# Statistical Modeling of the Human Body as an Extended Antenna

Thomas Wilding41, Erik Leitinger4, Ulrich Muehlmann3 Klaus Witrisal41, This
research was partly funded by the Austrian Research Promotion Agency (FFG)
within the project UBSmart (project number: 859475). The financial support by
the Christian Doppler Research Association, the Austrian Federal Ministry for
Digital and Economic Affairs and the National Foundation for Research,
Technology and Development is gratefully acknowledged. 4Graz University of
Technology, Austria  3NXP Semiconductors, Austria  1Christian Doppler
Laboratory for Location-aware Electronic Systems<EMAIL_ADDRESS>
###### Abstract

In this paper we investigate the possibility of modeling a single antenna
alone and in close proximity to a physical object by means of discrete point
source scatterers. The scatter point model allows joint modeling of a physical
antenna and the human body as a single extended object with direction
dependent scattering coefficients for the scatter points. We introduce the
term extended antenna describing antenna and human body together. To
investigate the identifiability of the model parameters we make use of
ultrawideband channel measurements and accurate ground truth position and
orientation measurements obtained with an optical tracking system. By
comparing measurements of the antenna attached directly to the user with
measurements for the antenna without the user nearby, we show the shadowing
and scattering effects of the human body and the antenna.

###### Index Terms:

channel modeling, ultrawideband, off-body measurements.

##  I Introduction

For positioning applications, an accurate model of the received signal is
needed. This model should include any environment-related effects that can
affect an estimator of position related signal parameters from the received
signal. Such effects can be specular multipath components (SMCs) and a diffuse
multipath component (DMC) [1] or scattering objects such as mobile users [2]
or trees, e.g., for GNSS signals [3]. Especially blockage of the line-of-sight
propagation path can introduce a strong bias in delay estimates, a survey of
which is presented in [4]. This is especially true in off-body communications,
where the anchor or the agent are in close proximity to the user [5, 6]. The
effects of wearable antennas on narrowband signals were already investigated
in [7, 8] highlighting the strong shadowing effect introduced by the human
body. As shadowing alone is not sufficient to accurately model effects on
wideband signals, it is of interest to develop suitable channel models
incorporating both shadowing and dispersion in the delay-angle domain.

In [9] we have shown that a user equipped with an agent can be modeled as an
extended object (EO) using a measurement based, empirically chosen scattering
point distribution and a gain function. In this paper we investigate the
possibility of calibrating the statistical model and show that the joint
description of scattering object and non-ideal antenna is implicitly possible
with the proposed model from [9]. We model agent and user as one EO under the
term extended antenna (EA), adding components via point source scatterers
jointly modeling the dispersive effects of antenna and user. The general setup
under investigation is shown in Fig. 1, with the EA denoted by $\mathcal{S}$
and the reference direction for the angle of arrival (AOA) $\varphi$ included
as well.

Figure 1: Representation of an extended antenna $\mathcal{S}$ modeling a user
in close proximity to the receive antenna at $\bm{p}=[x,y]^{\mathsf{T}}$ with
a signal impinging from an anchor located at
$\bm{a}=[x_{a},y_{a}]^{\mathsf{T}}$. Scattering points are indicated as
$\bm{q}=[x_{q},y_{q}]^{\mathsf{T}}$. An exemplary received signal $r(t)$ for
the receiver at on-body position $\mathsf{C}$ is shown for visualization
purpose.

For estimation and tracking of the shape of EOs, a large number of algorithms
exist [10, 11, 12, 13], which usually require a large number of (spatial)
measurements (e.g., obtained using LIDAR sensors [11]) to accurately estimate
the object’s state. As highly accurate spatial measurements using radio
signals require large antenna arrays and a large measurement bandwidth, state
of the art algorithms are not directly applicable. This paper shows that a
suitable measurement setup allows estimation of scattering points to calibrate
the model.

The rest of the paper is structured as follows: Section II describes the
system and signal model, Section III introduces the measurement campaign and
the estimation procedure. Results are shown in Section IV and Section V
concludes the paper.

##  II System Model and Signal Model

###  II-A System Model

We consider a single anchor at known position
$\bm{a}=[x_{a},y_{a}]^{\mathsf{T}}\in\mathbb{R}^{2}$ and an agent at position
$\bm{p}=[x,y]^{\mathsf{T}}\in\mathbb{R}^{2}$, each equipped with a single
antenna (see Fig. 1). The EA $\mathcal{S}=\\{\bm{\mu},\bm{\Sigma},Q\\}$ is
described by mean position
$\bm{\mu}=[x_{\mathcal{S}},y_{\mathcal{S}}]^{\mathsf{T}}\in\mathbb{R}^{2}$
with the matrix $\bm{\Sigma}\in\mathbb{R}^{2\times 2}$ denoting the shape in
terms of a covariance matrix. Scattering points
$\bm{q}_{j}=[x_{j},y_{j}]^{\mathsf{T}}\in\mathbb{R}^{2}$ belonging to the EA
$\mathcal{S}$ are modeled as a marked Poisson point process (MPPP) denoted by
$Q=\\{(\bm{q}_{1},\beta_{1}),(\bm{q}_{2},\beta_{2}),\dots\\}=\\{(\bm{q}_{j},\beta_{j})\\}$
where $j$ is an index solely used to distinguish points and
$\beta_{j}\in\mathbb{C}$ is the complex mark of the corresponding scattering
point $\bm{q}_{j}$. Per definition of an MPPP, the number of points in $Q$ is
Poisson distributed with mean $N(Q)$, with the points spatially distribution
in the EA shape according to an intensity function $\rho(\bm{q})$. A thorough
treatment of point process theory can be found in [14]. The extension of the
model to 3-dimensions is straightforward.

###  II-B Signal Model

The signal at the agent is modeled to consist of a deterministic and a
stochastic part. The former contains $K$ specular multipath components (SMCs),
e.g., modeled by an image source model [15]. The latter contains a scattered
component (SC) and a diffuse multipath component (DMC) modeling the
environment randomness. The SC is modeled by the MPPP $Q$ described above and
the DMC by a random process $\nu(\tau)$ with a suitable delay-angle power
spectral density, see, e.g., [1, 16]. We are only interested in effects on the
line-of-sight (LOS) path between anchor and agent, i.e., without loss of
generality we set $K=1$ and neglect the DMC. This gives the radio channel as

$\displaystyle
h(\tau;\bm{p})=\alpha\delta(\tau-\tau(\bm{p}))+\sum_{j}\beta_{j}\delta(\tau-\tau(\bm{q}_{j})-\tau(\bm{p}-\bm{q}_{j}))$

where $\alpha\in\mathbb{C}$ is the complex LOS amplitude and
$\beta_{j}\in\mathbb{C}$ the complex scattering coefficient of scattering
point $\bm{q}_{j}$. The path delays are
$\tau(\bm{p})=\frac{1}{c}\|\bm{p}-\bm{a}\|$ where $\|\cdot\|$ denotes the
vector norm and $c$ is the speed of light.

The signal $r(t)$ received at the agent when transmitting a baseband pulse
$s(t)$ at carrier frequency $f_{c}$ is given by

$\displaystyle r(t)$ $\displaystyle=\int s(t-\tau)e^{-i2\pi
f_{c}\tau}h(\tau;\bm{p})d\tau+n(t)$ (1) $\displaystyle=\alpha
s(t;\bm{p})+\sum_{j}\beta_{j}s(t;\bm{p},\bm{q}_{j})+n(t)$ (2)

where $s(t;\bm{p})=s(t-\tau(\bm{p}))e^{-i2\pi f_{c}\tau(\bm{p})}$ is the pulse
received at position $\bm{p}$ and
$s(t;\bm{p},\bm{q})=s(t-\tau(\bm{p}-\bm{q});\bm{q})e^{-i2\pi
f_{c}\tau(\bm{q}-\bm{q})}$ the pulse received at $\bm{p}$ after scattering at
$\bm{q}$. The measurement noise $n(t)$ is modeled as zero-mean complex
additive white Gaussian noise (AWGN) with double-sided power spectral density
$\frac{N_{0}}{2}$. Synchronous sampling of (2) with frequency
$f_{s}=\frac{1}{T_{s}}$ over an observation interval $T$ and stacking the
samples in the vector
$\bm{r}\triangleq[r(0),r(T_{s}),\dots,r((N-1)T_{s})]^{\mathsf{T}}\in\mathbb{C}^{N}$
and $N=T/T_{s}$ yields the discrete-time signal model

$\displaystyle\bm{r}$
$\displaystyle=\alpha\bm{s}(\bm{p})+\sum_{j}\beta_{j}\bm{s}(\bm{p},\bm{q}_{j})+\bm{n}~{}\in\mathbb{C}^{N}$
(3)
$\displaystyle\triangleq\alpha\bm{s}(\bm{p})+\bm{S}(\bm{p},\bm{Q})\bm{\beta}+\bm{n}$
(4)

where $\bm{\beta}=[\beta_{1},\dots,\beta_{J}]^{\mathsf{T}}\in\mathbb{C}^{J}$
and
$\bm{S}(\bm{p},\bm{Q})=[\bm{s}(\bm{p},\bm{q}_{1}),\dots,\bm{s}(\bm{p},\bm{q}_{J})]\in\mathbb{R}^{N\times
J}$ where $J$ denotes the number of components for a specific realization of
$Q$. Scattering point positions are combined into
$\bm{Q}=[\bm{q}_{1}^{\mathsf{T}},\dots,\bm{q}_{J}^{\mathsf{T}}]$ and the
columns of $\bm{S}(\bm{p},\bm{Q})$ are defined as
$\bm{s}(\bm{p})\triangleq[s(0;\bm{p}),\dots,s((N-1)T_{s};\bm{p})]^{\mathsf{T}}\in\mathbb{C}^{N}$
and
$\bm{s}(\bm{p},\bm{q})\triangleq[s(0;\bm{p},\bm{q}),\dots,s((N-1)T_{s};\bm{p},\bm{q})]^{\mathsf{T}}\in\mathbb{C}^{N}$.
The vector $\bm{n}$ contains the AWGN noise samples with covariance matrix
$\bm{C}\triangleq\mathsf{E}[\bm{n}\bm{n}^{\mathsf{H}}]=\sigma_{w}^{2}\bm{I}_{N}$,
$\sigma_{w}^{2}=\frac{N_{0}}{T_{s}}$ and $\bm{I}_{N}$ is an $N\times N$
identity matrix.

The model in (4) can be interpreted as a standard multipath model with $J+1$
components: the LOS path directly arriving at the agent $\bm{p}$ from the
anchor $\bm{a}$ and $J$ components that are paths scattered at locations
$\bm{q}_{j}$. The according likelihood function of the model is

$\displaystyle
f(\bm{r};\bm{p},\alpha,\bm{Q},\bm{\beta})\propto\mathrm{exp}(-\|\bm{r}-\alpha\bm{s}(\bm{p})-\bm{S}(\bm{p},\bm{Q})\bm{\beta}\|^{2}).$
(5)

Due to the large number of model parameters, estimation of all parameters
might not be possible, depending on the measurement setup. The following
section describes a measurement procedure to allow calibration of $\bm{\beta}$
and $\bm{Q}$ by forming a synthetic array assuming a known agent position
$\bm{p}$.

##  III Measurement Setup and Model Calibration

To calibrate the model ultrawideband measurements were performed alongside
ground truth position and orientation measurements of the agent and parts of
the user, obtained using an optical tracking system (OTS) with position and
orientation accuracy below $1\,\mathrm{mm}$ and $1^{\circ}$ respectively. This
setup allows using a synthetic array of anchors to formulate a non-coherent
estimation procedure of scattering points and coefficients.

###  III-A Measurement Setup

The radio measurements were performed with a correlative channel sounder (CS)
covering a frequency range of $f=3.8-10.2\,\mathrm{GHz}$ with
$f_{c}=6.95\,\mathrm{GHz}$ allowing fully coherent measurements. The
measurement system was calibrated, including CS, cables and connectors (but
not the antennas). The antennas used were a taoglas FXUWB10 patch antenna as
agent (see Fig. 2a) and a dipole antenna [17, App. B.3] as anchor, both
suitable for the measurement frequency range. The dipole antenna has a
sufficiently flat pattern in the horizontal plane with strong attenuation
towards floor and ceiling, reducing floor and ceiling reflections. Both
devices are positioned at similar height.

The anchor position was fixed throughout the measurements, with the agent
positioned at various on-body positions (see Fig. 3a) on a user performing a
full rotation on the spot with a static posture and static on-body position of
the agent, while recording $M=200$ channel responses. An overview of the on-
body positions is shown in Fig. 3a alongside the coordinate system used in the
evaluation. The floorplan of the measurement setup is shown in Fig. 2b,
indicating the user position and rotation direction. Using the OTS data it is
possible to reverse the setup and synthetically generate a static agent and
multiple anchors around the agent. This procedure allows to jointly use all
$M=200$ measurements obtained during a rotation of the user, simulating a
synthetic array by fixing the agent device (and thus the EA) and moving the
anchor at (synthetic) positions $\bm{a}_{m}$ on a circle centered around the
agent (see Fig. 2b). Note that this is only valid for investigation of the
LOS.

(a) mobile at $\mathsf{R}$

(b) floorplan

Figure 2: The agent at on-body position $\mathsf{R}$ with reflective markers
for optical tracking system is shown in 2a. The floorplan shows the position
$\bm{a}$ of the anchor (blue x) and the approximate agent $\bm{p}$ (red dot).

(a) body and ref.

(b) $\mathsf{C}$

(c) $\mathsf{L}$

(d) $\mathsf{R}$

Figure 3: Used on-body positions of the mobile device are shown in 3a.
Reference coordinate system for the AOA $\varphi$ (positive AOAs are defined
counter clockwise with the mobile position $\bm{p}$ (red dot) as reference
point) and the tracked body part positions of upper back (back), right lower
arm and left lower arm (right and left) and a cap (front) are shown in 3b-3d.
The dashed areas give a rough outline of the user.

###  III-B Model Calibration

To calibrate the model parameters we make use of the available ground truth
anchor and agent positions and apply a maximum likelihood (ML) estimator to
directly estimate scatter point positions. The underlying assumption is that
due to the static posture and on body position, only one realization of the EA
is observed during the $M$ measurements. Assuming furthermore that all
measurements $\bm{r}_{m}$ are independent with identical noise variance111This
is a valid assumption as the measurements were performed in a short time frame
with the measurement device in continuous use. $\sigma_{w}^{2}$, allows using
(5) to write the joint likelihood of all measurements as

$\displaystyle
f(\underline{\bm{r}};\bm{p},\underline{\bm{\alpha}},\bm{Q},\underline{\bm{\beta}})$
$\displaystyle=\prod_{m=1}^{M}f(\bm{r}_{m};\bm{p},\alpha_{m},\bm{Q},\bm{\beta}_{m})$
(6)

where
$\underline{\bm{r}}=[\bm{r}_{1}^{\mathsf{T}},\dots,\bm{r}_{M}^{\mathsf{T}}]^{\mathsf{T}}$,
$\underline{\bm{\alpha}}=[\alpha_{1},\dots,\alpha_{M}]^{\mathsf{T}}$,
$\underline{\bm{\beta}}=[\bm{\beta}_{1}^{\mathsf{T}},\dots,\bm{\beta}_{M}^{\mathsf{T}}]^{\mathsf{T}}$
are the stacked signals, LOS amplitudes and scattering coefficients for all
$M$ measurement. Based on (6) the resulting ML estimator is obtained as

$\displaystyle\underline{\hat{\bm{\alpha}}},\underline{\hat{\bm{\beta}}},\hat{\bm{Q}}=\operatorname*{arg\,max}_{\bm{\alpha},\bm{\beta},\bm{Q}}\prod_{m=1}^{M}f(\bm{r}_{m};\bm{p},\alpha_{m},\bm{Q},\bm{\beta}_{m}).$
(7)

To optimize (7), the LOS amplitudes and scattering coefficients are estimated
separately for each measurement $m$ as linear least squares (LS) estimates
[18] denoted by $\hat{\underline{\bm{\alpha}}}$ and
$\hat{\underline{\bm{\beta}}}$ using the known ground truth agent position
$\bm{p}_{t}$ available from the OTS.222Anchor positions $\bm{a}_{m}$ are also
obtained by the OTS and assumed known in the signal model as usually the case
in positioning applications. These estimates can be interpreted as non-
parametric estimates of the user shadowing and agent antenna pattern, modeling
the full gain pattern of the EA. Delay dispersion of the user and the agent
antenna is in turn modeled by the scattering points. To find estimates of
scattering points $\hat{\bm{Q}}$ we employ a simple procedure which greedily
maximizes (6) by adding scattering points consecutively (i.e., without joint
optimization) as long as the likelihood value can be increased. The chosen
stopping criterion is by no means optimal and was chosen heuristically. Even
though more sophisticated algorithms could be used, the effect of the human
body and the antenna alone can be compared nonetheless as we apply the same
algorithm to measurements of the agent with and without user, with (possible)
estimation artifacts arising in both cases. An overview of the described
iterative ML procedure is given in Algorithm 1.

Initialization: •
$\hat{\underline{\bm{\alpha}}}=[\hat{\alpha}_{1},\dots,\hat{\alpha}_{M}]^{\mathsf{T}}~{}\leftarrow~{}\hat{\alpha}_{m}=\frac{\bm{s}(\bm{p}_{t})^{\mathsf{H}}\bm{r}_{m}}{\bm{s}(\bm{p}_{t})^{\mathsf{H}}\bm{s}(\bm{p}_{t})}~{}~{}\forall\,m$;

* •

$j=0$;

Iterations:

do

$j~{}\leftarrow~{}j+1$;

find
$\hat{\bm{q}}_{j}=\operatorname*{arg\,max}\limits_{\bm{q}}\sum_{m=1}^{M}\text{log}\,f(\bm{r}_{m,j};\bm{p}_{t},\hat{\alpha}_{m},\bm{q},\hat{\beta}_{m,j})$;

using
$\bm{r}_{m,j}=\bm{r}_{m}-\bm{s}(\bm{p}_{t})\hat{\alpha}_{m}-\sum_{l=1}^{j-1}\bm{s}(\bm{p}_{t},\hat{\bm{q}}_{l})\hat{\beta}_{m,l}$;

using
$\hat{\beta}_{m,j}=\|\bm{s}(\bm{p}_{t},\hat{\bm{q}}_{k})\|^{-2}\bm{s}(\bm{p}_{t},\hat{\bm{q}}_{k})^{\mathsf{H}}\bm{r}_{m,j}$;

$\underline{\hat{\bm{\beta}}}^{(j)}~{}\leftarrow~{}\\{\hat{\beta}_{1,1},\dots,\hat{\beta}_{M,j-1}\\}~{}\leftarrow~{}\hat{\beta}_{m,j}$;

${\hat{\bm{Q}}}^{(j)}~{}\leftarrow~{}\\{\hat{\bm{q}}_{1},\dots,\hat{\bm{q}}_{j-1}\\}~{}\leftarrow~{}\hat{\bm{q}}_{j}$;

while
_$\mathrm{log}f(\underline{\bm{r}};\bm{p}_{t},\underline{\hat{\bm{\alpha}}},\hat{\bm{Q}}^{(j)},\hat{\underline{\bm{\beta}}}^{(j)})
<\mathrm{log}f(\underline{\bm{r}};\bm{p}_{t},\underline{\hat{\bm{\alpha}}},\hat{\bm{Q}}^{(j-1)},\hat{\underline{\bm{\beta}}}^{(j-1)})$_;

Algorithm 1 iterative estimation procedure

##  IV Results

Fig. 4 shows the scattering point estimates $\hat{\bm{Q}}$ (denoted by blue x)
obtained using Algorithm 1 for the agent at different on-body positions (see
Fig. 3a). The size of the scattering point visualizes the respective average
scattering coefficient magnitude $\bar{\beta}_{j}$ of the $j$th scattering
point, defined as

$\displaystyle\bar{\beta}_{j}=\frac{1}{M}\sum_{m=1}^{M}|\hat{\beta}_{m,j}|$
(8)

with $\hat{\beta}_{m,j}$ being the estimated scattering coefficient of
measurement $m$ and scattering point $j$. An identical definition is used for
the average LOS magnitude
$\bar{\alpha}=\frac{1}{M}\sum_{m=1}^{M}|\hat{\alpha}_{m}|$. We include the
scattering point covariance matrices for each on-body position computed as

$\displaystyle\hat{\bm{\Sigma}}=\frac{1}{J}\sum_{j=1}^{J}(\bm{q}_{j}-\hat{\bm{\mu}})(\bm{q}_{j}-\hat{\bm{\mu}})^{\mathsf{T}}~{}~{}\text{with}~{}~{}\hat{\bm{\mu}}=\frac{1}{J}\sum_{j=1}^{J}\bm{q}_{j}$

as simple estimates of the EA shape and position parameters respectively.
Comparing the results at different on-body agent positions with the reference
agent shows a strong increase in the spatial spread of the scattering points,
attributable to the human body. The orientation of the covariance ellipses
shows a slight correspondence to the employed on body position, with a larger
spread towards the sides when at an off-center on-body position ($\mathsf{L}$
and $\mathsf{R}$) and larger spread towards the the body for the roughly
symmetric position $\mathsf{C}$. The fact that also the reference agent
exhibits a non-zero number of estimated scattering points, but with much lower
spatial spread, supports the proposed joint model of agent and user as an EA.

(a) $\mathsf{L}$ (left upper torso)

(b) $\mathsf{C}$ (center upper torso)

(c) $\mathsf{R}$ (right upper torso)

(d) $\mathsf{0}$ (reference)

Figure 4: Estimated scatter points using the MLE algorithm described in
Algorithm 1 with signals recorded at the different on-body positions and the
agent alone as shown in Fig. 3a. The covariance ellipses (2-fold enlarged) of
the scattering points are shown as EO shape estimate.

Fig. 5 shows the squared LOS magnitude and squared scattering coefficient in
dB as functions of the AOAs $\varphi_{m}$ of each measurement
$\alpha_{m,j}\triangleq\alpha_{j}(\varphi_{m})$ and
$\beta_{m,j}\triangleq\beta_{j}(\varphi_{m})$ respectively, with the
$0^{\circ}$-direction indicated in Fig. 4. Positive angles are defined counter
clockwise (see also Fig. 1). For better visualization all quantities are
averaged in 36 uniformly spaced sectors, and only the curves for the LOS and
the scattering points with the largest and second largest average scattering
coefficient magnitudes are shown. Comparing the results for the on-body
positions with the reference agent shows the strong influence of the human
body when in between anchor and agent, i.e., when completely blocking the LOS
in $180^{\circ}$ direction. The notches at roughly $\pm 45^{\circ}$ in the
reference antenna LOS magnitude in Fig. 5d are attributable to increased pulse
dispersion in these directions, which is directly modeled by the larger
scattering coefficient magnitudes, observable in Fig. 5d. When comparing the
on-body positions, the pattern of LOS magnitude and the scattering coefficient
magnitude is of similar shape but slightly rotated. Note that this rotation is
in the opposite direction w.r.t. the rotation of the agent position from the
center upper torso. By observing Fig. 4a and 4c a possible explanation for
this effect can be given by the following: Moving the agent towards the left
side of the upper torso and thus closer to the left hand/shoulder results in
earlier shadowing by the left hand/shoulder, corresponding to earlier
shadowing from positive AOAs. The same reasoning holds for position
$\mathsf{R}$. For increased AOAs the scattering coefficient magnitudes
decrease much faster than the LOS magnitude supporting the effect of lower
pulse dispersion with AOAs close to $90^{\circ}$ before total blockage as
observed in [9].

(a) $\mathsf{L}$ (left upper torso)

(b) $\mathsf{C}$ (center upper torso)

(c) $\mathsf{R}$ (right upper torso)

(d) $\mathsf{0}$ (reference)

Figure 5: Squared LOS magnitude (black) and squared average scattering
coefficient magnitude of strongest (red) and second strongest (blue) average
scattering coefficient magnitude plotted over angle of arrival $\varphi$.
Quantities shown are averaged in 36 non-overlapping uniform angular sectors.

Fig. 6 shows the average magnitude ratio (AMR) and the peak to average ratio
(PAR) in dB. The former is the ratio of average LOS magnitude or average
scattering coefficient magnitude to the maximum value of the LOS magnitude for
the reference agent
$\alpha^{(\mathsf{0})}_{\mathrm{max}}=\max_{m}|\alpha^{(\mathsf{0})}_{m}|$,
i.e.,

$\displaystyle\mathrm{AMR}_{\text{LOS}}^{(\mathsf{X})}$
$\displaystyle=\frac{\bar{\alpha}^{(\mathsf{X})}}{\alpha^{(\mathsf{0})}_{\mathrm{max}}}$
or
$\displaystyle\mathrm{AMR}^{(\mathsf{X})}_{j}=\frac{\bar{\beta}^{(\mathsf{X})}_{j}}{\alpha^{(\mathsf{0})}_{\mathrm{max}}}.$
(9)

The latter is the ratio of maximum value to average value of LOS magnitude or
scattering coefficient magnitude for each on-body position, i.e.,

$\displaystyle\mathrm{PAR}_{\text{LOS}}^{(\mathsf{X})}$
$\displaystyle=\frac{\max\limits_{m}|{\alpha}^{(\mathsf{X})}_{m}|}{\bar{\alpha}^{(\mathsf{X})}}$
or
$\displaystyle\mathrm{PAR}^{(\mathsf{X})}_{j}=\frac{\max\limits_{m}|{\beta}^{(\mathsf{X})}_{m,j}|}{\bar{\beta}^{(\mathsf{X})}_{j}}$
(10)

where $\mathsf{X}\in\\{\mathsf{0},\mathsf{C},\mathsf{L},\mathsf{R}\\}$
indicates the on-body position.

The AMR curves in Fig. 6a show that for the reference agent $\mathsf{0}$ the
scattering coefficients AMRs exhibit a steeper drop relative to the LOS AMR,
while showing a similar trend as for the on-body positions. The AMR curves
indicate that higher pulse dispersion can be expected with the agent at an on-
body positions, due to the lower difference between LOS and scattering point
AMRs. The plots also show the number of estimated scattering points for the
different setups as $J_{\mathsf{0}}=J_{\mathsf{L}}=5$ and
$J_{\mathsf{C}}=J_{\mathsf{R}}=9$. $J_{\mathsf{X}}$ is the number of
scattering points estimated for on-body position $\mathsf{X}$.

The PAR curves in Fig. 6b show that even though the reference agent is far
from an isotropic antenna, i.e., with $\mathrm{PAR}=0\,\mathrm{dB}$, it
exhibits a generally lower PAR compared to the curves for the agent at
different on-body positions. This leads to the conclusion of an increased
directionality of the agent when at an on-body position.

(a) average magnitude ratio (AMR)

(b) peak to average ratio (PAR)

Figure 6: Ratio of average LOS magnitude or average scattering coefficient
magnitude over the average LOS magnitude of the reference agent.

##  V Conclusion

In this paper we investigate joint modeling of an antenna and the human body,
which add scattered components in addition to the line of sight path, as an
extended antenna (EA). We briefly describe a stochastic EA model based on our
previous work [9] and similar to [3], using a marked Poisson point process
(MPPP) to model the delay dispersion introduced by antenna and human body.
Using measurements of a received ultrawideband signal with the agent
positioned at different on-body positions on a static user, we calibrate the
model for the different on-body positions, highlighting the differences of the
on-body positions in comparison with the reference antenna. In all
investigated setups the description using scattering points is feasible, with
the different properties in terms of spatial spread of the scattering points
and angular magnitude patterns of scattering points and LOS giving useful
insights for modeling. The results function as a starting point for a deeper
analysis of the stochastic model parameter distributions.

Possible directions for future work are an in depth analysis of the model
parameters in dynamic scenarios (based on different measurements of the same
measurement campaign) and the application of this model in positioning and
tracking algorithms with the goal of reducing the influence of the human body
on the positioning quality.

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