5g short=5G, long= fifth generation, 6g short=6G, long= sixth generation, 3d
short=3D, long= three-dimensional, ace short=ACE, long= array calibration
error, alb short=ALB, long= absolute lower bound, aod short=AOD, long= angle-
of-departure, aoa short=AOA, long= angle-of-arrival, adc short=ADC, long=
analog to digital converter, aeb short=AEB, long= angle error bound, age
short=AGE, long= array gain error, ade short=ADE, long= antenna displacement
error, bs short=BS, long= base station, cfo short=CFO, long= carrier frequency
offset, ceb short=CEB, long= clock error bound, coa short=COA, long=
curvature-of-arrival, crb short=CRB, long= Cramér-Rao bound, ccrb short=CCRB,
long= constrained Cramér-Rao bound, crlb short=CRLB, long= Cramér-Rao lower
bound, cp short=CP, long= cyclic prefix, dac short=DAC, long= digital to
analog converter, dftsofdm short=DFT-s-OFDM, long= discrete-Fourier-transform
spread OFDM, dl short=DL, long= deep learning, gps short=GPS, long= global
positioning system, gnss short=GNSS, long= global navigation satellite system,
hwi short=HWI, long= hardware impairment, isac short=ISAC, long= integrated
sensing and communication, iqi short=IQI, long= in-phase and quadrature
imbalance, kpi short=KPI, long= key performance indicator, kf short=KF, long=
Kalman filter, ekf short=EKF, long= extended Kalman filter, ukf short=UKF,
long= unscented Kalman filter, ckf short=CKF, long= cubature Kalman filter, pf
short=PF, long= particle filter, lb short=LB, long= lower bound, lse
short=LSE, long= least-square estimator, lo short=LO, long= local oscillator,
los short=LOS, long= line-of-sight, mc short=MC, long= mutual coupling, mac
short=MAC, long= medium access control, meb short=MEB, long= mapping error
bound, ml short=ML, long= machine learning, mcrb short=MCRB, long=
misspecified Cramér-Rao bound, mimo short=MIMO, long= multiple-input-multiple-
output, mm short=MM, long= mismatched model, mpc short=MPC, long= multipath
component, mmwave short=mmWave, long= millimeter wave, mmle short=MMLE, long=
mismatched maximum likelihood estimation, mems short=MEMS, long= micro-
electro-mechanical system, mle short=MLE, long= maximum likelihood estimation,
nlos short=NLOS, long= none-line-of-sight, ofdm short=OFDM, long= orthogonal
frequency-division multiplexing, oeb short=OEB, long= orientation error bound,
otfs short=OTFS, long= orthogonal time-frequency space, pdf short=PDF, long=
probability density function, papr short=PAPR, long= peak-to-average-power
ratio, pan short=PAN, long= power amplifier nonlinearity, pa short=PA, long=
power amplifier, ps short=PS, long= phase shifter, pn short=PN, long= phase
noise, poa short=POA, long= phase-of-arrival, pwm short=PWM, long= planar wave
model, pdoa short=PDOA, long= phase-difference-of-arrival, prs short=PRS,
long= positioning reference signals, peb short=PEB, long= position error
bound, qam short=QAM, long= quadrature amplitude modulation, rnn short=RNN,
long= recurrent neural network, rl short=RL, long= reinforcement learning, rfc
short=RFC, long= radio-frequency chain, rf short=RF, long= radio frequency,
ris short=RIS, long= reconfigurable intelligent surface, sa short=SA, long=
subarray, ser short=SER, long= symbol error rate, simo short=SIMO, long=
single-input-multiple-output, sota short=SOTA, long= state-of-the-art, swm
short=SWM, long= spherical wave model, slam short=SLAM, long= simultaneous
localization and mapping, tm short=TM, long= true model, toa short=TOA, long=
time-of-arrival, tof short=TOF, long= time-of-flight, tdoa short=TDOA, long=
time-difference-of-arrival, thz short=THz, long= terahertz, ue short=UE, long=
user equipment, ummimo short=UM-MIMO, long= ultra-massive multi-input-multi-
output, vlp short=VLP, long= visible light positioning, veb short=VEB, long=
velocity error bound, vlc short=VLC, long= visible light communication, ula
short=ULA, long= uniform linear array, upa short=UPA, long= uniform planar
array, wlan short=WLAN, long= wireless local area network,

# Modeling and Analysis of 6G Joint Localization and Communication under
Hardware Impairments

Hui Chen, , Musa Furkan Keskin, , Sina Rezaei Aghdam, , Hyowon Kim, , Simon
Lindberg, , Andreas Wolfgang, , Traian E. Abrudan, , Thomas Eriksson, , and
Henk Wymeersch, H. Chen, M. F. Keskin, S. R. Aghdam, H. Kim, T. Eriksson and
H. Wymeersch are with the Department of Electrical Engineering, Chalmers
University of Technology, 412 58 Gothenburg, Sweden (email: hui.chen; furkan;
sinar; hyowon; thomase; henkw@chalmers.se).S. Lindberg and A. Wolfgang are
with Qamcom Research & Technology, Gothenburg, Sweden (email: simon.lindberg;
andreas.wolfgang@qamcom.se).T. E. Abrudan is with Nokia Bell Labs, Finland
(email: traian.abrudan@nokia-bell-labs.com).This work was supported, in part,
by the European Commission through the H2020 project Hexa-X (Grant Agreement
no. 101015956) and by the MSCA-IF grant 888913 (OTFS-RADCOM).

###### Abstract

Localization (position and orientation estimation) is envisioned as a key
enabler to satisfy the requirements of communication and context-aware
services in the sixth generation (6G) communication systems. User localization
can be achieved based on delay and angle estimation using uplink or downlink
pilot signals. However, hardware impairments (HWIs) distort the signals at
both the transmitter and receiver sides and thus affect the localization
performance. While this impact can be ignored at lower frequencies where HWIs
are less severe, and the localization requirements are not stringent, modeling
and analysis efforts are needed for high-frequency 6G bands (e.g., sub-THz) to
assess degradation in localization accuracy due to HWIs. In this work, we
model various types of impairments for a sub-THz multiple-input-multiple-
output communication system and conduct a misspecified Cramér-Rao bound
analysis to evaluate HWI-induced performance losses in terms of angle/delay
estimation and the resulting 3D position/orientation estimation error.
Complementary to the localization analysis, we also investigate the effect of
individual and overall HWIs on communication in terms of symbol error rate
(SER). Our extensive simulation results demonstrate that each type of HWI
leads to a different level of degradation in angle and delay estimation
performance. The prominent factors on delay estimation (e.g., phase noise and
carrier frequency offset) will have a dominant negative effect on SER, while
the impairments affecting only the angle estimation (e.g., mutual coupling and
antenna displacement) induce slight degradation in SER performance.

###### Index Terms:

Localization, 6G, hardware impairment, THz communications, CRB, MCRB, MIMO.

##  I Introduction

Localization refers to the process of estimating the position and orientation
of a connected device or ue, which is expected to have a tight interaction
with communication in future wireless systems [1]. Localization can benefit
from a large array dimension and wide bandwidth of high-frequency signals
(e.g., mmWave and sub-THz) [2]. In return, the position and orientation
information can improve spatial efficiency and optimize resource allocation
for communication [3]. As a result, high-accuracy context-aware applications
such as the tactile Internet, augmented reality, and smart cities will be
supported in next-generation wireless networks [4, 5, 6].

In gnss and traditional cellular networks, range-based algorithms, such as
trilateration, are usually applied for estimating position. When moving to
higher carrier frequencies, more antennas can be packed in a single array due
to shorter wavelengths. As a consequence, in addition to delay estimation, aoa
and aod information can be exploited for localization, and a variety of new
localization techniques have recently emerged in the fifth/sixth generation
(5G/6G) systems, e.g., [7, 8, 9, 10], considering localization with minimal
infrastructure requirements. mpc, which are usually considered as destructive
signals, can be resolved in the emerging wireless systems, thereby enabling
single-bs positioning and mapping [7] as well as slam [8]. When the ue is
equipped with an antenna array, orientation estimation is also possible [9].
In Doppler-assisted localization, although new unknowns (e.g., velocity) are
introduced, localization performance can be improved because mobility forms a
virtual array with a large aperture compared to the stationary scenarios [10].
Most localization works rely on idealized models of the received signals as a
function of the channel parameters (angles, delays, Dopplers) induced by the
propagation environment, based on the assumption of deterministic and sparse
channels in high-frequency systems [11, 12, 1, 13, 14, 15]. However, in sub-
THz bands for 6G communications, pilot signals can be distorted due to the
presence of hwi such as pn, cfo, mc, pan, age, ade, iqi, etc [16].
Consequently, when algorithm derivation is based on a mismatched model (i.e.,
without considering the hwi in the channel model), the localization
performance is unavoidably affected.

The effect of hwi on communication have been studied extensively in the
literature [16, 17, 18, 19, 20]. In [16], different types of impairments have
been accurately modeled and the effects on a mimo-ofdm system are discussed.
In [17], an aggregate statistical HWI model considering pan, local oscillators
with pn, and finite-resolution adc is derived and validated with numerical
simulations. The residual additive transceiver hardware impairments, caused by
direct current offset, mc, iqi and quantization noise, are discussed in [18],
with the derived spectral efficiency to quantify the degradation caused by the
HWIs. In addition to modeling and analysis of the HWIs, research has also been
conducted on impairment mitigation algorithms. By incorporating signal
distortions caused by hardware impairments, beamforming optimization is
performed to maximize the received SNR at the destination [19]. A channel
estimation algorithm is designed by taking into account the transceiver
impairments in [21], showing a better bit error rate and normalized mean-
squared-error performance than the conventional algorithms. Contrary to model-
based solutions, channel estimation under HWI can also be formulated as a deep
learning problem [20, 22]. Nevertheless, these works focus only on
communication performance.

Research on localization and sensing (here, sensing includes detection, angle,
and delay estimation, as well as tracking of passive targets) considering hwi
is recently drawing attention. The effect of pn on automotive radar [23, 24,
25], mc on aoa estimation [26], iqi on mmWave localization [27], and pan on
joint radar-communication systems [28] have been studied. However, these works
only consider one or two types of impairments and cannot provide a thorough
analysis in real scenarios. In [29, 30], the impairments are modeled as
additional Gaussian noise, with the variance determined by an ad hoc HWI
factor, from which the error bounds for 3D localization are discussed.
However, this approach fails to capture the contribution of each individual
HWI. In [31], which forms the basis of the current paper, a simplified
synchronized simo uplink system is considered for 2D positioning, and the
results show that different types of impairments affect angle and delay
estimation in different ways. Nevertheless, the perfect synchronization
assumption is impractical, and the impairments such as array calibration error
and iqi are not considered. Besides analyzing the effect of HWIs on
localization or communication alone, more recent works consider the HWIs in
joint localization and communication systems and use learning-based methods to
mitigate the performance degradation [32, 33]. Nevertheless, only a limited
number of impairment types are discussed (mc and ade in [32], iqi and DC
offset in [33]). In addition, no theoretical analysis is performed in these
works, and the relative importance of each HWI on communication compared to
localization is unknown. Hence, there is a need for a more systematic study
that evaluates the effect of different types of HWI on both communication and
localization performance.

In this paper, we investigate the problem of estimating the 3D position and 3D
orientation of a multiple-antenna UE using several multiple-antenna BSs in a
realistic uplink scenario for a sub-THz communications system under a wide
variety of hwi. Specifically, we consider an ofdm-based system by rigorously
modeling the impact of various hwi on the received observations, and assume
that the corresponding channel estimation and localization algorithms have no
knowledge about these hwi, resulting in degradation of localization and
communication performance. The mcrb [34, 35, 36] is employed to quantify the
estimation performance loss due to model mismatch. In addition, the effect of
HWI on communication is evaluated numerically in terms of ser based on the
developed model for a hardware-impaired channel under the same HWI levels,
which allows a fair comparison of the impact of HWI on communication and
localization. The contributions are summarized as follows:

  * •

Channel model with multiple HWIs: Based on the ideal MIMO model (mm) with
perfect hardware, we develop a more general channel model for the considered
sub-THz system (tm) that can accommodate a variety of HWI types (including pn,
cfo, mc, pan, age, ade, and iqi) in a 3D environment. To the best of the
authors’ knowledge, this is the first study to derive a comprehensive and
realistic signal model for localization and communications that provides
explicit modeling of major HWIs that are likely to affect 6G communication
systems at high-frequency operation (e.g., mmWave and sub-THz bands).

  * •

Analytical performance prediction of channel parameter estimation and
localization under HWIs: We leverage MCRB analysis to evaluate the effect of
individual and combined HWIs on the estimation of channel parameters (aod, aoa
and delay estimation) and on the corresponding localization performance (3D
position and 3D orientation estimation). More specifically, the bounds provide
the best performance of estimators using a MM to process the TM data.

  * •

Performance evaluation and comparison with communication: Extensive
simulations are performed to verify the performance analysis of the effect of
HWI on localization and communication. For communication, we approximate the
HWIs as additive noise and evaluate the effect of individual and aggregated
HWIs on communication performance in terms of ser using a 16-qam modulation
scheme. In addition, the effect of different HWIs on localization and
communication is evaluated with dominant factors identified. We notice that
the dominant factors that affect delay estimation will also affect
communication, whereas the impairments that only affect aoa, aod have a
limited impact on communication.

The rest of this paper is organized as follows. Section II reviews the system
models with and without HWIs. Section III describes the channel estimation and
localization algorithms. Theoretical performance analysis is carried out in
Section IV. Next, the simulation results are presented in Section V, followed
by the concluding remarks in Section VI.

Notations and Symbols: Italic letters denote scalars (e.g. $a$), bold lower-
case letters denote vectors (e.g. ${\bf a}$), and bold upper-case letters
denote matrices (e.g. ${\bf A}$). $(\cdot)^{\top}$, $(\cdot)^{\sf H}$,
$(\cdot)^{-1}$, ${\hbox{tr}}(\cdot)$, and $\|{\cdot}\|$ represent the
transpose, Hermitian transpose, inverse, trace, and $\ell$-2 norm operations,
respectively; ${\bf A}\\!\odot\\!{\bf B}$, ${\bf A}\\!\otimes\\!{\bf B}$,
${\bf a}\circ{\bf b}$ are the Hadamard product, Kronecker product, and outer
product, respectively; $[\cdot,\ \cdot,\ \cdots,\cdot]^{\top}$ denotes a
column vector; ${\hbox{tr}}(\cdot)$ returns the trace of a matrix,
$[\cdot]_{i,j}$ is the element in the $i$th row, $j$th column of a matrix, and
$[\cdot]_{a:b,c:d}$ is the submatrix constructed from the $a$th to the $b$th
row, and the $c$th to $d$th column of a matrix.

UE${\bf x}_{g}$$l$th BS${\bf y}_{g}$ Channel ${\bf H}_{l}$ Estimated UE state:
${\bf s}=[{\bf p}_{\text{U}}^{\top},B_{\text{U}},\text{vec}({\bf
R}_{\text{U}})^{\top}]^{\top}$ $\hbox{\boldmath$\eta$}_{1}$ $\cdots$
$\hbox{\boldmath$\eta$}_{L}$ Channel parameter extraction:
$\hbox{\boldmath$\eta$}_{l}=[\phi_{\text{B}},\theta_{\text{B}},\phi_{\text{U}},\theta_{\text{U}},\tau,\rho,\xi]^{\top}$
Estimated channel: $\hat{\bf H}_{l}$ Received symbol: $\hat{\bf y}_{l}=[{\bf
y}_{1}^{\top},\ldots,{\bf y}_{g}^{\top}]^{\top}$ Figure 1: Block diagram of
the hardware impairments considered at transmitter and receiver (highlighted
in shaded regions). When the localization algorithm does not have perfect
knowledge of the generative model, it operates under model mismatch. PN (phase
noise), CFO (carrier frequency offset), MC (mutual coupling), PAN (power
amplifier non-linearity), AGE (array gain error), ADE (antenna displacement
error), and IQI (in-phase and quadrature imbalance) are considered.

##  II System Model

In this section, we start with a mimo channel model (hwi-free model) and then
describe the model considering the hwi.

###  II-A Geometric Model

The block diagram of considered HWIs and localization procedures are shown in
Fig. 1. An uplink mimo system consisting of a ue and $L$ bs is considered. The
BSs and UE are equipped with an upa (antennas lie on the local YZ plane)
driven by a single rfc. The number of antenna elements at the $l$-th BS and
the UE arrays is denoted as $N_{\text{B},l}=N_{\text{B},l,z}\times
N_{\text{B},l,y}$ and $N_{\text{U}}=N_{\text{U,z}}\times N_{\text{U,y}}$ where
$N_{\text{z}}$ and $N_{\text{y}}$ are the number of antennas on the Z and Y
axes, respectively. The BSs are perfectly synchronized while a clock offset
$B_{\text{U}}$ exists between the UE and the BSs. We denote the array center
and orientation of the $l$-th BS as ${\bf p}_{\text{B},l}\in\mathbb{R}^{3}$
and ${\bf o}_{\text{B},l}\in\mathbb{R}^{3}$ in the global coordinate system.
Similarly, the position and orientation of the UE can be denoted as ${\bf
p}_{\text{U}},{\bf o}_{\text{U}}$. Since the orientation represented by an
Euler angle vector is not unique, we use rotation matrices ${\bf
R}_{\text{B},l}\in\text{SO(3)}$ and ${\bf R}_{\text{U}}\in\text{SO(3)}$ in
orientation estimation (more details can be found in [12, 1]). In
localization, channel estimations are performed at each BS, and all estimates
are combined to find the UE _state parameter vector_ ${\bf s}=[{\bf
p}_{\text{U}}^{\top},B_{\text{U}},\text{vec}({\bf
R}_{\text{U}})^{\top}]^{\top}\in\mathbb{R}^{13}$, containing the UE position
${\bf p}_{\text{U}}$, clock offset $B_{\text{U}}$, and rotation matrix ${\bf
R}_{\text{U}}$, as shown in Fig. 1.

###  II-B Hardware Impairment-free Model

Considering the transmitted ofdm symbol111For positioning, constant modulus
pilots are typically used. For communication, different modulations (e.g.,
16-QAM) can be adopted. at the $g$-th transmission and $k$-th subcarrier,
$x_{g,k}$ with an average transmit power
$\mathbb{E}\\{|x_{g,k}|^{2}\\}=P/{N_{\text{U}}}$, its observation at a
specific bs (the index $l$ is omitted for convenience) can be formulated as

$y_{g,k}={\bf w}_{\text{}g}^{\top}{\bf H}_{k}{\bf
v}_{\text{}g}x_{g,k}+n_{g,k},$ (1)

where ${\bf w}_{\text{}g}\in\mathbb{C}^{N}$ is the combiner at the BS for the
$g$-th transmission and ${\bf v}_{\text{}g}\in\mathbb{C}^{N}$ is the precoder
at the UE, both with unit amplitude entries, $n_{g,k}\in\mathcal{CN}(0,{\bf
w}_{\text{}g}^{\sf H}{\bf w}_{\text{}g}\sigma_{n}^{2})$ is the noise vector
with each entry following a complex normal distribution, with
$\sigma_{n}^{2}=N_{0}W$ ($N_{0}$ is the noise power spectral density (PSD) and
$W=K\Delta_{f}$ is the total bandwidth with $K$ subcarriers and subcarrier
spacing $\Delta_{f}$). We assume ${\bf H}_{k}$ remains the same during ${G}$
transmissions (within the coherence time). The channel matrix at subcarrier
$k$ is given by

$\displaystyle{\bf H}_{k}$ $\displaystyle=\underbrace{\alpha d_{k}(\tau){\bf
a}_{\text{B}}(\hbox{\boldmath$\varphi$}_{\text{B}}){\bf
a}^{\top}_{\text{U}}(\hbox{\boldmath$\varphi$}_{\text{U}})}_{\text{LOS~{}path}}$
(2) $\displaystyle+\underbrace{\sum_{p=1}^{P}\alpha_{p}d_{p,k}(\tau_{p}){\bf
a}_{\text{B}}(\hbox{\boldmath$\varphi$}_{\text{B},p}){\bf
a}^{\top}_{\text{U}}(\hbox{\boldmath$\varphi$}_{\text{U},p})}_{\text{NLOS~{}paths}},$

where for the LOS path, $\alpha=\rho e^{-j\xi}$ is the complex channel gain
assumed to be identical for different subcarriers, $d_{k}(\tau)=e^{-j2\pi
k\Delta_{f}\tau}$ ($\Delta_{f}$ is the subcarrier spacing) as a function of
the path delay $\tau$, while ${\bf
a}_{\text{B}}(\hbox{\boldmath$\varphi$}_{\text{B}})$ and ${\bf
a}_{\text{U}}(\hbox{\boldmath$\varphi$}_{\text{U}})$ are the receiver and
transmitter steering vectors as a function of the aoa
$\hbox{\boldmath$\varphi$}_{\text{B}}=[\phi_{\text{B}},\theta_{\text{B}}]^{\top}$
(azimuth angle $\phi_{\text{B}}$ and elevation angle $\theta_{\text{B}}$), and
aod
$\hbox{\boldmath$\phi$}_{\text{U}}=[\phi_{\text{U}},\theta_{\text{U}}]^{\top}$.
A steering vector ${\bf a}(\hbox{\boldmath$\varphi$})$ of an $N$-element array
is a function of the direction of the (incoming or outgoing) signal and the
locations of the antenna elements, which can be expressed as [1]

$\displaystyle{\bf a}(\hbox{\boldmath$\varphi$})$
$\displaystyle=e^{j\frac{2\pi f_{c}}{c}{\bf Z}^{\top}{\bf
t}(\boldsymbol{\varphi})},$ (3)

where we apply the $\exp$ operator element-wise, ${\bf
Z}\in\mathbb{R}^{3\times N}$ is the matrix containing the position of the $N$
antennas in the local coordinate system (all zeros in the first row of ${\bf
Z}$) and ${\bf
t}(\hbox{\boldmath$\varphi$})=[\cos(\theta)\cos(\phi),\cos(\theta)\sin(\phi),\sin(\theta)]^{\top}$.
For the NLOS paths, each path can correspond to single or multi-bounce
reflections, or diffuse scattering. Hence, the NLOS path will not be utilized
for the positioning of the UE in this work. We further make the assumption
that the LOS path is resolvable with respect to the NLOS paths (though the
NLOS paths may be mutually unresolved). This is a reasonable assumption222For
example, with a bandwidth of 1 GHz and $8\times 8$ BS arrays, a delay
resolution of 30 cm and an angle resolution of 22 degrees is achievable.
Unless the UE is very close to a reflector, multipath can be resolved in the
combined range-angle domain. for 6G systems, due to large bandwidth and a
large number of antennas [11]. Without significant loss of generality, the
channel matrix for the $k$th subcarrier can thus be simplified as

${\bf H}_{k}=\alpha d_{k}(\tau){\bf
a}_{\text{B}}(\boldsymbol{\varphi}_{\text{B}}){\bf
a}^{\top}_{\text{U}}(\hbox{\boldmath$\varphi$}_{\text{U}}).$ (4)

Correspondingly, the _channel geometric parameter vector_ of the los path
between a BS and the UE is defined as
$\hbox{\boldmath$\eta$}_{\text{ch}}=[\hbox{\boldmath$\eta$}_{1}^{\top},\ldots,\hbox{\boldmath$\eta$}_{L}^{\top}]^{\top}$
with
$\hbox{\boldmath$\eta$}_{l}=[\hbox{\boldmath$\varphi$}_{\text{B},l}^{\top},\hbox{\boldmath$\varphi$}_{\text{U},l}^{\top},\tau_{l},\rho_{l},\xi_{l}]^{\top}\in\mathbb{R}^{7}$
for the $l$th BS. For later analysis, we define a vector by removing all the
nuisance parameters (i.e., complex channel gain for each path) as ${\bf
c}_{\text{ch}}=[{\bf c}_{1}^{\top},\ldots,{\bf c}_{L}^{\top}]^{\top}$ with
${\bf
c}_{l}=[\hbox{\boldmath$\varphi$}_{\text{B},l}^{\top},\hbox{\boldmath$\varphi$}_{\text{U},l}^{\top},\tau_{l}]^{\top}\in\mathbb{R}^{5}$.
The geometric relationships between the channel parameters vector ${\bf c}$
and the state parameters ${\bf s}$ can be expressed as

$\displaystyle\boldsymbol{\varphi}_{\text{B}}$
$\displaystyle=\begin{bmatrix}\phi_{\text{B}}\\\
\theta_{\text{B}}\end{bmatrix}=\begin{bmatrix}\arctan
2(t_{\text{B},2},t_{\text{B},1})\\\ \arcsin(t_{\text{B},3})\end{bmatrix},$ (5)
$\displaystyle\boldsymbol{\varphi}_{\text{U}}$
$\displaystyle=\begin{bmatrix}\phi_{\text{U}}\\\
\theta_{\text{U}}\end{bmatrix}=\begin{bmatrix}\arctan
2(t_{\text{U},2},t_{\text{U},1})\\\ \arcsin(t_{\text{U},3})\end{bmatrix},$ (6)
$\displaystyle\tau$ $\displaystyle=\frac{\|{\bf p}_{\text{U}}-{\bf
p}_{\text{B}}\|}{c}+B_{\text{U}},$ (7)

where $c$ is the speed of light, ${\bf
t}_{\text{B}}=[t_{\text{B,1}},t_{\text{B,2}},t_{\text{B,3}}]^{\top}$ and ${\bf
t}_{\text{U}}=[t_{\text{U,1}},t_{\text{U,2}},t_{\text{U,3}}]^{\top}$ are the
direction vectors in the local coordinate system that can be expressed using
global direction vectors and rotation matrices as

$\displaystyle{\bf t}_{\text{B}}$ $\displaystyle={\bf
R}_{\text{B}}^{-1}\frac{{\bf p}_{\text{U}}-{\bf p}_{\text{B}}}{\|{\bf
p}_{\text{U}}-{\bf p}_{\text{B}}\|},$ (8) $\displaystyle{\bf t}_{\text{U}}$
$\displaystyle={\bf R}_{\text{U}}^{-1}\frac{{\bf p}_{\text{B}}-{\bf
p}_{\text{U}}}{\|{\bf p}_{\text{B}}-{\bf p}_{\text{U}}\|}.$ (9)

Finally, by concatenating all the received symbols into a column, we obtain
the received symbol block ${\bf y}\in\mathbb{R}^{{G}K}$ as ${\bf y}=[{\bf
y}_{1}^{\top},\ldots,{\bf y}_{g}^{\top},\ldots,{\bf y}^{\top}_{G}]^{\top}$,
where ${\bf y}_{g}=[y_{g,1},\ldots,y_{g,K}]^{\top}$ can be expressed as

$\displaystyle{\bf y}_{g}=\alpha({\bf w}_{\text{}g}^{\top}{\bf
a}_{\text{}}(\hbox{\boldmath$\varphi$}_{\text{B}}){\bf
a}^{\top}_{\text{}}(\hbox{\boldmath$\varphi$}_{\text{U}}){\bf v}_{g}){\bf
d}(\tau)\odot{\bf x}_{g}+{\bf n}_{g},$ (10)

in which ${\bf d}(\tau)=[d_{1}(\tau),\ldots,d_{K}(\tau)]^{\top}$, ${\bf
x}_{g}=[x_{g,1},\ldots,x_{g,K}]^{\top}$, and ${\bf
n}_{g}=[n_{g,1},\ldots,n_{g,K}]^{\top}$.

###  II-C Hardware Impairments

In this work, several types of HWIs are considered as shown in Fig. 1. We
study the effects of mc, pan, age, ade, pn, cfo, and iqi. Note that the
impairments such as pn, cfo, mc, age, ade and iqi exist both at the
transmitter and the receiver, while the PAN appears only at the transmitter.
The HWIs are usually compensated offline during calibration or online with
dedicated signals and routines, depending on whether the impairment is static
or time-variant. Both the offline and the online methods will have _residual
errors_ , which can be modeled as random perturbations around the nominal
values. This work focus on the impact of these residual errors after
calibration. For online methods, these random realizations correspond to
_different times_ for a specific device, while for offline methods, these
random realizations should be interpreted as corresponding to an _ensemble of
devices_.

The imperfections of adc, dac, low-noise amplifier and mixer are not
considered.

####  II-C1 Phase Noise and Carrier Frequency Offset

Imperfect lo in the up-conversion and down-conversion processes add PN to the
carrier wave phase. In addition, when the down-converting lo in the receiver
does not perfectly synchronize with the received signal’s carrier [37], CFO
occurs. In general, both PN and CFO are estimated and compensated by the
receiver [38], so we only consider the residual PN and residual CFO at the
receiver. With PN and CFO, the observation, $y_{g,k}$, is modified as in [39]

$\displaystyle y_{g,k}$ $\displaystyle\to{\bf f}^{\top}_{k}{\bf
E}_{g}\hbox{\boldmath$\Xi$}_{g}{\bf F}^{\mathsf{H}}{\bf y}_{g},$ (11)
$\displaystyle{\bf E}_{g}$ $\displaystyle=e^{j\frac{2\pi\epsilon
gK_{\text{tot}}}{{K}}}\text{diag}([1,e^{j\frac{2\pi\epsilon}{K}},\ldots,e^{j\frac{2\pi(K-1)\epsilon}{K}}]),$
(12) $\displaystyle\hbox{\boldmath$\Xi$}_{g}$
$\displaystyle=\text{diag}([e^{j\nu_{g,1}},\ldots,e^{j\nu_{g,K}}]),$ (13)

where ${\bf y}_{g}$ is the received signals of the ideal model without PN or
CFO (i.e., from (1)), ${\bf F}=[{\bf f}_{1},{\bf f}_{2},\ldots,{\bf f}_{K}]$
is the FFT matrix. The CFO matrix ${\bf E}_{g}$ considers both inter-OFDM
symbol phase changes as well as inter-carrier interference [39, 40]. More
specifically, $K_{\text{tot}}=K+K_{\text{cp}}$ with $K_{\text{cp}}$ as the
length of the cyclic prefix, and $\epsilon$ is the residual CFO with
$\epsilon\sim\mathcal{N}(0,\sigma_{\text{CFO}}^{2})$.
$\hbox{\boldmath$\Xi$}_{g}$ is the residual333Note that $\nu_{g,k}$ and
$\epsilon$ represent residual PN and CFO that remains after the carrier
synchronization process processing (e.g., [41, 42]). Hence, $\nu_{g,k}$ is
assumed to be independent across time. pn matrix with
$\nu_{g,k}\sim\mathcal{N}(0,\sigma_{\text{PN}}^{2})$. In (11), the vector
${\bf y}_{g}$ is converted to the time domain by ${\bf F}^{\mathsf{H}}{\bf
y}_{g}$, where the successive pn samples, as well as the CFO, are applied.
Finally, ${\bf f}^{\top}_{k}$ extracts the $k$-th subcarrier after applying an
FFT to ${\bf E}_{g}\hbox{\boldmath$\Xi$}_{g}{\bf F}^{\mathsf{H}}{\bf y}_{g}$.
Note that the residual CFO $\epsilon$ is fixed for each realization (e.g., one
localization measurement with $G$ transmission), while the pn $\nu_{g,k}$ is
different for all the subcarriers and OFDM symbols.

####  II-C2 Mutual Coupling

mc refers to the electromagnetic interaction between the antenna elements in
an array [26]. For a upa, we adopt the mc model as in [43] by assuming the
antenna is only affected by the coupling of the surrounding elements. As a
result, the MC matrix can be expressed as

${\bf C}_{\text{}}=\begin{bmatrix}{\bf C}_{1}&{\bf C}_{2}&0&\cdots&0\\\ {\bf
C}_{2}&{\bf C}_{1}&0&\cdots&0\\\ \vdots&\ddots&\ddots&\ddots&\vdots\\\
0&\cdots&{\bf C}_{2}&{\bf C}_{1}&{\bf C}_{2}\\\ 0&\cdots&0&{\bf C}_{2}&{\bf
C}_{1}\end{bmatrix}.$ (14)

Here, ${\bf C}_{\text{}}\in\mathbb{C}^{N_{\text{z}}N_{\text{y}}\times
N_{\text{z}}N_{\text{y}}}$ is the MC matrix with sub-matrices ${\bf
C}_{1}=\text{Toeplitz}([1,c_{x},0\ldots,0])\in\mathbb{C}^{N_{\text{y}}\times
N_{\text{y}}}$ and ${\bf
C}_{2}=\text{Toeplitz}([c_{x},c_{xy},0,\ldots,0])\in\mathbb{C}^{N_{\text{y}}\times
N_{\text{y}}}$ [43]. For convenience, we use one variable $\sigma_{\text{MC}}$
to denote the severity of the MC such that
$c_{x}\sim\mathcal{CN}(0,\sigma_{\text{MC}}^{2})$ and
$c_{xy}\sim\mathcal{CN}(0,\sigma_{\text{MC}}^{2}/4)$. The MC leads to the
following substitution of the channel matrix

${\bf H}_{k}\to{\bf C}_{\text{B}}{\bf H}_{k}{\bf C}_{\text{U}}^{\top}.$ (15)

####  II-C3 Power Amplifier Nonlinearity

For the PA nonlinearity, we consider a $Q$-th order memoryless polynomial
nonlinear model with a clipping point $x_{\text{clip}}\in\mathbb{R}$ as [16]

${h}_{\text{PA}}(\check{{x}}_{t})=\begin{cases}\sum_{q=0}^{Q-1}\beta_{q+1}\check{{x}}|\check{{x}}|^{q}&|\check{x}|\leq
x_{\text{clip}},\\\
\sum_{q=0}^{Q-1}\beta_{q+1}\frac{\check{{x}}}{|\check{{x}}|}|{{x}_{\text{clip}}}|^{q+1}&|\check{x}|>x_{\text{clip}},\end{cases}$
(16)

where $\check{x}_{t}=x_{t}/R$ denotes the voltage of the transmitted time-
domain signal ($R$ is the load impedance in Ohms) in the time domain and
$\beta_{1},\dots,\beta_{Q}$ are complex-valued parameters. We assume that (16)
models both the effect of digital pre-distortion and power amplifier, and we
use non-oversampled signals as input to PA for tractable localization
performance analysis444In order to fully characterize the effect of PAN, an
oversampled model is needed, which also captures the intersymbol interference
introduced by the nonlinearity, in addition to the symbol distortion (see (25)
in [44]).. Note that the PA affects the time domain signals and each antenna
at the Tx has a separate PA, and the PA model in (16) does not consider the
out-of-band emissions (only the in-band distortion). For simplicity, the
models are the same for different PAs and ${\bf h}_{\text{PA}}(\check{{\bf
x}}_{t})$ returns the time domain signal vector (by operating point-wise on
each of the elements) with PA nonlinearity introduced.

####  II-C4 Array Calibration Error

The age and ade are considered in the array calibration error. We define the
complex excitation coefficient of the $n$-th antenna at direction $\varphi$ as
[45]

$b_{n}(\hbox{\boldmath$\varphi$})=(1+\delta_{a})e^{j\delta_{p}},$ (17)

where $\delta_{a}\in\mathcal{N}(0,\sigma_{\text{AA}}^{2})$, and
$\delta_{p}\in\mathcal{N}(0,\sigma_{\text{AP}}^{2})$ are the relative
amplitude error and phase error, respectively. Regarding the displacement
error, we assume the $n$-th antenna position has a displacement on the 2D
plane of the local coordinate system as

$\tilde{\bf z}_{n}={\bf z}_{n}+[0,\delta_{n,y},\delta_{n,z}]^{\top},$ (18)

with ${\bf d}_{n}\in\mathbb{R}^{3}$ is the ideal position of the $n$th antenna
in the local coordinate system,
$\delta_{n,y},\delta_{n,z}\in\mathcal{N}(0,\sigma^{2}_{\text{ADE}})$ are the
displacement error. The steering vector is then modified as

${\bf a}(\hbox{\boldmath$\varphi$})\rightarrow{\bf
b}(\hbox{\boldmath$\varphi$})\odot e^{j\frac{2\pi}{\lambda}\tilde{\bf
Z}^{\top}{\bf t}},$ (19)

where $\tilde{\bf Z}=[\tilde{\bf z}_{1},\ldots,\tilde{\bf z}_{N}]$ contains
the geometry information of all the antennas. The array calibration error is
fixed for a certain array and varies across different devices.

####  II-C5 In-phase and quadrature imbalance

iqi operates on the time domain signal and the transmitted symbol vector is
modified as [27, 46]

${\bf x}_{g}\rightarrow{\bf F}(\alpha_{\text{U}}{\bf F}^{\sf H}{\bf
x}_{g}+\beta_{\text{U}}{\bf F}^{\sf H}{\bf x}_{g}^{*})=\alpha_{\text{U}}{\bf
x}_{g}+\beta_{\text{U}}{\bf x}_{g}^{*},$ (20)

where the FFT matrix ${\bf F}$ and IFFT matrix ${\bf F}^{\sf H}$ switch
between time and frequency domain,
$\alpha_{\text{U}}=\frac{1}{2}+\frac{1}{2}m_{\text{U}}e^{j\psi_{\text{U}}}$,
$\beta_{\text{U}}=\frac{1}{2}-\frac{1}{2}m_{\text{U}}e^{j\psi_{\text{U}}}$
with $m_{\text{U}}$ and $\psi_{\text{U}}$ as the amplitude and phase imbalance
parameters at the UE side. We assume that the IQI is compensated in the
system, leading to a residual impairment and the imbalance parameters can be
modeled as $m_{\text{U}}\sim\mathcal{N}(1,\sigma_{\text{IA}}^{2})$ and
$\phi_{\text{U}}\sim\mathcal{N}(0,\sigma_{\text{IP}}^{2})$. Similarly, the IQI
at the receiving BS can be expressed as

${\bf y}_{g}\rightarrow\alpha_{\text{B}}{\bf y}_{g}+\beta_{\text{B}}{\bf
y}_{g}^{*}.$ (21)

More accurate frequency-dependent IQI models can be found in [47, 48], which
is beyond the scope of this work.

###  II-D Hardware-impaired Model

Considering all types of hwi described in Sec. II-C and substituting (11)–(21)
into (10), the observation can be rewritten in the frequency domain.

####  II-D1 Transmit Signal under HWI

The precoded transmitted signal across subcarriers and antennas is modified
from ${\bf X}_{g}=\mathbf{x}_{g}{\bf v}^{\top}_{g}\in\mathbb{C}^{K\times
N_{\text{U}}}$ to

$\displaystyle\check{{{\bf X}}}_{g}$ $\displaystyle={\bf
F}\mathbf{h}_{\text{PA}}(\underbrace{\mathbf{E}_{\text{U}}\mathbf{\Xi}_{\text{U}}({\alpha}_{\text{U}}\mathbf{F}^{\sf
H}\mathbf{x}_{g}+{\beta}_{\text{U}}\mathbf{F}^{\sf H}\mathbf{x}_{g}^{*}){{\bf
v}}^{\top}_{g}}_{\text{precoded time domain signal before PA}}).$ (22)

####  II-D2 Channel under HWI

The channel is modified from ${\bf H}_{k}=\alpha d_{k}(\tau){\bf
a}_{\text{}}(\hbox{\boldmath$\varphi$}_{\text{B}}){\bf
a}^{\top}_{\text{}}(\hbox{\boldmath$\varphi$}_{\text{U}})\in\mathbb{C}^{N_{\text{B}}\times
N_{\text{U}}}$ in (4) to

$\displaystyle\begin{split}\check{\mathbf{H}}&=\alpha
d_{k}(\tau)\mathbf{C}_{\text{B}}(\underbrace{\bm{b}_{\text{B}}(\bm{\varphi}_{\text{B}})\odot
e^{j\frac{2\pi}{\lambda}{\tilde{\bf{Z}}_{\text{B}}^{\top}{{\bf
t}}_{\text{B}}(\bm{\varphi}_{\text{B}})}}}_{\text{steering vector\ }\tilde{\bf
a}_{\text{B}}(\boldsymbol{\varphi}_{\text{B}})})\\\
&\times(\underbrace{\bm{b}_{\text{U}}(\bm{\varphi}_{\text{U}})\odot
e^{j\frac{2\pi}{\lambda}{\tilde{\bf{Z}}_{\text{U}}^{\top}{{\bf
t}}_{\text{U}}(\bm{\varphi}_{\text{U}})}}}_{\text{steering vector\ }\tilde{\bf
a}_{\text{U}}(\boldsymbol{\varphi}_{\text{U}})})\mathbf{C}_{\text{U}}^{\top}.\end{split}$
(23)

####  II-D3 Received Signal under HWI

The received signal is modified from ${\bf y}_{g}\in\mathbb{C}^{K\times 1}$ to
(24).

$\displaystyle\check{\mathbf{y}}_{g}$
$\displaystyle=\mathbf{F}(\alpha_{\text{B}}(\mathbf{E}_{\text{B},g}\mathbf{\Xi}_{\text{B},g}\mathbf{F}^{\sf
H}({\check{{\bf X}}_{g}\check{\mathbf{H}}^{\top}{\bf w}_{g}\odot{\bf
d}(\tau))}))+\beta_{\text{B}}(\mathbf{E}_{\text{B},g}\mathbf{\Xi}_{\text{B},g}\mathbf{F}^{\sf
H}(\check{{\bf X}}_{g}\check{\mathbf{H}}^{\top}{\bf w}_{g}\odot{\bf
d}(\tau)))^{*})+{\mathbf{n}_{g}}.$ (24)

###  II-E Summary of the Models

To summarize, we have defined a mm in (1) without considering the hwi, which
will be used for algorithm development. With hwi introduced, the impaired
model defined in (24) will be used as the tm. In the following section, we
will evaluate the impact of using the mm to process data generated by tm on
localization performance. For the sake of convenience in performance analysis,
we use $\hbox{\boldmath$\mu$}_{g}(\hbox{\boldmath$\eta$})$ and
$\bar{\hbox{\boldmath$\mu$}}_{g}(\hbox{\boldmath$\eta$})$ to denote the noise-
free observation of (1) and (24), respectively.

##  III Localization Algorithm

Based on the models described above, a two-stage localization555In contrast,
the direct localization estimates the state vector ${\bf s}$ from the observed
signal vector ${\bf y}$ directly. Considering the high complexity involved, we
adopt two-stage localization in this work. problem can be formulated such that
the channel parameter vectors
$\hat{\hbox{\boldmath$\eta$}}_{\text{ch}}=[\hbox{\boldmath$\eta$}_{1}^{\top},\ldots,\hbox{\boldmath$\eta$}_{L}^{\top}]^{\top}$
are firstly estimated based on the observation vector $\hat{\bf
y}_{1},\ldots,\hat{\bf y}_{L}$ from all the BSs, and then the stage vector
$\hat{\bf s}$ is determined from $\hat{\hbox{\boldmath$\eta$}}_{\text{ch}}$.

###  III-A Mismatched Maximum Likelihood Estimator

The mle can be employed when the observation ${\bf y}$ is generated from the
same model used by the algorithm. If ${\bf y}\sim f_{\text{TM}}({\bf
y}|\bar{\hbox{\boldmath$\eta$}})$, the mle of the UE position and channel gain
is

$\displaystyle\hat{\hbox{\boldmath$\eta$}}_{\text{MLE}}$
$\displaystyle={\hbox{arg}}\max_{\bar{\boldsymbol{\eta}}}\ln
f_{\text{TM}}({\bf y}|\bar{\hbox{\boldmath$\eta$}}),$ (25)

where $\ln f_{\text{TM}}({\bf y}|\bar{\hbox{\boldmath$\eta$}})$ is the log-
likelihood of the tm. However, if ${\bf y}\sim f_{\text{TM}}({\bf
y}|\bar{\hbox{\boldmath$\eta$}})$, but the estimator uses $f_{\text{MM}}({\bf
y}|\hbox{\boldmath$\eta$})\neq f_{\text{TM}}({\bf
y}|\bar{\hbox{\boldmath$\eta$}})$, the mmle is given by

$\displaystyle\hat{\hbox{\boldmath$\eta$}}_{\text{MMLE}}$
$\displaystyle={\hbox{arg}}\max_{\boldsymbol{\eta}}\ln f_{\text{MM}}({\bf
y}|\hbox{\boldmath$\eta$}).$ (26)

More specifically, equation (26) formulates the mmle for channel parameters
extraction, which can also be implemented in position and orientation
estimation with known or approximated likelihood function. A practical
approach is to use the gradient descent method with an initial point, which
will be detailed in the following subsections.

###  III-B Channel Parameters Estimation

The channel parameters estimation will be performed with a coarse estimation
using ESPRIT, which provides a good initial point for a refined estimation
using (26).

####  III-B1 Coarse Estimation using ESPRIT

We aim to obtain an initial estimate of the channel parameters with a low
complexity, which can be solved using tensor-based beamspace ESPRIT666While
this work considers only the LOS channel, the ESPRIT also works for the
scenarios with NLOS paths. algorithm [13]. To implement the beamspace ESPRIT
algorithm, we reformulate a beamspace channel matrix ${\bf H}^{(b)}$ based on
the signal model in (1) as

${\bf H}^{(b)}_{k}=\alpha d_{k}(\tau){\bf W}^{{\sf H}}{\bf
a}_{\text{B}}(\hbox{\boldmath$\varphi$}_{\text{B}}){\bf
a}^{\top}_{\text{U}}(\hbox{\boldmath$\varphi$}_{\text{U}}){\bf V}$ (27)

where ${\bf W}={\bf T}_{1}\otimes{\bf T}_{2}\in\mathbb{C}^{N_{1}N_{2}\times
M_{1}M_{2}}$ and ${\bf V}=({\bf T}_{3}\otimes{\bf
T}_{4})^{*}\in\mathbb{C}^{N_{3}N_{4}\times M_{3}M_{4}}$ are the combining
matrix and precoder matrix and the total number of transmissions
${G}=M_{1}M_{2}M_{3}M_{4}$. Since the first row of the antenna position matrix
$\tilde{\bf Z}$ is all zeros (see Sec. II-A and equation (3)), we can express
the steering vector in (3) as

${\bf a}_{\text{B}}(\hbox{\boldmath$\varphi$}_{\text{B}})={\bf
a}^{(M_{1})}(\omega_{1})\otimes{\bf a}^{(M_{2})}(\omega_{2}),$ (28)

with

$\displaystyle\omega_{1}$
$\displaystyle=\pi\sin(\phi_{\text{B}})\cos(\theta_{\text{B}}),\ \
\omega_{2}=\pi\sin(\theta_{\text{B}}),$ (29) $\displaystyle{\bf
a}^{(M_{1})}_{\text{B}}(\omega_{1})$ $\displaystyle=e^{j\frac{2\pi
f_{c}\sin(\phi_{\text{B}})\cos(\theta_{\text{B}})}{c}{\tilde{\bf
z}}_{\text{B,2}}}=e^{j\frac{2}{\lambda_{c}}\omega_{1}{\tilde{\bf
z}}_{\text{B,2}}},$ (30) $\displaystyle{\bf
a}^{(M_{2})}_{\text{B}}(\omega_{2})$ $\displaystyle=e^{j\frac{2\pi
f_{c}\sin(\theta_{\text{B}})}{c}{\tilde{\bf
z}}_{\text{B,3}}}=e^{j\frac{2}{\lambda_{c}}\omega_{2}{\tilde{\bf
z}}_{\text{B,3}}}.$ (31)

Here, ${\tilde{\bf z}}^{\top}_{\text{B,2}}\in\mathbb{C}^{1\times
N_{\text{B}}}$ and ${\tilde{\bf z}}^{\top}_{\text{B,3}}\in\mathbb{C}^{1\times
N_{\text{B}}}$ are the second and third row of the matrix $\tilde{\bf Z}$,
respectively. The combining matrix can then be defined in terms of a grid of
the spatial frequencies
$\bar{\hbox{\boldmath$\omega$}}_{1}=[\bar{\omega}_{1,1},\ldots,\bar{\omega}_{1,M_{1}}]$
and
$\bar{\hbox{\boldmath$\omega$}}_{2}=[\bar{\omega}_{2,1},\ldots,\bar{\omega}_{2,M_{2}}]$
as

$\displaystyle{\bf T}_{1}$ $\displaystyle=[{\bf
a}^{(N_{1})}(\bar{\omega}_{1,1}),\ldots,{\bf
a}^{(N_{1})}(\bar{\omega}_{1,M_{1}})]^{\top}\in\mathbb{C}^{N_{1}\times
M_{1}},$ (32) $\displaystyle{\bf T}_{2}$ $\displaystyle=[{\bf
a}^{(N_{2})}(\bar{\omega}_{2,1}),\ldots,{\bf
a}^{(N_{2})}(\bar{\omega}_{2,M_{2}})]^{\top}\in\mathbb{C}^{N_{2}\times
M_{2}},$ (33)

where $\bar{\omega}_{1,m}$ and $\bar{\omega}_{2,m}$ are decided by beamforming
directions (detailed in Sec. V). A similar reasoning applies to the steering
vectors ${\bf a}^{(M_{3})}_{\text{U}}(\omega_{3})$ and ${\bf
a}^{(M_{4})}_{\text{U}}(\omega_{4})$ at UE to define ${\bf T}_{3}$ and ${\bf
T}_{4}$, with

$\omega_{3}=\pi\sin(\phi_{\text{U}})\cos(\theta_{\text{U}}),\ \
\omega_{4}=\pi\sin(\theta_{\text{U}}).$ (34)

We further define ${\bf b}^{(M_{n})}(\omega_{n})={\bf T}_{n}^{\sf H}{\bf
a}^{N_{n}}(\omega_{n})\in\mathbb{C}^{M_{n}}$ for $n\in\\{1,2,3,4\\}$ and ${\bf
b}^{(M_{5})}(\omega_{5})={\bf d}(\tau)$ ($\omega_{5}=2\pi\Delta_{f}\tau$), and
the beamspace channel matrix in (27) can be represented by a tensor
$\boldsymbol{{\cal H}}\in\mathbb{C}^{M_{1}\times M_{2}\times\cdots\times
M_{5}}$ as [14]

$\boldsymbol{{\cal H}}^{(b)}=\alpha{\bf
b}^{(M_{1})}(\omega_{1})\circ\ldots\circ{\bf b}^{(M_{5})}(\omega_{5}).$ (35)

In practice, the estimated beamspace channel matrix can be estimated with
known pilot signals as $\text{vec}(\hat{\bf
H}_{k}^{(b)})=[\hat{y}_{1,k}/x_{1,k},\ldots,\hat{y}_{{G},k}/x_{{G},k}]^{\top}$.
By rearranging the estimated channel into a tensor $\hat{\boldsymbol{{\cal
H}}}^{(b)}$ shown in (35), the beamspace tensor-based ESPRIT method can then
be used to estimate $\omega_{1}$ to $\omega_{5}$ and obtain the AOA, AOD, and
delay accordingly [14, 13].

####  III-B2 Fine Estimation using MMLE

From ESPRIT, we can obtain an initial estimate of the channel parameters
$\hat{\hbox{\boldmath$\eta$}}_{0}$. The refinement of the initial estimate can
be formulated as an optimization problem, based on (26), as

$\hat{\hbox{\boldmath$\eta$}}={\hbox{arg}}\min_{\boldsymbol{\eta}}\|{\bf
y}-\hbox{\boldmath$\mu$}(\hbox{\boldmath$\eta$})\|^{2}.$ (36)

Since $\alpha$ appears linearly in the noise-free observation $\mu$, we
further define
$\hbox{\boldmath$\gamma$}(\hbox{\boldmath$\eta$})=\hbox{\boldmath$\mu$}({\bf
c})/\alpha$ with ${\bf
c}=[\hbox{\boldmath$\varphi$}_{\text{B}}^{\top},\hbox{\boldmath$\varphi$}_{\text{U}}^{\top},\tau]^{\top}$.
By setting $\partial\|{\bf
y}-\hbox{\boldmath$\mu$}(\hbox{\boldmath$\eta$})\|^{2}/\partial\alpha=0$, we
can have

$\hat{\bf c}={\hbox{arg}}\min_{{\bf c}}\|{\bf
y}-\frac{\hbox{\boldmath$\gamma$}^{\sf H}({\bf c}){\bf
y}}{\|\hbox{\boldmath$\gamma$}^{\sf H}({\bf
c})\|^{2}}\hbox{\boldmath$\gamma$}({\bf c})\|^{2}.$ (37)

In this way, the nuisance parameters can be removed, which reduces the
dimension of the unknown parameters.

###  III-C Localization Algorithm

####  III-C1 Coarse Estimation

Given the estimated geometric parameter vector ${\bf c}_{l}$ ($1\leq l\leq L$)
for all the channels, the least squares solution for coarse estimation of
position and orientation as [49]

$\hat{\mathbf{R}}_{\text{U,LS}}=\left\\{\begin{array}[]{rl}\mathbf{U}\mathbf{V}^{\mathsf{T}},&\text{if
det}(\mathbf{U}\mathbf{V}^{\mathsf{T}})=1,\\\
\mathbf{U}\mathbf{J}\mathbf{V}^{\mathsf{T}},&\text{if
det}(\mathbf{U}\mathbf{V}^{\mathsf{T}})=-1,\end{array}\right.$ (38) $[\hat{\bf
p}_{\text{U,LS}},\hat{B}_{\text{U,LS}}]^{\top}=({\bf Q}_{3}^{\top}{\bf
Q}_{3})^{-1}{\bf Q}_{3}^{\top}{\bf q},$ (39)

where $\mathbf{J}=\text{diag}([1,1,-1])$, ${\bf U}$ and ${\bf V}$ are the
unitary basis matrices of the singular value decomposition of the matrix ${\bf
Q}_{1}{\bf Q}_{2}^{\top}$, together with ${\bf Q}_{3},{\bf q}$ are given by
[49]

$\displaystyle{\bf Q}_{1}$ $\displaystyle=-[{\bf R}_{\text{B},1}{\bf
t}(\hat{\hbox{\boldmath$\varphi$}}_{\text{B},1}),\ldots,{\bf
R}_{\text{B},L}{\bf t}(\hat{\hbox{\boldmath$\varphi$}}_{\text{B},L})],$ (40)
$\displaystyle{\bf Q}_{2}$ $\displaystyle=[{\bf
t}(\hat{\hbox{\boldmath$\varphi$}}_{\text{U},1}),\ldots,{\bf
t}(\hat{\hbox{\boldmath$\varphi$}}_{\text{U},L})],$ (41) $\displaystyle{\bf
Q}_{3}$ $\displaystyle=\begin{bmatrix}\mathbf{I}_{3}&{\bf R}_{\text{B},1}{\bf
t}(\hat{\hbox{\boldmath$\varphi$}}_{\text{B},1})\\\ \vdots&\vdots\\\
\mathbf{I}_{3}&{\bf R}_{\text{B},L}{\bf
t}(\hat{\hbox{\boldmath$\varphi$}}_{\text{B},L})\end{bmatrix},$ (42)
$\displaystyle{\bf q}$ $\displaystyle=\begin{bmatrix}{\bf
p}_{\text{B}}^{(1)}+{\bf R}_{\text{B},1}\hat{\tau}_{1}{\bf
t}(\hat{\hbox{\boldmath$\varphi$}}_{\text{B},1})\\\ \vdots\\\ {\bf
p}_{\text{B},L}+{\bf R}_{\text{B},L}\hat{\tau}_{L}{\bf
t}(\hat{\hbox{\boldmath$\varphi$}}_{\text{B},L})]^{\top}\end{bmatrix}.$ (43)

Different from the algorithm in [49], the estimator for position and clock
offset in (39) does not require the orientation of the UE ${\bf
R}_{\text{U}}$, which is still sufficient as a coarse estimate, as will be
shown in the simulation section.

####  III-C2 MMLE

Once the initial position and orientation results are obtained, joint position
and orientation estimation using MMLE can be formulated as

${\hat{\bf s}}=\operatorname*{arg\,min}_{{\bf s}}\sum_{l=1}^{L}({\bf
c}_{l}({\bf s})-\hat{\bf
c}_{l})^{\top}\hbox{\boldmath$\Sigma$}^{-1}_{c_{l}}({\bf c}_{l}({\bf
s})-\hat{\bf c}_{l}),$ (44)

which can be solved using the manifold optimization toolbox Manopt [50]. Note
that the covariance matrix may not be accurately obtained in practice. We
formulate localization as an MMLE problem with two purposes: (a) to evaluate
the performance improvement with known covariance matrices compared to the
coarse estimation; (b) to validate the derived bound, which will be detailed
in Sec. IV.

##  IV Lower Bound Analysis

In the next, we derive the CRB for MM, as well as the mcrb for the mismatched
estimator in (26).

###  IV-A CRB Analysis for the Mismatched Model

Based on the defined channel parameter vector $\eta$ and state vector ${\bf
s}$, the signal model in (1) and ${\bf y}\sim f_{\text{\ac{mm}}}({\bf
y}|\hbox{\boldmath$\eta$})$, the channel estimation CRB of the mm for the
$l$th channel can be obtained as
$\bm{\mathcal{I}}(\hbox{\boldmath$\eta$}_{l})^{-1}\in\mathbb{R}^{7\times 7}$
with [51]

$\displaystyle\bm{\mathcal{I}}({\boldsymbol{\eta}_{l}})$
$\displaystyle=\frac{2}{\sigma_{n}^{2}}\sum^{G}_{g=1}\sum^{K}_{k=1}\mathrm{Re}\left\\{\left(\frac{\partial\mu_{g,k}}{\partial{\boldsymbol{\eta}}_{l}}\right)^{\mathsf{H}}\left(\frac{\partial\mu_{g,k}}{\partial{\boldsymbol{\eta}}_{l}}\right)\right\\}.$
(45)

Here, $\mathrm{Re}\\{\cdot\\}$ extracts the real part of a complex variable.
Consequently, the FIM of all the channel parameters
$\hbox{\boldmath$\eta$}_{\text{ch}}$ can be formulated as

$\bm{\mathcal{I}}({\boldsymbol{\eta}_{\text{ch}}})=\text{blkdiag}(\bm{\mathcal{I}}({\boldsymbol{\eta}_{1}}),\ldots,\bm{\mathcal{I}}({\boldsymbol{\eta}_{L}})).$
(46)

where $\text{blkdiag}(\cdot)$ returns the block diagonal matrix created by
aligning the input matrices. The FIM of the state vector
$\bm{\mathcal{I}}({\bf s})\in\mathbb{R}^{13\times 13}$ can then be formulated
as

$\bm{\mathcal{I}}({\bf s})={\bf M}({\bf M}^{\top}\ {\bf
J}_{\mathrm{S}}^{\top}{\bm{\mathcal{I}}}({{\bf c}_{\text{ch}}}){\bf
J}_{\mathrm{S}}\ {\bf M})^{-1}{\bf M}^{\top},$ (47)

where ${\bm{\mathcal{I}}}({{\bf c}}_{\text{ch}})\in\mathbb{R}^{5L\times 5L}$
is the EFIM of non-nuisance parameters ${\bf c}_{\text{ch}}$ obtained from
$\bm{\mathcal{I}}({\boldsymbol{\eta}}_{\text{ch}})$, ${\bf
J}_{\mathrm{S}}\triangleq\frac{\partial{{\bf c}_{\text{ch}}}}{\partial{\bf
s}}$ is the Jacobian matrix using a denominator-layout notation, ${\bf
M}=\text{blkdiag}(\bm{\text{I}}_{4\times 4},\bar{\bf M})$ with $\bar{\bf M}$
as [9]

$\bar{\bf M}=\frac{1}{\sqrt{2}}\begin{bmatrix}-{\bf r}_{3}&\mathbf{0}_{3\times
1}&{\bf r}_{2}\\\ \mathbf{0}_{3\times 1}&-{\bf r}_{3}&-{\bf r}_{1}\\\ {\bf
r}_{1}&{\bf r}_{2}&\mathbf{0}_{3\times 1}\end{bmatrix},$ (48)

where ${\bf r}_{1}$, ${\bf r}_{2}$, and ${\bf r}_{3}$ are the first, second,
and third columns of the UE rotation matrix ${\bf R}_{\text{U}}$.

Based on $\bm{\mathcal{I}}({\boldsymbol{\eta}})$ in (45), we can define the
aod error bound (ADEB), aoa error bound (AAEB), and delay error bound (DEB) of
the link between the UE and the $l$th BS) as

$\displaystyle\mathrm{AAEB}$
$\displaystyle=\sqrt{{\hbox{tr}}([\bm{\mathcal{I}}({\boldsymbol{\eta}}_{l})^{-1}]_{1:2,1:2})},$
(49) $\displaystyle\mathrm{ADEB}$
$\displaystyle=\sqrt{{\hbox{tr}}([\bm{\mathcal{I}}({\boldsymbol{\eta}}_{l})^{-1}]_{3:4,3:4})},$
(50) $\displaystyle\mathrm{DEB}$
$\displaystyle=\sqrt{([\bm{\mathcal{I}}({\boldsymbol{\eta}}_{l})^{-1}]_{5,5})}.$
(51)

Similarly, based on $\bm{\mathcal{I}}({{\bf s}})$, we can define the position
error bound (PEB), clock offset error bound (CEB) and orientation error bound
(OEB) as

$\displaystyle\mathrm{PEB}$
$\displaystyle=\sqrt{{\hbox{tr}}([\bm{\mathcal{I}}({{\bf
s}})^{-1}]_{1:3,1:3})},$ (52) $\displaystyle\mathrm{CEB}$
$\displaystyle=\sqrt{([\bm{\mathcal{I}}({{\bf s}})^{-1}]_{4,4})},$ (53)
$\displaystyle\mathrm{OEB}$
$\displaystyle=\sqrt{{\hbox{tr}}([\bm{\mathcal{I}}({{\bf
s}})^{-1}]_{5:13,5:13})}.$ (54)

The bounds from (49)–(54) will be used to evaluate the channel estimation and
localization performance. In the next subsections, we will first formulate the
MCRB for channel estimation, and then the mismatched lower bound for position
and orientation estimation will be derived.

###  IV-B Misspecified CRB of Channel Parameters

For a given channel model, the model is said to be mismatched or misspecified
when ${\bf y}\sim f_{\text{TM}}({\bf y}|\hbox{\boldmath$\eta$})$, while the
estimation is based on the assumption that ${\bf y}\sim f_{\text{MM}}({\bf
y}|\hbox{\boldmath$\eta$})$), where $f_{\text{TM}}({\bf
y}|\hbox{\boldmath$\eta$})\neq f_{\text{MM}}({\bf y}|\hbox{\boldmath$\eta$})$.

The lb of using a mismatched estimator can be obtained as [35]

$\displaystyle\text{LB}(\bar{\boldsymbol{\eta}},{\boldsymbol{\eta}}_{0})$
$\displaystyle=\underbrace{{\bf A}_{{\boldsymbol{\eta}}_{0}}^{-1}{\bf
B}_{{\boldsymbol{\eta}}_{0}}{\bf
A}_{{\boldsymbol{\eta}}_{0}}^{-1}}_{=\text{MCRB}({\boldsymbol{\eta}}_{0})}+\underbrace{(\bar{\boldsymbol{\eta}}-{\boldsymbol{\eta}}_{0})(\bar{\boldsymbol{\eta}}-{\boldsymbol{\eta}}_{0})^{\top}}_{=\text{Bias}({\boldsymbol{\eta}}_{0})},$
(55)

where $\bar{\hbox{\boldmath$\eta$}}$ is the true channel parameter vector,
${\boldsymbol{\eta}}_{0}$ is the pseudo-true parameter vector (which will be
introduced soon), and ${\bf A}_{{\boldsymbol{\eta}}_{0}},{\bf
B}_{{\boldsymbol{\eta}}_{0}}$ are two possible generalizations of the FIMs.
The LB is a bound in the sense that

$\displaystyle\mathbb{E}\\{(\hat{\boldsymbol{\eta}}_{\text{MMLE}}-\bar{\hbox{\boldmath$\eta$}})(\hat{\boldsymbol{\eta}}_{\text{MMLE}}-\bar{\hbox{\boldmath$\eta$}})^{\top}\\}\succeq\text{LB}(\bar{\boldsymbol{\eta}},{\boldsymbol{\eta}}_{0}),$
(56)

where the expectation is with respect to $f_{\text{TM}}({\bf
y}|\boldsymbol{\eta})$. What remains is the formal definition and computation
of the pseudo-true parameter $\boldsymbol{\eta}_{0}$ and ${\bf
A}_{{\boldsymbol{\eta}}_{0}},{\bf B}_{{\boldsymbol{\eta}}_{0}}$.

####  IV-B1 Pseudo-true Parameter

Assume the pdf of the tm, where the observation data come from, is
$f_{\text{TM}}({\bf y}|\bar{\boldsymbol{\eta}})$, where ${\bf y}$ is the
received signals and $\bar{\boldsymbol{\eta}}\in\mathbb{R}^{7}$ (7 unknowns
for this case) is the vector containing all the channel parameters. Similarly,
the pdf of the mm for the received signal ${\bf y}$ can be noted as
$f_{\text{MM}}({\bf y},{\boldsymbol{\eta}})$. The pseudo-true parameter vector
is defined as the point that minimizes the Kullback-Leibler divergence between
$f_{\text{TM}}({\bf y}|\bar{\boldsymbol{\eta}})$ and $f_{\text{MM}}({\bf
y}|{\boldsymbol{\eta}})$ as

$\displaystyle{\boldsymbol{\eta}}_{0}={\hbox{arg}}\min_{\boldsymbol{\eta}}D_{\text{KL}}(f_{\text{TM}}({\bf
y}|\bar{\boldsymbol{\eta}})\|f_{\text{MM}}({\bf y}|{\boldsymbol{\eta}})).$
(57)

We define
$\hbox{\boldmath$\epsilon$}(\hbox{\boldmath$\eta$})\triangleq\bar{\hbox{\boldmath$\mu$}}(\bar{\boldsymbol{\eta}})-\hbox{\boldmath$\mu$}({\boldsymbol{\eta}})$,
and the pseudo-true parameter can be obtained as [36]

${\boldsymbol{\eta}}_{0}={\hbox{arg}}\min_{{\boldsymbol{\eta}}}\|\hbox{\boldmath$\epsilon$}(\hbox{\boldmath$\eta$})\|^{2}={\hbox{arg}}\min_{{\boldsymbol{\eta}}}\|\bar{\hbox{\boldmath$\mu$}}(\bar{\boldsymbol{\eta}})-\hbox{\boldmath$\mu$}({\boldsymbol{\eta}})\|^{2}.$
(58)

Hence, ${\boldsymbol{\eta}}_{0}$ can be found by solving (36) with the
observation ${\bf y}=\bar{\hbox{\boldmath$\mu$}}(\bar{\boldsymbol{\eta}})$,
which can be accomplished using the same algorithm in Sec. III, initialized
with the true value $\bar{\boldsymbol{\eta}}$.

####  IV-B2 MCRB Component Matrices

The matrices ${\bf A}_{{\boldsymbol{\eta}}_{0}}$ and ${\bf
B}_{{\boldsymbol{\eta}}_{0}}$ can be obtained based on the pseudo-true
parameter vector ${\boldsymbol{\eta}}_{0}$ as [36]

$\displaystyle[{\bf
A}_{{\boldsymbol{\eta}}_{0}}]_{i,j}=\left.\int\frac{\partial^{2}\text{ln}f_{\text{MM}}({\bf
y}|{\boldsymbol{\eta}})}{\partial{\eta}_{i}\partial{\eta}_{j}}f_{\text{TM}}({\bf
y}|\bar{\boldsymbol{\eta}})\text{d}{\bf
y}\right|_{{\boldsymbol{\eta}}={\boldsymbol{\eta}}_{0}}$
$\displaystyle=\left.\frac{2}{\sigma_{n}^{2}}\text{Re}\left[\frac{\partial^{2}\hbox{\boldmath$\mu$}({\boldsymbol{\eta}})}{\partial\eta_{i}\partial\eta_{j}}\hbox{\boldmath$\epsilon$}({\boldsymbol{\eta}})-\frac{\partial\hbox{\boldmath$\mu$}({\boldsymbol{\eta}})}{\partial\eta_{j}}\left(\frac{\partial\hbox{\boldmath$\mu$}({\boldsymbol{\eta}})}{\partial\eta_{i}}\right)^{\mathsf{H}}\right]\right|_{{\boldsymbol{\eta}}={\boldsymbol{\eta}}_{0}}$
(59)

and

$\displaystyle[{\bf
B}_{{\boldsymbol{\eta}}_{0}}]_{i,j}=\left.\int\frac{\partial\text{ln}f_{\text{MM}}({\bf
y}|{\boldsymbol{\eta}})}{\partial{\eta}_{i}}\frac{\partial\text{ln}f_{\text{MM}}({\bf
y}|{\boldsymbol{\eta}})}{\partial{\eta}_{j}}f_{\text{TM}}({\bf
y}|\bar{\boldsymbol{\eta}})\text{d}{\bf
y}\right|_{{\boldsymbol{\eta}}={\boldsymbol{\eta}}_{0}}$
$\displaystyle=\frac{4}{\sigma_{n}^{4}}\text{Re}\left[\frac{\partial^{2}\hbox{\boldmath$\mu$}({\boldsymbol{\eta}})}{\partial\eta_{i}}\hbox{\boldmath$\epsilon$}({\boldsymbol{\eta}})\right]\text{Re}\left[\frac{\partial^{2}\hbox{\boldmath$\mu$}({\boldsymbol{\eta}})}{\partial\eta_{j}}\hbox{\boldmath$\epsilon$}({\boldsymbol{\eta}})\right]$
$\displaystyle\left.+\frac{2}{\sigma_{n}^{2}}\text{Re}\left[\frac{\partial\hbox{\boldmath$\mu$}({\boldsymbol{\eta}})}{\partial\eta_{j}}\left(\frac{\partial\hbox{\boldmath$\mu$}({\boldsymbol{\eta}})}{\partial\eta_{i}}\right)^{\mathsf{H}}\right]\right|_{{\boldsymbol{\eta}}={\boldsymbol{\eta}}_{0}}.$
(60)

###  IV-C Absolute Lower Bound (ALB) for Localization

Another way to interpret the LB specified in (55) is that the estimated
channel parameter vector from an efficient estimator follows a nonzero-mean
multi-variable Gaussian distribution as

$\hat{\hbox{\boldmath$\eta$}}_{l}\sim\mathcal{N}({\hbox{\boldmath$\eta$}_{0,l}},{\bf
A}_{{\boldsymbol{\eta}}_{0,l}}^{-1}{\bf B}_{{\boldsymbol{\eta}}_{0,l}}{\bf
A}_{{\boldsymbol{\eta}}_{0,l}}^{-1}),$ (61)

while the assumed distribution of the MMLE is

$\hat{\hbox{\boldmath$\eta$}}_{l}\sim\mathcal{N}(\hbox{\boldmath$\eta$}_{l}(\bar{\bf
s}),\mathbf{I}(\hbox{\boldmath$\eta$}_{l})^{-1}),$ (62)

where $\bar{\bf s}$ is the true state of the UE. As a result, the position and
orientation estimation (from the channel parameter vectors of all the paths)
of the two-stage localization problem is another mismatched problem and the
bound follows as

$\text{LB}(\bar{\bf s},{\bf s}_{0})=\text{MCRB}({\bf
s}_{0})+\underbrace{(\bar{\bf s}-{\bf s}_{0})(\bar{\bf s}-{\bf
s}_{0})^{\top}}_{\text{Absolute lower bound (ALB)}}.$ (63)

Similar to (55), $\bar{{\bf s}}$ is the true state parameter vector, ${{\bf
s}}_{0}$ is the pseudo-true state parameter vector.

It is possible to derive the localization LB constrained MCRB [52]; however,
considering the high complexity when involving 3D orientation estimation, we
will focus on the bias term, defined as the absolute lower bound (ALB) of the
localization performance as $\text{ALB}=(\bar{\bf s}-{\bf s}_{0})(\bar{\bf
s}-{\bf s}_{0})^{\top}$, which can sufficiently evaluate the effect of HWIs on
localization as will be shown in Sec. V-C2 Following a similar derivation in
(58). The pseudo-true parameters for state vector ${\bf s}$ can be obtained as

${{\bf s}}_{0}=\operatorname*{arg\,min}_{\bar{\bf
s}}\sum_{l}(\hbox{\boldmath$\eta$}_{0,l}-\hbox{\boldmath$\eta$}_{l}(\bar{\bf
s}))^{\top}\bm{\mathcal{I}}(\hbox{\boldmath$\eta$}_{l})(\hbox{\boldmath$\eta$}_{0,l}-\hbox{\boldmath$\eta$}_{l}(\bar{\bf
s})),$ (64)

where
$\hbox{\boldmath$\eta$}_{0,l}={\hbox{arg}}\min_{{\boldsymbol{\eta}}}\|\bar{\hbox{\boldmath$\mu$}}(\bar{\boldsymbol{\eta}}_{l})-\hbox{\boldmath$\mu$}({\boldsymbol{\eta}_{l}})\|^{2}$
is the biased mapping obtained by calculating the pseudo-true parameters of
the $l$th channel from (58), and
$\bm{\mathcal{I}}(\hbox{\boldmath$\eta$}_{l})$ is the inverse of the
covariance matrix that can be obtained from (45).

###  IV-D Summary of Different Bounds

In this section, we introduced different types of lower bounds. For channel
geometric parameters, the CRB and LB are derived for AOA, AOD, and delay
estimations. For state parameters, the CRB and ALB are derived for the
position, orientation, and clock offset estimations. All types of the lower
bounds are summarized in Table I, which will be used in Sec. V Numerical
Results.

TABLE I: Summary of Different Lower Bounds
---
Types | Parameters | Remarks
| AOA | AOD | Delay | Channel Parameters
CRB | AAEB | ADEB | DEB |  (49)-(51)
LB | AALB | ADLB | DLB |  (55)
| Position | Orientation | Clock Offset | State Parameters
CRB | PEB | OEB | CEB |  (52)-(54)
ALB | PALB | OALB | CALB |  (63)

##  V Numerical Results

###  V-A Default Parameters

We consider a 3D MIMO uplink scenario with one UE and two BSs, and the
simulation parameters777The PA parameters are estimated from the measurements
of the RF WebLab, which can be remotely accessed at www.dpdcompetition.com.
Part of the parameters come from the Hexa-X Deliverable 3.1. can be found in
Table II. We utilize 10% of the total number of subcarriers $K_{\text{com}}$
for localization, resulting in $K=100$ subcarriers as pilot signals. The
amplitude of the channel gain is calculated as $\rho=\frac{\lambda}{4\pi
c\tau}$. The selection of these parameters is to show the performance of the
estimator in comparison to the derived bound. The analysis of each HWI type is
also discussed in the simulation results.

Regarding the evaluation of communication performance, only the first BS is
considered, and 16-qam modulation is adopted. Different from localization,
where BS-UE beam sweeping is needed, we evaluate the effect on communication
with fixed precoder and combiner vectors across different transmissions. By
considering all HWIs, we assume the channel can be perfectly estimated (with a
sufficient number of pilots) as $\hat{\bf H}=\check{\bf H}=\hat{\bf
a}_{\text{B}}\hat{\bf a}_{\text{U}}$ with $\hat{\bf
a}_{\text{B}}=\sqrt{\alpha}{\bf C}_{\text{B}}\tilde{\bf
a}_{\text{B}}(\hbox{\boldmath$\varphi$}_{\text{B}})$ and $\hat{\bf
a}_{\text{U}}=\sqrt{\alpha}\tilde{\bf
a}_{\text{U}}(\hbox{\boldmath$\varphi$}_{\text{U}}){\bf C}_{\text{U}}^{\top}$
from (23). In order to maximize the SNR with the amplitude constraints of the
precoder and combiner, we choose ${\bf w}$ and ${\bf v}$ respectively as the
conjugate of $\hat{\bf a}_{\text{B}}$ and $\hat{\bf a}_{\text{U}}$ with each
of the elements normalized to a unit amplitude. For each realization, 20 OFDM
symbols are sent with data drawn randomly from 16-qam, and ser is used to
evaluate the effect of HWIs on communication.

For localization, the pilot signal $x_{g,k}$ is chosen with random phase and a
constant amplitude $|x_{g,k}|^{2}={P/{N_{\text{U}}}}$. To assist the beamspace
ESPRIT algorithm, we set the number of sweeping beams as $M_{1}=4$, $M_{2}=4$,
$M_{3}=3$, $M_{4}=3$ with a total number of transmission $G=144$. For a
specific spatial frequency vector $\bar{\hbox{\boldmath$\omega$}}_{n}$
($n\in\\{1,2,3,4\\}$), we assume the sweeping range as
$(M_{n}-1)\Delta_{\omega}$ centered at the location prior
$\mathring{\omega}_{n}=\omega_{n}+\delta_{\omega}$, where $\omega_{n}$ is
defined in (29), (34), and $\delta_{\omega}$ is the error). More specifically,
we choose
$\bar{\omega}_{n,m}=\omega_{n}+\delta_{\omega}+\frac{2m-M_{n}-1}{2}\Delta_{\omega}$,
with $\Delta_{\omega}=0.15$ and $\delta_{\omega}=0.05$ in the simulation. The
sweeping priority is set to ‘BS-first’ by default, which means that the UE can
change its precoder vector when the BS finishes the $M_{1}M_{2}=16$ different
sweeping beams. Different error bounds (i.e., CRBs, LBs, ALBs from Table I)
are utilized as localization performance metrics.

TABLE II: Default Simulation Parameters
---
Parameters | True Model | Mismatched Model
BS |  ${\bf p}^{1}_{\text{B}}=[0,0,3]^{\top}$, ${\bf p}^{2}_{\text{B}}=[0,5,3]^{\top}$
BS Orientations |  ${\bf o}^{1}_{\text{B}}=[0^{\circ},15^{\circ},0^{\circ}]^{\top}$, ${\bf o}^{2}_{\text{B}}=[-30^{\circ},15^{\circ},0^{\circ}]^{\top}$
BS Antennas | $N^{1}_{\text{B}}=N^{2}_{\text{B}}=8\times 8$
UE Position | ${\bf p}_{\text{U}}=[4,2,1.5]^{\top}$
UE Orientation | ${\bf o}_{\text{U}}=[180^{\circ},0^{\circ},0^{\circ}]^{\top}$
UE Antennas | $N_{\text{U}}=4\times 4$
RFC at BS/UE | $1$
Carrier Frequency | $f_{c}=140\,\mathrm{GHz}$
Bandwidth | $W=1\,\mathrm{GHz}$
Transmissions | $G=4\times 4\times 3\times 3=144$
Subcarriers (Total) |  $K_{\text{com}}=1040$ ($\Delta_{f}=960\,\mathrm{kHz}$)
Subcarriers (Pilots) | $K=100$
Length of the CP | $K_{\text{cp}}=7$
Load Impedance | $R=50\,\mathrm{\Omega}$
Noise PSD | $N_{0}=-173.855\,\mathrm{dBm/Hz}$
Noise Figure | $10\,\mathrm{dB}$
Phase Noise | $\sigma_{\text{PN}}=2.5^{\circ}\;\;\;\;\;\;\;$ | $\sigma_{\text{PN}}=0^{\circ}$
Carrier Freq. Offset |  $\sigma_{\text{CFO}}=5e{-4}$ (0.036 ppm) | $\sigma_{\text{CFO}}=0$
Mutual Coupling | $\sigma_{\text{MC}}=0.002$ | $\sigma_{\text{MC}}=0$
|  $\beta_{1}=$ $0.9798$+$0.0286j$ |
Power Amplifier |  $\beta_{2}=$ $0.0122$-$0.0043j$ | n/a
|  $\beta_{3}=$ $-0.0007$+$0.0001j$ |
PA Clipping Voltage | $x_{\text{clip}}=1\,\mathrm{V}$ | n/a
Array Gain Error | $\sigma_{\text{GA}}=\sigma_{\text{GP}}=0.002$ | $\sigma_{\text{RA}}=\sigma_{\text{RP}}=0$
Antenna Disp. Error |  $\sigma_{\text{AD}}=5\,\mathrm{um}$ ($2.3e{-3}\lambda$) | $\sigma_{\text{AD}}=0$
IQ Imbalance | $\sigma_{\text{IA}}=\sigma_{\text{IP}}={0.02}$ | $\sigma_{\text{IA}}=\sigma_{\text{IP}}=0$

###  V-B The Effect of HWIs on Communication

####  V-B1 The Effect of HWIs on SER

We approximate the effect of HWIs on communication as the random noise and
evaluate the effect on SER based on numerical and analytical results888The SER
of M-QAM can be calculated as
$\text{SER}_{M}=1-(1-\frac{2\sqrt{M}-1}{\sqrt{M}}\text{Q}(\sqrt{\frac{3\text{SNR}}{M-1}}))^{2}$
[53, (6,23)], where $\text{Q}(\cdot)$ is the Q-function and SNR is effective
SNR considering both approximated HWI noise and background noise.. Considering
that the effects of some HWIs depend on the amplitude of the symbol (e.g.,
PAN), we also obtain the minimum and maximum noise levels across different
symbols to evaluate the lower bound and upper bound of the SER. The SERs of
16-QAM with different transmit power for different HWI coefficients are
visualized in Fig. 2, where the black solid curve is the benchmark SER without
HWIs. By default, $c_{\text{HWI}}=1$, and the HWI level is the same as the
parameters in Table II. A value of $c_{\text{HWI}}=10$ indicates that the
standard deviations (e.g., $\sigma_{\text{PN}},\sigma_{\text{CFO}}$) of all
the impairments (except for PAN) are multiplied by 10. We can see from the
figure that the analytical SERs with approximated noise levels (red, blue, and
green markers) are close to the numerical SERs (solid red, blue and green
curves), and both are within the lower and upper bounds (shaded areas). We can
also see from Fig. 2 that the selected impairment level (i.e.,
$c_{\text{HWI}}=1$) has limited effects on communication. However, we will
show the localization performance will be affected by the same level of HWIs
in Sec. V-C.

$-10$$-5$$0$$5$$10$$15$$10^{-7}$$10^{-5}$$10^{-3}$$10^{-1}$$P$ [dBm]SER
(16-QAM)Anal. without HWINumer. HWI ($c_{\text{HWI}}=0.1$)Anal.-Approx. HWI
($c_{\text{HWI}}=0.1$)Numer. HWI ($c_{\text{HWI}}=1$)Anal.-Approx. HWI
($c_{\text{HWI}}=1$)Numer. HWI ($c_{\text{HWI}}=2$)Anal.-Approx. HWI
($c_{\text{HWI}}=2$) Figure 2: The effect of different HWI levels on SER.
Numerical results for 100 realizations and analytical results calculated with
approximated equivalent HWI noise. The boundaries of the shadow areas indicate
the upper and lower bounds for SER.
$-10$$-5$$0$$5$$10$$15$$10^{-7}$$10^{-5}$$10^{-3}$$10^{-1}$$P$ [dBm]SER
(16QAM)PNPANCFOIQIMC+AGE+ADEWithout
HWI$-10$$-5$$0$$5$$10$$15$$10^{-7}$$10^{-5}$$10^{-3}$$10^{-1}$$P$ [dBm]SER
(16QAM)PNPANCFOIQIMC+AGE+ADEWithout HWI Figure 3: The effect of individual
HWIs on SER using approximated equivalent HWI noise. Under current simulation
parameters, the PN, PAN, CFO and IQI increase the SER, whereas the MC, AGE and
ADE have negligible effects on communication.

####  V-B2 The Effect of Individual HWIs on SER

We are also interested in the effect of individual HWIs on communication. By
considering PN, CFO, PAN, and IQI one by one, the SERs under HWI are shown in
Fig. 3. Benchmarked by the solid black curve without HWIs, these factors
degrade SERs. We also performed simulations by including MC, AGE, ADE at the
same time, as shown in the dashed curve with cross markers, and found their
effects on communication are negligible under the current simulation setup.

####  V-B3 Insights into the Impact of HWI on Communication

To gain further insight into the effect of HWI on communication, we separate
the overall system noise into equivalent HWI noise and background noise. We
can see from Fig. 4 that the equivalent HWI noise is determined by the HWI
level and has an approximately linear relationship with the transmit power
(when working within the linear region of the PA). In addition to the fixed
background noise, the overall equivalent noise level keeps increasing and is
dominated by the HWIs at high transmit power.

$-10$$-5$$0$$5$$10$$15$$-110$$-100$$-90$$-80$$-70$$-60$$P$ [dBm]Equivalent
Noise Level [dBm]Overall Noise ($c_{\text{HWI}}=2$)HWI Noise
($c_{\text{HWI}}=2$)Overall Noise ($c_{\text{HWI}}=1$)HWI Noise
($c_{\text{HWI}}=1$)Overall Noise ($c_{\text{HWI}}=0.1$)HWI Noise
($c_{\text{HWI}}=0.1$)Background Noise Figure 4: Visualization of overall
system noise, equivalent HWI, and background noise with different transmit
power $P$. The background noise has a large effect on communication in low
transmit power, whereas the HWIs contribute more in high transmit power.
$-10$$0$$10$$20$$30$$40$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$P$
[dBm]AOA $[^{\circ}]$ / AOD $[^{\circ}]$ / Delay $[\text{m}]$AOA-ESPRITAOD-
ESPRITDelay-ESPRITAOA-MMLEAOD-MMLEDelay-MMLEAAEBADEBDEBAALBADLBDLB Figure 5:
Comparison between channel parameters estimation results (ESPRIT and MMLE) and
different lower bounds (CRB of the MM and the LB of the mismatched estimator)
in terms of AOA, AOD and delay. Due to the HWIs, the performance starts to
saturate when the transmit power exceeds 30 dBm. Although the performance of
the coarse estimation using ESPRIT (using a mismatched model) may not attain
the theoretical bounds (especially for delay estimation), the refined results
using MMLE can reach the LB (solid curves align well with the cross-marked
dotted curve).
$-10$$0$$10$$20$$30$$40$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$P$
[dBm]Pos $[\text{m}]$ / Ori $[^{\circ}]$ / Clock $[\text{m}]$POS-LSORI-
LSClock-LSPOS-MMLEORI-MMLEClock-MMLEPEBOEBCEBPALBOALBCALB Figure 6: Comparison
between localization results (position, orientation, and clock offset
estimation) and different lower bounds (CRB of the MM and the LB of the
mismatched estimator). We noticed that the LS estimators are sufficient for
this 2-BS scenario, and the refined results using MMLE attain the ALBs.

###  V-C The Effect of HWIs on Localization

Before analyzing the HWIs in detail, we first establish the validity of the
derived bounds by comparing them against the performance of practical
algorithms.

####  V-C1 Channel Estimation Results

For convenient analysis, we adopt one specific realization of the HWIs for the
system. The results of channel parameters estimation using ESPRIT (circle,
square, and diamond markers) and MMLE (solid curves) are shown in Fig. 5. The
estimators are benchmarked by the CRBs of the ideal/mismatched model (CRB-MM,
dashed curves) and the LB using a mismatched model (dotted curves with cross
markers). Note that the average transmit power $P$ is calculated without
considering the nonlinearity of the power amplifier (calculated before the
PA). When the transmit power $P$ is low, the LB is determined by the MCRB
(since the bias part is constant, see (55)) and has a similar performance as
CRBs. This indicates that in low transmit power, the mismatched model will not
significantly affect the performance, as the expected accuracy is low and
limited by the noise. With the increase of transmit power, the contribution of
MCRB decreases due to an increased SNR, and eventually, the mismatched
localization is lower bounded by the alb (bias part in (55)). This indicates
that the localization performance can no longer be improved by increasing
transmit power, which cannot be ignored in scenarios requiring high-accuracy
localization performance999Note that the analysis here is under the same level
of residual noise (e.g., PN, CFO, IQI). In practice, the impairment levels
depend on specific HWI calibration algorithms and transmit power.. Regarding
the estimators, the ESPRIT (using a mismatched model) provides low-complexity
results with limited performance in delay estimation. However, the refined
results using MMLE can reach the LB (solid curves align well with the dotted
curve).

####  V-C2 Localization Results

Based on the estimated channel parameters, we are able to estimate the UE
position and orientation. Similar to the channel estimation results, two
estimators (LS and MMLE) and two bounds (CRB and LB) are evaluated. The
results for localization are shown in Fig. 6. From the figure, we can see that
at low transmit powers, the LB and CRBs coincide, implying that the HWIs are
not the main source of error. At higher transmit powers ($10\,\mathrm{dBm}$
for OEB, and $20\,\mathrm{dBm}$ for PEB), LB deviates from the CRBs, and the
positioning performance is thus more severely affected by HWIs. The MMLE in
high SNR is close to the ALB, indicating the validity of the MCRB analysis.

$0$$5$$10$$15$$20$$25$$30$$35$$40$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$P$
[dBm]AALB (Average)AALB (Multi)AAEBADLB (Average)ADLB (Multi)ADEBDLB
(Average)DLB (Multi)DEB

(a) PN

$0$$5$$10$$15$$20$$25$$30$$35$$40$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$P$
[dBm]

(b) CFO

$0$$5$$10$$15$$20$$25$$30$$35$$40$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$P$
[dBm]

(c) MC

$0$$5$$10$$15$$20$$25$$30$$35$$40$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$P$
[dBm]

(d) AGE

$0$$5$$10$$15$$20$$25$$30$$35$$40$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$P$
[dBm]

(e) ADE

$0$$5$$10$$15$$20$$25$$30$$35$$40$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$P$
[dBm]

(f) IQI

Figure 7: LBs of channel parameter estimation under different types of
impairment with multiple realizations: (a) Phase noise, (b) Carrier frequency
offset, (c) Mutual coupling, (d) Array gain error, (e) Antenna displacement
error, (f) IQ-imbalance.

Now that the validity of the bounds has been established, we rely solely on
the bounds to evaluate the effect of HWIs on localization. First, the
impairments are studied individually, then the impact of the waveform type is
evaluated, and finally, the impairment levels are varied.

####  V-C3 The Effect of Individual Impairments

To understand the effect of different types of hwi, we study the LB for AOA,
AOD, and delay estimation by considering one type of hwi at a time. The
results are shown in Fig. 7 for (a) PN, (b) CFO, (c) MC, (d) AGE, (e) ADE and
(f) IQI. The effect of PA will be separately discussed in Sec. V-C4.
Considering we define the HWIs as random variables with a fixed impairment
level as shown in Table II, we perform multiple hardware realizations with a
fixed pilot signal and plot all the resultant LBs in the shaded regions. We
can see that different types of the hwi affect angle and delay estimation
differently. The PN, CFO, and IQI introduce noise on the symbols across
different subcarriers and hence affect delay estimation101010If multiple RFCs
or several local oscillators are adopted in the array, PN may have a larger
effect on angle estimation.. Since the phase change introduced by CFO affects
the phase changes across beams, angle estimation will also be affected.
Instead of affecting the phase changes between different subcarriers, the MC,
AGE, and ADE distort the steering vectors and therefore have a more
significant effect on the angle estimation. For all the HWIs, the negative
effect on the performance occurs when the transmit power is high.

One special observation is that the effect of CFO on the AOA is less
pronounced than on AOD in Fig. 7 (b). This is because the sweeping strategy is
‘BS-first’. For a system with analog arrays, the estimation of AOA/AOD relies
on phase shifts across consecutive beams over time, meaning the angle cannot
be estimated from a single receive beam, like in a digital array. If the BS
sweeps across different beams while the UE is using a fixed beam, the AOA can
be estimated in one BS sweep, and the effect of CFO will be minor. However,
the AOD estimation requires multiple BS sweeps, which increases the effect of
CFO. To verify the explanation, we further changed the sweeping strategy from
‘BS-first’ to ‘UE-first,’ and the results with different array sizes can be
found in Fig. 8. We can see that the AOA is less affected if the sweeping is
‘BS-first’ (blue curves in (a)) as shown in (12). Similarly, the AODs are less
affected if the sweeping is ‘UE-first’ (dashed red curves in (b)) with a large
UE array size. However, when the array size is small, sweeping order will have
less impact (i.e., the gaps are small between the dashed curves in (a) and the
solid curves in (b)).

$0$$10$$20$$30$$40$$50$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$P$ [dBm]Angle Error
[∘]BS 8x8, UE 4x4, BS firstBS 8x8, UE 4x4, UE firstBS 4x4, UE 8x8, BS firstBS
4x4, UE 8x8, UE first

(a) AALB (average)

$0$$10$$20$$30$$40$$50$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$P$ [dBm]Angle Error
[∘]

(b) ADLB (average)

Figure 8: The effect of CFO on channel geometrical parameters with different
sweeping strategies. The ‘BS first’ strategy (blue curves) works better for
AOA estimation, while the ‘UE first’ strategy (red curves) works better for
AOD estimation.

####  V-C4 The Effect of PA with Different Pilot Signals

High papr is one of the critical issues in implementing the OFDM signals, and
a promising alternative is to use DFT-S-OFDM [54]. When increasing the
transmit power, the PAN is more likely to happen, as can be seen in Fig. 9
(a). The delay estimation suffers more from the nonlinear distortion because
of the clipping of transmit signal, which distorts the uniformity of phase
changes across the subcarriers. The effect on angle estimation is less
pronounced (at the same level of transmit power) since different antenna
elements experience similar distortions with identical PAs adopted in this
work. We compare using the random OFDM symbols and the FFT version of the
benchmark symbols (a special case of DFT-S-OFDM by choosing an identity
mapping matrix [54]), and the results are shown in Fig. 9. Due to the reduced
papr by DFT-S-OFDM, the localization performance can be improved, as shown in
the right figure.

$20$$25$$30$$35$$40$$45$$50$$55$$60$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$P$
[dBm]AALB (Average)AALB (Multi)AAEBADLB (Average)ADLB (Multi)ADEBDLB
(Average)DLB (Multi)DEB

(a) OFDM

$20$$25$$30$$35$$40$$45$$50$$55$$60$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$P$
[dBm]

(b) DFT-S-OFDM

Figure 9: The effect of PA on channel parameters estimation using (a) OFDM,
and (b) DFT-S-OFDM.

$-1$$-0.5$$0$$0.5$$1$$10^{-6}$$10^{-3}$$10^{0}$$10\text{log}(c_{\text{HWI}})$PALB
[m]ALLPNCFOMCAGEADEIQI

(a) PALB

$-1$$-0.5$$0$$0.5$$1$$10^{-6}$$10^{-3}$$10^{0}$$10\text{log}(c_{\text{HWI}})$OALBALLPNCFOMCAGEADEIQI

(b) OALB

Figure 10: An example of ALB with different levels of impairments: (a) PALB,
(b) OALB. The ALBs of the position and orientation affected by the HWIs
increase with $c_{\text{HWI}}$ (reflecting the impairment level).

####  V-C5 Evaluation of HWIs with Different Impairment Levels

We further evaluate the position and orientation alb with different levels of
HWIs by defining a HWI coefficient $c_{\text{HWI}}$. With different value of
$c_{\text{HWI}}$, the position ALB and orientation ALB by considering all the
HWIs, and individual HWIs, are shown in Fig. 10 (a) and (b). All the results
indicate the 75th percentile of the total 100 realizations. We notice that the
effect of PN, MC, AR, AG, and IQI on the localization increases approximately
in a linear trend with impairment level. The CFO has a larger effect in high
impairment level as the error residue accumulates over time. Based on Fig. 10,
we can quantize the contribution of individual HWIs (e.g., if the ALBs are
much smaller than current CRB, the negative contribution of HWI on
localization is negligible). In addition, it can also identify dominant
impairment factors for further compensation (e.g., ADE is one of the dominant
factors under current system parameters).

###  V-D Summary

From the simulation, we found that the HWIs affect both localization and
communication, especially at high transmit power. The equivalent noise is
mainly contributed by HWIs for communication, and the localization performance
will saturate due to model mismatch. However, different types of HWIs affect
localization and communication differently. The effect of the individual
impairment on angle/delay estimation and communication (i.e., SER) is
summarized in Table III, with two levels of impacts H/L to denote High/Low.
Note that in this uplink scenario, the position estimation is mainly affected
by AOA and TOA information, while the orientation estimation is mainly
affected by AOD.

As for the angle estimation for localization, the performance is strongly
affected by CFO, MC, AGE, and ADE. When talking about the TOA, it is mainly
affected by PN, CFO and IQI. Since communication does not exploit the phase
relationship between antennas (e.g., no sweeping is needed once the
communication link is established), SER will be affected by the same factors
as delay estimation, which are verified in Fig. 7. It should be noted that the
effect of CFO on AOA and AOD estimation depends on the sweeping order and
number of transmissions, while the effect of PA depends on the transmit power
and the nonlinear region of the amplifier.

TABLE III: Summary of the effects of HWIs on localization and communication

Type of HWI | AOD | AOA | TOA | SER
---|---|---|---|---
Phase Noise | L | L | H | H
Carrier Frequency Offset | H∗ | H∗ | H | H
Mutual Coupling | H | H | L | L
Power Amplifier Nonlinearity | H∗ | H∗ | H∗ | H∗
Array Gain Error | H | H | L | L
Antenna Displacement Error | H | H | L | L
IQ Imbalance | L | L | H | H

∗The effect of CFO on angle estimations depends on the sweeping order and
number of transmissions. The effect of PAN depends on the transmit power and
the nonlinear region of the amplifier.

##  VI Conclusion

As the requirements on localization and communication performance are more
stringent to support new applications, hwi become a prominent factor affecting
the performance in 6G systems. We have modeled different types of hwi and
utilized the mcrb to evaluate the localization error caused by model-mismatch.
The effects of HWIs on angle/delay and position/orientation estimation are
evaluated. We found that PN and IQI have a stronger effect on delay
estimation, while MC, AGE, and ADE have a more significant effect on angle
estimation. The CFO and PAN affect both angle and delay, where the former one
depends on the sweeping strategy and number of transmissions, and the latter
factor is determined by the transmit power (or amplitude) of the signals.
Furthermore, we evaluated the effect of individual HWIs on the communication
performance in terms of ser. The dominant impairments that degrade SER (i.e.,
PN, CFO, PA, and IQI) are in good agreement with the factors that affect delay
estimation.

In summary, the localization and communication performance that improves with
transmit power in an ideal model will saturate due to the effect of HWIs. To
fully realize the potential of 6G joint localization and communication system,
a dedicated pilot signal design and algorithms for estimating and mitigating
HWI are needed. Further works can consider the effect of HWIs in multipath and
reconfigurable intelligent surface-aided scenarios, as well as learning-based
methods for mismatch mitigation.

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