathcal{y} = [y_{\alpha_0}, y_{\alpha_1}, \dots, y_{\alpha_{N_b-1}}]^T$ is the vector consisting of the unknown PCE coefficients to be determined; $\varepsilon_{\text{PCE}}$ is the vector consisting of the $N_{\text{ED}}$ residual errors. Then, the initial problem of solving the PCE coefficients can be transformed into an optimization problem stated as: find $\hat{\mathbf{y}} \in \mathbb{R}^{N_b}$ to minimize the mean square error $\mathbb{E}[\varepsilon_{\text{PCE}}^2]$:

$$\hat{\mathbf{y}} = \arg \min_{\mathbf{y} \in \mathbb{R}^{N_b}} \mathbb{E}[\varepsilon_{\text{PCE}}^2] = \arg \min_{\mathbf{y} \in \mathbb{R}^{N_b}} \mathbb{E}[(\mathcal{Y} - \Phi(\mathcal{X})\mathbf{y})^2] \quad (3.28)$$

The OLS algorithm solves the optimization problem by:

$$\hat{\mathbf{y}} = (\Phi^T \Phi)^{-1} \Phi^T \mathcal{Y} \quad (3.29)$$

where $(\Phi^T \Phi)^{-1} \Phi^T$ is the Moore-Penrose pseudoinverse of $\Phi(\mathcal{X})$.
To ensure the good-conditioning of the least square regression problem, a minimum number of pre-evaluations are required for “training” the model. As a rule of thumb, the number of realizations is often chosen to be 2~3 times larger than $N_b$. We can see that the concern of high computational cost occurs again: in the situation where the input variables are numerous (i.e. high dimensionality) and/or where a high expansion degree $N_{PCE}$ is indispensable for achieving satisfactory precision, the number of pre-evaluations becomes very large.

To circumvent this limitation, a possible way is to use a truncation basis scheme during the expansion. For example, Blatman and Sudret proposed “hyperbolic truncation scheme” (also named “q-norm scheme”) [110] where only a subset of the full basis scheme $\mathcal{A}^{M,N_{PCE}}$ are retained. The set of multi-indices of the retained basis is defined as:

$$ \mathcal{A}^{M,N_{PCE},q} = \{\alpha \in \mathbb{N}^M \mid \|\alpha\|_q \le N_{PCE}\} \quad (3.30) $$

with $\|\alpha\|_q = (\sum_{i=1}^M (\alpha_i)^q)^{1/q}$, $0 < q < 1$ being the q-quasi-norm. It can be seen that $q = 1$ corresponds to the full scheme $\mathcal{A}^{M,N_{PCE}}$ whereas a lower value of $q$ would exclude high order interaction terms from the basis set (i.e. penalize the high-rank indices), thereby, the number of unknown PCE coefficients are reduced and then the required number of pre-evaluations can be reduced accordingly.

Furthermore, in [111] Blatman proposed a similar solution named “low-rank scheme” where only those basis of which the multi-indices $\alpha$ has at most $r$ ($\le M$) non-zero elements are retained:

$$ \mathcal{A}^{M,N_{PCE},r} = \{\alpha \in \mathcal{A}^{M,N_{PCE}} \mid \|\alpha\|_0 \le r\} \quad (3.31) $$

with $\|\alpha\|_0 = \sum_{i=1}^M 1_{\{\alpha_i > 0\}}$ being the rank of the multi-indices $\alpha$. This scheme has the same objective as the hyperbolic one of excluding (more or less) high order interaction terms, i.e. to favor the main effects which are more probably significantly represented by low order interactions [111]. However, such selection techniques are based on the assumption that “the system response usually has less projection weight on those interaction terms”. This assumption cannot be always guaranteed for all realistic applications but is only a rule of thumb. In the situation that strong interaction intervenes, such sparse truncation schemes would be less efficient or even worse than a full scheme.

## ✧ Least Angle Regression Sampling (LARS)
With the previous discussions in mind, we are aware that on the one hand, sparse basis truncation schemes are essential for circumventing the prohibitive barrier of “curse of dimensionality”; on the other hand, crudely exclude those interaction terms which “seem to be” unimportant is not a credible strategy since their real significance i.e. their actual influence on the system response, can be determined only after a thorough investigation. Therefore, more “intelligent” strategies which permit to take into account the real influence of each basis element while constructing the truncation scheme are urgently needed. In 2004, Efron et al. proposed a less greedy selection strategy named Least Angle Regression Sampling (LARS) [112]. LARS selects iteratively those regressors (i.e. basis elements) that have the greatest impact factor on the system response. The impact factor is evaluated in terms of their correlation with the residue after subtracting the current model prediction from the initial response. Precisely, LARS algorithm consists of the following steps:

1. Initialization:

a) Choose a starting scheme $\mathcal{A}^{M,N_{PCE}}$ (e.g. full scheme, hyperbolic, low-rank);

b) Generate an initial ED $\mathcal{X} = \{\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \dots, \mathbf{x}^{(N_{ED})}\}^T$;

c) Evaluate the corresponding system responses $\mathcal{Y} = \{Y^{(1)}, Y^{(2)}, \dots, Y^{(N_{ED})}\}^T$, subtract the empirical mean response $\hat{\mu}_{\gamma}$ from $\mathcal{Y}$ and obtain zero-mean responses $\underline{\mathcal{Y}}$;

d) Set the initial LARS scheme as $\mathcal{A}_{0}^{\text{LARS}}=\emptyset$, the initial PCE model as $\hat{\mathcal{M}}_{0}(\mathbf{X})=\mathbf{0}$, and the initial residue as $\mathbf{R}_{0}=\underline{\mathcal{Y}}-\hat{\mathcal{M}}_{0}(\mathcal{X})=\underline{\mathcal{Y}}$.

2. Find the regressor in $\{\Phi_{\alpha} | \alpha \in \mathcal{A}\}$ which has the most correlation with the residue $\mathbf{R}_0$; denote this regressor as $\Phi_{\alpha_1}$ and the 1st LARS scheme as $\mathcal{A}_1^{\text{LARS}}=\{\alpha_1\}$.

3. For $1 \le k \le m = \min\{\text{card}(\mathcal{A}), N_{ED}\} - 1$, construct the $(k+1)$-th LARS scheme $\mathcal{A}_{k+1}^{\text{LARS}}$ as follows:

a) Construct the k-th PCE model in such way: $\hat{\mathcal{M}}_k(\mathbf{X}) = \hat{\mathcal{M}}_{k-1}(\mathbf{X}) + \gamma_k u_k(\mathbf{X})$ where $u_k(\mathbf{X})$ is the so-called “equiangular vector” of length $N_{ED}$ and of unit norm ($\|u_k(\mathbf{X})\|^2 = 1$), and $\gamma_k$ is the smallest step such that we can find in the rest candidate set $\{\Phi_{\alpha} | \alpha \in \mathcal{A} \setminus \mathcal{A}_k^{\text{LARS}}\}$ another regressor which has as much correlation with the k-th residue $\mathbf{R}_k = \underline{\mathcal{Y}} - \hat{\mathcal{M}}_k(\mathcal{X})$ as all selected regressors $\{\Phi_{\alpha} | \alpha \in \mathcal{A}_k^{\text{LARS}}\}$ do; denote the new selected regressor as $\Phi_{\alpha_{k+1}}$.
b) The $(k+1)$-th LARS scheme is set as $\mathcal{A}_{k+1}^{\text{LARS}} = \mathcal{A}_k^{\text{LARS}} \cup \{\alpha_{k+1}\}$.

4. Continue this way until $\mathcal{A}_m^{\text{LARS}}$. Meanwhile, we obtain a series of PCE models $\{\hat{\mathcal{M}}_k(\mathbf{X}) + \hat{\mu}_Y, k = 1, \dots, m\}$.

Figure 3.20 illustrates the principle of LARS in the case of $M = 2$ covariates.

<figure><img src="image_37.png" /><figcaption>Figure 3.20: The LARS algorithm in the case of $M = 2$ covariates. $\bar{y}_2$ is the projection of $y$ into $\mathcal{L}(\bar{x}_1, \bar{x}_2)$. Beginning at $\hat{M}_0 = 0$, the residual vector $\bar{y}_2 - \hat{M}_0$ has greater correlation with $\bar{x}_1$ than $\bar{x}_2$; the next LARS estimate is $\hat{M}_1 = \hat{M}_0 + \gamma_1 \bar{x}_1$, where $\gamma_1$ is chosen such that $\bar{y}_2 - \hat{M}_1$ bisects the angle between $\bar{x}_1$ and $\bar{x}_2$; then $\hat{M}_2 = \hat{M}_1 + \gamma_2 \bar{u}_2$, where $\bar{u}_2$ is the unit bisector; $\hat{M}_2 = \bar{y}_2$ in the case of $M = 2$, but not for higher dimensional case (i.e. $M > 2$).</figcaption></figure>

### ♦ Hybrid LARS (HLARS) and adaptive HLARS

Based on Efron's LARS logic, Blatman and Sudret proposed a Hybrid LARS (HLARS) algorithm [113] in which LARS serves only to select the sparse truncation scheme and the PCE models are reconstructed using OLS method, instead of directly adopting the models obtained alongside in LARS. The first 4 steps of the HLARS are the same as in the LARS algorithm, and the 3 additional steps are briefly recalled herein:

5. For $k=1, \dots, m$, extract the PCE model $\hat{\mathcal{M}}_k(\mathbf{X})$ based on the truncation scheme $\mathcal{A}_k^{\text{LARS}}$ by applying OLS over $\{\mathcal{X}, \mathcal{Y}\}$. Then, estimate the model accuracy in terms of the Leave-One-Out Error (LOOE) $e_{\text{LOO}}$ (refer to § 3.4.2.4 for definition).

6. Choose the truncation scheme $\mathcal{A}_{k_{\text{min}}}^{\text{LARS}}$ which has the smallest $e_{\text{LOO}}$ value.

7. Set the HLARS scheme as: $\mathcal{A}^{\text{HLARS}} = \mathcal{A}_{k_{\text{min}}}^{\text{LARS}} \cup \{\alpha_0\}$ where $\alpha_0$ is the index of the constant regressor $\Phi_{\alpha_0}=1$. Note that $\Phi_{\alpha_0}$ has never been selected in any truncation scheme $\mathcal{A}_k^{\text{LARS}}$ since a constant vector has zero correlation with any residue $\mathbf{R}_k$. Then, extract the final PCE model based on $\mathcal{A}^{\text{HLARS}}$ by applying OLS on $\{\mathcal{X}, \mathcal{Y}\}$.

It is worth mentioning that the choice of the departure scheme has great influence on the efficiency of the HLARS method. A small departure scheme (i.e. low $N_{\text{PCE}}$,
and/or small $q/r$ values) may be not sufficient for an accurate approximation; a large departure scheme (i.e. high $N_{PCE}$, and high $q$ or $r$ values) may increase the ensuing computational cost for searching the most important regressors. To mitigate this trouble, Blatman and Sudret proposed furthermore an adaptive HLARS method. The principle is to begin with a starting scheme of small size, extract the PCE model using the HLARS method; then repeat this process for gradually enriched starting schemes until the model accuracy cannot be further improved. Precisely, the whole process of an adaptive HLARS is:

1. Choose a series of nested starting schemes: $\{\mathcal{A}^{M,d_i}, i=1,...,i_{Max}\}$ (usually, this can be done by choosing a series of increasing expansion degrees $N_{PCE}$).

2. For the smallest starting scheme $\mathcal{A}^{M,d_1}$, construct the PCE model using the HLARS method. Denote its LOOE as $e_{LOO}^1$ (refer to § 3.4.2.4 for definition), and also denote the current smallest LOOE as $e_{LOO\_min} = e_{LOO}^1$.

3. Then, for $i=2,...,i_{Max}$:

a) Construct the PCE model with $\mathcal{A}^{M,d_i}$ as departure scheme using the HLARS method. Denote its LOOE as $e_{LOO}^i$.

b) If $e_{LOO}^i < e_{LOO\_min}$, set the current smallest LOOE as $e_{LOO\_min} = e_{LOO}^i$; otherwise, stop the iterative process.

4. Choose the PCE model corresponding to $e_{LOO\_min}$ as the final model.

Comparing to the projection methods and the OLS method, the LARS-based methods can provide much better sparsity in basis scheme, which makes them more computationally efficient for treating complex realistic applications with higher dimensionality and higher expansion degree. Particularly, the adaptive HLARS achieves good performance both in terms of computational cost and model accuracy. Hereafter, we will adopt this method to build up our methodology for extracting surrogate models for variable antennas.

### 3.4.2.4 Error indicator for assessing the model accuracy

The accuracy of a PCE model $\hat{M}(X)$ can be characterized by the expectation of the square residual error $e_{MSR} \triangleq \mathbb{E}[\epsilon_{PCE}^2(X)]$. In practice, an intuitive way for estimating $e_{MSR}$ is by means of the empirical error $e_{EMP}$:

$$e_{EMP} = \frac{1}{N_{ED}} \sum_{n=1}^{N_{ED}} [M(\mathbf{x}^{(n)}) - \hat{M}(\mathbf{x}^{(n)})]^2 \quad (3.32)$$
However, despite its simplicity, $e_{EMP}$ risks to encounter severe “overfitting” problem – $e_{EMP}$ can be systematically reduced by increasing the complexity (or the degree of freedom) of the PC approximation (i.e. the number of regressors) whereas $e_{MSR}$ may increase. Indeed, a model of too high complexity to exhaustively fit a particular set of observations may lose its ability for predicting future observations. Figure 3.21 illustrates this phenomenon on a simple quadratic population with noise pollution, where polynomial fittings of order $N = 2$ and 10 are applied separately on a sample of 10 realizations (red diamond points) driven from the population (black round points). We can see that the high-order fitting curve (blue line) passes exactly through all those sample points (hence $e_{EMP}$ almost equals to zero), but obviously it is not the appropriate approximation to the theoretical curve (black line) seeing that it predicts so badly in other zones (i.e. with a high $e_{MSR}$). In contrast, the low-order fitting curve (cyan line) which does not exactly pass through all the sample points (i.e with a higher $e_{EMP}$) but predicts averagely well over all the domain of definition (i.e. with a lower $e_{MSR}$) should be chosen as the appropriate approximation.

<figure><img src="image_38.png" /><figcaption>Figure 3.21: An example showing the overfitting problem when approximating an empirical sample with high order polynomials.</figcaption></figure>

A more robust indicator against overfitting is the so-called Leave-One-Out Error (LOOE) $e_{LOO}$ [114], which cross-validates the model’s prediction capability for new input values. Its definition is recalled herein:

$$e_{LOO} = \frac{N_{ED} - 1}{N_{ED}} \frac{\sum_{n=1}^{N_{ED}} [M(\mathbf{x}^{(n)}) - \hat{M}^{(n)}(\mathbf{x}^{(n)})]^2}{\sum_{n=1}^{N_{ED}} (Y^{(n)} - \hat{\mu}_Y)^2} \quad (3.33)$$

where $\mathbf{X} = \{\mathbf{x}^{(1)}, \dots, \mathbf{x}^{(N_{ED})}\}^T$ and $\mathbf{Y} = \{Y^{(1)}, Y^{(2)}, \dots, Y^{(N_{ED})}\}^T$ are respectively the ED and the corresponding system responses; $\hat{M}^{(n)}$ is the PCE model derived from the whole set $\{\mathbf{X}, \mathbf{Y}\}$
except the n-th realization {$\mathbf{x}^{(n)}, Y^{(n)}$} ; and $\hat{\mu}_Y = \sum_{n=1}^{N_{ED}} Y^{(n)} / N_{ED}$ is the empirical mean response. It is proven that in the context of linear regression, the LOOE can be calculated in a much more efficient way [115]:

$$e_{\text{LOO}} = \frac{1}{N_{\text{ED}}} \sum_{n=1}^{N_{\text{ED}}} \left[ \frac{\mathcal{M}(\mathbf{x}^{(n)}) - \hat{\mathcal{M}}(\mathbf{x}^{(n)})}{1 - h_n} \right]^2 \quad (3.34)$$

where $h_n$ is the n-th diagonal element of the matrix $\mathbf{\Phi}(\mathbf{\Phi}^T\mathbf{\Phi})^{-1}\mathbf{\Phi}^T$. Consequently, the construction of $N_{ED}$ PCE models $\hat{\mathcal{M}}^{(n)}$ can be avoided.

### 3.4.2.5 Post-processing with PCE model

Once the PCE model is extracted, it can serve as an efficient surrogate model for various types of statistical analysis and sensitivity analysis. Particularly, the first and second moments of the system response have analytical expression in terms of the PC coefficients as:

$$\hat{\mu}_{Y,N_{\text{PCE}}} = y_{\alpha_0} \quad (3.35)$$

$$\hat{\sigma}_{Y,N_{\text{PCE}}}^2 = \sum_{\alpha \in \{\mathcal{A}^{M,N_{\text{PCE}}} \setminus \alpha_0\}} y_{\alpha}^2 \quad (3.36)$$

The sensitivity of the system response to the input variables can be investigated by means of the distribution of the total variance of the response, in other words, which amount of the total variance is due to the uncertainty in which input variables. Among others, the Sobol's indices are commonly adopted indicators to assess the sensitivity. The total Sobol's indices of the input variable $X_i$ can be quickly calculated from the PCE coefficients [116]:

$$S_{T}^{X_i} = \frac{1}{\hat{\sigma}_{Y}^{2}} \sum_{\alpha \in \mathcal{I}_{i}^{\dagger}} y_{\alpha}^{2} \quad (3.37)$$

where $\mathcal{I}_i^\dagger = \{\alpha \in \mathcal{A}^{M,N_{\text{PCE}}} | \alpha_i \neq 0\}$ contains all the multi-indices with a non-zero i-th component.

### 3.4.2.6 Experimental design (ED)

#### ♦ Typical space-filling techniques

All the least-square-regression based algorithms resolve the unknown PCE coefficients by investigating the system responses over a specific set of points in the input variables space $\mathcal{D}_X$, referred as the Experimental Design (ED). Therefore, the strategy for building up the ED has great impact on the performance of the derived PCE model, in terms of both convergence rate and model accuracy. Given that our objective is to model the variation of the system response caused by uncertainties in input variables, it is crucial that the ED reflects well the
uncertainty characteristics of the input variables. In other words, the ED points should follow as well as possible the distribution of the input variables.

Now take the situation where the input variables are independent and are uniformly distributed in the space $D_x \triangleq [0, 1]^M$ for example, then the hypercube $[0, 1]^M$ should be sampled as uniformly as possible (remind that in the situation of other types of distribution, they can be mapped to uniform distribution using iso-probabilistic transforms as explained in § 3.4.2.2, hence there is no loss of generality within this example). Various types of space-filling techniques can be adopted in such context. Without trying to be exhaustive, we can mention the crude Monte Carlo sampling (MCS), the quasi MCS techniques e.g. Halton's [117], Faure's [118], and Sobol's sequences [119], or the Latin Hypercube sampling (LHS) [120], etc. Among all these sampling techniques, MCS is the simplest one for implementation, e.g. using a random number generator (available in many mathematical tools and programming languages e.g. Matlab, Excel, Python, C, etc.). However, the problem of “clustering” arises in case of small-sized samples. That's to say, some zones of $D_x$ are more densely covered than the other zones. Therefore, to guarantee a good uniformity in the entire $D_x$, a substantial amount of points are often needed, hence inducing high computational cost especially when the pre-evaluation is time-consuming (such as EM simulations). The quasi MCS techniques aim at placing points at deterministic positions in order to uniformly cover the $D_x$, hence mitigate the “clustering” problem for small sized samples. LHS is a widely used technique which retains both benefits of uniformity and “random ability” (hence the “uncertainty” characteristics of the inputs). Precisely, to select $N_{ED}$ points from the input variables space $D_x$ four steps are to be followed:

i. the definition domain of each input variable $X_j$ ($j = 1, 2, \dots, M$) is divided into $N_{ED}$ segments of equal probability, thereby, the entire space $D_x$ is divided into $(N_{ED})^M$ sub-hypercubes;

ii. $N_{ED}$ random permutations of the sequence $\{1, 2, \dots, N_{ED}\}$ are carried out, denote the $i$-th element of the $j$-th permutation reordered sequence as $\pi_{i,j}$;

iii. $N_{ED}$ sub-hypercubes are selected in the way that the $i$-th sub-hypercube ($i = 1, 2, \dots, N_{ED}$) is framed by the $\pi_{i,j}$-th segment of the interval of $X_j$ ($j = 1, 2, \dots, M$);

iv. In each selected sub-hypercube, one point is randomly drawn with respect to the uniform distribution. Thereby, an ED of $N_{ED}$ points is built up.

Mathematically, this reads as:

$$ \mathcal{X}_{\text{LHS}} = \{x_j^{(i)} = \frac{\pi_{ij} - 1 + u_{ij}}{N_{ED}}, i = 1, \dots, N_{ED}; j = 1, \dots, M\} \quad (3.38) $$
where $[\pi_{ij}]$ is a $N_{ED} \times M$ matrix where each column is a random permutation of the sequence $\{1, 2, ..., N_{ED}\}$, and where $u_{ij}$ is a realization of a uniform random variable $U \sim U(0,1)$.

In practice, we usually generate a large number of independent LHS designs for a desired size, then choose the one which best satisfies a favorable space-filling criterion, such as the “Maximin distance” [121] which aims at maximizing the smallest Euclidean distance between ED points. Thereby, the uniformity over the $D_x$ can be optimized.

### ♦ Enrichment of the ED

Abundant experience of researchers in computational engineering shows that instead of directly taking a large sized ED, it is more efficient to begin with an initial ED of small size, construct the surrogate model, assess the model performance, and then if necessary, enrich iteratively the ED until reaching a satisfactory precision. The enrichment of an existing ED can be achieved by adding new points with respect to some favorable space-filling criterion. The aforementioned “Maximin distance” is a popular criterion suitable for this context, and has been shown to be particularly efficient in the situations where the boundary zones have to be properly covered [122]. In practice, we usually generate a huge number (e.g. $10^5$) of random points, and choose the one which has the “maximin distance” for the enrichment. Besides, other criteria include e.g. the so-called “$D$-optimality” [123] and “$S$-optimality” [124]. The underlying principle is to maximize the “good-conditioning” of the information matrix $\Phi^T\Phi$ in terms of: i) maximized determinant of $\Phi^T\Phi$; ii) maximized column orthogonality of $\Phi^T\Phi$ (hence minimized correlation between the ED points).

Furthermore, the feedback from accuracy assessment of the currently constructed model can also provide useful guidelines on the selection of new points for the next iteration. For example, regions where the current model predicts poorly should be sampled more densely.

### 3.4.3 Methodology based on VSH expansion and PCE

We propose an adaptive statistical methodology to extract surrogate models for variable antennas. The basic idea is to apply the HLARS-based PCE onto the VSH coefficients of the ATFs. For sake of simplicity, we adopt the notation of single condensed index $p = 2\{n(n+1)+m-1\}+u$ as proposed in [70], and then (3.8) rewrites as:

$$ \mathcal{H}(f, \hat{\mathbf{r}}) = \sum_{p=1}^{P} H_p(f) \hat{\psi}_p(\hat{\mathbf{r}}) \quad (3.39) $$

with $P = 2N_{VSH}(N_{VSH} + 2)$.
Assume that there are *M* random variables, denoted as a random vector **X** = (X₁, X₂, ..., Xₘ)ᵀ with known joint PDF fₓ, the variable ATF as well as the complex-valued VSH coefficients Hₚ become random functions of **X**:

$$ \mathcal{H}(\mathbf{X}, f, \hat{\mathbf{r}}) = \sum_{p=1}^{p} H_{p}(\mathbf{X}, f) \hat{\psi}_{p}(\hat{\mathbf{r}}) \quad (3.40) $$

To exploit the quantitative dependence of the ATF on **X**, we apply the HLARS-based PCE on the real and imaginary part of the random functions Hₚ(**X**, f) separately:

$$ H_{p}^{\text{Re/Im}}(\mathbf{X}, f) = \sum_{\alpha \in \mathcal{A}_{p}^{\text{HLARS}}} y_{\alpha}^{p, \text{Re/Im}}(f) \Phi_{\alpha}(\mathbf{X}) \quad (3.41) $$

where $\mathcal{A}_{p}^{\text{HLARS}}$ is the sparse basis scheme built up with the HLARS method (cf. § 3.4.2.3).

Thereby, we get a surrogate model $\hat{\mathcal{H}}(\mathbf{X}, f, \hat{\mathbf{r}})$ relating the ATF to the random input variables by a set of scalar coefficients $\{y_{\alpha}^{p, \text{Re/Im}}(f)\}$, as:

$$ \hat{\mathcal{H}}(\mathbf{X}, f, \hat{\mathbf{r}}) = \sum_{p=1}^{p} \sum_{\alpha \in \mathcal{A}_{p}^{\text{HLARS}}} (y_{\alpha}^{p, \text{Re}} + j y_{\alpha}^{p, \text{Im}})(f) \Phi_{\alpha}(\mathbf{X}) \hat{\psi}_{p}(\hat{\mathbf{r}}) \quad (3.42) $$

At this moment, we still only consider the antenna's mono-frequency behavior, hence the parameter "f" can be omitted in equation (3.42).

### 3.4.3.1 Iterative procedure

The entire procedure for extracting the surrogate model $\hat{\mathcal{H}}(\mathbf{X}, \hat{\mathbf{r}})$ comprises steps of:

1. Initialization:

a. Initialize the ED $\mathcal{X}$ of a desired size with the LHS design;

b. Determine the maximal degree $N_{\text{VSH}}$ for VSH expansion;

c. Choose a LOOE threshold ($e_{\text{LOO\_th}}$) for assessing the accuracy of PCE models.

2. Evaluate the ATFs over the ED. Precisely, for each input value $\mathbf{x}^{(n)}, n = 1, \dots, N_{\text{ED}}$, configure the corresponding antenna, carry out the EM simulations or measurements, and compute the ATF $\mathcal{H}(\hat{\mathbf{r}}, \mathbf{x}^{(n)})$.

3. Apply the VSH expansion to each ATF and obtain the VSH coefficients $H_{p}(\mathcal{X})$, $p = 1, \dots, P$.

4. Apply the HLARS-based PCE method separately to the real and imaginary part of each VSH coefficient $H_{p}(\mathcal{X})$ and finally get PCE models as in (3.41).

5. Assess the LOOE $e_{\text{LOO}}$ of the PCE models. If $e_{\text{LOO}} \le e_{\text{LOO\_th}}$ is satisfied for all models, or if $N_{\text{ED}}$ exceeds an acceptable value (i.e. too costly to continue) then stop the
modeling procedure; otherwise, enrich the ED (i.e. $X=[X;X_{add}]}$ with respect to the
“Maximin distance criterion”, and return to Step 2.

Eventually, a surrogate model relating quantitatively and explicitly the random variables
$X$ and the ATF via the whole set of polynomial coefficients $\{y_{\alpha}^{p,Re/Im} | \alpha \in A_{p}^{HLARS}, p = 1, \dots, P\}$
is established, allowing the ATF generation for any value of $X$ (within the variation ranges).

### 3.4.3.2 Validation on a canonical dipole antenna

To demonstrate the feasibility of our proposed methodology, we first apply it to a lossless
half-wave wire dipole, which is a simple and very well-known canonical antenna, hence an
ideal “test antenna” to verify systematically the relevance, accuracy and robustness of the
methodology. The same dipole design as was used in § 3.3.2.1 is re-used.

#### a. Random input parameters

The dipole is assumed to be deformable using 3 independent parameters, namely
stretching (with stretch ratio $\xi$), bending (with curvature $\kappa$ or bending radius $R_b = 1/\kappa$), and
torsion (with torsion rate $\alpha$), see Figure 3.22 (b)-(d). We suppose also that the input
parameters ($\xi$, $\kappa$, $\alpha$) are uniformly distributed over respectively [1, 1.05], [0, 0.1] mm$^{-1}$, and
[0, 0.1] rad·mm$^{-1}$. Figure 3.23 shows the joint effects of deformations on the reflection
coefficient $|S_{11}|$ and on the radiation patterns (at $f = 2.3$ GHz). Compared to the original dipole,
variations of more than 15 dB in $|S_{11}|$ and 13.5 dB in $G_r$, i.e. 92.6 % in terms of RRMSE are
observed. These large variations justify to resort to an accurate quantitative approach. At the
first stage, we consider the relatively simple case where $f$ is fixed at 2.3 GHz, thus the index
“f” will be omitted in all physical quantities.

<figure><img src="image_39.png" /><figcaption>Figure 3.22: Illustration of a dipole undergoing different deformations: (a) original dipole; (b) stretched dipole; (c) curved dipole; (d) curved and twisted dipole.</figcaption></figure>
<figure><img src="image_40.png" /><figcaption>Figure 3.23: Joint effect of stretching, bending, and torsion (ξ, κ, α) on the dipole: (a) reflection coefficient $|S_{11}|$; (b) radiation patterns (in E/H plane) at $f = 2.3$ GHz.</figcaption></figure>

b. Modeling description and results

The modeling approach described in § 3.4.3.1 is applied to the dipole. Firstly, the ED is initialized with a sample of 100 realizations using the LHS technique. The desired LOOE threshold $e_{LOO_th}$ is set to 4 %. Note that the choice of $e_{LOO_th}$ is subjective, depending on the required accuracy and the convergence rate. A typical value of < 5 % is popularly adopted in many publications e.g. [94]. Secondly, the evaluations of the ATFs are carried out in CST® MWS (with the FEM solver) to obtain radiation fields which are sampled every 5° in both elevation and azimuth. The full truncation degree of the VSH expansion is set to $N_{VSH} = 3$ (i.e. $P = 2N_{VSH} (N_{VSH} + 2) = 30$), for which the collected power is proved to be very close to the total power of the EMF in reality. The “Maximin distance” technique is used for iteratively enriching the initial ED. At last, we use the UQLab, a toolbox<fnref n="1" /> for the uncertainty quantification developed by the team of B. Sudret [125], to implement the HLARS-based PCE algorithm.

<figure><img src="image_41.png" /><figcaption>Figure 3.24: $e_{LOO}$ of the PCE models derived from the iteratively enriched EDs.</figcaption></figure>

<footnote n="1">¹ Available for free for academics. B. Sudret must be warmly thanked for offering this opportunity.</footnote>
Figure 3.24 shows the evolution of the $e_{LOO}$ of the PCE models. It can be seen that when $N_{ED}$ increases the error decreases for all PCE models and the $e_{LOO\_th}$ is satisfied for all models for $N_{ED} = 260$, for which we get a surrogate model with about 2000 polynomial coefficients.

In order to verify its accuracy and robustness, the constructed surrogate model has been used to generate the most deformed dipole, i.e. ($\xi, \kappa, \alpha$) = (1.05, 0.1, 0.1), and a comparison of radiation patterns between the initial field, the reconstructed field with full ($P = 30$) VSH modes, and the generated field with the surrogate model is presented in Figure 3.25 (a). It is noted that the maximal difference in $G_r$ is only 0.7 dB and the RRMSE is less than 8 %.

Furthermore, the surrogate model has been used to generate dipoles for 500 random values of ($\xi, \kappa, \alpha$), which are compared to their simulated (with CST® MWS) counterparts, as shown in Figure 3.25 (b). This constitutes actually a test of Monte Carlo (MC). We see that the RRMSE of the generated antennas remains below 2 % for 88 % of the cases and the extreme error is still less than 10 %, which confirms the accuracy and the robustness of the surrogate model. It is worth mentioning that there is always a trade-off between the model accuracy and the cost to achieve that precision – the higher the accuracy of the model, the higher the cost of the modeling procedure. One can choose the most appropriate trade-off according to each practical needs. Another conclusion we can draw is that the surrogate model also verifies high parsimony. Compared to the initial antennas representation by a huge amount of sampled EMF data (~ $10^6$), the surrogate model requires only a limited number of parameters (i.e. ~ $10^3$), reaching a data compression rate of about $10^3$. It is actually much higher as input data (random input parameters or angle variables) can be interpolated as desired.

<figure><img src="image_42.png" /><figcaption>Figure 3.25: (a) Radiation patterns (in E/H plane) of the initial, reconstructed and generated fields for the most deformed case at $f = 2.3$ GHz; (b) RRMSE of 500 generated dipoles for random values of ($\xi, \kappa, \alpha$).</figcaption></figure>

### c. Conclusion
The proposed methodology combining the VSH expansion and the HLARS-based PCE methods allows to develop compact and precise surrogate models of the ATF with a minimized cost. The feasibility, efficiency and high accuracy of this methodology have been preliminarily tested by means of modeling a canonical deformable dipole. The derived model is computationally costless for generating deformable dipole configurations, usable as surrogate models in various types of analyses where the *complete* field knowledge, including polarization and phase, are required.

However, although this methodology has been validated conceptually, there still remain some important issues that should be carefully solved before generalizing the methodology to much more complicated realistic applications, where robustness and efficiency are highly demanded. This aspect will be discussed in the next section.

## 3.5 Improvements of the statistical method

In this section, we further exploit the potential of our proposed method by: i) taking into account the frequency behavior; ii) enhancing the parsimony of the surrogate model; iii) more efficiently controlling the iterative modeling procedure with appropriate performance metrics. In particular, a more realistic application with larger number of input variables and more complicated random deformations will also be addressed.

### 3.5.1 Frequency behavior

Up to now the frequency-dependency of the ATF has not yet been modeled seeing that a single frequency (e.g. the nominal frequency) was considered in all previous examples. However, including the frequency behavior is fundamental to make the antenna model *complete*. To give an analogy, this is equivalent to switch from the modeling of a static to a dynamic problem, e.g. in mechanics (see e.g. [126]). For narrow-band (NB) antennas, taking into account the frequency allows to discriminate between mismatch/detuning effects (cf. Chapter 5) and directional effects due to the input variability in the FF behavior. A single frequency model is hence incomplete, and actually insufficient. For multi-band, wideband – and a fortiori for UWB – antennas, their frequency response is mandatory for obvious reasons. More generally, at least for moderate band (and beyond) or multi-band systems, if the model is used jointly with a stochastic frequency-dependent directional channel model in order to assess the radio links performance – e.g. for different PHY layer schemes – the antenna frequency response is mandatory (see e.g. [127]).

To address the frequency behavior, the simplest way is to consider the frequency as it is, a deterministic variable, i.e. to sample the frequency axis and built surrogate models separately for each frequency sample. Then, an interpolation technique can be used for covering the
entire frequency band. The cost is obviously a very significant reduction of the parsimony, and moreover, the interpolation may introduce additional errors.

Alternatively, the frequency can be considered somehow as a “random” input parameter for the problems for which the frequency behavior of the output quantity (represented by the vector-valued model $Y = [y_{\alpha}^{q}(f)]$ in our case) is sufficiently regular with respect to the input parameters variability. Indeed, the final derived surrogate model is actually deterministic, because it allows to recover the exact ATF for any given values in the input space.

The “regularity” here refers to such situation, for which the frequency response (in the statistical sense) does not vary abruptly (locally) with respect to the random input variables. Note indeed that polynomials are rather “smooth” functions (compared to rational functions), thus, in some situations, the PCE is not the most appropriate technique (polynomials are not suitable bases) if the local variations of the output quantities with respect to some input parameters are too considerable. It should be pointed out that the assumption of “smoothness” does not necessarily lead to insignificant global sensitivities – which measure how the variance (i.e. range of variation) of the output quantities depends on the input variables (see Sobol’s indices in § 3.4.2.5). In practice, this “regularity” corresponds to antennas which are not too much resonant. If the considered antenna is resonant, the question whether the frequency can be treated as another random input variable or not must be specifically addressed. In other words, some preliminary tests must be driven. If the frequency behavior appears too difficult to model this way, alternative approaches must be envisaged. For resonant systems, at least an alternative method proposed by Sudret’s team exists, based on preliminary affine transformations of the frequency axis, applied prior to PCEs [128]. However, this technique is significantly more sophisticated, and requires the realization of two independent EDs, which means a significant computational cost.

After including the frequency parameter “f” in the random input vector $X$, the complete surrogate model (3.42) is simplified into:

$$ \hat{\mathcal{H}}(X, \hat{\mathbf{r}}) = \sum_{p=1}^{P} \sum_{\alpha \in \mathcal{A}^{\text{HLARS}}}^{|\alpha| \le N_{\text{PCE}}} (y_{\alpha}^{p, \text{Re}} + j y_{\alpha}^{p, \text{Im}}) \Phi_{\alpha}(X) \hat{\psi}_{p}(\hat{\mathbf{r}}) \quad (3.43) $$

where $X$ includes now the frequency “f” as an additional input variable, and $y_{\alpha}^{q, \text{Re}/\text{Im}}$ are purely constant real coefficients to be determined.

### 3.5.2 Enhanced parsimony

The benefit of a parsimonious surrogate model is twofold: first, the computational cost for constructing the model can be reduced, since the number of unknown parameters thereby the number of required pre-evaluations (which are often time-consuming) becomes smaller.
Second, the computational cost for generating ATFs with the surrogate model can be reduced, since there will be less algorithmic operations in a more compact model. Such type of saving could become impressive when a substantial amount of realizations are demanded, such as for a Monte Carlo analysis, in particular when multiple statistics must be combined, e.g. in joint antenna/channel statistical modeling. We discussed in § 3.4.2 that the HLARS algorithm helps to obtain a sparse PCE expression. In this section, we try to exploit the possibility of sparsity in the VSH expansion.

### 3.5.2.1 Sparse VSH expansion

We express the complex-valued VSH coefficient $H_p$ in (3.39) explicitly by its real and imaginary parts in order to highlight their individual contribution.

$$ \hat{\mathcal{H}}(\hat{\mathbf{r}}) = j \sum_{q=1}^{Q} (-j)^{\text{mod}(q, 2)} H_q \hat{\psi}_{\lfloor q/2 \rfloor}(\hat{\mathbf{r}}) \quad (3.44) $$

where $Q = 2P$; $H_q$ are scalar coefficients representing either the real or imaginary part of $H_p$ according to the following rule: $H_q = \Re\{H_p\}$ (or $\Im\{H_p\}$) for $t = 1$ (or 2) with $q = 2(p-1)+t$; “mod(a, n)” means “a modulo n”; “$\lfloor \cdot \rfloor$” is the ceiling round function; and $j = \sqrt{-1}$.

We note that the radiated power $P_{\text{rad}}$ (which is normalized to an incident power of 1 Watt) of the antenna can be calculated easily from (3.44), thanks to the orthonormality of the basis:

$$ P_{\text{rad}}(\mathbf{X}) = \frac{1}{4\pi} \iint_{\Omega} \|\mathcal{H}(\mathbf{X}, \hat{\mathbf{r}})\|^2 d\Omega = \frac{1}{4\pi} \sum_{q=1}^{Q} H_q(\mathbf{X})^2 \quad (3.45) $$

By defining the modal power weight as:

$$ W_q(\mathbf{X}) = \frac{1}{4\pi} \frac{H_q(\mathbf{X})^2}{P_{\text{rad}}(\mathbf{X})} \quad (3.46) $$

we can construct a sparse VSH expansion: all $H_q(\mathbf{X})$ are sorted in descending order according to their power weight, and then only the first dominant ones with their collected weights amounting to a predefined threshold, denoted as $E_{c\_th}$ (e.g. 99.9 %), will be retained. This allows either to reduce the number of VSH coefficients for a given truncation degree $N_{\text{VSH}}$ (affecting only marginally the accuracy), or to increase $N_{\text{VSH}}$ for a given maximal number of coefficients (hence enabling to reduce the truncation error).

Denoting $I_D$ the set of index for all selected dominant modes, and $N_D$ the cardinal of $I_D$, equation (3.44) is simplified to:

$$ \hat{\mathcal{H}}(\hat{\mathbf{r}}) = j \sum_{q \in I_D}^{q \le Q} (-j)^{\text{mod}(q, 2)} H_q \hat{\psi}_{\lfloor q/2 \rfloor}(\hat{\mathbf{r}}) \quad (3.47) $$
### 3.5.2.2 Correlation analysis on the selected dominant VSH modes

The influence of input randomness may induce detuning, impedance mismatch, power loss, and variation of the radiation pattern shape. The first two effects change the antenna total efficiency, such that all dominant VSH modes are influenced in the same way, and the last effect may lead to power transfer among different modes. In other words, we think it is possible that the random effects may increase the modal energy of one VSH mode, and at the same time, decrease that of another mode. As a result, one can anticipate that evolutions of these VSH modes are not totally independent each other, and this can be efficiently exploited towards further parsimony. This can be for example observed in the parametric approach presented in §3.3.2.3: see e.g. the behavior of the modal coefficients $\Re(H_{1,1}^1)$ and $\Re(H_{3,0}^2)$ in Figure 3.12.

Only the strong linear correlations between modes have been exploited here in order to preserve quantitative explicitness and precision of the surrogate model. From the sparse VSH expansions of the $N_{ED}$ ATFs we calculate the $N_D \times N_D$ correlation matrix $\rho = [\rho_{m,n}]$ of which the element $\rho_{m,n}$ is the correlation coefficient of the m-th and the n-th column vector of the matrix $H$ (3.48). Then, the VSH modes having a correlation coefficient greater than a pre-defined threshold (e.g. 0.995) are considered as linearly dependent and arranged in disjoint subsets $G_l$ ($l = 1, ..., N_G$), each containing at least one mode. Each $G_l$ forms an equivalent class represented by its “generator” (the lower order VSH coefficient), denoted $H_{q_{G_l}}$. As a result, instead of $N_D$, only $N_G$ ($N_G < N_D$) VSH coefficients have to be modeled with the PCE method; the other VSH coefficients of each class are represented by their linear regression parameters and can be simply computed from them once the generator is modeled. This correlation procedure reduces significantly the model complexity, i.e. the number of PCE coefficients and the computational cost, resulting in favorable compactness of the surrogate model.

$$ H = \begin{bmatrix} H_{q_1}(\mathbf{x}^{(1)}) & H_{q_2}(\mathbf{x}^{(1)}) & \cdots & H_{q_{N_D}}(\mathbf{x}^{(1)}) \\ H_{q_1}(\mathbf{x}^{(2)}) & H_{q_2}(\mathbf{x}^{(2)}) & \cdots & H_{q_{N_D}}(\mathbf{x}^{(2)}) \\ \vdots & \vdots & \ddots & \vdots \\ H_{q_1}(\mathbf{x}^{(N_{ED})}) & H_{q_2}(\mathbf{x}^{(N_{ED})}) & \cdots & H_{q_{N_D}}(\mathbf{x}^{(N_{ED})}) \end{bmatrix} \quad (3.48) $$

After integrating all compaction techniques (i.e. frequency behavior modeling, sparse VSH expansion, and correlation analysis) the complete surrogate model (3.42) finally reduces to:
$$
\left\{
\begin{aligned}
\hat{\mathcal{H}}(\mathbf{X}, \hat{\mathbf{r}}) &= j \sum_{q \in I_D}^{q \le Q} (-j)^{\text{mod}(q, 2)} H_q(\mathbf{X}) \hat{\psi}_{[q/2]}(\hat{\mathbf{r}}) \\
H_q(\mathbf{X}) &= \begin{cases}
\sum_{\alpha \in \mathcal{A}_q^{\text{HLARS}}} |\alpha|^{\alpha} \Phi_{\alpha}(\mathbf{X}), & \text{if } q \in I_G = \{q_{G_1}, \dots, q_{G_{N_G}}\} \\
\mathcal{L}_q[H_{q_{G_l}}(\mathbf{X})], & \text{if } H_q \in \mathcal{G}_l \text{ and } q \neq q_{G_l}
\end{cases}
\end{aligned}
\right.
\quad (3.49)
$$

where X is the random vector containing the input random parameters and the frequency “f”
as an additional parameter; I<sub>D</sub> is the subset containing the index of the selected dominant VSH
modes; I<sub>G</sub> is the subset containing the index of the N<sub>G</sub> “generator” modes; A<sup>HLARS</sup><sub>q</sub> &sub; N<sup>M</sup> is
the sparse HLARS scheme adopted for the H<sub>q</sub>(X) expansion; L<sub>q</sub> is the linear operator relating
the mode H<sub>q</sub> to the generator H<sub>qG<sub>l</sub></sub> of the class G<sub>l</sub> it belongs to, which can be easily obtained
by linear regression analysis.

**3.5.3 Appropriate metrics**

New metrics for controlling the algorithm convergence and evaluating the accuracy of
such vector-valued surrogate models are defined: the Weighted Total Leave-One-Out error,
denoted e<sub>WTLOO</sub>, and the Supremum error, denoted e<sub>Sup</sub>. These indicators play a major role for
assessing the convergence of the iterative approach and ensuring the reliability and the
robustness of the deduced surrogate model.

1) The e<sub>Sup</sub>, defined in (3.50), assesses the absolute difference between the reconstructed and the initial ATFs in terms of the realized gain G<sub>r</sub> over an appropriate solid angle.

$$
e_{\text{Sup}}(\mathbf{x}) = \max_{(\theta, \varphi) \in \Omega_m(\mathbf{x})} \left\{ \left| G_r^{\text{init}}(\mathbf{x}, \theta, \varphi) - G_r^{\text{rcs}}(\mathbf{x}, \theta, \varphi) \right| \right\} \quad (3.50)
$$

where $G_r^{\text{init}}$ and $G_r^{\text{rcs}}$ denote the $G_r$ (in “dB”) of the initial and the reconstructed fields respectively; $\Omega_m(\mathbf{x}) \triangleq \{(\theta, \varphi) | G_r^{\text{init}}(\mathbf{x}, \theta, \varphi) \ge \max_{(\theta, \varphi) \in [0, \pi] \times [0, 2\pi]} [G_r^{\text{init}}(\mathbf{x}, \theta, \varphi)] - \Delta G_r\}$ is the so-called main radiation zones, with $\Delta G_r > 0$ an adjustable margin.

2) The e<sub>WTLOO</sub>, an analogue of the LOOE when taking into consideration the modal weight, is defined for our multi-input/multi-output problem as:

$$
e_{\text{WTLOO}} = \sum_{l=1}^{N_G} e_{\text{LOO}}^{l} W_{l} \qquad (3.51)
$$

where $e_{\text{LOO}}^l$ is the $e_{\text{LOO}}$ of the PCE model of the “generator” $H_{q_{G_l}}$ of the $l$-th group $\mathcal{G}_l$,
and $W_l$ is the total power weight of $\mathcal{G}_l$. This indicator takes into account the power
proportion of each scalar PCE model. This “penalty” applied to modes with a lower
power contribution (but with a significant $e_{LOO}$) is aiming to avoid any artificial overestimation of the error.

### 3.5.4 Iterative modeling procedure

The improved algorithm for constructing antenna surrogate models is summarized below:

1. Initialization:

a. Build up the initial ED $\mathcal{X}_0$ of a desired size with the LHS design;

b. Determine the maximal degree $N_{VSH}$ for VSH expansion;

c. Assign a WTLOOE threshold ($e_{WTLOO\_th}$) for assessing the accuracy of derived PCE models;

d. Set an initial power threshold $E_{c\_th}$ for sparse VSH expansion.

2. Construct the surrogate model $\hat{\mathcal{H}}^{(0)}$ by applying the VSH/HLARS-based PCE method:

a. Evaluate the ATFs over the ED: $\mathcal{H}(\mathbf{x}^{(n)}, \hat{\mathbf{r}}), n = 1, \dots, N_{ED}$;

b. Apply the sparse VSH expansion to each ATF under the power criterion $E_{c\_th}$ according to (3.47) and obtain the index set of dominant modes $I_d^{(n)}$;

c. Define the aggregated index set as $I_D = \bigcup_{n=1}^{N_{ED}} I_d^{(n)}$;

d. Re-decompose all ATFs into modes $\{\hat{\psi}_{[q/2]}, q \in I_D\}$ and obtain the VSH coefficients $\{H_q(\mathcal{X}), q \in I_D\}$. Note that the purpose of Step 2.b – 2.d is to guarantee that all VSH expansions are based on the same set of modes, whereas the power criterion $E_{c\_th}$ is also satisfied;

e. Reconstruct the ATFs from the selected dominant modes $\{(H_q(\mathcal{X}), \hat{\psi}_{[q/2]}), q \in I_D\}$ and assess the error of reconstruction: if $e_{Sup}$ is satisfied then pass to Step 2.f; otherwise increase $E_{c\_th}$ and go back to Step 2.b;

f. Group all the $H_q$ into equivalent classes $\mathcal{G}_i$ using correlation analyses;

g. Apply the HLARS-based PCE on the “generator” modes $H_{q\mathcal{G}_i}$, and finally get the surrogate model $\hat{\mathcal{H}}^{(0)}$ as in (3.49).

3. Improve iteratively the model accuracy. For $i=1: I_{max}$:

a. Enrich the ED using the “Maximin” technique: $\mathcal{X}_i = [\mathcal{X}_{i-1}; \mathcal{X}_{add}]$;

b. Repeat Step 2 to construct the new surrogate $\hat{\mathcal{H}}^{(i)}$;

c. Predict the ATFs over $\mathcal{X}_i$ using the latest two successive surrogates, i.e. $\hat{\mathcal{H}}^{(i-1)}(\mathcal{X}_i)$ and $\hat{\mathcal{H}}^{(i)}(\mathcal{X}_i)$; and investigate the accuracy and convergence:
✓ is the $e_{WTLOO}$ of the surrogate model $\hat{\mathcal{H}}^{(i)}$ sufficiently small, i.e. $e_{WTLOO} \le e_{WTLOO\_th}$?

✓ are the $e_{Sup}$ of the predicted ATFs $\hat{\mathcal{H}}^{(i)}(\mathcal{X}_i)$ sufficiently small?

✓ is the $e_{WTLOO}$ still decreasing (rapidly) when passing from $\hat{\mathcal{H}}^{(i-1)}$ to $\hat{\mathcal{H}}^{(i)}$?

✓ does the distribution of the $e_{Sup}$ converge passing from $\hat{\mathcal{H}}^{(i-1)}(\mathcal{X}_i)$ to $\hat{\mathcal{H}}^{(i)}(\mathcal{X}_i)$?

If the $e_{WTLOO}$ and $e_{Sup}$ criteria are satisfied or the $e_{Sup}$ converges, i.e. no remarkable improvement is achieved by further enrichment, we stop the algorithm and the actual model $\hat{\mathcal{H}}^{(i)}$ is chosen; if not, continue until reaching the maximum loop number $I_{max}$.

The final surrogate model, represented by a set of scalar coefficients $\{y_{\alpha}^q | \alpha \in \mathcal{A}_q^{HLARS}, q \in I_G\}$, allows to quantitatively predict the ATFs taking into account the random variabilities.

### 3.5.5 Application to a realistic deformable textile patch antenna

We test the performance of the improved methodology on a rather complex and realistic antenna other than the dipole antenna. Consider now a microstrip-fed rectangular textile patch antenna, as shown in Figure 3.26 (a). Its substrate is a thin layer of weakly compressible jeans tissue, and the patch and ground plane are made with a thin copper tape (thickness ≈ 25 µm). This patch could be integrated into clothing, for example, on or into the sleeve of a jacket or coat, for transmitting to an access point vital signs data measured by sensors (such as body temperature, blood pressure, pulse rate, etc), in the context of off-body communications. The patch being deformable is subject to various sources of random variability such as crumpling, bending, etc. as shown in Figure 3.26 (b) which could seriously impact its radiation performance. Our proposed methodology is now applied to extract a surrogate model of this antenna example in order to quantitatively account for those statistical effects.

<figure><img src="image_43.png" /><figcaption>Figure 3.26: Wearable textile patch antenna: (a) un-deformed plane patch; (b) patch undergoing the joint effect of crumpling, bending and thickness variation.</figcaption></figure>
### 3.5.5.1 Design of the textile patch antenna

The antenna dimensions are optimized for a good impedance matching over the Industrial Scientific Medical (ISM) frequency band of 2.4 – 2.5 GHz. The obtained design parameters are, $L_p = 44.96$ mm, $W_p = 40$ mm, $L_s = W_s = 70$ mm, $W_0 = 3$ mm, $g = 2$ mm, $y_{in} = 3$ mm, $h = 0.85$ mm, $\epsilon_r = 1.776$, $\tan\delta = 0.05$ (Figure 3.26 (a)). In order to illustrate the modeling principle with a problem of already high but tractable complexity, we choose to restrict the antenna random variability to classical deformations encountered in practice, namely bending, crumpling, and thickness change, and to material property variations. Precisely,

- The bending around the z'Oz axis is quantified by its curvature $\kappa$ (or its bending radius $R_b = 1/\kappa$).

- The crumpling along the z'Oz axis is modeled in a very realistic way (instead of purely periodic variations used e.g. in [129]) with surfaces of revolution created by rotating a bandlimited Weierstrass function $W(z)$ (3.52) [130] around the z'Oz axis. The amplitude $C$ and the spatial pseudo-frequency $s$ are used to adjust the crumpling. The other parameters are fixed to $N_2 = 3$, $b = 2e/3 \sim 1.8$, and the dimension $D$ to 1 corresponding to rather smooth curves.

$$W(z) = C \sum_{n=1}^{N_2=3} b^{(D-2)n} \cos\left\{2\pi s b^n \left(\frac{L_s}{2} + z\right) + \frac{8}{5}\pi\right\}, z \in \left[-\frac{L_s}{2}, \frac{L_s}{2}\right] \quad (3.52)$$

- The thickness change is quantified by the variation of substrate height $h$.

- The tissue property is characterized by the variation of its relative permittivity $\epsilon_r$, the variation of its loss tangent $\tan\delta$ is not considered here, since the effect is only of second-order compared to that of the permittivity [51].

- Additionally, the frequency “f” is also considered as a “random variable” so as to account for the frequency behavior.

### 3.5.5.2 Joint distribution of the random input parameters

We assume that these 6 random parameters are independent each other and are uniformly distributed over their proper ranges listed in Table III.

<table><caption>Table III: Random input parameters of the textile patch.</caption><thead><tr><th>Random parameter</th><th>Unit</th><th>Nominal value</th><th>Support</th></tr></thead><tbody><tr><td>κ</td><td>mm<sup>-1</sup></td><td>0</td><td>[0, 1/30]</td></tr><tr><td>C</td><td>mm</td><td>0</td><td>[0, 5]</td></tr><tr><td>s</td><td>-</td><td>1</td><td>[1, 1/0.6]</td></tr><tr><td>h</td><td>mm</td><td>0.85</td><td>[0.8, 0.9]</td></tr><tr><td>ε<sub>r</sub></td><td>-</td><td>1.776</td><td>[1.6, 1.9]</td></tr><tr><td>f</td><td>GHz</td><td>2.45</td><td>[2.4, 2.5]</td></tr></tbody></table>
This assumption is a simplification from the complex reality. In general, global geometric deformations, in particular complex ones such as the combination of bending and crumpling, induce intricate local mechanical stress over the substrate (compression – or even tension – often anisotropic, possible shear stress, etc.), causing various types of local and often anisotropic deformations such as thinning, stretching, and possibly thickening, inducing some variations of the substrate permittivity. As pointed out in Boeykens’ work [49], [131] and Locher’s work [132], bending tends to reduce the substrate thickness when the patch is not stretchable, causing an increase of the permittivity by densification. Therefore, the permittivity is correlated to the curvature, and for a given curvature, the thicker the substrate, the stronger the “compression effect”, and the stronger the correlation between $ε_r$ and $h$. Indeed, the combination of bending and crumpling is significantly more complex than the bending alone, so that the induced influence on the permittivity is certainly more complicated than the possible “compression effect”. As a result, in general but in particular for fabrics of high thickness and high compressibility, the substrate permittivity is certainly more or less correlated with the global geometrical parameters. In the general case, correlations between the input parameters should of course be taken into account. This could be done either using classical decorrelation techniques based on appropriate isoprobabilistic transforms such as the one used e.g. in [51], before performing the PCE expansion, or using gPCE, as presented in [133]. However, one of the difficulties is of course to assess these possible correlations (i.e. the covariance matrix) for complex deformations such as the combination of bending and crumpling. To our knowledge, to date, no data are available in the literature, neither even a possible methodology, for this case (notably in the context of our crumpling modeling with surfaces based on Weierstrass curves, which is new). Nevertheless, this does not call into question the very principle of the proposed modeling (once the decorrelation is performed, if needed, or if these correlations can be neglected as a first approximation).

Indeed, we assume deliberately independency between the input parameters in order to simplify the modeling complexity. That is a reason why we have chosen, for the substrate material, a type of thin and weakly compressible jeans tissue. Various advantages of using these types of textile which have relatively “stable” mechanical and electrical properties under geometrical deformations, including “large” ones are e.g. reported in Osman’s work [24]. In this context, although the assumption of independency is an approximation, it can be considered as a reasonable simplification. The consistency of our assumption on the joint distribution (independency and uniform distribution) of the input parameters in our considered context is broadly justified by the following qualitative and quantitative arguments.

# 1. Justification of the assumption of independency of the input parameters
First, irrespective of any other potential physical effects, it is obvious that global geometrical deformations induce a variation of the current density distribution, affecting both the input impedance (hence the reflection coefficient $S_{11}$, the center frequency $f_c$, etc.) and the radiation characteristics. This effect on the $f_c$ will be called here “geometric effect”. Second, the eventual effects induced by local stress (compression, etc.) on the substrate (variability of its thickness and permittivity) will be called the “substrate effect”. Our point of view is that the simplifying assumption neglecting the correlation between $\epsilon_r$ and ($h, R_b$) due to bending is a reasonable approximation if the two following conditions are met:

C1: The “geometrical effect” is dominant compared to the “substrate effect”;

C2: The “substrate effect” is sufficiently small compared to the considered variability range of $\epsilon_r$.

In order to verify quantitatively (or semi-quantitatively) whether these two conditions are verified in our case, we carried out a set of experiments and simulations to bring out the impact of bending around the z'Oz axis on the shift of the center frequency $f_c = \arg \min_f |S_{11}(f)|$.

<figure><img src="image_44.png" /><figcaption>Figure 3.27: Experimental setup (a) and simulation design (b) for bending analysis.</figcaption></figure>

## 1.1. Experimental measurements

A flat patch is bent around a hollow carton tube of 31.75 mm radius (close to the considered minimal value $R_{b,min} = 30$ mm) and 1.9 mm thickness (Figure 3.27 (a)). The resonance frequencies before and after bending were measured with a Vector Network Analyzer (VNA) ROHDE&SCHWARZ ZVA 40. Prior to the measurement, we had verified that the carton tube (placed directly under the ground plane) has no influence on the antenna behavior. This is due to the isolation effect of the large ground plane of the antenna, the low permittivity of the carton and the thinness of the tube. Three prototypes were fabricated and each one was measured 3 times following complete “flat-bent-flat” cycles in order to ensure the reliability of the measurements. The result found is that the maximal bending causes on average a relative $f_c$ shift of $\delta f_{c,meas} = -0.54 \% \pm 0.06 \%$. Of course, as this shift value comes from measurements, it accounts for the combination of all possible effects.
## 1.2. Simulations using CST MWS (with adaptive meshing for good precision)

* The flat patch with nominal design parameters ($W_{p0}$, $h_0$, $\varepsilon_{r0}$) is simulated. The center frequency is denoted $f_{c0}$.

* Only the pure “geometric effect” of bending is simulated, for which the parameters are assigned as: $W_p = W_{p0}$, $h = h_0$, $\varepsilon_r = \varepsilon_{r0}$, and the bending radius is $R_b = 31.75$ mm. The center frequency is denoted $f_{c,g}$. The obtained frequency shift $\Delta f_{c,g} = f_{c,g} - f_{c0}$ is of $\delta f_{c,g} = \Delta f_{c,g}/f_{c0} = -0.37\%$ relative to $f_{c0}$.

* Both the “geometric effect” and the entire “substrate effect” of bending are included in simulation. To simulate this case, we need to assign reasonable values for the parameters $W_p$, $h$, and $\varepsilon_r$. For this, we followed Boeykens’ work [131] in which was proposed a phenomenological model relating the “compressed permittivity” $\varepsilon_{r,comp}$ to the “initial” $\varepsilon_{r0}$, the “initial” height $h_0$ and the bending radius $R_b$, for a textile patch antennas under bending:

$$ \varepsilon_{r,comp} = \varepsilon_{r,0}(1 + \eta \frac{h_0(d - 0.5)}{R_b}) $$

where $d$ and $\eta$ are empirical parameters; $d \in [0.5, 1]$ indicates the stretchability of the patch, and the factor $\eta$ is dependent on the substrate material and can be derived from experimental measurements.

As our prototype is made of copper foil & jeans (> 99 % cotton) and is bent symmetrically with respect to the feeding point, we first considered the “most possible correlated” situation, for which $d = 1$, and $\eta = 0.654$ (which is the largest value among the three cotton-based prototypes presented in [131]), and $R_b = 31.75$ mm. Therefore, the “real” permittivity value is given by
$$ \varepsilon_r = \varepsilon_0(1 + 0.654 \frac{0.85(1 - 0.5)}{31.75}) = 1.792 \quad (\text{corresponding to a relative variation of} \\ \Delta\varepsilon_r/\varepsilon_{r0} = 0.88 \%) $$
$W_p$ is set to $W_{p0}$ since our patch is considered perfectly non-stretchable. However, we do not have exact information about the value of $h$, except that we know it decreases more or less due to the local compression effect. Hence, we performed simulations with a set of values of $h$, decreasing from $h_0$ down to 50 % of $h_0$, in order to compute the center frequency denoted $f_{c,g+\varepsilon}$. The blue line in Figure 3.28 shows the frequency shift of $f_{c,g+\varepsilon}$ with respect to $f_{c0}$. We see that the shift is typically constant at -0.81 % up to 33 % of compression. In addition, we carried out measurements of the substrate thickness, under the patch, of two antenna prototypes (appropriately cut for that purpose) bent around another (smaller height) cylinder of about 30.7 mm radius, with a microscope and a Neltec/Micro-control micrometric precision linear translation platform (of less than 10 µm resolution): on average, the measured thicknesses are about 0.71 mm,
i.e. a relative compression of ~ 17 %, that is about half the “threshold” of about
33 % found in simulation. Thereby, the obtained $f_c$ shift value of –0.81 %
represents well the case of “geometric effect” + entire “substrate effect” for the
considered curvature (and for the considered value of $\eta$).

<figure><img src="image_45.png" /><figcaption>Figure 3.28: Center frequency shift under bending ($R_b$ = 31.75 mm) for increasing substrate compression $\Delta h/h_0$ (by simulation).</figcaption></figure>

* As the simulated frequency shift is clearly beyond the measured one ($\delta f_{c,meas} = -0.54 \%$), new simulations were performed in order to seek for a more appropriate value of the empirical parameter $\eta$. The target ($\delta f_{c,g+\epsilon} = \delta f_{c,meas}$) is reached for $\eta = 0.2145$ (corresponding to $\epsilon_r = 1.781$, i.e. $\Delta\epsilon_r/\epsilon_r = 0.28 \%$). This value is significantly smaller than the previous one, when assuming a stronger correlation. As a result, in reality the correlation is indeed rather moderate. Note also that this value of $\eta$ is very close to the smallest value obtained in [131] which gives some confirmation of the reliability of our combined measurements and simulations approach. New simulations were performed for $\eta = 0.2145$ and for the same set of decreasing values of $h$. The $f_c$ shift with regard to $h$ is shown by the red line in Figure 3.28. It can be seen that $f_c$ shift is constant at –0.54 % up to 30 % (significantly larger than 17 %) of compression.

### 1.3. Interpretation and conclusion

* The condition C1 can be typically considered as verified, since:

* ✓ The comparison between simulations and experiments show that the “geometric effect” (–0.37 %) is the dominant “explicative contribution” to the measured shift (–0.54 %), whereas the remainder (–0.17 %) is due to rather complicate physical mechanisms including the “substrate effect”.
✓ These results show that the resulting “substrate effect” obtained assuming a rather significant correlation between $ε_r$ and $h$ (with $η = 0.654$), i.e. –0.44 % (–0.81–(–0.37)) %) is 2.6 times larger than expected (–0.17 %).

* The condition C2 is verified, since:

✓ Even if we took full consideration of the “most appropriate” correlation between $ε_r$ and the global bending deformation, the variation of $ε_r$ due to this deformation would be only of $Δε_r/ε_{r0}$ ($η = 0.2145$) = +0.28 % (corresponding to the residual variation of –0.17 % of $f_c$), and this for the “extreme” bending case. Comparing to the variability range of 17.1 % assigned to $ε_r$, this amount is very small (about 1.6 % of this range) and could be neglected to first order in a simplified statistical model.

* Conclusion:

✓ The observed correlation is not as strong as “theoretically” predicted (following Boeykens’s model), as the “substrate effect” explicative contribution appears to correspond to an equivalent value of $η$ smaller than expected (about 0.2145 instead of about 0.654). A possible interpretation is that the thin layer of copper tape glue between the patch and the substrate (which is probably stretchable) reduces to some extent the stress due to the copper tape during bending. This would correspond to a lower value of parameter $d$: if the empirical value $η \sim 0.654$ is a correct order of magnitude for the considered jean textile, this would correspond to a value of $d \sim 0.66$ (=$0.5 + 0.2145 \times 0.5/0.654$).

We also performed a simulation corresponding to a perfectly stretchable patch (for which $W$ increases from 40 mm to 40.53 mm for $R_b$ = 31.75 mm, i.e. by 1.3 %), all other parameters remaining constant ($d$ = 0.5 in this case). The obtained simulated frequency shift is –0.41 % (in between $\delta f_{c,g}$ = –0.37 % and $\delta f_{c,meas}$ = –0.54 %).

✓ The result that the influence of $h$ on $ε_r$ under bending is typically marginal is due to the fact that the type of jean fabrics considered in the model (and used in practice for prototyping) is: (i) really thin, (ii) moderately compressible, and (iii) the variation range of $h$ (although significant in relative scale; about 11.8 %) is rather small (0.1 mm).

✓ For the same reasons, there is no particular argument to think that the crumpling could have a noticeable influence on $ε_r$, in the considered context. Indeed, crumpling is mainly due to a global constraint exerted “laterally”, i.e. along the antenna plane. The resulting local stresses can thus induce as well
local thickening than local thinning. Although local curvatures larger than the largest bending curvature considered above can be encountered, local curvatures far lower are necessarily present as well. It is consequently difficult to anticipate – and even more difficult to quantitatively assess – the crumpling influence on average, over $\epsilon_r$. However, all the preceding arguments, in particular the substrate thinness, suggest that the overall effect should be weak.

* As a conclusion, we consider that the assumption of independency between the input parameters, notably between $\epsilon_r$ and the geometric parameters (h on the one hand, and the deformation parameters $R_h$, C and $s$, on the other hand) is a reasonable approximation, simplifying the modeling process without significant loss of accuracy.

## 2. Justification of the choice of the uniform distribution

We have chosen to address a set (or category) of thin jeans-based patch antennas, with some common substrate features (typically “dense” woven jeans) but with variable nominal permittivities, rather than simply considering the variability of $\epsilon_r$ due to fabrication tolerances. In such context, it is more appropriate to assume a uniform distribution for $\epsilon_r$ (rather than a normal one for example) since there is no particular reason that one value of $\epsilon_r$ be privileged from another. The above mentioned “common features” are notably their mechanical properties (weakly “compressible” or stretchable, compared to other fabrics), the type of fiber used (mainly cotton-based, in particular without elastane), and the type of fabrication process (mainly woven, rather than e.g. embroidered, knitted, etc.). The chosen range is based on a survey of the open literature [129], [134], [135]: typically, $\epsilon_r$ takes values between 1.4 and 2 for jeans or denim tissues, and commonly used values are around 1.7 – 1.8. After considering the trade-off between a large enough representative “population” and a moderate enough variation for limiting the model complexity (maximal degree of the PCE and required ED size), we choose to limit $\epsilon_r$ in the range [1.6, 1.9].

Regarding the other parameters:

* The frequency $f$ is a deterministic parameter which would be evenly sampled if it were not considered as an input “random” parameter. The uniform distribution choice is hence obvious.

* For the reasons already discussed, the nominal thickness of the chosen substrate ($h$ = 0.85 mm), as well as its absolute variability range ([0.8, 0.9] mm), are small. Hence, although its relative range of variability is noticeable (~ 11.8 %), it was pointed out above that its overall influence is limited (if not marginal), so that its
distribution appears not really critical. It is why we chose a uniform distribution
for sake of simplicity. For example, we think that a possible alternative choice of a
truncated Gaussian (in particular with a significant σ) would only marginally
affect the results.

* Regarding the crumpling deformation (controlled by parameters *C* and *s*) an amplitude of 5 mm is chosen for the “Weierstrass-based” surfaces. This corresponds to a maximal “vertical” variation of 2 × 5 = 10 mm of crumpling corresponding to a ratio of 1/7 with respect to the overall antenna side length. Parameter *s* is set between 1 and 1/0.6. These ranges seem to represent well the real crumples. Considering that, (i) to our knowledge there are no available data in the open literature with regards to the crumpling statistics (in particular in the context of our crumpling model based on Weierstrass curves), (ii) the experimental assessment of such statistics is far from obvious and would require a significant dedicated work, and (iii) the relation between the “overall crumpling shape” (modeled by the Weierstrass curves) and the parameter *s* is not simple (actually nonlinear), we chose a uniform distribution for these two parameters for sake of simplicity as well.

* The bending is actually a particularly “application-dependent” deformation. Even in the context of wearable devices, the bending statistics (distribution and variability range) depends in practice on “usages”. First, these statistics depend on the considered *location(s)* of the antenna on the body (arm, wrist, thigh, ankle, chest, etc.). Second, the *way* the device/antenna is placed on the body – e.g. directly on the skin or in garments – plays as well a role in the choice of the statistics (in particular the variation ranges). For a specific “scenario” (e.g. an antenna embedded in a garment at the arm level), one could resort to some public anthropometric statistical databases. However, even for such specific scenario we observed that the bending deformation does not necessarily follow exactly the anthropometric parameters. In addition, these statistics depend on the considered population. For example, if we refer to the arm circumference database 2015-2016 provided by the US National Center for Health Statistics (from which the “medium” arm radius can be deduced; note that this is already an approximation), and if the entire population is considered, it appears that the radius statistics is not normal, but bimodal (close to a Gaussian mixture), as shown in Figure 3.29.
<figure><img src="image_46.png" /><figcaption>Figure 3.29: Statistic of the medium human arm radius, derived from the database 2015-2016 provided by the US National Center for Health Statistics.</figcaption></figure>

Moreover, since we consider only the “symmetrical” bending (whose effect is less than that of the bending along the patch length), its overall impact is moderate, at least over the reflection coefficient. All these elements suggest that the choice of the bending distribution must have a moderate influence on the output statistics. We decided also for sake of simplicity to assume a uniform distribution for the curvature $κ$. Note that this assumption leads to the fact that the bending radius $R_b$ follows random distribution of PDF as $1/R_b^2$, which favors the smallest radii, hence, e.g. a part of adults wrist statistics is somehow represented, and extend the radius range to infinity with a lower probability, hence e.g. suited for a location on the chest, or for some loose clothes whose deformations don’t necessarily follow precisely the morphology.

### 3. Conclusion regarding the assumption about the joint distribution of the input parameters

Our simplifying assumption of a uniform and independent joint distribution has been broadly justified as reasonable in first approximation for the considered thin and weakly compressible Jean fabrics. The advantage of assuming the same distribution for all the parameters is of course to simplify the process – Legendre polynomials are appropriate for all parameters for the PCE; otherwise, hybrid of different categories of polynomials (e.g. Legendre or Hermite) would be necessary which would complicate the solution
process without bringing too much added value for illustrating the principle of our methodology.

Nevertheless, it is clear that our simplifying assumption is an approximation that cannot be generalized to other types of fabrics which can be far more compressible or stretchable. In that case, a dedicated work should be done to quantify the actual joint distribution of the input parameters (and their potential correlations) in advance. Then, it is generally possible to resort to decorrelation techniques based on appropriate isoprobabilistic transformations prior to building the PCE expansion, in order to go back to independent parameters following the same type of distribution (uniform, normal, for which the PCE bases are known – respectively Legendre or Hermit polynomials according to the Wiener-Askey scheme [48]) before applying our methodology. For example, if one wishes to address some specific scenario, one can drive dedicated analyses of the input statistics (distributions, interdependencies) if the operational conditions are clearly defined. In the context of wearables e.g., the antenna positioning – on the arm or wrist, etc., directly placed on the skin or into garments, etc. – would be a key information allowing to assess the bending or crumpling statistics.

### 3.5.5.3 Pre-evaluation of the ATFs

We use the full-wave simulator CST® MWS which is controlled by a Matlab routine for the automatic evaluation of ATFs over the ED. First, a parametrical patch antenna is designed and optimized at nominal parameters values; second, for each ED sample the patch is reconfigured accordingly and the transient solver with adaptive meshing is run to compute the FF. Once we get the evaluations a process of data quality check procedure is applied before further exploitation.

The reliability of the pre-evaluations depends on the quality of each operation during the automatic routine, notably: i) the precision of the geometric model of the antenna, i.e. the ability to transform the input parameters values to the corresponding geometrical dimension; ii) the accuracy of the solver; iii) eventual unexpected and abnormal software behavior (e.g. w.r.t the importation, exportation, etc.). Therefore, some preliminary tests (such as, parametric study in case of only one variable, see e.g. [69]) should be carried out in order to: i) find a robust automatic shaping process (analytical curve, lofting, extrusion, layer discretization, conforming...) for generating accurate antenna models (note that in CST, crumplings of “Weierstrass” form & bending curves are not so easy to realize on a patch antenna composed of three thin layers, not to mention that we need to repeat the process a large number of times for different input parameters values); ii) adjust the control parameters of the adaptive meshing technique in order to obtain a good balance between the
stability/accuracy and the computational cost; iii) overcome possible software unexpected and non-physical behaviors.

### 3.5.5.4 Prototyping and control measurements

The comparison between the simulated and measured results (i.e. $S_{11}$ and $G_r$) for three representative patch configurations was achieved (Figure 3.31). The 3 prototypes were all made of jeans tissue ($h = 0.85$ mm, $\varepsilon_r = 1.776$, characterized using the Resonant-Patch method proposed in [136]) and 0.025 mm thick copper tape. Patch #1 is the un-deformed patch, considered as the reference; Patch #2 corresponds to a “severely bent & moderately crumpled” configuration, with $C = 5$ mm, $s = 1$, $\kappa = 1/30$ mm⁻¹; Patch #3 corresponds to a “severely bent & crumpled” configuration, with $C = 5$ mm, $s = 1/0.6$, $\kappa = 1/30$ mm⁻¹. The crumpling and bending were realized with two complementary extruded polystyrene matrices.

<figure><img src="image_47.png" /><figcaption>Figure 3.30: Experimental setup for measuring the ATFs.</figcaption></figure>

Measurements of the ATFs (along two principal plane cuts) were carried out in a Siepel anechoic chamber (see the setup in Figure 3.30) equipped with an ETS-Lindgren two-axis Positioning System (controlled by a PC through a dedicated driver), and a Double-Ridged Waveguide Horn Lindgren-3117 (BW of 1 – 18 GHz) as reference antenna (positioned on a fix mast). The Antenna Under Test (AUT) and the reference antenna were placed 3.5 m away, and aligned using a multi-beam Laser system. The deformed prototypes #2 and #3 were measured without the foam matrices as the ground plane and patch copper tape used rigidifies the antennas sufficiently to avoid any deformation during rotations of the positioner. The transmission coefficient $S_{21}$ was measured with the R&S ZVA40 VNA. The ATF is computed using the deconvolution procedure described in [57] based on the “complex Friis equation” [59]. The reflection coefficients $S_{11}$ were measured with the same VNA.
<figure><img src="image_48.png" /><figcaption>Figure 3.31: Three representative patches for which: the prototypes, the simulated/measured reflection coefficient $|S_{11}|$, and the simulated/measured $G_r$ at 2.45 GHz in principal cut-planes for the co-polarization and cross-polarization components are presented in columns.</figcaption></figure>

Figure 3.31 depicts for each patch: the picture of the prototype; the simulated and measured reflection coefficient ($|S_{11}|$); the simulated and measured realized gain patterns $G_r$ at 2.45 GHz for the co-polarization (CP) in two principle cut-planes (xOy and xOz planes) and for the cross-polarization (XP) in one principle cut-plane (yOz plane). We see that for Patch #1 the simulations are fully consistent with the measurements for both the $|S_{11}|$ and the $G_r$. For Patch #2 and #3, some deviations of the center frequency are observed (resp. ~ 37 and 34 MHz) but globally the measured and simulated $|S_{11}|$ remain consistent. Regarding $G_r$, a good agreement within the measurement error margins (typically ±0.5 dB) is observed for the CP, and also for the XP in the main radiation directions; the discrepancies observed in the XP patterns around the zeros (±90°) are due to the perturbation of the turntable mast. In addition, since the XP component is significantly lower than the CP component, this error can be tolerated. Briefly, the full-wave simulator is reliable for ATF evaluations.

### 3.5.5.5 Preliminary study: sensitivity analysis

We first investigated roughly the sensitivity of the ATF to the random parameters. The parameters of least influence can be discarded so as to reduce the model complexity (and modeling process) without weakening its representativeness. Based on the total Sobol's