# An EM Body Model for Device-Free Localization with Multiple Antenna
Receivers: A First Study ††thanks: Funded by the European Union. Views and
opinions expressed are however those of the author(s) only and do not
necessarily reflect those of the European Union or European Innovation Council
and SMEs Executive Agency (EISMEA). Neither the European Union nor the
granting authority can be held responsible for them. Grant Agreement No:
101099491.

Vittorio Rampa1, Federica Fieramosca2, Stefano Savazzi1, Michele D’Amico2 1
_Consiglio Nazionale delle Ricerche_ _,__IEIIT_ institute, Piazza Leonardo da
Vinci 32, I-20133, Milano, Italy.
2 __ DEIB, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133,
Milano, Italy

###### Abstract

Device-Free Localization (DFL) employs passive radio techniques capable to
detect and locate people without imposing them to wear any electronic device.
By exploiting the _Integrated Sensing and Communication_ paradigm, DFL
networks employ Radio Frequency (RF) nodes to measure the excess attenuation
introduced by the subjects (i.e., human bodies) moving inside the monitored
area, and to estimate their positions and movements. Physical, statistical,
and ElectroMagnetic (EM) models have been proposed in the literature to
estimate the body positions according to the RF signals collected by the
nodes. These body models usually employ a single-antenna processing for
localization purposes. However, the availability of low-cost multi-antenna
devices such as those used for WLAN (Wireless Local Area Network) applications
and the timely development of array-based body models, allow us to employ
array-based processing techniques in DFL networks. By exploiting a suitable
array-capable EM body model, this paper proposes an array-based framework to
improve people sensing and localization. In particular, some simulations are
proposed and discussed to compare the model results in both single- and multi-
antenna scenarios. The proposed framework paves the way for a wider use of
multi-antenna devices (e.g., those employed in current IEEE 802.11ac/ax/be and
forthcoming IEEE 802.11be networks) and novel beamforming algorithms for DFL
scenarios.

###### Index Terms:

Electromagnetic body models, device-free passive radio localization,
integrated sensing and communication, array processing.

##  I Introduction

Device-Free Localization (DFL), also known as Passive Radio Localization
(PRL), is an opportunistic set of methods capable to detect, locate, and track
people in a monitored area covered by ambient Radio Frequency (RF) signals. By
exploiting the _Integrated Sensing and Communication_ paradigm [1], DFL
systems are able to transform each RF node of the wireless network covering
the monitored area into a _virtual sensor_ able perform sensing operations as
well.

For instance, the RF signals emitted for communication purposes by a wireless
network over the monitored area, can be usefully employed to estimate people
location and status information from the received ElectroMagnetic (EM) field.
In fact, it is well known that the presence, and the movements, of people, or
objects, generically indicated as _targets_ , induce modifications and
alterations of the EM field [2] collected by each wireless device of the
communication network. These perturbations not only impair the radio channel,
but can be also measured and processed to estimate information about the
presence, location, and status (e.g., posture, movements, size, etc.) of the
targets [3].

Body models for DFL applications have been mostly evaluated for single-antenna
DFL systems for both single-target [4, 5, 6, 7] and multi-targets [8]
scenarios. These models have been applied to different radio channel
measurements, such as Channel State Information (CSI), Received Signal
Strength (RSS), Angles of Arrival (AoA) and Time of Flight (ToF) [1], and
processed by different methods such as radio imaging [4], Bayesian tracking
[9], fingerprinting methods [1], Compressive Sensing algorithms [10], and
Machine Learning/Deep Learning (ML/DL) systems [3, 11], to evaluate the
connection between targets location and the corresponding measured
perturbations.

At the same time, the wide diffusion of multi-antenna WLAN devices (e.g., WiFi
devices built according to the IEEE 802.11n/ac/ax standards and the
forthcoming IEEE 802.11be one), and the concomitant availability of CSI
extraction tools [12, 13, 14], have stimulated the research activities with
the adoption of multi-antenna CSI-based processing systems for DFL
applications [15, 16, 17].

Unfortunately, only a few body models deal with antenna arrays [18, 20, 19]
since almost all array-based DFL systems hold the attention on the processing
steps instead of focusing on the body model. However, reference [18] employs a
very simple propagation model while [19] exploiting a computational-intensive
Ray Tracing (RT) approach. Of course, EM simulators can be also used, but they
are too complex to be of practical use for on-line DFL systems as based on RT
techniques, uniform theory of diffraction (UTD), or full-wave methods as shown
in [8] (and included references).

The paper extends the physical-statistical array-based model proposed by the
authors in [20] to predict the body-induced propagation losses using WLAN
nodes equipped with multiple antennas at the receiver node. The body model is
then used, in conjunction with array processing techniques, to focus on the
AoA and the position of the target. The paper is organized as follows: in
Sect. II, the array-based electromagnetic body model for applications in
multi-antennas DFL systems is briefly recalled and discussed. Sect. III
proposes the array-based processing tools for DFL applications while Sect. IV
presents some preliminary simulation results comparing the array-based model
with the single-antenna one. Finally, in Sect. V, we draw some preliminary
conclusions and propose future activities.

Figure 1: 3-D deployment of the radio link using an ULA-like array of $2M+1$
antennas. The segment where the array is deployed is placed at distance
$d=d_{0}$ from the TX and it is orthogonal to the LoS path connecting the $TX$
node with the $RX_{0}$ one.

##  II EM body model with multiple antenna receiver

In this section we briefly recall the body model proposed in [20] for a single
link scenario where an antenna array is used at the receiver while only one
transmitter (TX) is used. Fig. 1 shows the layout for an Uniform Linear Array
(ULA) of $2M+1$ isotropic receiver antennas RXm, with$-M\leq m\leq M$ and $M$
an integer, when a single target $S$ is in the monitored area near the radio
link.

The 2-D footprint of the 3-D deployment of Fig. 1 is also shown in Fig. 2,
where each _m_ -th antenna RXm of the array is uniformly deployed at mutual
distance $d_{a}$ along a segment orthogonal to the Line-of-Sight (LoS) at
distance $d=d_{0}$ from the TX and horizontally placed w.r.t. the floor. The
central antenna is indicated by the index $m=0$.

Without any loss of generality, we assume here that the floor has no influence
on the radio propagation but it only needed to support the standing target.
However, if needed, the EM influence of the floor can be easily included as
shown in [21]. The target is sketched [7] as a vertical standing absorbing 2-D
sheet $S$ of height $2a_{z}$ and traversal size $2a_{y}$ that is rotated of
the angle $\theta$ w.r.t. the $\xi_{2}$ axis. Excluding mutual antenna
coupling (approximately valid for $d_{a}>\lambda/4$), the electric field
$E^{\left(m\right)}$ received by the _m_ -th antenna of the array, is:

$\begin{array}[]{ll}\frac{E^{\left(m\right)}}{E_{R}^{\left(m\right)}}=&1-j\frac{d_{m}}{\lambda}\intop_{S}\frac{1}{r_{1,m}r_{2,m}}\cdot\\\
&\cdot\exp\left\\{-j\frac{2\pi}{\lambda}\bigl{(}r_{1,m}+r_{2,m}-d_{m}\bigr{)}\right\\}d\xi_{2}\,d\xi_{3},\end{array}$
(1)

where $E_{R}^{\left(m\right)}$ is the EM field received by the same $RX_{m}$
node in the reference condition i.e. the free-space scenario characterized by
the absence of any target in the link area. The term $d_{m}$ indicates the
distance of the _m_ -th antenna $RX_{m}$ of the array from the TX while
$d_{1,m}$ and $d_{2,m}$ are the distances of the projection point
$O_{m}^{{}^{\prime}}$ (of the barycenter $P$ of the 2-D surface $S$) from the
TX and $RX_{m}$ nodes. Likewise, $r_{1,m}$ and $r_{2,m}$ are the distances of
the generic elementary area $dS$ of the target $S$ from the TX and $RX_{m}$,
respectively. In (1), the integration operations are performed over the
squared domain $S$ having height $2a_{z}$ and traversal size $2a_{y}$.

Notice that, for $M=0$, equation (1) reduces to the single-antenna case where
RX0 coincides with the RX antenna at distance $d=d_{0}$ from the TX. The
excess attenuation [7] measured by the m-th antenna of the array, that is due
to the target w.r.t. the free-space scenario, is thus given by
$A_{T}^{\left(m\right)}=P_{R}^{\left(m\right)}/P^{\left(m\right)}=\left|E_{R}^{\left(m\right)}/E^{\left(m\right)}\right|^{2}$
where $P_{R}^{\left(m\right)}$ and $P^{\left(m\right)}$ are the _m_ -th
received power in without and with a target in the link area, respectively.
Usually, the excess attenuation is given in dB as
$A_{T,dB}^{\left(m\right)}=10\,\log_{10}\left|E_{R}^{\left(m\right)}/E^{\left(m\right)}\right|^{2}$.
It is worth noticing that, for short arrays, the excess attenuation due to the
target is almost constant along the receiving antennas as shown in [20].

In free-space conditions, the term
$E_{R}^{\left(m\right)}/E_{R}^{\left(0\right)}$ is given by:

$\begin{array}[]{ll}\frac{E_{R}^{\left(m\right)}}{E_{R}^{\left(0\right)}}=&\frac{d_{0}}{d_{m}}\,\exp\left\\{-j\frac{2\pi}{\lambda}\bigl{(}d_{m}-d_{0}\bigr{)}\right\\},\end{array}$
(2)

where $E_{R}^{\left(0\right)}$ is the electric field received by the central
antenna of the array of index $m=0$ that is on the LoS at distance $d_{0}$
from the TX. Thus, (1) can be rearranged as:

$\begin{array}[]{ll}\frac{E^{\left(m\right)}}{E_{R}^{\left(0\right)}}=&\frac{d_{0}}{d_{m}}\,\exp\left\\{-j\frac{2\pi}{\lambda}\left(d_{m}-d_{0}\right)\right\\}\bigl{[}1-j\frac{d_{m}}{\lambda}\intop_{S}\frac{1}{r_{1,m}r_{2,m}}\cdot\bigr{.}\\\
&\bigl{.}\cdot\exp\left\\{-j\frac{2\pi}{\lambda}\left(r_{1,m}+r_{2,m}-d_{m}\right)\right\\}d\xi_{2}\,d\xi_{3}\bigr{]}.\end{array}$
(3)

For additional details about the array-based model (3) and its validation with
the commercial EM simulator FEKO [22], the interested reader can refer to
[20].

Figure 2: 2-D layout of the radio link of Fig. 1. The point
$O_{m}^{{}^{\prime}}$ is the projection of the barycenter $P$ of the target
$S$ over the _m_ -th LoS path having length $d_{m}$.

##  III Array processing with model predictions

In the following sections, we consider the ULA-arranged layout of $2M+1$
antennas of Figs. 1 and 2. Classical array processing, such as conventional
beamforming [23], is designed to _steer_ the antenna array in one direction by
forming a linear combination of each antenna contribution. Being
$\left(\cdot\right)^{T}$ the vector transposition operator, the output of the
beamforming processing, considering $2M+1$ antennas and the vector
$\mathbf{w}=\left[w_{-M}\,...\,w_{-1}\,w_{0}\,w_{1}\,...\,w_{M}\right]^{T}$ of
size $2M+1$ of linear beamforming coefficients, is thus given in general by:

$y\left(t,\mathcal{S}\right)=\sum_{m=-M}^{+M}w_{m}^{*}\,r_{m}\left(t,\mathcal{S}\right)=\mathbf{w}^{H}\mathbf{r}\left(t,\mathcal{S}\right),$
(4)

where $\left(\cdot\right)^{*}$ indicates the conjugate operator and
$\left(\cdot\right)^{H}$ the Hermitian operator, while
$r_{m}\left(t,\mathcal{S}\right)$ is the received EM field at the $RX_{m}$
antenna. The column vector $\mathbf{r}\left(t,\mathcal{S}\right)$ of size
$2M+1$, collects all the received electric field terms
$\mathbf{r}\left(t,\mathcal{S}\right)=\left[r_{-M}\left(t,\mathcal{S}\right)\,...r_{-1}\left(t,\mathcal{S}\right)\,r_{0}\left(t,\mathcal{S}\right)\,r_{1}\left(t,\mathcal{S}\right)\,...\,r_{M}\left(t,\mathcal{S}\right)\right]^{T}$.
Focusing on passive sensing applications, two main scenarios are adopted for
processing: the first one refers to the empty environment (i.e., the free-
space case, namely $\mathcal{S}=0$) while, in the second scenario, the target
$S$ is present in the monitored area ($\mathcal{S}=1$). According to this
assumption, the _m_ -th component of the received vector
$\mathbf{r}\left(t\right)$ is defined as:

$r_{m}\left(t,\mathcal{S}\right)=\left\\{\begin{array}[]{ll}E_{R}^{\left(m\right)}+n_{m}=E_{R}^{\left(0\right)}\,\frac{E_{R}^{\left(m\right)}}{E_{R}^{\left(0\right)}}+n_{m}&\textrm{if
$\mathcal{S}=0$}\\\
E^{\left(m\right)}+n_{m}=E_{R}^{\left(0\right)}\,\frac{E_{R}^{\left(m\right)}}{E_{R}^{\left(0\right)}}\,\frac{E^{\left(m\right)}}{E_{R}^{\left(m\right)}}+n_{m}&\textrm{if
$\mathcal{S}=1$}\end{array}\right.$ (5)

where $n_{m}$ is the _m_ -th component of the Additive White Gaussian Noise
(AWGN) complex vector
$\mathbf{n}\left(t\right)=\left[n_{-M}\left(t\right)\,...n_{-1}\left(t\right)\,n_{0}\left(t\right)\,n_{1}\left(t\right)\,...\,n_{M}\left(t\right)\right]^{T}$
of size $2M+1$, that is assumed to be spatially white with zero mean and
covariance $\sigma_{n}^{2}\mathbf{I}$, while
$E_{R}^{\left(m\right)}/E_{R}^{\left(0\right)}$ and
$E^{\left(m\right)}/E_{R}^{\left(0\right)}$ are given by (2) and (3),
respectively. We assume here that the noise distribution does not change
according to the presence or absence of the target. For discussions about this
topic, the interested reader can refer to [7, 9, 8]. As analyzed in the
following sections, different beamforming methods correspond to different
choices of the weighting vector $\mathbf{w}$.

Considering the passive radio sensing problem, we are interested in evaluating
the power $P_{y}\left(\mathcal{S}\right)$ of the signal $y$ with or without
target $S$. This is given by:

$\begin{array}[]{ll}P_{y}\left(\mathcal{S}\right)&=\mathbb{E}\left[y\left(t,\mathcal{S}\right)\,y^{*}\left(t,\mathcal{S}\right)\right]\\\
&=\mathbf{w}^{H}\,\mathbf{R}\left(\mathcal{S}\right)\,\mathbf{w}\end{array}$
(6)

with
$\mathbf{R}\left(\mathcal{S}\right)=\mathbf{R}=\mathbb{E}\left[\mathbf{r}\left(t\right)\,\mathbf{r}^{H}\left(t\right)\right]$
of size $\left(2M+1\right)\times\left(2M+1\right)$, being the autocorrelation
matrix of $y\left(t,\mathcal{S}\right).$

###  III-A Impact of beamforming on body induced attenuation

In what follows, we model the received EM field
$r_{m}\left(t,\mathcal{S}\right)$ in (4) using the diffraction model
considerations and exploiting the equations (1)-(3) for $\mathcal{S}=1$, and
(2) for $\mathcal{S}=0$. For conventional ULA scenarios, that assume planar
wavefront propagation, the steering (column) vector
$\mathbf{a}=\left[a_{m}\right]$ of size $2M+1$ for the considered array is
given by [23]:

$\begin{array}[]{ll}\mathbf{a}=&\left[\exp\left\\{-jM\frac{2\pi}{\lambda}d_{a}\cos\gamma\right\\}\right.\\\
&\>\exp\left\\{-j\left(M-1\right)\frac{2\pi}{\lambda}d_{a}\cos\gamma\right\\}\\\
&...\>1\>...\\\
&\>\exp\left\\{+j\left(M-1\right)\frac{2\pi}{\lambda}d_{a}\cos\gamma\right\\}\\\
&\left.\exp\left\\{+jM\frac{2\pi}{\lambda}d_{a}\cos\gamma\right\\}\right]^{T},\end{array}$
(7)

where $\gamma$ is also known as DoA (Direction Of Arrival): this is the
direction of propagation of the impinging wavefront w.r.t. the axis of the
array, and $d_{a}$ is the inter-element antenna distance. According to (7), it
is also $\left|\mathbf{a}\right|^{2}=\mathbf{a}^{H}\mathbf{a}=(2M+1)$.
However, due to the fact that most DFL applications are employed in indoor
scenarios with relatively short links compared with the usual far-field
hypothesis, the planar wavefront assumption no longer holds and the new
steering vector is:

$\begin{array}[]{ll}\mathbf{a}=&\left[\frac{d_{0}}{d_{-M}}\,\exp\left\\{-jM\frac{2\pi}{\lambda}d_{a}\frac{\cos\left\\{\left(\gamma+\varphi_{-M}\right)/2\right\\}}{\cos\left\\{\left(\gamma-\varphi_{-M}\right)/2\right\\}}\right\\}\right.\\\
&\frac{d_{0}}{d_{-M+1}}\,\exp\left\\{-j\left(M-1\right)\frac{2\pi}{\lambda}d_{a}\frac{\cos\left\\{\left(\gamma+\varphi_{-M+1}\right)/2\right\\}}{\cos\left\\{\left(\gamma-\varphi_{-M+1}\right)/2\right\\}}\right\\}\\\
&\>...\>1...\\\
&\frac{d_{0}}{d_{M-1}}\,\exp\left\\{+j\left(M-1\right)\frac{2\pi}{\lambda}d_{a}\frac{\cos\left\\{\left(\gamma+\varphi_{M-1}\right)/2\right\\}}{\cos\left\\{\left(\gamma-\varphi_{M-1}\right)/2\right\\}}\right\\}\\\
&\left.\frac{d_{0}}{d_{M}}\,\exp\left\\{+jM\frac{2\pi}{\lambda}d_{a}\frac{\cos\left\\{\left(\gamma+\varphi_{M}\right)/2\right\\}}{\cos\left\\{\left(\gamma-\varphi_{M}\right)/2\right\\}}\right\\}\right]^{T},\end{array}$
(8)

where the angle $\varphi_{m}$, formed by the LoS of the _m_ -th array antenna
with the $\xi_{2}$ axis, also verifies $\forall m=0,\pm 1,...,\pm M$ the
following equation
$\varphi_{m}=\arccos\left[\left(d_{0}/d_{m}\right)\,\cos\gamma-m\,\left(d_{a}/d_{m}\right)\right]$
with $\varphi_{0}=\gamma$. Moreover, it is also
$\left|\mathbf{a}\right|^{2}=\sum_{m=-M}^{M}\left(d_{0}/d_{m}\right)^{2}$. It
is easy to check that (8) converges to (7) for planar wavefronts i.e. when
$\forall m$, it is $\varphi_{m}=\gamma$. According to (8), the received field
$\mathbf{r}\left(t,\mathcal{S}\right)$ is given by:

$\mathbf{r}\left(t,\mathcal{S}\right)=\left\\{\begin{array}[]{ll}E_{R}^{\left(0\right)}\,\mathbf{a}+\mathbf{n}&\textrm{if
$\mathcal{S}=0$}\\\
E_{R}^{\left(0\right)}\,\textrm{diag$\left(\mathbf{a}\right)$}\,\mathbf{E}_{r}+\mathbf{n}=&\textrm{if
$\mathcal{S}=1$,}\end{array}\right.$ (9)

where the column vector $\mathbf{E}_{r}$ of size $2M+1$ represents the
electric field ratio (1) received by the antenna array for $\mathcal{S}=1$.
The $m$-th element of this vector is equal to
$E^{\left(m\right)}/E_{R}^{\left(m\right)}$.

According to diffraction model of Sect. II, and by rearranging the terms in
(9) with (8) and (1), the received field $y\left(t,\mathcal{S}\right)$ after
linear beamforming becomes as in eq. (12). Similarly as for the single antenna
case [7], the mean excess attenuation $A_{T}$ due to the target $S$ w.r.t. the
empty environment, corresponds to the ratio between
$P_{y}\left(\mathcal{S}=0\right)$ and $P_{y}\left(\mathcal{S}=1\right)$, by
neglecting the noise terms:

$\begin{array}[]{ll}A_{T}=&\frac{P_{y}\left(\mathcal{S}=0\right)}{P_{y}\left(\mathcal{S}=1\right)}=\frac{\mathbf{w}^{H}\,\mathbf{a}\,\mathbf{a}^{H}\mathbf{w}}{\mathbf{w}^{H}\,\textrm{diag}\left(\mathbf{a}\right)\,\mathbf{R}_{E}\,\textrm{diag}\left(\mathbf{a}^{H}\right)\mathbf{w}}.\end{array}$
(10)

In particular, $R_{E}$ is the autocorrelation matrix of
$E^{\left(m\right)}/E_{R}^{\left(m\right)}$, of size
$\left(2M+1\right)\times\left(2M+1\right)$, given by (11).

Equation (10) can be also written as in (13), neglecting the noise terms.

$\mathbf{R}_{E}=\left[\begin{array}[]{lcr}\left|\frac{E^{\left(-M\right)}}{E_{R}^{\left(-M\right)}}\right|^{2}&...&\frac{E^{\left(-M\right)}}{E_{R}^{\left(-M\right)}}\,\left(\frac{E^{\left(M\right)}}{E_{R}^{\left(M\right)}}\right)^{*}\\\
\frac{E^{\left(-M+1\right)}}{E_{R}^{\left(-M+1\right)}}\,\left(\frac{E^{\left(-M\right)}}{E_{R}^{\left(-M\right)}}\right)^{*}&...&\frac{E^{\left(-M+1\right)}}{E_{R}^{\left(-M+1\right)}}\,\left(\frac{E^{\left(M\right)}}{E_{R}^{\left(M\right)}}\right)^{*}\\\
...&...&...\\\
\frac{E^{\left(M-1\right)}}{E_{R}^{\left(M-1\right)}}\,\left(\frac{E^{\left(-M\right)}}{E_{R}^{\left(-M\right)}}\right)^{*}&...&\frac{E^{\left(M-1\right)}}{E_{R}^{\left(M-1\right)}}\,\left(\frac{E^{\left(M\right)}}{E_{R}^{\left(M\right)}}\right)^{*}\\\
\frac{E^{\left(M\right)}}{E_{R}^{\left(M\right)}}\,\left(\frac{E^{\left(-M\right)}}{E_{R}^{\left(-M\right)}}\right)^{*}&...&\left|\frac{E^{\left(M\right)}}{E_{R}^{\left(M\right)}}\right|^{2}\end{array}\right]$
(11)

$y\left(t,\mathcal{S}\right)=\left\\{\begin{array}[]{ll}E_{R}^{\left(0\right)}\,\sum_{m=-M}^{+M}w_{m}^{*}\,\frac{d_{0}}{d_{m}}\,\exp\left\\{-j\left(M-1\right)\frac{2\pi}{\lambda}d_{a}\frac{\cos\left\\{\left(\gamma+\varphi_{m}\right)/2\right\\}}{\cos\left\\{\left(\gamma-\varphi_{m}\right)/2\right\\}}\right\\}+n_{m}&\textrm{if
$\mathcal{S}=0$}\\\
E_{R}^{\left(0\right)}\,\sum_{m=-M}^{+M}w_{m}^{*}\,\frac{E^{\left(m\right)}}{E_{R}^{\left(m\right)}}\,\frac{d_{0}}{d_{m}}\,\exp\left\\{-j\left(M-1\right)\frac{2\pi}{\lambda}d_{a}\frac{\cos\left\\{\left(\gamma+\varphi_{m}\right)/2\right\\}}{\cos\left\\{\left(\gamma-\varphi_{m}\right)/2\right\\}}\right\\}+n_{m}&\textrm{if
$\mathcal{S}=1$}\end{array}\right.$ (12)

$\begin{array}[]{l}A_{T}=\frac{\left|\sum_{m=-M}^{+M}w_{m}^{*}\,\frac{d_{0}}{d_{m}}\,\exp\left\\{-j\left(M-1\right)\frac{2\pi}{\lambda}d_{a}\frac{\cos\left\\{\left(\gamma+\varphi_{m}\right)/2\right\\}}{\cos\left\\{\left(\gamma-\varphi_{m}\right)/2\right\\}}\right\\}\right|^{2}}{\left|\sum_{m=-M}^{+M}w_{m}^{*}\,\frac{d_{0}}{d_{m}}\,\exp\left\\{-j\left(M-1\right)\frac{2\pi}{\lambda}d_{a}\frac{\cos\left\\{\left(\gamma+\varphi_{m}\right)/2\right\\}}{\cos\left\\{\left(\gamma-\varphi_{m}\right)/2\right\\}}\right\\}\,\left[1-j\frac{d_{m}}{\lambda}\intop_{S}\frac{1}{r_{1,m}r_{2,m}}\exp\left\\{-j\frac{2\pi}{\lambda}\bigl{(}r_{1,m}+r_{2,m}-d_{m}\bigr{)}\right\\}d\xi_{2}d\xi_{3}\right]\right|^{2}}\end{array}$
(13)

Figure 3: Array factor for an ULA composed by $9$ antennas ($M=4$) uniformly
spaced from $0.1\,\lambda$ up to $0.5\,\lambda$.

###  III-B Array factor analysis

The array factor $F_{a}=\mathbf{w}^{T}\mathbf{a}\left(\gamma\right)$ defines
the response of the array as a function of the selected beamforming
coefficients $\mathbf{w}$ and of the incident angle $\gamma$ of the steering
vector $\mathbf{a}\left(\gamma\right)$ that characterizes the impinging
wavefront. For instance, if (7) holds true with the weighting vector
$\mathbf{w}$ such that, for all components $w_{m}$, it is $w_{m}=1/(2M+1)$,
then the array factor is given by:

$\begin{array}[]{ll}F_{a}&=\mathbf{w}^{T}\mathbf{a}\left(\gamma\right)=\frac{1}{2M+1}\sum_{m=-M}^{+M}\exp\\{{jm\frac{2\pi}{\lambda}d_{a}\cos\gamma}\\}\\\
&=\frac{1}{2M+1}\cdot\frac{\sin\bigl{\\{}\bigl{(}2M+1\bigr{)}\frac{\pi
d_{a}}{\lambda}\cos\gamma\bigr{\\}}}{\sin\bigl{\\{}\frac{\pi
d_{a}}{\lambda}\cos\gamma\bigr{\\}}}.\end{array}$ (14)

Fig. 3 shows the modulus of the array factor in dB for an ULA of $9$ antennas
($M=4$) uniformly spaced with $d_{a}$ from $0.1\,\lambda$ up to
$0.5\,\lambda$. It is apparent the effect due to the side lobes. The width of
the first lobe $\Delta_{BW}$ is given by:

$\Delta_{BW}=2\left|\arccos\left\\{\frac{\lambda}{\left(2M+1\right)d_{a}}\right\\}\right|\approx\frac{2\lambda}{\left(2M+1\right)d_{a}},$
(15)

where the approximation holds for long arrays i.e. for
$L=\left(2M+1\right)\,d_{a}\gg\lambda$.

Figure 4: Link layout used for the simulations: the antenna array is composed
by 5 antennas ($M=2$) while the target $S$, with size $a_{z}=0.9$ m and
$a_{y}=0.275$ m, is placed in different positions along the line having
distance $x=1$ m from the TX. Figure 5: From top to bottom: a) target
placement used for this test; b) excess attenuation expressed in terms of the
DoA $\gamma$ of the array when the target $S$ is placed along the segment
having distance $x=1$ m from the TX. Three different positions of the target
$S$ are simulated at $(x,y)$: $(1.0,-0.25)$, $(1.0,0.0)$, and $(1.0,0.25)$; c)
excess attenuation as observed on each single antenna without any array
processing and for corresponding positions of the targets as above.

##  IV Preliminary results

In this section, we provide some preliminary results concerning the use of the
array model of Sect. II vs the single-antenna model [7].

The carrier frequency $f_{c}$ is set to $f_{c}=2.4868$ GHz, and we used an UL
array of $5$ omni-directional antennas ($M=2$) spaced at $d_{a}=\lambda/2$ as
shown in the layout sketch described in Fig. 4. The length of the central link
of the array (i.e., for $m=0$) is equal to $d=d_{0}=4$ m while all links of
the array are horizontally placed at height $h=0.9$ m from the ground.

As far as the body model is concerned, the absorbing 2-D sheet that represents
the target has size $a_{z}=0.9$ m and $a_{y}=0.275$ m (i.e. a total size of
1.80 m x 0.55 m). The target stands vertically on the floor, that is used only
to support the target and does not have any EM influence on the radio links.
The LoS of the central link will be the reference LoS line for the target
positions with the origin of the axes placed in the TX as shown in Fig. 4.

Figs. 5, 6, and 7 show the behavior of the entire array, namely the array
response, in terms of the excess attenuation
$20\,\log_{10}|E_{R}^{(0)}/E^{(m)}|$ as a function of the DoA $\gamma$ and for
different values of the $y$ displacement of the target $S$ (w.r.t. the central
LoS). The locations of the target are limited to belong to the line at
$\xi_{1}=x=1$ m that is orthogonal to the central LoS. Processing of the array
signals to extract the response for varying DoA $\gamma$ is based on Fast
Fourier Transform (FFT) and employs $N_{FFT}=257$ points.

For positions of the target $S$ near the LoS as in Figs. 5.a and 6.a, the
extra attenuation for off-LoS positions (blue and yellow lines) are comparable
with the one (red line) representing the target on the LoS (i.e., $\sim 15$ dB
for $y=0$ m compared with $\sim 7$ dB for $y=\pm 0.25$ m). In addition, in
Fig. 5.b the three positions, spaced by $0.25$ m, are clearly separable since
they result in different array responses. Considering the closed-spaced
positions (i.e., $5$ cm) of Fig. 6.b, the resulting extra attenuation values
($\sim 17$ dB) are now much closer to the one of target placed on the LoS,
thus making more difficult to visually separate these locations. On the other
hand, the targets are still distinguishable as they are associated with
separable DoAs.

Figure 6: From top to bottom: a) target placement used for this test; b)
excess attenuation of the array expressed in terms of the DoA $\gamma$ when
the target $S$ is placed along the segment having distance $x=1$ m from the
TX. Three very close positions of the target $S$ are simulated at $(x,y)$:
$(1.0,-0.05)$, $(1.0,0.0)$, and $(1.0,0.05)$. Figure 7: From top to bottom: a)
target placement used for this test; b) excess attenuation of the array
expressed in terms of the DoA $\gamma$ when the target $S$ is placed along the
segment having distance $x=1$ m from the TX. Three widely spaced positions of
the target $S$ are simulated at $(x,y)$: $(1.0,-1.0)$, $(1.0,0.0)$, and
$(1.0,1.0)$.

To compare the excess attenuation results for the antenna array of Fig. 5.b vs
the ones of the single antennas, Fig. 5.c show the excess attenuation for each
antenna. Comparing for instance the results for the target on the LoS (red
line), it is apparent that, Fig. 5.c, the attenuation ranges from $\sim 14$ dB
to $\sim 16$ dB depending on the selected antenna, while in Fig. 5.b the
maximum value of the excess attenuation is the averaged value of $\sim 15$ dB.

For positions of the target $S$ outside the link area (i.e., outside the first
Fresnel’s ellipsoid, whose minor axis is $\sim 0.7$ m for the considered
$f_{c}$), the influence due to the target is negligible. Some minor effects
are visible in the nulls among the main and the side lobes, but they may be
too weak to be exploited to identify the presence of the target in a real-
world scenario, where there is always a certain degree of multipath.

We have also performed some preliminary evaluations of conventional [23] and
MVDR (Minimum Variance Distortionless Response) [24] beamforming algorithms.
However, for space constraints, the results will not be discussed in this
paper. Although preliminary, the results are promising and highlight the
possibility of using EM body models to develop ad-hoc array processing tools
adapted for passive DFL applications.

##  V Conclusions

In this paper, we presented a tool for the simulation of the Electro-Magnetic
(EM) effects of different body (i.e., target) positions inside a link area
covered by an array of closely spaced omni-directional antennas. In the
proposed preliminary study, we assessed the effects of the body on the antenna
array response. In particular, we showed that for different target positions
in the monitored area, the array is able to separate the contribution of the
target when it is inside the first Fresnel’s ellipsoid. The effects of the
target can be represented both in terms of excess attenuation, which depends
on body relative position, and as alterations of the array response pattern
which loosely depends on target Direction of Arrival (DoA), relative to the
Line-of-Sight (LoS) path. The proposed method paves the way for future
applications of multiple antenna processing techniques in device-free
localization applications.

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