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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. XX, NO. X, XXX 2022 1
Efficient Joint DOA and TOA Estimation for Indoor
Positioning with 5G Picocell Base Stations
Mengguan Pan, Peng Liu, Shengheng Liu, Senior Member, IEEE, Wangdong Qi, Member, IEEE,
Yongming Huang, Senior Member, IEEE, Xiaohu You, Fellow, IEEE, Xinghua Jia, Xiaodong Li
Abstract—The ubiquity, large bandwidth, and spatial diversity
of the fifth generation (5G) cellular signal render it a promis-
ing candidate for accurate positioning in indoor environments
where the global navigation satellite system (GNSS) signal is
absent. In this paper, a joint angle and delay estimation (JADE)
scheme is designed for 5G picocell base stations (gNBs) which
addresses two crucial issues to make it both effective and
efficient in realistic indoor environments. Firstly, the direction-
dependence of the array modeling error for picocell gNB as
well as its impact on JADE is revealed. This error is mitigated
by fitting the array response measurements to a vector-valued
function and pre-calibrating the ideal steering-vector with the
fitted function. Secondly, based on the deployment reality that
5G picocell gNBs only have a small-scale antenna array but
have a large signal bandwidth, the proposed scheme decouples
the estimation of time-of-arrival (TOA) and direction-of-arrival
(DOA) to reduce the huge complexity induced by two-dimensional
joint processing. It employs the iterative-adaptive-approach to
resolve multipath signals in the TOA domain, followed by a
conventional beamformer to retrieve the desired line-of-sight
DOA. By further exploiting a dimension-reducing pre-processing
module and accelerating spectrum computing by fast Fourier
transforms, an efficient implementation is achieved for real-
time JADE. Numerical simulations demonstrate the superiority
of the proposed method in terms of DOA estimation accuracy.
Field tests show that a triangulation positioning error of 0.44
m is achieved for 90% cases using only DOAs estimated at two
separated receiving points.
Index Terms—Direction-of-arrival, time-of-arrival, JADE, ar-
ray modeling error, efficient implementation, positioning, wireless
localization, 5G.
I. INTRODUCTION
LOCATION awareness plays a paramount role in a wealth
of scenarios, such as autonomous driving [2], intelligent
transportation [3], emergency relief [4], assisted living [5],
etc., in the era of Internet-of-everything (IoE). In outdoor
environments, the global navigation satellite systems (GNSS)
provide robust and accurate positioning information, while
in deep urban canyons and indoor environments, they are
unreliable owing to the severe blockage of the line-of-sight
(LOS) signals.
Manuscript received XXX XX, 2022. This research was supported in part by
the National Natural Science Foundation of China under Grant Nos. 62001103
and U1936201. Part of this work was presented at the 2021 CIE International
Radar Conference [1]. (Corresponding author: P. Liu)
The authors are with the Purple Mountain Laboratories, Nanjing
211111, China. (Emails: panmengguan@pmlabs.com.cn, herolp@gmail.com,
s.liu@seu.edu.cn, qiwangdong@pmlabs.com.cn, huangym@seu.edu.cn,
xhyu@seu.edu.cn)
P. Liu is also with the Nanjing University of Aeronautics and Astronautics,
and the Army Engineering University of PLA, Nanjing 210007, China.
S. Liu, W. Qi, Y. Huang, and X. You are also with the National Mobile
Communications Research Laboratory, Southeast University, Nanjing 210096,
China.
Recently, considerable attention has been devoted to employ
other radio frequency (RF) signals, such as ultra-wideband
(UWB) signals [6], [7], Bluetooth signals [8], [9], Wi-Fi
signals [10], [11], radio frequency identification (RFID) sig-
nals [12], [13], or signals from cellular networks [14], [15],
for high-accuracy indoor positioning. Among them, the fifth-
generation new radio (5G NR) signal is particularly notewor-
thy from the following aspects. First, since the 5G picocell
base stations (a.k.a gNodeBs, or gNBs) are being, or will
be densely deployed for indoor coverage, 5G signals will be
abundant in indoor scenarios [16]. Second, the large bandwidth
and high received power of 5G signals and the widely em-
ployed antenna array technology for 5G transmission/reception
points (TRPs) benefit the localization parameter estimation
especially [17]. Lastly, although 5G networks are designed
for the purpose of communications, there exists reference
signals dedicated for positioning and sensing in the up-to-date
3rd generation partnership project (3GPP) standard, i.e. the
sounding reference signal (SRS) for positioning in the uplink
and the positioning reference signal (PRS) for positioning
in the downlink [18]. This renders it viable to implement
integrated localization and communications (ILAC) by 5G
infrastructure without modifying the underlying hardware and
the upper-layer protocol [19].
This paper focuses on the joint estimation of the direction-
of-arrival (DOA) and the time-of-arrival (TOA), which is also
known as the joint angle and delay estimation (JADE), based
on 5G picocell gNBs for the purpose of indoor positioning.
The main challenges are twofold. First, the small-scale antenna
arrays established for picocell gNBs suffer from much lower
spatial resolution and much severer array modeling errors
than large-scale antenna arrays, both of which limit the DOA
estimation accuracy. Second, the parameter space for JADE
is huge and the dimension for the vectorized space-time or
space-frequency data is high, which make JADE algorithms
unfavorable for real-time implementations. Neither of these
challenges has been fully addressed in previous work.
A. Related Works
The JADE problem was first addressed in [20] for the global
system for mobile communications (GSM) signal, also known
as the second-generation (2G) cellular signal, in which the
two-dimensional (2-D) multiple signal classification (MUSIC)
and the maximum likelihood (ML) method are utilized to esti-
mate the DOA and TOA jointly. After that, the MUSIC-based
solution becomes popular for JADE problems in different
scenarios [21]–[23]. Since the MUSIC algorithm is originally
derived on the assumption of non-coherent impinged signals,
This article has been accepted for publication in IEEE Transactions on Instrumentation and Measurement. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
© 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
2 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. XX, NO. X, XXX 2022
additional pre-processing techniques must be developed for
coherent sources, which is the case for an indoor environment
with dense multipath reflections. In particular, a similar JADE
approach which combines the 2-D MUSIC algorithm with an
elaborately designed spatial-frequential smoothing technique
is proposed for Wi-Fi signals and general orthogonal frequency
division multiplexing (OFDM) signals, respectively in [21]
and [22]. However, the eigen value decomposition (EVD) of
the high-dimensional covariance matrix and the 2-D spectrum
search on the DOA-TOA plane performed by MUSIC-based
JADE methods are both computational intensive.
To reduce the overhead, search-free subspace-based meth-
ods which utilize the shift-invariance properties of the re-
ceived signal in spatial and temporal/frequential domains
were proposed in the JADE literature, such as that based
the estimation of signal parameters via rotational invariant
technique (ESPRIT) [24], [25] and that based on the matrix-
pencil method [26]–[28]. Specifically, [24] and [25] derive
the ESPRIT-based JADE method for GSM signals and UWB
signals, respectively; [26], [27], and [28] derive the matrix-
pencil-based JADE method for general OFDM signals, Wi-Fi
signals, and the fourth-generation (4G) long-term evolution
(LTE) signals, respectively. When in the presence of coherent
sources, ESPRIT-based JADE method must employ additional
pre-processing technique to recover the rank of the covariance
matrix, similar to the MUSIC-based JADE method, while the
matrix-pencil-based one can be directly applied to the orig-
inal data. Although these search-free subspace-based JADE
methods obviate the 2-D spectrum search and achieve a much
lower complexity than the MUSIC-based solution, the EVD or
the singular value decomposition (SVD) of a high-dimensional
matrix is still needed.
Compared with the aforementioned subspace-based JADE
methods, the ML-based JADE methods achieve a better statis-
tical performance when the number of snapshots is small and
can be directly applied for coherent sources [29]. However,
they suffer from exhaustive computational burden as they
require a pK -dimensional search, where Kand prepresent
the number of impinged sources and the number of signal
parameters to be estimated in each source. There have been
several research efforts on developing computational attractive
solutions for ML-based JADE methods [23], [30]–[33]. The
first category transforms the complicated high-dimensional
ML search into several successive low-dimensional search
based on the idea of alternating minimization [30], [31] or
expectation-maximization (EM) [23], [32]; while the second
category derives a close-form solution for each ML iteration
based on polynomial parameterizations [33]. The most widely
used ML implementation in the first category is the space-
alternating generalized EM (SAGE) method [34], which has
been employed in [32] and [23] for JADE based on Wi-Fi
signals and 5G signals, respectively. As an example for the
second category, [33] derives the 2-D iterative quadratic ML
(IQML) algorithm [35] for JADE based on OFDM signals.
Other polynomial parameterization methods for ML estimators
can also be extended to solve the JADE problem [36]. How-
ever, computation times for these algorithms can still be long
owing to the large number of iterations before convergence.
There are also several efforts in modeling the JADE as a
sparse inverse problem and seeking the solution according to
the compressed sensing (CS) theory [37]–[40]. When certain
conditions are met, it is possible for sparse recovery methods
to recover DOAs and TOAs from fewer measurements than
those needed by the subspace-based and ML-based methods.
In addition, CS-based estimators do not require the knowledge
of the number of signal sources. However, CS-based JADE so-
lutions also exhibit high computational burdens since they have
to deal with the huge-scale 2-D overcomplete dictionary and,
depending on the underlying sparse recovery algorithms, they
usually have to solve a large-dimensional convex optimization
problem [37], [38] or perform complicated matrix operation
iteratively [39], [40].
These existing JADE algorithms all assume the perfect
antenna response. However, the actual receiving signals from
real-world antenna arrays are inevitably impaired by mutual
coupling effects, sensor location perturbations, the gain-phase
mismatch of the array multichannel receiver and other unpre-
dictable effects [41]. Deviations of the actual array manifold
from the ideal one are referred as the array modeling errors
and they can degrade the estimation performance severely
[42]. Further, the actual manifold cannot preserve the Vander-
monde structure even for uniform arrays, which renders many
aforementioned algorithms infeasible, such as the spatial-
frequential smoothing technique and the ESPRIT-based, the
matrix-pencil-based, and the IQML-based JADE algorithms.
B. Contributions
In this paper, an efficient JADE scheme for 5G picocell
gNBs is proposed to address the aforementioned two main
challenges. First, the array modeling error is counteracted
by employing the actual spatial steering-vector function for
JADE, which is obtained by offline pre-calibrating the ideal
steering-vector function using the array response measured
in a multipath-free environment. Then based on the fact that
picocell gNBs exhibit a much better TOA resolution than DOA
resolution owing to its large signal bandwidth and small-scale
antenna array, the proposed scheme achieves a largely reduced
complexity by decoupling the subcarrier and space domains
of the multichannel OFDM signal and performing TOA and
DOA estimation cascadingly. Specifically, a TOA spectral
estimator based on the iterative adaptive approach (IAA) [43]
is first applied to each receiving channel to segregate multipath
components, followed by a conventional beamformer (CBF) to
extract DOA information. After that, two enhancements to the
proposed scheme are further presented to facilitate its real-
time implementation. The first one is a pre-processing module
which reduces the number of the frequency-domain samples
based on the fact that the coverage of a picocell gNB is
several orders lower than its unambiguous measurable range.
The latter is the accelerating of the IAA-based TOA spectral
estimator by exploiting the fast Fourier transform (FFT) based
on the evenly spaced subcarrier pattern of the 5G reference
signal.
Part of this work was presented in the 2021 CIE Inter-
national Radar Conference [1], which did not address the
This article has been accepted for publication in IEEE Transactions on Instrumentation and Measurement. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
© 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
PAN et al.: EFFICIENT JOINT DOA AND TOA ESTIMATION FOR INDOOR POSITIONING WITH 5G PICOCELL BASE STATIONS 3
problem of array modeling errors. The technical contributions
of this work are summarized below.
1) We investigate the array modeling errors of picocell
gNBs based on real-measured data from commercial
5G equipment. To our best knowledge, this is the first
work to reveal their direction-dependent characteristics
and their impact on the JADE performance.
2) We characterize the direction-dependent antenna errors
as a vector-valued function, which fully captures all sorts
of array modeling errors and is more expressive than the
popular approach used in the DOA estimation literature
which describes each error source separately by a limited
number of parameters [44]. Based on this model, we
also propose a calibration approach which pre-calibrates
the steering-vector function at offline stage, and when
this pre-calibrated function is used at online stage, the
antenna errors are counteracted for signals from any
direction.
3) We propose a JADE scheme whose real-time implemen-
tation is guaranteed. Its complexity has been reduced
by three approaches: a cascading estimation scheme
(Section IV-B), dimension reduction by pre-processing
(Section V-A) and FFT accelerating (Section V-B).
Its averaged running time is 10.1 ms on a personal
computer (PC) for typical picocell gNB configurations,
which is nearly three orders lesser than that of the
MUSIC-based JADE method.
In addition, to evaluate the proposed JADE scheme, compre-
hensive experiments, including numerical simulations based
on the signal model and based on 3GPP 5G indoor channel
models [45] and field tests in an anechoic chamber and in
a realistic indoor environment, are conducted. Experiments
in a multipath-free anechoic chamber demonstrate that the
proposed JADE scheme can substantially reduce the DOA bias
caused by the direction-dependent antenna errors, especially
in large incident directions. For example, the averaged DOA
estimation error can be reduced from 8.11◦to 1.28◦at an
incident angle of +60◦. According to numerical simulation
results, when compared with existing 2-D super-resolution
JADE methods, the proposed method has a significant im-
provement for the DOA estimation performance at a cost of a
slightly reduced TOA estimation performance. Field test in an
indoor environment shows that in 90% cases, a triangulation
positioning accuracy of 0.44 m can be achieved using the
DOAs estimated by the proposed method from only two TRPs,
which meets the 3GPP R17 requirements for commercial use
cases [18].
C. Paper Outline and Notations
This paper is organized as follows. First, the system model
for 5G-signal-based positioning is presented and the problem
of JADE is formulated in Section II. It is followed by real-data-
based array modeling error analysis in Section III. Then the
JADE signal model in the presence of array modeling errors
is re-formulated in this section. Afterwards, details of the
proposed array modeling error calibration and JADE scheme
are presented in Section IV. Its computational complexity and
storage requirement are also analyzed. Next, in Section V,
enhancements to the proposed scheme are presented for further
complexity reduction, which yields an efficient implementation
for the proposed method. In Section VI, in-depth performance
evaluations, including numerical simulation and field tests,
are provided. Furthermore, Section VII presents discussions
regarding the adaptability of the proposed method to other
potential scenarios. Finally, Section VIII concludes the paper.
In the rest of this paper, vectors and matrices are de-
noted by boldface lowercase and boldface uppercase letters,
respectively, where vectors are by default in column orienta-
tion. Italic English letters and lowercase Greek letters denote
scalars. Blackboard-bold characters denote number sets, in
particular, Rand Crepresent the sets of real and complex
numbers, respectively. =√−1denotes the imaginary unit.
x=O(a)for a>0denotes that ∃k1,k2>0, such that
k2·a≤x≤k1·a. TABLE I lists all the other notations used in
this paper and their meanings. For convenience, abbreviations
used in this paper are summarized in TABLE II.
TABLE I
NOTATIO NS
(·)Ttranspose of a vector or matrix
(·)Hconjugate transpose of a vector or matrix
(·)−1inverse of a square matrix
|a|absolute value of the scalar a
|a|a vector with absolute values of entries of vector a
[a]nn-th element of vector a
[A]m,nelement at m-th row and n-th column of matrix A
[a]i:jsubvector formed by i-th to j-th entries
0Na zero vector of length N
the Hadamard (element-wise) matrix product
CN(µ, σ2)complex Gaussian distribution parameterized by µand σ2
U(a,b]uniform distribution from ato b
diag(a)diagonal matrix whose main diagonal entries are a
FFT[a,P]P-point FFT on a
Toeplitz(a)a Hermitian Toeplitz matrix with elements of a
a(x)a scalar-valued function with input variable of x
a(x)a vector-valued function with input variable of x
II. SY ST EM MO DE L
As shown in Fig. 1, we consider the positioning scenario
in which the 5G TRPs receive the up-link SRS transmitted by
the user equipment (UE) and determine the UE position by
jointly estimating the DOA and TOA. The SRS is an OFDM
signal with a pre-defined Zadoff-Chu sequence for each unique
terminal [46]. It is designed for uplink channel sounding,
while the sounding result in the frequency domain, which is
also denoted as the channel frequency response (CFR) of the
wireless channel, contains the DOA and TOA information of
both the direct path and reflected paths.
In this section, we will setup the multichannel CFR-based
JADE signal model based on the following hypothesis:
1) Since the main focus of this paper is 2-D parameter
estimation for positioning in a 2-D space, we assume that
the UE antenna and the TRP antennas are at the same
height and the TRP antenna arrays are all horizontally
This article has been accepted for publication in IEEE Transactions on Instrumentation and Measurement. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
© 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
4 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. XX, NO. X, XXX 2022
TABLE II
ABBREVIATIONS
Abbreviation Definition Abbreviation Definition
1-D/2-D/3-D one/two/three-dimensional IQML iterative quadratic ML
2G/4G/5G 2nd/4th/5th-generation mobile communications technol-
ogy
JADE joint angle and delay estimation
3GPP 3rd generation partnership project LTE long-term evolution
5G NR 5G new radio LS least-squares
BBU baseband unit LOS line-of-sight
CBF conventional beamformer ML maximum likelihood
CDF cumulative distribution function MUSIC multiple signal classification
CFR channel frequency response NLOS non-line-of-sight
CIR channel impulse response NR new radio
CS compressed sensing OFDM orthogonal frequency division multiplexing
CSI channel state information PC personal computer
dB decibel PRS positioning reference signal
DOA direction-of-arrival RF radio frequency
EM expectation-maximization RFID radio frequency identification
ESPRIT estimation of signal parameter via rotational invariant
technique
RMSE root mean square error
EVD eigen value decomposition RRU radio remote unit
FFT fast Fourier transform SAGE space-alternating generalized EM
FR1 frequency range 1 SMV single measurement vector
gNB next generation Node-B SNR signal-to-noise ratio
GNSS global navigation satellite systems sps symbols per second
GSM global system for mobile communications SRS sounding reference signal
IAA iterative adaptive approach SVD singular value decomposition
IF intermediate frequency TDOA time difference of arrival
IFFT inverse FFT TOA time-of-arrival
i.i.d. identically and independently distributed TRP transmission/reception point
ILAC integrated localization and communications UE user equipment
InF indoor factory ULA uniform linear array
InO indoor office UWB ultra-wideband
IoE Internet-of-everything VNA vector network analyzer
IQ in-phase and quadrature VSWR voltage standing wave ratio
SRS
SRS
UE 1
UE 2
5G TRP (Antenna
array + RRU)
FFT Channel
estimator
XJADE &
Positioning
BBU
Raw IQ data
transmission via
optical fiber
IF
processing
1
N
H
Positioning
server
N
1
RF
signal
UE
position
CFR matrix
transmission via
Ethernet
Fig. 1. Positioning scenario based on 5G cellular network with a picocell
gNB. The abbreviations of BBU, RRU, IF and IQ data stand for baseband
unit, radio remote unit, intermediate frequency and in-phase and quadrature
data, respectively.
placed. Therefore, the DOA refers to the azimuth angle
and is defined as the angle with respect to the array
broadside in this paper.
2) We only consider the problem of positioning a single
user terminal, while the positioning of multiple termi-
nals can be easily decomposed into this case owing to
the orthogonality between uplink SRS sequences from
different terminals.
3) We consider the problem of JADE using a single SRS
symbol, i.e. JADE in a single measurement vector
(SMV) scenario, which is more promising for moving
UE positioning. Therefore, the time index is omitted in
the following signal model.
Considering an SRS symbol which occupies Msubcarri-
ers is transmitted by the UE and impinges on a 5G TRP
which is equipped with an array of Nantenna elements.
The received time-domain multichannel SRSs are first trans-
formed to the frequency-domain via the FFT. The frequency-
domain received signal of channel nis denoted as xn=
X1,n,X2,n, . . . , XM,nT∈CM×1, in which Xm,nrepresents the
SRS component received at n-th receiving channel and m-th
subcarrier. The frequency-domain signals from all channels are
arranged into a single matrix X=[x1,x2, . . ., xN]∈CM×N.
Assuming that the SRS arrives at the TRP via Kpaths with
DOAs of {˜
θk}K
k=1, TOAs of {˜τk}K
k=1, and attenuation coeffi-
cients of {˜γk}K
k=1, and denoting the index of the LOS path as
kLOS, then the received signal matrix can be represented as
X=
K
Õ
k=1˜γkS·aτ(˜τk)aT
θ(˜
θk)+W.(1)
Symbols used in equation (1) are clarified as follows. First,
S=diag [S[1],S[2], . . . , S[M]]Tis the SRS data matrix,
where S[m],m=1,2, . . ., Mis the transmitted SRS sequence.
This article has been accepted for publication in IEEE Transactions on Instrumentation and Measurement. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
© 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
PAN et al.: EFFICIENT JOINT DOA AND TOA ESTIMATION FOR INDOOR POSITIONING WITH 5G PICOCELL BASE STATIONS 5
Next, aτ(·) :T→CM×1is the delay signature function
which represents the frequency-domain structure of the SRS.
It is a known function whose input is the path delay, or the
TOA of the signal, and the output is the delay signature vector.
The interested range of delay T⊆Rcan be determined by the
power coverage of the TRP. For example, assuming that τmin
and τmax represent the minimum and maximum measurable
path delay, respectively, then T=ττmin ≤τ≤τmax. The
m-th element of the output delay signature vector aτ(˜τk)
represents the shift factor caused by the path delay ˜τkin the
m-th subcarrier, and can be denoted as
[aτ(˜τk)]m=exp (−2π(m−1)∆f˜τk),m=1,2, . . . , M,(2)
where ∆fis the subcarrier spacing.
Then, aθ(·) :U→CN×1is the steering-vector function
determined by the spatial structure of the antenna array. It is
also a known function with DOA as its input and steering-
vector at that specific direction as the output. The interested
range of DOA U⊆Ris determined by the angular coverage
of the TRP antenna array. Also as an example, when θmin
and θmax are the minimum and maximum measurable angles,
respectively, we have U=θθmin ≤θ≤θmax . Assuming that
the receiving array is an ideal uniform linear array (ULA), then
the n-th element of the output steering-vector for the impinged
DOA of ˜
θkis
aθ˜
θkn=exp 2π(n−1)dsin ˜
θk
λ,n=1,2, . . . , N,
(3)
where dis the array element spacing and λis the wavelength.
Lastly, matrix W∈CM×Nin equation (1) is the noise
matrix, whose element [W]m,nrepresents the noise component
at n-th receiving channel and m-th subcarrier.
As shown in Fig. 1, after the FFT operation, a channel esti-
mator is followed in the BBU to estimate the wireless channel
response. Assuming that the least-squares (LS) algorithm is
applied using the a prior of the SRS sequence, the estimated
CFR matrix H∈CM×Ncan be derived as
H=S−1X=
K
Õ
k=1˜γk·aτ(˜τk)aT
θ(˜
θk)+W0,(4)
where W0=S−1W∈CM×Nrepresents the noise components
in the CFR matrix H. Entries of W0are identically and inde-
pendently distributed (i.i.d.) complex-valued Gaussian noise
with zero-mean and variance σ2
w, i.e. [W0]m,n∼ CN 0, σ2
w.
The basic signal model for the JADE problem addressed in
this paper is expressed by equation (4) and is formulated as:
Given a single space-frequency CFR matrix H, es-
timate the LOS DOA ˜
θkLOS and TOA ˜τkLOS.
This ideal signal model without considering the impairments
induced by the array modeling errors is quite similar to
the JADE models for Wi-Fi [21], UWB [25], GSM [20],
or LTE [28] signals in the literature. The impact of array
modeling errors of picocell gNBs will be analyzed and will
be incorporated into the signal model in Section III.
III. ARR AY MODELING ERROR ANALYS IS BA SE D ON
REA L-DATA
The array modeling errors can be roughly divided into two
parts: the part induced by the multichannel receiver and the
part induced by the antenna elements, which are referred as RF
channel errors and antenna errors in this paper, respectively.
The RRU and the antenna array of a picocell gNB employed
in experiments of this paper are evaluated for exemplifying the
spatial-frequential characteristics of these errors. The antenna
array is a six-element ULA with an inter-element spacing
of 3 cm, as shown in Fig. 2. Among these six antenna
elements, those two at both sides are dummy elements which
are designed for alleviating the mutual coupling effect of the
whole array and are not connected to any RF channels; while
the middle four antennas are valid elements for conducting the
receiving signals to the according RF channels. Each antenna
element is a microstrip antenna with linear vertical polarization
working in the frequency range of 4.80 −4.90 GHz, which is
within the 5G NR frequency range 1 (FR1). At the frequency
of 4.85 GHz, each antenna provides a nominal gain of 5 dBi
with a 3 dB beamwidth of 120◦. In its working frequency
range, the voltage standing wave ratio (VSWR) is below 1.5,
while the isolation is above 16 dB.
Fig. 2. RRU and antenna array of the 5G picocell gNB employed in
experiments of this paper.
A. RF Channel Error Analysis
Wideband SRS is conducted to the receiving RF channels of
this four-channel picocell RRU directly via coaxial cables and
a RF power divider to measure the RF channel errors, which
is also referred as the amplitude and phase responses of the
RF channels. The SRS is configured according to TABLE III,
which is also the default SRS configuration used throughout
this paper.
The measured channel errors are shown in Fig. 3, where
in each subfigure, the solid lines are the average values of
500 repeated RF channel amplitude and phase measurements
and the shadow areas illustrate the upper and lower bounds
of those measurements. It can be inferred from Fig. 3 that:
First, the initial phase of the channel responses differ distinctly,
which will lead to the DOA estimation bias; Second, in the
frequency-domain, the non-linear components predominate,
which will distort the signal spectrum in the TOA domain.
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content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
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6 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. XX, NO. X, XXX 2022
TABLE III
WAVEFORM PARAMETERS FOR UPLINK SRS ASSU MED I N TH IS ST UDY
Parameter Value
Working frequency range FR1 (4.80 −4.90 GHz)
Numerology µ1
Subcarrier spacing 30 kHz
Number of resource blocks 272
Number of subcarriers 3264
SRS pattern Comb-two [46]
SRS transmission bandwidth 100 MHz
SRS periodicity 80 ms
Cyclic prefix mode Normal
FFT size 4096
Sampling rate 122.88 Mspsa
asps: abbreviation for symbols per second.
4.80 4.85 4.90
Frequency (GHz)
0
1000
2000
3000
4000
Amplitude
-100
-50
0
50
100
Phase (degree)
RF channel 1
4.80 4.85 4.90
Frequency (GHz)
0
1000
2000
3000
4000
Amplitude
-100
-50
0
50
100
Phase (degree)
RF channel 2
4.80 4.85 4.90
Frequency (GHz)
0
1000
2000
3000
4000
Amplitude
-100
-50
0
50
100
Phase (degree)
RF channel 3
4.80 4.85 4.90
Frequency (GHz)
0
1000
2000
3000
4000
Amplitude
-100
-50
0
50
100
Phase (degree)
RF channel 4
Fig. 3. RF channel error measurements of the four-channel RRU.
B. Antenna Error Analysis
Different from RF channel errors, owing to the mutual cou-
pling effects, location perturbations and beampattern errors of
antenna elements, the antenna errors are direction-dependent.
As the DOA is obtained from phase shifts across receiving
antennas, this paper mainly focuses on the antenna phase
errors.
Phase responses of the four-element antenna array of the
picocell gNB are measured in a far-field anechoic chamber by
Approach (a) shown in Fig. 4 (Approach (b) is used to collect
CFR matrices for performance evaluation, as will be presented
later in Section VI-A).
Here, single tone signals are transmitted by the horn antenna
with center frequency sweeping from 4.80 GHz to 4.90 GHz
with a step size of 10 MHz. The receiving arrays are placed
on a swiveling pedestal, which swivels from −60◦to +60◦in
an angular step of 5◦. Phase responses of this antenna array
are then measured by a vector network analyzer (VNA), from
which the phase shifts caused by free-space propagation are
subtracted, deriving the desired phase errors caused by the
imperfect array response.
Antenna phase error is first investigated in the frequency
domain in Fig. 5. Different lines in each subfigure of Fig.
0.5 0.5 0.5 0.5 0.5
Antenna 1 to 4 Dummy
element
Dummy
element
Swiveling pedestal
14.5 m
Transmitting signal of
Approach (a): single tone
signal with carrier
frequency sweeping
Signal
Generator Test UE
select
Transmitting signal of
Approach (b): SRS
(Wideband OFDM signal)
VNA
RRU BBU
Exported data of
Approach (a): array
response at discrete
frequency point
Exported data of
Approach (b):
CFR matrix
Tx
Rx
Fig. 4. Setups for anechoic chamber experiments, where blue and green colors
mark the components used for array response measuring (Section III-B) and
for field test in a multipath-free environment (Section VI-A), respectively.
5(a) illustrate the phase errors of a corresponding antenna
element when the signal impinges on the array from different
directions. It demonstrates that, contrary to the phase errors
caused by RF channels, those caused by antenna elements
are dominated by linear components. This phenomenon is
plausible since the antenna element proper is a passive device
which merely conducts the received wireless signal to the
RF port. Moreover, these phase error curves shown in Fig.
5(a) are nearly parallel to each other. By applying a linear
transformation to the slopes of these curves according to the
following equation:
Distance bias =Slope [rad/Hz] ·c
2π,(5)
in which cpresents the speed of light in vacuum, the distri-
butions of the distance biases caused by the antenna errors
when the incident direction varies are revealed in Fig. 5(b).
It can be inferred that the distance biases are all in the range
of [2.0,2.2]m, with the variance across antennas and across
incident directions less than 0.2m, which is much less than the
distance resolution of 3m. This means that, the antenna errors
cause a nearly identical TOA shift in all receiving channels
for signals impinging from any direction.
Then the antenna phase errors are investigated in the angular
domain in Fig. 6. Offsets of these errors relative to the
first antenna element averaged by measurements sampled at
different frequencies are shown in Fig. 6, with 95% confidence
intervals of these measurements also highlighted by the shad-
owed areas. It demonstrates that, to the contrary of channel
errors, variances of antenna phase errors across frequencies
are relatively small and the antenna phase error is highly
direction-dependent. The consequences are two-fold: First, to
reduce the complexity of calibration, the antenna phase error
can be regarded as a constant across different frequencies and
only spatial domain calibration is necessary; Second, since the
incident direction of the SRS is the parameter to be estimated
for uplink positioning, which is unknown to the gNB, the gNB
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content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
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PAN et al.: EFFICIENT JOINT DOA AND TOA ESTIMATION FOR INDOOR POSITIONING WITH 5G PICOCELL BASE STATIONS 7
(a) Antenna phase errors across fre-
quencies in different incident angles.
(b) Histogram of distance bias
caused by antenna errors.
Fig. 5. Antenna phase error characteristics in the frequency-domain.
is unable to determine the calibration coefficients for antenna
errors.
-60 -40 -20 0 20 40 60
Incident direction (degree)
-60
-40
-20
0
20
40
60
Phase (degree)
Antenna 2 - Antenna 1
-60 -40 -20 0 20 40 60
Incident direction (degree)
-60
-40
-20
0
20
40
60
Phase (degree)
Antenna 3 - Antenna 1
-60 -40 -20 0 20 40 60
Incident direction (degree)
-60
-40
-20
0
20
40
60
Phase (degree)
Antenna 4 - Antenna 1
95% confidence
interval
Averaged phase
error
Fig. 6. Antenna phase errors relative to the first antenna element.
C. Signal Model Refinement in the Presence of Array Model-
ing Errors
Based on the aforementioned analysis, the signal model of
equation (4) is revised by incorporating the array modeling
errors, including both the RF channel errors and the antenna
errors. The frequency-selective RF channel errors are modeled
as a matrix Γ∈CM×N, whose element [Γ]m,nrepresents
the amplitude and phase response of the RF channel in the
m-th subcarrier and in the n-th receiving channel. Then the
direction-dependent antenna error is formulated as a vector-
valued function ζ(θ):U→CN×1, which outputs the antenna
errors as a N-dimensional vector when a DOA value is given.
The n-th element of the output vector is [ζ(θ)]n=exp(φn(θ)),
where φn(θ):U→ [−π, π]is the phase error function of
the n-th antenna element. By this means, all kinds of errors
are gathered into a single direction-dependent antenna error
function and thereby equation (4) can be extended to
H=
K
Õ
k=1˜γk·aτ(˜τk)a0T
θ(˜
θk)Γ+W0,(6)
where a0
θ(·) represents the actual spatial steering-vector func-
tion. According to the aforementioned antenna error model,
it relates to the ideal steering-vector function and the antenna
error function as follows:
a0
θ(θ)=aθ(θ) ζ(θ).(7)
Finally, the problem addressed in this paper is concluded
as:
Given a single estimated space-frequency CFR ma-
trix Hand the pre-measured data set for the array
modeling error, estimate the desired positioning pa-
rameters ˜
θkLOS and ˜τkLOS .
IV. ARR AY MODELING ERRO R CA LI BR ATIO N AN D JADE
The flowchart of the proposed JADE scheme is shown in
Fig. 7. The proposed scheme is composed of two main parts:
calibration modules which cope with the RF channel errors and
the antenna errors separately and estimator modules consisting
of an IAA-based TOA spectral analyzer and a CBF.
A. Calibration Modules
As shown in Fig. 7, each entry of the received multichannel
CFR matrix is first divided by the measured RF channel
response matrix ˆ
Γto calibrate the RF channel phase and
amplitude inconsistency. The element at m-th row and n-th
column of the CFR matrix after calibrating the channel errors
is
[H0]m,n=[H]m,n/ˆ
Γm,n.(8)
On the contrary, the direction-dependent antenna error pre-
vents the receiver from calibrating it directly in the received
CFR. Therefore, in the proposed scheme, the steering-vector
function is calibrated at offline stage to counteract the antenna
error. Specifically, there are four main steps in the offline stage
for antenna error calibration as illustrated by the upper part
of Fig. 7. First, antenna phase errors in Rdiscrete angles
within the array coverage are measured using the approach
illustrated in Section III-B. It is noteworthy to mention that,
electromagnetic numerical simulations can be considered to
reduce the overheads of measuring. The angular sampling set
and the phase error measurements for antenna element non
this set are denoted as ¯
θrR
r=1and ¯
φn,rR
r=1,n=1, . . . , N,
respectively.
Then the phase error function of each antenna element
is estimated based on measured data sets. This function
estimation problem can be formulated as a regression problem,
for which LS-based polynomial curve fitting, support vector
machine, neural network, or other regression method, can be
applied here. Take the naive polynomial curve fitting method
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content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
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8 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. XX, NO. X, XXX 2022
IAA-based TOA
spectrum
estimation
IAA-based TOA
spectrum
estimation
Peak
Detection
LOS
Identification
LOS TOA
spectral
peaks
extraction
RF
channel
error
calibration
CFR
matrix in
TOA estimate out
CBF-based
DOA
estimation DOA
estimate out
Measuring RF
channel
response
Measuring
array phase
error
Estimating
phase error
function
Deriving
antenna error
function
Pre-calibrating
ideal steering-
vector function
Offline stage
Online stage
Calibration
modules
Estimator
modules
Fig. 7. Flowchart of the proposed JADE scheme.
as an example. The estimated function for the n-th antenna
element has the form of
φpoly,n(θ, g)=
I
Õ
i=1
giθi−1,n=1, . . . , N,(9)
where g∈RI×1presents the coefficient vector of a I-th-order
polynomial. According to the LS curve fitting principle, the
phase error function can be determined by
(ˆ
φn(θ)=φpoly,n(θ, ˆ
g),
ˆ
g=arg mingÍR
r=1φpoly,n(¯
θr,g) − ¯
φn,r
2.(10)
For the antenna array shown in Fig. 2, whose phase error
measurements have been demonstrated in Fig. 5 and Fig. 6,
its phase error functions for all elements estimated by fourth-
order polynomial curve fitting are illustrated in Fig. 8.
-60 -40 -20 0 20 40 60
Incident direction (degree)
20
40
60
80
100
120
140
Phase (degree)
Antenna 1
-60 -40 -20 0 20 40 60
Incident direction (degree)
20
40
60
80
100
120
140
Phase (degree)
Antenna 2
-60 -40 -20 0 20 40 60
Incident direction (degree)
20
40
60
80
100
120
140
Phase (degree)
Antenna 3
-60 -40 -20 0 20 40 60
Incident direction (degree)
20
40
60
80
100
120
140
Phase (degree)
Antenna 4
-60 -40 -20 0 20 40 60
Incident direction (degree)
20
40
60
80
100
120
140
Phase (degree)
Antenna 4
Measurement Estimated function
Fig. 8. Polynomial curve fitting for antenna phase error function estimation
based on real-measured phase error data set.
After that, based on the estimated phase error function
ˆ
φn(θ),n=1, . . . , N, the antenna error function ζ(θ)is deter-
mined, with the n-th element of the function output for signal
impinging from the direction of θequals to
ˆ
ζ(θ)n=exp(ˆ
φn(θ)).(11)
Lastly, the ideal steering-vector function is pre-calibrated
according to the estimated antenna error function as follows:
ˆ
a0
θ(θ)=aθ(θ) ˆ
ζ(θ).(12)
This offline-stage pre-calibrated steering-vector function
ˆ
a0
θ(θ)captures the antenna errors and delineate the actual
responses of the antenna array for signals impinging from
different directions more precisely than the ideal steering-
vector function. It is used by the estimator modules at online-
stage to improve the DOA estimation accuracy.
B. Estimator Modules
Conventional joint DOA-TOA estimators usually incorpo-
rate 2-D searching on the DOA-TOA plane or operations on
the huge-dimensional space-frequency data matrix, which are
both computational intensive. In the proposed scheme shown
in Fig. 7, TOA and DOA are estimated cascadingly with
an IAA-based spectral estimator and a CBF. The main idea
behind such a design is to fully exploit the abundant subcarrier
resources of the 5G signal and the super-resolution ability
of the IAA spectral estimator to separate multipaths by their
delays, then a naive CBF can be applied to the segregated
multipath components to retrieve their DOAs.
The reason of employing the IAA method for TOA spectrum
estimation is three-fold:
1) Compared with parametric spectral estimators which
need to tune their parameters for desired performance
[21], [28], [37], [39], the IAA method is a nonparametric
estimator which can be employed for data set from any
scenario without cumbersome parameter-tuning;
2) Compared with extensively used subspace-based spectral
estimators, which demand multiple snapshot or need to
be combined with the smoothing technique to construct
multiple measurements artificially [21], [22], the IAA
method can be applied in a SMV scenario directly;
3) Compared with the pseudo-spectrum obtained by the
MUSIC algorithm, whose spectral values do not repre-
sent signal amplitude, the IAA method is able to recover
the amplitude and phase of each signal component,
which benefits the subsequent LOS discriminator and
can be directly used by the following DOA estimator.
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content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
© 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
PAN et al.: EFFICIENT JOINT DOA AND TOA ESTIMATION FOR INDOOR POSITIONING WITH 5G PICOCELL BASE STATIONS 9
According to equation (6) and (8), the n-th column of the
channel-error calibrated CFR matrix H0can be derived as
h0
n=
K
Õ
k=1
˜
βn,kaT
τ(˜τk)+wn,(13)
where wn∈CM×1represents the noise components in h0
nand
˜
βn,k=˜γk·αn(˜
θk), in which αn(˜
θk)=a0
θ(˜
θk)n.
IAA method is a dense spectrum estimator which estimates
the power spectrum at a predefined grid set, hence the signal
model expressed by equation (13) is further extended to make
the following explanation concise. Suppose the TOA span of T
is covered by a uniform grid set {τp}P
p=1, and the attenuation
coefficient sounded at the p-th TOA grid by the n-th channel
is denoted as βn,p,p=1, . . . , P,n=1, . . . , N. Then βn,phas
the form of:
βn,p=(˜
βn,k, τp=˜τk,
0,elsewhere.(14)
Then equation (13) can be reformulated as
h0
n=Aτβn+wn,(15)
where βn=βn,1, βn,2, . . . , βn,PT∈CP×1is the atten-
uation coefficient vector of the n-th channel and Aτ=
[aτ(τ1), . . . , aτ(τP)]∈CM×Pis the over-complete delay sig-
nature matrix.
In the IAA framework, solution for each element of βnis
sought according to the weighted least squares criteria [43]:
ˆ
βn,p=arg min
βn,p
h0
n−βn,paτ(τp)
2
R0−1
n,p,p=1, . . ., P,(16)
in which kxk2
Q=xHQx represents the square of the weighted
`2-norm of the vector x. In (16), the weighted matrix R0−1
n,pis
the inverse of the interference covariance matrix R0
n,p, while
the interference here refers the signal components arrived the
receiver at delays other than the delay of current interest τp.
Therefore, R0
n,pcan be presented by
R0
n,p=Rn−βn,p
2aτ(τp)aH
τ(τp),(17)
where Rnrepresents the covariance matrix of the signal
received by the n-th channel.
Solution to equation (16) is derived to be
ˆ
βn,p=
aH
τ(τp)R0−1
n,ph0
n
aH
τ(τp)R0−1
n,paτ(τp).(18)
According to equation (17) and the matrix inversion lemma,
equation (18) can be simplified as
ˆ
βn,p=
aH
τ(τp)R−1
nh0
n
aH
τ(τp)R−1
naτ(τp).(19)
Since Rnin equation (19) is unknown, IAA solves the
problem by estimating βnand Rnalternatively and iteratively
using equation (20) to equation (22):
ˆ
βn,p=
aH
τ(τp)ˆ
R−1
nh0
n
aH
τ(τp)ˆ
R−1
naτ(τp),p=1, . . . , P,(20)
ˆ
Pn=diag hˆ
βn,1
2,ˆ
βn,2
2, . . . , ˆ
βn,P
2iT,(21)
ˆ
Rn=Aτˆ
PnAH
τ.(22)
The initialization can be performed via the classical pe-
riodogram method [47]. Usually, IAA method converges in
10 −15 iterations. After convergence, the TOA spectrum
estimates ˆ
βn,n=1, . . . , Nare obtained. According to Fig.
7, amplitudes of multichannel TOA spectra are averaged at
each TOA grid as follows:
¯
β=1
N
N
Õ
n=1ˆ
βn.(23)
A detection algorithm is then applied to vector ¯
βto retrieve
significant path components, followed with a LOS identifi-
cation module to discriminate the desired LOS component
from those path components. Since signal detection and LOS
path identification are not the main concern of this paper, we
will not discuss their details. Denoting the TOA of the LOS
path and the corresponding TOA index as ˆτLOS and pLOS,
respectively, then the LOS components from all channels can
be collected in a vector bLOS =ˆ
β1,pLOS,ˆ
β2,pLOS, . . ., ˆ
βN,pLOS T.
The last step is to estimate the DOA of the LOS path
from the vector bLOS. Since the LOS and non-line-of-sight
(NLOS) components have been separated in the TOA domain,
the proposed scheme uses the naive CBF to estimate the
LOS DOA. As stated in Section IV, to offset the direction-
dependent antenna errors, the pre-calibrated spatial steering-
vector function ˆ
a0(θ)is used instead of the ideal one. There-
fore, the LOS DOA is obtained by
ˆ
θLOS =arg max
θˆ
a0
θ(θ)HbLOS.(24)
Equation (24) is solved by an one-dimensional (1-D) searching
over a pre-defined uniform grid set {θq}Q
q=1which covers the
DOA span of U.
Afterward, one can utilize a positioning and tracking frame-
work, such as that based on the extended Kalman filter, the
unscented Kalman filter, or the particle filter [48], to exploit the
estimated LOS DOA ˆ
θLOS and LOS TOA ˆτLOS for positioning
the target UE in the 2-D plane.
C. Complexity Analysis
In this section, the computational complexity and storage
requirements of the proposed scheme are analyzed and com-
pared with JADE algorithms presented in the literature [20]–
[22], [26]–[28]. Conventional computational implementations
are considered throughout the complexity analysis.
The computational complexity is measured in terms of the
number of complex multiplications. According to the flowchart
illustrated in Fig. 7, the calibration of the antenna errors
is performed in offline-stage, adding no extra computational
burdens to the real-time JADE. Therefore, its complexity is
not analyzed. The calibration of channel errors expressed by
equation (8) can be implemented by multiplying the receiving
data with the pre-stored reciprocal of the RF channel responses
1/[Γ]m,n,m=1, . . . , M,n=1, . . . , N. The resultant com-
plexity is O(M N ). In the proposed scheme, CBF is applied
only to the extracted LOS path component, which has the
complexity of O(NQ). Hence, the TOA spectrum estimation
for each channel dominates the computational burden of the
proposed scheme.
This article has been accepted for publication in IEEE Transactions on Instrumentation and Measurement. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
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10 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. XX, NO. X, XXX 2022
According to equation (20) to equation (22), in each it-
eration, the most computational intensive steps of the IAA-
based TOA spectrum estimator are the calculation of the
covariance matrix ˆ
Rnand its inverse ˆ
R−1
n, and the grid-by-grid
searching of the IAA spectral value βn,p. Their complexities
are O(PM 2),O(M3), and O(PM2), respectively. Denoting
the iteration steps of the IAA method as ni, then its total
computational complexity is OniPM2+M3. As stated
before, the IAA method converges fast, thus the iteration
number niis small and is independent on the problem scale.
Therefore, the complexity of the IAA-based TOA spectrum
estimation for all receiving channels is O(NPM2+N M 3)
and the overall computational complexity for the proposed
method is ON PM2+N M3+M N +NQ, which is also on
the order of ON P M2+N M3. Since the number of TOA
search grids Pis usually much larger than the subcarrier
number M, the total computational complexity can also be
deducted to O(N PM2).
For the IAA-based TOA spectrum estimation, as the pro-
posed method estimates TOA spectrum for each channel
sequentially, only the intermediate results for one-channel need
to be taken into account for storage requirement analysis.
It can be inferred from equation (20) to equation (22) that
matrix Aτand Rnneed to be stored for the IAA method.
The resultant space complexity is O(M P +M2). Besides,
memory requirement for storing N-channel TOA spectrum is
O(NP). Since the actual spatial structure of the antenna array
is modeled as a function a0
θ(θ), only a few coefficients for that
function, not the entire manifold over the angular set {θq}Q
q=1,
need to be stored. Therefore, the total space complexity for
the proposed method is O(M P +M2+N P).
TABLE IV also lists the computational complexities and
storage requirements of two representative JADE methods
in the literature for comparison. The first one is the 2-D
MUSIC-based JADE method [20]–[22]. Although [21] and
[22] improve the original 2-D MUSIC method proposed in
[20] by employing the spatial-frequential smoothing tech-
nique, the dimension of the smoothed matrix is still linear
related to that of the original matrix. Therefore, they have
the same computational complexity and space requirement.
Their computational complexity is dominated by the EVD
of a M N ×M N covariance matrix and a 2-D parameter
search, whose complexities are OM3N3and OPQM2N2,
respectively. They all need to store the spatial-frequential
signature matrix and the covariance matrix, whose storage
requirements are O(PQ M N )and OM2N2, respectively. The
other one is the search-free 2-D matrix-pencil-based JADE
method [26]–[28]. Its most computational intensive step is the
SVD of the constructed enhanced-matrix, whose dimension is
determined by the frequency-domain and space-domain pencil
parameters which are linear related to the subcarrier number
Mand array element number N, respectively. Therefore,
its computational complexity and space requirement can be
denoted as OM3N3and OM2N2, respectively.
As shown in TABLE IV, the proposed method has a much
lower computational complexity and storage requirement com-
pared with these popular 2-D subspace-based JADE methods.
For CS-based JADE methods [37]–[40], their complexities are
TABLE IV
SUM MARY O F CO MPU TATION AL CO MP LEX ITY A ND S TOR AGE
REQUIREMENT
JADE algorithm Time complexity Space complexity
MUSIC-
based [20]–[22] OM3N3+PQ M2N2OPQ M N +M2N2
Matrix-pencil-
based [26]–[28] OM3N3OM2N2
Proposed ON P M2+N M 3O(M P +M2+N P)
dependent on the underlying sparse recovery methods. Usually,
they have to deal with a huge-scale 2-D overcomplete dictio-
nary, and incorporate a solver for a large-dimensional convex
optimization problem [37], [38] or an iterative procedure with
complicated matrix operations at each iteration [39], [40].
Therefore, they also exhibit much higher computational and
storage burden than the proposed method.
V. ENHANCEMENTS FOR EFFICIENT IMPLEMENTATION
As analyzed in Section IV-C and illustrated in TABLE IV,
although the computation and storage budget of the proposed
JADE scheme have been largely reduced compared with those
2-D joint estimation methods in the literature, challenges for
its real-time implementation still remain, especially in time-
sensitive applications. In this section, enhancements to the
proposed scheme are proposed to further reduce complexity.
Concretely, a CFR pre-processing module is prepended to
facilitate the following estimation procedures, and the IAA-
based TOA spectrum estimator, which dominates the compu-
tational complexity, is accelerated by employing FFTs.
A. Pre-Processing: CFR Denoising
As shown in TABLE IV, the computational complexity of
the proposed method increases with the number of subcarriers
Min an increasing rate larger than the quadratic function.
On the other hand, the demands of obtaining a finer charac-
terization of the wireless channel and a better resolution for
multipath components prompt the positioning system to use a
large-bandwidth SRS with a plentiful amount of subcarriers,
which gives rise to a large-dimensional CFR matrix. Therefore,
applying the proposed JADE method to this CFR matrix
directly will occupy substantial computing resources and lead
to an unacceptable processing latency.
However, if the wireless channel is investigated in the
perspective of the channel impulse response (CIR), i.e. the
time-domain counterpart of the CFR, the delay taps of the
CIRs that exceed the maximum delay covered by the gNB
make no contribution to JADE and bring extra noises into
the CFR [49]. For a picocell gNB with a coverage of several
tens of meters, the valid delay taps occupy only a small
proportion of the entire CIR. For example, assuming that
the SRS is configured in accordance with TABLE III, since
the comb-two structure is used, the spacing between two
adjacent SRS subcarriers is 60 kHz, which corresponds to
a maximum unambiguous delay of 16.67 µsin the CIR.
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content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
© 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
PAN et al.: EFFICIENT JOINT DOA AND TOA ESTIMATION FOR INDOOR POSITIONING WITH 5G PICOCELL BASE STATIONS 11
For a gNB coverage of 100 m, the first 2% delay taps of
the CIR already preserve all the information of the LOS
and multipath components. Hence, by truncating the CIR and
re-transforming it to the frequency domain, the CFR can
be recovered with noise suppressed and dimension reduced.
Based on this observation, a pre-processing scheme for the
purposes of denoising and dimension-reducing is proposed,
whose flowchart is shown in Fig. 9.
0 40 80 120 160 200
Distance (m)
-60
-50
-40
-30
-20
-10
Amplitude (dB)
Distance (m)
-60
-50
-40
-30
-20
-10
Amplitude (dB)
0 40 80 4920 5000
Random
peak shift
-100 -75 -50 -25 0 25 50 75 100
Distance (m)
-60
-50
-40
-30
-20
-10
Amplitude (dB)
1. CFR phase
slope removal 2. IFFT
3. Centering the zero
frequency (vector
swapping)
5. FFT
4. Windowing
the CIR
Folded
peaks
CIR
window
CFR out
TOA shift out
Symbol 1 Symbol 2 Symbol 3 Symbol 4 Symbol 5 Symbol 6
CFR in
Fig. 9. Flowchart of the proposed pre-processing scheme.
Fig. 9 also considers the random TOA offsets between
consecutive SRS symbols caused by the imperfect synchro-
nization between the UE and the gNB [50]. In the presence of
this additional offset, the desired CIR taps with propagation
information are randomly shifted from the zero point, and
it is improper to apply a low-pass filter with an identical
cutoff delay of τmax to the CIRs estimated from different SRS
symbols. The proposed pre-processing scheme deals with this
realistic issue by adding a CFR phase slope removing module
(Step 1) in the front of the denoising modules (Step 2 to 5).
The phase slope removal approach similar to those proposed
in [51] and [21] for the sanitization of Wi-Fi sensed channel
state information (CSI) can be used here. After this phase
slope removal operation, the main peak of the CIR is nearly
centered at the zero delay.
In the flowchart of Fig. 9, the CIR plots at crucial steps
are illustrated for consecutive SRS symbols transmitted by
a non-moving user terminal and collected by a 5G picocell
gNB in an indoor environment. Parameters of the SRS con-
form to TABLE III. Point for inverse FFT (IFFT) in Step
2 is set to be 1632 and the delay window in Step 4 is
selected to be [−166.67 ns,+166.67 ns], which corresponds to
[−50 m,+50 m]in distance. Then in this example, only 41
points in the CIR fall in this window, and the FFT point in
Step 5 can be set as 64 accordingly.
The effects of the proposed pre-processing scheme are
demonstrated from the aspects of denoising and complex-
ity reduction. First, the denoising effect is illustrated by
examining the CFR amplitudes for these six consecutive
SRS symbols shown in the example of Fig. 9. The IFFT
point, delay window, and FFT point are also set to be
1632,[−166.67 ns,+166.67 ns], and 64, respectively. CFR am-
plitudes before and after this pre-processing are respectively
shown in Fig. 10(a) and Fig. 10(b). They clearly show that
the output CFR is a smoothed and noise reduced version
of the input one. Second, denoting the subcarrier number
of the SRS after pre-processing as ˜
M, then the computa-
tional complexity of this pre-processing module is nearly
O(Mlog2M+˜
Mlog2˜
M). According to the analysis in Section
IV-C, the complexity of the proposed JADE method based
on the dimension-reduced CFR matrix is ON P ˜
M2+N˜
M3.
Similar analysis can be applied to its storage requirement.
Since ˜
MM, the overall time and space complexities can be
largely reduced when prepending the proposed pre-processing
module to the front of the JADE modules.
-50 -30 -10 10 30 50
Frequency (MHz)
0
0.2
0.4
0.6
0.8
1
CFR Amplitude
(a) Before pre-processing.
-50 -30 -10 10 30 50
Frequency (MHz)
0
0.2
0.4
0.6
0.8
1
CFR Amplitude
(b) After pre-processing.
-50 -40 -30 -20 -10 0 10 20 30 40 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Channel 1 Channel 2 Channel 3 Channel 4
Fig. 10. Impact of pre-processing on CFR amplitude.
B. TOA Spectrum Estimation: Accelerating Spectrum Comput-
ing by FFTs
In this section, based on the evenly-spaced comb pattern
of the SRS [46] and the resulting Vandermonde structure
of the delay signature matrix, as revealed by equation (2),
the computing of the TOA spectrum is accelerated by em-
ploying several FFT operations. Concretely, suppose the grid
set {τp}P
p=1for TOA spectrum searching spans the whole
unambiguous delay range of 0,(∆f)−1, then according to
equation (2), the delay signature matrix Aτis composed of
the first Mrows of the P-point discrete Fourier transform base
matrix. Therefore, Aτ-involved matrix multiplications can be
computed via FFTs. Note that whether FFT or IFFT is used
depends on the sign of the complex exponential for the delay
signature matrix, and we use FFT uniformly here.
First, denoting
Q=ˆ
R−1
n,(25)
ι=Qhn,(26)
then the numerators of equation (20) for all Psearching grids
can be obtained by applying a single FFT on vector ιand
gathered in a vector ρ. That is, ρ=FFT[ι,P].
Secondly, since the denominator of equation (20) is identical
to that of the classical Capon spectral estimator, then according
to the proof in [52], its values for all the Psearching grids
can also be computed in a batch using a FFT. Specifically,
collecting these values in a vector ξ, it can be computed by
ξ=FFT[u,P],(27)
where the vector u∈CP×1is:
u=u0, . . . , uM−1,01×(P+1−2M),u1−M, . . . , u−1T,(28)
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content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
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12 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. XX, NO. X, XXX 2022
in which for m=−(M−1), . . . , M−1,umis the summation
of the m-th diagonal of the matrix Q:
um=
min(M−1+m,M−1)
Õ
i=max(0,m)[Q]i+1,i−m+1.(29)
Thirdly, since Aτis Vandermonde and ˆ
Pnis a diagonal
matrix, ˆ
Rnrepresented by equation (22) is both Toeplitz and
Hermitian [53]. Therefore, it is fully determined by its first
row, which can be computed by another FFT on the diagonal
elements of ˆ
Pn[54]. Specifically, collecting elements of the
first row of ˆ
Rnin a vector r∈CM×1, then equation (22) can
be calculated as:
r0=FFT hˆ
βn,1
2,ˆ
βn,2
2, . . . , ˆ
βn,P
2iT
,P,(30)
r=[r0]1:M,(31)
ˆ
Rn=Toeplitz (r).(32)
The IAA method implemented with these three FFTs is
denoted as the FFT-IAA method herein. According to the
analysis above, by employing FFT-IAA, the total computa-
tional complexity of the proposed method can be reduced from
ONPM2+N M 3to ON P log2P+N M 3. Additionally,
since the FFT-IAA method does not need to store the delay
signature matrix, its storage requirement is also reduced from
OMP +M2+N Pto OM2+N P . Since for 5G picocell
gNBs, PMN, much lower time and space complexities
are achieved. Actually, the proposed scheme presented in
Section IV combined with the CFR denoising module and this
FFT-IAA method has already been implemented and deployed
in the positioning server for field test, as will be presented in
Section VI-D.
VI. PE RF OR MA NC E EVALUATI ON
In this section, the performance of the proposed method
is evaluated by the following series of elaborately designed
experiments:
1) First, the effectiveness of the calibration part of the
proposed scheme is validated by multipath-free signals
collected in an anechoic chamber;
2) Next, the resolution and accuracy of the proposed
method in a dense multipath environment are demon-
strated by simulated multipath CFR data conforming to
the signal model represented by equation (6);
3) After that, the performance of the proposed method
in typical indoor environments is further evaluated by
simulated 5G channel data whose parameters conform
to the standard 5G indoor channel models [45] and by
field test data collected in an underground parking lot,
respectively;
4) Lastly, we conclude the performance evaluations by
examining the running time of the proposed method.
Throughout these experiments, the proposed method is com-
pared with existing JADE methods, including those based on
the classical periodogram method [47], the smoothed-MUSIC
method [21], [22], and the matrix-pencil method [26]–[28].
The pre-processing scheme as shown in Fig. 9 is applied before
all these JADE methods to denoise and to reduce complexity.
The counts of IFFT points in Step 2 and FFT points in Step 5
of this pre-processing scheme are chosen to be 2048 and 64,
respectively. For the smoothed-MUSIC-based JADE method,
the smoothing orders in frequency-domain and space-domain
are set to be 6and 2, respectively; while for the matrix-
pencil-based JADE method, the pencil parameters in these two
domains are set to be 35 and 3, respectively.
During all the experiments, a picocell gNB configuration is
considered. When considering the estimation of the DOA and
TOA of the uplink SRS impinged at a specific TRP, the TRP is
assumed to locate at [0,0]min the XOY plane, and the UE is
always in the sector area with the azimuth angle spanning from
−60◦to 60◦and with the maximum range of 50 m. Then for
search-based JADE methods, such as the periodogram-based,
the smoothed-MUSIC-based, and the proposed, the DOA and
TOA are searched in the range of [−60◦,60◦]and [0,50]m
with a grid size of 0.2◦and 0.2 m, respectively.
The SRS parameters are all configured according to TABLE
III for experiments in this section. A batch of antenna arrays
and 5G picocell RRUs identical to that shown in Fig. 2 is fabri-
cated and established for experiments in the anechoic chamber
and indoor environment. Their calibration coefficients are pre-
measured using the approaches introduced in Section III. For
numerical simulation experiments, an identical four-element
ULA with a half-wavelength spacing is used. In addition, the
calibration coefficients of a randomly selected antenna array
are used to generate received signals with direction-dependent
antenna errors, while the RF channel errors are assumed to be
perfectly compensated.
Besides, the time synchronization error between the UE
and the gNB induces a severe offset in the TOA estimates,
which is hard to be separated from the absolute propagation
delays in free-space [50]. Hence, for simulations, we assume
perfect timing, while for field test in the indoor environment,
differentials of the TOA estimates from two TRPs, i.e. the
time differences of arrival (TDOAs), are evaluated instead.
A. Field Test in an Anechoic Chamber
The experimental setup in the anechoic chamber for per-
formance evaluation is illustrated by Approach (b) of Fig.
4. Different from Approach (a) which measures the array
response using dedicated instruments, Approach (b) estab-
lishes the whole 5G infrastructure in this anechoic chamber
and employs the 5G UE and gNB for signal transmitting and
receive signal processing, respectively. Concretely, similar to
Approach (a), the receiving ULA is still rotated from −60◦to
+60◦by an angular interval of 5◦during the experiments. The
difference is that, at each angle, 500 SRS symbols transmitted
by the 5G UE are conducted to the horn antenna and the
sounded CFRs are collected by the gNB.
To illustrate the repercussions caused by the array modeling
errors and demonstrate the effectiveness of the calibration
module in the proposed JADE scheme, the DOA estimation
errors of the proposed method with the ideal and calibrated
steering-vector functions are compared in Fig. 11.
Fig. 11 demonstrates the root mean square errors (RMSEs)
of the DOA estimates obtained by experiments with 12 differ-
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content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
© 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
PAN et al.: EFFICIENT JOINT DOA AND TOA ESTIMATION FOR INDOOR POSITIONING WITH 5G PICOCELL BASE STATIONS 13
ent antennas, each of which is calculated based on the DOA
measures from consecutive SRS symbols collected by that
antenna as follows:
RMSE =v
u
t1
L
L
Õ
l=1ˆ
θl−θtruth 2
,(33)
where ˆ
θlrepresents the estimate for l−th SRS, θtruth represents
the DOA ground truth, and Lis the count of measurements,
which equals to 500 for Fig. 11.
In Fig. 11, at each angle, the solid dot shows the average
value of RMSEs from these 12 antennas, and the lower and
upper bounds of the shadowed area mark the minimum and
maximum RMSE values, respectively. Fig. 11 indicates that,
estimating DOA using the calibrated steering-vector function
counteracts the direction-dependent antenna modeling errors
dramatically, especially when the signal impinges the receiving
array from directions larger than ±45◦. For example, the
averaged DOA estimation error can be reduced from 8.11◦
to 1.28◦at the incident direction of +60◦. This phenomenon
is attributable to the divergence of the antenna phase errors in
large directions, which has been evidently shown in Fig. 6.
-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60
Incident direction (degree)
0
2
4
6
8
10
12
RMSE of DOA estimate (degree)
Using ideal steering-vector function
Using calibrated steering-vector function
Fig. 11. RMSE statistics of DOA estimates obtained by 12 antennas in an
anechoic chamber when incident direction varies from −60◦to +60◦.
B. Numerical Experiments Using Simulated Multipath Data
CFR data with direction-dependent array modeling errors
is simulated based on equation (6). During experiments, the
multipath number Kvaries from three to five, which is repre-
sentative for typical indoor environments, and the attenuation
coefficients of these path components ˜γk,k=1, . . . , Kare
assumed to be identical. The signal-to-noise-ratio (SNR) in
the received signal varies from −10 decibels (dB) to 10 dB,
which is calculated as follows:
SNR [dB]=10 lg |˜γk|2
σ2
w
,(34)
in which |˜γk|2and σ2
wrepresent signal and noise powers on
a single subcarrier of a single receiving channel, respectively.
500 Monte Carlo trials are conducted for each scenario
with a specific multipath number and a specific SNR. In
each Monte Carlo trial, the DOA and TOA of each multipath
component and the realization of the wideband multichannel
noise component are generated randomly. Among them, the
DOAs and TOAs are generated according to the uniform dis-
tributions over −60◦to +60◦and over 0to 166.67 ns (50 m),
respectively, i.e. θk∼ U (−60,+60], τk∼ U (0,166.67 ns],
and the noise component on each subcarrier of each receiving
channel is generated independently according to a standard
complex Gaussian distribution CN(0,1). Then the attenuation
coefficient ˜γkin each experiment is derived by equation (34).
The simulation results are shown in Fig. 12. Firstly, Fig.
12(a) to 12(c) indicate that, although a relatively naive CBF
method is used for DOA estimation in the proposed scheme,
its DOA estimation performance is still superior to all the other
JADE methods. For example, from Fig. 12(c), when there are
five strong multipaths and the SNR is −10 dB, the proposed
method can achieve a reduction of at least 44% for the 80-th
percentile of the DOA estimation error (from 4.58◦to 2.58◦).
Secondly, since the proposed method performs TOA spectral
analysis for each channel individually and then averages the
amplitudes of these TOA spectra, it integrates all these multi-
channel information non-coherently. On the contrary, the other
three JADE methods employ 2-D spectral estimators, which
process the multichannel signals coherently. Therefore, it is
plausible that these methods outperform the proposed method
in the TOA domain. But instead, the 2-D periodogram and
2-D matrix-pencil methods exhibit performance degradation:
for the 2-D periodogram, this owes to its limited resolving
ability; while for the 2-D matrix-pencil, this attributes to the
destruction of the shift-invariance property by the direction-
dependent array modeling error. From Fig. 12(f), when there
are five strong multipaths and the SNR is −10 dB, the 80-
th percentile of the TOA estimation error increases only by
0.3 ns (0.09 m)for the proposed method when compared
with the 2-D smoothed-MUSIC-based JADE method (From
1.45 ns (0.435 m)to 1.75 ns (0.525 m)).
Lastly, the 80-th percentiles of the positioning errors based
on these DOA and TOA estimates are shown in Fig. 12(g)
to Fig. 12(i). The location of the UE is derived from the
DOA and TOA estimated by a single TRP, and the positioning
error is calculated as the Euclidean distance between the
derived position and the ground truth. It can be inferred that
a much better positioning accuracy can be achieved based
on the proposed method than those based on the existing
JADE methods. Also according to the simulation, when the
path number is no more than 5and the receiving SNR is
above −10 dB, the single-site positioning error achieved by
the proposed method is below 1.3 m for 80% cases.
C. Numerical Experiments Using Simulated 5G Wireless
Channel Data
The performance of the proposed method is then evalu-
ated with realistic wireless channel data generated by the
QuaDRiGa channel simulator [55]. The configurations of
3GPP_38.901_InF_LOS and 3GPP_38.901_Indoor_LOS
for FR1 band are selected for the simulator, which represent
the indoor factory LOS (InF-LOS) and the indoor office LOS
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content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
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14 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. XX, NO. X, XXX 2022
10-1 100101
DOA estimation error (degree)
0
0.2
0.4
0.6
0.8
1
Empirical CDF
(a) DOA errors (path number: 3, SNR: −10 dB).
10-1 100101
DOA estimation error (degree)
0
0.2
0.4
0.6
0.8
1
Empirical CDF
(b) DOA errors (path number: 4, SNR: −10 dB).
10-1 100101
DOA estimation error (degree)
0
0.2
0.4
0.6
0.8
1
Empirical CDF
(c) DOA errors (path number: 5, SNR: −10 dB).
0.01 0.1 1 2 3
TOA (Distance) estimation error (m)
0
0.2
0.4
0.6
0.8
1
Empirical CDF
(d) TOA errors (path number: 3, SNR: −10 dB).
0.01 0.1 1 2 3
TOA (Distance) estimation error (m)
0
0.2
0.4
0.6
0.8
1
Empirical CDF
(e) TOA errors (path number: 4, SNR: −10 dB).
0.01 0.1 1 2 3
TOA (Distance) estimation error (m)
0
0.2
0.4
0.6
0.8
1
Empirical CDF
(f) TOA errors (path number: 5, SNR: −10 dB).
-10 -5 0 5 10
SNR (dB)
0
0.5
1
1.5
2
2.5
3
3.5
4
Positioning error (m)
(g) Positioning errors (path number: 3).
-10 -5 0 5 10
SNR (dB)
0
0.5
1
1.5
2
2.5
3
3.5
4
Positioning error (m)
(h) Positioning errors (path number: 4).
-10 -5 0 5 10
SNR (dB)
0
0.5
1
1.5
2
2.5
3
3.5
4
Positioning error (m)
(i) Positioning errors (path number: 5).
10-1 100101
Azimuth angle error (degree)
0
0.2
0.4
0.6
0.8
1
Empirical CDF
Periodogram
Smoothed-MUSIC
Matrix-pencil
Proposed
Fig. 12. Performance comparisons of JADE methods by simulated multipath data with path number varies from three to five. (a)-(c) and (d)-(f) show the
empirical cumulative distribution function (CDF) curves under the SNR of −10 dB for DOA estimation and TOA (distance) estimation errors, respectively.
(g)-(i) show the 80-th percentiles of the positioning errors based on the DOA and TOA estimates when the SNR varies from −10 dB to 10 dB.
(InO-LOS) scenarios, respectively [45]. During simulations,
the transmitting power of the UE and the noise figure of the
gNB are fixed to Pt=200 mW and F=5 dB, respectively.
Then the power of the receiving signal is derived by the
QuaDRiGa simulator according to the propagation model and
the noise power is calculated as follows:
Pn=kBT0BF,(35)
where kB,T0,Bare the Boltzmann’s constant, standard noise
temperature, and the measurement bandwidth, respectively.
They are respectively fixed to 1.38 ×1023,290 Kelvin, and
100 MHz in simulations.
The default Ricean K-factors for 3GPP 38.901 InF-LOS
and InO-LOS channels are both 7 dB [45]. To evaluate the
parameter estimation performance when the multipath power
varies, the Ricean K-factors for both channels are configured
to vary from 0 dB to 7 dB during the experiments. For each
scenario with a specific Ricean K-factor, 1000 Monte Carlo
trials are conducted, and similar to the approach stated in
Section VI-B, the ground truths of the DOAs and TOAs are
randomly generated in each trial.
The Monte Carlo evaluation results for DOA, TOA and 2-
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content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
© 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
PAN et al.: EFFICIENT JOINT DOA AND TOA ESTIMATION FOR INDOOR POSITIONING WITH 5G PICOCELL BASE STATIONS 15
D position estimations are summarized in Fig. 13. Similar
to the results presented in Section VI-B, in realistic indoor
environments, the proposed method has a significant improve-
ment for the DOA estimation performance at the cost of a
slightly reduced TOA estimation performance. Furthermore,
when the DOA and TOA are exploited to locate the UE, the
proposed method also achieves the best single-site positioning
performance. For example, for the default InF-LOS scenario,
a reduction of at least 72% (from 1.8◦to 0.5◦) for DOA
estimation error is achieved, while the TOA estimation error
only increases by 0.02 m (from 0.06 m to 0.08 m) at most,
and the total positioning error is also reduced by at least 67%
(from 1.54 m to 0.51 m).
Further, according to Fig. 13(c), when the Ricean K-factor
is above 0 dB, positioning errors of 0.99 m and 1.39 m can
be achieved by the proposed method for the standard 3GPP
38.901 InF-LOS and InO-LOS scenarios, respectively. It can
be also inferred from Fig. 13 that, all algorithms perform better
in the InF-LOS scenario than in the InO-LOS scenario. This
is attributable to the smaller delay spread parameter of the
InO-LOS channel, which generates more densely distributed
multipath components [45].
D. Field Test in an Indoor Environment
To evaluate the performance of the proposed JADE scheme
in a real indoor environment, a field test is conducted in an
underground parking lot with a commercial 5G picocell gNB
established. As illustrated by the experiment layout of Fig. 14,
this gNB contains two TRPs, which are separated by 7.6 m,
and have an overlapping coverage to communicate to the UE
simultaneously. Each of these TRPs is composed of the six-
element ULA and the picocell RRU as shown in Fig. 2. The
5G UE is deployed on an autonomous vehicle, which can
determine its location at the accuracy of several centimeters.
Our proposed JADE method has been implemented and
deployed in the positioning server and is able to estimate
the DOA and TOA of the uplink SRS in realtime. DOA and
TOA estimates and the ground truths given by the autonomous
vehicle are exported for offline analysis. As shown in Fig.
14, during the experiments, the UE is placed at 18 different
locations in the lane, and at some locations, the UE is rotated
and retested to enlarge the dataset. Therefore, 30 cases are
tested in total. For each case, the real-time results from 300
SRS symbols are collected. Also note that, seven typical test
points are marked out in Fig. 14 and results at those individual
points will be presented later.
First, the JADE performance is evaluated. As stated before,
errors for the differentials of the TOA estimates at these two
TRPs, a.k.a the TDOA estimation errors, are calculated instead
of the TOA estimation errors. The ensemble empirical CDF
curves for DOA and TDOA estimation errors are shown in
Fig. 15, along with empirical CDFs for the seven selected test
points. The total 80-th percentiles for DOA estimation errors
at these two RRUs are 0.88◦and 1.49◦, respectively, while
the 80-th percentile for TDOA estimation error is 0.31 m.
The estimation error is highly location-dependent, which is
attributed to the varied multipath effects at different locations.
Then the position of the UE is determined by triangulation
based on the DOA estimates of these two TRPs, resulting in
the positioning errors illustrated in Fig. 16. Fig. 16(a) indicates
that the positioning results fall within the 0.8 m error bounds in
most cases. Similar to Fig. 15, Fig. 16(b) depicts the empirical
CDF curves of the positioning errors for the seven selected
positions and the ensemble CDF curve. It demonstrates that
the total 90-th percentile of positioning error for all these 9000
samples is 0.44 m, which meets the 3GPP R17 requirement for
commercial use cases (1 m for 90% cases) [18]. In addition,
while the current positioning results are obtained via trian-
gulation for each individual SRS symbol, hybrid positioning
methods which combine the tracking frameworks with both
DOA and TDOA estimates, such as the one proposed in [56],
can be employed for further performance improvement.
E. Running Time Evaluation
The running time of the proposed method is evaluated
for the above picocell gNB and SRS configurations and is
compared with that of the smoothed-MUSIC-based JADE
method. The running times are all recorded by the timer of
MATLAB on a PC with a 2.60 GHz Intel i7 CPU and a 16 GB
memory. As stated at the start of Section VI, the proposed pre-
processing scheme with same parameters is applied to all algo-
rithms. Therefore, their input CFR matrices have the identical
dimension. The averaged running times are shown in TABLE
V. It shows that, by using the cascading estimation scheme
for JADE, the computation time is reduced by two orders
of magnitude when compared with the popular smoothed-
MUSIC-based JADE method. In addition, its running time
can be further reduced by nearly an order of magnitude by
employing FFTs for fast IAA spectrum computing.
TABLE V
RUNNING TIMES ON A PC WITH AN IN TEL I 7 CPU @2.60 GHZ AN D A
16 G B MEMORY
JADE algorithm Averaged running time
Smoothed-MUSIC-based [21], [22] 8777.0 ms
IAA + CBF (proposed, Section IV-B) 88.1 ms
FFT-IAA + CBF (proposed, Section V-B) 10.1 ms
VII. DISCUSSION
This work proposes a JADE method in the presence
of direction-dependent antenna modeling errors for picocell
gNBs with ULAs established. However, the framework of the
proposed method is quite adaptable to different scenarios. In
this section, discussions about the extensions of the proposed
framework to other potential scenarios are presented.
A. Extension to Arbitrary Array Structures
Many JADE methods require the Vandermonde array man-
ifold and cannot be directly applied in non-uniform array
configurations, such as those relied on the spatial smoothing
technique [21], [22], the array shift invariance property [24],
[26], or the polynomial rooting technique [33]. In contrast, the
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content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
© 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
16 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. XX, NO. X, XXX 2022
01234567
Ricean K-factor (dB)
0
0.5
1
1.5
2
2.5
3
3.5
4
DOA estimation error (degree)
(a) DOA estimation errors.
0 1 2 3 4 5 6 7
Ricean K-factor (dB)
0
0.05
0.1
0.15
0.2
0.25
TOA estimation error (m)
(b) TOA (Distance) estimation errors.
01234567
Ricean K-factor (dB)
0
0.5
1
1.5
2
2.5
3
Positioning error (m)
(c) Positioning errors.
01234567
Ricean K-factor (dB)
0
0.5
1
1.5
2
2.5
3
Localization error (m)
Periodogram (InF-LOS) Proposed (InF-LOS)
Periodogram (InO-LOS)
Smoothed-MUSIC (InF-LOS)
Smoothed-MUSIC (InO-LOS)
Matrix-pencil (InF-LOS)
Matrix-pencil (InO-LOS)
Proposed (InO-LOS)
Fig. 13. Performance comparisons of JADE methods by simulated 5G InF-LOS and InO-LOS channel data whose parameters conform to the 3GPP 38.901
report [45] except their Ricean K-factors, which vary from zero to seven decibels to generate multipath components with different powers. (a), (b), and (c)
show the 80-th percentiles for the DOA, TOA (distance) and position estimation errors, respectively.
Fig. 14. Experiment layout for positioning experiments in an underground
parking lot.
proposed method is applicable for arbitrary array structures
since it employs the CBF for DOA estimation. That is, it can
handle the space-domain non-uniformity.
B. Extension to Nonuniformly Distributed Subcarriers
Thanks to the IAA spectral estimator, which achieves a
clear and accurate spectrum even when the underlying signa-
ture structure is nonuniform [57], [58], the proposed method
can also handle the frequency-domain non-uniformity. This
characteristic is promising in the scenario with co-channel
interference, in which the contaminated subcarriers can be
discarded and the remaining subcarriers with nonuniformly
distributed pattern are exploited for TOA spectrum estimation.
Additionally, in this situation, the IAA method can also be
accelerated by FFTs using the approach presented in Section
V-B. Specifically, denoting the uncontaminated CFR for n-th
channel as ˘
hnand the number of uncontaminated subcarriers
as ˘
M, then ˘
hnrelates to the full CFR h0
nvia a selection matrix
J∈R˘
M×Mas follows:
˘
hn=Jh0
n,n=1, . . . , N,(36)
where when the m-th subcarrier is not contaminated, Jhas an
element equals to 1at the m-th column, otherwise its m-th
column is 0˘
M. Similarly, the delay signature matrix for the
uncontaminated CFR also relates to that for the original CFR
by the same selection matrix J:
˘
Aτ=JAτ.(37)
Accordingly, the IAA iterations of equation (20) and (22)
for uncontaminated CFR can be updated to
ˆ
βn,p=
aH
τ(τp)hJT·˘
R−1
n˘
hni
aH
τ(τp)JT˘
R−1
nJaτ(τp),p=1, . . . , P,(38)
˘
Rn=JAτˆ
PnAH
τJT,(39)
where Aτ-related operations can still be implemented by FFTs
according to Section V-B, and J-related operations can be
implemented by matrix selection and matrix zero-padding,
which have little impact on the computational complexity.
Therefore, the proposed method implemented with FFT-
IAA can be extended effortlessly to the CFR model with
nonuniformly distributed subcarriers.
C. Extension to 3-D Localization Parameter Estimation
This paper emphasizes on the estimation of the propagation
delay τand azimuth angle θof the arrived signal for 2-D
positioning based on a linear array. In fact, since the proposed
method separates the temporal and spatial processing, it can
also be extended to estimate the delay τ, azimuth angle θ, and
elevation angle φfor positioning in a three-dimensional (3-D)
space based on an arbitrary planar array. Concretely, in this
situation, the 1-D CBF search expressed by equation (24) is
This article has been accepted for publication in IEEE Transactions on Instrumentation and Measurement. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
© 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
PAN et al.: EFFICIENT JOINT DOA AND TOA ESTIMATION FOR INDOOR POSITIONING WITH 5G PICOCELL BASE STATIONS 17
10-2 10-1 100101
DOA estimation error (degree)
0
0.2
0.4
0.6
0.8
1
Empirical CDF
(a) Empirical CDFs for DOA estimation errors of
TRP-1.
10-2 10-1 100101
DOA estimation error (degree)
0
0.2
0.4
0.6
0.8
1
Empirical CDF
(b) Empirical CDFs for DOA estimation errors of
TRP-2.
10-3 10-2 10-1 100
TDOA estimation error (m)
0
0.2
0.4
0.6
0.8
1
Empirical CDF
(c) Empirical CDFs for errors of the differentials
of TOA estimates at these two TRPs.
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
Point index: (1) Point index: (2) Point index: (3) Point index: (4)
Point index: (5) Point index: (6) Point index: (7) Total
Fig. 15. Empirical CDF curves for DOA and TDOA estimation errors with data collected in the underground parking lot.
-20 -15 -10 -5 0 5 10 15 20
X coordinate (m)
-10
-5
0
5
10
15
20
25
Y coordinate (m)
TRP 1
TRP 2
TRP
Test point
Positioning result
0.8 m error bound
(1)
(2)
(6)
(4)
(3)
(7)
(5)
(a) Scatter plot for positioning results of all 300 SRS symbols
at all test points.
0.01 0.1 1 2
Positioning error (m)
0
0.2
0.4
0.6
0.8
1
Empirical CDF
Point index: (1)
Point index: (2)
Point index: (3)
Point index: (4)
Point index: (5)
Point index: (6)
Point index: (7)
Total
(b) Empirical CDFs for errors of triangulation positioning based on DOA estimates of
two TRPs.
Fig. 16. Performance evaluation for triangulation positioning based on DOAs at the two established TRPs estimated by the proposed JADE method.
replaced with the following 2-D CBF search to determine the
optimal θand φof the LOS path:
ˆ
θLOS,ˆ
φLOS=arg max
(θ,φ)hˆ
a0
θ,φ (θ, φ)iH
bLOS,(40)
where ˆ
a0
θ,φ (θ, φ)is the estimated actual steering-vector func-
tion of the specific planar array for input pair of (θ, φ).
VIII. CONCLUSION
This study has demonstrated the potential of 5G picocell
gNB for indoor positioning by investigating its array modeling
errors specifically and designing an efficient JADE scheme to
calibrate these errors and to provide DOA and TOA estimates
in real-time. First, based on the characteristics of the array
modeling errors of typical picocell gNBs, a vector-valued func-
tion for the direction-dependent antenna error is incorporated
in the signal model to capture all sorts of antenna errors. Then
the antenna error function is estimated and compensated to
the ideal steering-vector to derive the actual array steering-
vector for DOA estimation in the proposed JADE scheme.
The proposed scheme achieves largely reduced computational
complexity and storage requirement by employing a cascading
processing scheme with the IAA-based TOA spectral analyzer
and a CBF-based DOA estimator, which fully exploits the
fact that the 5G picocell gNB only has a small-scale antenna
array but has a large signal bandwidth. By further proposing
a dimension-reducing pre-processing method and deriving the
FFT-based version of the IAA method, an averaged running
time of 10.1 ms is achieved for the proposed method in a
typical gNB configuration, which is nearly three-orders lower
than that of the MUSIC-based JADE method. Some attentions
have also drawn to the adaptability of the proposed scheme,
which has shown to be able to cope with both the irregularities
of the antenna array and the subcarrier distribution, and be
This article has been accepted for publication in IEEE Transactions on Instrumentation and Measurement. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
© 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
18 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. XX, NO. X, XXX 2022
scalable to the scenario of planar-array-based 3-D positioning.
A series of experiments was designed for performance vali-
dation. First, a field test in the anechoic chamber demonstrated
the effectiveness of the calibration scheme. Then numerical
simulations based on simulated multipath data and 5G channel
data showed the superiority of the proposed method for DOA
estimation at the cost of a slightly reduced TOA estima-
tion performance when compared with popular 2-D super-
resolution JADE methods. Lastly, according to the field test
in an indoor environment, the proposed method achieved a
positioning error of 0.44 m for 90% cases by employing a
minimum gNB configuration with only two separate TRPs. In
summary, we think the proposed method is a viable option for
performing real-time JADE in future 5G and beyond network
for the purpose of high-accuracy indoor positioning.
ACKNOWLEDGMENT
The authors would like to thank Yan Li and Haipeng Xu
for providing antenna parameters, Jia Xu for field test data
collection, and Wang Zheng, Yuqing Li, and Zihuan Mao for
helpful discussions. We would also like to thank the associate
editor and the anonymous reviewers for their early feedbacks
and constructive suggestions which greatly helped to improve
the quality of this paper.
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content may change prior to final publication. Citation information: DOI 10.1109/TIM.2022.3191705
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