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Channel Parameter Estimation and TX Positioning
with Multi-Beam Fusion in 5G mmWave Networks
Mike Koivisto, Jukka Talvitie Member, IEEE, Elizaveta Rastorgueva-Foi, Yi Lu Student Member, IEEE, and
Mikko Valkama Senior Member, IEEE
Abstract—Since the beginning of the fifth generation (5G)
standardization process, positioning has been considered as a
key element in future cellular networks. In order to perform
accurate positioning, solutions for estimating and processing
location-related measurements such as direction of arrival (DoA)
and time of arrival (ToA) for various use-cases need to be
developed. In this paper, building on the existing 5G new radio
(NR) specifications and millimeter wave frequencies, we propose
a novel estimation and tracking solution of the DoA and ToA such
that only analog/RF beamforming-based observations are utilized.
In addition to the proposed extended Kalman filter (EKF)-
based estimation and tracking approach, we derive Crame´r-
Rao lower bounds (CRLBs) for the considered RF multi-beam
system, and propose an information-based criterion for selecting
the necessary beams for the estimation process in order to
provide highly accurate performance with feasible computational
complexity. The performance of the proposed method is evaluated
using extensive ray-tracing simulations and numerical evalua-
tions, and the results are compared with other estimation and
beam-selection approaches. Based on the obtained results, beam-
selection at the receiver can have a significant impact on the DoA
and ToA estimation performance as well as on the subsequent
positioning accuracy. Finally, we demonstrate the highly accurate
performance of the methods when extended to joint multi-
receiver-based device positioning and clock synchronization.
Index Terms—5G, beamforming, Crame´r-Rao lower bound,
direction of arrival, extended Kalman filter, millimeter waves,
positioning, time of arrival, tracking
I. INTRODUCTION
IN addition to the demanding communication requirements ofthe fifth generation (5G) mobile radio networks, it has been
envisioned that accurate positioning will play a key role not
only in personal navigation and various vertical applications but
also in empowering network functionalities in terms of location-
aware communications [1]–[4]. Especially the 5G millimeter
wave (mmWave) networks will employ large bandwidths as
well as antenna arrays with increasing number of antenna
elements, thus creating a convenient environment for device
positioning based on, e.g., direction of arrival (DoA) and time
of arrival (ToA) measurements, which are already supported
This work was supported by the Doctoral School of Tampere University,
The Finnish Foundation for Technology Promotion, the Nokia Foundation,
the Emil Aaltonen fund, Tuula and Yrjo¨ Neuvo fund, the Academy of
Finland under the projects LOCALCOM (323244) and ULTRA (328214), and
Business Finland under the projects ”5G VIIMA”, and ”5G FORCE”.
The authors are with the Department of Electrical Engineering, Tampere
University, FI-33720 Tampere, Finland (e-mail: mike.koivisto@tuni.fi;
jukka.talvitie@tuni.fi; elizaveta.rastorgueva-foi@tuni.fi; yi.lu@tuni.fi;
mikko.valkama@tuni.fi).
This manuscript contains multimedia material, available at
https://research.tuni.fi/wireless/research/positioning/TWC2021/.
in 5G new radio (NR) specifications [5]. Furthermore, the
mmWave networks rely highly on line of sight (LoS) conditions
which in turn implies even more densified deployments [6]
as well as more accurate positioning performance if designed
properly. However, when the number of antenna elements
increases, thus theoretically improving the DoA estimation
accuracy [7], [8], the overall positioning performance may
suffer from signal-to-noise ratio (SNR) degradation due to
possible beam-misalignments in a fixed-beam system [9].
In order to compensate for such occasional performance
degradation, we propose a novel extended Kalman filter (EKF)-
based DoA and ToA estimation solution that is able to fuse
information from several available beams in a polarimetric 5G
mmWave system with analog/RF beamforming. Importantly,
facilitating efficient positioning and tracking through RF-only
beamforming is even further emphasized when the networks
evolve towards the sub-THz regime, paving the way for future
6G systems [10].
In the recent studies, channel variables such as DoA and
ToA are typically estimated by utilizing compressed sensing
(CS)-based estimation approaches [11]–[13], through subspace-
based methods [14], or by employing sequential estimation
solution building on, e.g., EKFs [15]–[18]. While the sequential
estimation methods can be found more attractive compared
to classical estimation approaches due to more elaborate
fusion of the prior information of the system [19], [20], the
authors in [15], [16] assume that each antenna element in
the arrays is equipped with a radio frequency (RF) chain.
This, however, may not be feasible in 5G mmWave networks
with large antenna arrays due to the cost and complexity of
the hardware [21]. Despite several works on actual mmWave
positioning being available in the existing literature [4], [7],
[8], [17], [22], [23], the methods either consider different
location-related measurements with a known transmitter (TX)-
side antenna and beamforming information [17], [22], [23],
provide only theoretical performance bounds [4], [7], [8] or
consider obtaining the location-related measurements from a
pre-defined statistics [24], [25] without practical estimators.
Building on the existing 5G NR specifications and dual-
polarized antenna arrays with inexpensive RF-beamforming
capabilities, we therefore propose a novel EKF-based solution
for estimating and tracking the azimuth and co-elevation DoA
angles and ToA through sophisticated and adaptive beam-
selection in 5G mmWave networks. Despite various beam-
selection methods are available in the existing literature [26],
[27], the proposed beam-selection method is analyzed through
its theoretical and practical positioning performance instead of
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communication metrics [27]–[29] or only theoretical limits [30]
typically considered in the available literature. In addition
to the normal communication between the TX and receiver
(RX), the proposed solution does not require any information
about the TX-side antenna array or additional communication
overhead in terms of, e.g., dedicated positioning reference
signals or two-way communications, thus being fundamentally
different to the methods, e.g., in [17], [18]. Despite the fact that
theoretical bounds for DoA and ToA estimates and subsequent
positioning performance are derived, e.g., in [7], [31]–[35],
the Crame´r-Rao lower bounds (CRLBs) for the considered
RF multi-beam formulation are not readily available in the
existing literature. Besides the proposed estimation method,
we therefore derive such theoretical bounds for the DoA and
ToA estimates based on multi-beam fusion and compare the
results with those obtained using a digital beamformer at the
RX-side. Finally, the performance of the proposed method
and its extension to multi-RX and joint TX synchronization
scenarios are evaluated using extensive ray-tracing simulations
in realistic environments while considering the latest 5G NR
numerology.
The contributions of this article are summarized as follows:
• We propose a novel and adaptive beam-selection method
for analog beamforming systems where the TX-side
antenna and beamforming information is assumed to be
unknown.
• We derive CRLBs for the DoA and ToA estimates as
well as for the subsequent positioning for the proposed
beam-selection method.
• We propose a practical EKF-based solution for DoA and
ToA estimation by employing multiple RX-beams while
assuming that the TX-side information is unknown.
• We evaluate the performance of the proposed method
building on the uplink (UL) sounding reference signal
(SRS) transmissions by employing the latest 5G NR
specifications and extensive ray-tracing simulations at
28 GHz in realistic urban canyon and open area scenarios.
The rest of the paper is organized as follows. In Section II,
we provide a general concept and necessary assumptions
considered in the proposed framework including the assumed
signal and channel models. In Section III, we first derive the
theoretical lower bounds for the considered DoA and ToA
estimates and then provide the notion of position error bound
(PEB) for the subsequent TX location estimate. In Section IV,
we then derive the proposed estimation and tracking algorithm
employing a beam-selection scheme that is stemming from the
available information of the received beam-based observations.
The ray-tracing environment as well as the considered scenarios
are described in Section V, after which the results are provided
and analyzed in Section VI, where the proposed method is
also extended to cover multi-RX and joint TX synchronization
scenarios. Finally, conclusions are drawn in Section VII.
II. SYSTEM MODEL AND GENERAL ASSUMPTIONS
In this section, we shortly describe the general system
model and assumptions regarding the problem formulation
in the context of the considered mmWave 5G NR networks.
Thereafter, we derive the signal and channel models with proper
polarimetric antenna models, which are then revised for the
assumed scenario by reformulating the models with respect
to the desired DoA and ToA channel parameters. Finally, the
proposed beam-selection method is described.
A. General System Model and Assumptions
Building on the premises of 5G mmWave networks, an
active TX at an unknown location pTX = [xTX, yTX, zTX]
T is
assumed to transmit beamformed reference signals (RSs) in
a periodic manner exploiting orthogonal frequency division
multiplexing (OFDM) waveforms. In particular, we do not
set any further constraints for the number of transmitted
beams, thus implying that such RSs can be transmitted by
the TX either through a single beam based on, e.g., beam-
correspondence, or by using a set of fixed TX-beams BT in
a beam-sweep process. After propagating through the LoS-
dominant multi-path channel, which is typically considered
at high mmWave frequency communications, the transmitted
RSs are then received by the RX through a set of its own
fixed beams BR during a beam-sweep process as illustrated
in Fig. 1. In this work, we assume that the given RX knows
its exact location pRX = [xRX, yRX, zRX]
T as well as the dual-
polarized antenna array structure that is exploited at the RX. For
practicality and improved cost-efficiency, we also assume that
the beamforming at both ends is carried out using inexpensive
RF-beamforming by the means of phase-shifters instead of
digital or hybrid beamformers.
In the proposed channel parameter estimation and subsequent
positioning solution, the received beam-based RS observations
are first deployed to extract the information of the most
powerful TX-beam. Given the most feasible TX-beam index,
the RX then determines the candidate RX-beams MB ⊆ BR
using the proposed beams-selection method that employs the
available information of the received beam-based observations
in an adaptive manner. After determining the set of candidate
RX-beams MB, the beam-based observations corresponding to
the candidate RX-beams are facilitated in the proposed EKF-
based solution that estimates and tracks the DoA angles and the
ToA of the LoS-path (see Fig. 1). Such estimates can be then
fused in the subsequent positioning stage into the sequential
location estimates similar to, e.g., in [16], [18].
Finally, it is to note that the beamformer as well as the
antenna model of the TX are both assumed to be unknown at
the RX side in the proposed beam-selection and subsequent
two-stage EKF-based estimation solution, which in turn results
in the fact that we cannot estimate the direction of departure
(DoD) angles from the received signals, for instance. Moreover,
we do not essentially restrict the role of the TX and RX in
the considered system, thus implying that the communication
direction can be either UL or downlink (DL) assuming that
the orientation of the RX is known or jointly estimated.
However, in the numerical evaluations, we consider a network-
centric approach where the network elements are receiving
beamformed UL SRS having a form of an extended Zadoff-Chu
sequence [36]. Such a choice is stemming from the fact that
the UL SRSs are beamformed, they are considered as potential
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LMC
LMC/LMF
RX
Beam-sweep
process
TX
TX location p(pTX[n])
Uplink SRS
x
y
⊙
z
iˆ jˆ
Set of candidate beamsMB
Currently active beams
Sets of all beams BT and BR
RF-processing
Store beam-
based
measurements
Beam-selection
DoA and ToA
estimation
TX positioning
y(i,j)[n]
Y[n]
MB
N
(
Θˆ[n], Cˆ[n]
)
p(pTX[n])
Fig. 1: Proposed two-fold estimation and positioning framework, with an adaptive beam-selection method. The proposed approach first determines the set of
candidate RX-beams MB, which are then facilitated in joint DoA and ToA estimation and subsequent TX positioning. The processing can run in a local
location management component (LMC), or alternatively at a centralized location management function (LMF) allowing to fuse the DoA and ToA estimates
from several RX nodes
positioning-related RSs in 5G NR, and a specific SRS-for-
positioning information element configuration is currently under
consideration in the specification process [36]. In the network-
centric approach, locations of the RXs are also readily available,
whereas in the device-centric positioning, such information
is not typically available. In order to reduce the positioning
latency, it also expected that the logical positioning functionality
can be brought closer to the edge by the means of LMC [37].
B. Signal and Channel Models
Let us assume that at a given time-instant n, a TX equipped
with an antenna array containing MT antenna elements,
transmits beamformed OFDM symbols by using one of its TX-
beams. In particular, the transmitted complex-valued symbol
allocated at the mth subcarrier is denoted as x(i)[m,n] ∈ C,
where the superscript i ∈ BT indicates the index of the utilized
TX-beam. As mentioned in Section II-A, we assume that the
beamformers at both TX and RX side are designed purely
using RF phase-shifters and such a TX-side RF-beamformer
is denoted as F(i)T [n] ∈ CMT×1 for which ‖F(i)T [n]‖2 =MT.
However, the design of the RF-beamformers is not restricted
to the considered assumptions, and other similar antenna array
and beamforming designs such as those in [38], [39] can be
considered as well.
After propagating through a LoS-dominant multiple-input
multiple-output (MIMO) multi-path channel, the transmitted
signal is then received at the RX through its current RF-
beamformer with an index j ∈ BR. In a similar manner as with
the TX, the RF-beamformer of the jth RX-beam is denoted as
F
(j)
R [n] ∈ CMR×1 for which we have ‖F(j)R [n]‖2 =MR. The
received complex-valued sample between the TX-RX beam-pair
(i, j) after taking the fast Fourier transform (FFT) is denoted
as y(i,j)[m,n] ∈ C and it is given as
y(i,j)[m,n]=F
(j)
R [n]
H(H[m]F
(i)
T [n]x
(i)[m,n]+n[m,n]), (1)
where the noise term n[m,n] ∼ CN (0, σ2IMR), representing
the noise at RX antenna elements, is modeled as the complex-
Gaussian white-noise with a power spectral density of σ2.
Moreover, H[m] ∈ CMR×MT is the double-directional polari-
metric MIMO channel matrix, which according to [15], [40],
can be written as a superposition of P + 1 propagation paths
as
H[m]=
P∑
p=0
AR(ϕR,p, ϑR,p)ΓpA
H
T (ϕT,p, ϑT,p)e
−j2pifmτp , (2)
where p = 0 denotes the LoS-path, and e−j2pifmτp is a phase-
shift introduced by the propagation delay τp ∈ R of the
pth propagation path and the mth subcarrier with a baseband
frequency of fm. In addition, Γp ∈ C2×2 is a polarimetric
complex-valued path-gain having the radio wave polarization
coefficients of the considered pth path as its elements [15], [40].
In this work, it is assumed that the whole beam-sweep process
occurs within a coherence time of the channel, thus implying
that the response H[m] remains almost constant during a single
beam-sweep process, and therefore, we omit the time-index n
from its definition.
Finally, the polarimetric antenna array responses of the TX
and the RX in (2) are given as
AT(ϕT,p, ϑT,p)=[AT,V(ϕT,p, ϑT,p),AT,H(ϕT,p, ϑT,p)] (3)
AR(ϕR,p, ϑR,p)=[AR,V(ϕR,p, ϑR,p),AR,H(ϕR,p, ϑR,p)], (4)
where AT,X(ϕT,p, ϑT,p) ∈ CMT×1 and AR,X(ϕR,p, ϑR,p) ∈
CMR×1 denote the antenna array responses of the TX and the
RX for the considered polarization component X ∈ {H, V},
respectively. Moreover, the angle-pair (ϕT,p, ϑT,p) denotes the
DoD angle-pair and (ϕR,p, ϑR,p) is the DoA angle-pair of the
pth propagation path. Here, the azimuth angles are denoted
as ϕ ∈ [0, 2pi) and the co-elevation angles are denoted as
ϑ ∈ [0, pi] and both of them are defined in a global coordinate
system.
C. Considered Antenna and Single-path Channel Models
Since the proposed estimation approach estimates the DoA
angles and the ToA corresponding to the LoS-path, we first
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re-formulate the signal model in (1) accordingly. Due to the
considered LoS-path assumption, we ignore the path-index p
from the channel variables throughout the rest of the paper.
However, despite the simplified single-path channel model
used in the theoretical derivations as well as in the algorithm
derivations, all the essential multi-path components are modeled
and included in the final numerical evaluations using extensive
ray-tracing simulations.
Let us first collect all the received frequency-domain samples
at the RX into a single multicarrier measurement vector denoted
as y(i,j)[n] ∈ CMf×1, where Mf is the number of active
subcarriers and (i, j) is the current beam-pair between the TX
and the RX. Applying the results in [15], [40] into the signal
model (1), the multicarrier observation at the RX between the
considered beam-pair (i, j) can be written in a simplified form
as
y(i,j)[n] = B(i,j)(Θ[n])ψ(i)[n] + n(i,j)[n], (5)
where Θ[n] = [τ [n], ϕR[n], ϑR[n]]T denotes the channel
parameters of the LoS-path. Moreover,
ψ(i)[n] = [ψ
(i)
V [n], ψ
(i)
H [n]]
T (6)
= ΓAHT (ϕT[n], ϑT[n])F
(i)
T [n] ∈ C2×1 (7)
is a combination of the unknown path-gains and the TX side
antenna response including the beamforming gain, whereas the
matrix-valued function B(i,j)(Θ[n]) ∈ CMf×2 reads
B(i,j)(Θ[n])=(F
(j)
R [n]
HAR(ϕR[n], ϑR[n]))Bf (τ [n]), (8)
where denotes the Khatri-Rao product, i.e., a column-
wise Kronecker product, and Bf (τ [n]) = X(i)[n]Gfd(τ) ∈
CMf×1 is a combination of the transmitted signal and the
combined frequency response of the RF-chains, in which [40]
d(τ) = e
−j2pif0τ
[
− (Mf−1)2 ,...,
(Mf−1)
2
]
, (9)
where f0 is the subcarrier spacing. Furthermore, X(i)[n] ∈
CMf×Mf is a matrix containing the transmitted multicarrier
SRS reference symbols of the ith TX-beam on the diagonal
with equally distributed energy over the subcarriers. Finally,
the noise-term in (5) after the RX-side beamformer is now
given as n(i,j)[n] ∼ CN (0, σ2MRIMf ), and it is assumed
to be independent of the RX beamformer. In particular, the
structure of B(i,j)(Θ[n]) in (8) is assumed to be known at the
RX whereas ψ(i)[n] in (7) is unknown and therefore, it needs
to be estimated jointly with the unknown model noise variance
σ2 and the desired channel parameters Θ[n].
In the proposed estimation method as well as in the
conducted theoretical derivations, the polarimetric antenna
array responses are modeled by employing effective aperture
distribution functions (EADFs) which can be considered as a
numerically efficient solution in representing the antenna array
manifold using Fourier series expansion [40], [41]. Hence, the
antenna array responses in (3)-(4) can be given as
AT,X(ϕT,p, ϑT,p) = GT,Xd(ϕT,p, ϑT,p) (10)
AR,X(ϕR,p, ϑR,p) = GR,Xd(ϕR,p, ϑR,p) (11)
where GT,X ∈ CMT×MaMe and GR,X ∈ CMR×MaMe
denote the EADFs of the TX and the RX to the given
polarization direction, respectively. Moreover, d(ϕT,p, ϑT,p)
and d(ϕR,p, ϑR,p) denote the antenna domain sampling vectors
which are formed such that d(ϕ·,p, ϑ·,p) = d(ϑ·,p)⊗ d(ϕ·,p),
where ⊗ denotes the Kronecker product and [16], [40], [41]
d(ϑ·,p) = ejϑ·,p[−
(Me−1)
2 ,...,
(Me−1)
2 ] ∈ CMe×1 (12)
d(ϕ·,p) = ejϕ·,p[−
(Ma−1)
2 ,...,
(Ma−1)
2 ] ∈ CMa×1 (13)
denote the Vandermonde-structured vectors. Here, Ma and
Me are the number of modes, i.e., spatial harmonics of the
antenna response in the azimuth and co-elevation directions,
respectively [40], [42].
D. Proposed Information-based Beam-Selection Scheme
Since the information from several available RX-beams is
fused in the proposed estimation process, we shortly present
the notation for the combined beam-based observations. After
the whole beam-sweep process and storing the beam-based
multicarrier observations, the RX needs to then determine the
set of beam-based observations that are the most significant
in the actual estimation phase. Therefore, the RX first de-
termines the best TX side beam-index iˆ ∈ BT based on the
received beam-based reference signal received power (RSRP)
measurements [43] over all the RX-beams such that
iˆ = argmax
i
∑
j∈BR
1
Mf
Mf∑
m=1
|y(i,j)[m,n]|2
. (14)
In particular, the TX-beam corresponding to the obtained index
iˆ is the most powerful beam from the RX perspective. If the
TX transmits the signals using a single beam, e.g., through
beam-correspondence, the best TX beam-index iˆ is thus the
index of such beam.
In order to determine the suitable RX side beam-set MB ⊆
BR, different approaches based on, e.g., noise-threshold or some
heuristics can be applied [17], [22]. In order to find the best
possible set of RX-beams, solutions relying on, e.g., Bayesian
information criterion (BIC) could be applied, but finding such
a beam-set requires evaluating the BIC for all the possible
RX beam-combinations, which is computationally extremely
demanding. In this work, the set MB is defined as a unique
set of beam-indices over all the desired channel parameters for
which the cumulative sum of the normalized Fisher information
matrix (FIM) values exceeds a certain predefined threshold
αthr ∈ (0, 1]. At each time-instant, the set of RX beam-indices
can be thus determined using a mathematical set notation as
MB =
⋃
z∈IΘ
j : ∑
j∈BR,sort
(
J(j)z (Θˆ[n])∑
j∈BR J
(j)
z (Θˆ[n])
)
>αthr
, (15)
where J(j)z (Θˆ[n]) denotes the FIM of the state-variable z ∈
IΘ = {ϕR, ϑR, τ} which can be calculated as in (19) by con-
sidering only the jth beam-based observation y(ˆi,j)[n] instead
of the whole concatenated set of beam-based observations. In
addition, BR,sort is a set of RX beam-indices after ordering
them based on the normalized FIM values, thus ensuring
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that the most informative beam-indices are taken into account
in MB. In practice, αthr can be interpreted as a percentage
threshold describing the information portion (i.e., a cumulative
sum of normalized beam-based information values) that needs
to be exceeded for a given state-variable z. As a concrete
example, if αthr = 1, it would imply that all the RX-beams
are selected for the estimation process. As shortly discussed
in Section IV, the computational complexity of the proposed
method is proportional to the number of facilitated RX-beams
while the performance of the method can be improved by
increasing the value of αthr. Hence, the proposed beam-selection
method allows for adjusting the computational complexity and
the estimation performance through tuning αthr even between
state-variables. In the numerical evaluations, we compare the
proposed information-based beam-selection scheme with the
one used in [18] by analyzing the estimation and subsequent
positioning performance with both beam-selection approaches.
Finally and for the notational convenience, given the set of
concatenated beam-based observations Y[n] ∈ C|MB|Mf×1,
we define the combined signal model for the corresponding
set of MB candidate RX-beams simply as
Y[n] = B(Θ[n])ψ[n] + n[n], (16)
where the signal model function B(Θ[n]) ∈ CM×2, with
M = |MB|Mf , reads
B(Θ[n]) = (FR[n]
HAR(ϕR[n], ϑR[n]))Bf (τ [n]), (17)
in which FR[n] ∈ CMR×|MB| is the RF-beamforming matrix
containing all the selected RX-beams. Furthermore, n[n] ∼
CN (0, σ2IM) is the combined noise-term over the chosen
RX-beams.
III. CRAME´R-RAO LOWER BOUNDS
In this work, we seek to estimate the DoA angles and the
ToA by fusing information fromMB available RX-beams given
the best TX-beam with an index iˆ. In order to demonstrate
the impact of such a measurement fusion on the accuracy of
the desired channel parameter estimates and on the subsequent
positioning performance, we first derive the CRLBs for the
DoA and ToA under the considered system model. Thereafter,
we provide PEBs for analyzing the instantaneous positioning
performance at a theoretical level. In particular, the CRLB
provides a theoretical lower-bound for the covariance matrix
of an unbiased estimator xˆ such that [19]
CRLB(x) = (J(x))−1 ≤ E{(xˆ− x)(xˆ− x)T}, (18)
where J(x) is the FIM of the vector-valued variable x. By
considering the ith and jth scalar-valued elements of x, denoted
as xi and xj , respectively, the FIM is obtained through [19],
[40]
[J(x)]i,j = −E
{
∂2`(Y|x)
∂xi∂xj
}
, (19)
where [·]i,j denotes the element of the matrix argument located
at the index-pair (i, j), and `(Y|x) is the log-likelihood of
the observations given the variable x. Throughout this section,
we omit the time-index n from the derivations for notational
simplicity.
A. Theoretical Bounds of the Channel Variables
In general, the CRLBs of the channel variables can be derived
similar to e.g., in [8], [32], [40], however, we derive the CRLBs
such that the estimators as well as the CRLBs employ the same
models where the impact of existing multi-path components
is also incorporated. Due to the fact that we are not actually
interested in estimating the unknown noise variance σ2 or
the combined channel and TX information ψ in the proposed
framework, we employ a subspace fitting technique through
variable projection [44] in a similar manner as, e.g., in [16],
[18]. Building on such a variable projection approach, we thus
derive the CRLBs for the DoA and the ToA estimates in order
to emphasize the role of multi-beam fusion in the estimation
process.
For the notational convenience, let us first define a vector
x that contains all the unknown variables in the considered
beam-based signal model (16), as
x = [ΘT,ψT, σ2]T ∈ C6×1, (20)
where Θ = [τ, ϕR, ϑR]T denotes the desired channel variables.
Given the model for the complex-valued beam-based multi-
carrier observations in (16), the likelihood function for the
observations Y ∈ CM×1 can be expressed as [40]
p(Y|x) = p(Y|τ, ϕR, ϑR,ψ, σ2) (21)
=
1
piM det(σ2IM)
e−
1
σ2
‖Y−B(ϕR,ϑR,τ)ψ‖2 , (22)
which can be transformed into the log-likelihood function in a
straightforward manner such that
`(Y|x) = log(p(Y|x)) (23)
= −M log(piσ2)− 1
σ2
‖Y−B(ϕR, ϑR, τ)ψ‖2, (24)
where log denotes the natural logarithm.
In particular, since (24) is separable with respect to the
nuisance variables σ2 and ψ, and they can be solved in
a closed-form, we first solve them from the obtained log-
likelihood function. Based on (24), the maximum likelihood
(ML) estimates of σ2 and ψ are then given as
σˆ2 =
‖P⊥(Θ)Y‖2
M , and ψˆ = (B(Θ))
†Y, (25)
where (·)† denotes the Moore-Penrose pseudo-inverse, and
P⊥(Θ) = IM − B(Θ)(B(Θ))† denotes the orthogonal
projection matrix onto the null-space of B(Θ). Substituting
the ML estimators (25) into the log-likelihood (24), we obtain
the concentrated log-likelihood function
ˆ`(Y|Θ) = ˆ`(Y|ϕR, ϑR, τ) ∝ −M log
(‖P⊥(Θ)Y‖2) , (26)
where the proportionality takes into account only the parts
that play a role in the following FIM determination. Given
the definition (19) and after straightforward manipulation,
the observed FIM for the proposed model can be calculated
according to
J(Θ) =
2
σˆ2
<
{(
∂P⊥(Θ)Y
∂Θ
)H
∂P⊥(Θ)Y
∂Θ
}
+c(Θ), (27)
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where σˆ2 has the form as given in (25), and where the term
c(Θ) vanishes after taking the expectation value (see, e.g., [40]).
Utilizing the differentiation results for projection matrices
from [44], [45], the partial derivatives in (27) with respect
to each channel variable have the form
∂P⊥(Θ)Y
∂τ
= − (gτ (Θ) + gτ (Θ)H)Y, (28)
where the auxiliary variable is given as
gτ (Θ) = P⊥(Θ)
∂B(Θ)
∂τ
B(Θ)†. (29)
The corresponding formulation can be conducted in a similar
manner for the remaining partial derivatives as well.
Since we model the antenna array responses by the means
of the EADFs, the partial derivatives of B(Θ) in (28) with
respect to each desired channel variable can be given as
∂B(Θ)
∂τ
=(FRAR(ϕR, ϑR))(ΞτBf (τ)) (30)
∂B(Θ)
∂ϕR
=(FR[GR,V,GR,H](Ξϕd(ϕR, ϑR)))Bf (τ) (31)
∂B(Θ)
∂ϑR
=(FR[GR,V,GR,H](Ξϑd(ϕR, ϑR)))Bf (τ), (32)
where the auxiliary matrices are
Ξτ = diag
(
−j2pif0
[
−Mf − 1
2
, . . . ,
Mf − 1
2
])
(33)
Ξϕ = diag
(
j
[
−Ma − 1
2
, . . . ,
Ma − 1
2
])
⊗ IMe (34)
Ξϑ = IMa ⊗ diag
(
j
[
−Me − 1
2
, . . . ,
Me − 1
2
])
, (35)
where Mf denotes the number of active subcarriers, and Ma
and Me are the spatial modes of the EADFs, respectively.
Finally, the overall FIM can be expressed as a real-valued
matrix
J(Θ)=
Jτ,τ (Θ) Jτ,ϕR(Θ) Jτ,ϑR(Θ)JϕR,τ (Θ) JϕR,ϕR(Θ) JϕR,ϑR(Θ)
JϑR,τ (Θ) JϑR,ϕR(Θ) JϑR,ϑR(Θ)
∈R3×3, (36)
while the root mean squared errors (RMSEs) of the estimates
can be obtained as
RMSE τˆ =
√
CRLBτ (Θ) =
√
J−1τ,τ (Θ) (37)
RMSE ϕˆR =
√
CRLBϕR(Θ) =
√
J−1ϕR,ϕR(Θ) (38)
RMSE ϑˆR =
√
CRLBϑR(Θ) =
√
J−1ϑR,ϑR(Θ). (39)
These RMSE values are used to analyze the theoretical
estimation performance of the individual channel variables
in the conducted simulations and numerical evaluations. We
also note that the derived FIM in (27) contains the stochastic
signal properties due to the variable projection and hence, the
final FIM utilized in the theoretical results is averaged over
several Monte Carlo realizations. Finally, in order to allow
for proper comparisons, we compare the theoretical lower
bounds of the proposed approach to those obtained using a
digital beamformer at the RX similar to the available existing
literature [7], [31]–[35].
B. Positioning Error Bounds
Since the DoA angles and the ToA can be related to the
target position using simple geometric relations, the obtained
FIMs can be translated to the PEBs [19], [32]. In particular, the
PEB gives the theoretical lower bound for the instantaneous
position estimate through position FIM. In this work, the PEB is
analyzed in order to provide better insights into how the CRLBs
of the DoA and ToA jointly affect the theoretical positioning
performance. Now, by denoting the unknown location of the TX
as pTX = [xTX, yTX, zTX]
T and the corresponding known RX
location as pRX = [xRX, yRX, zRX]
T, the relation between the
device location pTX and the DoA angles and the ToA is given
with a differentiable function [τ, ϕR, ϑR]T = h(pTX) ∈ Rm×1,
where [32]
h(pTX) =
‖pRX − pTX‖
c
arctan
(
yTX − yRX
xTX − xRX
)
arccos
(
zTX − zRX
‖pRX − pTX‖
)
, (40)
where c is the speed of light.
Considering then the mapping in (40), the FIMs of the
azimuth and the co-elevation DoA angles and the ToA can be
transformed into the 3D PEB according to
PEB =
√
trace(J−1pTX(pTX)) (41)
=
√
trace((H(pTX)TJ(Θ)H(pTX))−1), (42)
where H(pTX) ∈ R3×3 denotes the Jacobian matrix of
the measurement model function h(pTX) with respect to
the TX location pTX. While mutually synchronized TX and
RX as well as known RX orientation are assumed in the
theoretical evaluations in this work, we note that such effects
can also be taken into account in the considered CRLB and
PEB derivations in a similar manner as, e.g., in [8], [32].
Furthermore, in the conducted simulations, we extend and
demonstrate the performance of the proposed method also in
joint TX positioning and synchronization scenarios.
IV. PROPOSED EKF-BASED ESTIMATION AND TRACKING
SOLUTION
The proposed sequential estimation framework consists of
two stages, as illustrated at a general level in Fig. 1. The
fundamental idea behind the two-fold approach is that the
output of the first-stage estimation process allows for efficient
positioning through Gaussian approximations without the need
of communicating the raw channel measurements between
the network elements, e.g., when several network nodes are
participating in device positioning. In this section, we derive
the proposed first-stage EKF where the beam-based multicarrier
observations are first fused into the DoA angle and the ToA
estimates. Thereafter, we shortly describe the second-stage EKF
where the obtained DoA and ToA estimates are facilitated in the
final device positioning, thus providing essential information
on the performance of the proposed method through one and
easily understandable performance metric. It is to note that the
second-stage positioning EKF is not considered as a novelty
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D
oA
&
To
A
E
K
F
Y[n]∈CM×1 N
(
Θˆ+[n], [Cˆ+[n]](1:3,1:3)
)
Po
si
tio
ni
ng
E
K
F
N
(
xˆ+[n], Pˆ+[n]
)Prediction step of the DoAand ToA EKF using (48)-(49) Prediction step of thePositioning EKF using (58)-(59)
Update step of the DoA and
ToA EKF using (51)-(52)
Update step of the Positioning
EKF using (60)-(62)
Fig. 2: Schematic of the proposed estimation and tracking framework, where the beam-based measurements are fused into the TX location estimate and
the corresponding location uncertainty after employing the information-based beam-selection method. The dashed boxes indicate the possibility to fuse
measurements from several RX nodes, as shortly illustrated in Section VI-D.
of the work as such, since similar works on fusing DoA and
ToA measurements in positioning are readily available in the
existing literature. The more detailed schematic of the proposed
two-fold estimation framework is depicted in Fig. 2.
A. Joint DoA and ToA Estimation and Tracking EKF
After obtaining the most significant beams according to (15),
the DoA angles and the ToA are then jointly estimated by fusing
the beam-based measurements in an EKF-based solution. In
general, an EKF is a sequential Bayesian estimation approach
for the systems with non-linear state and measurement models
where the noise is assumed to be Gaussian distributed [20],
[46]–[48]. In contrast to classical estimation methods, an EKF
is able to incorporate prior information of the state to the
estimation process which in turn can improve the overall
estimation performance [19], [20], while being computationally
more attractive compared to other Bayesian estimation methods,
e.g., particle filters [46].
In this work, a constant white-noise acceleration (CWNA)
model is considered as a state evolution model for the system,
implying that the considered state variables, that is, the DoA
angles and the ToA, are assumed to evolve with almost a
constant rate over time. At a time-instant n, the state of the
system can be written as
s[n]=
[
ΘT[n],∆ΘT[n]
]T
(43)
=[ϕR[n], ϑR[n], τ [n],∆ϕR[n],∆ϑR[n],∆τ [n]]
T
, (44)
where ∆Θ[n] = [∆ϕR[n],∆ϑR[n],∆τ [n]]T denote the rate-of-
change in the azimuth and co-elevation DoA angles, and in the
ToA, respectively. Given the CWNA state-transition model and
the considered multi-beam observation model, the state-space
model can be thus written as
s[n] = Fs[n− 1] + w[n] (45)
Y[n] = B(s[n])ψ[n] + n[n], (46)
where the measurement model is as shown in (17), while the
state-transition matrix F ∈ R6×6 with ∆t denoting the time
between two consecutive estimation time-instances reads
F =
[
1 ∆t
0 1
]
⊗ I3. (47)
Furthermore, the state-process noise is given as w[n] ∼
N (0,Q[n]), where the state-model driving noise covariance
matrix can be obtained through discretization of the continuous
time model [46]. Using a similar notation as in [20], the a
priori estimates of the mean sˆ−[n] and the covariance matrix
Cˆ
−
[n] can be then obtained such that
sˆ−[n] = Fsˆ+[n− 1] (48)
Cˆ
−
[n] = FCˆ
+
[n− 1]FT + Q[n], (49)
where sˆ+[n−1] and Cˆ+[n−1] denote the previous a posteriori
estimates of the mean and the covariance matrix, respectively.
After the prediction step, the measurements can be then
incorporated to the estimation process in the update step of
the EKF. Instead of a more common Kalman-gain form of the
EKF, the a posteriori estimates are updated in this work in
the information form1 by employing the gradient and observed
FIM of the measurement model function [15]. Such a decision
is stemming from the fact that the information form of the
update step can be found computationally more attractive than
the Kalman-gain form especially in a case of having large
number of complex-valued measurements [16], [49].
Next, in order to derive the gradient and the observed FIM
for the proposed EKF, we assume that the TX beam-index iˆ
as well as the set of RX-beams MB have been determined
and available at the RX. As described in Section III, we
are not interested in tracking the noise variance σ2 or the
combined channel and TX information ψ[n] in the proposed
estimation framework. Therefore, we employ a subspace fitting
technique through variable projection [44] and concentrate the
log-likelihood function with respect to the nuisance variables in
a similar manner as in the CRLB derivations in Section III. In
particular, we deploy the observed FIM defined in (27), and thus
we only need to determine the gradient of the log-likelihood
function for the proposed EKF. Given the concentrated log-
likelihood function in (26), we can thus calculate the gradient,
i.e., the score-function, by taking the partial derivative of the
concentrated log-likelihood with respect to the related channel
parameters Θ[n] according to [19], [40]
q(Θ[n]) =
∂
∂Θ[n]
ˆ`(Y[n]|Θ[n]) (50)
= − 2
σˆ2
<
{(
∂P⊥(Θ[n])Y[n]
∂Θ[n]
)H
P⊥(Θ[n])Y[n]
}
,
1In contrast to the information filter in [20], the exploited information form
of the update step follows the notion used, e.g., in [15] and it is essentially
the same as a single update iteration of the iterated Kalman filter [49].
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TABLE I: APPROXIMATED NUMBER OF FLOPS REQUIRED IN THE DOA AND
TOA ESTIMATION AND TRACKING EKF (pM).
Prediction step 4p3 + 3p2
Update step (18 + 82p)M2+196pM+ 4|MB|M2R+ 8MRM2f
where σˆ2 is given in (25), and the partial derivatives can be
obtained as described in Section III.
Finally, given the gradient (50) and the observed FIM (27),
the a posteriori covariance matrix and mean estimates denoted
as Cˆ
+
[n] and sˆ+[n], respectively, can be obtained using the
information form of the update step [15] as
Cˆ
+
[n] = ((Cˆ
−
[n])−1 + J(sˆ−[n]))−1 (51)
sˆ+[n] = sˆ−[n] + Cˆ
+
[n]q(sˆ−[n]), (52)
where q(sˆ−[n]) and J(sˆ−[n]) are the gradient and the observed
FIM, both evaluated at a priori mean estimate s−[n]. Since the
log-likelihood function does not depend on the state-variables
∆Θ[n], the corresponding values in the gradient and in the
observed FIM are zero. The computational complexity of the
proposed DoA and ToA EKF is in the order of O(M2) as
presented in Table I by the means of approximate floating
point operations (FLOPs), thus implying that the complexity
is dominated by the utilized bandwidth and the number of
facilitated beams.
In this work, the initialization of the DoA and ToA estimation
EKF is assisted by first selecting the most powerful beam-
indices in a similar manner as in [18] and then calculating
the ML estimates for the DoA angles and ToA based on the
selected beams. Such a choice is stemming from the fact that
the prior information of the TX location is not assumed to be
available in the initialization and thus the information-based
beam-selection approach cannot be facilitated as such. However,
other initialization methods can be applied to the considered
estimation problem as well. Specifically, we initialize the DoA
and ToA estimates according to
Θˆ
+
[0] = argmax
Θ
ˆ`(Y[0]|Θ) (53)
[Cˆ
+
[0]]1:3,1:3 = (J(Θˆ
+
[0]))−1, (54)
where [·](k:l,k:l) denotes the submatrix between the indices k
and l. Moreover, J(Θˆ
+
[0]) is the observed FIM as given in (27)
and it is evaluated at the obtained ML estimate. Without the
loss of generality, the velocity components of the state-vector
can be initialized after two consecutive state-estimates.
B. Positioning EKF
We next express the corresponding positioning EKF utilizing
the estimated DoA angles and ToA values as measurements. In
the considered positioning EKF, we assume that the velocity
of the target is almost constant and therefore, the linear CWNA
state-transition model is employed. Given the considered state-
model, we can write the state-vector of the positioning EKF
as
x[n]=[x[n], y[n], z[n],∆x[n],∆y[n],∆z[n]]T ∈ R6×1, (55)
TABLE II: APPROXIMATED NUMBER OF FLOPS REQUIRED IN THE POSI-
TIONING EKF.
Prediction step 4`3 + 3`2
Update step 6m`2 + 4`m2 +m3 + `3 + 2`2 + 2m`
where pTX[n] = [x[n], y[n], z[n]]
T denotes the 3D Cartesian
coordinates of the target, and [∆x[n],∆y[n],∆z[n]]T denotes
the corresponding velocity components. Due to the considered
non-linear measurement model, through which the estimated
DoA angles and the ToA are related to the state-vector, the
state-space model of the positioning EKF can be thus written
as
x[n] = Fx[n− 1] + u[n] (56)
Θˆ
+
[n] = h(x[n]) + v[n], (57)
where the measurement Θˆ
+
[n] = [τ+[n], ϕ+[n], ϑ+[n]]T is
the a posteriori estimate of the joint DoA and ToA estimation
and tracking EKF, and the state-transition model matrix F
is identical to that in (47). For the sake of simplicity, we
here assume that the orientation of the RX is known and
the RXs and the TX are synchronized, thus allowing us to
analyze the pure positioning performance of the proposed
method. However, such assumptions can be relaxed through
joint estimation of the involved variables [8], [13], [17], [18], by
employing a time difference of arrival (TDoA)-based approach,
or by using a two-way communications [50]. Hence, the
considered measurement model function h(x[n]) relates the
obtained DoA and ToA estimates to the current state of the
positioning EKF as given in (40). Moreover, the state-model
driving noise u[n] ∼ N (0,U[n]), and the measurement noise
v[n] ∼ N (0, [Cˆ+[n]](1:3,1:3)), where [Cˆ
+
[n]](1:3,1:3) is the
covariance matrix of the DoA and ToA estimates obtained
using the first-stage EKF.
For a given state-space model, the prediction step of the
positioning EKF can be expressed as
xˆ−[n] = Fxˆ+[n− 1] (58)
Pˆ
−
[n] = FPˆ
+
[n− 1]FT + U[n] (59)
after which the DoA and ToA estimates can be fused in the
update step of the proposed positioning EKF by employing
the Kalman-gain form of the EKF. This is expressed as
K[n] = Pˆ
−
[n]H[n]T(H[n]Pˆ
−
[n]H[n]T + Cˆ
+
[n])−1 (60)
P+[n] = (I−K[n]H[n])Pˆ−[n] (61)
x+[n] = x−[n] + K[n](Θˆ
+
[n]− h(x−[n])), (62)
where H[n] is the Jacobian matrix of the measurement model
function (40) evaluated at the a priori mean estimate x−[n].
The computational complexity of the positioning EKF is in the
order of O(`3) as given in Table II, where ` and m denote the
dimensions of the state and observation vectors, respectively.
In this work, the positioning EKF is initialized in a similar
manner as the proposed DoA and ToA EKF by adopting
applicable ML estimates. After the two consecutive time-
instants, the corresponding velocity components of the state
can be initialized based on the difference between the first TX
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(a) Urban canyon scenario
(b) Open area scenario
Fig. 3: Considered a) urban canyon and b) open area scenarios on top of
the METIS Madrid grid environment. In both scenarios, the TX is moving
from left to right with 30 km/h, thus attempting to imitate a realistic vehicle
movement in an urban environment. Furthermore, the TX location utilized in
CRLB evaluation results in Fig. 5 is indicated in b).
location estimates. In addition to the state-vector, the covariance
estimate corresponding to the velocity components is initialized
as a relatively large diagonal matrix.
V. EVALUATION SCENARIOS AND ASSUMPTIONS
In order to evaluate the performance of the proposed solution
in terms of the channel parameter estimation and the subsequent
positioning accuracy, simulations employing a full ray-tracing
channel model [51], [52] are carried out on top of the METIS
Madrid grid [53] with two different deployment scenarios,
namely the street canyon and the open area scenarios depicted in
Fig. 3a and Fig. 3b, respectively. The system-level numerology
for the baseline evaluation scenarios is given in details in
Table III. In particular, we assume that the TX is equipped
with a uniform linear array (ULA) whereas the RX is equipped
with a uniform rectangular array (URA), and the array elements
at both link ends are modeled as dual-polarized patch-elements
as described in [51], [54]. Moreover, the number of analog
RF beams is assumed to be limited to the number of antenna
array elements while it is noted that a denser grid of beams
could also be considered in the estimation process with a cost
of an increasing communication overhead.
In general, the network is considered to operate at 28 GHz
carrier frequency with a subcarrier spacing of 60 kHz. In the
evaluations, the TX moving with 30 km/h along the predefined
trajectory transmits SRS employing a single OFDM symbol in
time-domain at every 100 ms (every 400th slot). In this work,
we implement the comb2-configuration indicating that the SRS
TABLE III: BASELINE EVALUATION NUMEROLOGY AND ASSUMPTIONS
Parameter Value
Carrier frequency 28GHz
Subcarrier spacing 60 kHz
Bandwidth 25MHz
Antenna elements 3GPP patch-elements [51]
TX array model ULA (4 dual-polarized elements)
RX array model URA (4×4 dual-polarized elements)
TX velocity 30 km/h
RX height 7.5m
TX height 1.5m
TX power +10dBm
SRS periodicity periodic, every 100ms
SRS resource allocation one OFDM symbol, comb2 [36]
occupies every second subcarrier in the frequency domain,
whereas various bandwidth allocations can be considered
depending on the selected resource block (RB) configura-
tion [36], [55]. The user equipment (UE) transmit power is set
to +10 dBm, and the noise spectral density and the base-station
RX noise figure are defined to be −174 dBm/Hz and 5 dB,
respectively. Such choices result in SNRs between 15-35 dB
depending on the TX location as well as the chosen bandwidth
and antenna array configurations. In the conducted simulations,
the beam-selection threshold of the proposed beam-selection
method is selected from the set αthr ∈ {0.3, 0.5, 0.9}.
In evaluating the theoretical lower bounds and PEB of the
considered multi-beam fusion, we utilize various configurations
of the TX and RX antenna arrays while considering only
the open area scenario in order to allow for more intuitive
analysis by the means of active beam-directions and observed
SNRs. Thereafter, the proposed EKF-based solution facilitating
the proposed beam-selection method is compared with the
similar EKF-based solution employing an BRSRP-based beam-
selection method presented in [18] in the urban area scenario
with more severe multi-path characteristics. The DoA and
ToA estimation performance of the EKF-based methods are
compared with the similar ML-based approaches which can
be obtained in a straightforward manner by facilitating the
provided gradient and Hessian matrix. In order to avoid
divergence when employing consecutive estimates in ML
algorithm, an initial guess in ML is drawn from the Gaussian
distribution using the reference channel parameters as the mean
at each time-instant. Besides the DoA and ToA estimation
performance, the EKF-based solutions are finally compared by
the means of 3D positioning performance in order to analyze
the joint performance of the DoA and ToA estimates with a
more convenient metric.
VI. NUMERICAL EVALUATIONS AND RESULTS
A. CRLBs Along Open Area Trajectory
We first analyze the impact of the proposed beam-selection
method on the theoretical ToA and DoA angle estimation perfor-
mance and the subsequent PEB along the open area trajectory
as depicted in Fig. 3b. Besides illustrating the performance
of the proposed method, we compare the theoretical bounds
to those obtained using a digital beamformer at the RX side.
Evaluating and analyzing the theoretical bounds only with the
presented open area scenario is stemming mainly from the
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Fig. 4: Theoretical lower bounds for the DoA elevation and azimuth estimates (top row), ToA estimates (bottom row on the left), and positioning performance
(bottom row on the right) along the trajectory in the considered open area scenario. Digital RX beamformer based bounds are also shown for reference.
fact that it allows for easier performance evaluation due to the
more intuitive and symmetric SNR behavior along the trajectory.
However, the performance of the proposed EKF-based solutions
are still evaluated and compared to the performance of the
reference ML estimator in the urban canyon scenario with
more severe multipath propagation characteristics.
Since the theoretical ToA and DoA estimation performance is
in general highly dependent on the observed SNR, which in turn
reflects the current TX location as well as the deployed antenna
configurations with the assumed bandwidth, we first provide the
theoretical lower bounds along the TX trajectory with couple
of antenna configurations and with 25 MHz RS bandwidth.
The obtained theoretical ToA estimation performance of the
proposed multi-beam fusion is illustrated in Fig. 4 together
with an estimator that considers only a single RX-beam in the
ToA estimation. Despite the fact that the performance of the
single-beam estimator is close to those of the proposed multi-
beam and the digital beamformer approaches, e.g., in the RX
boresight direction, a clear difference in the performance can
be observed when the beam-directions between the TX and RX
are not perfectly aligned. Such occasions can be observed in
the case of TX having two dual-polarized elements (blue line),
e.g., before and after the boresight direction of the RX, and in
the beginning and in the end of the trajectory. Based on a rather
straightforward derivation from (30), such an improvement in
the ToA estimation performance of the proposed multi-beam
method is proportional to the sum of beam-based SNRs over
the selected RX-beams. This in turn implies that when the
TX and RX beams are perfectly aligned and almost all the
received signal power is in the single beam-pair, there is no
larger difference in the ToA estimation between the single-
beam and the proposed multi-beam approach. In general, we
can observe that the performance of the proposed multi-beam
fusion is extremely close to that obtained by employing a digital
beamformer at the RX. The small difference in the results is
stemming from the limited spatial sampling of the channel in
the assumed analog/RF beamforming scheme compared to that
available with the digital beamformer.
Moreover, a similar SNR-dependency can be observed also
in the theoretical DoA estimation performance as depicted in
Fig. 4, where the same beam-selection threshold of αthr = 0.5
is employed. Occasional peaks in the theoretical estimation
performance are stemming from bad realizations and the
utilized projection approach, which in turn can be filtered
out with proper sequential estimation solution, for instance.
Based on the detailed analysis of the obtained results, at
least two RX-beams are required in both azimuth and co-
elevation direction in order to ensure theoretically feasible
estimation accuracy without divergence in the variance of
the corresponding estimators. Therefore, we are not able to
provide the theoretical DoA estimation performance for a single-
beam approach. In contrast to the ToA estimation performance
that is related to the beams with high beam reference signal
received powers (BRSRPs), the DoA estimation accuracy is
also significantly affected by the set of chosen RX-beams as
expected. For instance, the CRLB of the azimuth DoA becomes
smaller when the RX beam-set contains various beams in the
azimuth plane, whereas improvement in co-elevation estimation
accuracy is depending on the number of beams in the elevation
domain. In order to analyze the impact of theoretical ToA
and DoA estimation errors on the corresponding positioning
performance, we translate the angular and temporal bounds
to the PEB, and the obtained results are depicted in Fig. 4.
Based on the PEB results, the proposed two-stage estimation
approach can in theory facilitate extremely accurate positioning
performance especially with high SNRs, while being able to
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provide almost as good performance as the reference system
employing a digital beamformer at the RX-side.
Based on a more detailed analysis regarding the number
of selected RX-beams, we observed that the number of the
beams is related to the observed SNR in two ways. First,
in a case of poor SNR due to a possible beam-mismatch,
the proposed solution combines more RX-beams in order to
improve the estimation performance as expected. Second, for
almost perfect TX-RX beam-alignment, the number of RX-
beams in the proposed beam-selection method increases in
order to gather enough information to keep the DoA estimation
performance at a reasonable level. This observation can be
explained with the chosen orthogonal set of RX-beams which
in turn means that the gain of secondary RX-beams is small
in such beam-alignment occasions.
B. CRLBs with Different Beam-Selection Thresholds and TX-
RX Antenna Configurations
We next provide theoretical lower bounds for the considered
channel parameters as a function of various TX and RX
configurations given a single TX location on the open area
trajectory. In Fig. 5a, the bounds for the ToA and DoA estimates
are shown with respect to the number of TX antenna elements.
As expected, the theoretical estimation performance increases
when the number of antenna elements at the TX is increased.
This is mainly due to the improved TX-gain which in turn
improves the received SNR. Based on a more detailed analysis
of the obtained results, the number of selected RX-beams
is essentially independent of the number of elements in the
TX ULA. Such an observation is relatively intuitive, since
only the best TX-beam is selected in the proposed beam-
selection process for the subsequent estimation. With the
considered 4 × 4 URA at the RX, the number of selected
RX-beams in the proposed beam-selection at such TX location
is either 4, 6, or 12 depending only on αthr ∈ {0.3, 0.5, 0.9},
correspondingly. As expected, the performance of the proposed
method increases when the number of RX-beams increases,
that is, when the value of αthr is increased. However, the
computational complexity of the algorithm is also proportional
to the number of RX-beams as indicated in Section IV, which
implies that the higher performance is achieved at the cost
of added computational complexity. Here, it is to note that
the provided results in Fig. 5a-5b are obtained in a single
TX-location and instantaneous SNRs may have an impact
on the observed results. For instance, the slightly degraded
performance in Fig. 5a and 5b in the case of 8 TX elements
compared to those of 5-7 TX-side elements is due to the bad
geometry of the TX-beams, i.e., the TX with 4 and 8 elements
have the same notch in the considered TX location due to
beam-mismatch, whereas the cases with 5-7 TX elements are
more favorable in this given TX location.
Finally, the performance bounds with varying number of
antenna elements at the RX-side are depicted in Fig. 5b.
Because of the better resolution in angular domain and the
improved RX beamforming gain, the estimation performance
of both ToA and DoA increases when the number of antenna
elements at the RX increases. However, the proposed beam-
selection method adapts to the increasing number of antenna
elements, and therefore, the number of facilitated beams at the
RX also increases as expected. Based on the overall results
shown in Fig. 5a-5b, the estimation performance of the co-
elevation angle is slightly better than that of the azimuth angle.
This is mainly due to the fact that the considered antenna system
at the TX results in wider beams in the elevation domain and
the height of the TX is resulting in a favorable elevation-angle
that is not typically aligned with the employed RX-beams, thus
distributing the received signal power to some extent across
several beams in elevation domain.
C. Estimation Performance of Proposed Two-Stage EKF
In addition to assessing the theoretical performance, we also
evaluate and provide the actual estimation performance results
with the proposed two-stage EKF employing the proposed
beam-selection method in the more complex urban canyon
scenario with more severe multi-path propagation compared
to the previous open area evaluations. The proposed method
is compared with the same EKF-based solution that utilizes
the beam-selection method where the candidate RX-beams
are determined based on a fixed number of most powerful
BRSRP measurements as used, e.g., in [18]. The number of
RX-beams in the compared beam-selection method is selected
based on αthr such that the number of beams is on average
at the same level or higher compared to the proposed beam-
selection method. Moreover, the obtained EKF-based results
are compared with the similar ML-based solutions in order to
see the impact of sequential filtering on the achieved estimation
performance. In the conducted simulations, we also compared
the proposed method against two adaptive beam-selection
methods proposed in [17], [22]. However, the number of
facilitated RX-beams in [22], relying on a likelihood-ratio
test, was all the time significantly higher than in the proposed
and the non-adaptive [18] methods, thus making the fair
comparison unfeasible. In addition, the method in [17] resulted
in a divergence of the first-stage EKF due to extremely small
covariance estimates and the fact that the RX-side beamformer
is not designed as a function of angles. Hence, the adaptive
beam-selection methods in [17], [22] are not comparable with
the proposed method. The DoA and ToA estimation results for
all the considered simulation scenarios are depicted in Fig. 6
and Fig. 7, respectively.
Based on the obtained DoA estimation results depicted
in Fig. 6, the proposed EKF-based method is outperforming
the ML-based approach not only in terms the median error
(black horizontal line), but also in terms of the 50% and
99% confidence intervals (illustrated with boxes and whiskers).
Such observations are caused by better noise filtering of
the EKFs which take into account also the available prior
information in the estimation process. In all the considered
simulation cases, the average number of selected RX-beams
in the case of the proposed beam-selection method (denoted
as Info. in the corresponding figures) is lower compared to
the BRSRP-based beam-selection method, while the estimation
performance of the proposed method is still better than that
of the BRSRP-based approach. This in turn highlights the
importance of considering the available observed information
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(a) (b)
Fig. 5: Theoretical lower bounds for DoA angles and ToA as a function of the number of dual-polarized antenna elements a) at the TX-side and b) at the
RX-side. Again, digital RX beamformer based bounds are shown for reference.
in selecting the candidate RX-beams in an adaptive manner
for DoA estimation and subsequent positioning. However, the
performance differences between the considered beam-selection
methods are not significant with higher number of selected
beams (case with 50% of beams or αthr = 0.5) due to the fact
that the most informative beams are most likely included in
the candidate beam-sets of both beam-selection methods.
Furthermore, similar characteristics can be also observed
from the obtained ToA estimation results, depicted in Fig. 7.
At some time-instances with 100 MHz bandwidth, the BRSRP-
based ML-estimator is not able to fully converge during
the allowed Gauss-Newton iterations, thus resulting in the
maximum ToA estimation errors around 4 m. Overall, the
observed ToA and DoA estimation results are extremely
accurate. In general, increasing the bandwidth and the number
of antenna elements at the RX side improves the estimation
performance as expected. However, increasing the number
of antenna elements at the RX also increases the number
of available beams and therefore, also the number of selected
candidate beams intuitively increases. Based on a more detailed
analysis, we observed that at some time-instances the existing
multi-path components cause a minor degradation in the
estimation performance. This in turn highlights the role of
proper data-association, where beams corresponding to different
propagation paths could be distinguished even better. Such a
data-association and properly distinguished non-LoS (NLoS)
paths could potentially improve the estimation performance of
the considered system [56].
Finally, we evaluate the 3D positioning performance using
the proposed two-stage EKF with both presented beam-
selection methods, namely the proposed information-based
method and the BRSRP-based method that is used in [18].
For the sake of simplicity, the performance of the previously
presented ML estimators are ignored in the following po-
sitioning evaluations due to generally lower DoA and ToA
estimation performance compared to the EKF-based solutions
as seen in Figures 6-7. Based on the obtained 3D positioning
results in Figure 8, the proposed beam-selection method is
able to provide more accurate and consistent positioning
performance than the reference method with the BRSRP-based
beam-selection approach. In particular, significant positioning
performance differences can be observed between the beam-
selection methods, when the number of selected RX-beams
is relatively low. This performance improvement is stemming
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Fig. 6: Azimuth and elevation angle estimation performance of the proposed first-stage EKF-based solution with the proposed beam-selection method (denoted
as Info.) in comparison with the non-adaptive beam-selection method (denoted as BRSRP) with the same EKF-based solution, and similar ML-based solutions.
Three different simulation numerologies are considered in the urban canyon scenario while showing also two different beam-selection thresholds of αthr = 0.3
and αthr = 0.5. The average number of facilitated RX-beams with the considered beam-selection method are illustrated on top of the corresponding box-plots.
The black horizontal line in each box-plot indicates the median value, whereas the wider box and the whiskers represent approximately the 50% and 99%
confidence intervals, respectively.
Fig. 7: ToA estimation performance of the proposed first-stage EKF-based
solution with the proposed beam-selection method compared to another beam-
selection method and to the similar ML-based solutions.
Fig. 8: 3D positioning performance results with the proposed beam-selection
method and the proposed two-stage EKF compared with the same EKF-based
solution under a different beam-selection method.
from fact that at some time-instances, the proposed beam-
selection method can facilitate one or two beams more than
the non-adaptive beam-selection method in order to improve
the estimation performance at those unfavorable situations.
However, the average number of utilized beams in the proposed
beam-selection method is still lower than in the BRSRP-
based method, which implies that the proposed beam-selection
method is able to select more suitable RX-beams even when
the number of selected beams is low.
D. Extension to Multi-RX Fusion and Joint TX Synchronization
Due to the cascaded structure of the proposed framework,
the method can be extended in a straightforward manner to
cover situations where several network nodes are in a favorable
LoS condition towards a given TX. Such an extension enables
not only improving the positioning performance through
information fusion but also allows for estimating other closely
associated parameters and variables including TX orientation
and clock offsets [16]–[18] or employing TDoA-based position-
ing approach to compensate for the mutually asynchronous TX
and RXs clocks. In order to enable positioning in a practical and
asynchronous network, we shortly present an extension of the
proposed method to a joint TX positioning and synchronization
where not only the position but also the clock offset and
skew of the TX/UE are estimated by extending the state-
vector (55) and by including the TX clock offset term in (57)
in a similar manner as, e.g., in [16]. Furthermore, we assume
that the RX nodes are mutually synchronized, as e.g. in time-
sensitive networks [57]. Since extending the proposed solution
to the multi-RX and joint TX synchronization scenarios is
conceptually straightforward, we omit the further mathematical
derivation for such cases in this paper. In short, the channel
parameters are first estimated at individual LoS-RX nodes after
which these estimates are then fused into the TX position and
clock offset estimates in a sequential manner, conceptually
similar to [16], [18].
In the considered multi-RX fusion scenario, a TX is assumed
to be attached on top of a vehicle such that the orientation of
the TX’s antenna panel (8 ULA with 3GPP patch-elements)
is parallel to the vehicle’s heading direction. In particular, the
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Fig. 9: Illustration of a multi-RX positioning scenario, with a TX attached to a vehicle moving according to a realistic acceleration profile (maximum speed
of 40 km/h) along a predefined trajectory at the METIS Madrid Map environment (left). The obtained positioning results (right) are illustrated for the
positioning-only EKF (Pos EKF) and for the joint positioning and synchronization EKF (Pos&Sync EKF) in terms of cumulative distribution functions (CDFs).
TX transmits 8 SRS-beams using a 50 MHz bandwidth every
100 ms which are then received at LoS-RXs and processed
using the proposed EKF-based scheme. In addition, the clock of
the TX is modelled using the first-order auto-regressive model
in a similar manner as described in [16], [58]. In order to
model the propagation environment as realistically as possible,
the channel is modelled using extensive ray-tracing simulations
on top of the METIS Madrid Map environment depicted in
Fig. 9.
As expected, based on the results shown in Fig. 9, the 3D
positioning performance significantly increases when the DoA
and ToA measurements from multiple available RX nodes are
fused for TX positioning. Even with the joint positioning and
synchronization case (denoted as Pos&Sync EKF in Fig. 9),
better than 2 m positioning accuracy can be reached in more
than 90% of the cases, while being able to obtain as accurate
as 3.6 ns and 2.0 ns RMSEs for the TX clock offsets in the
two and three LoS-RXs scenarios, respectively. For the sake of
better visualization, we provide the same simulation scenario as
a complementary online multimedia material at https://research.
tuni.fi/wireless/research/positioning/TWC2021/.
VII. CONCLUSIONS
Building on the existing 5G NR specifications, we proposed
a novel estimation and tracking solution for the DoA and
ToA using an EKF-based solution for an mmWave system
with emphasis on analog/RF beamforming capabilities. The
proposed approach consists of a novel beam-selection method,
which is able to determine a set of candidate RX-beams based
on the available observed information in an adaptive manner. In
order to evaluate the theoretical estimation performance of such
multi-beam approaches, we derived and evaluated the CRLBs
for the considered estimation problem, while also comparing
to the corresponding bounds of digital RX beamforming-based
system. Based on the extensive ray-tracing based evaluations
covering both open area and street canyon type propagation
scenarios on the METIS Madrid grid at 28 GHz, the proposed
solution was shown to outperform the reference methods almost
in all of the cases, while the most significant benefit was
observed when the number of available RX-beams is low. In
general, the ability to perform efficient positioning and tracking
through RF-only beamforming is emphasized further when the
network center-frequencies are increasing towards the sub-THz
regime, paving the way for future 6G systems.
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Mike Koivisto was born in Rauma, Finland, in
1989. He received the M.Sc. degree in mathematics
from the Tampere University of Technology (TUT),
Finland, in 2015, where he is currently pursuing the
Ph.D. degree. From 2013 to 2016, he was a Research
assistant with TUT. He is currently a Researcher with
the Electrical Engineering unit at Tampere University.
His research interests include positioning with an
emphasis on network-based positioning and the
utilization of location information in future mobile
networks.
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TWC.2021.3119227, IEEE
Transactions on Wireless Communications
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 16
Jukka Talvitie was born in HyvinkA˜¤A˜¤, Finland,
in 1981. He received the M.Sc. degree in automation
engineering and the Ph.D. degree in computing and
electrical engineering from the Tampere University
of Technology (TUT), Finland, in 2008 and 2016,
respectively. He is currently a University Lecturer at
Tampere University, Finland. In addition to academic
research, he has involved several years in industry-
based research and development projects on wide
variety of research topics, including radio signal
waveform design, network based positioning and next
generation WLAN, and cellular system design. His main research interests
include signal processing for communications, wireless locations techniques,
radio signal waveform design, and radio network system level development.
Elizaveta Rastorgueva-Foi received M.Sc. in elec-
trical engineering from Tampere University (TAU),
Finland, in 2019, where she is currently working
towards Ph.D. degree. Her research interests include
statistical signal processing, positioning and location-
aware communications in mobile networks with an
emphasis on 5G and beyond.
Yi Lu received the M.Sc degree in mobile communi-
cations (with distinction) from Heriot-Watt University,
UK, in 2012. From 2014 to 2017, he worked as
a Research Assistant at Universitat Autonoma de
Barcelona (UAB), Spain. Since March 2018, he
has been employed as a Doctoral Researcher at the
Electrical Engineering unit of Tampere University,
Finland. In addition, he serves as a TPC member of
IEEE CIC ICCC since 2019. His research interests in-
clude the development of network-centric positioning
system and location-aware communications scheme
for the Industrial IoT system.
Mikko Valkama was born in Pirkkala, Finland,
in 1975. He received the M.Sc. (Tech.) and D.Sc.
(Tech.) degrees (Hons.) in electrical engineering
from Tampere University of Technology (TUT),
Finland, in 2000 and 2001, respectively. In 2002,
he received the Best Ph.D. Thesis Award from the
Finnish Academy of Science and Letters for his
dissertation entitled Advanced I/Q Signal Processing
for Wideband Receivers: Models and Algorithms. In
2003, he was a Visiting Post-Doc Researcher with
the Communications Systems and Signal Processing
Institute, SDSU, San Diego, CA, USA. He is currently a Full Professor
and the Head of the Electrical Engineering Unit at the newly established
Tampere University (TAU), Finland. His general research interests include radio
communications, radio localization, and radio-based sensing, with emphasis
on 5G and beyond mobile radio networks.