# 5G Multi-BS Positioning:
A Decentralized Fusion Scheme

Sharief Saleh*  — , Qamar Bader*  — , Mohamed Elhabiby*  — , Aboelmagd
Noureldin*  — , Sharief Saleh, Qamar Bader, and Aboelmagd Noureldin are with
the Department of Electrical and Computer Engineering, Queen’s University,
Kingston, ON K7L 3N6, Canada, and also with the Navigation and Instrumentation
(NavINST) Lab, Department of Electrical and Computer Engineering, Royal
Military Collage of Canada, Kingston, ON K7K 7B4, Canada (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>nourelda@queensu.ca).Mohamed
Elhabiby is with Micro Engineering Tech. Inc., and also with the Public Works
Department, Ain Shams University, Cairo 11566, Egypt (e-mail:
mmelhabi@ucalgary.ca).

###### Abstract

Fifth-generation (5G) networks are expected to provide high-precision
positioning estimation utilizing mmWave signals in urban and downtown areas.
In such areas, 5G base stations (BSs) will be densely deployed, allowing for
line-of-sight (LOS) communications between the user equipment (UE) and
multiple BSs at the same time. Having access to a plethora of measurement
sources grants the need for optimal integration between the BSs to have an
accurate and precise positioning solution. Traditionally, 5G multi-BS fusion
is conducted via an extended Kalman filter (EKF), that directly utilizes range
and angle measurements in a centralized integration scheme. Such measurements
have a non-linear relationship with the positioning states of the filter,
giving rise to linearization errors. Counter to the common belief, an
unscented Kalman filter (UKF) will fail to totally eradicate such
linearization errors. In this paper, we argue that a decentralized integration
between 5G BSs would fully avoid linearization errors within the filter and
significantly enhance the positioning performance. This is done by fusing
position measurements instead of directly fusing range and angle measurements,
which inherently leads to a linear measurement model by design. The proposed
decentralized KF method was evaluated in a quasi-real simulation setup
provided by Siradel using a real trajectory in Downtown Toronto. The
experiments compared the performance of decentralized KF integration to that
of centralized EKF and UKF integration schemes. It was shown that the proposed
method outperformed both UKF and EKF implementations in multiple scenarios as
it significantly decreased the RMS and maximum 2D positioning errors,
achieving decimeter-level accuracy for $93.9\%$ of the time.

###### Index Terms:

5G; angle of departure (AOD); autonomous vehicles (AVs); decentralized sensor
fusion; Kalman filter (KF); localization; mm-Wave; positioning; round trip
time (RTT).

##  I Introduction

Precise positioning and navigation solutions are becoming of considerable
importance in recent years due to the growing research in highly-intelligent
internet of things (IoT) applications, context-aware services, and location-
based services (LBS). Additionally, precise position information is necessary
for safety-critical applications including self-driving cars; which require
higher levels of autonomy. Such systems compel centimeter-level of absolute
and relative positioning accuracy, low latency, and high reliability[1].
Wireless-based positioning has been utilized for decades to provide
positioning solutions for applications where high accuracy is subordinate.
Such technologies utilize time-, angle-, and power-based measurements to find
the user equipment’s (UE) position by means of trilateration, triangulation,
and hybrid positioning techniques [2]. Trilateration employs ranges acquired
by three or more base stations (BSs) to obtain the position of the UE.
Triangulation, on the other hand, exploits multiple relative angle
measurements to estimate the position of the UE. Lastly, hybrid positioning
approaches utilize both range and angle measurements to get the location of
the user, which can be achieved through a single BS. Global navigation
satellite systems (GNSS) for instance, can provide a few-meter level of
accuracy in standalone operation and down to cm-level of accuracy when aided
by real-time kinematic (RTK) technology [3]. However, due to multi-path and
signal blockage, GNSS-based positioning solutions deteriorate in urban canyon
environments. On the other hand, WiFi and ultra-wideband (UWB) are capable of
attaining sub-meter level of accuracy, yet, both technologies suffer from
short-range operation ranging between $50$ m to $200$ m[4] [5], rendering them
advantageous only in indoor environments. On the contrary, 5G new radio (NR)
small-cells are expected to be densely deployed in deep-urban for
communication purposes with an inter-cell distance of $200$ to $500$ meters
[6]. This will increase the chance of line-of-sight (LoS) communications with
the user, which is important for LOS-based positioning algorithms.
Furthermore, 5G mmWave positioning signals will span a bandwidth of up to
$400$ MHz, yielding accurate time-based measurements estimation, such as time
of arrival (TOA), round trip time (RTT), and time difference of arrival
(TDOA), as well as high multi-path (MP) resolvability in the time domain.
Additionally, the 5G frequency range 2 (FR2) operates on high-frequency mmWave
signals spanning from $28$ GHz to $52$ GHz, allowing for the emergence of
massive MIMO. Thus, 5G receivers will comprise a large number of antennas
enabling accurate angle-based measurements such as angle of arrival (AOA) and
angle of departure (AOD), as well as high MP resolvability in the angle
domain.

In order to optimize the estimation of the UE’s position considering noisy
range and angle measurements, optimal state estimators are needed. The most
commonly used filters are the Kalman filter (KF) [7], the extended Kalman
filter (EKF) [8] and the unscented Kalman filter (UKF) [9]. While KF assumes
the linearity of both the state transition and the observation models, the
remaining stated filters do not oblige the same requirement. Additionally, KF,
EKF, and UKF assume normally distributed process noise and measurement noise
to function optimally. Breaching any of these requirements will revoke the
filters’ optimality. The aforementioned filters can be utilized in centralized
or decentralized state estimation schemes, also known as tightly coupled (TC)
and loosely coupled (LC) integration schemes, respectively. TC and LC
integration schemes are well-known in the GNSS/INS community. Centralized/TC
schemes integrate all technologies/BSs on the measurement level within a
single-stage filter. Decentralized/LC schemes, on the other hand, involve
multi-stage positioning where each technology/BS computes its own estimate of
the position, and then integration on the position level takes place in a
cascaded manner. As will be seen in Section II, all methods proposed in the
literature utilize a centralized integration scheme. Such a scheme inherently
revokes the optimality of the filters, in the case of 5G positioning, as the
relationship between the states and the measurements is non-linear.
Simulations of the detrimental effects of centralized integration will be
shown in Section V-B. It is then shown that the decentralized integration of
multi-BSs totally avoids the linearization issues as it adapts a linear
measurement model by design. Hence, the contributions of this paper are as
follows:

  1. 1.

We present an extensive analysis of the drawbacks of centralized/TC
integration schemes.

  2. 2.

A novel decentralized/LC multi-BS 5G positioning approach is proposed, which
will avoid the disadvantages of centralized integration schemes.

  3. 3.

For validation, we assess three Bayesian estimators, namely, LC-KF, TC-EKF,
and TC-UKF in a quasi-real testing environment.

The remainder of the paper is organized as follows: Section II presents
related works on 5G positioning using optimal estimators. Section III
establishes the system model and Section IV exhibits the various
implementations of Kalman filters. Section V thoroughly analyzes the
centralized fusion scheme errors and proposes 5G decentralized integration as
a strong contender. Section VI provides details on the experimental setup and
presents the results along with a discussion. Finally, Section VII concludes
the paper.

##  II Literature Review

Various works have investigated LOS 5G positioning using different estimation
techniques. Yet, a common theme that is prevalent throughout the literature is
the use of centralized integration schemes. It is worth noting that methods
that rely on trilateration or triangulation are considered centralized by
nature. Authors in [10] used EKF to estimate the UE’s position employing AOD
and TOA measurements, giving rise to high linearization errors due to the use
of highly non-linear observation models, achieving sub-meter accuracy for
$75\%$ of the time. In [11, 12] an accuracy of the sub-meter level was
accomplished for $90\%$ of the time through utilizing TDOA trilateration.
However, this was achieved assuming that the UE is connected to three BSs
simultaneously at all times, which is an impractical assumption. Authors in
[13] proposed a hybrid positioning scheme based on TDOA, and AOA using an EKF,
achieving a sub-3-meter level of accuracy for $95\%$ of the time.
Nevertheless, it still endures the same limitations as observed in previous
works. In [13], authors have integrated TDOA and AOA measurements in a
centralized manner and achieved sub-3 meters of accuracy $95\%$ of the time.
The work in [14] has mitigated some of these limitations by endorsing a non-
linear state transition model (NLSTM) as an alternative to the constant
velocity model. This model makes use of the forward velocity of the vehicle
and its heading angle to estimate the vehicle’s position. The proposed NLSTM
surpassed former works, attaining sub-2.3 meters of accuracy for $95\%$ of the
time. Inspired by [14], authors in [15] utilized the same NLSTM model but with
centralized integration of TOA, AOA, and heading measurements instead. The
heading of the vehicle measurement was artificially computed by combining AOA,
AOD, and TOA measurements. The use of TOA measurements without proper
estimation of the clock bias between the UE and the BS is not practical.
Additionally, the proposed method utilizes correlated measurements without
proper manipulation of the measurement covariance matrix, which leads to the
sub-optimality of the EKF. The proposed method was able to achieve $0.34$ m
error for $95\%$ of the time by using two BSs only. Authors in [16] integrate
rough TOA estimates in a centralized fashion to directly position the UE via a
compressed sensing approach. The proposed method achieved a sub-meter level of
accuracy for around $90\%$ of the time. An EKF-based triangulation scheme
based on three AOD measurements was proposed by [17]. The method also
estimates the orientation bias of the BS due to errors in installation. The
proposed triangulation method was able to achieve a sub-meter level of
accuracy for $88\%$ of the time. Yet, the assumption of having access to three
BSs that are in LOS with the UE is usually impractical. The work reported in
[18] proposes a least-squares approach to directly integrate TDOA and AOD
measurements from multiple BSs. The minimum number of LOS BSs needed for such
an approach is two BSs, which is not usually available to the UE. Moreover, LS
approaches neither have the optimal weighting capabilities nor the tracking
capability of KF implementations. The proposed LS method was reported to have
the ability to estimate the position with a sub-meter level of accuracy
without reporting the confidence interval on such an assumption. Authors in
[19] propose two centralized EKF-based trilateration solutions that utilize
TDOA measurements. The first method proposes a modified implementation of the
EKF, where they added additional noise to the measurement’s covariance matrix
and the state covariance matrix to account for the non-linearities and the
non-gaussian distribution of the noises. The second method proposes a deep-EKF
scheme, which utilizes neural networks (NN) to replace the innovation sequence
of the modified EKF to better capture non-linearities. The results of the
proposed method show a clear improvement in the RMS error compared to
conventional EKF implementations, yet, it did not fully report the statistics
of the solution. The work described in [20] directly fuses TDOA and AOA
measurements to estimate the position of the UE via a Gauss-Newton LS method.
The proposed method also includes a BS exclusion mechanism based on NLOS
detection and synchronization anomalies. The proposed method shows poor
positioning performance due to the use of cm-waves. Limited work was done on
the use of UKF for 5G standalone positioning. In [21] a UKF-based estimator
was employed to simultaneously estimate AOA, TOA, and the position of the UE
through a centralized scheme, accomplishing a sub-meter level of accuracy.
Nevertheless, they have assumed BSs are deployed every $50$ m, which does not
follow the standard for small cells deployed in urban areas as seen in [22].
It is worth noting that other works proposed joint positioning and
synchronization schemes like [23, 24], and others have proposed joint
estimation of UE position and positioning measurables as in [25, 26].
Moreover, works in [21, 27] have proposed joint estimation of UE position,
positioning measurables as well as UE-BS synchronization. Works that jointly
estimate positioning measurables usually utilize a multi-stage filtering
approach, where positioning measurables are first estimated through an EKF/UKF
filter, then positioning takes place in a separate filter. Such schemes are
not to be considered decentralized schemes, as the positioning solution is
still computed within a single centralized filter. Works related to UE-BS
synchronization conduct time bias estimation concurrently with UE position and
orientation estimation, and are also conducted in a centralized fashion. In
conclusion, all of the aforementioned works utilize centralized integration
schemes to integrate the BSs’ various raw time and angle measurements. This
will prove detrimental to the accuracy of the 5G positioning solution.

##  III System Model

5G NR has enhanced the LTE’s downlink (DL) positioning reference signal (PRS)
and uplink (UL) sounding reference signal (SRS) to allow for more precise
positioning measurements. With the aid of these signals, TOA, RTT, and TDOA
are computed through correlation with the configured reference signal.
Moreover, AOD is inherently computed within the beamforming training sequence
by comparing the received power of multiple PRS signals. In this paper, the
authors consider positioning at the UE side, and thus, DL-PRS and UL-SRS
signals are utilized to compute DL-AOD and RTT measurables. Such measurables
are then used to compute the relative distance between the UE and the BS
$(r)$, as well as the azimuth angle $(\theta)$ as seen in (1) and (2).

$\begin{split}r&=\sqrt{\Delta x^{2}+\Delta y^{2}+\Delta z^{2}}\\\ \end{split}$
(1)

$\theta=\arctan\left(\frac{\Delta y}{\Delta x}\right)$ (2)

Where $(\Delta x,\Delta y,\Delta z)$ denote the difference between the UE’s
position $(x,y,z)$ and the BS position $(x_{BS},y_{BS},z_{BS})$. Next, the
range and angle measurements offered by each connected BS can either be fused
in a centralized or decentralized fashion. In centralized fusion, all ranges
and angles are fused in a single filter to compute the UE’s position as seen
in Fig. 1. On the other hand, in decentralized fusion, the UE first computes
multiple position estimates by using individual BS measurables through an
open-loop geometry-based calculation. Subsequently, an optimal estimator fuses
these position estimates into a single position estimate as seen in Fig. 2.
Finally, to explore the true potential of 5G-NR’s small-cell densification,
the setting of autonomous driving was set to be in downtown areas where urban
canyons and high-rise buildings surround the vehicle. The simulation setup,
shown in Section VI-A, was designed as per 3GPP release 16 standards for deep-
urban environments, which entails that BSs will be deployed $200-500$ m apart
[28]. Designing such a setting showed that a UE might be connected to a
maximum of three BSs at a time with a probability of $30\%$. It also showed
that the UE will have access to two BSs and sometimes a single BS for $46\%$
and $21\%$ of the time, respectively. Finally, a total loss of LOS connection
to all BSs occurs in rare cases with a probability of $3\%$ of the time. Thus,
an NLOS detection algorithm based on range comparison between RTT and RSS
measurements, proposed in [29], will be utilized to exclude NLOS BSs. In a
nutshell, the method exploits discrepancies between time-based and power-based
measurements to detect NLOS communications. This is done by comparing the
difference in range computed by TOA and RSS measurements to a preset threshold
as seen in (3).

$\text{LOS}=\begin{cases}\text{True},&r_{TOA}-r_{RSS}\leq\varepsilon\\\
\text{False},&\text{otherwise.}\end{cases}$ (3)

Where $r_{TOA}$ and $r_{RSS}$ are the time-based and power-based range
measurements, respectively, and $\varepsilon$ is the preset threshold set to
$80$ m, as per [29].

Figure 1: Block diagram of centralized 5G multi-BS integration. Figure 2:
Block diagram of decentralized 5G multi-BS integration.

##  IV Kalman Filter Realizations

The Kalman filter is an optimal iterative estimator for linear systems with
Gaussian noise, illustrated in Alg. 1. With these two assumptions, the KF is
able to propagate, transform, and manipulate the states’ probability density
function (PDF) by only tracking their mean and covariance expressed by
$\boldsymbol{x}$ and $\boldsymbol{P}$, respectively. The KF consists of two
consecutive stages of state prediction and correction. In the state prediction
stage, the KF propagates the posterior state estimate
$\boldsymbol{x}_{k-1}^{+}$ and the covariance matrix
$\boldsymbol{P}_{k-1}^{+}$ from the previous epoch, $k-1$, to the current
epoch, $k$, via the transition matrix
$\boldsymbol{\Phi}=\boldsymbol{I}+\boldsymbol{F}$. Where $\boldsymbol{I}$ is
an identity matrix and $\boldsymbol{F}$ is the transition model. The outputs
of such transformation are the a priori state estimate
$\boldsymbol{x}_{k}^{-}$ and the covariance matrix $\boldsymbol{P}_{k}^{-}$.
Noise caused by external sources within the transition model is described by
the process covariance matrix $\boldsymbol{Q}$. The process noise is added to
the a priori covariance matrix with the aid of the noise coupling matrix
$\boldsymbol{G}$. The state correction stage utilizes the measurement vector
$\boldsymbol{z}_{k}$ along with the Kalman gain $\boldsymbol{K}_{k}$ to update
the a priori state and covariance estimate into a posteriori estimates
$\boldsymbol{x}^{+}_{k}$ and $\boldsymbol{P}^{+}_{k}$. The Kalman gain
$\boldsymbol{K}_{k}$ is computed with the aid of $\boldsymbol{P}_{k}^{-}$ and
the measurement covariance matrix $\boldsymbol{R}$. Moreover, the measurement
observation matrix $\boldsymbol{H}_{k}$ is used to transform
$\boldsymbol{P}_{k}^{-}$ and $\boldsymbol{x}^{-}_{k}$ from the state domain to
the measurement domain and vice versa.

It is worth noting that the main difference between linear KF and EKF is that
EKF deals with non-linear transition models and non-linear state-measurement
models. In case of encountering non-linearity in the transition model,
$f(\boldsymbol{x}^{+}_{k-1})$ is used to translate the state instead of
$\boldsymbol{\Phi}\boldsymbol{x}^{+}_{k-1}$, and $\boldsymbol{F}$ is computed
by finding the Jacobian of $f(\boldsymbol{x})$. Likewise, in case of
encountering non-linearity between the states and the measurements, then
$h(\boldsymbol{x}^{-}_{k})$ is used instead of
$\boldsymbol{H}_{k}\boldsymbol{x}^{-}_{k}$ to transform the states to
measurements, and $\boldsymbol{H}_{k}$ is computed by finding the Jacobian of
$h(\boldsymbol{x})$. It is worth noting that $\boldsymbol{\Phi}$ and
$\boldsymbol{H}_{k}$ are still utilized to transition and transform the state
covariance matrix $\boldsymbol{P}$, respectively.

The use of UKF resolves the non-linear approximation drawbacks of the EKF by
transforming the state’s PDFs through non-linear functions. The UKF conducts
PDF transformation by generating $2N+1$ sigma points $\boldsymbol{\chi}_{k}$,
where $N$ is the number of connected BSs. Then, the UKF propagates the
individual $\boldsymbol{\chi}_{k}$ points through the non-linear function and
finally computes the mean and covariance of the propagated points to find
$\boldsymbol{x}$ and $\boldsymbol{P}$, respectively. The UKF algorithm is
illustrated in Alg. 2.

initialization:
$\boldsymbol{x}^{+}_{0},\boldsymbol{P}^{+}_{0},\boldsymbol{Q},\boldsymbol{R},k=0$

while  _positioning_ do

$k=k+1$

$\boldsymbol{x}^{-}_{k}=\boldsymbol{\Phi}\boldsymbol{x}^{+}_{k-1}$

$\boldsymbol{P}^{-}_{k}=\boldsymbol{\Phi}\boldsymbol{P}^{+}_{k-1}\boldsymbol{\Phi}^{T}+\boldsymbol{GQG}^{T}$

$\boldsymbol{K}_{k}=\boldsymbol{P}^{-}_{k}\boldsymbol{H}^{T}_{k}(\boldsymbol{H}_{k}\boldsymbol{P}^{-}_{k}\boldsymbol{H}^{T}_{k}+\boldsymbol{R})$

$\boldsymbol{x}^{+}_{k}=\boldsymbol{x}^{-}_{k}+\boldsymbol{K}_{k}(\boldsymbol{z}_{k}-\boldsymbol{H}_{k}\boldsymbol{x}^{-}_{k})$

$\boldsymbol{P}^{+}_{k}=\boldsymbol{P}^{-}_{k}-\boldsymbol{K}_{k}\boldsymbol{H}_{k}\boldsymbol{P}^{-}_{k}$

end while

Algorithm 1 Kalman Filter

initialization:
$\boldsymbol{x}^{+}_{0},\boldsymbol{P}^{+}_{0},\boldsymbol{Q},\boldsymbol{R},W_{m}^{i},\lambda,k=0$

while  _positioning_ do

$k=k+1$

$\boldsymbol{\chi}_{k}^{-(i)}=\left[\hat{\boldsymbol{x}}_{k-1}^{+}\pm\sqrt{\lambda+N}\left[\sqrt{\boldsymbol{P}_{k-1}^{+}}\right]\right]$

$\boldsymbol{x}_{k}^{(i)}=f\left(\boldsymbol{\chi}_{k}^{-(i)}\right);i=0,1,\dots,2N$

$\hat{\boldsymbol{x}}_{k}^{-}=\sum\limits_{i=0}^{2N}W_{m}^{i}\boldsymbol{x}_{k}^{(i)}$

$\boldsymbol{P}_{k}^{-}=\sum\limits_{i=0}^{2N}W_{m}^{i}\left(\boldsymbol{x}_{k}^{(i)}-\hat{\boldsymbol{x}}_{k}^{-}\right)\left(\boldsymbol{x}_{k}^{(i)}-\hat{\boldsymbol{x}}_{k}^{-}\right)^{T}+\boldsymbol{Q}$

$\boldsymbol{z}_{k}^{(i)}=h\left(\boldsymbol{x}_{k}^{(i)}\right)$

$\hat{\boldsymbol{z}}_{k}=\sum\limits_{i=0}^{2N}W_{m}^{i}\boldsymbol{z}_{k}^{(i)}$

$\boldsymbol{P}_{z}=\sum\limits_{i=0}^{2N}W_{m}^{i}\left(\boldsymbol{z}_{k}^{(i)}-\hat{\boldsymbol{z}}_{k}^{-}\right)\left(\boldsymbol{z}_{k}^{(i)}-\hat{\boldsymbol{z}}_{k}^{-}\right)^{T}+\boldsymbol{R}$

$\boldsymbol{P}_{xz}=\sum\limits_{i=0}^{2N}W_{m}^{i}\left(\boldsymbol{x}_{k}^{(i)}-\hat{\boldsymbol{x}}_{k}^{-}\right)\left(\boldsymbol{z}_{k}^{(i)}-\hat{\boldsymbol{z}}_{k}^{-}\right)^{T}$

$\boldsymbol{K}_{k}=\boldsymbol{P}_{xz}\boldsymbol{P}_{z}^{-1}$

$\hat{\boldsymbol{x}}^{+}_{k}=\boldsymbol{x}^{-}_{k}+\boldsymbol{K}_{k}\left(\boldsymbol{z}_{k}+\hat{\boldsymbol{z}}_{k}\right)$

$\boldsymbol{P}^{+}_{k}=\boldsymbol{P}^{-}_{k}+\boldsymbol{K}_{k}\boldsymbol{P}^{-}_{k}\boldsymbol{K}_{k}^{T}$

end while

Algorithm 2 Unscented Kalman Filter

##  V 5G Multi-BS Sensor Fusion

In this section, both the traditional centralized integration scheme and the
proposed decentralized fusion scheme will be presented. The transition model
is shared between both schemes, hence, it will be introduced first. Next, the
linearization errors of the centralized scheme will be displayed and
thoroughly discussed. Finally, the proposed decentralized scheme will be
presented.

###  V-A States and State Transition Process

The navigation states considered in all methodologies presented in this paper
comprise the estimated 2D position and velocity of the UE, i.e.,
$\boldsymbol{x}=[x,y,{v_{x}},{v_{y}}]^{T}$. Likewise, the state transition
process for all KF implementations in this work considers a constant velocity
model, yielding a linear state transition model as seen in (4).

$\boldsymbol{\Phi}=\begin{bmatrix}1&0&\Delta t&0\\\ 0&1&0&\Delta t\\\
0&0&1&0\\\ 0&0&0&1\end{bmatrix}$ (4)

The process covariance matrix $\boldsymbol{Q}$ comprises the errors arising
due to violating the constant velocity model by either changing the
orientation of the vehicle or by accelerating / decelerating. Thus,
$\boldsymbol{Q}$ was set by empirically computing the average variance of the
velocity of vehicles driving in a downtown setting. The experimental setup
used for such empirical tuning of $\boldsymbol{Q}$ is detailed in Section VI.
On the other hand, $\boldsymbol{R}$ comprises measurement errors. For
centralized integration, $\boldsymbol{R}$ comprises the noise variance of the
range and angle measurements, which were empirically tuned to $1$ mm and
$0.1\degree$, respectively. The empirical tuning was done based on the
resulting positioning RMS error from the whole trajectory. In decentralized
integration, however, $\boldsymbol{R}$ is the noise variance of the position
estimates from each BS. Hence, $\boldsymbol{R}$ was set to be $1$ cm for both
$x$ and $y$ measurements, based on empirical tuning.

###  V-B Centralized Integration Linearization Errors

In centralized 5G multi-BS sensor fusion, the raw range and angle measurements
are directly fused together in the fusion filter as seen in Fig. 1. The
measurement vector $\boldsymbol{z}$ for both EKF and UKF centralized
implementations is as follows:
$\boldsymbol{z}~{}=~{}[r_{1},\theta_{1},\dots,r_{N},\theta_{N}]^{T}$. The non-
linear measurement model of the system for both EKF and UKF is expressed in
(5). Furthermore, the Jacobian matrix
$\boldsymbol{H}_{k}\in\mathbb{R}^{2N\times 4}$ is computed using Taylor series
expansion as seen in (6). It is worth noting that $\boldsymbol{H}_{k}$ is only
utilized for centralized EKF implementation.

$h(\boldsymbol{x})=\begin{bmatrix}\sqrt{(\Delta x_{k}[n])^{2}+(\Delta
y_{k}[n])^{2}}\\\ \arctan\left(\frac{\Delta y_{k}[n]}{\Delta
x_{k}[n]}\right)\end{bmatrix}$ (5)

$\boldsymbol{H}_{k}=\begin{bmatrix}\frac{dx_{1}}{r_{1}}&\frac{dy_{1}}{r_{1}}&0&0\\\
\frac{-dy_{1}}{r_{1}^{2}}&\frac{dx_{1}}{r_{1}^{2}}&0&0\\\
\vdots&\vdots&\vdots&\vdots\\\
\frac{dx_{N}}{r_{N}}&\frac{dy_{N}}{r_{N}}&0&0\\\
\frac{-dy_{N}}{r_{N}^{2}}&\frac{dx_{N}}{r_{N}^{2}}&0&0\end{bmatrix}$ (6)

Linearization errors are inevitable to occur in the centralized integration of
5G BSs using raw range and angle measurements. This is mainly caused due to
the elimination of the higher-order terms during the linearization of
$h(\boldsymbol{x})$. The linearized $\boldsymbol{H}$ is used in multiple
places in Kalman filter implementations as seen in Alg. 1. It is therefore
important to study the negative effects imposed by each use of such a
linearized version of the measurement observation matrix. It is worth pointing
out that a common mistake that is usually encountered in the literature while
implementing EKF is the use of $\boldsymbol{H}$ instead of $h(\boldsymbol{x})$
within the innovation sequence, i.e., the use of
$\boldsymbol{z}-\boldsymbol{H}\boldsymbol{x}$ instead of
$\boldsymbol{z}-h(\boldsymbol{x})$. Such a mistake will bias the rather
unbiased filter, leading to extra unnecessary errors. With that being said, in
EKF implementations, the observation matrix is solely used in the computation
of the Kalman gain $\boldsymbol{K}$, which has two purposes in the correction
process. First, it optimally weights the individual measurement errors in the
innovation sequence. Second, it transforms the innovation sequence from the
measurement domain to the state domain. The observation matrix
$\boldsymbol{H}$ has a role in both functions. First, $\boldsymbol{H}$ is
utilized to find the innovation covariance matrix
$(\boldsymbol{H}\boldsymbol{P}^{-}\boldsymbol{H}^{T}+\boldsymbol{R})$. The
innovation covariance matrix is essential in the weighting process of the EKF.
Therefore, linearization errors in $\boldsymbol{H}$ would lead to a filter
that is either overconfident or underconfident about the innovation sequence.
It is worth noting that the UKF implementation is free of such errors due to
using sigma points to compute the innovation covariance matrix. This will lead
to a more accurate weighting of the measurements in UKF implementations. The
second use of the observation matrix is to help with transforming the
innovation residual errors from the measurement domain to the state domain.
This is done through the use of $\boldsymbol{H}^{T}$. Meaning, the non-linear
relationship between the states and the measurements is reduced to a linear
multiplication between $\boldsymbol{H}^{T}$ and
$\boldsymbol{z}-h(\boldsymbol{x})$. Such simplification of the non-linear
relationship will induce high positioning errors in centralized EKF
implementations. It is worth noting that the UKF centralized implementation
will also suffer from such transformation errors, despite the usage of sigma
points to compute the Kalman gain. That is due to the fact that the UKF still
utilizes a linear multiplication between $\boldsymbol{K}$ and
$\boldsymbol{z}-h(\boldsymbol{x})$ to perform the non-linear transformation.
Thus, counter to the common belief, the UKF is not totally immune to
linearization errors. As a matter of fact, the authors claim that all KF
implementations and variations will suffer from linearization errors to some
degree due to the inherent fundamental design of Kalman filters.

To further study the discussed linearization errors, multiple theoretical
simulations were conducted. The simulations address both weighting and
transformation errors for both range and angle measurements. In order to
simulate weighting errors, a random number generator was utilized to generate
a cloud of normally distributed UE positions with a set mean and variance. The
point cloud was then passed once through the non-linear measurement model
$h(\boldsymbol{x})$ and through the linearized measurement model
$\boldsymbol{Hx}$. Finally, the resulting transformed PDFs are displayed for
comparison. The PDF resulting from the non-linear transformation acts as the
ground truth for what the distribution should ideally look like. Of course,
the non-linearly transformed distributions are not expected to be Gaussian as
opposed to a linearly transformed distribution. Yet, it is still expected that
the linearly transformed distribution would be as close as possible to the
non-linearly transformed distribution to have a fair weighting of
measurements. Figures 3 and 4 show the simulation for range measurement’s PDF
transformation at close range and far range, respectively. It can be seen that
at close ranges, the linearly transformed PDF does not mimic the non-linearly
transformed PDF, which was expected according to [30]. More importantly, it
can be seen that the linearly transformed PDF possesses higher variance
compared to the real variance of non-linear transformation, meaning that the
filter will underestimate the correction update. On the other hand, it can be
seen that at far ranges, both PDFs are almost identical, resulting in the
proper weighting of the measurements.

Figure 3: Transformation of position PDF to range PDF at close BS-UE range
($\Delta x=\Delta y=0.1$ m). Figure 4: Transformation of position PDF to range
PDF at far BS-UE range ($\Delta x=\Delta y=50$ m).

Figures 5 and 6 present the simulation for the transformation of the PDFs of
angle measurements at close range and far range, respectively. It can be seen
that the linearly transformed PDF is heavily overconfident in both scenarios
compared to their non-linearly transformed counterparts. Such overconfidence
in measurements is dangerous in real-life navigation especially if the update
is expected to be biased as well due to the transformation errors.

Figure 5: Transformation of position PDF to angle PDF at close BS-UE range
($\Delta x=\Delta y=0.1$ m). Figure 6: Transformation of position PDF to angle
PDF at far BS-UE range ($\Delta x=\Delta y=50$ m).

In order to simulate transformation errors, a mesh of UE positions was created
and utilized to compute the range and angle through the non-linear measurement
model $h(\boldsymbol{x})$ and through the linearized measurement model
$\boldsymbol{Hx}$. The difference between the two transformations is then
presented, which is directly indicative of the linearization error arising
from the Taylor series approximation. Figures 7 and 8 show the linearization
errors of range measurements at close and far BS-UE ranges, respectively. It
is clear that the linearization errors due to linearized transformation do
increase in magnitude the closer the UE drives near the BS. Additional
analysis on the errors can be found in [30]. Fig. 9 depicts the transformation
errors in angle when the linearization was done around $\theta=45\degree$. The
figure shows a couple of findings. First, linearizing an arc-tan function
through the Taylor series approximation would result in a considerable amount
of errors that can reach up to $240\degree$. Second, it can be seen that the
linearization error was at its lowest when $\Delta x$ and $\Delta y$ are
equal, which is expected as the linearization was performed around
$45\degree$. Third, linearization of the arc-tan function is very sensitive to
any errors in the innovation sequence, as linearization errors would
dramatically increase when the innovation sequence errors cease to be very
close to zero. Hence, it is not advised to linearize such a function if other
alternative solutions are available, which is the case with decentralized
integration schemes.

Figure 7: Range linearization error at close BS-UE range ($\Delta x=\Delta
y=1$ m).

###  V-C Proposed Decentralized Integration

Decentralized integration of multiple BSs is viewed by the author as one of
the main advantages of 5G-NR. Decentralized integration schemes were not
attainable for GNSS-based systems in the past, as individual satellites do not
have the ability to estimate the position of the UE using a single range
measurement. Thus, GNSS systems were forced to use centralized fusion schemes
along with their inevitable linearization errors. On the contrary, individual
5G BSs are capable of providing independent estimates of the position of the
UE, enabling decentralized integration of BSs, where the relationship between
the states and the measurements is linear. In order to compute the 3D position
of a vehicle for decentralized fusion using a single BS, the range, the
azimuth angle, and the elevation angles are required as seen below:

$\begin{split}&x=r\cdot\sin(\theta)\cos(\phi)+x_{BS}\\\
&y=r\cdot\cos(\theta)\cos(\phi)+y_{BS}\\\
&z=r\cdot\sin(\phi)+z_{BS},\end{split}$ (7)

where $\phi$ is the elevation angle. To measure the elevation angle along with
the azimuth angle of departure, a 2D uniform rectangular array (URA) of
antennas would be required. Alternatively, if the height of the vehicle is
known to be constant, then the requirement for knowledge of elevation angle
can be alleviated. Thus, requiring only a 1D uniform linear array (ULA). Such
an assumption usually holds for land vehicles, as changes in the z-axis are
generally minute. The position of the vehicle can thus be computed as follows:

Figure 8: Range linearization error at far BS-UE range ($\Delta x=1m,\Delta
y=50$ m). Figure 9: Angle linearization error at $45\degree$ orientation.

$\begin{split}&x=r_{2D}\cdot\sin(\theta)+x_{BS}\\\
&y=r_{2D}\cdot\cos(\theta)+y_{BS}\\\ &r_{2D}=\sqrt{r^{2}-\Delta
z^{2}},\end{split}$ (8)

where $r_{2D}$ is the 2D range between the BS and the UE, and $\Delta z$ is
the known height difference between the BS and the UE. It can be seen that the
computations in (8) would still utilize non-linear models. Yet, such non-
linear models are utilized outside of the KF implementation. Hence, no
linearization is required.

In the proposed decentralized 5G multi-BS sensor fusion, the measurement
vector $\boldsymbol{z}$ will be as follows:
$\boldsymbol{z}~{}=~{}[x_{1},y_{1},\dots,x_{N},y_{N}]^{T}$. It can be seen
that the linear relationship between the states and measurements is indeed
linear. Therefore, the $\boldsymbol{H}_{k}$ matrix for a decentralized fusion
of 5G BSs, presented in (9), can fully describe the relationship between the
state vector $\boldsymbol{x}$ and the measurement vector $\boldsymbol{z}$,
while not incurring any linearization errors.

$\boldsymbol{H}=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\
\vdots&\vdots&\vdots&\vdots\\\ 1&0&0&0\\\ 0&1&0&0\end{bmatrix}$ (9)

It is worth noting that despite the simplistic and elegant design of the
proposed method, the 5G positioning literature has failed to propose such a
scheme. Thus, in this paper, we propose adopting decentralized integration
schemes for 5G multi-BS positioning as it eliminates linearization errors
without incurring any downsides in return.

##  VI Experimental Setup and Results

###  VI-A Experimental Setup

In order to verify the superiority of decentralized/LC integration over
centralized/TC integration, a quasi-real 5G simulation setup that was provided
by Siradel was utilized. Siradel 5G_Channel comprises LiDAR-based maps of the
buildings, vegetation, and water bodies in downtown areas of cities like
Toronto, New York, and Paris as seen in Fig. 10. The simulation tool requires
the position of the UE and the virtually connected BSs to compute required
positioning measurables like RSS, TOA, AOA, and AOD via its ray-tracing
capabilities and propagation models. To mimic a true urban navigation
scenario, a vehicle was equipped with a high-end positioning solution provided
by NovAtel; that includes KVH 1700, a tactical grade IMU, along with NovAtel’s
tactical grade GNSS receiver, and was driven in Downtown Toronto. The vehicle
was driven for $1$ hr and $13$ mins, and the trajectory was $9$ km long as
seen in Fig. 11. The trajectory was conducted during rush hour, and thus, many
sudden car stopping/acceleration dynamics were encountered. Moreover, the
trajectory features many turns and challenging maneuvering dynamics. Next, a
code was developed to place BSs along the driven trajectory that are $250$ m
apart, as per 3GPP’s Release 16 guidelines [28]. Finally, the generated BS
positions and NovAtel’s reference solution were imported to Siradel to
generate the needed 5G measurables. Siradel was set up to utilize mmWave
signals with a carrier frequency around $28$ GHz and a bandwidth of $400$ MHz.
The BSs were equipped with $8\times 1$ ULAs while the UE was equipped with an
omnidirectional antenna.

Figure 10: Downtown Toronto, ON, Google Earth (Top) vs Siradel simulation tool
(Bottom). Figure 11: Downtown Toronto Trajectory (Red), and 5G BSs (Yellow
circles).

###  VI-B Results and Discussions

In this section, the performance of the centralized schemes using EKF and UKF
will be compared to the performance of the proposed decentralized scheme using
a KF. The comparison will be done in three scenarios. First, perfect range and
angle measurements are supplied to all filters. Positioning errors arising
from this test will be solely attributed to linearization errors. The second
test scenario will incorporate quasi-real measurements from Siradel’s
simulations and NLOS communications will be permitted without NLOS detection
functionalities. Such a test will show the need for NLOS detection
capabilities as a prerequisite to lane-level positioning. Finally, the third
test scenario will include quasi-real measurements and NLOS detection
capabilities discussed in [29]. This will act as an accurate benchmark of the
expected performance of the compared filters and integration schemes. For all
test scenarios, a trilateration variant solution will also be presented for
EKF and UKF solutions; referred to as EKF-R and UKF-R, to gauge the advantages
and disadvantages of adding angle measurements to centralized fusion.

Fig. 12 depicts the Cumulative Distribution Function (CDF) of all implemented
filters and integration schemes using perfect measurements. It is evident that
the decentralized KF implementation trumped all other schemes, as it had an
almost perfect performance. On the other hand, it can be seen that the EKF
hybrid implementation had the lowest performance, even compared to its
trilateration counterpart. This proves that linearizing angle-based
measurements hinder the performance of the filter as it adds more errors than
extra information. Additionally, both UKF implementations had almost similar
performance, meaning that adding angle measurements did not add extra value in
centralized fusion schemes. Moreover, it is apparent that the UKF had more
errors at the decimeter level compared to its EKF counterpart, yet, it was
able to suppress errors above $0.5$ m. To have a deeper insight into how
linearization errors affect measurements, two close-up test samples are shown
in Figures 13 and 14, where the UE drives close to BSs. It can be seen that
both decentralized KF and centralized UKF did not sustain much error while
passing by nearby BSs. On the contrary, the centralized EKF implementation has
sustained considerable positioning errors due to linearization errors.

Figure 12: CDF of the 2D positioning errors while utilizing perfect range and
angle measurements. Figure 13: First close-up to showcase the performance of
filters while driving close to a BS using perfect measurement Figure 14:
Second close-up to showcase the performance of filters while driving close to
a BS using perfect measurements.

Fig. 15 illustrates the CDF of all implemented filters and integration schemes
using real range and angle measurements from Siradel without NLOS detection
capabilities. Patently, the proposed decentralized KF implementation has
outperformed all other schemes by a remarkable margin. Yet, such performance
is not enough to achieve lane-level accuracy, as it was not able to suppress
high levels of errors. Meanwhile, all other implementations failed to provide
presentable estimations, as they sustained sub-2m level of accuracy less than
26% of the time. This is expected, as linearization errors have a higher
impact when the innovation sequence’s residual error is high in value, which
is the case for NLOS measurements. This finding shows the importance of the
development of NLOS detection algorithms, which are not given enough attention
in the community. Also, it proves the superiority of the proposed
decentralized approach in case the NLOS detector failed to capture an NLOS
communication link.

Figure 15: CDF of the 2D positioning errors while utilizing quasi-real Siradel
range and angle measurements without NLOS detection.

Fig. 16 shows the CDF of the positioning error for all aforementioned schemes
while using real Siradel range and angle measurements along with NLOS
detection capabilities found in [29]. Evidently, the proposed decentralized KF
implementation was able to outperform other implementations in both decimeter-
level accuracy and meter-level accuracy statistics. The proposed method was
able to navigate with a sub-meter level of accuracy for around $91\%$ of the
time. Both UKF implementations were able to maintain a sub-1m level of
accuracy for around $82\%$ of the time. It is also noticeable that the
trilateration-based UKF was slightly better than its hybrid counterpart due to
the avoidance of extra linearization errors imposed by the angle measurements.
Finally, it is conspicuous beyond doubt that both EKF implementations are
unsuitable for lane-level positioning, albeit the trilateration version had
remarkably better performance for the aforementioned reasons.

Figure 16: CDF of the 2D positioning errors while utilizing quasi-real Siradel
range and angle measurements with NLOS detection.

Figures 17-19 show close-up samples of the positioning solution of both
decentralized KF and centralized UKF in three different scenarios. The first
scenario, shown in Fig. 17, depicts the performance of the filters when close
to a BS while connected to multiple BSs at the same time. It can be seen that
the proposed decentralized KF method did not sustain errors while the
centralized UKF implementation sustained lane-level errors for a short time
around the close BS due to linearization errors. The second scenario, shown in
Fig. 18, depicts the performance of the filters when they are connected to a
single BS. The superiority of the proposed method over centralized UKF can be
clearly seen, as the UKF implementation suffered from positioning errors for a
longer period of time, due to the lack of supporting BSs. Finally, the third
scenario, shown in Fig. 19, illustrates the performance when the UE is in
total NLOS communication with all nearby BSs. It can be seen that both
implementations have sustained large amounts of errors. Such a result is
expected to occur from time to time while using 5G standalone positioning.
Thus, the authors see a crucial need for integration between 5G and other
onboard motion sensors to bridge such gaps. Investigation of novel integration
methodologies between 5G and other sensors is to be conducted in future works.
A summary of the performance statistics of all algorithms and schemes is found
in Table I.

Figure 17: First close-up to showcase the performance of filters while driving close to a BS while connected to multiple BSs and using quasi-real measurements. Figure 18: Second close-up to showcase the performance of filters while driving close to a BS while connected to a single BS and using quasi-real measurements. Figure 19: Third close-up to showcase the performance of filters while driving close to a BS while connected to NLOS BSs and using quasi-real measurements. TABLE I: 2D Positioning Error Statistics Summary Statistics |  LC-KF |  TC-UKF |  TC-UKF-R |  TC-EKF |  TC-EKF-R
---|---|---|---|---|---
RMS |  9 m |  47.7 m |  12.8 m |  881.2 m |  115.7 m
Max |  89.3 m |  885.7 m |  145.3 m |  5.8e4 m |  2.3e3 m
$<2$ m |  95.2% |  84.5% |  85.7% |  3.5% |  23.3%
$<1$ m |  94.7% |  81.9% |  82.4% |  2.5% |  17.1%
$<30$ cm |  93.9% |  71.4% |  68.9% |  1.7% |  12.7%

##  VII Conclusion

In conclusion, it was shown that 5G multi-BS integration schemes for
positioning purposes in the literature are heavily inspired by GNSS
trilateration centralized schemes. It was proven that such schemes suffer from
linearization errors due to the direct use of range and angle measurements.
Such errors were necessary for GNSS-based algorithms, as the states and the
measurements do not have a one-to-one function, as they rely on range
measurements only. In 5G, however, it was shown that with the aid of angle-
based measurements, a one-to-one function between the states and the
measurements can be achieved, opening the way for integration on the position
level rather than on the measurement level. Therefore, it was proposed to
adopt a decentralized KF-based integration scheme, which totally eliminates
linearization errors. To test the proposed integration scheme, a quasi-real
simulation system was utilized to simulate 5G operation in downtown Toronto
for a real trajectory that is $1$ hour and $13$ minutes long. The tests took
place in a perfect measurement scenario and in a real measurement scenario
with/without NLOS detection capabilities. It was shown that the decentralized
positioning scheme was able to outperform its centralized counterpart schemes
that utilize EKF-based and UKF-based filters by a considerable margin in all
scenarios.

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|  Sharief Saleh (Graduate Student Member, IEEE) received the B.Sc. and M.Sc.
degrees in electrical engineering from Qatar University, Doha, Qatar, in 2016
and 2018, respectively, and is currently pursuing a PhD degree in electrical
engineering at Queen’s University, Canada. He was awarded a Graduate Assistant
position at Qatar University during his master’s studies and was then
appointed as a Research Assistant at Qatar University, Doha, Qatar. He is
currently a member of the Navigation and Instrumentation (NavINST) Research
Lab, RMCC. His current research interests include 5G positioning and
navigation, sensor fusion, sensors and instrumentation, estimation theory, and
reinforcement learning.
---|---
|  Qamar Bader (Graduate Student Member, IEEE) received her B.Sc. degree in
electrical engineering from Qatar University, Doha, Qatar, in 2016 and is
currently pursuing an MSc degree in electrical engineering at Queen’s
University, Canada. She is currently a member of the Navigation and
Instrumentation (NavINST) Research Lab, RMCC. Her current research interests
include 5G positioning and navigation, sensor fusion, deep learning, computer
vision, and environment mapping.
---|---
|  Mohamed Elhabiby was the Treasurer of the Geodesy Section at the Canadian
Geophysical Union from 2008 to 2014. He is currently an Associate Professor
with the Faculty of Engineering, Ain Shams University, Cairo, Egypt. He is
also the Executive Vice President and a Co-Founder of Micro Engineering Tech
Inc., Calgary, AB, a high-tech international company, specialized in high
precision engineering and instrumentation, mobile mapping, laser scanning,
deformation monitoring, and GPS/INS integrations. He is a Leader of an
Archaeological Mission in the Area of Great Pyramids, Cairo. He received the
Astech Awards. He is named by Avenue Magazine as one of the Top 40 under 40.
He is the Chair of WG 4.1.4: Imaging Techniques, Sub-Commission 4.1:
Alternatives and Backups to GNSS. He chaired the Geocomputations and Cyber
Infrastructure Oral Session at the Canadian Geophysical Union annual meeting
from 2008 to 2012.
---|---
|  Aboelmagd Noureldin (Senior Member, IEEE) received the B.Sc. degree in
electrical engineering and the M.Sc. degree in engineering physics from Cairo
University, Egypt, in 1993 and 1997, respectively, and the Ph.D. degree in
electrical and computer engineering from the University of Calgary, AB,
Canada, in 2002. He is currently a Professor with the Department of Electrical
and Computer Engineering, Royal Military College of Canada (RMCC) with Cross
Appointment at the School of Computing and the Department of Electrical and
Computer Engineering, Queen’s University. He is also the Founder and the
Director of the Navigation and Instrumentation (NavINST) Research Lab, RMCC.
He has published two books, four book chapters, and more than 270 papers in
journals, magazines, and conference proceedings. His research interests
include global navigation satellite systems, wireless location, and
navigation, indoor positioning, and multi-sensor fusion targeting applications
related to autonomous systems, intelligent transportation, road information
services, crowd management, and the vehicular Internet of Things. His research
work led to 13 patents and several technologies licensed to the industry in
the area of position, location, and navigation systems.
---|---