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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. XX, NO. X, XXX 2023 1
In-Situ Calibration of Antenna Arrays for
Positioning With 5G Networks
Mengguan Pan, Member, IEEE, Shengheng Liu, Senior Member, IEEE, Peng Liu, Wangdong Qi, Member, IEEE,
Yongming Huang, Senior Member, IEEE, Wang Zheng, Qihui Wu, Senior Member, IEEE,
Markus Gardill, Member, IEEE
Abstract—Owing to the ubiquity of cellular communication
signals, positioning with the fifth generation (5G) signal has
emerged as a promising solution in global navigation satellite
system-denied areas. Unfortunately, although the widely em-
ployed antenna arrays in 5G remote radio units (RRUs) facilitate
the measurement of the direction of arrival (DOA), DOA-based
positioning performance is severely degraded by array errors.
This paper proposes an in-situ calibration framework with a
user terminal transmitting 5G reference signals at several known
positions in the actual operating environment and the acces-
sible RRUs estimating their array errors from these reference
signals. Further, since sub-6GHz small-cell RRUs deployed for
indoor coverage generally have small-aperture antenna arrays,
while 5G signals have plentiful bandwidth resources, this work
segregates the multipath components via super-resolution delay
estimation based on the maximum likelihood criteria. This differs
significantly from existing in-situ calibration works which resolve
multipaths in the spatial domain. The superiority of the proposed
method is first verified by numerical simulations. We then
demonstrate via field test with commercial 5G equipment that, a
reduction of 46.7% for 1-σDOA estimation error can be achieved
by in-situ calibration using the proposed method.
Index Terms—5G positioning, angle-of-arrival (AOA), array
calibration, direction-of-arrival (DOA), field test, in-situ calibra-
tion, multipath, wireless localization.
I. INTRODUCTION
PRECISE positioning is the key enabler for a wide range of
emerging applications such as indoor navigation [1], au-
tonomous driving [2], healthcare [3], intelligent transportation
[4], industrial internet of things [5], etc. With the progressive
deployment of the 5G small-cell (i.e. microcell, picocell, or
femtocell) base stations (a.k.a. gNodeBs, or gNBs) in GNSS
challenging scenarios [6], such as deep urban canyons, tun-
nels, undergrounds, or indoor environments, the abundant 5G
Manuscript received 29 December 2022; revised 16 February 2023; ac-
cepted xx xxx 2023. Date of publication xx xxx 2023; date of current version
xx xxx 2023. This research was supported in part by the National Natural
Science Foundation of China under Grant Nos. 62001103 and U1936201.
(Corresponding authors: Shengheng Liu and Peng Liu.)
Mengguan Pan and Wang Zheng are with the Purple Mountain Laboratories,
Nanjing 211111, China (e-mail: panmengguan@outlook.com).
Shengheng Liu, Wangdong Qi, and Yongming Huang are with the National
Mobile Communications Research Laboratory, Southeast University, Nanjing
210096, China, and also with the Purple Mountain Laboratories, Nan-
jing 211111, China (e-mail: s.liu@seu.edu.cn, qiwangdong@pmlabs.com.cn,
huangym@seu.edu.cn).
Peng Liu is with the College of Electronic and Information Engineering,
Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China,
and also with the Department of Network Engineering, Army Engineering
University of PLA, Nanjing 210007, China (e-mail: herolp@gmail.com).
Qihui Wu is with the College of Electronic and Information Engineering,
Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China.
Markus Gardill is with Brandenburg University of Technology Cottbus–
Senftenberg, Cottbus, 03046, Germany.
signals become a promising candidate for achieving accurate
and reliable positioning in these areas [7]–[9].
The widespread employment of antenna array technique
for small-cell 5G RRUs has attracted growing interest in
exploiting the DOA information for 5G positioning [10]–
[12], as it not only obviates the need for precise timing-
synchronizations but is also an indispensable measurement
for implementing single site positioning [10]. However, the
DOA estimation performance is inevitably impaired by the
nonideal responses of the antenna arrays and the RF channels
of the receiver [13], [14]. The majority of existing works on
DOA-based wireless positioning either assume an ideal array
model [10], [11] or merely take into account the gain-phase
errors introduced by the RF channels [15]–[17]. However, the
antenna array per se is also suffered from severe imperfections
(a.k.a. array errors), resulting in the deviations of the real
array manifold from the theoretic one. Therefore, precise array
calibration, which amounts to estimating the array error and
deriving the real array manifold, is pivotal to achieving high-
accuracy DOA-based positioning.
According to the adopted model for array errors, calibra-
tion can be achieved by using either parametric methods or
non-parametric methods. Parametric methods only consider
typical array errors, i.e. gain-phase errors, mutual couplings,
and element location perturbations, and model them with a
small number of direction-independent parameters [18]–[27].
However, apart from these common array errors, real-world
antennas are generally impaired by other unpredictable and
more complicated imperfections, such as the electromagnetic
interactions between the array and nearby structures, manufac-
turing inaccuracy, etc., which can hardly be captured by the
parametric models, as verified by field experiments in [27].
On the contrary, non-parametric methods, which gather all the
array nonidealities into a direction-dependent [28] (also known
as scan-dependent in [29]) error function [28]–[37], can depict
arbitrary array error patterns.
Calibration techniques can also be categorized as chamber
calibration, in-situ calibration, and self-calibration methods.
Measuring the array response in an anechoic chamber is the
standard way for array calibration [35]–[37]. However, the
array manifold in its working environments is hardly the same
as the nominal manifold measured in the anechoic chamber,
owing to installation errors, scatterings from array mounting
structure and nearby objects, coupling behavior changes, etc
[26], [27]. Also, calibrating every antenna array in an anechoic
chamber and reinstalling them in their working positions is
costly and time-consuming.
Self-calibration circumvents the aforementioned drawbacks
This article has been accepted for publication in IEEE Transactions on Microwave Theory and Techniques. 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/TMTT.2023.3256532
© 2023 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 MICROWAVE THEORY AND TECHNIQUES, VOL. XX, NO. X, XXX 2023
of chamber calibration by estimating simultaneously both
the wavefield DOA and the array errors based on online
measurements [18]–[20]. It commonly employs the parametric
array error model to reduce the number of unknowns and
optimizes a cost function for joint estimation. However, the
global optimality of this multi-dimensional estimation problem
is not always guaranteed and ambiguities arise for some array
geometries [21], [38]. Moreover, it also suffers from high
computational complexity owing to the vast parameter space.
In-situ calibration measures the array error in the actual op-
erating environment using auxiliary calibration sources whose
positions, which are also denoted as the CPPs, are known [21]–
[27], [29]–[34]. It reaches a reasonable compromise between
chamber calibration and self-calibration as it obtains the exact
in-field array response with significantly lower complexity and
much better accuracy than self-calibration.
However, the in-field calibration signal is inevitably in-
terfered by the multipath effect caused by reflections or
scatterings of surrounding objects. The most popular solution
in current literature is to approximate the array steering vector
with the PE of the spatial covariance matrix of the received
signal [29]–[32]. Its performance is guaranteed only when
the LOS signal power dominates all the NLOS propagated
signal powers. Besides, the coherency between these multipath
components (including both LOS and NLOS paths) also dete-
riorates the approximation performance. Leshem and Wax [33]
and Yamada et al. [23] explicitly consider the multipaths in the
signal model for in-situ calibration. However, the former work
[33] depends on a complex calibration scheme that includes
physically rotating the array and transmitting calibration sig-
nals in two different locations. The latter work [23] assumes
a parametric array error model with only gain-phase errors
and mutual couplings. Moreover, all the aforementioned in-situ
calibration methods work on the premise that the multipaths
are separable by the spatial resolution of the antenna array.
Besides, other research efforts in counteracting the mul-
tipath effect for in-situ calibration include: Pan et al. [34],
who solve it from the statistical perspective by treating the
summation of the multipath components as Gaussian noise
based on the Rayleigh fading assumption; and Sippel et al.
[26], who propose to choose the CPPs within the array’s near
field to elude multipath effects.
As clearly demonstrated in our prior work [37] with real-
measured data from 5G sub-6GHz picocell RRUs, the array
errors of a small-aperture antenna array exhibit noticeable
dependency on incident directions. This arbitrary direction-
dependent pattern can hardly be decomposed into parametric
models. On the other hand, as discussed above, although some
solutions exist for multipath mitigation in array calibration
literature, they rely entirely on the array’s spatial aperture to
resolve multipath components and are inapplicable to small-
scale arrays equipped by the small-cell 5G RRUs. Therefore,
this paper attempts to solve the in-situ array calibration prob-
lem with explicit modelings of both the direction-dependent
array errors and the multipath effects, and aims at providing
a universal and pragmatic solution to in-situ calibrate the
pervasively established 5G small-cell RRUs to support accu-
rate DOA estimation and positioning. Specifically, the main
contributions of this work are summarized as follows.
1) We design a comprehensive scheme for accurate ar-
ray calibration by non-parametrically modeling the
direction-dependent array errors and explicitly consider-
ing the multipath effects. Existing works either consider
the non-parametric array model in an ideal environment
or tackle the multipaths with a simplified parametric
model assumed.
2) We propose an in-situ array calibration framework that is
easily deployable on existing commodity 5G infrastruc-
ture by exploiting the standard 5G reference signal as the
calibration source and using the ready-to-use baseband
channel estimates for array manifold estimation. As a
result, it obviates any modification to the hardware or
protocol and is scalable to calibrate these pervasively
installed 5G RRUs.
3) For small-cell RRUs whose apertures are extremely
small, we propose to segregate the multipaths by their
TOAs in estimating the array manifold, which is superior
to the conventional way of resolving them in the spatial
(angular) domain. In this vein, a joint array response
and TOA estimation problem is formulated using the
maximum likelihood criterion and solved via the com-
putationally efficient EM approach. To the best of our
knowledge, this is the first attempt to leverage signal
bandwidth for multipath resolution in in-situ calibration.
4) We prototype a 5G positioning system with commercial
5G picocell RRUs and conduct extensive indoor field
tests in an area of nearly 1125 m2. We demonstrate:
(i) the disparity between the in-situ and the nominal
array manifold, and (ii) a reduction of 46.7% for 1-σ
DOA estimation error achieved by calibrating with the
proposed method.
The rest of this paper is organized as follows. Section II
presents the in-situ array calibration framework for 5G RRUs
and describes the signal model for the array manifold estima-
tion. Then the algorithm for estimating the array manifold in-
situ is proposed in Section III. Next in Section IV and Section
V, the calibration performance of the proposed method in
multipath environments is evaluated by numerical simulations
and field tests, respectively. Finally, Section VI concludes the
paper.
Notations: Boldface lowercase and uppercase letters re-
spectively denote vectors and matrices, where vectors are by
default in column orientation. Italic English letters and lower-
case Greek letters denote scalars. Blackboard-bold characters
denote number sets, in particular, Rand Crepresent the sets
of real and complex numbers, respectively. For convenience,
remaining notations and abbreviations used in this article are
explained in the Nomenclature Section.
II. IN-SI TU CALIBRATION FRAMEWORK AND SIGNAL
MOD EL
A. In-Situ Calibration Framework
An in-situ array error calibration framework is proposed
for 5G RRU and is illustrated in Fig. 1. All the measurement
procedures shown in Fig. 1 are conducted in a real working
This article has been accepted for publication in IEEE Transactions on Microwave Theory and Techniques. 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/TMTT.2023.3256532
© 2023 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.: IN-SITU CALIBRATION OF ANTENNA ARRAYS FOR POSITIONING WITH 5G NETWORKS 3
...
Coverage
of RRU
...
……
UL-SRS
Antenna array
of RRU
RF & IF
processing FFT Channel
estimation
...
CPP 1 CPP kCPP K
…… ...
RF
signal
θ
1
2N
Array broadside
BB
signal Y
(k)
y
(k)
[m]
x
(k)
(t)
CFR H
(k)
θ
k
θ
K
1
Array Manifold
Estimation
Angular
set
Ideal array
manifold
{θ
k
}
K
k = 1
â
θ
(θ)
RF channel error
compensation
Positioning
Position
estimate
CFR H
(k)
Calibration procedure
Positioning and
communications
procedures
Span of
θ
RF channel
coefficients
RRU
...
Used as uplink CSI, as well as downlink
CSI (via reciprocity), for communications
UT
Fig. 1. Illustration of measurement setup for in-situ calibration of array errors
for 5G RRU.
environment with a gNB installed. The UT is placed at K
known positions (CPPs) at which it sends the UL-SRS to the
RRU. CFRs perceived from these UL-SRSs, i.e {H(k)}K
k=1 in
Fig. 1, are the data source for array calibration. To calibrate
the direction-dependent errors for signals impinged from all
possible directions, the CPPs should be distributed over the
entire coverage area. Further, it is noticeable from Fig. 1 that
the calibration procedure shares the identical transmit wave-
form and front-end signal processing chain with the standard
5G communications and positioning procedures [9], [39]. This
indicates that the proposed in-situ calibration framework not
only eludes extra hardware and protocol overheads, but also
precisely delineates all the hardware impairments suffered by
the positioning signals.
It is worthwhile emphasizing that, this paper mainly con-
siders the imperfect array response induced by the antennas,
while that induced by the RF channels is compensated with
pre-measured RF channel coefficients in both calibration and
positioning procedures, as demonstrated in Fig. 1. The RF
channel calibration coefficients are commonly measured by
the means of internal calibration [40]. For small-cell RRUs
whose transceivers lack internal calibration circuits, such as
the dedicated calibration channel and the calibration network
[41], the RF channel coefficients can be measured by directly
conducting the calibration signal from a 5G test UT or a signal
generator to the receiving RF channels of the RRU via coaxial
cables and an RF power splitter [37].
B. Signal Model
As shown in Fig. 1, after the RF and IF processing, the
wireless channel response is estimated from the BB UL-SRS.
Assuming that a UL-SRS with Msubcarriers is transmitted
at the frequency of f(c)and impinges on an RRU via L
propagation paths (i.e., Lmultipath components), then the
CFR sensed by an N-element antenna array can be represented
as [37]
H=
L
X
l=1 h˜γl·aτ(˜τl)aT
θ(˜
θl)i+W,(1)
where ˜
θl,˜τl, and ˜γlare the incident direction (a.k.a. DOA),
propagation delay (a.k.a. TOA), and complex gain of the l-th
path. We assume a LOS scenario for in-situ calibration, hence
the existence of lLOS ∈ {1,2, . . . , L}which indicates the LOS
path index. H∈CM×Nand W∈CM×Nin (1) represent the
CFR matrix and the noise, respectively. Entries of Ware i.i.d.
complex-valued Gaussian noise with zero-mean and variance
σ2
w, i.e. [W]m,n ∼ CN 0, σ2
w.
Further, aτ(·)is the delay signature function whose value
for the input delay of τis
aτ(τ) = exp −2π[f1, . . . , fM]Tτ,(2)
where fm=f(c)+(m−M
2)∆f,∆fis the subcarrier spacing.
aθ(·)denotes the real array manifold which is actually a
function of the continuous DOA. Its value at a specific DOA
of θis also known as the steering vector which, according to
the direction-dependent array error model, is represented by
aθ(θ) = a0
θ(θ)ζ(θ),(3)
in which ζ(·) : U→CN×1denotes the array modeling error
function, where U⊆Ris the interested range of DOA. The
n-th element of ζ(θ)is [ζ(θ)]n=gn(θ)·exp (ϕn(θ)), where
gn(θ)and ϕn(θ)represent the array gain and phase errors
suffered by the n-th element for signals from the direction of
θ.a0
θ(·)is the ideal array manifold and for a linear array, the
value of a0
θ(θ)is given by
a0
θ(θ) = exp 2π
λ[d1, . . . , dN]Tsin θ,(4)
where dnis the position for the n-th antenna element and λ
is the wavelength1.
Array calibration amounts to the process of estimating the
array error function ζ(·)and deriving the actual manifold
aθ(·). Since DOA is essentially obtained from the phase shifts
between antenna elements for a far-field signal model, this
paper only considers the phase errors. Then the crucial issue
of array calibration is reduced to the estimation of phase error
functions ϕn(θ), n = 1, . . . , N based on measured CFRs,
which will be studied in Section III.
III. ARRAY MANIFOLD ESTIMATION
This section presents the proposed in-situ array calibration
algorithm, i.e. algorithm for array manifold estimation shown
1Note that the received CFR signal model of (1) has separated signature
functions for delay and angular domains. This indicates that the far-field
model is adopted throughout this paper. For wireless positioning with 5G
small-cell RRUs operated in the sub-6GHz frequency band, this assumption
is reasonable, as the corresponding Fraunhofer distance is usually below 1 m
[42].
This article has been accepted for publication in IEEE Transactions on Microwave Theory and Techniques. 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/TMTT.2023.3256532
© 2023 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 MICROWAVE THEORY AND TECHNIQUES, VOL. XX, NO. X, XXX 2023
in the framework of Fig. 1. To give an insight, we derive the
CFR model of each receiving channel from equation (1) as
follows:
hn=Aτ(τ)ξn+wn, n = 1, . . . , N, (5)
where hnand wnare the n-th columns of Hand W
respectively; τ= [˜τ1,...,˜τL]T,ξn= [ξn1, . . . , ξnL ]T, and
Aτ(τ) = [aτ(˜τ1),...,aτ(˜τL)] are the collections of path
delays, path gains, and delay signature vectors of all paths
respectively, in which ξnl = ˜γl·haθ(˜
θl)indenotes the complex
gain of the l-th path observed by the n-th antenna.
In the process of in-situ calibration, hnand aτ(·)are known
while ξnand τare unknown parameters to be solved. The
problem of joint estimating the path gain ξnand path delay
(TOA) τis an inverse problem, i.e. the model parameters ξn
and τthat produce the observations hnneed to be determined
[43]. Further, since the complex gain induced by the wireless
propagation (namely ˜γl) is the same for all antenna elements,
it can be easily canceled by retrieving the phase differences
between ξn, n = 1, . . . , N . Then the remaining term of ξnl is
the array response haθ(˜
θl)in. Therefore, this inverse problem
is also denoted as the joint array response and TOA estimation
problem. Based on this idea, the detailed calibration procedure
is designed as shown in Fig. 2.
LOS path
identification
Compensation for
phase rotation caused
by propagation
Phase
difference
calculation
Refinement
by combining
multiple
symbols
Array phase
error function
estimation
Array
manifold
calibration
MLE problem
formulation
EM algorithm
for MLE
CFR
matrix
Path number
estimation
Array
geometry
Array
geometry
Fig. 2. Flowchart for the proposed array manifold estimation algorithm.
First, to resolve and segregate the multipath components,
an MLE problem is formulated. It is a multiple measurement
vector problem with N“spatial snapshots” of the wireless
channel and has 2LN +Lreal unknown parameters, which
can be gathered in a vector Θ= [τT,ξT
1,...,ξT
N]T. Denoting
the joint probability density of the CFR measurements as
f(H;Θ), which is also the likelihood function of parameters
Θ, then according to the signal model represented by equation
(5) and the assumed i.i.d. Gaussian distribution of the noise
components, the likelihood function is
f(H;Θ) = 1
(πσ2
w)MN exp
−
N
P
n=1 khn−A(τ)ξnk2
σ2
w
.
(6)
Taking the negative logarithm of this likelihood function and
ignoring the terms that do not depend on any element in Θ,
the MLE problem for the joint estimation of parameters Θis
equivalent to
ˆ
Θ= min
Θg(H;Θ),(7)
g(H;Θ) =
N
X
n=1 khn−Aτ(τ)ξnk2.(8)
The objective function g(H;Θ)is highly non-linear and no
closed-form solution exists. Also, brute-force searching in this
R2LN+Lspace is computationally intensive. Therefore, the
idea of EM is employed here to seek the solution iteratively.
Before the presentation of the EM approach for this prob-
lem, two issues need to be clarified:
1) As indicated by Fig. 2, at each CPP, the CFR can be
sensed multiple times by consecutive UL-SRS symbols
to improve the calibration accuracy. Here the symbol
index, the number of total symbols, and each CFR mea-
surement are denoted as q,Q, and H(kq), respectively.
However, the following EM derivation drops the suffices
of qand kfor notational simplicity, which introduces no
ambiguity since it is applied to each H(kq)individually.
2) During the following derivation, we assume that the
model order Lis known. However, it is determined
by the number of multipath components and is usually
unavailable in real scenarios. That is why a path number
estimation module is prepended to the estimation pro-
cedure as shown in Fig. 2, which can be implemented
based on the information-theoretic criteria [44] or the
sequential hypothesis-testing [45].
Based on the notation of the EM algorithm [46], the
observed CFRs hn, n = 1, . . . , N are named as the incom-
plete data since they are the amalgamation of Lmultipath
components. Then it is intuitive to choose the observations of
each segregated path component as the complete data, which
are in the form of
ynl =aτ(τl)ξnl +znl, l = 1, . . . , L, n = 1, . . . , N , (9)
where znl denotes the noise components in the complete
data, whose entries are i.i.d. with CN 0, βlσ2
w, in which
PL
l=1 βl= 1 and we choose βl=1
Lfor simplicity.
The EM algorithm iteratively decomposes the observed
incomplete signal hninto these segregated complete signals
ynl and applies the MLE to obtain estimates of Θfrom ynl.
These two steps are performed sequentially and iteratively
based on the last estimates and are respectively referred to
as the Expectation Step and Maximization Step. Denoting the
estimates at the p-th iteration as ˆ
Θ(p), then the (p+ 1)-th
iteration is carried out as follows.
Expectation Step:
ˆ
y(p+1)
nl =aτ(ˆτ(p)
l)ˆ
ξ(p)
nl +1
Lhhn−Aτ(ˆ
τ(p))ˆ
ξ(p)
ni,
l= 1, . . . , L, n = 1, . . . , N. (10)
This article has been accepted for publication in IEEE Transactions on Microwave Theory and Techniques. 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/TMTT.2023.3256532
© 2023 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.: IN-SITU CALIBRATION OF ANTENNA ARRAYS FOR POSITIONING WITH 5G NETWORKS 5
Maximization Step:
ˆτ(p+1)
l= arg max
τl(N
X
n=1 aH
τ(τl)ˆ
y(p+1)
nl
2), l = 1, . . . , L,
(11)
ˆ
ξ(p+1)
nl =1
MaH
τˆτ(p+1)
lˆ
y(p+1)
nl , l = 1, . . . , L, n = 1, . . . , N.
(12)
As an iterative algorithm, the initial value of Θand the
stopping criteria have to be determined. Specifically, to reduce
iteration numbers, the successive interference cancellation
approach is adopted for initialization [47]. Its main idea is
to estimate the parameters of Lpaths successively and when
estimating those of path l, the interference caused by the
previously estimated paths is calculated and subtracted from
the CFR H. Each time when the interference caused by first
l−1paths is canceled, the initial value for the parameters of
path lis determined by correlating the interference-canceled
CFR Ywith the delay signature function aτ(τ)and searching
the peaks, as illustrated by the pseudo-code in Algorithm 1. To
determine whether the EM iteration converges, the difference
between consecutive estimates of Θis calculated. When
ˆ
τ(p)−ˆ
τ(p−1)
< δτ and
ˆ
ξ(p)
n−ˆ
ξ(p−1)
n
ˆ
ξ(p−1)
n
< , n = 1, . . . , N
are met, the iteration stops, where δτ is the search grid size
for path delay and is a pre-defined threshold.
Algorithm 1 Successive interference cancellation for initial-
ization of EM algorithm
Input: CFR matrix H∈CM×Nand path number L.
1: Y←H;
2: for l=1:Ldo
3: if l > 1then
4: b= [ˆ
ξ1(l−1),..., ˆ
ξN(l−1)]T;
5: Y←Y−aτ(ˆτ(0)
l−1)·bT;
6: end if
7: ˆτ(0)
l= arg maxτ
aH
τ(τ)·YT
;
8: hˆ
ξ(0)
1l,...,ˆ
ξ(0)
Nl i=1
MaH
τ(ˆτ(0)
l)·Y;
9: end for
Output: ˆ
Θ(0).
As indicated by equations (10) to (12), the EM algorithm
simplifies the multi-dimensional maximum-likelihood search
into iterative one-dimensional searches. Assuming Jsearching
grids in the delay domain, then the computational complexities
for the Expectation Step and the Maximization Step in each
EM iteration are O(MNL2)and O(J M NL) + O(MNL),
respectively. Since there are only 6-8significant reflection
paths in normal indoor environments [48], [49] and the RF
channel numbers of small-cell base stations are usually no
more than 8(e.g. 2or 4for picocell RRUs) [6], Land
Nare small. Although wideband 5G signals occupy a large
number of subcarriers, the CIR (inverse Fourier transform of
the CFR) can be gated [37] to reduce the effective subcarrier
number based on the fact that 5G small-cell base stations have
restricted power coverage. According to the analysis in our
prior work (Section V-A of [37]), for a coverage of 100 m, the
subcarrier number Mcan be lowered to 64 after CIR gating.
Therefore, it can be seen that N≈LMJand the
computational complexity of the EM iteration is dominated
by the delay searching in the Maximization Step, which is
O(JMNL). Furthermore, the searching grid number Jcan
also be substantially reduced by employing a coarse-to-fine
searching strategy.
The EM iteration has been proven to be monotonically
decreasing and has a fast convergence rate [50]. Specifically,
the EM solution for the MLE problem of (7)-(8) generally
converges within 10 iterations, according to our evaluations in
typical multipath environments.
Denoting the estimates of path delays and gains of the q-th
CFR measurement at the k-th pilot position as ˜τ(kq)
land ˜
ξ(kq)
nl ,
respectively, then the shortest path with its gain larger than a
threshold is picked as the LOS path. Here the paths with too
small gains are filtered out by this threshold to guard against
false local minima detected by the EM algorithm.
After that, as illustrated by Fig. 2, fixed phase rotations
caused by path differences among antenna elements are sub-
tracted from the path gain of the LOS path ˜
ξ(kq)
n,LOS. This fixed
phase rotation is determined by the array geometry and the
known LOS DOA θkof the calibration signal emitted from
the k-th CPP, and after compensated, the path gain at the n-th
antenna element is ξ(kq)
n=˜
ξ(kq)
n,LOS ·[a0
θ(θk)]∗
n.
Next, the phase of ξ(kq)
nis extracted as φ(kq)
n=∠ξ(kq)
n,
which represents the antenna phase error measurement at the
n-th antenna for the direction of θk. Taking into account that
there exist sample timing offset, carrier frequency offset, and
carrier phase offset in typical RF front-ends of commercial
wireless communication equipment, the initial phase of UL-
SRS varies across symbols [51]. Therefore, the differences
between φ(kq)
nand φ(kq)
1are further calculated to derive a
coherent measurement sequence over Qconsecutive symbols.
These phase differences are denoted as ∆φ(kq)
n(obviously
∆φ(kq)
1= 0) and they are combined to reduce the phase
fluctuation caused by noise. For example, an outlier removal
algorithm can be applied to this measurement sequence first,
followed by retrieving the average value of the filtering results.
Up to this point, array phase error measurements at the
discrete angular set {θk}K
k=1 have been obtained (∆¯
φ(k)
n).
Since DOA is a continuous variable, not necessarily an element
in this angular set, the phase error function ϕn(·)which can
output the phase error value at any DOA needs to be inferred.
For this purpose, one can employ a parametric regression
method, such as the polynomial curve fitting, to derive parame-
ters of ϕn(θ)directly, or a non-parametric regression method,
such as the kernel regression or the interpolation, to obtain
function values of ϕn(θ)at pre-defined dense searching grids.
Lastly, based on the estimated phase error function
ˆϕn(θ), n = 1, . . . , N , the array calibration process is per-
formed by compensating the ideal array manifold a0
θ(θ)with
the estimated array modeling error function ˆ
ζ(θ)as follows
ˆ
aθ(θ) = a0
θ(θ)ˆ
ζ(θ).(13)
Since the calibrated array manifold ˆ
aθ(θ)captures different
types of array errors and delineates the true array response for
This article has been accepted for publication in IEEE Transactions on Microwave Theory and Techniques. 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/TMTT.2023.3256532
© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
6 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. XX, NO. X, XXX 2023
signals from any direction, DOA estimation using this matched
array manifold achieves better performance than using the
ideal but mismatched one (a0
θ(θ)).
A wealth of searching-based DOA estimation algorithms,
such as the conventional beamformer, the Capon method, the
multiple signal classification method, the maximum likelihood
estimators, and the compressive sensing-based methods, can
directly employ this calibrated array manifold. They share a
general DOA estimation procedure as follows:
1) Compute the calibrated array steering vectors on the
predefined searching grids {θu}U
u=1, which are actually
the values of calibrated array manifold ˆ
aθ(θ)at the
DOAs of {θu}U
u=1.
2) Compute the spatial spectrum P(θu), u = 1, . . . , U at
these discrete grids using the calibrated steering vector
set {ˆ
aθ(θu)}U
u=1.
3) Search the dominant spectral peaks and find the DOAs.
Furthermore, in wireless positioning applications, to im-
prove the degrees-of-freedom and resolution ability, JADE
methods are usually employed to estimate the DOA and TOA
simultaneously rather than separately [52]. Similar to the idea
of DOA estimation presented above, they can also use the
calibrated array manifold for spatial processing to counteract
array errors. One can refer to our prior works [37] and [53]
for detailed discussions about how to use the calibrated array
manifold in different JADE methods.
IV. NUMERICAL SIMULATIONS
A. Simulation Setup
In this section, the effectiveness of the proposed in-situ
calibration framework is demonstrated with simulated 5G
wireless channel data. The system and waveform parameters
used in simulations are shown in TABLE I.
TABLE I
CON FIGU RATI ONS F OR 5G SYSTEM AND UL-SRS
Parameter Value
Type of gNB Picocell gNB
Number of antenna elements in an RRU 4
Array type ULA
Carrier frequency 4.85 GHz
Subcarrier spacing 30 kHz
Number of subcarriers 3264
UL-SRS pattern Comb-two [54]
UL-SRS transmission bandwidth 100 MHz
Sampling frequency 122.88 MHz
UL-SRS temporal interval 80 ms
To test in-situ antenna array calibration methods in typical
multipath environments, simulations are conducted with realis-
tic wireless channel data generated by the QuaDRiGa channel
simulator [55]. The indoor factory LOS channel at sub-6GHz
working frequency (3GPP 38.901 InF LOS) whose parame-
ters conform to [56] is chosen for QuaDRiGa throughout the
experiments. The antenna element pattern is configured ac-
cording to the default antenna modeling parameters defined in
this same 3GPP report [56]. Specifically, its 3 dB beamwidths
in both azimuth and elevation directions are set to 65◦and the
directional antenna gain is set to 8 dBi, as illustrated by its 3-D
radiation pattern in Fig. 3. To simulate the direction-dependent
array errors, we modulate each multipath component generated
by the QuaDRiGa simulator with an additional phase offset,
whose value is determined by its DOA and a look-up-table
with antenna phase measurements of a realistic four-element
antenna array equipped by a 5G RRU. During simulations,
the transmit power of the UT 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 Pris derived by the
QuaDRiGa simulator according to the propagation model and
the noise power is calculated as Pn=kBT0B, in which kB,
T0, and Brepresent the Boltzmann’s constant, standard noise
temperature, and measurement bandwidth, respectively. The
noise component on each subcarrier of each receiving channel
is generated independently according to the complex Gaussian
distribution of CN(0, Pn).
-5 0 5 10 15 20
X coordinate (m)
-10
-5
0
5
10
15
Y coordinate (m)
RRU center
RRU antennas
Calibration pilot positions
120º
-0.05
0
0.05
Azimuth
Elevation
(dBi)
Fig. 3. Visualization of simulated scenario for in-situ calibration.
The performance of the proposed EM-based array manifold
estimation method is compared against: (i) the widely adopted
PE-based approach [29]–[32], and (ii) the direct measuring
approach [35], [37]. The latter approximates the array phase
response by the measured phases of the multi-channel CFR
at the center frequency. Since a clean one-path wireless
environment is assumed for this method, it is commonly used
in chamber calibration. These three array manifold estimation
methods are all applied in the proposed in-situ calibration
framework as presented in Section II-A. Their performance
comparisons are presented in this section according to the
metrics of (i) the accuracy of the estimated array manifold
and (ii) the DOA estimation error of the calibrated array.
Moreover, to evaluate the in-situ calibration performance in
different multipath environments, the Ricean K-factors for the
simulated wireless channels are configured to vary from 0 dB
to 7 dB during simulations.
B. Evaluating Accuracy of Estimated Array Manifold
We first evaluate the accuracy of the estimated array mani-
fold by comparing it to the true manifold. We place the UT at
25 evenly distributed pilot positions on an arc centered at the
RRU, as indicated by cross marks in Fig. 3. Their distances to
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content may change prior to final publication. Citation information: DOI 10.1109/TMTT.2023.3256532
© 2023 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.: IN-SITU CALIBRATION OF ANTENNA ARRAYS FOR POSITIONING WITH 5G NETWORKS 7
the RRU are 10 m and they cover the sector of 120 degrees
(−60◦to +60◦) with an angular separation of 5degrees. That
is θk=−60 + 5(k−1), k = 1, . . . , K, where the total CPP
number Kis 25. Then with the Ricean K-factor and CPP
fixed, 500 UL-SRS symbols are simulated by the QuaDRiGa
channel simulator.
Channel responses sensed by the RRU for these 25 pilot
positions are used for array manifold estimation. The angle
between the estimated manifold ˆ
aθ(θ)and the true manifold
aθ(θ)at a specific direction θ, which is essentially the angle
between two vectors, represents the manifold mismatch and is
used as the metric for evaluating the accuracy of the estimated
array manifold. It is calculated as
α(θ) = arccos "ˆ
aH
θ(θ)aθ(θ)
kˆ
aθ(θ)k·kaθ(θ)k#.(14)
Two sets of experiments are conducted to investigate the
impacts of the multipath condition and the number of accu-
mulated UL-SRS symbols on the array manifold estimation
performance, respectively. Their results are shown in Fig. 4.
First, to assess the estimation performance under the ex-
treme single snapshot scenario, the array manifold is estimated
based on a single CFR measurement. We use the standard box
plot (a.k.a. the box-and-whisker plot) to visualize the statistics
of α(θ)as it is more informative when used for error analysis
than single-metric evaluations such as the N-th percentile of
the error set or the root mean square error [57]. The resulting
box plots of α(θk), k = 1, . . . , K at each Ricean K-factor are
shown in Fig. 4(a) for the proposed and benchmark methods.
As stated above, each box plot demonstrates the statistics of
α(θ)from 25 ×500 = 12500 realizations. As exemplified by
Fig. 4(a), the minimum (Q0or 0-th percentile), first quartile
(Q1or 25-th percentile), median (Q2or 50-th percentile), third
quartile (Q3or 75-th percentile), and maximum (Q4or 100-th
percentile) of the α(θ)set are illustrated in the corresponding
box plot, as respectively represented by the lower limit of the
lower whisker, the lower edge of the box, the middle line of
the box, the upper edge of the box, and the upper limit of the
upper whisker.
Fig. 4(a) shows that the direct measuring approach performs
poorly in the presence of multipaths. It also clearly indicates
the superiority of the proposed EM-based manifold estimation
method over the PE-based approach, especially in dense mul-
tipath environments. For example, when the Ricean K-factor
is 0 dB, which denotes a severe multipath condition with the
averaged signal power from scattered paths equals to the LOS
signal power, a reduction of 69% of the median of α(θ)is
achieved by the EM-based approach (from 5.60◦to 1.71◦);
While when the Ricean K-factor is 7 dB, the performance
improvement is only 39% (median error reduces from 0.89◦
to 0.54◦). This implies that, limited by the spatial resolution
of the small-scale antenna array, conventional spatial-domain-
only in-situ calibration methods exhibit unsatisfactory perfor-
mance in the presence of multipath reflections. In contrast,
the proposed approach resolves multipaths via delay-domain
super-resolution, and the large bandwidth of 5G signals guar-
antees an accurate estimation of the real antenna manifold even
in a multipath-rich environment.
Then we examine the performance improvement for these
array manifold estimation algorithms when multiple UL-SRS
symbols are used. To fully utilize these multiple measure-
ments, the outlier rejection algorithm based on the Hampel
identifier [58] is applied to the corresponding phase estimates
and the filtering results are averaged to derive the final array
manifold estimate at this CPP. In this experiment, the Ricean
K-factor of the wireless channel is fixed to 3 dB and the
number of accumulated symbols varies from 1to 256. Fig. 4(b)
shows the box plots of manifold mismatches for the three array
manifold estimation methods, with each box demonstrating the
statistic of α(θk)at all 25 CPPs.
We readily observe from Fig. 4(b) that all methods benefit
from multiple measurements and the EM-based approach is
superior to both the benchmark algorithms in all these multi-
snapshot scenarios. It also shows that, even when there are
only 4symbols, the median estimation error is below 0.9◦for
the proposed EM-based approach. This implies that the time
and effort required for measuring during in-situ calibration
can be saved by reducing the dwell time at each CPP, or
more conveniently, a UT travels across the angular coverage
of the RRU can be used to provide continuous measurements
during which the samplings of the UT trajectory at the UL-
SRS transmitting instants form the CPPs.
Further, Fig. 5 demystifies the underlying antenna phase
error estimates when the Ricean K-factors are 0 dB,3 dB, and
7 dB, respectively. Here, we only show the estimates obtained
by the PE-based and the proposed EM-based approaches as
the direct measuring method exhibits much higher estimation
variance, which is obvious from Fig. 4. The mean and variance
of array phase error estimates from 500 CFR measurements
at each CPP are shown in Fig. 5. It can be observed from
Fig. 5 that the proposed EM-based array manifold estimator
outperforms that based on the PE in terms of stability and
accuracy. The increased 1-σbounds for both methods in large
incident directions, as shown in Fig. 5, are attributed to the
decreased receiving power of the directive antenna elements
for signals impinged from these directions.
C. Evaluating DOA Estimation Error of Calibrated Array
We then demonstrate the performance improvements for
DOA estimation of different in-situ calibration techniques. To
this end, DOA estimation is performed on the simulated 5G
channel data with the estimated array manifolds. We conduct
1000 Monte Carlo trials under each multipath condition and
in each trial, the path DOA, TOA, and the wireless chan-
nel response are generated randomly. Specifically, the DOA
and TOA of the LOS path conform to U(−60◦,+60◦]and
U(0,333.33 ns], respectively, which means the UT locates in
a sector with a central angle of 120◦and a radius of 100 m
centered at the RRU.
While Section IV-B only investigates the array manifold
estimates at the discrete CPPs, a continuous or a finer array
manifold is needed for DOA estimation. Therefore, following
the flowchart of Fig. 2, we smooth the discrete phase error es-
timates with the local weighted regression [59] and interpolate
the regression results with the Akima spline [60] to derive a
This article has been accepted for publication in IEEE Transactions on Microwave Theory and Techniques. 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/TMTT.2023.3256532
© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
8 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. XX, NO. X, XXX 2023
0 1 2 3 4 5 6 7
10-1
100
101
102
0.540.89
5.60 1.71
Q0
Q4
Q3
Q1
Q2
(a) Manifold mismatches in different multipath conditions when a
single UL-SRS symbol is used.
1 2 4 8 16 32 64 128 256
10-1
100
101
102
(b) Manifold mismatches when multiple UL-SRS symbols are used
(Ricean K-factor fixed to 3 dB).
Fig. 4. Demonstration of manifold mismatches of the proposed EM-based array manifold estimation algorithm and benchmark algorithms under different
multipath conditions and different numbers of accumulated UL-SRS symbols.
-60 -40 -20 0 20 40 60
Incident direction (degree)
-40
-30
-20
-10
0
10
20
30
40
Phase (degree)
(a) Ricean K-factor: 0 dB (PE approach).
-60 -40 -20 0 20 40 60
Incident direction (degree)
-40
-30
-20
-10
0
10
20
30
40
Phase (degree)
(b) Ricean K-factor: 3 dB (PE approach).
-60 -40 -20 0 20 40 60
Incident direction (degree)
-40
-30
-20
-10
0
10
20
30
40
Phase (degree)
(c) Ricean K-factor: 7 dB (PE approach).
-60 -40 -20 0 20 40 60
Incident direction (degree)
-40
-30
-20
-10
0
10
20
30
40
Phase (degree)
(d) Ricean K-factor: 0 dB (EM approach).
-60 -40 -20 0 20 40 60
Incident direction (degree)
-40
-30
-20
-10
0
10
20
30
40
Phase (degree)
(e) Ricean K-factor: 3 dB (EM approach).
-60 -40 -20 0 20 40 60
Incident direction (degree)
-40
-30
-20
-10
0
10
20
30
40
Phase (degree)
(f) Ricean K-factor: 7 dB (EM approach).
-60 -40 -20 0 20 40 60
Incident direction (degree)
-40
-30
-20
-10
0
10
20
30
40
Phase (degree)
Fig. 5. Demonstration of antenna phase errors at the directions of CPPs estimated by the PE-based approach ((a)-(c)) and the proposed EM-based approach
((d)-(f)).
This article has been accepted for publication in IEEE Transactions on Microwave Theory and Techniques. 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/TMTT.2023.3256532
© 2023 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.: IN-SITU CALIBRATION OF ANTENNA ARRAYS FOR POSITIONING WITH 5G NETWORKS 9
continuous phase error curve, which is then utilized to obtain
the continuous manifold. Besides, to handle the multipath
effects during DOA estimation, we employ the JADE method
proposed in our prior work [37] for parameter estimation. It
composes of an iterative-adaptive-approach-based delay spec-
trum estimator and a conventional beamformer. In addition, the
DOA estimates with non-calibrated and perfectly calibrated
manifolds are also investigated for comparison.
Similar to the methodology adopted in Section IV-B, the
DOA estimation errors with different estimated array mani-
folds are compared by varying the multipath condition and the
number of accumulated UL-SRS symbols. The corresponding
results are summarized in Fig. 6(a) and Fig. 6(b), respectively.
First, Fig. 6(a) shows the 80-th percentiles of DOA estima-
tion errors when the Ricean K-factor varies from 0 dB to 7 dB.
The count of UL-SRS symbols accumulated for array manifold
estimation is fixed to 8throughout this experiment. To sim-
ulate the scenario that in-situ calibration and the subsequent
positioning process are carried out in the same environment,
the array manifolds used for DOA estimation are estimated
under the same Ricean K-factor.
Fig. 6(a) illustrates that, in terms of DOA estimation error,
the performance of the proposed EM-based in-situ calibration
method approximates that of the perfect calibration when the
Ricean K-factor is no less than 0 dB, while a similar effect
is achieved by the traditional PE-based method only when the
Ricean K-factor is above 5 dB. Also according to the results,
calibrated with the directly in-situ measured manifold facili-
tates DOA estimation only when the measurement process is
conducted in a clear wireless environment (when the Ricean
K-factor is above 6 dB as shown in Fig. 6(a)).
Next in Fig. 6(b), we fix the Ricean K-factor to 3 dB
and demonstrate the DOA estimation errors when using ar-
ray manifolds estimated with different numbers of UL-SRS
symbols. Results shown in Fig. 6(b) confirm the superiority
of the proposed EM-based method over both the benchmark
algorithms when multiple snapshots exist. We can also observe
that it achieves nearly identical performance as the perfect
calibration when at least 16 CFR measurements are utilized
for manifold estimation. By contrast, although the estimation
errors of both the benchmark in-situ calibration methods
decrease as the snapshot number increases, they can hardly
approach that of perfect calibration.
V. INDOOR FI EL D TES TS
A. Experimental Setup
To further verify the effectiveness of the proposed in-situ
calibration method and demonstrate its performance improve-
ment for DOA estimation, field tests are conducted in an
underground parking lot of an office building2. Fig. 7 depicts
the experimental environment and hardware setups. During
experiments, we only use a single RRU and a single UT, whose
working parameters also conform to TABLE I. As shown in
Fig. 7, this area has a lot of metallic plumbing pipes, poles, and
thick pillars, which cause harsh multipath effects for wireless
signals.
2The field test data used in performance evaluations is available in [61].
Main devices used in experiments have been illustrated in
Fig. 7, and they are summarized as follows:
1) UT: The 5G UT with a single omnidirectional cylindrical
antenna is mounted on an autonomous vehicle, which
is also equipped with various active sensors, includ-
ing an inertia measurement unit and multiple lidars,
cameras, and ultrasonic distance sensors. Measurements
from these active sensors are fused via a simultaneous
localization and mapping algorithm to generate ground-
truth locations with an accuracy of several centimeters.
2) RRU: We use a commercial four-channel picocell RRU
in field tests, which incorporates the RF and IF process-
ing modules shown in Fig. 1. Specifically, RF signals are
first filtered, amplified, down-converted, and sampled to
digital IF signals. Then digital down converters are fol-
lowed to generate BB in-phase and quadrature signals.
The original antennas equipped by this RRU are dis-
persed at four corners with element spacing much larger
than half-wavelength, preventing it from supporting the
DOA estimation function. Therefore, we designed and
fabricated a six-element ULA to replace the existing
RRU antennas. Its middle four antennas connect to
the RRU RF channels accordingly, while those at both
sides are dummy elements, which guarantee the same
boundaries seen by the central four elements of the array
[62]. The element spacing is 3 cm. Fig. 8(a) presents the
structure of this antenna array. The simulated 3-D and 2-
D radiation patterns of the antenna element are shown in
Fig. 8(b) and Fig. 8(c), respectively. The corresponding
specifications are also listed in TABLE II.
3) BBU: A commercial 5G BBU is employed, whose
physical layer modules are fully compatible with the
3GPP standard. As shown in Fig. 1, during in-situ
calibration, it processes BB UL-SRS signals and outputs
CFR measurements. It also needs to mention that the
BBU is placed in the equipment room rather than in the
experimental field and is connected to the RRU via a
long optical fiber.
TABLE II
SPE CIFI CATI ONS O F AN TEN NA EL EM ENT S OF TH E DE SIG NE D ANT EN NA
AR RAY
Parameter Value
Type Microstrip antenna
Working frequency range 4.80-4.90 GHz
Polarization Vertical polarization
Gain 5.20 dBi at 4.85 GHz
Efficiency 93% at 4.85 GHz
H-plane HPBW 122◦at 4.85 GHz
E-plane HPBW 72◦at 4.85 GHz
VSWR less than 1.5in 4.80-4.90 GHz
B. Experiment Results
In the field test, the RRU is fixed in position
(−34.4 m,8.5 m) and the UT moves in a rectangular area
of nearly 1125 m2([−35 m,10 m] ×[8 m,33 m]) and stops
at 476 coordinates in this area with an interval of about
This article has been accepted for publication in IEEE Transactions on Microwave Theory and Techniques. 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/TMTT.2023.3256532
© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
10 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. XX, NO. X, XXX 2023
01234567
Ricean K-Factor (dB)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
DOA estimation error (degree)
Non-calibrated
Calibrated with direct measuring
Calibrated with PE approach
Calibrated with EM approach
Perfect calibration
(a) 80-th percentiles of DOA estimation errors under different multi-
path conditions (Number of UL-SRS symbols used in array manifold
estimation is 8).
1 2 4 8 16 32 64 128 256
Number of accumulated symbols for manifold estimation
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
DOA estimation error (degree)
Non-calibrated
Calibrated with direct measuring
Calibrated with PE approach
Calibrated with EM approach
Perfect calibration
(b) 80-th percentiles of DOA estimation errors using manifolds esti-
mated with different numbers of UL-SRS symbols (Ricean K-factor
fixed to 3 dB).
Fig. 6. DOA estimation errors when using array manifolds estimated by the proposed EM-based algorithm and benchmark algorithms under different multipath
conditions and when different numbers of UL-SRS symbols are accumulated for manifold estimation.
6-element ULA with
2 dummy elements
4 valid antennas
4-channel picocell
RRU
Dummy elements
Lidars
Ultrasonic
sensor
Camera
Autonomous vehicle
with 5G UT
5G
antenna
BBU
UL-SRS
Digital BB signal
transmission via
optical fiber
Fig. 7. Experimental environment and hardware setups for field test in an underground parking lot.
1.5-2.5 m, as depicted in Fig. 9. Among them, 48 LOS
positions around the RRU are chosen as the CPPs and channel
measurements at those positions are used for array manifold
estimation. Also as shown in Fig. 9, ten evenly spaced pillars
exist in this rectangular area, which give rise to 95 NLOS
positions. Therefore, CFRs collected at the remaining 333
normal LOS positions are used for DOA estimation perfor-
mance evaluations. At each position, the UT sends 100 UL-
SRS symbols.
Fig. 10 demonstrates the phase error estimates obtained
by the proposed EM-based method at these 48 CPPs. Only
10 UL-SRS symbols are used at each CPP for phase error
estimation. To derive the continuous phase error function
for a specific antenna element, the same approach stated in
Section IV-C is also used here, which encompasses the local
weighted regression [59] and the Akima spline interpolation
[60]. The deviation of the in-field array manifold from the
nominal manifold measured in an anechoic chamber is clearly
illustrated.
The ideal array manifold is calibrated by the chamber-
measured and in-situ estimated phase error functions shown in
Fig. 10 to obtain the nominal and estimated array manifolds.
Then the ideal, nominal, and estimated array manifolds are
used to estimate the DOAs of signals transmitted from the
333 normal LOS positions (The total number of samples
is 333 ×100 = 33300). Their corresponding results are
respectively denoted as non-calibration, chamber calibration,
and in-situ calibration results. Similar to simulations presented
in Section IV-C, the JADE method proposed in [37] is also
adopted here for DOA estimation. The DOA estimation error
datasets for these three approaches are derived by comparing
their estimation results to the true DOAs and the resulting
This article has been accepted for publication in IEEE Transactions on Microwave Theory and Techniques. 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/TMTT.2023.3256532
© 2023 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.: IN-SITU CALIBRATION OF ANTENNA ARRAYS FOR POSITIONING WITH 5G NETWORKS 11
Dummy
element Radiator Reflector Isolation
plate
Dummy
element
(a) Layout and photograph of the designed antenna array.
Azimuth 0
Elevation 90
Azimuth 0
Elevation 0
Azimuth 90
Elevation 0
-30
-25
-20
-15
-10
-5
0
5
10
Gain (dBi)
(b) 3-D radiation pattern of the antenna element.
-150
-120
-90
-60
-30
0
30
60
90
120
150
180
-30
-20
-10
0
10
(dBi)
H-plane (Azimuth)
E-plane (Elevation)
(c) 2-D radiation patterns of the antenna
element in H-plane and E-plane.
Fig. 8. Layout, photograph, and radiation patterns of the antenna array.
-40 -35 -30 -25 -20 -15 -10 -5 0 5 10
X coordinate (m)
5
10
15
20
25
30
35
40
Y coordinate (m)
RRU LOS (normal) NLOS
LOS (CPP) Pillar
Fig. 9. Experiment layout and visualizations for data-collecting positions.
error statistics are depicted and compared in Fig. 11.
First, Fig. 11(a) presents the empirical CDF curves for
these three DOA estimation error datasets. It shows that, for
this antenna array, since its real manifold after installation
severely deviates from the nominal manifold, calibrating it
using chamber measurements even slightly deteriorates the
DOA estimation performance. On the contrary, the in-situ
estimated array manifold captures the real array responses
more precisely by utilizing the post-established in-field mea-
surements. Specifically, as illustrated by empirical CDF curves
in Fig. 11(a), through in-situ calibration, a reduction of 46.7%
(from 3.0◦to 1.6◦) for the 68-th percentile (1-σ) error and a
reduction of 23.1% (from 5.2◦to 4.0◦) for the 90-th percentile
error are achieved.
Then, to reveal more details of the error statistics, we divide
the data into five adjacent sets according to their true DOAs
and present the box plots of DOA estimation errors at each set
in Fig. 11(b). It shows that, the non-calibration and chamber
calibration results are obviously biased, while the in-situ
calibration errors are all nearly centered at zero. This clearly
demonstrates that, array errors offset the DOA estimates, and
by in-situ calibration, these offsets are corrected.
-60 -45 -30 -15 0 15 30
Incident direction (degree)
-30
-20
-10
0
10
20
30
40
Phase (degree)
(a) Phase error at antenna 2.
-60 -45 -30 -15 0 15 30
Incident direction (degree)
-30
-20
-10
0
10
20
30
40
Phase (degree)
(b) Phase error at antenna 3.
-60 -45 -30 -15 0 15 30
Incident direction (degree)
-30
-20
-10
0
10
20
30
40
Phase (degree)
(c) Phase error at antenna 4.
-60 -45 -30 -15 0 15 30
Incident direction (degree)
-30
-20
-10
0
10
20
30
40
Phase (degree)
Chamber measurement
Estimate at CPP
Estimated function
Fig. 10. Demonstration of array phase errors estimated by the proposed EM-
based approach at CPPs and the estimated phase error functions ϕn(θ), n =
2,3,4. Since ϕ1(θ)≡0, (a), (b), and (c) respectively show the corresponding
estimates for the antenna elements 2,3, and 4.
VI. CONCLUSION
An in-situ calibration framework and an array manifold
estimation algorithm have been proposed in this work to
support high-accuracy 5G positioning. This framework re-
duces calibration costs by using off-the-shelf 5G devices
and obviating extra hardware and protocol modifications, and
improves calibration accuracy by capturing all kinds of in-
field array errors, including those induced after installations,
in a direction-dependent array error function. The proposed
estimation algorithm fully exploits the bandwidth resources
provided by 5G signals and the super-resolution ability of the
EM algorithm to resolve the multipaths in the delay domain,
whose calibration accuracy is demonstrated to be superior to
This article has been accepted for publication in IEEE Transactions on Microwave Theory and Techniques. 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/TMTT.2023.3256532
© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
12 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. XX, NO. X, XXX 2023
012345678910
DOA estimation error (degree)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Cumulative probability
Non-calibration
Chamber calibration
In-situ calibration
(a) Empirical CDF curves for DOA estimation errors.
[-65, -40) [-40, -20) [-20, 0) [0, 20) [20, 40)
Range of incident directions (degree)
-10
-8
-6
-4
-2
0
2
4
6
8
10
DOA estimation error (degree)
Non-calibration
Chamber calibration
In-situ calibration
(b) Box plots of DOA estimation errors in different sections of
incident directions.
Fig. 11. Comparisons of DOA estimation performance with non-calibrated,
chamber calibrated, and in-situ calibrated array manifolds.
methods that only utilize the spatial aperture for multipath
resolving. We believe this paper provides a low-cost and
scalable solution to calibrate these pervasively installed RRUs
in-situ, thereby enabling the 5G network to provide high-
precision positioning and sensing services.
Although in this paper we have set the focus on 5G posi-
tioning, the proposed calibration scheme can also be applied
to other similar wireless positioning systems, such as Wi-Fi
or ultra-wideband, to improve their DOA estimation accuracy.
Possible future research directions include: (i) investigating
the in-situ calibration method for planar arrays to support 3-
D positioning, and (ii) studying near-field array calibration to
support near-field or mixed far-field and near-field positioning
as the far-field condition may not always be guaranteed if the
array aperture has been further improved by, for example, the
sparse array design or the massive multiple-input multiple-
output configuration.
NOMENCLATURE
Abbreviations
2-D/3-D Two-/three-dimensional.
3GPP Third generation partnership project.
5G Fifth-generation mobile communications
technology.
BB Baseband.
BBU Baseband unit.
CDF Cumulative distribution function.
CFR Channel frequency response.
CIR Channel impulse response.
CPP Calibration pilot position.
CSI Channel state information.
DOA Direction-of-arrival.
EM Expectation-maximization.
FFT Fast Fourier transform.
gNB Next-generation Node-B.
HPBW Half-power beamwidth.
IF Intermediate frequency.
i.i.d. Independent and identically distributed.
LOS Line-of-sight.
MLE Maximum likelihood estimation.
NLOS Non-line-of-sight
PE Principal eigenvector.
RRU Remote radio unit.
TOA Time-of-arrival.
UL-SRS Uplink-sounding reference signal.
ULA Uniform linear array.
UT User terminal.
VSWR Voltage standing wave radio.
Notations
Imaginary unit (√−1).
(·)TTranspose operator.
(·)HConjugate transpose operator.
(·)∗Conjugate operator.
(·)−1Inverse of a square matrix.
|a|Modulus of the complex number a.
∠aPhase (a.k.a. argument) of the complex num-
ber a.
k·k `2-norm of a vector.
[a]nn-th element of vector a.
[A]m,n Element at m-th row and n-th column of
matrix A
Hadamard (element-wise) matrix product
CN(µ, σ2)Complex Gaussian distribution parameterized
by µand σ2
U(a, b]Uniform distribution from ato b.
a(x)Scalar-valued function with the input variable
of x.
a(x)Vector-valued function with the input variable
of x.
x=O(a)∃k1, k2>0, such that k2·a≤x≤k1·a.
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© 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
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