# ELAA Near-Field Localization and Sensing with Partial Blockage Detection

Hui Chen1, Pinjun Zheng2, Yu Ge1, Ahmed Elzanaty3, Jiguang He4, Tareq Y. Al-
Naffouri2, Henk Wymeersch1
1Chalmers University of Technology, Sweden 2King Abdullah University of
Science and Technology, KSA
3University of Surrey, UK 4Technology Innovation Institute, UAE

###### Abstract

High-frequency communication systems bring extremely large aperture arrays
(ELAA) and large bandwidths, integrating localization and (bi-static) sensing
functions without extra infrastructure. Such systems are likely to operate in
the near-field (NF), where the performance of localization and sensing is
degraded if a simplified far-field channel model is considered. However, when
taking advantage of the additional geometry information in the NF, e.g., the
encapsulated information in the wavefront, localization and sensing
performance can be improved. In this work, we formulate a joint
synchronization, localization, and sensing problem in the NF. Considering the
array size could be much larger than an obstacle, the effect of partial
blockage (i.e., a portion of antennas are blocked) is investigated, and a
blockage detection algorithm is proposed. The simulation results show that
blockage greatly impacts performance for certain positions, and the proposed
blockage detection algorithm can mitigate this impact by identifying the
blocked antennas.

###### Index Terms:

Near field, localization, sensing, partial blockage, extremely large aperture
array.

##  I Introduction

Localization is expected to be integrated into the communication systems due
to the increased bandwidth and array size with mmWave/THz band signals [1]. A
by-product of localization is the estimated position of surrounding objects or
scattering points from non-line-of-sight (NLOS) paths, which is usually called
(bi-/multi-static) sensing. In the 3rd Gneration Partnership Project (3GPP)
project, positioning in the 5G new radio (NR) has been studied in TR38.855
[2], followed by an expanded study in TR38.895 [3]. In the meantime, many
theoretical analysis and experimental studies have also been carried out, both
showing huge potentials of localization and sensing (L&S) in communication
systems [4, 5, 6]. Consequently, such integration can provide vast
opportunities for Internet-of-Things applications, augmented reality, smart
cities, and other new use cases [7].

Due to the preferable simplicity, the mmWave channels are usually modeled with
the far field (FF) assumption, based on which multi-dimensional signal
processing algorithms such as ESPRIT can be adopted to extract the channel
parameters of interest [8, 9]. The angle/delay information (from phase changes
across antennas/subcarriers) of the line-of-sight (LOS) path and NLOS paths
provide the position information of the target and the incident points (e.g.,
scatter points or objects), respectively [8]. However, the FF assumption is no
longer valid for extremely-large antenna array (ELAA) and wideband systems
[10, 11]. Communications, as well as L&S tasks that are performed in the near
field (NF) experience channel features such as spatial non-stationarity (SNS),
beam squint effect (BSE), and spherical wave model (SWM) [10].

When adopting a FF model that ignores these features, model mismatch happens,
and L&S performance will be affected [12]. However, when considering these
features, extra geometrical information can be obtained, although at the cost
of increased channel model complexity. For example, localization in the NF can
be performed with narrowband signals by exploiting the curvature of arrival
(COA) [13], even under LOS blockage [14, 15]. When considering wideband
signals, joint localization and synchronization can be performed [16]. The BSE
can also be utilized for coverage enhancement thanks to the spatial diversity
spanned by different subcarriers (i.e., squinted beams) [17]. Hence, involving
more accurate NF channel models and developing corresponding signal processing
algorithms are needed.

Figure 1: Illustration of localization and sensing under partial blockage.

In addition to the NF model, another issue accompanying the large array is the
partial blockage of the antennas. Unlike the FF model where the antenna array
is treated as a point target, and the whole antenna’s blockage is considered
if an object exists between the transmitter and the receiver, partial blockage
indicates that a portion of antennas is affected. Similar research works have
studied antenna diagnosis [18] and RIS pixel failures [14], where only few
antennas are impaired. However, a larger portion of antennas might be affected
by blockage, depending on the position and shape of the obstacles between the
transmitter and receivers, making the detection problem challenging for analog
arrays.

In this work, we study the L&S in the NF and consider partial blockage of the
ELAA. The contributions of this work can be summarized as follows: (i) a NF
joint synchronization, localization, and sensing problem under a stationary
scenario is formulated in an uplink single-input-multiple-output (SIMO) system
with a single base station (BS), which was previously not possible using a FF
model; (ii) the effect of partial blockage on localization is evaluated based
on the Kullback–Leibler divergence (KLD) metric; (iii) a partial blockage
detection (PBD) algorithm is developed to detect blocked antennas in an analog
or hybrid array with a limited number of transmissions.

##  II System Model

This section describes the signal and channel models of the localization
system. We consider a 2D uplink SIMO system with $L$ SPs. An $N$-antenna BS is
located along the $y$-axis with its center denoted by ${\bf p}_{\text{B}}$,
and each antenna is located at ${\bf b}_{n}$ ($n=1,\ldots,N$). The user
equipment (UE) and the $\ell$-th SP are located at ${\bf p}_{0}$ and ${\bf
p}_{\ell}$, respectively. The obstacle may appear between the transmitter and
the receiver and block part of the array in the LOS and NLOS paths, as seen in
Fig. 1. For the convenience of performance analysis, we start with a simple
scenario by adopting a point-target model of the SP. More realistic and
complicated models, such as circular targets or extended targets [19] can be
considered in future works.

###  II-A Signal Model

An orthogonal frequency division multiplexing (OFDM)-based transmission is
adopted in this system with $K$ subcarriers and $G$ transmissions. Given the
transmitted signal symbol for the $k$-th subcarrier and $g$-th transmission
$x_{g,k}$ ($|x_{g,k}|=\sqrt{P}$ with $P$ as the average transmit power), the
received signals can be expressed as

${\bf y}_{g,k}={\bf W}_{g}{\bf h}_{k}x_{g,k}+{\bf W}_{g}{\bf n}_{g,k}.$ (1)

Here, ${\bf y}_{g,k}\in\mathbb{C}^{M}$ with $M$ as the number of radio
frequency chains, ${\bf h}_{k}\in\mathbb{C}^{N}$ is the channel vector of the
$k$-th subcarrier that is assumed to be coherent during the $G$ transmissions,
${\bf n}_{g,k}\in\mathcal{CN}(\mathbf{0}_{N},N_{0}W\mathbf{I}_{N})$ is the
noise vector with $N_{0}$ as the noise power spectral density (PSD) and $W$ as
the bandwidth, ${\bf W}_{g}\in\mathbb{C}^{M\times N}$ is the combining matrix
that differs depending on array architectures.111For digital arrays, ${\bf
W}_{g}=\mathbf{I}_{N}$ ($M=N$) and can thus be removed. For fully connected
hybrid arrays consisting $M_{\text{R}}$ RFCs, ${\bf
W}_{g}\in\mathbb{C}^{S\times N}$ with each element
$|w_{i,j}|=\frac{1}{\sqrt{N}}$. For array-of-subarray (AOSA) structures and
assume $N_{\text{S}}$ is the number of antennas for all the $S$ sub-arrays
($N=SN_{\text{S}}$), the combiner can be expressed as ${\bf
W}_{g}=\text{blkdiag}({\bf w}_{g,1}^{\top},\ldots,{\bf
w}_{g,S}^{\top})\in\mathbb{C}^{S\times N}$ with ${\bf
w}_{g,s}\in\mathbb{C}^{S}$ ($s=1,\ldots,S$) as the combiner vector of each SA,
$(\cdot)^{\top}$ as the transpose operation, and $\text{blkdiag}(\cdot)$
indicating the block diagonal operation. In this work, we adopt the AOSA
structure with $S$ SAs with each of them connected to a specific RFC (the
number of RFCs $M$ equals to $S$).

###  II-B Far Field Channel Model

When considering the FF scenario, the channel vector for the $k$-th subcarrier
${\bf h}_{k}^{\text{FF}}\in\mathbb{C}^{{N}}$ can be formulated as

$\displaystyle{\bf h}_{k}^{\text{FF}}=$ $\displaystyle\sum_{\ell=0}^{L}{\bf
h}_{\ell,k}^{\text{FF}}=\sum_{\ell=0}^{L}\alpha_{\ell}\,{\bf
a}_{\ell}(\vartheta_{\ell})D_{k}(\tau_{\ell}),$ (2)

where $L$ is the number of paths, $\ell=0$ indicates the LOS path and $\ell>0$
corresponds to the $\ell$-th NLOS path. For the $\ell$-th path
($\ell=0,\ldots,L$), $\alpha_{\ell}$, ${\bf a}_{\ell}(\vartheta_{\ell})$,
$D_{k}(\tau_{\ell})$ are the complex channel gain, FF steering vector, and
delay component at the $k$-th subcarrier, respectively, with
$\vartheta_{\ell}$ and $\tau_{\ell}$ as the angles-of-arrival (AOA) and delay
of the $\ell$-th path. More specifically,

$\displaystyle\alpha_{\ell}=$
$\displaystyle\rho_{\ell}e^{-j\xi_{\ell}}=\begin{cases}\frac{\lambda_{c}e^{-j\xi_{\ell}}}{4\pi\|{\bf
p}_{{}_{0}}-{\bf p}_{{}_{\text{B}}}\|}&\ell=0,\\\
\sqrt{\frac{c_{\ell}}{4\pi}}\frac{\lambda_{c}e^{-j\xi_{\ell}}}{4\pi\|{\bf
p}_{{}_{0}}-{\bf p}_{\ell}\|\|{\bf p}_{\ell}-{\bf
p}_{{}_{\text{B}}}\|}&\ell>0,\end{cases}$ (3) $\displaystyle{\bf
a}_{\ell}(\vartheta_{\ell})=$
$\displaystyle[e^{-j\pi\frac{N-1}{2}\sin(\vartheta_{\ell})},\ldots,1,\ldots,e^{-j\pi\frac{1-N}{2}\sin(\vartheta_{\ell})}]^{\top},$
(4) $\displaystyle D_{k}(\tau_{\ell})=$ $\displaystyle
e^{-j\frac{2\pi}{\lambda_{k}}\tau_{\ell}}=\begin{cases}e^{-j\frac{2\pi}{\lambda_{k}}(\|{\bf
p}_{{}_{0}}-{\bf p}_{{}_{\text{B}}}\|+\beta)}&\ell=0,\\\
e^{-j\frac{2\pi}{\lambda_{k}}(\|{\bf p}_{{}_{0}}-{\bf p}_{\ell}\|+\|{\bf
p}_{\ell}-{\bf p}_{{}_{\text{B}}}\|+\beta)}&\ell>1,\end{cases}$ (5)

where $\lambda_{c}$ and $\lambda_{k}$ are the wavelengths corresponding to the
central frequency and the $k$-th subcarrier, and $c_{\ell}$ is the Radar cross
section of the $\ell$-th path. Due to the unknown clock offset $\beta$ in
$\tau_{\ell}$, localization cannot be performed in a SIMO system with a single
BS under FF scenarios, even with wideband signals.

###  II-C Near-field Channel Model

By adopting a NF model, the channel vector ${\bf h}_{k}^{\text{NF}}$ can then
be formulated as

$\displaystyle\begin{split}{\bf
h}_{k}^{\text{NF}}=\sum_{\ell=0}^{L}\hbox{\boldmath$\alpha$}_{\ell,k}({\bf
p}_{\ell})\odot{\bf m}_{\ell}\odot{\bf d}_{\ell,k}({\bf
p}_{\ell})D_{\ell,k}({\bf p}_{0},{\bf p}_{\ell},\beta),\end{split}$ (6)

where ${\bf m}_{\ell}\in\mathbb{R}^{N}$ is the masking vector with each
element as either 1 (unblocked) or 0 (blocked). Diffraction may happen
depending on the shape and materials of the object and cause non-zero
coefficients for the blocked antennas [20], which will be evaluated in the
simulation. The frequency-dependent NF channel gain, steering vector, and
delay component $\hbox{\boldmath$\alpha$}_{\ell,k}$, ${\bf d}_{\ell,k}$, and
$D_{\ell,k}$ can be expressed as

$\displaystyle\alpha_{\ell,k,n}({\bf p}_{\ell})=$
$\displaystyle\alpha_{\ell}c_{\ell,k,n}({\bf p}_{\ell}),\ \ c_{\ell,k,n}({\bf
p}_{\ell})=\frac{\lambda_{k}\|{\bf p}_{\ell}-{\bf
p}_{{}_{\text{B}}}\|}{\lambda_{c}\|{\bf p}_{\ell}-{\bf b}_{n}\|},$ (7)
$\displaystyle d_{\ell,k,n}({\bf p}_{\ell})=$ $\displaystyle
e^{-j\frac{2\pi}{\lambda_{k}}(\|{\bf p}_{\ell}-{\bf b}_{n}\|-\|{\bf
p}_{\ell}-{\bf p}_{{}_{\text{B}}}\|)},$ (8) $\displaystyle D_{\ell,k}({\bf
p}_{0},{\bf p}_{\ell},\beta)=$
$\displaystyle\begin{cases}e^{-j\frac{2\pi}{\lambda_{k}}(\|{\bf
p}_{{}_{0}}-{\bf p}_{{}_{\text{B}}}\|+\beta)}&\ell=0,\\\
e^{-j\frac{2\pi}{\lambda_{k}}(\|{\bf p}_{{}_{0}}-{\bf p}_{\ell}\|+\|{\bf
p}_{\ell}-{\bf p}_{{}_{\text{B}}}\|+\beta)}&\ell>1.\end{cases}$ (9)

Note that equations (9) and (5) are identical, and the near-field features are
reflected on (7) and (8).

##  III Localization and Sensing

In this section, we describe the L&S algorithms that ignore the blockage of
the antennas (i.e., assuming no blockage with ${\bf
m}_{\ell}=\mathbf{1}_{N}$), and the PBD is detailed in Sec. IV. The L&S
problem can be formulated as estimating the state vector ${\bf
s}_{\text{N}}=[\alpha_{0},{\bf p}_{0}^{\top},\ldots,\alpha_{L},{\bf
p}_{L}^{\top},\beta]^{\top}$ from the observed signal vector ${\bf y}=[{\bf
y}^{\top}_{1},\ldots,{\bf y}^{\top}_{S}]^{\top}\in\mathbb{C}^{SGK}$ of an AOSA
system with ${\bf
y}_{s}=[y_{1,1,s},\ldots,y_{1,K,s},y_{2,1,s},\ldots,y_{G,K,s}]^{\top}\in\mathbb{C}^{GK}$.
Usually, the complex channel gains $\alpha_{\ell}$ ($\ell=0,\ldots,L$) are
treated as nuisance parameters, and the nuisance-free state vector can be
expressed as ${\bf s}=[{\bf p}_{0}^{\top},\ldots,{\bf
p}_{L}^{\top},\beta]^{\top}$. Next, we propose a multi-stage L&S algorithm to
provide coarse positions of the UE and SPs, and then perform position
refinement using maximum likelihood estimator (MLE).

###  III-A Coarse Localization Algorithm

####  III-A1 Localization with only LOS Path

By segmenting the whole large array into $S$ SAs, we can adopt available FF
algorithms to estimate the AOA $\theta_{s}$ and delay $\tau_{s}$ of the $s$-th
SA separately [21]. For the $s$-th SA, the estimation problem can be
formulated as [22]

$[\hat{\theta}_{s},\hat{\tau}_{s}]=\operatorname*{arg\,min}_{\theta,\tau}\left\|{\bf
y}_{s}-\frac{{\bf u}_{s}^{\sf H}(\theta,\tau){\bf y}_{s}}{\|{\bf
u}_{s}(\theta,\tau)\|^{2}}{\bf u}_{s}(\theta,\tau)\right\|,$ (10)

where ${\bf u}_{s}(\theta,\tau)=\text{vec}({\bf
U}_{s}(\theta,\tau))\in\mathbb{R}^{GK}$ with each entry of ${\bf
U}_{s}(\theta,\tau)$ as $U_{s,g,k}={\bf w}_{g}^{\top}{\bf
a}(\theta)D_{k}(\tau)x_{g,k}$ defined in (4) and (5), and $(\cdot)^{\sf H}$ is
the Hermitian operation. Note that the AOA $\theta$ and delay $\tau$ are
calculated based on the center of each SA denoted as ${\bf p}_{\text{SA},s}$.

Once the channel parameters for each path are estimated, we can then obtain a
coarse location estimate by minimizing the distances between the UE position
and all the AOA direction vectors, as

$\hat{\bf p}_{0}=\operatorname*{arg\,min}_{{\bf p}}\sum_{s=1}^{S}\|{\bf
p}_{\text{SA},s}+{\bf t}(\hat{\theta}_{s})^{\top}({\bf p}-{\bf
p}_{\text{SA},s}){\bf t}(\hat{\theta}_{s})-{\bf p}\|.$ (11)

Here, $t(\theta)=[\cos(\theta),\sin(\theta)]^{\top}$. A coarse clock offset
$\hat{\beta}$ can also be obtained as

$\hat{\beta}=\frac{1}{S}\sum_{s=1}^{S}(\hat{\tau}_{s}-\|{\bf
p}_{\text{SA},s}-\hat{\bf p}_{0}\|).$ (12)

Finally, based on the coarse estimation from (10) and (11), an MLE problem for
LOS-only NF localization can be formulated as

$[\hat{\bf p}_{0},\hat{\beta}]=\operatorname*{arg\,min}_{{\bf p},\beta}\|{\bf
y}-\frac{{\bf v}_{0}^{\sf H}({\bf p},\beta){\bf y}}{\|{\bf v}_{0}({\bf
p},\beta)\|^{2}}{\bf v}_{0}({\bf p},\beta)\|,$ (13)

with ${\bf v}_{0}({\bf p},\beta)=[\text{vec}({\bf V}_{0,1}({\bf
p},\beta))^{\top},\ldots,\text{vec}({\bf V}_{0,S}({\bf p},\beta))^{\top}]$ as
the normalized noise-free received signal in a similar structure as ${\bf y}$,
where each entry of ${\bf V}_{0,s}({\bf p},\beta)$ is $V_{0,s,g,k}={\bf
w}_{g}^{\top}({\bf c}_{k}({\bf p})\odot{\bf d}_{k}({\bf p}))D_{k}({\bf
p},\beta)x_{g,k}$.

####  III-A2 Localization and Sensing with Strong NLOS Paths

When signals from different paths are resolvable, multi-dimensional algorithms
such as ESPRIT can be applied to estimate the channel parameters of each path
directly [23]. By picking up the shortest path (i.e., minimum delay) as the
LOS path, an initial estimate of UE position and clock bias can be obtained
based on (11) and (12). Then, the position of the SPs (assuming correct
channel parameters association of each SP) can be obtained as

$\hat{\bf p}_{\ell}=\operatorname*{arg\,min}_{\bf
p}\begin{bmatrix}\hat{\hbox{$\bm{\theta}$}}_{\ell}-\hbox{$\bm{\theta}$}_{\text{SA}}({\bf
p})\\\
\hat{\hbox{\boldmath$\tau$}}_{\ell}-\hbox{\boldmath$\tau$}_{\text{SA}}({\bf
p})-\hat{\beta}\end{bmatrix}\bm{\mathcal{I}}_{\bm{\eta}_{\ell}}\begin{bmatrix}\hat{\hbox{$\bm{\theta}$}}_{\ell}-\hbox{$\bm{\theta}$}_{\text{SA}}({\bf
p})\\\ \hat{\tau}_{\ell}-\hbox{\boldmath$\tau$}_{\text{SA}}({\bf
p})-\hat{\beta}\end{bmatrix}^{\top},$ (14)

where
$\hat{\hbox{$\bm{\theta}$}}_{\ell}=[\hat{\theta}_{\ell,1},\ldots,\hat{\theta}_{\ell,S}]^{\top}$
and
$\hat{\hbox{\boldmath$\tau$}}_{\ell}=[\hat{\tau}_{\ell,1},\ldots,\hat{\tau}_{\ell,S}]^{\top}$
are the estimated channel parameters of the $\ell$-th path. Vectors
$\hbox{$\bm{\theta}$}_{\text{SA}}({\bf p})=[\theta_{\text{SA},1}({\bf
p}),\ldots,\theta_{\text{SA},S}({\bf p})]$ with $\theta_{\text{SA},s}({\bf
p})=\text{atan2}(p_{1}-p_{\text{SA},s,1},p_{2}-p_{\text{SA},s,1})$, and
$\hbox{\boldmath$\tau$}_{\text{SA}}({\bf p})=[\tau_{\text{SA},1}({\bf
p}),\ldots,\tau_{\text{SA},S}({\bf p})]$ with $\tau_{\text{SA},s}({\bf
p})=\|{\bf p}_{\text{SA},s}({\bf p})-{\bf p}-\hat{\beta}\|$ are the angle and
delay information at SAs for a given ${\bf p}$. The weighting matrix
$\bm{\mathcal{I}}_{\bm{\eta}_{\ell}}$ can be selected based on the statistical
information of the channel parameters
$[\hat{\hbox{$\bm{\theta}$}}^{\top}_{\ell},\hat{\hbox{\boldmath$\tau$}}^{\top}_{\ell}]$.
However, the NLOS paths are usually much weaker than the LOS path, and the
corresponding L&S algorithm will be presented next.

####  III-A3 Localization and Sensing with Weak NLOS Paths

Considering the NLOS path is much weaker, we need to remove the LOS path from
the signal and estimate SPs from the residual signals defined as

$\tilde{\bf y}={\bf y}-\frac{{\bf u}^{\sf
H}(\hat{\hbox{$\bm{\theta}$}}_{0},\hat{\hbox{\boldmath$\tau$}}_{0}){\bf
y}_{s}}{\|{\bf
u}(\hat{\hbox{$\bm{\theta}$}}_{0},\hat{\hbox{\boldmath$\tau$}}_{0})\|^{2}}{\bf
u}(\hat{\hbox{$\bm{\theta}$}}_{0},\hat{\hbox{\boldmath$\tau$}}_{0}),$ (15)

where
$\hat{\hbox{$\bm{\theta}$}}_{0}=[\hat{\theta}_{0,1},\ldots,\hat{\theta}_{0,S}]$
and $\hat{\tau}_{0}=[\hat{\tau}_{0,1},\ldots,\hat{\tau}_{0,S}]$ contains
channel parameters of the LOS path that can be obtained based on (10) to (13),
${\bf u}=[{\bf u}_{1}^{\top},\ldots,{\bf u}_{S}^{\top}]$ is the concatenation
of the normalized noise-free received signal at each SA. The estimation of SPs
can be performed based on (14) using the residual signals obtained in (15).

###  III-B Maximum Likelihood Estimator

Once a coarse estimation of the state vectors ${\bf s}$ is obtained, we can
then formulate a MLE problem similarly to (13) as

$\hat{\bf s}=\operatorname*{arg\,min}_{\bf s}\|{\bf y}-\bm{\Upsilon}({\bf
s}){\bf g}({\bf s})\|,$ (16)

where ${\bf g}({\bf s})=(\bm{\Upsilon}^{\sf H}({\bf s})\bm{\Upsilon}({\bf
s}))^{-1}\bm{\Upsilon}^{\sf H}({\bf s}){\bf y}\in\mathbb{C}^{L+1}$ contains
the gain of each identified path, $\bm{\Upsilon}({\bf s})=[{\bf v}_{0}({\bf
p}_{0},\beta),\ldots,{\bf v}_{L}({\bf p}_{0},{\bf
p}_{\ell},\beta)]\in\mathbb{C}^{SGK\times(L+1))}$. The $\ell$-th ($\ell>0$)
vector in $\bm{\Upsilon}({\bf s})$ can be expressed as ${\bf v}_{\ell}({\bf
p}_{0},{\bf p}_{\ell},\beta)=[\text{vec}({\bf V}_{\ell,1}({\bf p}_{0},{\bf
p}_{\ell},\beta))^{\top},\ldots,\text{vec}({\bf V}_{\ell,S}({\bf p}_{0},{\bf
p}_{\ell},\beta))^{\top}]$, where each entry of the matrix ${\bf
V}_{\ell,s}({\bf p}_{0},{\bf p}_{\ell},\beta)$ is given by
$V_{\ell,s,g,k}={\bf w}_{g}^{\top}({\bf c}_{k}({\bf p}_{\ell})\odot{\bf
d}_{k}({\bf p}_{\ell}))D_{k}({\bf p}_{0},{\bf p}_{\ell},\beta)x_{g,k}$. The
Cramér-Rao bound (CRB) can be used to benchmark the algorithm performance as
detailed in [24, Sec. 3.4], which will not be detailed in this work.

##  IV Partial Blockage Detection

Once the position of UE and SP are estimated, the PBD algorithm can be
performed. In this section, we focus on the partial blockage of LOS path only
and propose a heuristic algorithm to detect the blocked antennas.

###  IV-A The Effect of Partial Blockage

Partial blockage affects localization and sensing performance differently
depending on the knowledge of the blockage. If we know the blocked antennas,
the L&S performance loss is mainly due to the reduced array size (i.e., only
visible antennas provide information). However, if we ignore the blockage and
perform L&S tasks by assuming all the antennas are visible to the UE, model
mismatches are introduced, resulting in severe performance degradation. More
specifically, the estimation error can be lower bounded by [25]

$\text{LB}(\bar{\bf s},{\bf s}_{0})=\text{MCRB}({\bf s}_{0})+(\bar{\bf s}-{\bf
s}_{0})(\bar{\bf s}-{\bf s}_{0})^{\top},$ (17)

which contains an MCRB term and a biased term characterized by pseudotrue
parameters ${\bf s}_{0}$, with $\bar{\bf s}$ as the true state [25]. The
pseudotrue parameter ${\bf s}_{0}$ is defined as the vector that minimizes the
KLD between the probability density function of the blocked model $f_{B}$
(signal model from (1) and (6)) and the assumed model $f_{U}$ (i.e., by
assuming no blockage with ${\bf m}_{\ell}=\mathbf{1}_{N}$) as

${\bf s}_{0}=\operatorname*{arg\,min}_{\bf s}D_{\text{KL}}(f_{\text{B}}({\bf
y}|\bar{\bf s})||f_{\text{U}}({\bf y}|{\bf s})).$ (18)

At high SNR, MCRB is ignorable and the biased term dominates, which limits the
fundamental mismatched error. In other words, the estimation that ignores the
effect of partial blockage will introduce a biased term, resulting in a result
${\bf s}_{0}$ with mismatch error instead of the true state $\bar{\bf s}$.

###  IV-B Partial Blockage Problem Formulation

####  IV-B1 Blockage Detection with Sufficient Observations

It is easy to detect the partial blockage of the LOS path by simple
thresholding in digital arrays. For analog and hybrid arrays, similar
approaches can be applied to recover the beamspace channel to the element-
space channel with sufficient observations (i.e., no less than the number of
antennas in analog arrays). Take the DFT codebook in an analog array as an
example; concatenating the observed signals as $\tilde{\bf y}=[{\bf
y}_{1},\ldots,{\bf y}_{G}]^{\top}$ and multiplying by an IFFT matrix will
result in an element space channel vector (with noise). When other combiner
vectors are adopted (e.g., ${\bf W}_{g}\in\mathcal{R}^{M\times N}$, $M>1$),
this process can be done by multiplying a pseudoinverse matrix of $\tilde{\bf
W}=[{\bf W}_{1},\ldots,{\bf W}_{G}]$ to the concatenated observations
$\tilde{\bf y}=[{\bf y}_{1,1},\ldots,{\bf y}_{1,M},\ldots,{\bf
y}_{G,M}]^{\top}$ to obtain an element-space channel vector, with ${\bf
y}_{g,m}$ as the received signal of the $g$-th transmission at the $m$-th RFC.

####  IV-B2 PBD with a Limited Number of Observations

When there is only a limited number of observations (i.e., less than the
number of antennas in analog arrays), the PBD task becomes challenging. If we
assume the blocked antennas are adjacent to each other, the detection problem
becomes feasible by implementing a heuristic way. Specifically, we want to
find the masking vector where a continuous sequence of antennas is all zeros.
By considering the blockage of the LOS path (given the fact that the NLOS
paths are much weaker), the problem can be formulated as finding the masking
vector ${\bf m}$ that minimizes the objective function $J(\hat{\bf s},{\bf
m})$ given by

$\hat{\bf m}=\operatorname*{arg\,min}_{{\bf m}}J(\hat{\bf s},{\bf
m})=\operatorname*{arg\,min}_{{\bf m}}||{\bf y}-\tilde{\bm{\Upsilon}}(\hat{\bf
s},{\bf m}){\bf g}(\hat{\bf s})||,$ (19)

where $\tilde{\bm{\Upsilon}}=[{\bf u}_{0}({\bf p},\beta)\odot{\bf m},{\bf
u}_{1},\ldots,{\bf u}_{L}]$ and ${\bf g}(\hat{\bf s})$ is defined in (16). The
detection accuracy can be defined as

$\text{Accuracy}=1-\frac{\|\hat{\bf m}-\bar{\bf m}\|^{2}}{N}.$ (20)

###  IV-C The Proposed Algorithm

The detection problem in (19) is an NP-hard integer programming. In the next,
We propose a low computational complexity method for blockage detection. At
the first step, we assume only one antenna is blocked, resulting in
$N_{\text{S}}+1$ different candidate masking vectors ${{\bf
m}_{<0>},\ldots,{\bf m}_{<N_{\text{S}}>}}$ (where ${\bf m}_{<i>}$ has all ones
except the $i$-th element as zero, and ${\bf m}_{<0>}=\mathbf{1}_{N}$). By
calculating the cost based on (19), the masking vector with the lowest value
will be considered, and the blockage index is denoted as $i_{C}$. Note that if
the vector $m_{<0>}$ has the lowest cost, no blockage happens. Starting from
${\bf m}_{<i_{C}>}$, we evaluate the masking vectors by including left
elements ${\bf m}_{<i:i_{C}>}$ (where ${\bf m}_{<i:j>}$ has all ones except
the $i$-th to $j$-th entries), until the cost function is not decreasing or
reaching the first element and we denote the leftmost blocked antenna as
$i_{L}$. Similar can be performed for ${\bf m}_{<i_{L}:i>}$ by incorporating
the right antennas, and ${\bf m}_{<i_{L}:i_{R}>}$ will be taken as the masking
vector. Once the masking vector has been obtained, the localization and
sensing results can also be refined iteratively. The algorithm can be found in
Algorithm 1.

Algorithm 1 Partial Blockage Detection

1:— Heuristic Detection —

2:Input: $\tilde{\bf Y}$, $\hat{\bf s}$

3:$i_{\text{C}}=\operatorname*{arg\,min}_{i}J(\hat{\bf s},{\bf m}_{<i>})$
($i=0,1,\ldots,N$) using (19)

4:if $i_{\text{C}}\neq 0$ then

5: $i_{\text{L}}\leftarrow i_{\text{C}}$, $i_{\text{R}}\leftarrow
i_{\text{C}}$

6: if $i_{\text{L}}\neq 1$ then

7: $i_{\text{L}}=\operatorname*{arg\,min}_{i}J(\hat{\bf s},{\bf
m}_{<i:i_{\text{C}}>})$ ($i=1,\ldots,i_{\text{C}}-1$)

8: if $i_{\text{R}}\neq N$ then

9: $i_{\text{R}}=\operatorname*{arg\,min}_{i}J(\hat{\bf s},{\bf
m}_{<i_{\text{L}}:i>})$ ($i=i_{\text{C}}+1,\ldots,N$) return $\hat{{\bf
m}}\leftarrow{\bf m}_{<i_{\text{L}}:i_{\text{R}}>}$

10:— Iterative Refinement (Optional) —

11:Estimate the state vector $\hat{\bf s}$ by solving (16) with ${\bf y}$,
$\hat{{\bf m}}$ as inputs.

12:Repeat steps 1-8 for a refined $\hat{\bf m}$

##  V Simulation

We consider a $28\,\mathrm{GHz}$ system with a bandwidth
$W=200\,\mathrm{MHz}$, and $K=10$ subcarriers for localization. A single-
antenna UE located at ${\bf p}_{0}=[2,4]^{\top}\mathrm{m}$ and one SP located
at ${\bf p}_{1}=[2,-2]^{\top}\mathrm{m}$, and the array size is set as
$N=100$. For reference, the Fresnel distance and Fraunhofer distance are
$2.2\,\mathrm{m}$ and $50\,\mathrm{m}$, respectively. We assume $M=4$ RFCs are
connected to $S=4$ SAs (e.g., the first SA contains antennas $1$-$25$), and
the number of transmissions is set as $G=25$. The RCS coefficient for the SP
is set as $c_{\text{RCS}}=0.5\,\mathrm{m^{2}}$, noise figure is set as
$13\,\mathrm{dBm}$, and the noise PSD is set as $-173.855\,\mathrm{dBm/Hz}$.

###  V-A Evaluation of the Estimator

We evaluate the root mean squared error (RMSE) of the estimated UE and SP
positions with different estimators, as shown in Fig. 2. The figure shows that
the SA-based solution in (11) provides limited accuracy. With coarse
estimation based on (13), ${\bf p}_{\text{U}}$ estimation can be improved but
will deviate from the bound at high SNR due to the ignorance of weak NLOS
path. Similar can be found in SP position results, where the deviation happens
because of the residual error of LOS removal. However, with joint optimization
of both paths based on (16), the RMSEs can attain the bound. Note that we
assume the number of SPs is known, and a wrong number of $L$ will degrade
localization and sensing performance. Compared with the RMSE of UE position
that can provide good estimations at $-10\,\mathrm{dBm}$, satisfactory sensing
results can only be obtained at high frequencies due to the weak signals that
are easy to be interfered with by the LOS signal and background noise.
Finally, the CRB of ${\bf p}_{\text{U}}$ and ${\bf p}_{\text{S}}$ are plotted
in the black curve to benchmark the estimators.

$-10$$0$$10$$20$$30$$40$$50$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$Transmit
Power [dBm]RMSE/Error Bounds [m]${\bf p}_{\text{U}}$-SA${\bf
p}_{\text{U}}$-Coarse${\bf p}_{\text{S}}$-Coarse${\bf p}_{\text{U}}$-Fine${\bf
p}_{\text{S}}$-Fine${\bf p}_{\text{U}}$-CRB${\bf p}_{\text{S}}$-CRB

Figure 2: RMSE of the estimated UE and SP positions.

###  V-B Effect of Partial Blockage

To evaluate the effect of partial blockage on localization in terms of the
biased term. The UE is located on a grid within a $5\times 16\,\mathrm{m^{2}}$
with an $1\,\mathrm{m}$ interval. The pseudotrue locations are plotted in red,
with the segment representing the Biased term. In (a), where only one antenna
is blocked, we find that the effect is ignorable. When more antennas are
blocked, we can see that the target at a large angle of AOA will be affected
more severely, and the biased term at different locations shows a different
pattern. When a partial blockage happens at different locations, we can see
that the blockage of middle antennas has less effect on localization,
indicating less bias introduced. As a result, detecting partial blockage and
identifying blocked antennas can help in localization and sensing.

(a)

(b)

(c)

(d)

Figure 3: The effect of partial blockage on localization under different
scenarios (blue circle indicates the ground truth and the red cross indicates
the biased estimation): (a) antenna $100$ is blocked; (b) antennas $96$-$100$
are blocked; (c) antennas $76$-$80$ are blocked; (d) antennas $56$-$60$ are
blocked.

###  V-C Evaluation of Partial Blockage Detection Algorithm

The cost function after performing steps $1$-$9$ in Algorithm 1 is shown in
Fig. 4 (averaged with $100$ times Monte Carlo simulations). From the benchmark
scenario with a limited number of transmission and transmit power (blue curve
with circle markers), the averaged cost values are close to each other,
indicating an inaccurate detection is likely to happen. With increased
transmissions (red curves with square markers), two local minima (antenna
index $6$ and $10$) are more visible, indicating the boundary of the blocked
antennas. The averaged cost functions with higher transmit power for different
blocked antennas (green curve with diamond markers and black curve with
triangle markers) are also shown in the figure, indicating these setups are
likely to be successfully detected.

$0$$5$$10$$15$$20$$25$$0$$0.1$$0.2$Antenna IndexCost
Value$G=5,P=20\,\mathrm{dBm}$ (6-10)$G=20,P=20\,\mathrm{dBm}$
(6-10)$G=20,P=30\,\mathrm{dBm}$ (6-10)$G=20,P=30\,\mathrm{dBm}$ (11-15)

Figure 4: Cost function of the heuristic algorithm (with partial blockage in
antennas $6$-$10$ and $11$-$15$).

$0$$5$$10$$15$$20$$25$$30$$0.6$$0.8$$1$Number of Transmissions
$G$AccuracyThresholdingHeuristicLow Transmit PowerMore BlockageNon-zero
MasksBiased ${\bf p}_{\mathrm{U}}$

Figure 5: The evaluation of the PBD methods under different scenarios.

We take the first SA with antenna index $1$-$25$) to evaluate two different
partial blockage methods with different known UE locations (i.e., a true
location and a biased location). The default parameters are set as
$P=20\,\mathrm{dBm}$, $c=0.5\,\mathrm{m^{2}}$, with antennas $5$-$10$ in the
SA blocked as the benchmark. Other scenarios such as lower power
($P=0\,\mathrm{dBm}$), non-zero masks (the coefficient follows
$\mathcal{CN}(0.2,0.2)$ instead of $0$) and more blockage (antennas $5$-$15$
are blocked), biased ${\bf p}=[2.0691,4.1377]^{\top}\mathrm{m}$ (based on
(18)) are also evaluated. We can see that the thresholding method (black
curve) only works for a sufficient number of transmissions ($G\geq 25$), and
accuracy decreases with a reduced value of $G$. When $G=1$, no blocked
antennas can be detected, resulting in a constant value of $1-6/25=0.76$.
However, the heuristic method (blue curve) shows a much-improved performance.
The results also show that a lower transmit power, more blockages, and a wrong
assumption of the masking vector decrease detection accuracy. However, a
biased position estimate has a limited effect on the accuracy due to the fact
that the biased position minimizes the KLD, providing similar signal
observations as the ground truth positions.

##  VI Conclusion

In this work, we studied a SIMO uplink joint synchronization, localization,
and sensing problem in the NF scenario with a single BS, which was previously
impossible with a far-field model. A subarray-based coarse localization and
sensing algorithm is performed, followed by an MLE-based refinement. We also
analyzed the impact of localization under partial blockage and proposed a
heuristic blockage detection algorithm to mitigate the effect of blockage.
However, the reflection and diffraction of the blockage itself are not
considered, and joint localization and PBD in 3D space need to be considered
in future work.

## Acknowledgment

This work has been supported by the SNS JU project 6G-DISAC under the EU’s
Horizon Europe research and innovation programme under Grant Agreement No
101139130.

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  *[NLOS]: non-line-of-sight
  *[3GPP]: 3rd Gneration Partnership Project
  *[NR]: new radio
  *[L&S]: localization and sensing
  *[FF]: far field
  *[LOS]: line-of-sight
  *[ELAA]: extremely-large antenna array
  *[NF]: near field
  *[SNS]: spatial non-stationarity
  *[BSE]: beam squint effect
  *[SWM]: spherical wave model
  *[COA]: curvature of arrival
  *[SIMO]: single-input-multiple-output
  *[BS]: base station
  *[KLD]: Kullback–Leibler divergence
  *[PBD]: partial blockage detection
  *[SPs]: scattering point
  *[UE]: user equipment
  *[SP]: scattering point
  *[OFDM]: orthogonal frequency division multiplexing
  *[PSD]: power spectral density
  *[RFCs]: radio frequency chain
  *[AOSA]: array-of-subarray
  *[SA]: sub-array
  *[SAs]: sub-array
  *[RFC]: radio frequency chain
  *[AOA]: angles-of-arrival
  *[MLE]: maximum likelihood estimator
  *[CRB]: Cramér-Rao bound
  *[RMSE]: root mean squared error