indices (TSI) in (3.37), we define the weighted TSI, a more appropriate indicator to quantify the influence of each input parameter on our multi-output system:

$$S_{WT}^{(m)} = \sum_{l=1}^{N_{G_l}} S_{T}^{(m,l)} W_l \qquad (3.53)$$

where $S_{WT}^{(m)}$ is the weighted TSI of the m-th input parameter for the whole antenna system; $S_{T}^{(m,l)}$ is the TSI of the m-th input parameter for the “generator” $H_{q_l}$ of the l-th class $G_l$; and the weight $W_l$ is the total power contribution of the class $G_l$.

As shown by (3.37) for a PCE-modeled system the TSI of its input parameters can be obtained directly from the polynomial coefficients of the model. For this purpose, we constructed a preliminary model for the six-input system using our proposed statistical methodology. Then, the $S_{WT}$ of all the 6 random parameters were calculated, as shown in Figure 3.32.

<figure><img src="image_49.png" /><figcaption>Figure 3.32: $S_{WT}$ of the 6 random input parameters.</figcaption></figure>

It can be seen that the curvature $\kappa$ and the thickness $h$ have the lowest impact on the ATF. Therefore, one reasonable choice is to ignore these two parameters and to concentrate on the remaining dominant factors for a fast analysis. However, bending is by essence a common random phenomenon for wearable devices and should have some effect on the beamwidth in azimuth, thence is useful to be included in the surrogate model. In contrast, the thickness $h$ is not a so timely varying factor. Having this in mind, we decided to discard only the parameter $h$ and to keep the 5 remaining input parameters in our model.

### 3.5.5.6 Modeling procedure and results

The algorithm in § 3.5.4 was adopted to extract the surrogate model for the actual five-input system. First, the ED was initialized with 200 points using the LHS design. Second, the ATFs were evaluated over the ED and then decomposed into full VSH modes with a maximal degree of $N_{VSH} = 5$, from which the most dominant modes were selected. The error of
reconstruction $e_{\text{Sup}}$ (taking $\Delta G_r = 10$ dB as the main radiation zones margin) for different power thresholds $E_{c\_th}$ is shown in Figure 3.33. The blue (+) curve corresponds to the “full decomposition” ($N_{\text{VSH}} = 5$), used as reference; the black (◊), magenta (□) and green (x) curves correspond to $E_{c\_th}$ values of 99 %, 99.5 % and 99.9 % respectively. We see that the $e_{\text{Sup}}$ decreases progressively as $E_{c\_th}$ is growing, but at the cost of an increase of the number of dominant modes (from 27 to 44), which degrades the model compactness. We finally chose the value of 99.9 % for $E_{c\_th}$, since it offers a good compromise between accuracy and compactness ($e_{\text{Sup}} < 0.2$ dB for all dominant modes).

<figure><img src="image_50.png" /><figcaption>Figure 3.33: ATF reconstruction errors for different power threshold $E_{c\_th}$.</figcaption></figure>

The coefficients of the selected dominant VSH modes were then expanded onto polynomials where a maximal degree $N_{\text{PCE}} = 10$ was adopted. Finally a first surrogate model $\hat{\mathcal{H}}^{(0)}$ was constructed. The model accuracy was then progressively improved following Step 3, in which the “Maximin” technique is used to enrich iteratively the ED (by 100 new points each time). It is worth mentioning that the “Maximin” technique enriches preferentially the regions near boundaries. This property is suitable in our considered context since the boundaries correspond to more severe deformations thence must be sampled densely.

The convergence of the algorithm is analyzed as follows. As shown in Figure 3.34, by enriching the ED from 200 to 694 effective points, the prediction error $e_{\text{WTLOO}}$ is reduced from 10.9 % to 4.9 %. Furthermore, we take a deeper look at the distribution of the estimation errors by classifying them into three categories: A – “very satisfactory” ($e_{\text{Sup}} < 1$ dB), B – “satisfactory” ($1 < e_{\text{Sup}} < 1.5$ dB), and C – “poor” ($1.5 < e_{\text{Sup}} < 2.5$ dB), as shown in Figure 3.35. For the *i*-th ED, the left and right color bars are respectively the $e_{\text{Sup}}$ distribution for the predicted ATFs using the (*i*−1)-th and the *i*-th surrogate model $\hat{\mathcal{H}}^{(i-1)}(\mathcal{X}_i), \hat{\mathcal{H}}^{(i)}(\mathcal{X}_i)$. We can see that at the beginning, the error distribution of $\hat{\mathcal{H}}^{(0)}(\mathcal{X}_1)$ is quite different from that of $\hat{\mathcal{H}}^{(1)}(\mathcal{X}_1)$, which implies that the 1st surrogate model $\hat{\mathcal{H}}^{(0)}$ is not reliable enough as predictor
for new sets of random input values. This difference decreases gradually as the ED is enriched, and finally stabilizes when passing from $\hat{\mathcal{H}}^{(4)}$ to $\hat{\mathcal{H}}^{(5)}$. For $\hat{\mathcal{H}}^{(5)}(\mathcal{X}_5)$, the $e_{\text{Sup}}$ is below 1.5 dB for 98.41 % of the cases, and the average and maximal value of $e_{\text{Sup}}$ are respectively 0.52 dB and 1.96 dB, which can be considered relatively satisfactory as it meets almost the precision level of usual FF measurements in anechoic chamber. As a result, we first stopped the algorithm and chose $\hat{\mathcal{H}}^{(5)}$ as a “final” surrogate model offering a reasonable compromise between cost and accuracy. It comprises 3217 scalar coefficients, resulting from a 1st data compression rate of 65.71 % by sparse VSH expansion (i.e. 48 dominant modes among 140 full modes), followed by an “ultra-compression” of 43.75 % by correlation analysis (i.e. 48 dominant modes sorted into 27 equivalent classes).

<figure><img src="image_51.png" /><figcaption>Figure 3.34: Prediction error $e_{\text{WTLOO}}$ of surrogate models iteratively constructed by enriching the ED.</figcaption></figure>

<figure><img src="image_52.png" /><figcaption>Figure 3.35: Comparison between the errors of two successive surrogate models for different EDs.</figcaption></figure>

### 3.5.5.7 Validation of the surrogate model

Note first that the criterions chosen in § 3.5.4 regarding the algorithm convergence and the trade-off between the model accuracy and complexity ($E_{\text{c-th}}$, $N_{\text{PCE}}$, and the stop criterion of the enrichment of the ED) presents some arbitrariness, but they must be put into perspective with the other sources of uncertainty and/or errors in practical usages. In particular, in a real
radio communication system, a stochastic antenna model with an excessive precision would be actually useless in comparison with the added noise, channel variability, interferences, etc.

In any case, in order to verify the prediction performance of the extracted models, a MC analysis with a new statistical sample of 2000 realizations (i.e. 2000 new EM simulations) was realized. The ATFs computed with the models from the MC sample were compared with those directly obtained from the EM simulations, used as reference. Note that the MC sample was limited to 2000 realizations for practical reasons (e.g. the simulation cost of 10 000 realizations would have been prohibitive, about 104 days!).

The $e_{Sup}$ for the generated ATFs are shown in Figure 3.36. The average and maximal value are respectively 0.53 and 2.2 dB; the proportions of category A, B and C are respectively 89.32 %, 8.16 %, and 2.52 %, which coincides with the accuracy level reached by our final surrogate model $\hat{\mathcal{H}}^{(5)}$. Figure 3.37 and 3.38 show the 3D $G_r$ patterns for representative average-error (0.53 dB) and maximal-error (2.2 dB) configurations. We see that for the former, the generated patterns, both in CP and XP, are close to the simulated ones; for the latter, the generated patterns still have very similar appearance as the simulated, except for a moderate attenuation of ~ 2 dB in amplitude. Nevertheless, the accuracy and robustness of the derived surrogate model can be considered satisfactory.

<figure><img src="image_53.png" /><figcaption>Figure 3.36: Error of the model-generated ATFs w.r.t. the reference MC sample.</figcaption></figure>
<figure><img src="image_54.png" /><figcaption>Figure 3.37: EM-simulated (left column) and model-generated (right column) $G_r$ patterns for a representative average-error configuration.</figcaption></figure>

<figure><img src="image_55.png" /><figcaption>Figure 3.38: EM-simulated (left column) and model-generated (right column) $G_r$ patterns for a representative maximal-error configuration.</figcaption></figure>

### 3.5.5.8 Application examples of the surrogate model

Once the surrogate model of the radiated FF is built up, any “secondary observable” can be easily computed, and the computational cost of the statistical analysis of such “secondary observable” is low, even if slow convergence techniques such as the MC method are used.

For example purpose, we consider here, first, the –10 dB beamwidth of the principal cutplanes ($xOy$ and $xOz$) for the CP component, and second, the maximal $G_r$ of the CP
component. Their statistics, computed from the EM simulations and from the surrogate model (from the MC sample), are compared in Figure 3.39 using the CDF. It can be seen that for the beamwidths, the CDFs deviations are within 2.2 degrees, whereas the deviations of the maximal $G_{r\theta}$ are within 0.25 dB, in agreement with the $e_{Sup}$ values shown in Figure 3.35 and 3.36.

<figure><img src="image_56.png" /><figcaption>Figure 3.39: Comparison between the CDFs of: (a) -10 dB beamwidth, (b) maximal $G_{r\theta}$ computed from the model and EM-simulations from the MC sample.</figcaption></figure>

### 3.5.5.9 Discussion about the model accuracy

Certainly, the accuracy of the surrogate model could be further improved in case of necessity. That could be achieved by including more VSH modes in the model (i.e. increasing the power threshold $E_{c\_th}$), and by offering more information to train the surrogate model (i.e. enriching the ED). For example, if we enrich the ED up to 1470 realizations (denoted $ED_{13}$) and adopt a higher PCE degree of $N_{PCE} = 13$, the prediction error $e_{WTLOO}$ of the derived model $\hat{H}^{(13)}$ falls to 2.7 % (Figure 3.40). In Figure 3.41, the CDFs of those dominant VSH coefficients computed with the model $\hat{H}^{(13)}$ from the MC sample were compared with those directly obtained from the EM simulations. The Kolmogorov-Smirnov (K-S) test was performed to quantitatively verify the consistency between the model-generated and the EM-simulated CDFs. The maximum distances $D$ between the two are shown in TABLE II. We see that the K-S test is satisfied with a significance level $\alpha = 0.01$ (i.e. $D \le D_{\alpha=0.01} = 0.0515$) for all VSH coefficients, and is still satisfied with $\alpha = 0.05$ (i.e. $D \le D_{\alpha=0.05} = 0.043$) for the vast majority of these VSH modes (except the mode 1, 6, 17; note that the distance $D$ is however close to the threshold for mode 1, which is often the most energetic).

However, the cost for a more accurate model is obviously a high computation time and a degradation in model compactness. In brief, there is always a trade-off between the cost/compactness and the accuracy, and our proposed modeling methodology is versatilely scalable according to application-specific requirements.
<figure><img src="image_57.png" /><figcaption>Figure 3.40: Prediction error $e_{WTLOO}$ of surrogate models iteratively constructed by enriching the ED till large cardinal.</figcaption></figure>

<figure><img src="image_58.png" /><figcaption>Figure 3.41: Comparison between the CDFs of the model-generated and the EM-simulated dominant VSH coefficients from the MC sample.</figcaption></figure>

<table><caption>Table IV: K-S test on the 27 dominant VSH coefficients.</caption><tbody><tr><th scope="row">VSH Coeff</th><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td></tr><tr><th scope="row">D</th><td>0.0439</td><td>0.0216</td><td>0.043</td><td>0.0309</td><td>0.0222</td><td>0.0513</td><td>0.0175</td><td>0.0284</td><td>0.0286</td></tr><tr><th scope="row">VSH Coeff</th><td>10</td><td>11</td><td>12</td><td>13</td><td>14</td><td>15</td><td>16</td><td>17</td><td>18</td></tr><tr><th scope="row">D</th><td>0.0279</td><td>0.0172</td><td>0.0357</td><td>0.0255</td><td>0.0252</td><td>0.022</td><td>0.0284</td><td>0.0515</td><td>0.0219</td></tr><tr><th scope="row">VSH Coeff</th><td>19</td><td>20</td><td>21</td><td>22</td><td>23</td><td>24</td><td>25</td><td>26</td><td>27</td></tr><tr><th scope="row">D</th><td>0.0323</td><td>0.0244</td><td>0.0288</td><td>0.0206</td><td>0.0198</td><td>0.0399</td><td>0.0354</td><td>0.0192</td><td>0.0222</td></tr></tbody></table>

### 3.5.5.10 Discussion about the computational cost

A high performance computing server equipped with a dual-core CPU (Intel Xeon @ 2.6 GHz, 256 GB RAM) and 2 GPU (Graphical Processor Unit) accelerators (NVIDIA Tesla K80)
was used to perform the EM simulations with CST. It takes averagely 15 minutes for a single
ATF computation. Therefore, a total elapsed time of about 175 hours is necessary for
computing the 700 pre-evaluations of ED<sub>5</sub>, and 367.5 hours for ED<sub>13</sub>. It should be emphasized
that such rather high simulation cost is the result of the demanding constraints imposed to the
EM solver so as to guarantee a sufficient precision, because of the complexity of the
considered geometries, involving in particular thin layers. The computation time of the
surrogate models (including Step 2.b-2.g in § 3.5.4) varies from 25 seconds (for model $\mathcal{H}^{(0)}$)
to 153 seconds (for model $\mathcal{H}^{(5)}$) and 17 minutes (for model $\mathcal{H}^{(13)}$). The computation of the
ATF with the model $\mathcal{H}^{(5)}$ or $\mathcal{H}^{(13)}$ for any values of the input parameters takes only 7
milliseconds corresponding to a speedup factor of $(15 \times 60)/(7 \times 10^{-3}) = 1.3 \times 10^{5}$ with respect
to using EM simulations.

**3.6 Conclusion**

In this chapter, we presented methodologies for quantitatively characterizing the *complete*
ATF of antennas undergoing random uncertainties due to deformations or disturbances from
fluctuating environments. To mitigate the prohibitive complexity of a direct modeling, we
proposed to split the modeling procedure into two steps, i.e. a first step of “*field parsimony*”
followed with a second step of “*surrogate model extraction*”. The first step aims to get a
sparse/compact expression of the ATF, and is effectively achieved by means of the VSH
expansion method which transforms the spatial 3-dimentional complex vector-valued ATF
into a finite set of scalar parameters. The second step aims to quantitatively and explicitly
interpret the dependency of the sparse ATF on those random input parameters. Both
conventional deterministic and advanced statistical approaches have been investigated.

At the first stage, a deterministic approach based on the conventional parametric study
was proposed. This approach was applied to a canonical bendable dipole for the purpose of
modeling the single effect of bending, or the joint effect of bending & torsion, on the dipole's
ATF. The modeling results justified the relevance of this simple approach. However, this
approach is suited only for relatively simple univariate or bivariate systems, and cannot be
generalized to complex systems subject simultaneously to a higher number of random
uncertainties. Nevertheless, this preliminary study has demonstrated the feasibility of our
“*Two-Step*” idea and thence paved the way for the exploration of more sophisticated modeling
approaches.

At the second stage, we presented a statistical methodology based on the HLARS-PCE
method. Comparing to the deterministic approach, the statistical methodology is much more
computationally efficient, flexible and robust for treating multivariate problems. This
approach was firstly applied to a canonical deformable dipole (with 3 input parameters, i.e. stretching, bending and torsion) to demonstrate its effectiveness. Then, the methodology was improved/enhanced, in particular by taking into account the frequency behavior and by further exploiting the parsimony. The improved approach was tested on a more realistic complex wearable textile patch antenna (with 4 geometrical random parameters and the frequency as an additional “random” parameter) for which accurate and robust surrogate models were successfully derived.

In the following chapter, we will present the utility of such accurate *complete* antenna surrogate models in the performance evaluation of radio links where they are implemented jointly with propagation channel models.
# Chapter 4

## 4. Antenna surrogate models in radio links analysis

### 4.1 Introduction

The quality of service (QoS) provided by a wireless communication system is notably determined by the real-time and *in situ* performance of the radio link, which is a combination of the base station (or access point), the propagation channel and the user end terminal antenna (see Figure 4.1). In practice, various sources of uncertainties may randomly arise in each of the three parts (notably the last two parts) and drastically degrade the radio link quality. Today, with the increasing demands on reliability and robustness in many application scenarios, accurately assess the radio link performance while taking into account those randomness becomes crucially important. In recent years, a lot of efforts have been made to develop empirical models to account for random effects (mainly the large-scale fading effect due to reflection, diffraction, scattering, etc., and medium-scale fading or *shadowing*) on the propagation channel. Conversely, the random characteristics of the terminal antenna have not been fully incorporated in such kind of radio link analysis. The main reasons are, firstly, the large-scale fading effect of the propagation channel is most often decisively dominant in front of the variabilities of the terminal antenna; secondly, the “effective” performance of the terminal antenna which suffers from both inherent uncertainties (especially true for variable antennas, e.g. deformable ones) and external uncertainties from its complex local environment, is not easy or straightforward to model efficiently. Consequently, those random effects on the terminal antenna side are often roughly taken into account or even ignored in a traditional radio link analysis, where a simplistic and fixed antenna model (e.g. “isotropic” antenna of 0 dBi gain) is used, to which an ad hoc margin is added in order to roughly account for those randomness that may be encountered in real environments [53]. However, resorting to finer modeling approaches in place of such kind of rough approximation can be profitable for many application contexts, notably in the wake of emerging networks, standards and usages (5G, IoT, femtocells, etc.), which require more precise performance analyses.

This problem could not be effectively tackled only if we can efficiently model on the one side, the effect of inherent uncertainties of the terminal antenna itself, and on the other side, the effect of external uncertainties from the propagation environment. The first goal can be
achieved by using our proposed ATF modeling methodology; and the second aspect could be described by some existing advanced channel models, such as the well-known WINNER2/+ model [137], [138]. In this chapter we will present a preliminary approach to show the advantage of using accurate antenna surrogate models instead of a simplistic fixed antenna for radio links analysis.

<figure><img src="image_59.png" /><figcaption>Figure 4.1: Illustration of the radio link of a wireless communication system.</figcaption></figure>

## 4.2 Effective performance of terminal antenna

In practical usage, the “effective” performance of the terminal antenna depends on both the radio-electric properties of the antenna itself – ATF or radiation pattern, and its local propagation environment – the spatial and polar distribution of the incoming waves. The Mean Effective Gain (MEG) is a popular parameter for quantifying such “effective” performance of an in situ antenna. It is defined as the ratio of the average power received by the antenna under study and the sum of the average power that would be received by two isotropic antennas which are respectively vertically and horizontally polarized [139], given by:

$$ 
\begin{aligned}
G_{\text{MEG}} = \frac{P_{\text{rec}}}{P_{\text{V}} + P_{\text{H}}} &= \oint \left( \frac{P_{\text{V}} G_{\theta}(\theta, \varphi) P_{\theta}(\theta, \varphi) + P_{\text{H}} G_{\varphi}(\theta, \varphi) P_{\varphi}(\theta, \varphi)}{P_{\text{V}} + P_{\text{H}}} \right) d\Omega \\
&= \oint \left( \frac{\chi G_{\theta}(\theta, \varphi) P_{\theta}(\theta, \varphi) + G_{\varphi}(\theta, \varphi) P_{\varphi}(\theta, \varphi)}{\chi + 1} \right) d\Omega
\end{aligned}
\quad (4.1) $$

where,

* $\chi = P_V / P_H$ is called the cross-polarization power ratio (XPR), defined as the ratio of the average power available in the vertical polarization $P_V$ to the average power available in the horizontal polarization $P_H$. This parameter characterizes the depolarization effect of the propagation channel on the transmitted waves.

* $P_{\theta}(\theta, \varphi)$ ($P_{\varphi}(\theta, \varphi)$) is the angular power distribution density of the vertical (horizontal) polarization component. These two parameters characterize the spatial distribution of the incoming waves in the vicinity of the terminal antenna.
* $G_{\theta}(\theta, \phi)$ and $G_{\phi}(\theta, \phi)$ are the polarimetric gain (or realized gain) of the terminal antenna.

For sake of simplicity, we adopt herein a normalized form of the MEG for evaluating the effective performance of a terminal antenna when it is combined with a discrete multipath channel model – the Effective Gain (EG) [140] as defined:

$$ 
\begin{aligned}
G_{EG} &= \sum_{n} |A_{nH}|^2 G_{nH} + |A_{nV}|^2 G_{nV} \\
&= \sum_{n} |A_{nH}|^2 (G_{nH} + \chi_n G_{nV})
\end{aligned} 
\quad (4.2) $$

where $|A_{nH}|$ ($|A_{nV}|$) is the normalized amplitude of the horizontal (vertical) polarization component of the n-th path satisfying the normalization relation: $\sum_{n} |A_{nH}|^2 + |A_{nV}|^2 = 1$; $\chi_n$ is the cross-polarization ration (XPR) for the n-th path: $\chi_n = |A_{nV}|^2 / |A_{nH}|^2$. Note that for an ideal lossless isotropic antenna with equal polarization components (i.e. $G_{\theta}(\theta, \phi) = G_{\phi}(\theta, \phi) \equiv 0.5$), its EG is constantly equal to -3 dB regardless of the variability of the propagation environment. It should also be noted that the interference between multipath, i.e. the fast fading (or small scale fading) effect is neglected in the computation of the EG, seeing that the phases of multipath do not appear in (4.2).

Clearly, for a terminal antenna which is subject to inherent random uncertainties $X$ (a common case by essence for wearable textile antennas and other types of deformable antennas), its gain patterns $G_{nH}$ and $G_{nV}$ change randomly, and so do the $G_{EG}$ as well as the final radio link budget. It is in such situations that an accurate antenna surrogate model $\hat{H}(X, f, \hat{r})$ taking into account those inherent random effects plays a beneficial role for precise performance evaluation.

In the following we will implement the derived antenna surrogate model for the wearable textile patch antenna (cf. § 3.5.5) in some classical local propagation environments to show its influence with regards to only considering the “nominal” flat antenna. We use a simplified version of the WINNER2/+ model to consider different types of propagation environments.

## 4.3 Local propagation environments

The WINNER2/+ is a cluster-based channel model which describes a propagation environment consisting of multiple Non-Line-Of-Sight (NLOS) paths by Rayleigh distribution, and a propagation environment consisting of several NLOS paths plus a LOS path by Rice distribution. We adopted a simplified version of WINNER2/+ [36] by assuming: i) a vertically-polarized base station; ii) only one path per cluster; and iii) ignored phase for all paths (since they are not used for computing the EG). Precisely, the random generation of
the directions of arrival (DoAs) and amplitudes of multipath is carried out with respect to the following rules [141]:

For NLOS paths:

* The number of clusters (i.e. paths) obeys a Gaussian distribution, and at least one cluster (path) is required:

$$N = \text{Max}[1, \text{Int}(\mathcal{N}(\mu_c, \sigma_c))]$$

with $(\mu_c, \sigma_c)$ being the mean and standard deviation of the number of clusters for the considered propagation environment.

* The DoAs $(\theta_n, \varphi_n)$ of the paths are defined such that the elevation angles obey Laplacian distribution and the azimuth angles obey Gaussian distribution:

$$\begin{cases} \theta_n \sim \mathcal{L}(\mu_\theta, b_\theta) \\ \varphi_n \sim \mathcal{N}(\mu_\varphi, \sigma_\varphi) \end{cases}$$

where $\mu_\theta$ is the mean position in elevation; $b_\theta$ is the spread in elevation, which obeys a Log-normal distribution of mean and standard deviation $(\mu_b, \sigma_b)$; $\mu_\varphi$ is the mean position in azimuth, assumed to be uniformly distributed in a specified angular range; $\sigma_\varphi$ is the spread in azimuth, which obeys a Gaussian distribution of mean and standard deviation $(\mu_{\sigma_\varphi}, \sigma_{\sigma_\varphi})$.

* The vertical polarization amplitudes are assumed to obey Rayleigh distribution:

$$|A_{nV}| \sim \mathcal{R}(\sigma_{nV})$$

where the variance $\sigma_{nV}^2$ of the Rayleigh distribution is a Laplacian function of the azimuth angle of the n-th path $\varphi_n$, and the stochastic parameters of the azimuth distribution $(\mu_\varphi, \sigma_\varphi)$:

$$\sigma_{nV}^2 = C \cdot \exp(-\sqrt{2} |\varphi_n - \mu_\varphi| / \sigma_\varphi)$$

* The horizontal polarization amplitudes are related to the vertical one through the cross polarization ration XPR: $|A_{nH}| = |A_{nV}|/\chi_n$, where $\chi_n$ is assumed to be lognormally distributed with mean and standard deviation $(\mu_\chi, \sigma_\chi)$, and C is a normalization constant satisfying $\sum_n |A_{nH}|^2 + |A_{nV}|^2 = 1$.

For LOS paths: The DOA of the LOS path is taken as the closest one to the mean direction $(\mu_\theta, \mu_\varphi)$, and its power relative to the total power of all NLOS paths is defined by the K factor, which is assumed to be lognormally distributed, with mean and standard deviation $(\mu_K, \sigma_K)$.
In our case, we chose four types of WINNER2/+ propagation environment, namely:

- Env. 1: Indoor (small office/ residential), NLOS

- Env. 2: Indoor (small office/ residential), LOS

- Env. 3: Typical Urban (hot spot), NLOS from base station

- Env. 4: Typical Urban Macro-cell, NLOS from base station

The standard stochastic characteristics of these environments are summarized in Table V.

<table><caption>Table V: Characteristics of 4 WINNER2/+ propagation environments.</caption><thead><tr><th rowspan="2">Env.</th><th rowspan="2">K factor (dB) μ<sub>K</sub> / σ<sub>K</sub></th><th rowspan="2">XPR (dB) μ<sub>χ</sub> / σ<sub>χ</sub></th><th colspan="2">Azimuth</th><th colspan="2">Elevation*</th><th rowspan="2"># Paths μ<sub>c</sub> / σ<sub>c</sub></th></tr><tr><th>Mean (°) μ<sub>φ</sub></th><th>Spread (°) μ<sub>σ<sub>φ</sub></sub> / σ<sub>σ<sub>φ</sub></sub></th><th>Mean (°) μ<sub>θ</sub></th><th>Spread (log<sub>10</sub>([°])) μ<sub>b</sub> / σ<sub>b</sub></th></tr></thead><tbody><tr><td>1</td><td>-</td><td>10/4</td><td rowspan="4">U([0, 360°])</td><td>49/7</td><td>91.6</td><td>1.10/0.17</td><td>16/4.5</td></tr><tr><td>2</td><td>7/6</td><td>11/4</td><td>45/9</td><td>91.6</td><td>0.94/0.26</td><td>12/6</td></tr><tr><td>3</td><td>-</td><td>8/3</td><td>35.5/35</td><td>92</td><td>0.88/0.16</td><td>16/3</td></tr><tr><td>4</td><td>-</td><td>7/3</td><td>52.5/20.5</td><td>100</td><td>1.26/0.16</td><td>20/4</td></tr></tbody></table>

$$ * \theta_n \sim \mathcal{L}(\mu_\theta, b_\theta), \quad b_\theta = 10^{\lambda(\mu_b, \sigma_b)}. $$

# 4.4 Effective performance and discussion

## 4.4.1 Relative Effective Gain (REG)

We can anticipate that taking into account or not of those inherent randomness of the terminal antenna will result in more or less difference in antenna's effective performance. The more the antenna patterns change dramatically with the inherent randomness, the more significant this difference will be, and the more justifiable it is to use a precise antenna model for the radio links analysis. To visualize the dispersion of $G_{EG}$, we resort to the quantity "relative Effective Gain (REG)" which is the ratio between the $G_{EG}$ of a specified antenna realization and that of the fixed reference antenna, i.e. the original non-deformed flat patch operating at the nominal frequency ($C = 0, s = 1, \kappa = 0, \varepsilon_r = 1.75, f = 2.45$ GHz):

$$ G_{REG} \triangleq G_{EG} / G_{EG}^{ref} \qquad (4.3) $$

Note that the choice to use the nominal flat patch as the reference antenna is to highlight the antenna variability in the channel context. Another possible choice would have been to resort, e.g., to an "ideal" isotropic radiator as was done in [36]. However, this would have over-represented the directive character of the patch antenna, which is not the main objective of these analyses.

We generated 1000 antenna realizations using our surrogate model for 1000 random values of ($C, s, \kappa, \varepsilon_r, f$). For each considered propagation environment (Env. 1-4), we generated randomly 100 channel realizations using the simplified WINNER2/+ model and:
i. For each of these 100 channel realizations, we combined it with the 1000 random antennas and computed the 1000 REGs. Then we drew the corresponding CDF curve, referred to as “partial CDF” (color thin lines in Figure 4.2) and exhibited its variability range and standard deviation. The average value of variability ranges and standard deviations for the 100 channels were summarized in Table VI.

ii. We combined all these 100 channel realizations with the 1000 random antennas and computed the 100 × 1000 = 10⁵ REGs. Then we drew the CDF curve, referred to as “global CDF” (red thick line in Figure 4.2) and exhibited its variability range and standard deviation, see Table VI.

<figure><img src="image_60.png" /><figcaption>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.</figcaption></figure>

<table><caption>Table VI: Statistics of the relative effective gains.</caption><thead><tr><th rowspan="2">Env.</th><th colspan="2">Partial CDFs</th><th colspan="2">Global CDF</th></tr><tr><th>Average Variability range (dB)</th><th>Average Std (dB)</th><th>Variability range (dB)</th><th>Std (dB)</th></tr></thead><tbody><tr><td>1</td><td>7.57</td><td>1.12</td><td>17</td><td>1.54</td></tr><tr><td>2</td><td>10.24</td><td>1.61</td><td>35.6</td><td>3.23</td></tr><tr><td>3</td><td>9.27</td><td>1.43</td><td>38.8</td><td>2.76</td></tr><tr><td>4</td><td>8.1</td><td>1.21</td><td>23.6</td><td>1.73</td></tr></tbody></table>
Observations and discussions are given herein:

1. The “partial CDFs” (see color thin lines in Figure 4.2 and Table VI) show that for whichever considered channel in whichever Env., the REG always retains a wide dispersion – the average variability range is between 7.57 ~ 10.24 dB and the average standard deviation is larger than 1.1 dB. This dispersion is a direct reflection of the inherent randomness (caused by *X*) of the terminal antenna. In other words, the effective performance of a randomly deformed antenna could dramatically differ from that of the nominal flat design. Therefore, in the application contexts where precise performance analyses are required, those random effects must not be ignored and accurate antenna models in place of a fixed (original) antenna must be used.

2. Env. 1 and 2 are of similar “Indoor” type, except that the former only has NLOS paths while the latter has both NLOS paths and a LOS path. From their “global CDF” (see red thick line in Figure 4.2 and Table VI) we note that the latter has wider variability range – 35.6 dB vs. 17 dB, and more significant standard deviation – 3.23 dB vs. 1.54 dB. Our explanation is that: a channel containing only NLOS paths is not quite “directional” because the NLOS paths are of statistically equal mean power weight and their DoAs are more or less “ubiquitous” in the space. In such situation, the EG is determined by the average “matching degree” between the antenna’s gain pattern and all the NLOS paths, thereby, the effect of the antenna’s inherent randomness is weakened (or “randomized”) to some extent by this global averaging aggregation. In contrast, a channel containing both LOS and NLOS paths is somewhat more “directional” because the LOS path is energetically dominant in front of all NLOS paths. In such situation, the antenna’s effective performance will be mainly determined by the “matching degree” between its gain pattern and the single LOS path (e.g. DoA and polarization). Therefore, if the randomness causes significant variation in the LOS direction, this would be immediately reflected (in an obvious manner) on the statistics of the REG.

3. Env.3 and 4 have similar NLOS parameters, except that the former has narrower spreads in DoAs – 35.5° vs. 52.5° in azimuth, and 0.88 vs. 1.22 (in $log_{10}([°])$) in elevation. Comparing their “global CDF” (see red thick lines in Figure 4.2 and Table VI), we note that the former has wider variability range – 38.8 dB vs. 23.6 dB, and more significant standard deviation – 2.76 dB vs. 1.73 dB. It is not difficult to figure out the reason since narrower spreads in DoAs imply indeed higher “directionality” of the propagation channel, where the variation in terminal antenna can be more evidently captured.
### 4.4.2 Relative Average EG (RAEG) in sub-bands

In the previous example, we have considered the frequency “f” as an ordinary random input variable. However, in practical applications, a terminal antenna never operates at discrete frequencies but over an allowable sub-band. For example, in the 2.4 – 2.5 GHz range for the WiFi standard there are 14 sub-bands designated, as shown in Figure 4.3. Therefore, it is indeed its “equivalent” effective performance over a considered operating sub-band that determines the final performance of the terminal antenna. Having this in mind, we choose to consider hereby the EG averaged over a considered sub-band ($\bar{G}_{EG}$) as in (4.4):

<figure><img src="image_61.png" /><figcaption>Figure 4.3: Allowable sub-bands for WiFi standard in the 2.4 – 2.5 GHz range.</figcaption></figure>

$$ 
\begin{aligned}
\bar{G}_{EG} &= \sum_{n} |A_{nH}|^2 \bar{G}_{nH} + |A_{nV}|^2 \bar{G}_{nV} \\
&= \sum_{n} |A_{nH}|^2 (\bar{G}_{nH} + \chi_n \bar{G}_{nV})
\end{aligned}
\qquad (4.4) $$

where $\bar{G}_{nV}$ and $\bar{G}_{nH}$ are the polarimetric realized gain averaged over the considered sub-band:

$$ \bar{G}_{nV,H} = \frac{1}{N_f} \sum_{i=1}^{N_f} G_{nV,H}(f_i) \qquad (4.5) $$

The relative average Effective Gain (RAEG) is defined as the ratio between the $\bar{G}_{EG}$ of a specified antenna realization and that of the fixed reference antenna, i.e. the original non-deformed flat antenna operating over the same sub-band ($C = 0, s = 1, \kappa = 0, \varepsilon_r = 1.75$):

$$ G_{RAEG} \triangleq \bar{G}_{EG} / \bar{G}_{EG}^{ref} \qquad (4.6) $$

We generated randomly 1000 antenna realizations. For each realization:

1) A value of ($C, s, \kappa, \varepsilon_r$) was randomly drawn according to their joint distribution.

2) One of the 14 sub-bands (see Figure 4.3) was randomly chosen, in which a series of frequencies (spaced by 1 MHz) spanning the sub-band were chosen. For instance, if the sub-band #1 is chosen, the frequency samples are $F_{sub} = [2401:1:2423]$ MHz.

3) The ATFs were generated for the input parameters ($C, s, \kappa, \varepsilon_r, F_{sub}$).

4) The average realized gain patterns $\bar{G}_{nV}$ and $\bar{G}_{nH}$ of this specified antenna realization as well as those of the reference antenna were computed according to (4.5).
For each Env., we generated randomly 100 channel realizations using the simplified WINNER2/+ model. Finally, we computed the RAEG of each channel-antenna combination according to (4.4) and (4.6), and derived the “partial CDFs” and “global CDFs” of the RAEGs in the same manner as for the REGs in § 4.4.1. The CDFs are shown in Figure 4.4 and the variability ranges and the standard deviations are summarized in Table VII.

<figure><img src="image_62.png" /><figcaption>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.</figcaption></figure>

<table><caption>Table VII: Statistics of the relative average effective gains.</caption><thead><tr><th rowspan="2">Env.</th><th colspan="2">Partial CDFs</th><th colspan="2">Global CDF</th></tr><tr><th>Average variability range (dB)</th><th>Average std (dB)</th><th>Variability range (dB)</th><th>Std (dB)</th></tr></thead><tbody><tr><td>1</td><td>3.74</td><td>0.28</td><td>12.1</td><td>1.03</td></tr><tr><td>2</td><td>4.99</td><td>0.45</td><td>28.27</td><td>2.68</td></tr><tr><td>3</td><td>4.53</td><td>0.39</td><td>28.45</td><td>2.32</td></tr><tr><td>4</td><td>4.1</td><td>0.32</td><td>18.15</td><td>1.26</td></tr></tbody></table>

Remarks and discussions are given herein.

If we look at the “global CDF” which aggregates all channel-antenna combinations, we draw some conclusions similar to those drawn for the REGs, e.g. concerning the directional
effect of the considered propagation environment. Effectively, more “directional” channels
lead to wider dispersion in RAEG: larger variability ranges – 28.27 (Env. 2) vs. 12.1 dB (Env.
1), and 28.45 (Env. 3) vs. 18.15 dB (Env. 4); and larger standard deviations – 2.68 (Env. 2) vs.
1.03 dB (Env. 1), and 2.32 (Env. 3) vs. 1.26 dB (Env. 4).

Moreover, when we take a deeper look at those “partial CDFs”, we note some significant
difference comparing to those of the REGs:

1. The “partial CDFs” of RAEGs have much smaller dispersions: average variability range between 3.74 ~ 5 dB (vs. 7.57 ~ 10.24 dB for REGs), and average standard deviation between 0.28 ~ 0.45 dB (vs. > 1.1 dB for REGs).

2. The “partial CDFs” seem to be somehow “parallel” spaced one to another, each occupies a small range of variability, and together contribute to a wide “global” variability range.

This phenomenon implies the fact that the random effect on the terminal antenna side is
small in front of that on the propagation channel side – the latter determines the first-order
moment and the former describes the second-order moment of the partial distributions.
Therefore, using the REG in § 4.4.1 which evaluates the performance of the terminal antenna
operating at discrete frequencies is actually inappropriate since it exaggerates the dispersion
of the antenna’s effective gain, as well as the random effect of antenna’s inherent randomness.
On the contrary, the RAEG appear as a more appropriate and relevant indicator for exhibiting
the random effects.

4.4.3 RAEG for restricted azimuthal DoAs

However, even if we resort to the RAEG indicator, the quantitative conclusions drawn in
the above section should still be taken with caution. In the previous examples, the
WINNER2/+ model suggests that the azimuthal mean angle $\mu_{\phi}$ obeys uniform distribution in
$[0, 360^{\circ}]$. This assumption would be reasonable when considering more or less
omnidirectional terminal antennas or an application context where the 360° fully free azimuth
rotation of the terminal antenna is required. Yet, for directional antennas that are dedicated for
directional usages, such as our patch antenna, it seems more appropriate to evaluate its
effective performance in “realistic scenarios” more consistent with the practical antenna
usage<fnref n="2" />, i.e. with DoAs restricted in only the broadside hemisphere. Here, by assuming that $\mu_{\phi}$
~ $\mu$([-90°, 90°]) we re-computed the statistics of the RAEGs. The CDFs are shown in Figure
4.5 and the variability ranges and the standard deviations are summarized in Table VIII.

<footnote n="2">² For example, downloading a file more or less in front of a hotspot terminal in a shopping mall.</footnote>
<figure><img src="image_63.png" /><figcaption>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.</figcaption></figure>

<table><caption>Table VIII: Statistics of the relative average effective gains ($\mu_{\phi} \sim U([-90^{\circ}, 90^{\circ}])$).</caption><thead><tr><th rowspan="2">Env.</th><th colspan="2">Partial CDFs</th><th colspan="2">Global CDF</th></tr><tr><th>Average variability range (dB)</th><th>Average std (dB)</th><th>Variability range (dB)</th><th>Std (dB)</th></tr></thead><tbody><tr><td>1</td><td>2.65</td><td>0.2</td><td>5.52</td><td>0.66</td></tr><tr><td>2</td><td>2.84</td><td>0.21</td><td>8.26</td><td>0.68</td></tr><tr><td>3</td><td>2.73</td><td>0.2</td><td>5.58</td><td>0.55</td></tr><tr><td>4</td><td>2.75</td><td>0.2</td><td>5.17</td><td>0.51</td></tr></tbody></table>

Remarks and discussions are given herein.

1. The “global CDFs” now have much narrower dispersions than in § 4.4.2 – smaller variability ranges and smaller standard deviations. Therefore, the assumption of identical possibility of the mean azimuth angle in the whole range of [0, 360°] will overestimate the dispersion of the RAEG of our textile patch antenna.

2. The variability ranges are of comparable level (~ 5 dB) for the 3 NLOS propagation environments Env. 1, 3, 4, and they are all smaller than that of the LOS channel Env. 2 (~ 8 dB). This is because the LOS path plays a decisive role in the channel
directionality while the effect due to azimuthal spreads is not very significant, since
the DoAs have been largely restricted, i.e. in the broadside hemisphere.

3. Generally, such large variability (> 5 dB) must not be neglected for precise radio links analysis, and the adoption of accurate antenna models in place of fixed reference antenna is highly required.

**4.5 Conclusion**

In this chapter, we showed the utility of antenna surrogate models for evaluating radio links performance. The latter depends strongly, among others, on the effective performance of the terminal antenna. In practical usage, the terminal antenna generally suffers from inherent random uncertainties as well as external random uncertainties from its surrounding environment (especially true for e.g. wearable textile antennas), which may drastically perturb the performance of the terminal antenna, hence the radio links performance. In such context, antenna surrogate models which account for antenna inherent randomness are highly advantageous over either a conventional simplistic “isotropic model” or a practical but fixed reference model (whether simulated or measured).

To show the benefit of antenna surrogate models, we resorted to some performance
indicators (i.e. REG and RAEG) based on the MEG – a commonly used indicator e.g. in
mobile radio communications. We computed firstly the variability of the REG, and then that
of the RAEG of our textile patch antenna in some classical propagation environments. A
significant variability in REG and RAEG was observed in all propagation environments,
which justifies the necessity for adopting antenna surrogate models if one requires accurate
radio links analysis. This is particularly demanding when both the terminal antenna and the
propagation environment are more “directional”. In such context, we can construct accurate
surrogate models for the terminal antenna (by using our proposed modeling methodology) in
the first step, and then implement the antenna surrogate model in appropriate propagation
environments to carry out accurate radio links analysis in the second step.

This approach of joint antenna/channel modeling can be further refined to be able to
address e.g. the received signal as shown in [142]. In particular, if the phase is taken into
account (which was not the case here – see a discussion about this matter e.g. in [36]), it is
possible to perform simulations at some system level, including e.g. the physical layer so as to
test various modulation schemes (see e.g. [127]) and assess system performance indicators
such as the Bit Error Rate. This kind of approach, more elaborate and finer, which would
highlight the interest of developing surrogate models of the complete radiated field (vector
and phased), is left as future development.
# Chapter 5

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

### 5.1 Introduction

As introduced in Chapter 2, the *overall* FF behavior of a transmitting antenna (or of a receiving antenna because of the reciprocity theorem) can be completely described by two complex quantities: the *complete* radiated FF or equivalently, the ATF $\mathcal{H}(f, \hat{r})$, and the reflection coefficient $S_{11}(f)$, which describe respectively the transmission and the reflection properties of the considered antenna. In the previous chapters we presented effective methodologies for modeling the ATF; and in this chapter we will concentrate on the modeling of the $S_{11}(f)$, thereby to complete the *overall* modeling scheme for variable antennas.

In general, $S_{11}(f)$ is a quantity quite sensitive to random uncertainties – notably for NB antennas – since any type of deformations, material properties change, or environmental perturbation could easily induce impedance mismatches, hence causing variations in $S_{11}(f)$. In recent years, the problem of modeling the $S_{11}(f)$ or its equivalent – the input impedance $Z_{in}(f)$ due to tunable design parameters has been widely addressed by means of relatively sophisticated methods such as ANNs [45], [46], Gaussian Process Regression (GPR) [143], Support Vector Regression (SVR) [39], [41], and gPCE [50], etc. In [39], [45], [46] and [50], the $Z_{in}$ of the considered antennas were modeled only at the resonant frequency; in [41] and [143], the $S_{11}(f)$ were modeled over a wide frequency band by means of taking the frequency “f” as one of the random input parameters. These methods have been proved to be effective for handling highly non-linear problems. However, on the one hand, some of these methods are not always straightforward for implementation; on the other hand, taking the frequency as an additional random input parameter would weaken the space-filling property of the other input parameters, hence larger-sized ED might be required inducing higher computational cost. Hereby, we propose a simpler alternative approach for modeling the $S_{11}(f)$ of NB antennas. Instead of directly considering the frequency as a random parameter in the input space, we split the modeling procedure into two steps: first, we use an
identification technique to convert the frequency-dependent $S_{11}(f)$ to a finite set of frequency-independent characteristic parameters. This is analogue to the “separation of variables” for solving partial differential equations. Second, we extract surrogate models by using the HLARS-based PCE method to quantify the impact of random variables on those frequency-independent characteristic parameters derived in the 1st step.

## 5.2 Identification

The identification of a system represented by its transfer function has been widely addressed since the 1950s. The objective is to approximate the transfer function with a rational function either in the Laplace, frequency or Z domain. Numerous algorithms have been proposed for identifying the corresponding polynomial coefficients, among which some ones are directly applicable in the frequency domain, and some others are applicable in the time domain (i.e. on the impulse response), which requires a prior inverse Fourier transform if the data are available in the frequency domain. Without being exhaustive, we can mention the Cauchy's method [144], the Levy's complex curve fitting method [145], and the Kumaresan's Frequency-domain Prony's method [146] for the former category; and the famous Prony's method [147], the Steiglitz-McBride's method [148], and the Matric-Pencil method [149] for the latter category. Hereby, we investigate two identification techniques as a first attempt for verifying the Two-Step modeling methodology. The first technique is the classical polynomial fitting method, which relies on the continuity of the real and imaginary parts of $S_{11}(f)$ and approximates them separately with polynomial functions. This rather simple method has shown good accuracy and robustness while applied to NB antennas. Besides, the Levy's algorithm has also been tested, in the purpose of verifying the relevance of the rational interpolation.

### 5.2.1 Technique 1: Polynomial fitting

The real and imaginary parts of the $S_{11}(f)$, respectively denoted $S_R(f)$ and $S_I(f)$, are both continuous functions of the frequency. They can be fitted with polynomial functions as:

$$S_R(f) = \sum_{m=0}^{M_R} R_m f^m + \varepsilon_R(f), \quad S_I(f) = \sum_{n=0}^{N_I} I_n f^n + \varepsilon_I(f) \quad (5.1)$$

where $M_R(N_I)$, $\varepsilon_R(\varepsilon_I)$ are respectively the maximal polynomial degree and the residual error for the real (imaginary) part function. For NB antennas, usually the $S_{11}(f)$ is only considered over a very limited band around the resonant frequency. In such cases the variability of the $S_R$ and $S_I$ is rather smooth, which implies that low or moderate fitting order could be sufficient for a good precision. Once the polynomial degrees are determined, the unknown polynomial coefficients can be calculated by solving linear equations while minimizing the square error
$\sum_{k=1}^{N_f} |\epsilon_{R,I}(f_k)|^2$. Therefore we get a parsimonious representation for $S_{11}(f)$, i.e. a finite set of frequency-independent characteristic parameters $P=\{R_0, R_1, \dots, R_{M_R}, I_0, I_1, \dots, I_{N_1}\}$.

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

Denoting $S_{11}(z)$ the discrete transfer function of the reflection coefficient, where $z = \exp(j2\pi f/F_s)$, $F_s$ being the sampling frequency, $S_{11}(z)$ can be approximated by a rational function as:

$$S_{11}(z) = \sum_{m=0}^{M_b} b_m z^{-m} / (1 + \sum_{n=1}^{N_b} a_n z^{-n}) + \epsilon(z) \quad (5.2)$$

where $M_b$ and $N_b$ are respectively the order of the numerator and the denominator functions, and $\epsilon$ is the residue. Again for NB and resonant antennas, the order could not be too high. The Levy's algorithm combined with the damped Gauss-Newton method [150] allow to calculate the unknown coefficients while minimizing the square error $\sum_{k=1}^{N_f} |\epsilon(z_k)|^2$. Therefore, $S_{11}(z)$ is compacted to a set of frequency-independent characteristic parameters $P=\{b_0, \dots, b_{M_b}, a_1, \dots, a_{N_b}\}$.

## 5.3 Surrogate model extraction

We apply the HLARS-based PCE method for mapping between the random uncertainties and the parsimonious $S_{11}(f)$. Assume the input variables are represented by $X = (X_1, X_2, \dots, X_M)^T$ of which the joint PDF is known, then the modeling of the $S_{11}$ comprises the following steps:

1. Initialize the ED $\mathcal{X} = \{x^{(1)}, x^{(2)}, \dots, x^{(N_{ED})}\}^T$, and set the LOOE threshold ($e_{LOO_th}$) for the HLARS-based PCE algorithm.

2. Pre-evaluate the frequency responses over the ED set. Precisely, for each value $x^{(n)}, n = 1, \dots, N_{ED}$, design the corresponding antenna configuration, carry out the EM simulations or measurements to get the frequency data $S_{11}(f, x^{(n)})$.

3. Apply an identification technique to obtain the characteristic parameters $P=\{P_1, P_2, \dots, P_{N_P}\}$ for each $S_{11}(f)$ function.

4. Apply the HLARS-based PCE method to each characteristic parameter, and finally get surrogate models $P_i = \sum_{\alpha \in A^{HLARS}} y_{\alpha}^i \Phi_{\alpha}(X), \{i=1, \dots, N_P\}$.

5. Assess the $e_{LOO}$. If $e_{LOO} \le e_{LOO_th}$ for all models, or if $N_{ED}$ exceeds an acceptable value then stop the procedure; otherwise enrich the ED and return to Step 2.
Eventually, a surrogate model mapping quantitatively the input-output relation between these random variables $X$ and $S_{11}(f)$ via a set of polynomial coefficients $\{y_i^j|\alpha \in \mathcal{A}_i^{\text{HLARS}}, i=1,...,N_p\}$ is established, allowing the reproduction of $S_{11}(f)$ for any value of $X$.

## 5.4 Applications

We applied the proposed modeling methodology on two NB antennas to verify its performance. Firstly, the same deformable dipole (undergoing three independent deformations) used in § 3.4.3.2 was reconsidered; secondly, a wearable textile patch subject to four sources of uncertainties was considered.

To evaluate the accuracy of the derived surrogate model, we chose the maximal difference between the model-evaluated and the original frequency responses as the error indicator:

$$err = \max_{k=1,...,N_f} \{|dB[\hat{S}_{11}(f_k)] - dB[S_{11}(f_k)]|\} \quad (5.3)$$

Note that when both the initial and the model-evaluated values are less than -15 dB, the difference is considered as negligible and will not be accounted for when calculating the err.

### 5.4.1 Application to a canonical dipole

Recall that the dipole was assumed to be deformable considering three independent sources of deformation: stretching (with stretch ratio $\xi$), bending (with curvature $\kappa$) or bending radius $R_b = 1/\kappa$), and torsion (with torsion rate $\alpha$); and the input variables $(\xi, \kappa, \alpha)$ are uniformly distributed over respectively [1, 1.05], [0, 0.1] mm$^{-1}$, and [0, 0.1] rad·mm$^{-1}$. It was shown in Figure 3.23 that the joint effect of deformation on the dipole is significant: variations of $|S_{11}|$ up to more than 15 dB are possible, which justifies the necessity of an accurate quantitative model.

We applied the iterative procedure in § 5.3 with the LOOE threshold $e_{LOO_th}$ being chosen as 4 %, and the polynomial degrees for the two identification techniques set as $M_R = N_I = 3$, and $M_b = 2, N_a = 3$ respectively. We took advantage of the already obtained pre-evaluations in § 3.4.3.2 for training the surrogate model in the current framework. The modeling results are summarized below. Figure 5.1 and 5.2 show the results for the Approach I: “Technique 1 + PCE”. In Figure 5.1 (a), it is noted that the model error decreases for all characteristic parameters when enriching progressively the ED. The predefined LOOE threshold is finally satisfied for $N_{ED} = 260$, for which a statistical model with about 570 polynomial coefficients is derived. To illustrate visually the performance of the constructed model, we compared the initial, the identified (with polynomial fitting), and the model-generated frequency responses
for two cases: the worst case (i.e. with the maximal value of *err*) and the median case (with the median value of *err*), as in Figure 5.2. We see that the responses overlap well for the median case, and remain satisfactory for the worst case, which justify the accuracy and the robustness of this approach.

Similar results are shown in Figure 5.3 and 5.4 for the Approach II: “Technique 2 + PCE”. It is noted that this approach requires fewer pre-evaluations (only 120 compared to 260 in Approach I), without degrading the accuracy of the model. As a result, it is more computationally efficient.

<figure><img src="image_64.png" /><figcaption>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 $N_{ED} = 260$.</figcaption></figure>

<figure><img src="image_65.png" /><figcaption>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*).</figcaption></figure>
<figure><img src="image_66.png" /><figcaption>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 $N_{ED} = 120$.</figcaption></figure>

<figure><img src="image_67.png" /><figcaption>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 $err$).</figcaption></figure>

## 5.4.2 Application to a textile patch

A wearable textile patch antenna designed for the ISM band was chosen as the second application example. The antenna was assumed to subject to variations in length, bending curvature, substrate thickness and relative permittivity when attached around a human arm or leg, as shown in Figure 5.5. For sake of simplicity, we made simplifying assumption of independent uniform distribution of the four random variables. Their nominal values and variation ranges are listed in Table IX.
<figure><img src="image_68.png" /><figcaption>Figure 5.5: Textile patch undergoing random variations in length, bending radius, substrate height and permittivity.</figcaption></figure>

<table><caption>Table IX: Design parameters and variability range for the textile patch.</caption><thead><tr><td>Patch parameter</td><td>Nominal value</td><td>Variability range</td></tr></thead><tbody><tr><td>Length, l</td><td>46.7 mm</td><td>± 1 %</td></tr><tr><td>Bending curvature, κ</td><td>0</td><td>[0, 1/40] mm<sup>-1</sup></td></tr><tr><td>Substrate thickness, h</td><td>2 mm</td><td>± 20 %</td></tr><tr><td>Permittivity, ε<sub>r</sub></td><td>1.67</td><td>[1.5, 1.9]</td></tr></tbody></table>

The proposed modeling methodology was applied to characterize the combined effect of deformations on the reflection coefficient. The ED was initialized with 120 points, and the LOOE threshold $e_{LOO_th}$ is chosen as 4 %. The polynomial orders for the two identification techniques were chosen as $M_R = N_I = 5$, and $M_b = N_a = 5$ respectively. Figure 5.6 and 5.8 show the results for the Approach I and Figure 5.7 and 5.9 show those for the Approach II.

<figure><img src="image_69.png" /><figcaption>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 $N_{ED} = 200$.</figcaption></figure>

<figure><img src="image_70.png" /><figcaption>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 $N_{ED} = 280$.</figcaption></figure>
<figure><img src="image_71.png" /><figcaption>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*).</figcaption></figure>

<figure><img src="image_72.png" /><figcaption>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*).</figcaption></figure>
It is noted that the former approach outmatches the latter in this application example: in Approach I, only 200 pre-evaluations are used to satisfy the PCE model precision (Figure 5.6 (a)), while in Approach II, 280 pre-evaluations are even not sufficient to obtain a correct accuracy (the computational cost is considered too high beyond). As a result, the final model obtained in Approach II is much less accurate than that in Approach I, see Figure 5.6 (b) and 5.7 (b). From Figure 5.8 and 5.9 we can see that the green lines coincide quite well with the red lines for all cases, which implies that the characteristic parameters derived with both identification techniques offer a very accurate parsimonious representation of the $S_{11}(f)$. The observed inaccuracy (mainly for the worst case) occurs during the PCE step. A possible reason is that the rational function is very sensitive to the accuracy of the poles. Therefore, if the PCE modeling procedure is not able to recover the poles with “perfect” accuracy, then the whole rational function also cannot be recovered within good accuracy. More generally, it is known that many transfer function identification methods in the frequency domain are prone to be more or less important sensitive to noise [71].

Briefly, the 1st approach works well for both the dipole and the textile patch, while the 2nd works well for the dipole but is not fully satisfactory for the patch. The advantages of Approach I are its robustness and its simplicity, but it is not necessarily the most efficient approach in terms of parsimony and computational cost. Moreover, the causality, i.e. the relations of Bayard-Bode or Kramers-Kronig are not guaranteed. The Approach II provides a deeper insight into the inherent resonance modes of the antennas – thanks to the zeros/poles derived from the rational functions, which is a natural and efficient way to characterize a linear time invariant (LTI) system, notably for resonant devices in NB as in our examples. In general, the *Two-Step* modeling methodology has been justified through the two application examples.

## 5.5 Conclusion

In this chapter, we proposed a relatively simple and fast statistical method to model the frequency-dependent reflection coefficient $S_{11}(f)$ of NB variable antennas. The main idea of this method is to realize a separation of the variable “f” of $S_{11}(f)$ by means of identification techniques, prior to the extraction of statistical models with the HLARS-base PCE approach in a second stage. Two identification techniques were tested, each having its own advantages and draw-backs. Once the surrogate model has been constructed, we can reproduce the reflection coefficient over the whole frequency band of interest for any values of the random input parameters, producing considerable computational savings compared to full-wave EM simulations.
In the future work, we will take into account the causality of the antenna system. Some existing methods which permit to impose the causality should be studied, for example the algorithm based on the Hilbert transform that was proposed by F. M. Tesche [151]. This algorithm was already used in the lab previously to improve UWB antenna measurements in the frequency domain. This actually results somehow in “noise filtering” by imposing causality to the noise contribution, which is not necessarily causal in this case.
Chapter 6

# 6. Conclusions and Perspectives

## 6.1 Conclusions

This thesis is dedicated to the exploration of effective methodologies for quantitatively modeling random effects on soft or flexible antennas in fluctuating conditions (referred to as “variable antennas”). The work mainly covers three parts: parsimonious modeling of the *complete* FF radiated by the antenna under study; deterministic and statistical modeling of those random effects on the antenna’s *complete* FF; statistical modeling of those random effects on the antenna’s reflection coefficient $S_{11}(f)$.

Starting from the state of the art, we have reviewed the development of soft or flexible antennas and their promising applications in the context of IoT and WBANs. Soft or flexible antennas have prominent advantages such as flexibility, low-profile, low-cost, etc. over traditional rigid antennas, but they generally suffer from severe disturbances due to enormous intrinsic and/or external random uncertainties. This complex context of randomness leads to the widespread awareness of the urgency to quantitatively characterize their effects on variable antennas. Most of the existing work have focused on the modeling of certain FoMs which are only a partial description of the antenna’s behavior. Yet, to our knowledge, not too much effort has been made to model the antenna’s *complete* radiated field. A first attempt was recently presented in Rossi’s work [58] where the field modeling was carried out in a rather different context (in Near-Field instead of FF). However, this method is not necessarily always computationally efficient for some complex conditions. This fact motivates us to explore more appropriate methodologies for quantitatively and efficiently modeling the *overall* behavior of variable antennas.

Regarding the FF parsimony, we reviewed the principles of two scalar transform methods, i.e. the DFS and SSH, and a vector transform method, i.e. the VSH. We have compared their performance for representing scalar and vector-valued spherical functions, and are finally convinced that to accurately and parsimoniously represent vector-valued spherical functions such as the FF or its equivalent, the ATF, it is more appropriate to resort to the VSH method.

Then, the relation between the random variables (representing those random uncertainties) and the parsimonious ATF has been investigated explicitly. A canonical deterministic
approach based on the principle of parametric study has been firstly proposed. This simple approach proves to be suited for univariate or at most bivariate systems, but suffers from limited expressiveness for more complicated problems where higher number of random inputs and larger range of variability are considered.

After that, we have investigated an advanced statistical methodology based on the HLARS-PCE, which is more robust and computationally efficient for dealing with multivariate and higher variability range problems. This approach has been applied to a canonical deformable dipole (with 3 input parameters, i.e. stretching, bending and torsion) to demonstrate its effectiveness. Then, the methodology has been further improved/enhanced, in particular by taking into account the frequency behavior and by exploiting extreme parsimony. The improved approach has been tested on a realistic wearable textile patch antenna (with 4 geometrical random parameters and the frequency as an additional “random” parameter) for which accurate and robust surrogate models have been derived, which confirms its effectiveness in terms of efficiency, robustness and scalability. The surrogate models derived in such way have been implemented in several classical local propagation environments (realized with WINNER2/+ model) to demonstrate their practical utility for joint channel/antenna modeling and accurate radio links analysis.

A statistical methodology has also been proposed in order to model the frequency-dependent reflection coefficient $S_{11}(f)$ of variable antennas. This approach aims to parsimoniously represent the reflection coefficient by means of some identification techniques (e.g. polynomial fitting, rational functions), prior to statistical modeling through the HLARS-PCE. This approach has shown good effectiveness for NB antennas.

We believe that our proposed methodology can be beneficial for various types of analyses, notably when the *complete* knowledge (including polarization, amplitude and phase of the field, as well as the reflection coefficient) is mandatory for the antenna under study. Such application contexts comprise for example, joint antenna & directional channel modeling, antenna plugged-in asymptotic simulators (e.g. ray tracing), end-to-end radio links analysis, MIMO systems characterization, or beamforming designs, etc.

## 6.2 Perspectives

The methodologies that we have proposed in this thesis open the way for an *overall* modeling of variable antennas. However, there are still some aspects that should be further improved, both from a methodological and an applicative point of view.

### 1. Improvements of the modeling methodology

To model the frequency behavior of the ATF, we have chosen to consider the frequency "f" as an ordinary "random" input variable. However, as we have already pointed out in §
3.5.1, this wise and flexible choice is viable only for antennas whose frequency behavior is not too sensitive to the other input variables, for example, non-resonant antennas. Obviously, for resonant antennas, alternative solutions must be envisaged. The idea of “separation of variables” that we adopted for modeling the reflection coefficient can be quite instructive in such context. For example, we may firstly apply the Singularity Expansions Method (SEM) to expand the ATF onto a sum of rational functions which are characterized by their poles and frequency-independent residues; then, those residues as a function of the input variables (except the frequency) can be modeled efficiently with the HLARS-PCE method.

Regarding the ED, we have adopted the LHS algorithm for the initialization, and the “Maximin” technique for the iterative enrichment. However, there is no guarantee that the “Maximin” technique enriches the ED always in the most efficient way. More “intelligent” or “educated” strategies which take into account feedback from the performance evaluation of the currently constructed model should be further developed. For example, regions where the current model predicts poorly should be sampled more densely in the next iteration.

## 2. Generalization to more realistic applications

The implementation of our proposed methodologies can be generalized to more realistic and complex context. This would include e.g. considering other types of antennas (e.g. IFA), materials (e.g. conductive fabrics which are more flexible than copper foil) as well as fluctuating environments e.g. the human body. This work is planned to be realized in the near future. A preliminary study has already been carried out. The purpose is to work out a low-profile, compact and highly flexible UWB integrated IFA (IIFA) antenna that is an alternative candidate to other antenna types, notably suitable for on-body communications, hence paving the way for the ensuing statistical modeling of the *in situ* antenna in the vicinity of human bodies. The detailed description of this work can be found in the Annex.
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# Publications

## Journal papers:

[1] J. Du and C. Roblin, “Stochastic surrogate models of deformable antennas based on vector spherical harmonics and polynomial chaos expansions: application to textile antennas,” *IEEE Transactions on Antennas and Propagation*, vol. 66, n° 7, 2018.

[2] J. Du and C. Roblin, “Statistical modeling of disturbed antennas based on the polynomial chaos expansion,” *IEEE Antennas and Wireless Propagation Letters*, (online Sept. 2016), vol. 16, pp. 1843-1846, 2017.

## International conference papers:

[1] J. Du and C. Roblin, “An UWB Coplanar Waveguide Fed Integrated IFA Design for Wearable Communications,” in *EuCAP*, London, England, April 2018.

[2] J. Du and C. Roblin, “Statistical Modeling of the Reflection Coefficient of Deformable Antennas,” in *EuCAP*, Paris, France, March 2017.

[3] J. Du and C. Roblin, “Parametric modeling of deformable antennas based on the spherical modes expansion method,” in *EuCAP*, Davos, Switzerland, April 2016.
# Annex

**Abstract** — An UWB CPW-fed Integrated “IFA-like” design is proposed. This design has a much longer short-ended arm and a much narrower gap between the radiating arms and the coplanar ground plane. The bandwidth increase is achieved thanks to the superposition of multi-resonances induced by the joint effects of a strong coupling between the radiating arms and the ground plane and the excitation of both even and odd modes in the feedline. The design has been tested for two types of substrate, FR4 and denim, for which –10 dB bandwidths of more than 60 % and 87 % are achieved respectively. Body proximity effects are analysed for the denim IIFA by simulation. It shows that the antenna remains working well in the presence of human body thanks to its UWB feature. This extremely simple, low-profile and easily integrable UWB IIFA-like design can be potentially used for wearable communications (on-body/off-body), supporting multi-standards.

## I. Introduction

Since recent years, wearable antennas are gaining exceptional research attention due to their vital roles in wearable communications such as the Wireless Body Area Network, WBAN. Appropriate wearable antennas should verify good properties such as low-profile, light weight, easy integration, etc. and particularly, should be robust to body proximity effects such as detuning, impedance mismatch, losses, etc. which are well known since long ago.

For minimizing the body influence, a simple and intuitive idea is to isolate the antennas from the human body by introducing a shielding layer. Large-size backing ground planes are the most commonly used for this purpose, for example, in patch antennas [A-1]. These antennas usually achieve excellent isolation, however, they have disadvantages such as either narrow bandwidth (≈ 6 %), or moderate [A-2] to large size, reduced flexibility and are mostly suitable for off-body (rather than on-body) radio links. Other solutions include metamaterial based, Artificial Magnetic Conductor (AMC) based or ferrite based screening layers [A-3]-[A-6]. They usually allow to have a good isolation while keeping a satisfactory to a relatively large bandwidth, but their thickness is generally significant and the overall size is large, which is quite inconvenient for integration into clothes. An alternative approach to improve the *resilience* of antennas to body proximity effects (notably the detuning effect) is to increase their bandwidth (BW) significantly – instead of resorting to isolation techniques (which often increase their complexity – design, size, cost, etc.). Of course, this approach does not prevent antenna/body Near-Field (NF) coupling effects, inducing extra losses and noticeable modifications of both NF and Far-Field (FF) patterns. In this context, one of the design objectives – considering as early as possible the body proximity effects – is notably to
“monitor” losses (trying to maximize the total efficiency as much as possible), in order to achieve an interesting compromise between performance and complexity suitable for some applications. Thereby, planar ultra-wideband (UWB) antennas are interesting candidates for wearable communications. The above-mentioned body proximity effects still exist for UWB antennas, but their bandwidth is so wide that the possible detuning effects appear somehow more “globally” regarding the matching over the BW of interest (although for multi-resonant antennas each resonance can be affected). Therefore, possible abrupt radio link outage due to the collapse of the total efficiency which often occurs for narrow band antennas in the vicinity of a human body (when the detuning is such that the mismatch becomes extremely severe) is generally avoided for UWB antennas [A-7].

Among various antenna types, Coplanar Waveguide fed integrated Inverted-F Antennas (CPW-fed IIFAs) present interesting features such as large bandwidth as well as easy integrability; therefore, it is a quite suitable candidate for wearable communications. In [A-8], a folded CPW-fed IIFA printed on FR4 substrate ($ε_r = 4.4, tanδ = 0.025, h = 1.6 mm$) is proposed. The radiating element of the antenna measures $25 \times 8 mm^2$, for an overall size including the ground plane as large as $50 \times 60 mm^2$. This antenna achieves a -10 dB bandwidth of about 20% around 2.045 GHz. In [A-9], a general design of a wideband CPW-fed IIFA, based on a design proposed in [A-7], is presented. The shorted end and open end horizontal strips being close enough to the CPW ground plane “patches”, their coupling is strong and they can be viewed as asymmetrical coplanar strip (ACPS) [A-10]. In addition, the IFA is asymmetrically fed by the CPW which induces an additional resonance of the ground plane (explicable by the theory of characteristic modes). This resonance is close enough to the classical IFA resonance so that the bandwidth is increased to more than 40% for a FR4 based IIFA and to about 25% for a denim ($ε_r = 1.7, tanδ = 0.05, h = 0.76 mm$) based IIFA.

Herein, we propose a new UWB small size CPW-fed “IIFA-like” design, as shown in Fig. 1. The design has quite similar form as classical IIFAs except that the short-ended arm is much longer and the gap between the radiating element and the coplanar ground plane is much narrower than conventionally. This special choice of the geometry leads to different mechanisms of radiation compared to classical IIFAs (hence the terminology “IIFA-like”). Indeed, it allows to create strong coupling between the radiating arms and the coplanar ground plane, and also to excite both the even mode (i.e. the CPW fundamental mode) and the odd mode (i.e. the asymmetrical slot-line mode) within the feedline. The combination of these factors results in close multi-resonances leading to a very significant increase of the bandwidth. We applied the new design with both FR4 and denim substrates. Full-wave simulations are performed for both antennas, and measurements are also carried out for the FR4 based IIFA. Results are presented and analysed below. It shows that the proposed design allows to achieve ultra-wide band for both antennas. Even in the presence of human body, the
denim based IIFA can maintain quite good performance: the reflection coefficient $|S_{11}|$ is below -6.5 dB over a bandwidth of about 3 GHz, which remains a completely acceptable matching level, avoiding any dramatic radio link outage, as pointed out in the above paragraph. This compact UWB CPW-fed IIFA design can be suitable for wearable applications.

<figure><img src="image_73.png" /><figcaption>Fig. 1. Geometry of the proposed CPW-fed IIFA design.</figcaption></figure>

## II. CPW-Fed IIFA on FR4 Substrate

### a. Design of the antenna

In the first stage, the proposed antenna (Fig. 1) was designed with a commonly used FR4 substrate (relative permittivity $ε_r = 4.4$, loss tangent $tanδ = 0.025$ and thickness $h = 1.6$ mm). Dimensions are chosen for operation in the ISM band at 2.45 GHz. The radiating element is a horizontal strip which is fed at the middle by a 50 Ω CPW feedline (linewidth $W = 6$ mm, slot width $s = 0.6$ mm, length $W_g = 21$ mm), short-ended to the left (arm length $L_1 = 19$ mm) and open-ended to the right (arm length $L_2 = 16$ mm). The width of the strip is $W_{strip} = 1$ mm and the gap between the horizontal strip and the CPW ground plane is very narrow: $g = 1$ mm. The CPW feedline is chamfered at the edge to facilitate the connection to a 50 Ω SMA connector by welding. The overall size of the antenna is $L_s × W_s = 50$ mm × 26 mm.

### b. Simulation and measurement results

The proposed antenna structure was firstly studied using the full-wave simulator CST® MWS. Note that the adaptive meshing functionality has been used for all simulations, so that they can be considered reliable and accurate. Then, a prototype was fabricated and measurements were carried out in a well-conditioned anechoic chamber. Fig. 2 shows the
reflection coefficient $|S_{11}|$ from simulation (blue solid line) and measurement (red dash line). We can see that the –10 dB impedance bandwidth is 61.3 % (1.835 GHz – 3.455 GHz) in simulation and 63.3 % (1.89 – 3.64 GHz) for the measurement – more than 10 % larger than that obtained in [A-9].

<figure><img src="image_74.png" /><figcaption>Fig. 2. Simulated and measured reflection coefficient $|S_{11}|$ of the proposed FR4 based CPW-fed IIFA.</figcaption></figure>

<figure><img src="image_75.png" /><figcaption>Fig. 3. Simulated and measured realized gain $G_r$ in principle cut-planes of the proposed FR4 based CPW-fed IIFA: (a) $G_{r\theta}$ in yOz and xOy planes; (b) $G_{r\phi}$ in xOz and xOy planes.</figcaption></figure>

In Fig. 3 the simulated and measured realized gain patterns ($G_r$) in principal cut-planes for both $\theta$ and $\phi$ polarizations are compared. It can be seen that measurements and simulations are in relatively good agreement. Fig. 4 (a) shows the simulated 3D radiation pattern of the antenna at 2.45 GHz. A maximal $G_r$ of 2.3 dBi is observed. We also point out that the antenna radiates significantly towards both the perpendicular broadside (+x) and the “end-fire” (+z) – i.e. in the antenna plane – directions, implying that both off-body and on-body communications can be supported. Fig. 4 (b) shows that the simulated total efficiency $\eta_t$ ($\eta_t = \eta_e(1-|S_{11}|^2)$ with $\eta_e$ the radiation efficiency) is between 70 % and 93 % over the whole –10 dB band.
<figure><img src="image_76.png" /><figcaption>Fig. 4. (a) 3D radiation pattern at 2.45 GHz; (b) total efficiency, of the proposed FR4 based CPW-fed IIFA in free space.</figcaption></figure>

The ultra-wide band is the result of superposition of multi-resonances close to each other. In fact, it can be seen from Fig. 5 that three resonances or anti-resonances (at which $Imag(Z_{in}) = 0$) are located within the –10 dB input BW (at 2.1, 3.05 and 3.36 GHz) and two others are very close (at 1.78 and 3.99 GHz). Conventional IIFA structures present generally only one resonance; in [A-9] an additional close resonance is created thanks to the technique already described, notably by coupling the CPW feedline and the ground plane. A thorough inspection of the current and E-field distributions allows to interpret the obtained multiple resonances as a multi-mode behaviour.

<figure><img src="image_77.png" /><figcaption>Fig. 5. Impedance ($Z_{in}$) of the FR4 based CPW-fed IIFA.</figcaption></figure>

Fig. 6 (a) shows the surface current density at the resonance of 3.05 GHz. We can see that there exists a strong coupling between the radiating arms and the CPW ground plane due to the high length of the short-ended arm and the narrow inter-gap. Fig. 6 (b) shows the E-field distribution (for two different phase values) in the vertical cut-plane at the SMA pin point. It can be seen that both the even mode and the odd mode are excited “in parallel” within the feedline. The same phenomenon has also been observed for almost all other resonances, except the one at 2.1 GHz for which only the fundamental even mode is excited. The latter
can be considered as the “fundamental IIFA mode”. In other words, our proposed design seems to behave as a parallel combination of a classical CPW-fed IIFA and an asymmetrical coplanar slot-line-fed IIFA. However, as the antenna is fed symmetrically, contrary to the design proposed in [A-7] or [A-9], no characteristic mode of the ground plane is excited.

<figure><img src="image_78.png" /><figcaption>Fig. 6. (a) Surface current density distribution; (b) E-field distribution, in the vertical cut-plane at the SMA pin point for the FR4 based CPW-fed IIFA (at 3.05 GHz).</figcaption></figure>

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

#### a. Design of the antenna

In the second stage, we applied the same principle for designing an UWB CPW-fed IIFA on denim ($ε_r$ =1.7, tanδ = 0.05, h = 0.85 mm). The dimensions taken are comparable to previous values: Ls = 50 mm, Ws = 25.5 mm, Wg = 20.5 mm, L1 = 19.4 mm, L2 = 15.6 mm, Wstrip = 1.2 mm, g = 1.3 mm, W = 6 mm, s = 0.6 mm. In particular, we took exactly the same dimensions for the CPW feedline after analysing the trade-off between the impedance matching and the antenna size. Actually, the design of a 50 Ω CPW feedline on denim would require a very high ratio of W/s, corresponding to either an excessively large strip or very narrow slots. However, none of them would be favourable for the practical fabrication and integration. Keeping the same dimensions increases the characteristic impedance to about Zcpw ~ 67 Ω for the Denim version. Nevertheless, we'll see next that this value of Zcpw operates an impedance transformation favourable to the overall performance of the proposed antenna.
<figure><img src="image_79.png" /><figcaption>Fig. 7. Simulated reflection coefficient $|S_{11}|$ of the proposed denim based CPW-fed IIFA in free space and on the phantom arm at different distances.</figcaption></figure>

b. Simulation results

In Fig. 7 the blue solid line shows the simulated reflection coefficient $|S_{11}|$ of the denim based IIFA in free space. It can be seen that the –10 dB input bandwidth is 2.99 GHz (1.92 GHz – 4.91 GHz), i.e. a relative BW of 87.6 %, which is very high for such kind of antenna, as far as we know. Fig. 8 (a) shows the simulated 3D radiation pattern at 2.45 GHz. A maximal $G_r$ of 2.24 dBi is obtained. The radiation pattern is comparable to the previous one so that the same comments still hold with regards to on and off body usage. Fig. 8 (b) shows that the simulated total efficiency is between 70 % and 93 % over the whole operating band.

<figure><img src="image_80.png" /><figcaption>Fig. 8. (a) 3D radiation pattern at 2.45 GHz; (b) total efficiency, of the proposed denim based CPW-fed IIFA in free space.</figcaption></figure>

Fig. 9 illustrates the multi-resonances phenomenon: four resonances are located in the passband (at 2.15, 2.29, 2.855 and 4.39 GHz) and two other ones in the vicinity (at 1.75 and 5.05 GHz), consequently resulting in a widely enhanced bandwidth. Similarly to the FR4 substrate case, analyses on the current and E-field distributions reveal the reason for the induced multi-resonances: a combined effect of, 1) a strong coupling between the radiating
arms and the ground plane, and 2) the excitation of both even mode (for all the 6 resonances) and odd mode (for the resonances at 2.15, 2.29 and 4.39 GHz) within the feedline.

<figure><img src="image_81.png" /><figcaption>Fig. 9. Impedance (Zin) of the denim based CPW-fed IIFA.</figcaption></figure>

<figure><img src="image_82.png" /><figcaption>Fig. 10. Homogeneous phantom: (a) “IIFA – phantom” view; (b) dispersion law of the effective medium of the phantom.</figcaption></figure>

### c. Test on a body phantom

The proposed denim based IIFA can be easily integrated into clothes for WBAN communications. To verify its *in situ* performance in the presence of human body, we carried out simulation tests with a canonical homogeneous phantom, as shown in Fig. 10 (a). The dispersion law (Fig. 10 (b)) of its effective medium is chosen to follow the measured values of a real body phantom used for experiments at our laboratory [A-11] (it is demonstrated in this article – amid others – that such simplified homogeneous phantom models designed with canonical geometries can be used satisfactorily in place of heterogeneous anthropomorphic phantoms for *external* electromagnetic problems). The antenna was placed on the phantom arm, near the shoulder, at a distance *d* of 10 mm, 5 mm and 2 mm.
In Fig. 7, the dash lines illustrate the evolution of the reflection coefficient as the distance decreases, which calls for the following comments. First, the low cut-off frequency decreases because the phantom behaves as an “additional substrate layer” for the antenna. Given that the phantom has a much higher effective permittivity than denim, the antenna equivalent electrical size becomes larger. Second, the input impedance is affected by the antenna-body coupling, inducing a degradation of the reflection coefficient $|S_{11}|$ over a part of the band. However, the matching remains acceptable (the $|S_{11}|$ remaining below –6.5 dB for the worst case), avoiding any dramatic radio link outage as pointed out in the introduction. Third, the BW is enlarged because the phantom operates as a dissipative load lowering the antenna quality factor. Meanwhile, the total efficiency of the antenna is decreased to values over the input band between 30 % and 55 %, 18 % and 35 %, 9 % and 17 % respectively for the 10, 5, 2 mm cases. These preliminary tests with a body phantom justify globally the applicability of the proposed CPW-fed IIFA for wearable communications.

Note finally that this antenna can be used for multi-standard/multi-band applications as it covers, when used on the body, the GSM 1800, UMTS, WiFi (2.45 GHz and possibly “5 GHz”), LTE/LTE-A 2100 and 2600, and WiMAX (2.5 GHz, 3.5 GHz and possibly 5.8 GHz) bands.

## IV. Conclusion

An extremely simple, compact, and small UWB CPW-fed IIFA design is investigated. A –10 dB bandwidth of more than 61 % and 87 % is achieved for FR4 and denim based realizations respectively. Tests with a phantom show that the body proximity effects are relatively moderate for the denim based IIFA. This antenna design can be suitable for wearable applications, including on-body and off-body communications and/or can support multi-standard applications. At the present time, the optimisation of the in-situ denim based IIFA (in order to improve the input matching and total efficiency) is still ongoing.

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# Résumé français

*Condensé du manuscrit en français*

## Analyse et modélisation statistique d'antennes souples ou flexibles dans des conditions fluctuantes

Jinxin DU

Soutenu : le 03/07/2018

Superviseur : 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
Introduction

Depuis la publication de la norme de communication mobile de première génération (1G)
(Système de Téléphonie Mobile Avancée, développé par Bell Laboratoire), les technologies
de communication sans fil ont connu une croissance fulgurante et ont considérablement
facilité la vie des gens. Aujourd'hui, les gens sont fortement dépendants de tous les types de
services sans fil tels que les réseaux cellulaires (2G/3G/4G), les systèmes mondiaux de
navigation par satellite, les réseaux locaux sans fil, et les réseaux corporels sans fil (par
exemple, Wi-Fi, Bluetooth, ZigBee). Le rythme de la révolution n'a jamais cessé et est encore
plus rapide que jamais. Nous sommes maintenant presque prêts à accueillir la technologie 5G
grâce à laquelle non seulement des personnes mais aussi de nombreuses "choses" (par
exemple des machines, des véhicules, des capteurs ...) seront connectées, formant un immense
« Internet of Things (IoT) ». L'IoT est censé contribuer à un monde "intelligent" (par exemple,
réseau intelligent, transport intelligent, soins de santé en temps réel, maison/ bureau/usine
intelligentes, etc.) et améliorer la qualité de vie des personnes considérablement. Avec
l'expansion généralisée des technologies sans fil, le besoin d'appareils électroniques sans fil
connaîtra certainement une croissance explosive dans les années à venir.

Étant l'interface unique pour communiquer avec l'extérieur, l'antenne est un composant
essentiel pour tous les appareils électroniques sans fil. Les succès dans l'industrie des
communications sans fil n'auraient pas été atteints sans les progrès continus de la technologie
des antennes. Ces dernières années, les antennes ont tendance à être hautement intégrées,
miniaturisées et organisées pour répondre à la demande de déploiement massif, de cycle de R
& D court, de faible coût et de haute performance. De plus, d'autres propriétés avantageuses
sont également indispensables dans des scénarios d'application spécifiques. En particulier,
dans le contexte des réseaux corporels sans fil où les antennes devraient être intégrées dans
des vêtements ou collées directement sur le corps humain, la souplesse ou la flexibilité leur
permettant de se conformer à diverses formes et tailles de corps serait vraiment utile.

Cependant, la souplesse ou la flexibilité implique une déformation géométrique (par
exemple, flexion, torsion, froissement ...) qui peut perturber de façon drastique le
comportement (par exemple, diagramme de rayonnement, impédance ...) des antennes. Les
antennes perturbées ne sont plus fixes ou statiques, mais variables ou dynamiques car leur
comportement varie de manière aléatoire au cours de l'opération. Outre les auto-déformations,
le caractère aléatoire externe de l'environnement proche (éventuellement fluctuant) peut
également perturber les antennes. Par exemple, la variation de la température et de l'humidité
peut entraîner des modifications des propriétés du matériau des antennes, telles que la
permittivité et la conductivité ; les environs proches, notamment le corps humain, pourraient
perturber la zone de proximité des antennes, entraînant une forte absorption, réflexion, diffusion ou l'effet conjoint de celles-ci. Cet effet devient remarquable lorsque les antennes deviennent miniaturisées.

De nos jours, avec l'utilisation répandue de l'électronique flexible (comme l'électronique textile) et la complexité croissante des environnements d'exploitation, la présence de aléas et d'incertitudes sur les antennes est plus réaliste que jamais dans presque tous les contextes d'application. Cela pose de sérieux problèmes dans la conception des antennes et dans l'analyse des liaisons radio. Comment caractériser la performance des antennes devient une préoccupation incontournable et attire de plus en plus l'attention. De plus, afin de garantir la fonctionnalité des antennes, la solution traditionnelle consiste à ajouter une marge de gain ad hoc pour rendre compte en gros des effets aléatoires pouvant être rencontrés dans le monde réel. Cependant, cela induit généralement un coût élevé et un gaspillage d'espace et/ou de ressources énergétiques précieux. Pour éviter des valeurs de marge excessives, il est indispensable de caractériser quantitativement et précisément le comportement in situ des antennes perturbées et d'attirer de plus en plus l'attention des ingénieurs des antennes.

Traditionnellement, la caractérisation d'une antenne déterministe classique (par exemple, antennes rigides ou de grande taille, ou clairement isolées, etc.) peut être réalisée au moyen de mesures ou de simulations. Cependant, cette approche devient inappropriée pour les problèmes d'antenne dans des conditions incertaines où un grand nombre de sources aléatoires interviennent. En effet, la combinatoire des situations peut être si explosive que l'exécution de mesures ou de simulations une par une serait trop longue et même physiquement impossible pour nous en ce moment. Dans telles circonstances, des approches de caractérisation plus efficaces seront requises de toute urgence.

Cette thèse vise à étudier des méthodologies efficaces pour analyser et modéliser quantitativement les effets aléatoires sur le comportement "global" des antennes souples ou flexibles dans des conditions fluctuantes. Ici le terme "global" signifie que le champ lointain rayonné (Far-Field en Anglais, FF) ainsi que le coefficient de réflexion des antennes considérées seront étudiés. Notamment, des méthodologies statistiques qui fournissent de la bonne précision tout en minimisant les coûts de modélisation sont présentées.

Le principe est d'essayer de ne réaliser qu'un nombre très limité de pré-évaluations (par mesure ou simulation) et d'en tirer pleinement parti pour extraire des modèles de substitution efficaces (aussi appelé Métamodèles) qui ensuite permettent de prédire le comportement des antennes considérées dans d'autres situations éventuelles. Les perspectives applicatives de tels metamodèles consistent à substituer des antennes réelles ou des simulations encombrantes lourdes dans différents types d'analyses de niveaux supérieurs, notamment lorsque la connaissance complète (y compris la polarisation, l'amplitude et la phase du champ) de l'antenne étudiée est obligatoire. Tels contextes d'application comprennent, par exemple, la
modélisation conjointe antenne/canal, puis celle du lien radio, l'intégration des antennes dans les simulateurs asymptotiques (lancé ou tracé de rayons), la caractérisation des systèmes Multi-Entrée & Multi-Sortie, ou la formation de faisceaux (illustré par Figure 10).

<figure><img src="image_83.png" /><figcaption>Figure 10: Modèles de substitution (Métamodèles) d'antenne pour l'analyse de niveau supérieur.</figcaption></figure>

## Organisation de la thèse

L'organisation de cette thèse est la suivante :

Le *Chapitre 2* présente l'état de l'art des antennes souples ou souples. Dans la première partie, les technologies de conception, les avantages et les grandes perspectives d'application, ainsi que les problèmes liés aux antennes souples ou souples, sont brièvement présentés. Dans la seconde partie, l'état de la technique de la modélisation des antennes est passé en revue et les objectifs que nous souhaitons atteindre dans cette thèse sont décrits.

Le *Chapitre 3* propose des méthodologies pour modéliser les FF ou ATF (Antenna Transfer Function) rayonnés par l'antenne. En premier lieu, la technique de parcimonie du champ basée sur l'expansion des harmoniques sphériques vectoriels (Vector Spherical Harmonics, VSH, en Anglais) est introduite. Ensuite, des méthodologies de modélisation déterministes ainsi que stochastiques sont proposées. Toutes les méthodologies ont un processus en deux étapes, c'est-à-dire la parcimonie suivie de l'extraction du métamodèle. La méthodologie déterministe est la combinaison de l'expansion VSH et de l'étude paramétrique conventionnelle ; en revanche, la méthodologie statistique est la combinaison de l'expansion VSH et de la méthode avancée d'expansion du chaos polynomial (Polynomial Chaos Expansion, PCE, en Anglais).

Le *Chapitre 4* étudie l'utilité des métamodèles d'antenne pour l'analyse des liaisons radio, où les performances effectives de l'antenne sont quantifiées en tenant compte des incertitudes inhérentes à l'antenne elle-même et à celles des environnements de propagation locaux stochastiques.
Le *Chapitre 5* présente une méthodologie statistique pour la modélisation du coefficient de réflexion complexe des antennes à bande étroite. Cette approche est divisée en deux étapes : identification et extraction du métamodèle en utilisant la méthode PCE.

Le *Chapitre 6* résume le travail réalisé au cours de cette thèse et présente quelques perspectives pour de futures recherches.

## Chapitre 2 : Etat de l'art

Ce chapitre présente le contexte de recherche de cette thèse. Premièrement, la technologie des antennes souples ou flexibles et leurs larges perspectives d'application sont brièvement présentées. Les problèmes liés aux caractéristiques stochastiques des antennes flexibles/souples sont également expliqués. Ensuite, la demande croissante de méthodes de modélisation efficaces est discutée. Enfin, l'état de la technique de la modélisation des antennes est passé en revue et les objectifs que nous souhaitons atteindre dans cette thèse sont décrits.

L'état de l'art est présenté en deux parties, l'une relative aux antennes flexibles et l'autre relative au traitement des fluctuations des paramètres d'entrée concourant à l'analyse des performances de telles antennes. La description d'antennes flexibles et de leurs caractéristiques (matériaux diélectriques et conducteurs) ainsi que leur utilisation dans un contexte des réseaux corporels (par exemple, Figure 11) permet de réaliser rapidement et concrètement où on souhaite en venir.

<figure><img src="image_84.png" /><figcaption>Figure 11: Antennes portables : (a) surveillance de la santé ; b) T-shirt intelligent pour les athlètes ; c) vêtements intelligents pour les pompiers ; d) des gants intelligents pour le contrôle à distance.</figcaption></figure>
En effet, les avantages liés à ces antennes (légèreté, flexibilité, procédures de fabrication, ...) sont associés à leur contrepartie : des aléas relatifs aux propriétés des matériaux inhomogènes employés, aux modifications géométriques, aux tolérances de fabrication, et enfin à la proximité du corps humain. Ces antennes sont donc significativement variables dans leur comportement pour justifier le caractère quasi-obligatoire d'une analyse stochastique.

Dans la seconde partie de ce chapitre, on évoque les méthodes habituelles de mise au point d'antennes par la simulation numérique. Cette dernière permet de réaliser une analyse paramétrique en vue de rechercher la configuration optimale au sens de certains facteurs de mérite (efficacité, bande passante, coefficient de réflexion, ...). Cependant, dans le contexte évoqué plus haut la combinaison de (trop) nombreux paramètres fluctuants rendent l'analyse paramétrique inaccessible. Le recours aux méthodes statistiques s'impose. Les différentes méthodes statistiques possibles ont été énumérés en partant de l'approche classique de Monte-Carlo. Les méthodes successivement évoquées sont ANOVA, analyse petits signaux et développement en série de Taylor associé, « Support vector machine », processus gaussiens (Kriging), réseaux de neurones et collocation stochastique notamment basée sur le chaos polynomial généralisé.

Des références sont données ayant pour application le domaine des antennes. Quelques arguments sont utilisés en vue de privilégier l'approche du chaos polynomial. Il est toujours difficile de comparer ces méthodes entre elles, aucune d'entre elles ne pouvant prétendre à être universelle. L'approche du chaos polynomial possède aussi ses limites.

# Chapitre 3 : Méthodologies de modélisation de l'ATF d'antennes variables

Ce chapitre volumineux comporte les principales contributions scientifiques de la thèse et décrit en particulier l'approche méthodologique innovante pour traiter de la question soulevée au chapitre précédent.

Le problème posé consiste à déterminer une estimation de la fonction de transfert d'une antenne qui dépend non seulement de la fréquence et des coordonnées du point d'observation mais d'un vecteur constitué d'un nombre plus ou moins important de variables aléatoires. Cependant, le champ électromagnétique (en se plaçant dans un système de coordonnées sphériques) s'écrit sous la forme d'une expansion de fonctions vectorielles de type harmoniques sphériques. Sa description est alors compacte ou parcimonieuse au moyen d'un nombre limité de coefficients complexes de pondération de cette base de fonctions (Figure
12). L'idée formulée consiste alors à extraire cette représentation parcimonieuse à partir de la simulation électromagnétique effectuée pour une réalisation particulière des variables aléatoires.

<figure><img src="image_85.png" /><figcaption>Figure 12: Principe de l'expansion VSH.</figcaption></figure>

Dans un second temps, les coefficients d'harmoniques sphériques sont eux-mêmes projetés sur une base de polynômes des variables aléatoires d'entrée. Ce sont donc ces coefficients de pondération d'harmoniques sphériques qui porteront l'information sur la propagation d'incertitude (Figure 13).

C'est ici l'idée phare de cette thèse dont la pertinence sera évaluée tout au long de ce chapitre. Ce principe de modélisation en deux étapes est illustré par Figure 14.

<figure><img src="image_86.png" /><figcaption>Figure 13: Projection des VSH modes sur des bases chaos polynomial.</figcaption></figure>
<figure><img src="image_87.png" /><figcaption>Figure 14: Construction des métamodèles en deux étapes.</figcaption></figure>

On distingue ensuite deux méthodes permettant de parvenir à cette représentation
polynomiale des variables aléatoires d'entrée. Si le nombre de variables aléatoires est faible,
une étude paramétrique menée sur chacune de ces variables permet d'adapter les coefficients
et l'ordre d'un polynôme pour chacune de ces variables de manière déterministe. Lorsque la
réponse d'intérêt est fonction d'un plus grand nombre de variables aléatoires, cette méthode
n'est plus réaliste et c'est alors une représentation sous la forme d'une base de polynômes du
chaos qui est proposée.

On expose dans le détail l'approche statistique qui forme le cœur de cette thèse.
L'historique du chaos polynomial ainsi que son principe est tout d'abord rappelé. Il s'agit de
décomposer la réponse d'intérêt sous la forme d'une somme pondérée de coefficients de
projection sur une base de polynômes orthonormés des variables d'entrée. Cette base est
orthogonale au sens d'un produit scalaire impliquant la loi de distribution des paramètres d'entrée. Ainsi, selon la nature de la distribution des variables aléatoires concernées, la famille de polynômes peut être choisie de manière optimale (exemple : polynômes de l'Hermite pour des processus gaussiens) selon le schéma d'Askey. On rappelle également qu'en cas de corrélations entre les variables aléatoires dont les distributions diffèrent, des transformations en des variables aléatoires équivalentes de distribution identiques peuvent être réalisées, ce qui rend l'application de la méthode très générale. Le calcul des coefficients peut être réalisé de différentes manières. Le processus retenu pour cette thèse est non intrusif et basé une approche de régression. Cette technique, appliquée sous sa forme classique est cependant limitée à l'analyse de quelques variables aléatoires. En effet, malgré la troncature opérée sur l'ordre des polynômes (troncature hyperbolique qui limite l'ordre total à un degré maximal P), le nombre de coefficients à estimer devient prodigieusement élevé et la méthode inexploitable car moins efficace qu'une simulation directe de type Monte Carlo. Pour contourner ce problème, différents auteurs ont proposé des solutions afin de sélectionner de manière itérative un nombre très limité de polynômes après troncature en privilégiant des degrés de polynômes peu élevés. C'est en particulier ce que permet la méthode LARS ou ses évolutions Hybrid-LARS, adaptative Hybrid LARS. C'est cette dernière option qui sera utilisée par la suite dans cette thèse. Elle aboutit à la sélection du meilleur ensemble de polynômes au sens du « leave-one-out error (LOOE) ».

<figure><img src="image_88.png" /><figcaption>Figure 15: Approche itérative pour la construction des métamodèles.</figcaption></figure>

On a proposé une approche itérative (Figure 15) pour extraire du métamodèle en minimisant au maximum le coût computationnel. Le métamodèle est ajusté au moyen d'un plan d'expérience dont l'objectif est d'obtenir une évaluation du modèle initial en certains
points. Ce plan d'expérience peut lui-même être enrichi après une première estimation du métamodèle.

Les performances des méthodes déterministes et stochastiques sont illustrées par un exemple canonique d'un dipôle à géométrie variable (Figure 16). En particulier, la méthode stochastique est évaluée pour 3 variables aléatoires. Après enrichissement du plan d'expérience à $N_{ED} = 260$ réalisations selon un tirage de type LHS (Latin Hypercube Sampling), l'ensemble des 30 métamodèles associés aux 30 modes (degré des harmoniques sphériques limité à $N_{VSH} = 3$) est alors estimé. Le métamodèle complet comporte environ 2000 coefficients polynomiaux pour une erreur de type LOOE de 4%.

<figure><img src="image_89.png" /><figcaption>Figure 16: Illustration d'un dipôle subissant différentes déformations : (a) dipôle d'origine ; b) dipôle étiré ; c) dipôle courbé ; (d) dipôle courbé et torsadé.</figcaption></figure>

A la suite de ce premier résultat, bien que positif, on ait cherché des améliorations potentielles. Plusieurs axes sont proposés et peuvent être employés simultanément. La fréquence peut être incorporée comme variable aléatoire si la réponse d'intérêt est suffisamment régulière vis-à-vis de ce paramètre. Une description plus parcimonieuse des modes sphériques peut être également adoptée, en ne retenant que les modes dominants (contribution significative à la puissance totale rayonnée). Plus encore, certains de ces modes dominants sont corrélés au sens de la corrélation de leur pondération. Dès lors l'examen préalable de la matrice de corrélation, permet de regrouper certains modes et donc de diminuer le nombre de métamodèles élémentaires.

La méthode stochastique initiale ainsi enrichie est appliquée au cas d'une antenne patch sur textile dans la bande 2.4 à 2.5 GHz pour un problème à 6 variables aléatoires uniformes (courbure, repliement, épaisseur du substrat, permittivité, fréquence), Figure 17. Il faut précise qu'il s'agit ici d'une approximation de la réalité puisque, notamment, ces variables sont supposées indépendantes les unes des autres. On suit une discussion concernant cette hypothèse ainsi qu'une analyse numérique permettant de conforter cette hypothèse d'indépendance. C'est évidemment une étape très délicate du processus et on comprend ici qu'il n'est pas aisé de détecter et quantifier d'éventuelles corrélations.
<figure><img src="image_90.png" /><figcaption>Figure 17: Antenne patch textile portable : (a) patch plan non déformé ; (b) patch subissant l'effet conjoint du froissement, de la flexion et de la variation d'épaisseur.</figcaption></figure>

Une analyse préliminaire conduit à ne plus considérer la hauteur comme variable aléatoire. Finalement, le métamodèle est ajusté pour 5 variables aléatoires. Le modèle final, résultat d'un bon compromis précision/complexité comporte 3217 coefficients scalaires, et est ensuite comparé à 2000 réalisations. L'erreur moyenne et maximale en terme du gain réalisé sont 0,53 et 2,2 dB (Figure 18 - 19). Le temps consommé par notre métamodèle est considérablement plus faible que celui quand on utilise simulation classique (150s vs 15 mins).

Les résultats reportés dans ce manuscrit confortent totalement la méthodologie en deux étapes (représentation parcimonieuse du rayonnement électromagnétique et chaos polynomial) qui permet d'aboutir à une représentation particulièrement compacte de la fonction de transfert de l'antenne. Les limites de la méthode sont également judicieusement mentionnées. La question de la connaissance a priori de la loi support est effectivement centrale et constitue à elle seule une voie de recherche future.

<figure><img src="image_91.png" /></figure>
<figure><img src="image_92.png" /><figcaption>Figure 19: Gr simulés par EM (colonne de gauche) et générés par modèle (colonne de droite) pour une configuration d'erreur maximale représentative.</figcaption></figure>

# Chapitre 4 : Métamodèles pour des analyses des liaisons radio

Dans ce chapitre, On rend compte d'une application originale du métamodèle développé au cours de cette thèse. Il s'agit d'évaluer le bilan de réception radio englobant la statistique du canal de propagation et son interaction avec le diagramme de rayonnement de l'antenne, Figure 20.

<figure><img src="image_93.png" /><figcaption>Figure 20: Illustration de la liaison radio d'un système de communication sans fil.</figcaption></figure>

Ce bilan de réception s'évalue classiquement selon le critère de gain effectif moyen. Ce gain est défini par la puissance reçue par l'antenne de réception sous test rapportée à la
somme des puissances qui auraient été collectées par deux antennes isotropes en polarisation horizontale et verticale. Cette grandeur intègre le gain de l'antenne physique mais également le canal de propagation décrit sous sa forme statistique éventuellement standardisée. Le modèle de canal exploité est de type multi-trajets avec station de base en polarisation verticale Winner 2/+.

Dans le contexte des antennes flexibles, significativement variables, l'introduction du comportement aléatoire des antennes semble justifiée. Afin de le démontrer dans le cas précis de l'antenne textile étudiée au chapitre précédent, on combine le métamodèle de cette antenne avec le calcul du gain effectif moyen. Concrètement, 1000 réalisations d'antennes sont produites à partir du métamodèle. Chacune de ces antennes est alors combinée avec 100 réalisations de canal différentes.

La statistique du gain effectif moyen peut alors être établie pour chaque réalisation d'antenne. Les résultats produits montrent qu'il existe une forte dispersion d'une réalisation d'antenne à l'autre par rapport à une antenne non déformée, comme monté par Figure 21. Ceci s'explique par la nature même de l'environnement traité. L'analyse est ensuite raffinée par sous-bandes de fréquence ou en restreignant le secteur angulaire azimutal des angles d'arrivée. Les dispersions observées initialement sont alors réduites.

<figure><img src="image_94.png" /></figure>
Figure 21: CDF du gain effectif moyen relatif de l'antenne textile patch dans différents environnements de propagation : lignes fines en couleur - CDF partielles ; ligne épaisse rouge - CDF global.

Pour conclure sur ce chapitre, la démonstration de l'utilité d'un tel modèle est claire et justifiée au regard des dispersions enregistrées lorsque le modèle de la liaison radio intègre la variabilité des antennes. Cette approche de modélisation conjointe antenne/canal peut être encore affinée pour pouvoir, par exemple, traiter le signal reçu. En particulier, si la phase est prise en compte, il est possible d'effectuer des simulations au niveau de certains systèmes, par ex. la couche physique de manière à tester différents schémas de modulation et à évaluer des indicateurs de performance du système tels que le taux d'erreur sur les bits. Ce type d'approche, plus élaboré et plus fin, qui mettrait en évidence l'intérêt de développer des modèles de substitution du champ rayonné complet (vecteur et phasé), reste un développement futur.

# Chapitre 5 : Modélisation statistique du coefficient de réflexion des antennes variables

Dans ce dernier chapitre, on traite de l'estimation du coefficient de réflexion d'une antenne variable. Tout d'abord, on signale que ce type de problème a déjà été traité par différentes méthodes (réseaux de neurones, processus gaussiens, Support Vector Machine et également par le chaos polynomial) en intégrant la fréquence comme variable aléatoire. A cette occasion, on énonce que ces méthodes sont en capacité de reproduire la réponse d'intérêt lorsque celle-ci possède un comportement non-linéaire (forte sensibilité de la réponse aux variations des paramètres aléatoires) mais au prix d'une certaine complexité.
<figure><img src="image_95.png" /><figcaption>Figure 22: Modélisation du coefficient de réflexion en deux étapes.</figcaption></figure>

On décide, compte tenu des difficultés, de ne pas considérer la fréquence comme variable aléatoire et de procéder de manière similaire à la description de la fonction de rayonnement d'antenne, en représentant dans le modèle la fréquence par une courbe de régression de type polynomiale ou rationnelle. Chacun des coefficients de régression fait alors l'objet d'une expansion de type chaos polynomial. Ce principe de modélisation en deux étapes est montré par Figure 22.

Deux approches sont appliquées pour l'antenne canonique et l'antenne textile. Il n'est pas possible de statuer sur l'avantage de l'une ou de l'autre mais les performances sont correctes au sens du seuil d'erreur toléré initialement. Il semble toutefois que pour l'antenne textile, il soit difficile de converger avec un modèle compact (notamment avec la seconde approche de Levy), les performances paraissent alors difficiles à garantir de manière systématique.

En bref, la première approche fonctionne bien tant pour le patch dipôle que pour le patch textile, tandis que la deuxième fonctionne bien pour le dipôle mais n'est pas totalement satisfaisante pour le patch. Les avantages de l'approche I sont sa robustesse et sa simplicité, mais ce n'est pas nécessairement l'approche la plus efficace en termes de parcimonie et de coût de calcul. De plus, la causalité, c'est-à-dire les relations de Bayard-Bode ou de Kramers-Kronig, n'est pas garantie. L'Approche II permet de mieux comprendre les modes de résonance inhérents aux antennes – grâce aux zéros/pôles dérivés des fonctions rationnelles, ce qui constitue un moyen naturel et efficace de caractériser un système invariant linéaire (LTI), notamment pour les dispositifs résonnants au NB comme dans nos exemples. En général, la méthodologie de modélisation en deux étapes a été justifiée par les deux exemples d'application.
# Chapitre 6 : Conclusions et perspectives

## Conclusions

En conclusion, cette thèse a été consacrée à l'exploration de méthodologies efficaces pour la modélisation quantitative des effets aléatoires sur les antennes souples ou flexibles dans des conditions fluctuantes. Les travaux couvrent principalement trois parties : la modélisation parcimonieuse du champ lointain (FF) rayonné par l'antenne considérée ; modélisation déterministe et statistique de ces effets aléatoires sur le FF de l'antenne ; modélisation statistique de ces effets aléatoires sur le coefficient de réflexion de l'antenne.

En partant de l'état de l'art, nous avons examiné le développement d'antennes souples ou souples et leurs applications prometteuses dans le contexte de l'IoT et des WBAN. Les antennes souples ou flexibles présentent des avantages notables tels que la flexibilité, le faible encombrement, le faible coût, etc., par rapport aux antennes rigides traditionnelles, mais elles souffrent généralement de perturbations graves dues à d'énormes incertitudes aléatoires intrinsèques et/ou externes. Ce contexte complexe d'aléa conduit à la conscience générale de l'urgence de caractériser quantitativement leurs effets sur les antennes variables. La plupart des travaux existants ont porté sur la modélisation de certaines caractéristiques qui ne sont qu'une description partielle du comportement de l'antenne. Cependant, à notre connaissance, peu d'efforts ont été faits pour modéliser le champ rayonné complet de l'antenne. Un premier essai a récemment été présenté dans les travaux de M. Rossi où la modélisation de champ a été réalisée dans un contexte assez différent (dans le champ proche au lieu de FF). Cependant, cette méthode n'est pas nécessairement toujours efficace pour certaines conditions complexes. Ce fait nous pousse à explorer des méthodologies plus appropriées pour modéliser quantitativement et efficacement le comportement global des antennes variables.

En ce qui concerne la parcimonie du FF, nous avons examiné les principes de deux méthodes de transformation scalaire, à savoir le DFS et le SSH, et une méthode de transformation vectorielle, à savoir le VSH. Nous avons comparé leurs performances pour représenter les fonctions sphériques scalaires et vectorielles et sommes finalement convaincus que pour représenter avec précision et parcimonie des fonctions sphériques vectorielles telles que le FF ou son équivalent, l'ATF, il est préférable de recourir au VSH méthode.

Ensuite, la relation entre les variables aléatoires (représentant ces incertitudes aléatoires) et l'ATF parcimonieuse a été explicitement étudiée. Une approche déterministe canonique basée sur le principe de l'étude paramétrique a d'abord été proposée. Cette approche simple s'avère adaptée aux systèmes univariés ou au maximum bivariés, mais souffre d'une expressivité limitée pour les problèmes plus complexes où l'on considère un plus grand nombre d'entrées aléatoires et une plus grande gamme de variabilité.
Après cela, nous avons étudié une méthodologie statistique avancée basée sur le HLARS-PCE, qui est plus robuste et plus efficace sur le plan des calculs pour traiter les problèmes de gamme multivariée et de variabilité plus élevée. Cette approche a été appliquée à un dipôle déformable canonique (avec 3 paramètres d'entrée, à savoir l'étirement, la flexion et la torsion) pour démontrer son efficacité. Ensuite, la méthodologie a été encore améliorée/améliorée, notamment en prenant en compte le comportement en fréquence et en exploitant l'extrême parcimonie. L'approche améliorée a été testée sur une antenne patch textile réaliste (avec 4 paramètres aléatoires géométriques et la fréquence comme paramètre « aléatoire » supplémentaire) pour lesquels des métamodèles précis et robustes ont été obtenus, ce qui confirme son efficacité en termes d'efficacité, robustesse et évolutivité. Les métamodèles ainsi obtenus ont été mis en œuvre dans plusieurs environnements de propagation locaux classiques (réalisés avec le modèle WINNER2+) pour démontrer leur utilité pratique pour la modélisation conjointe canal/antenne et l'analyse précise des liaisons radio.

Une méthodologie statistique a également été proposée pour modéliser le coefficient de réflexion en fonction de la fréquence des antennes variables. Cette approche vise à représenter de manière parcimonieuse le coefficient de réflexion au moyen de certaines techniques d'identification (par exemple, fitting polynomial, fonctions rationnelles), avant la modélisation statistique par le biais du HLARS-PCE. Cette approche a montré une bonne efficacité pour les antennes NB.

Nous pensons que notre méthodologie proposée peut être bénéfique pour différents types d'analyses, notamment lorsque les connaissances complètes (notamment la polarisation, l'amplitude et la phase du champ, ainsi que le coefficient de réflexion) sont obligatoires pour l'antenne étudiée. De tels contextes d'application comprennent, par exemple, la modélisation conjointe d'antennes et de canaux directionnels, les simulateurs asymptotiques enfichables dans l'antenne (par exemple, ray tracing), l'analyse de liaisons radio de bout en bout, la caractérisation de systèmes MIMO ou les conceptions de formation de faisceau, etc.

## Perspectives

En ce qui concerne les perspectives de recherche, d'une part, nous étudions des antennes textiles plus flexibles. Cela comprend trois parties : premièrement, la technologie de fabrication, pour laquelle nous n'avons pas trop d'expérience, et nous testons maintenant certains matériaux textiles conducteurs disponibles dans le commerce, tels que les deux prototypes présentés par Figure 23; d'autre part, nous devrions développer des méthodes plus fiables pour la quantification de la distribution des paramètres aléatoires, en particulier lorsqu'il existe de fortes corrélations entre les paramètres aléatoires. Et nous devrions faire très attention lors de l'hypothèse sur la distribution conjointe, ce qui n'est évidemment pas
une tâche facile. Troisièmement, appliquer l'approche de modélisation statistique pour l'extraction du modèle de substitution.

<figure><img src="image_96.png" /><figcaption>Figure 23: deux types d'antennes textiles sous investigation.</figcaption></figure>

D'un autre côté, nous allons inclure les effets aléatoires de proximité corporelle dans notre modèle de substitution, ce qui est particulièrement indispensable pour les antennes à profil bas qui n'ont pas de plan de masse de protection, telles que les antennes IIFA, Figure 24.

<figure><img src="image_97.png" /><figcaption>Figure 24: Modélisation de l'effet du corps sur une antenne de type IIFA.</figcaption></figure>
<figure><img src="image_98.png" /></figure>
## Analyse et modélisation statistique d'antennes souples ou flexibles dans des conditions fluctuantes

**RESUME** : Avec le déploiement massif des électroniques flexibles dans le contexte de l'Internet des objets et des réseaux corporels sans fil, la demande d'antennes souples ou flexibles est amenée à se développer. Contrairement aux antennes traditionnelles non déformables ou clairement isolées dont les caractéristiques sont déterministes, les antennes souples/flexibles, en particulier lorsqu'elles sont mises en œuvre dans des conditions fluctuantes, sont sujettes à divers types de variabilité par essence aléatoires : déformations géométriques, variations dans les propriétés des matériaux, interactions avec l'environnement proche (par ex. le corps humain), etc. Ces incertitudes aléatoires peuvent perturber significativement les performances in situ des antennes et doivent être prises en compte quantitativement. La caractérisation de ces effets aléatoires ne peut être efficacement réalisée au moyen de l'approche conventionnelle de "simulation électromagnétique – prototypage – mesure" en raison de son coût très élevé (pour chacune de ces étapes). Dans ce contexte, des modèles de substitution qui permettent de prédire efficacement ces effets aléatoires avec une bonne précision sont excessivement utiles. Ce mémoire de thèse se concentre sur l'étude des méthodologies efficaces dédiées à la construction des modèles de substitution pour les antennes subissant des déformations aléatoires. Deux objectifs principaux sont visés. D'une part, décrire quantitativement et parcimonieusement le comportement complet des antennes, c'est-à-dire le champ lointain rayonné ainsi que le coefficient de réflexion. D'autre part, proposer des méthodologies efficaces, fiables et polyvalentes pour l'extraction de modèles d'antennes à partir d'un nombre limité de pré-évaluations. Notamment, une méthodologie statistique fondée sur le chaos polynomial a été proposée et son efficacité a été démontrée en l'appliquant tout d'abord à une antenne canonique subissant divers types de déformations, puis à une antenne imprimée sur substrat en tissu dans un cadre applicatif plus réaliste. L'intérêt des modèles de substitution ainsi développés est de pouvoir disposer de réponses « d'antennes aléatoires » pour un très faible coût de calcul. Du point de vue applicatif, cela permet de les utiliser dans divers types d'analyses de niveaux supérieurs telles que la modélisation conjointe antenne/canal, puis celle du lien radio, l'intégration des antennes dans les simulateurs asymptotiques (lancé ou tracé de rayons), la caractérisation des systèmes Multi-entrée & Multi-sortie, ou la formation de faisceaux.

**Mots clés** : antennes souples/flexibles, antennes déformables, modèles de substitution, modélisation statistique d'antennes, harmoniques sphériques vectorielles, polynômes de chaos.

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

**ABSTRACT:** With the massive deployment of flexible electronics in the context of Internet of Things (IoT) and wireless body area networks (WBANs), the demand for soft or flexible antennas is substantially growing. Unlike traditional non-deformable or clearly isolated antennas whose characteristics are deterministic, soft/flexible antennas, especially when they are implemented in fluctuating conditions, are subject to various types of variability by essence random: geometric deformations, variations in the material properties, interactions with the surrounding environment (e.g. the human body), etc. These random uncertainties can significantly disturb the *in situ* performance of antennas and must be taken into account quantitatively. The characterization of these random effects cannot be effectively achieved using the conventional "electromagnetic simulation - prototyping - measurement" approach because of its very high cost (for each of these steps). In this context, surrogate models which can effectively predict these random effects with good precision are extremely useful. This dissertation focuses on the investigation of efficient methodologies dedicated to the construction of surrogate models for antennas undergoing random uncertainties. Two main objectives are targeted. On the one hand, describe quantitatively and parsimoniously the *overall* behavior of the antennas, that is to say, the *complete* radiated far-field (FF) as well as the reflection coefficient. On the other hand, propose efficient, reliable and versatile methodologies for the extraction of antenna models from a limited number of pre-evaluations. In particular, a statistical methodology based on polynomial chaos has been proposed and its effectiveness has been demonstrated by applying it firstly to a canonical antenna undergoing various types of deformations, then in a more realistic application framework, to an antenna printed on a fabric substrate. The benefit of the surrogate models thus developed is to have access to the responses of the "random antennas" at a very low calculation cost. From the application point of view, such surrogate models can be used in various types of higher-level analysis such as joint antenna/channel modeling, radio links analysis, antenna plugged-in asymptotic simulators (e.g. ray launching or tracing), Multi-Input & Multi-Output (MIMO) systems characterization, or beamforming.

**Keywords:** soft/flexible antennas, deformable antennas, surrogate models, statistical modeling, vector spherical harmonics, polynomial chaos.