5g short=5G, long= fifth generation, 6g short=6G, long= sixth generation, 3d
short=3D, long= three-dimensional, 2d short=2D, long= two-dimensional, aod
short=AOD, long= angle-of-departure, aosa short=AOSA, long= array-of-subarray,
adod short=ADOD, long= angle-difference-of-departure, aoa short=AOA, long=
angle-of-arrival, adc short=ADC, long= analog to digital converter, aeb
short=AEB, long= angle error bound, av short=AV, long= autonomous vehicle, bs
short=BS, long= base station, csi short=CSI, long= channel state information,
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, cmos
short=CMOS, long= complementary metal-oxide-semiconductor, crlb short=CRLB,
long= Cramér-Rao lower bound, cdf short=CDF, long= cumulative distribution
function, cp short=CP, long= cyclic prefix, dac short=DAC, long= digital to
analog converter, dfl short=DFL, long= device-free localization, dmimo
short=D-MIMO, long= distributed MIMO, dlprs short=DL-PRS, long= downlink
positioning reference signal, d2d short=D2D, long= device-to-device, dftsofdm
short=DFT-s-OFDM, long= discrete-Fourier-transform spread OFDM, dl short=DL,
long= deep learning, efim short=EFIM, long = equivalent Fisher information
matrix, gps short=GPS, long= global positioning system, fim short=FIM, long =
Fisher information matrix, hwi short=HWI, long= hardware impairment, hemt
short=HEMT, long= high electron mobility transistor, hbt short=HBT, long=
heterojunction bipolar transistors, iot short=IoT, long= internet of things,
imu short=IMU, long= inertial measurement unit, isac short=ISAC, long=
integrated sensing and communication, iqi short=IQI, long= in-phase and
quadrature imbalance, ip short=IP, long= incidence point, ia short=IA, long=
initial access, 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, mds short=MDS, long= multidimensional
scaling , mimo short=MIMO, long= multiple-input-multiple-output, siso
short=SISO, long= single-input-single-output, mm short=MM, long= mismatched
model, mpc short=MPC, long= multipath components, 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, 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, rfid short=RFID, long= radio frequency identification, ris
short=RIS, long= reconfigurable intelligent surface, rss short=RSS, long=
received signal strength, rtt short=RTT, long= round-trip time, sm short=SM,
long= standard model, sige short=SiGe, long= silicon-germanium, spp short=SPP,
long= surface plasmon polariton, sa short=SA, long= subarray, 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,

# Doppler-Enabled Single-Antenna Localization and Mapping Without
Synchronization

Hui Chen, Fan Jiang, Yu Ge, Hyowon Kim, Henk Wymeersch  Department of
Electrical Engineering, Chalmers University of Technology, Sweden
Email: {hui.chen; fan.jiang; yuge; hyowon<EMAIL_ADDRESS>
###### Abstract

Radio localization is a key enabler for joint communication and sensing in the
fifth/sixth generation (5G/6G) communication systems. With the help of
multipath components (MPCs), localization and mapping tasks can be done with a
single base station (BS) and single unsynchronized user equipment (UE) if both
of them are equipped with an antenna array. However, the antenna array at the
UE side increases the hardware and computational cost, preventing localization
functionality. In this work, we show that with Doppler estimation and MPCs,
localization and mapping tasks can be performed even with a single-antenna
mobile UE. Furthermore, we show that the localization and mapping performance
will improve and then saturate at a certain level with an increased UE speed.
Both theoretical Cramér-Rao bound analysis and simulation results show the
potential of localization under mobility and the effectiveness of the proposed
localization algorithm.

###### Index Terms:

Radio localization, mmWave, Doppler, multipath, CRB

##  I Introduction

Location and map information can assist communication in the the millimeter
wave/Terahertz (mmWave/THz) band 5G/6G systems, as well as a variety of
location-based applications such as autonomous driving [1], tactile robots
[2], etc. Due to the geometrical channel property of high frequency radio
signals [3], the localization and mapping tasks can be done by exploiting the
channel parameters (e.g., angles and delay) of each path from the estimated
channel matrix [4]. With the help of one or more reference points (e.g., bs or
ris), the position and orientation of the ue, as well as the position of the
ip, can be estimated [5].

Localization and mapping for ofdm-based communication systems have been
studied extensively; with the assistance of nlos paths in 2d or 3d scenarios
[3, 6]. Recent research considers single-snapshot localization and mapping
with a single-antenna receiver by assuming the system is synchronized [7].
However, only single snapshot localization111Even though multiple measurements
are performed, they are all within a channel coherence interval so that the
channel is assumed to be fixed, and we refer to this as single snapshot
localization. is discussed in these works, and no mobility is involved.
Nevertheless, UE mobility could be found in a lot of scenarios, such as
autonomous driving [1] and high speed trains [8], which degrades the
localization accuracy without considering the Doppler effect [9].

The traditional way of dealing with this issue is to reduce the coherence
time, within which the channel could be considered fixed. In addition, with a
properly designed signal, such as Zadoff-Chu sequence [10], the Doppler effect
can be mitigated. However, estimating the Doppler could be a better option
that can benefit localization and tracking [11, 12, 13, 14]. In [12], passive
localization using Doppler shift is discussed, proving that a unique position
can be estimated with at least 5 Doppler measurements in 2D scenarios. The
work in [13] shows that the Doppler effect increases the AOA information,
which can be interpreted as the enlargement of the virtual array aperture
brought by the movement. More recent work considers a single-BS mimo ofdm
mmWave system, showing the NLOS-only scenario can be significantly improved by
mobility, depending on the variance of the synchronization error [14].

In this work, we consider a more challenging MISO scenario to locate a single-
antenna ue with a single ula BS under unknown clock offset. The contributions
of this work is summarized as follows: (i) We show that with a sufficient
number of mpc, Doppler estimation can enable localization and mapping (which
is previously impossible for a stationary UE) by providing extra radial
velocity measurements, in addition to angles and delays; (ii) We prove that
the localization and mapping performance improves and then saturates with an
increased velocity, while the velocity estimation performance keeps
decreasing, which is determined by the geometrical relationship of different
paths; and (iii) We propose a simple 1-D search localization and mapping
algorithm, and use simulations and crb analysis to show the effectiveness of
the algorithm.

Figure 1: Illustration of Doppler-assisted localization. If the UE is
stationary, the localization problem is not solvable where ${\bf
p}_{0}^{\prime}$ is one of ambiguous positions that produces the same geometry
parameters as the true position ${\bf p}_{0}$ (i.e., AODs and time-difference-
of-arrivals).

##  II Problem Statement

In this section, we start with the system model and then describe the
relationship between the channel parameters and unknown states. How Doppler
can assist localization with a sufficient number of nlos paths are also
discussed.

###  II-A System Model

We consider a 2D downlink scenario with one BS equipped with a
$N_{\text{B}}$-antenna ula (with the center ${\bf p}_{\text{B}}$ located at
the origin of the coordinate system) and one UE with a single antenna [15].
The location of the UE at the $g$th transmission is
$\mathbf{p}_{0,g}=\mathbf{p}_{0}+\mathbf{v}gT_{\text{int}}$ where
$T_{\text{int}}$ is the measurement interval and ${\bf v}$ is the velocity,
assumed to be fixed within the observation duration $GT_{\text{int}}$ (where
$G$ is the number of total transmissions). We assume there exist one los path
($l=0$) and $L\geq 1$ nlos paths (with ip located at $\mathbf{p}_{l},1\leq
l\leq L$), as illustrated in Fig. 1. If only one rfc is adopted at both the bs
and ue side, the observation model (the extension to 3D scenarios are also
valid) can be formulated as

$\displaystyle y_{g,k}=$ $\displaystyle\mu_{g,k}+n_{g,k}={\bf
h}_{g,k}^{\top}{\bf w}_{\text{B},g}x_{g,k}+n_{g,k},$ (1)

where $\mu_{g,k}$ is the noise-free version of the received signal, ${\bf
w}_{\text{B},g}$ is the precoding matrix at the BS of the $g$th transmission
containing the phase-shifter coefficients with unit amplitude $|{\bf
w}_{\text{B},i}|=1$, $x_{g,k}$ is the transmitted signal symbol of the $g$th
transmission at the $k$th subcarrier with a constant average transmission
power $P$ ($|x_{g,k}|^{2}={P}$), $n_{g,k}\sim\mathcal{CN}(0,\sigma_{n}^{2})$
is the additive white Gaussian noise. The channel matrix ${\bf h}_{g,k}$ can
be expressed as

$\displaystyle{\bf h}_{g,k}=$ $\displaystyle\sum_{l=0}^{L}\rho_{l}{\bf
a}_{\text{B}}({\theta}_{\text{B},l})e^{-j2\pi\Delta_{f}k\tau_{l}}e^{j2\pi
gT_{\text{int}}v_{l}/\lambda},$ (2)

where ${\bf a}_{\text{B}}({\theta})$ is the steering vector at BS,
$\Delta_{f}$ is the subcarrier spacing, $\lambda$ is the wavelength, and
$\rho_{l}$, $\theta_{\text{B},l}$, $\tau_{l}$, $v_{l}$ ($0\leq l\leq L$) are
the channel parameters representing the complex channel gain, aod, delay, and
radial velocity of the $l$th path, respectively. Next, we describe the
relationship between the channel parameters and state parameters.

###  II-B Channel Parameters

Consider the sparse property of the channel, it is possible to use a limited
number of parameters to represent the channel matrix with a much larger
dimension as shown in (2). We define a _channel parameter vector_ as
$\hbox{\boldmath$\gamma$}=[\hbox{\boldmath$\gamma$}_{0}^{\top},\ldots,\hbox{\boldmath$\gamma$}_{L}^{\top}]^{\top}$
($\hbox{\boldmath$\gamma$}_{l}=[\theta_{l},\tau_{l},v_{l}]^{\top}$), and a
_state vector_ as ${\bf s}=[{\bf p}_{0}^{\top},{\bf p}_{1}^{\top},\ldots,{\bf
p}_{L}^{\top},B,{\bf v}^{\top}]^{\top}$. Note that for a complete FIM
calculation, the nuisance parameters (i.e., complex channel gain
$\rho_{l}=\alpha_{l}e^{-j\xi_{l}}$) should be considered such that
$\tilde{\hbox{\boldmath$\gamma$}}=[\hbox{\boldmath$\gamma$}^{\top},\hbox{\boldmath$\alpha$}^{\top},\hbox{\boldmath$\xi$}^{\top}]^{\top}$
and $\tilde{\bf s}=[{\bf
s}^{\top},\hbox{\boldmath$\alpha$}^{\top},\hbox{\boldmath$\xi$}^{\top}]^{\top}$
(where $\hbox{\boldmath$\alpha$}=[\alpha_{0},\ldots,\alpha_{L}]^{\top}$ and
$\hbox{\boldmath$\xi$}=[\xi_{0},\ldots,\xi_{L}]^{\top}$ contain the amplitude
and phase of the channel gain of the $L+1$ paths) are able to present all the
unknowns. In the following, we use $\gamma$ and ${\bf s}$ to indicate the
unknowns of interest for convenience. For a stationary scenario, the radial
velocity $v_{l}$ as well as the velocity unknown ${\bf v}$ can be removed. The
relationship between channel parameters and the state parameters can be
expressed as

$\displaystyle\theta_{l}$ $\displaystyle=\begin{cases}\arctan
2(t_{\text{B},0,2},t_{\text{B},0,1}),{\bf t}_{\text{B},0}=\frac{{\bf
p}_{0}-{\bf p}_{\text{B}}}{d_{0}}&l=0\\\ \arctan
2(t_{\text{B},l,2},t_{\text{B},l,1}),{\bf t}_{\text{B},l}=\frac{{\bf
p}_{l}-{\bf p}_{\text{B}}}{d_{l,1}}&l>0\end{cases},$ (3)
$\displaystyle\tau_{l}$
$\displaystyle=\frac{d_{l}+B}{c}=\begin{cases}\frac{{\|{\bf p}_{0}-{\bf
p}_{\text{B}}\|}+B}{c}&l=0\\\ \frac{{\|{\bf p}_{l}-{\bf
p}_{\text{B}}\|}+{\|{\bf p}_{l}-{\bf p}_{0}\|}+B}{c}&l>0\end{cases},$ (4)
$\displaystyle v_{l}$ $\displaystyle=\mathbf{v}^{\top}{\bf
t}_{\text{U},l}=\begin{cases}\mathbf{v}^{\top}\frac{{\bf p}_{\text{B}}-{\bf
p}_{\text{U}}}{d_{0}}&l=0\\\ \mathbf{v}^{\top}\frac{{\bf p}_{\text{l}}-{\bf
p}_{\text{U}}}{d_{l,2}}&l>0\end{cases},$ (5)

where $\alpha$ and $\xi$ are unknown vectors to be estimated, $d_{0}={\|{\bf
p}_{0}-{\bf p}_{\text{B}}\|}$ is the distance between the BS and UE,
$d_{l}=d_{l,1}+d_{l,2}={\|{\bf p}_{l}-{\bf p}_{\text{B}}\|}+{\|{\bf
p}_{l}-{\bf p}_{0}\|}$ is the actual signal propagation distance of the $l$th
path, $c$ is the speed of light (in [m/s]), $B$ is the clock offset in [m],
${\bf t}_{\text{B},l}$, and ${\bf t}_{\text{U},l}$ are the direction vector at
the BS and UE, respectively. The channel parameters contain the geometry
information of each path, which is assumed to be obtained by the channel
parameter estimation methods (e.g., MD-ESPRIT [4]). The lower bound of the
measurement parameter vector $\gamma$ can be obtained using crb, which will be
detailed in Section III-A.

###  II-C Doppler-assisted Localization

From the channel model and the relationship between the channel parameter
vector $\gamma$ and state vector ${\bf s}$, we are able to explain how Doppler
can assist in localization. For $L$ resolvable NLOS paths, the numbers of
unknown UE states and IP positions are $5$ (2D UE position, 2D UE velocity,
and clock offset) and $2L$ (2D IP position for $L$ paths), respectively. By
contrast, the number of channel parameters is $3(L+1)$ (AOD, delay, velocity)
with UE mobility, and $2(L+1)$ for stationary UE. If the number of channel
parameters is larger than the number of unknowns, the fim of the state
unknowns are of full rank, indicating the localization problem is
solvable.222This is valid for the scenarios described in this work with
resolvable paths LOS and NLOS paths. However, when the paths are unresolvable,
the rank calculation needs to be modified.

Obviously, localization and mapping tasks cannot be completed for the MISO
scenario with a stationary UE. However, when radial velocity can be estimated,
the number of channel parameters could be larger than the number of unknowns
with a sufficient number of NLOS paths. A summary of the minimum number of
NLOS paths needed for localization is shown in Table I. Note that with a
mobile UE, the localization and mapping can still be performed without the LOS
channel, which is not impossible even with synchronization [7]. This table
also provided the scenarios with known velocity information; the number of
unknowns is reduced by two. Note that the LOS path only provides 2 channel
parameter measurements with known velocity (due to the radial velocity can be
calculated directly with known AOD), which can be easily verified by 2D-LOS
(known ${\bf v}$) cannot be localized with $L=0$).

Table I: Summary of Minimum Number of NLOS Paths needed for Doppler-assisted Localization Localization Scenarios |  Unknown States |  Channel Parameters |  Min. # of NLOS Paths
---|---|---|---
With LOS (stationary) | $3+2L$ | $2+2L$ | Unsolvable
Without LOS (stationary) | $3+2L$ | $2L$ | Unsolvable
With LOS (mobile) | $5+2L$ | $3+3L$ | 2
Without LOS (mobile) | $5+2L$ | $3L$ | 5
With LOS (known ${\bf v}$) | $3+2L$ | $2+3L$ | 1
Without LOS (known ${\bf v}$) | $3+2L$ | $3L$ | 3

##  III Performance Analysis and Localization Algorithm

In this section, we briefly describe the crb of the unknown states (i.e., peb,
meb, ceb and veb) UE state by deriving the Fisher information matrix under UE
mobility. In addition, fim analysis is performed to discuss the effect of
speed on the estimation of unknowns. A simple localization and mapping
algorithm is also proposed with the estimated channel parameters.

###  III-A Cramér-Rao Bound

Based on the channel model defined in (1), the CRB of the state parameters can
be obtained as $\mathrm{CRB}\triangleq\left[\mathbf{I}({\bf
s})\right]^{-1}=\left[{\bf
J}_{\mathrm{S}}\mathbf{I}({\hbox{\boldmath$\gamma$}}){\bf
J}_{\mathrm{S}}^{\top}\right]^{-1}$, where $\mathbf{I}({\bf s})$ is the efim
[16] of the unknown state parameters ${\bf s}$,
$\mathbf{I}({\hbox{\boldmath$\gamma$}})$ is the efim of unknowns of interests
$\gamma$ from $\mathbf{I}({\tilde{\hbox{\boldmath$\gamma$}}})$ with

$\displaystyle\mathbf{I}({\tilde{\hbox{\boldmath$\gamma$}}})$
$\displaystyle=\frac{2}{\sigma_{n}^{2}}\sum^{G}_{g=1}\sum^{K}_{k=1}\mathrm{Re}\left\\{\left(\frac{\partial\mu_{g,k}}{\partial{\tilde{\hbox{\boldmath$\gamma$}}}}\right)^{\mathsf{H}}\left(\frac{\partial\mu_{g,k}}{\partial{\tilde{\hbox{\boldmath$\gamma$}}}}\right)\right\\}.$
(6)

Here, $\mathrm{Re}\\{\cdot\\}$ is getting the real part of a complex number,
and ${\bf J}_{\mathrm{S}}\in\mathbb{R}^{(2L+5)\times(3L)}$ is the Jacobian
matrix using a denominator-layout notation from the channel parameter vector
$\gamma$ to the state vector ${\bf s}$ as

${\bf
J}_{\mathrm{S}}\triangleq\frac{\partial{\hbox{\boldmath$\gamma$}}}{\partial{\bf
s}}=\begin{bmatrix}\frac{\partial\hbox{\boldmath$\gamma$}_{0}}{\partial{\bf
s}},&\ldots,&\frac{\partial\hbox{\boldmath$\gamma$}_{L}}{\partial{\bf
s}}\end{bmatrix}.$ (7)

For stationary scenarios, matrices $\mathbf{I}(\hbox{\boldmath$\gamma$})$ and
${\bf J}_{\text{S}}$ can be obtained similarly as [3]. With UE mobility, the
derivatives involving radial velocity $v_{l}$ of the $l$th path can be
expressed as

$\displaystyle\frac{\partial v_{l}}{\partial{\bf p}_{0}}$
$\displaystyle=\begin{cases}(\frac{\partial{{\bf
t}_{\text{U},0}}}{\partial{\bf p}_{0}})^{\top}{\bf v}=\frac{v_{0}{\bf
t}_{\text{U},0}-{\bf v}}{d_{\text{0}}}&l=0\\\ (\frac{\partial{{\bf
t}_{\text{U},l}}}{\partial{\bf p}_{0}})^{\top}{\bf v}=\frac{v_{l}{\bf
t}_{\text{U},l}-{\bf v}}{d_{l,2}}&l>0\end{cases},$ (8)
$\displaystyle\frac{\partial{v_{l}}}{\partial{{\bf p}_{l^{\prime}}}}$
$\displaystyle=\begin{cases}(\frac{\partial{{\bf
t}_{\text{U},l}}}{\partial{\bf p}_{l}})^{\top}{\bf v}=-\frac{v_{l}{\bf
t}_{\text{U},l}-{\bf v}}{d_{l,2}}&l=l^{\prime}>0\\\
0&\text{others}\end{cases},$ (9)
$\displaystyle\frac{\partial{v_{l}}}{\partial{B}}$
$\displaystyle=0,\quad\frac{\partial v_{l}}{\partial{\bf v}}={\bf
t}_{\text{U},l}.$ (10)

Finally, we can define the peb, meb, meb and veb as

PEB $\displaystyle=\sqrt{{\hbox{tr}}([\mathrm{CRB}]_{1:2,1:2})},$ (11)
$\displaystyle\text{MEB}_{l}$
$\displaystyle=\sqrt{{\hbox{tr}}([\mathrm{CRB}]_{(2l+1):(2l+2),(2l+1):(2l+2)})},$
(12) CEB $\displaystyle=\sqrt{[\mathrm{CRB}]_{(2L+3),(2L+3)}},$ (13) VEB
$\displaystyle=\sqrt{{\hbox{tr}}([\mathrm{CRB}]_{(2L+4):(2L+5),(2L+4):(2L+5)})},$
(14)

where ${\hbox{tr}}(\cdot)$ returns the trace of a matrix, and $[\cdot]_{i,j}$
is getting the element in the $i$th row, $j$th column of a matrix. The bounds
from (11)–(14) will be used to evaluate the localization and mapping
performance.

###  III-B FIM Analysis for Varying Speed

To evaluate the effect of speed on localization and mapping performance, we
re-order the rows and columns of the matrix ${\bf J}_{\text{S}}$ (take $L=2$
for illustration) as

$\displaystyle{\bf J}_{\mathrm{R}}=\left[\begin{array}[]{c : c}{\bf A}&{\bf
B}\\\ \hdashline{\bf O}&{\bf D}\end{array}\right]=$ (17)
$\displaystyle\left[\begin{array}[]{c c c c c c : c c
c}\vspace{1mm}\frac{\partial\theta_{0}}{\partial{\bf
p}_{0}}&\frac{\partial\tau_{0}}{\partial{\bf
p}_{0}}&\mathbf{0}_{2}&\frac{\partial\tau_{1}}{\partial{\bf
p}_{0}}&\mathbf{0}_{2}&\frac{\partial\tau_{2}}{\partial{\bf
p}_{0}}&\frac{\partial v_{0}}{\partial{\bf p}_{0}}&\frac{\partial
v_{1}}{\partial{\bf p}_{0}}&\frac{\partial v_{2}}{\partial{\bf p}_{0}}\\\
\vspace{1mm}\mathbf{0}_{2}&\mathbf{0}_{2}&\frac{\partial\theta_{1}}{\partial{\bf
p}_{\text{1}}}&\frac{\partial\tau_{1}}{\partial{\bf
p}_{\text{1}}}&\mathbf{0}_{2}&\mathbf{0}_{2}&\mathbf{0}_{2}&\frac{\partial
v_{1}}{\partial{\bf p}_{\text{1}}}&\mathbf{0}_{2}\\\
\mathbf{0}_{2}&\mathbf{0}_{2}&\mathbf{0}_{2}&\mathbf{0}_{2}&\frac{\partial\theta_{1}}{\partial{\bf
p}_{\text{2}}}&\frac{\partial\tau_{1}}{\partial{\bf
p}_{\text{2}}}&\mathbf{0}_{2}&\mathbf{0}_{2}&\frac{\partial
v_{2}}{\partial{\bf p}_{\text{2}}}\\\
\vspace{1mm}0&\frac{\partial\tau_{0}}{\partial
B}&0&\frac{\partial\tau_{1}}{\partial B}&0&\frac{\partial\tau_{2}}{\partial
B}&0&0&0\\\
\hdashline\vspace{1mm}\mathbf{0}_{2}&\mathbf{0}_{2}&\mathbf{0}_{2}&\mathbf{0}_{2}&\mathbf{0}_{2}&\mathbf{0}_{2}&\frac{\partial
v_{0}}{\partial{\bf v}}&\frac{\partial v_{1}}{\partial{\bf v}}&\frac{\partial
v_{2}}{\partial{\bf v}}\end{array}\right],$ (23)

where $\mathbf{0}_{2}$ is a $2\times 1$ zero vector. By defining a direction
vector ${\bf t}_{v}=[\cos(\theta_{v}),\sin(\theta_{v})]^{\top}$ and speed
$v=\|{\bf v}\|$ such that ${\bf v}=v{\bf t}_{v}$, we noticed that the
submatrices ${\bf A}$, ${\bf O}$ and ${\bf D}$ will not change with speed $v$
if the velocity direction ${\bf t}_{v}$ is fixed. Based on this observation,
we define a matrix $\bar{\bf B}={\bf B}/v$. In the following, we show that the
PEB, MEB and CEB will reduce and saturate with an increased $v$, showing the
velocity can improve these bounds to a certain level.

With reasonable assumptions, such as the AODs have no spatial correlation,
delays of different paths can be resolved, the FIM of the delays and AODs in
stationary scenario can be approximated as a diagonal matrix [16]. We make
further approximation by assuming the radial velocity is independent of other
channel parameters and the reordered FIM ${\bf F}$ can be formulated from
$\mathbf{I}(\hbox{\boldmath$\gamma$})$ as

${\bf
F}(\hbox{\boldmath$\gamma$})=\text{diag}(\mathbf{I}(\hbox{\boldmath$\gamma$})^{-1})^{-1},$
(24)

where $\text{diag}(\cdot)$ is the operation that keeps only the diagonal
elements of a matrix (i.e., to form a diagonal matrix). We further segment
${\bf F}(\hbox{\boldmath$\gamma$})$ into two diagonal matrix as ${\bf
F}(\hbox{\boldmath$\gamma$})=\text{blkdiag}({\bf F}_{1},{\bf F}_{2})$ and the
matrices ${\bf F}_{1}\in\mathbb{R}^{(2L+2)\times(2L+2)}$ and ${\bf
F}_{2}\in\mathbb{R}^{(L+1)\times(L+1)}$ contain the variance of all the
channel parameters that can be described as ${\bf
F}_{1}=\text{diag}(1/\sigma^{2}_{\theta_{0}},1/\sigma^{2}_{\tau_{0}},1/\sigma^{2}_{\theta_{1}},1/\sigma^{2}_{\tau_{1}},1/\sigma^{2}_{\theta_{2}},1/\sigma^{2}_{\tau_{2}})$,
and ${\bf
F}_{2}=\text{diag}(1/\sigma^{2}_{v_{0}},1/\sigma^{2}_{v_{1}},1/\sigma^{2}_{v_{2}})$.
The approximated FIM can then be formed as ${\bf J}_{\text{S}}{\bf
F}(\hbox{\boldmath$\gamma$}){\bf J}_{\text{S}}^{\top}$, and we will show that
the error of this approximation is in the simulation Section IV-B.

The EFIM of the state vector containing positions (both UE and IPs) and clock
offset can be expressed as

$\begin{split}{\bf E}_{\text{s}}=&\underbrace{{\bf A}{\bf F}_{1}{\bf
A}^{\top}}_{{\bf A}_{\text{S}}\text{(stationary
info)}}+\underbrace{v^{2}\bar{\bf B}{\bf F}_{2}\bar{\bf B}^{\top}}_{{\bf
B}_{\text{G}}=v^{2}\bar{{\bf B}}_{\text{G}}\ \text{(mobility gain)}}\\\
&-\underbrace{v^{2}\bar{\bf B}{\bf F}_{2}{\bf D}^{\top}({\bf D}{\bf F}_{2}{\bf
D}^{\top})^{-1}{\bf D}{\bf F}_{2}\bar{\bf B}^{\top}}_{{\bf
B}_{\text{L}}=v^{2}\bar{{\bf B}}_{\text{L}}\ \text{(mobility estimation
loss)}}.\end{split}$ (25)

The matrix ${\bf A}_{\text{S}}$ is the FIM for stationary UE, the matrix ${\bf
B}_{\text{G}}$ is the information gain with UE mobility if the velocity vector
${\bf v}$ is known, and ${\bf B}_{\text{L}}$ is the information loss with
unknown UE velocity.

Similarly, the EFIM of the UE velocity can be expressed as

${\bf E}_{\text{v}}=\underbrace{{\bf D}{\bf F}_{2}{\bf D}^{\top}}_{={\bf
D}_{0}\text{(velocity info)}}-\underbrace{v^{2}{\bf D}{\bf F}_{2}{\bf
B}^{\top}({\bf A}_{0}+v^{2}\bar{\bf B}_{\text{G}})^{-1}{\bf B}{\bf F}_{2}{\bf
D}^{\top}}_{={\bf D}_{\text{L}}=v^{2}\bar{{\bf D}}_{\text{L}}\text{(unknown
estimation loss)}}.$ (26)

where ${\bf D}_{0}$ contains the velocity from the radial velocity estimation
and ${\bf D}_{\text{L}}$ is the information loss due to other unknown
parameters.

######  Lemma 1.

The EFIM of the UE and IP positions ${\bf E}_{\text{p}}$ and the EFIM of clock
offset $E_{\text{c}}$ are given by

$\displaystyle{\bf E}_{\text{p}}$ $\displaystyle={\bf A}_{0}+v^{2}{\bf
B}_{0}={\bf A}_{1}-{\bf A}_{2}A_{4}^{-1}{\bf A}_{3}+v^{2}{\bf B}_{0},$ (27)
$\displaystyle E_{\text{c}}$ $\displaystyle=A_{4}-{\bf a}^{\top}_{2}({\bf
A}_{1}+v^{2}{\bf B}_{0})^{-1}{\bf a}_{2},$ (28)

where ${\bf A}_{1}=[{\bf A}_{\text{S}}]_{1:(2L+2),1:(2L+2)}$, ${\bf
a}_{2}=[{\bf A}_{\text{S}}]_{1:(2L+2),(2L+3)}$, $A_{4}=[{\bf
A}_{\text{S}}]_{(2L+3),(2L+3)}$ are the submatrices of the matrix ${\bf
A}_{\text{S}}$, and ${\bf B}_{0}=[\bar{\bf B}_{\text{G}}-\bar{\bf
B}_{\text{L}}]_{1:(2L+2),1:(2L+2)}$.

###### Proof.

By segmenting ${\bf B}$ into a $\tilde{\bf B}=[{\bf B}]_{1:(2L+2),1:(L+1)}$
and a zero vector $\mathbf{0}_{1\times(L+1)}$, we can see that the last row
and last column of matrices ${\bf B}_{\text{G}}$, ${\bf B}_{\text{L}}$ are all
zeros, and Lemma 1 can be obtained based on the matrix inverse lemma. ∎

######  Proposition 1.

When the speed $v\to 0$, the localization problem is not solvable.

###### Proof.

We notice that the matrix $\tilde{\bf A}=[{\bf A}]_{1:(2L+2),1:(2L+2)}$ (first
$2L+2$ rows, without the row containing clock offset) from (17) is a block
uppler diagonal matrix. Since the matrix $[{\partial\theta_{l}}/{\partial{\bf
p}_{l}},{\partial\tau_{l}}/{\partial{\bf p}_{l}}]$ is of rank 2 (nonzero
determinant), $\tilde{\bf A}$ has full rank. Then, ${\bf A}$ is of rank
$(2L+2)$ and hence, ${\bf A}_{\text{S}}={\bf A}{\bf F}_{1}{\bf A}^{\top}$ is
of rank $(2L+2)$. As a consequence, the matrix ${\bf A}_{\text{S}}$ is of rank
$2L+2$ (which is not full rank due to the unknown clock offset), indicating
the localization and mapping can not be performed without UE mobility.333Note
that the VEB does not grow with increasing UE/IP position estimation error.
Instead, it is more affected by the speed. For example, when the speed (norm
of the velocity) is small, the radial velocity estimations are also small,
resulting a small velocity estimation error regardless of how large the UE/IP
position estimation errors are (as shown in the simulation). However, this
velocity estimation (although with a small error bound) does not help in
solving the localization problem since the scenario is almost stationary. ∎

This proposition is clearly congruent with the observations from Section II-C.
Moreover, the scenario with $v\to 0$ is equivalent to the setup in [15, 17],
where the localization problem was solved by assuming perfect knowledge of the
clock bias $B$ [15] or a reconfigurable intelligent surface [17].

######  Proposition 2.

When the speed $v\to\infty$, the PEB, MEB, and CEB will converge to a constant
value, whereas the VEB keeps increasing with $v$.

###### Proof.

Based on the lemma derived in [18], stating if ${\bf Q}$ has rank one, ${\bf
E}$ and ${\bf E}+{\bf Q}$ are nonsingular, then

$({\bf E}+v{\bf Q})^{-1}={\bf E}^{-1}-\frac{v}{1+v\text{tr}({\bf Q}{\bf
E}^{-1})}{\bf E}^{-1}{\bf Q}{\bf E}^{-1}.$ (29)

Therefore, we can decompose ${\bf B}_{0}$ into a summation of several rank-1
matrices ${\bf B}_{1},{\bf B}_{2},\ldots,{\bf B}_{R_{B}}$ by using SVD, where
$R_{\text{B}}$ is the rank of ${\bf B}_{0}$. We can see the improvement of
localization and mapping performance will saturate as ${\bf
E}^{-1}_{\text{p}}(v\to{\infty})={\bf E}^{-1}_{\text{p},R_{\text{B}}}$, which
can be obtained recursively from (29) as

${\bf E}_{\text{p},i}^{-1}=\begin{cases}{\bf
A}_{0}^{-1}-\frac{1}{\text{tr}({\bf B}_{1}{\bf A}_{0}^{-1})}{\bf
A}_{0}^{-1}{\bf B}_{1}{\bf A}_{0}^{-1}&i=1,\\\ {\bf
E}_{\text{p},i-1}^{-1}-\frac{1}{\text{tr}({\bf B}_{i}{\bf
E}_{\text{p},i-1}^{-1})}{\bf E}_{\text{p},i-1}^{-1}{\bf B}_{i}{\bf
E}_{\text{p},i-1}^{-1}&i\neq 1,\end{cases}$ (30)

where ${\bf E}_{\text{p},i}={\bf A}_{0}+\sum_{j=1}^{i}v^{2}{\bf B}_{j}$. Since
we have shown in Lemma (2) that $({\bf A}_{0}+v^{2}{\bf B}_{0})^{-1}$ is
getting close to a constant matrix when $v\to\infty$, $E_{\text{c}}$ is also
close to a constant. Considering ${\bf A}_{0}+{\bf B}_{\text{G}}={\bf
A}_{0}+v^{2}\bar{\bf B}_{\text{G}}$ is a constant matrix with large $v$, we
can see the VEB keeps increasing with speed $v$ as the estimation loss ${\bf
D}_{\text{L}}$ in (26) increases linearly with $v^{2}$. Thus, Proposition 2 is
proved, which is also validated in the simulation results in Section IV-B. ∎

###  III-C Localization and Mapping Algorithm

With a high dimension of unknowns (size of state vector ${\bf s}$), it is not
practical to perform mle and estimate all the unknowns at the same time (e.g.,
using gradient descent). Here, similar to the ad-hoc estimator in [19], we
propose an algorithm that simplifies localization and mapping tasks into a 1D
search problem.

Consider the LOS channel has the strongest signal strength, we search along
the direction of the estimated aod $\hat{\theta}_{0}$ at BS, which is
equivalent to the direction vector $\hat{\bf t}_{\text{B},0}$ that can be
calculated from (3). For a given UE candidate position444We use the notation
$\tilde{\cdot}$ to indicate the variables depending on the candidate
$\tilde{d}_{0}$, and use the notation $\hat{\cdot}$ to represent the estimated
channel parameters. $\tilde{\bf p}_{0}=\tilde{d}_{0}\hat{\bf t}_{0}$ on the
line ${\bf p}=d\hat{\bf t}_{0}$, the clock offset (for this specific UE
candidate $\tilde{\bf p}_{0}$) can be estimated as
$\tilde{B}=c\hat{\tau}_{0}-\|\tilde{\bf p}_{0}\|$ and the propagation distance
of the NLOS channel can be obtained as
$\tilde{d}_{l}=c\hat{\tau}_{l}-\tilde{B}$. We further define an intermediate
variable $e_{l}$ for the $l$th path as

$\displaystyle\tilde{e}_{l}$ $\displaystyle=(\tilde{\bf p}_{0}-{\bf
p}_{\text{B}})^{\top}\tilde{\bf t}_{\text{B},l}=\tilde{\bf
p}_{0}^{\top}\tilde{\bf t}_{\text{B},l},$ (31)

where $\tilde{\bf t}_{\text{B},l}=\hat{\bf t}_{\text{B},l}$ is the estimated
direction vector from angle $\hat{\theta}_{l}$ based on (3). The position of
the $l$th ip can be obtained from
$\tilde{d}_{0}^{2}-\tilde{e}_{l}^{2}+(\tilde{d}_{l,1}-\tilde{e}_{l})^{2}=(\tilde{d}_{l}-\tilde{d}_{l,1})^{2}$
as

$\tilde{\bf p}_{l}={\bf p}_{\text{B}}+\tilde{d}_{l,1}\tilde{\bf
t}_{\text{B},l}=\tilde{d}_{l,1}\tilde{\bf t}_{\text{B},l},\ \
\tilde{d}_{l,1}=\frac{\tilde{d}_{0}^{2}-\tilde{d}_{l}^{2}}{2(\tilde{e}_{l}-\tilde{d}_{l})}.$
(32)

After obtaining all the position of the IPs, we can then obtain the direction
vector $\tilde{\bf t}_{\text{U},l}$ for each path based on (5) and hence the
velocity can be estimated using a least squares method as

$\tilde{\bf v}=(\tilde{\bf X}^{\top}\tilde{\bf X})^{-1}\tilde{\bf
X}^{\top}\hat{\bf b},$ (33)

where $\tilde{\bf X}=[\tilde{\bf t}_{\text{U},0},\ldots,\tilde{\bf
t}_{\text{U},L}]^{\top}$, $\hat{\bf
b}=[\hat{v}_{0},\ldots,\hat{v}_{L}]^{\top}$. The estimated velocity
$\tilde{\bf v}$ is the velocity that fits current UE candidate $\tilde{\bf
p}_{0}$ and radial velocity estimation $\hat{\bf b}$ the best. Since
$\tilde{\bf X}$ is a function of $d_{0}$, the residual error can be expressed
as $\varepsilon(d_{0})=\|\tilde{\bf X}\tilde{\bf v}-\hat{\bf b}\|$, from which

$d_{0}^{*}={\hbox{arg}}\min_{d_{0}}\varepsilon(d_{0}).$ (34)

Finally, the estimated position of the UE can be obtained as ${\bf
p}_{0}^{*}=d_{0}^{*}\hat{\bf t}_{0}$, and the remaining unknowns ${\bf
p}_{l}^{*}$, $B^{*}$ can be obtained similarly from (31) to (33). Since the
positions of the IPs are obtained from the estimated channel parameters
$d_{l}$ and $\theta_{l}$, equation (34) is identical to
$\operatorname*{arg\,min}_{d_{0}}||\tilde{\hbox{\boldmath$\gamma$}}(\tilde{\bf
p}_{0})-\hat{\hbox{\boldmath$\gamma$}}||$. Further improvement can adopt
weighted least square with the covariance matrix of the estimated radial
velocity at each path, or refine the results using gradient descent after
getting the initial result from the proposed 1D search.

##  IV Numerical Results

###  IV-A Simulation Parameters

We consider a 2D downlink scenario with a single-antenna UE and a BS with a
16-element ULA lies on the x-axis. The pilot signal $x_{g,k}$ is chosen with a
constant amplitude and random phase. The channel gain for each path is
calculated as $\rho_{0}=\frac{\lambda}{4\pi
d_{0}}e^{-j\frac{2\pi}{\lambda}d_{0}}$ for the LOS path and
$\rho_{l}=\sqrt{\frac{c_{l}}{4\pi}}\frac{\lambda}{4\pi
d_{l,1}d_{l,2}}e^{-j\frac{2\pi}{\lambda}d_{l}}$ for the $l$th NLOS path, where
$c_{l}$ is the radar cross-section (RCS). The default simulation parameters
can be found in Table II.

Table II: Default Simulation Parameters
---
Types | Simulation Parameters
BS Position | ${\bf p}_{\text{B}}=[0,0]^{\top}$
UE Position | ${\bf p}_{0}=[5,2]^{\top}$
BS Array Size | $N_{\text{B}}=16$
IP Positions |  ${\bf p}_{1}=[-6,8]^{\top}$, ${\bf p}_{2}=[8,6]^{\top}$
Measurement Interval | $T_{\text{int}}=1\,\mathrm{ms}$
RCS coefficients | $c_{l}=10\,\mathrm{m^{2}}$
Carrier Frequency | $f_{c}=28\,\mathrm{GHz}$
Bandwidth | $W=400\,\mathrm{MHz}$
Number of Transmissions | $G=20$
Number of Subcarriers | $K=20$
Average Transmission Power |  $P=30$ dBm
Noise PSD | $N_{0}=-173.855\,\mathrm{dBm/Hz}$
Noise Figure | $10\,\mathrm{dBm}$

###  IV-B Performance Bounds Results

Based on the analysis from Section II-C, the minimum number of IPs that can
support localization and mapping under UE mobility is 2. We use the parameters
provided in Table II and visualize PEB, MEB, CEB, and VEB with different UE
positions; the results are shown in Fig. 2. We can see that the PEB, CEB, and
OEB are showing a similar pattern, and a low error bound can be found in the
convex hull formed by BS and IPs. In contrast, low VEBs are seen in the UE
positions where the IP is aligned with the velocity direction.

(a) PEB

(b) MEB

(c) CEB

(d) VEB

Figure 2: Visualization of PEB, MEB, CEB, and VEB for different UE positions.
We can see that worse performance is shown in the area that lie between the BS
and scattering points due to the unresolvable paths. Regarding the VEB, when
the IPs is on the direction of the velocity ($[1,2]^{\top}$), more accurate
estimation can be achieved.

We evaluate how the PEB and VEB are effected by changing UE speed from
$0.001\,\mathrm{m/s}$ to $10\,\mathrm{m/s}$ with a fixed UE position ${\bf
p}_{0}=[5,2]^{\top}$. Two scenarios are evaluated, namely, velocity direction
$[1,2]^{\top}/\sqrt{5}$ (scenario 1) and $[2,1]^{\top}/\sqrt{5}$ (scenario 2).
The approximated error bound from (24) and the PEB when $v\to\infty$ are shown
in the figure. With the increase of speed $v$, the PEBs are getting lower and
saturate at around $3\,\mathrm{m/s}$. As for the estimation of velocity, VEB
keeps increasing with speed. Note that in reality the speed cannot be too
large as the channel may not be coherent.

$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{-1}$$10^{1}$UE Speed [m/s]PEB
and MEB [m]PEB-1PEB-1 (Approx.)PEB-1 ($v=\infty$)PEB-2PEB-2 (Approx.)PEB-2
($v=\infty$)

(a) PEB

$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{-4}$$10^{-2}$UE Speed [m/s]VEB
[m/s]VEB-1VEB-1 (Approx)VEB-2VEB-2 (Approx)

(b) VEB

Figure 3: PEB and VEB change with different UE speed. The velocity directions
are $[1,2]^{\top}/\sqrt{5}$ and $[2,1]^{\top}/\sqrt{5}$ for scenario 1 and 2,
respectively. We can see that the improvement on PEB and MEB with increased
speed saturates at around $0.1\,\mathrm{m/s}$ (which aligns well with the
theoretical analysis as dashed curves), whereas the VEB keeps increasing.

###  IV-C Evaluation of the Localization Algorithm

We evaluate the performance of the proposed localization algorithm with the
error bound discussed in Section III. Since channel estimation is not
considered in this work, we generate channel parameter vector (by assuming the
channel estimation is efficient) following a multi-variable Gaussian
distribution as
$\tilde{\hbox{\boldmath$\gamma$}}\in\mathcal{N}(\hbox{\boldmath$\gamma$},\mathbf{I}(\hbox{\boldmath$\gamma$})^{-1})$,
where $\mathbf{I}(\hbox{\boldmath$\gamma$})$ is the EFIM of the unknown
channel parameters obtained from obtained from (6). The results are shown in
Fig. 4 with $500$ simulations performed for each point. We can see that even
with a simple algorithm, the localization and mapping results are close to the
bound when the transmission power is above $15\,\mathrm{dBm}$. This is due to
the signal from LOS path is much stronger than the NLOS path and hence it is
reasonable to search along the LOS direction. Note that the localization and
mapping are done with a limited number of measurements within the coherence
time, better results can be obtained in tracking scenario with the assist of
estimated velocity.

$-20$$-10$$0$$10$$20$$30$$10^{-2}$$10^{0}$$10^{2}$Average Transmission Power P
[dBm]Estimation ResultsPE [m]PEB [m]ME [m]MEB [m]VE [m/s]VEB [m/s]

Figure 4: Comparison between simulation results and the derived lower bounds
(PEB, VEB, MEB). When $P\geq 5\,\mathrm{dBm}$, the estimation results using
the proposed localization and mapping algorithm attach the bounds.

##  V Conclusion

With a sufficient number of multipaths, UE mobility helps localization and
mapping by providing extra channel parameters. We have shown that mobility can
enable localization in a SIMO uplink scenario where BS and UE are not
synchronized. In addition, even though extra unknowns (i.e., velocity) are
introduced, mobility can enhance localization and mapping to some extent with
an increased speed. We also analyzed the system performance under different
scenarios and evaluated the performance of the proposed localization
algorithm. Future works could be the research on channel estimation under UE
mobility and Doppler-assisted simultaneous localization and mapping in
tracking scenarios.

## Acknowledgment

This work was supported, in part, by the European Commission through the H2020
project Hexa-X (Grant Agreement no. 101015956), and by the Wallenberg AI,
Autonomous Systems and Software Program (WASP) funded by Knut and Alice
Wallenberg Foundation.

## References

  * [1] H. Wymeersch, G. Seco-Granados, G. Destino, D. Dardari, and F. Tufvesson, “5G mmWave positioning for vehicular networks,” _IEEE Wireless Commun._ , vol. 24, no. 6, pp. 80–86, Dec. 2017.
  * [2] S. Haddadin, L. Johannsmeier, and F. D. Ledezma, “Tactile robots as a central embodiment of the tactile internet,” _Proc. IEEE_ , vol. 107, no. 2, pp. 471–487, Dec. 2018.
  * [3] A. Shahmansoori, G. E. Garcia, G. Destino, G. Seco-Granados, and H. Wymeersch, “Position and orientation estimation through millimeter-wave MIMO in 5G systems,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 3, pp. 1822–1835, Dec. 2017.
  * [4] F. Jiang, F. Wen, Y. Ge, M. Zhu, H. Wymeersch, and F. Tufvesson, “Beamspace multidimensional ESPRIT approaches for simultaneous localization and communications,” _arXiv preprint arXiv:2111.07450_ , 2021.
  * [5] H. Chen, H. Sarieddeen, T. Ballal, H. Wymeersch, M.-S. Alouini, and T. Y. Al-Naffouri, “A tutorial on terahertz-band localization for 6G communication systems,” _Accepted for publication in IEEE Commun. Surveys Tuts. arXiv preprint arXiv:2110.08581_ , 2022.
  * [6] Z. Abu-Shaban, X. Zhou, T. Abhayapala, G. Seco-Granados, and H. Wymeersch, “Error bounds for uplink and downlink 3D localization in 5G millimeter wave systems,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 8, pp. 4939–4954, May. 2018.
  * [7] A. Fascista, A. Coluccia, H. Wymeersch, and G. Seco-Granados, “Downlink single-snapshot localization and mapping with a single-antenna receiver,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 7, pp. 4672–4684, Mar. 2021\.
  * [8] J. Talvitie, T. Levanen, M. Koivisto, T. Ihalainen, K. Pajukoski, and M. Valkama, “Positioning and location-aware communications for modern railways with 5G new radio,” _IEEE Commun. Mag._ , vol. 57, no. 9, pp. 24–30, Sep. 2019.
  * [9] M. Z. Win, Y. Shen, and W. Dai, “A theoretical foundation of network localization and navigation,” _Proc. IEEE_ , vol. 106, no. 7, pp. 1136–1165, Jul. 2018.
  * [10] Y. J. Kim, M. Asim, and Y. S. Cho, “Preamble design technique for accurate timing/positioning in high Doppler environments,” _IEEE Trans. Veh. Technol._ , Mar. 2022.
  * [11] A. Amar and A. J. Weiss, “Localization of narrowband radio emitters based on Doppler frequency shifts,” _IEEE Trans. Signal Process._ , vol. 56, no. 11, pp. 5500–5508, Aug. 2008.
  * [12] I. Shames, A. N. Bishop, M. Smith, and B. D. Anderson, “Doppler shift target localization,” _IEEE Trans. Aerosp. Electron. Syst._ , vol. 49, no. 1, pp. 266–276, Jan. 2013.
  * [13] Y. Han, Y. Shen, X.-P. Zhang, M. Z. Win, and H. Meng, “Performance limits and geometric properties of array localization,” _IEEE Trans. Inf. Theory_ , vol. 62, no. 2, pp. 1054–1075, Dec. 2015.
  * [14] A. Kakkavas, M. H. C. García, R. A. Stirling-Gallacher, and J. A. Nossek, “Performance limits of single-anchor millimeter-wave positioning,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 11, pp. 5196–5210, Aug. 2019\.
  * [15] A. Fascista, A. Coluccia, H. Wymeersch, and G. Seco-Granados, “Millimeter-wave downlink positioning with a single-antenna receiver,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 9, pp. 4479–4490, Jul. 2019.
  * [16] R. Mendrzik, H. Wymeersch, G. Bauch, and Z. Abu-Shaban, “Harnessing NLOS components for position and orientation estimation in 5G millimeter wave MIMO,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 1, pp. 93–107, Oct. 2018.
  * [17] A. Fascista, M. F. Keskin, A. Coluccia, H. Wymeersch, and G. Seco-Granados, “RIS-aided joint localization and synchronization with a single-antenna receiver: Beamforming design and low-complexity estimation,” _Accepted for publication in IEEE J. Sel. Topics Signal Process.. arXiv preprint arXiv:2204.13484_ , 2022.
  * [18] K. S. Miller, “On the inverse of the sum of matrices,” _Math. Mag._ , vol. 54, no. 2, pp. 67–72, Mar. 1981.
  * [19] M. A. Nazari, G. Seco-Granados, P. Johannisson, and H. Wymeersch, “MmWave 6D radio localization with a snapshot observation from a single BS,” _arXiv preprint arXiv:2204.05189_ , 2022.


* Friedjung & Duerbeck (1993) Friedjung, M., & Duerbeck, H. W. 1993, Models of classical and recurrent novae, Vol. 507, 371–412
  * Gallagher & Starrfield (1978) Gallagher, J. S., & Starrfield, S. 1978, ARA&A, 16, 171, doi: 10.1146/annurev.aa.16.090178.001131
  * Gazak et al. (2015) Gazak, J. Z., Kudritzki, R., Evans, C., et al. 2015, ApJ, 805, 182, doi: 10.1088/0004-637X/805/2/182
  * Gehrels et al. (2004) Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, ApJ, 611, 1005, doi: 10.1086/422091
  * Gehrz (1988) Gehrz, R. D. 1988, ARA&A, 26, 377, doi: 10.1146/annurev.aa.26.090188.002113
  * Gehrz et al. (1980a) Gehrz, R. D., Grasdalen, G. L., Hackwell, J. A., & Ney, E. P. 1980a, ApJ, 237, 855, doi: 10.1086/157934
  * Gehrz et al. (1980b) Gehrz, R. D., Hackwell, J. A., Grasdalen, G. I., et al. 1980b, ApJ, 239, 570, doi: 10.1086/158143
  * Gehrz et al. (1998) Gehrz, R. D., Truran, J. W., Williams, R. E., & Starrfield, S. 1998, PASP, 110, 3, doi: 10.1086/316107
  * Gehrz et al. (2007) Gehrz, R. D., Roellig, T. L., Werner, M. W., et al. 2007, Review of Scientific Instruments, 78, 011302, doi: 10.1063/1.2431313
  * Gehrz et al. (2018) Gehrz, R. D., Evans, A., Woodward, C. E., et al. 2018, ApJ, 858, 78, doi: 10.3847/1538-4357/aaba81
  * Gimeno et al. (2016) Gimeno, G., Roth, K., Chiboucas, K., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9908, Proc. SPIE, 99082S, doi: 10.1117/12.2233883
  * Gonzaga et al. (2012) Gonzaga, S., Hack, W., Fruchter, A., & Mack, J. 2012, The DrizzlePac Handbook
  * Gordon et al. (2021) Gordon, A. C., Aydi, E., Page, K. L., et al. 2021, ApJ, 910, 134, doi: 10.3847/1538-4357/abe547
  * Groenewegen et al. (2009) Groenewegen, M. A. T., Sloan, G. C., Soszyński, I., & Petersen, E. A. 2009, A&A, 506, 1277, doi: 10.1051/0004-6361/200912678
  * Hachisu & Kato (2006) Hachisu, I., & Kato, M. 2006, ApJS, 167, 59, doi: 10.1086/508063
  * Hachisu & Kato (2007) —. 2007, ApJ, 662, 552, doi: 10.1086/517600
  * Hachisu & Kato (2018) —. 2018, ApJS, 237, 4, doi: 10.3847/1538-4365/aac833
  * Hachisu & Kato (2019) —. 2019, ApJS, 242, 18, doi: 10.3847/1538-4365/ab1b43
  * Hachisu et al. (2006) Hachisu, I., Kato, M., Kiyota, S., et al. 2006, ApJ, 651, L141, doi: 10.1086/509615
  * Hack et al. (2012) Hack, W. J., Dencheva, N., Fruchter, A. S., et al. 2012, in American Astronomical Society Meeting Abstracts, Vol. 220, American Astronomical Society Meeting Abstracts #220, 135.15
  * Hauschildt et al. (1994) Hauschildt, P. H., Starrfield, S., Austin, S., et al. 1994, ApJ, 422, 831, doi: 10.1086/173775
  * Hauschildt et al. (1995) Hauschildt, P. H., Starrfield, S., Shore, S. N., Allard, F., & Baron, E. 1995, ApJ, 447, 829, doi: 10.1086/175921
  * Henden & Munari (2014) Henden, A., & Munari, U. 2014, Contributions of the Astronomical Observatory Skalnate Pleso, 43, 518
  * Henden et al. (2016) Henden, A. A., Templeton, M., Terrell, D., et al. 2016, VizieR Online Data Catalog, II/336
  * Henze et al. (2015a) Henze, M., Darnley, M. J., Kabashima, F., et al. 2015a, A&A, 582, L8, doi: 10.1051/0004-6361/201527168
  * Henze et al. (2011) Henze, M., Pietsch, W., Haberl, F., et al. 2011, A&A, 533, A52, doi: 10.1051/0004-6361/201015887
  * Henze et al. (2015b) Henze, M., Ness, J. U., Darnley, M. J., et al. 2015b, A&A, 580, A46, doi: 10.1051/0004-6361/201526028
  * Hillman et al. (2015) Hillman, Y., Prialnik, D., Kovetz, A., & Shara, M. M. 2015, MNRAS, 446, 1924, doi: 10.1093/mnras/stu2235
  * Hillman et al. (2016) —. 2016, ApJ, 819, 168, doi: 10.3847/0004-637X/819/2/168
  * Hinkle et al. (2013) Hinkle, K. H., Fekel, F. C., Joyce, R. R., & Wood, P. 2013, ApJ, 770, 28, doi: 10.1088/0004-637X/770/1/28
  * Hjellming et al. (1986) Hjellming, R. M., van Gorkom, J. H., Taylor, A. R., et al. 1986, ApJ, 305, L71, doi: 10.1086/184687
  * Hobbs (1974) Hobbs, L. M. 1974, ApJ, 191, 381, doi: 10.1086/152976
  * Hook et al. (2004) Hook, I. M., Jørgensen, I., Allington-Smith, J. R., et al. 2004, PASP, 116, 425, doi: 10.1086/383624
  * Hounsell et al. (2010) Hounsell, R., Bode, M. F., Hick, P. P., et al. 2010, ApJ, 724, 480, doi: 10.1088/0004-637X/724/1/480
  * Hounsell et al. (2016) Hounsell, R., Darnley, M. J., Bode, M. F., et al. 2016, ApJ, 820, 104, doi: 10.3847/0004-637X/820/2/104
  * Hoyle & Lyttleton (1939) Hoyle, F., & Lyttleton, R. A. 1939, Proceedings of the Cambridge Philosophical Society, 35, 592, doi: 10.1017/S0305004100021368
  * Iłkiewicz et al. (2019) Iłkiewicz, K., Mikołajewska, J., Belczyński, K., Wiktorowicz, G., & Karczmarek, P. 2019, MNRAS, 485, 5468, doi: 10.1093/mnras/stz760
  * Ita & Matsunaga (2011) Ita, Y., & Matsunaga, N. 2011, MNRAS, 412, 2345, doi: 10.1111/j.1365-2966.2010.18056.x
  * Jencson (2020) Jencson, J. E. 2020, PhD thesis, California Institute of Technology
  * Jencson et al. (2019) Jencson, J. E., Kasliwal, M. M., Adams, S. M., et al. 2019, ApJ, 886, 40, doi: 10.3847/1538-4357/ab4a01
  * José et al. (2006) José, J., Hernanz, M., & Iliadis, C. 2006, Nucl. Phys. A, 777, 550, doi: 10.1016/j.nuclphysa.2005.02.121
  * Karambelkar et al. (2019) Karambelkar, V. R., Adams, S. M., Whitelock, P. A., et al. 2019, ApJ, 877, 110, doi: 10.3847/1538-4357/ab1a41
  * Kasliwal et al. (2017) Kasliwal, M. M., Bally, J., Masci, F., et al. 2017, ApJ, 839, 88, doi: 10.3847/1538-4357/aa6978
  * Kato & Hachisu (1994) Kato, M., & Hachisu, I. 1994, ApJ, 437, 802, doi: 10.1086/175041
  * Kato & Hachisu (2012) —. 2012, Bulletin of the Astronomical Society of India, 40, 393.  https://arxiv.org/abs/1212.2295
  * Kato et al. (2014) Kato, M., Saio, H., Hachisu, I., & Nomoto, K. 2014, ApJ, 793, 136, doi: 10.1088/0004-637X/793/2/136
  * Khan (2017) Khan, R. 2017, ApJS, 228, 5, doi: 10.3847/1538-4365/228/1/5
  * Kilpatrick et al. (2018) Kilpatrick, C. D., Foley, R. J., Drout, M. R., et al. 2018, MNRAS, 473, 4805, doi: 10.1093/mnras/stx2675
  * Kobulnicky et al. (2003) Kobulnicky, H. A., Nordsieck, K. H., Burgh, E. B., et al. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4841, Proc. SPIE, ed. M. Iye & A. F. M. Moorwood, 1634–1644, doi: 10.1117/12.460315
  * Kučinskas et al. (2006) Kučinskas, A., Hauschildt, P. H., Brott, I., et al. 2006, A&A, 452, 1021, doi: 10.1051/0004-6361:20054431
  * Kučinskas et al. (2005) Kučinskas, A., Hauschildt, P. H., Ludwig, H. G., et al. 2005, A&A, 442, 281, doi: 10.1051/0004-6361:20053028
  * Lagadec & Zijlstra (2008) Lagadec, E., & Zijlstra, A. A. 2008, MNRAS, 390, L59, doi: 10.1111/j.1745-3933.2008.00535.x
  * Li et al. (2017) Li, K.-L., Metzger, B. D., Chomiuk, L., et al. 2017, Nature Astronomy, 1, 697, doi: 10.1038/s41550-017-0222-1
  * Linford et al. (2017) Linford, J. D., Chomiuk, L., Nelson, T., et al. 2017, ApJ, 842, 73, doi: 10.3847/1538-4357/aa7512
  * Livio & Warner (1984) Livio, M., & Warner, B. 1984, The Observatory, 104, 152
  * Lloyd et al. (1992) Lloyd, H. M., O’Brien, T. J., Bode, M. F., et al. 1992, Nature, 356, 222, doi: 10.1038/356222a0
  * Lomb (1976) Lomb, N. R. 1976, Ap&SS, 39, 447, doi: 10.1007/BF00648343
  * Maoz et al. (2014) Maoz, D., Mannucci, F., & Nelemans, G. 2014, ARA&A, 52, 107, doi: 10.1146/annurev-astro-082812-141031
  * Martin & Dubus (2013) Martin, P., & Dubus, G. 2013, A&A, 551, A37, doi: 10.1051/0004-6361/201220289
  * Martin et al. (2018) Martin, P., Dubus, G., Jean, P., Tatischeff, V., & Dosne, C. 2018, A&A, 612, A38, doi: 10.1051/0004-6361/201731692
  * Mason et al. (2018) Mason, E., Shore, S. N., De Gennaro Aquino, I., et al. 2018, ApJ, 853, 27, doi: 10.3847/1538-4357/aaa247
  * McCully et al. (2018) McCully, C., Volgenau, N. H., Harbeck, D.-R., et al. 2018, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10707, Software and Cyberinfrastructure for Astronomy V, 107070K, doi: 10.1117/12.2314340
  * McLaughlin (1942) McLaughlin, D. B. 1942, ApJ, 95, 428, doi: 10.1086/144414
  * McLaughlin (1947) —. 1947, PASP, 59, 244, doi: 10.1086/125957
  * Merrill & Wilson (1938) Merrill, P. W., & Wilson, O. C. 1938, ApJ, 87, 9, doi: 10.1086/143897
  * Metzger et al. (2014) Metzger, B. D., Hascoët, R., Vurm, I., et al. 2014, MNRAS, 442, 713, doi: 10.1093/mnras/stu844
  * Mohamed & Podsiadlowski (2012) Mohamed, S., & Podsiadlowski, P. 2012, Baltic Astronomy, 21, 88, doi: 10.1515/astro-2017-0362
  * Mukai et al. (2008) Mukai, K., Orio, M., & Della Valle, M. 2008, ApJ, 677, 1248, doi: 10.1086/529362
  * Munari et al. (1990) Munari, U., Margoni, R., & Stagni, R. 1990, MNRAS, 242, 653, doi: 10.1093/mnras/242.4.653
  * Munari et al. (2011) Munari, U., Joshi, V. H., Ashok, N. M., et al. 2011, MNRAS, 410, L52, doi: 10.1111/j.1745-3933.2010.00979.x
  * Nelemans et al. (2016) Nelemans, G., Siess, L., Repetto, S., Toonen, S., & Phinney, E. S. 2016, ApJ, 817, 69, doi: 10.3847/0004-637X/817/1/69
  * Nelson et al. (2012) Nelson, T., Donato, D., Mukai, K., Sokoloski, J., & Chomiuk, L. 2012, ApJ, 748, 43, doi: 10.1088/0004-637X/748/1/43
  * Nelson et al. (2014) Nelson, T., Chomiuk, L., Roy, N., et al. 2014, ApJ, 785, 78, doi: 10.1088/0004-637X/785/1/78
  * Ney & Hatfield (1978) Ney, E. P., & Hatfield, B. F. 1978, ApJ, 219, L111, doi: 10.1086/182618
  * O’Brien et al. (2006) O’Brien, T. J., Bode, M. F., Porcas, R. W., et al. 2006, Nature, 442, 279, doi: 10.1038/nature04949
  * Orio et al. (2015) Orio, M., Rana, V., Page, K. L., Sokoloski, J., & Harrison, F. 2015, MNRAS, 448, L35, doi: 10.1093/mnrasl/slu195
  * Orio et al. (2001) Orio, M., Parmar, A., Benjamin, R., et al. 2001, MNRAS, 326, L13, doi: 10.1046/j.1365-8711.2001.04448.x
  * Orio et al. (2020) Orio, M., Drake, J. J., Ness, J. U., et al. 2020, ApJ, 895, 80, doi: 10.3847/1538-4357/ab8c4d
  * Osborne et al. (2011) Osborne, J. P., Page, K. L., Beardmore, A. P., et al. 2011, ApJ, 727, 124, doi: 10.1088/0004-637X/727/2/124
  * Ossenkopf et al. (1992) Ossenkopf, V., Henning, T., & Mathis, J. S. 1992, A&A, 261, 567
  * Page et al. (2010) Page, K. L., Osborne, J. P., Evans, P. A., et al. 2010, MNRAS, 401, 121, doi: 10.1111/j.1365-2966.2009.15681.x
  * Page et al. (2015) Page, K. L., Osborne, J. P., Kuin, N. P. M., et al. 2015, MNRAS, 454, 3108, doi: 10.1093/mnras/stv2144
  * Pagnotta et al. (2009) Pagnotta, A., Schaefer, B. E., Xiao, L., Collazzi, A. C., & Kroll, P. 2009, AJ, 138, 1230, doi: 10.1088/0004-6256/138/5/1230
  * Payne-Gaposchkin (1957) Payne-Gaposchkin, C. H. 1957, The Galactic Novae (Amsterdam: North-Holland Publishing)
  * Persson et al. (2013) Persson, S. E., Murphy, D. C., Smee, S., et al. 2013, PASP, 125, 654, doi: 10.1086/671164
  * Phillips et al. (2013) Phillips, M. M., Simon, J. D., Morrell, N., et al. 2013, ApJ, 779, 38, doi: 10.1088/0004-637X/779/1/38
  * Prochaska et al. (2020a) Prochaska, J., Hennawi, J., Westfall, K., et al. 2020a, The Journal of Open Source Software, 5, 2308, doi: 10.21105/joss.02308
  * Prochaska et al. (2020b) Prochaska, J. X., Hennawi, J., Cooke, R., et al. 2020b, pypeit/PypeIt: Release 1.0.0, v1.0.0, Zenodo, doi: 10.5281/zenodo.3743493
  * Reichart et al. (2005) Reichart, D., Nysewander, M., Moran, J., et al. 2005, Nuovo Cimento C Geophysics Space Physics C, 28, 767, doi: 10.1393/ncc/i2005-10149-6
  * Rest et al. (2005) Rest, A., Stubbs, C., Becker, A. C., et al. 2005, ApJ, 634, 1103, doi: 10.1086/497060
  * Rest et al. (2021) Rest, S., Wang, Q., Ridden-Harper, R., et al. 2021, in American Astronomical Society Meeting Abstracts, Vol. 53, American Astronomical Society Meeting Abstracts, 551.16
  * Ritchey (1917) Ritchey, G. W. 1917, PASP, 29, 210, doi: 10.1086/122638
  * Rodrigo & Solano (2020) Rodrigo, C., & Solano, E. 2020, in Contributions to the XIV.0 Scientific Meeting (virtual) of the Spanish Astronomical Society, 182
  * Rodrigo et al. (2012) Rodrigo, C., Solano, E., & Bayo, A. 2012, SVO Filter Profile Service Version 1.0, IVOA Working Draft 15 October 2012, doi: 10.5479/ADS/bib/2012ivoa.rept.1015R
  * Roming et al. (2005) Roming, P. W. A., Kennedy, T. E., Mason, K. O., et al. 2005, Space Sci. Rev., 120, 95, doi: 10.1007/s11214-005-5095-4
  * Sargent et al. (2011) Sargent, B. A., Srinivasan, S., & Meixner, M. 2011, ApJ, 728, 93, doi: 10.1088/0004-637X/728/2/93
  * Scargle (1982) Scargle, J. D. 1982, ApJ, 263, 835, doi: 10.1086/160554
  * Schaefer (2010) Schaefer, B. E. 2010, ApJS, 187, 275, doi: 10.1088/0067-0049/187/2/275
  * Schlafly & Finkbeiner (2011) Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ, 737, 103, doi: 10.1088/0004-637X/737/2/103
  * Schlegel et al. (1998) Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525, doi: 10.1086/305772
  * Schlegel (1990) Schlegel, E. M. 1990, MNRAS, 244, 269
  * Schwarz et al. (2011) Schwarz, G. J., Ness, J.-U., Osborne, J. P., et al. 2011, ApJS, 197, 31, doi: 10.1088/0067-0049/197/2/31
  * Seaquist & Bode (2008) Seaquist, E. R., & Bode, M. F. 2008, in ”Classical Novae”, ed. M. F. Bode & A. Evans, Vol. 43
  * Shafter (2017) Shafter, A. W. 2017, ApJ, 834, 196, doi: 10.3847/1538-4357/834/2/196
  * Shafter et al. (2009) Shafter, A. W., Rau, A., Quimby, R. M., et al. 2009, ApJ, 690, 1148, doi: 10.1088/0004-637X/690/2/1148
  * Shara et al. (2018) Shara, M. M., Prialnik, D., Hillman, Y., & Kovetz, A. 2018, ApJ, 860, 110, doi: 10.3847/1538-4357/aabfbd
  * Shore (2014) Shore, S. N. 2014, in Astronomical Society of the Pacific Conference Series, Vol. 490, Stellar Novae: Past and Future Decades, ed. P. A. Woudt & V. A. R. M. Ribeiro, 145
  * Shore et al. (2011) Shore, S. N., Wahlgren, G. M., Augusteijn, T., et al. 2011, A&A, 527, A98, doi: 10.1051/0004-6361/201015901
  * Skopal (2005) Skopal, A. 2005, A&A, 440, 995, doi: 10.1051/0004-6361:20034262
  * Smith et al. (2020) Smith, K. W., Smartt, S. J., Young, D. R., et al. 2020, PASP, 132, 085002, doi: 10.1088/1538-3873/ab936e
  * Smith et al. (2010) Smith, N., Chornock, R., Silverman, J. M., Filippenko, A. V., & Foley, R. J. 2010, ApJ, 709, 856, doi: 10.1088/0004-637X/709/2/856
  * Sokoloski et al. (2006) Sokoloski, J. L., Kenyon, S. J., Espey, B. R., et al. 2006, ApJ, 636, 1002, doi: 10.1086/498206
  * Srinivasan et al. (2011) Srinivasan, S., Sargent, B. A., & Meixner, M. 2011, A&A, 532, A54, doi: 10.1051/0004-6361/201117033
  * Srinivasan et al. (2009) Srinivasan, S., Meixner, M., Leitherer, C., et al. 2009, AJ, 137, 4810, doi: 10.1088/0004-6256/137/6/4810
  * Srivastava et al. (2015) Srivastava, M. K., Ashok, N. M., Banerjee, D. P. K., & Sand, D. 2015, MNRAS, 454, 1297, doi: 10.1093/mnras/stv2094
  * Starrfield et al. (2008) Starrfield, S., Iliadis, C., & R., H. W. 2008, in ”Classical Novae”, ed. M. F. Bode & A. Evans, Vol. 43
  * Starrfield et al. (1985) Starrfield, S., Sparks, W. M., & Truran, J. W. 1985, ApJ, 291, 136, doi: 10.1086/163048
  * Stetson (1987) Stetson, P. B. 1987, PASP, 99, 191, doi: 10.1086/131977
  * Strope et al. (2010) Strope, R. J., Schaefer, B. E., & Henden, A. A. 2010, AJ, 140, 34, doi: 10.1088/0004-6256/140/1/34
  * Tang et al. (2014) Tang, S., Bildsten, L., Wolf, W. M., et al. 2014, ApJ, 786, 61, doi: 10.1088/0004-637X/786/1/61
  * Tartaglia et al. (2018) Tartaglia, L., Sand, D. J., Valenti, S., et al. 2018, ApJ, 853, 62, doi: 10.3847/1538-4357/aaa014
  * Tatarnikova et al. (2003) Tatarnikova, A. A., Marrese, P. M., Munari, U., et al. 2003, MNRAS, 344, 1233, doi: 10.1046/j.1365-8711.2003.06911.x
  * Taylor et al. (1987) Taylor, A. R., Seaquist, E. R., Hollis, J. M., & Pottasch, S. R. 1987, A&A, 183, 38
  * Tody (1986) Tody, D. 1986, in Proc. SPIE, Vol. 627, Instrumentation in astronomy VI, ed. D. L. Crawford, 733, doi: 10.1117/12.968154
  * Tody (1993) Tody, D. 1993, in Astronomical Society of the Pacific Conference Series, Vol. 52, Astronomical Data Analysis Software and Systems II, ed. R. J. Hanisch, R. J. V. Brissenden, & J. Barnes, 173
  * Tonry et al. (2018) Tonry, J. L., Denneau, L., Heinze, A. N., et al. 2018, PASP, 130, 064505, doi: 10.1088/1538-3873/aabadf
  * Townsley & Bildsten (2005) Townsley, D. M., & Bildsten, L. 2005, ApJ, 628, 395, doi: 10.1086/430594
  * Ueta & Meixner (2003) Ueta, T., & Meixner, M. 2003, ApJ, 586, 1338, doi: 10.1086/367818
  * Uthas et al. (2010) Uthas, H., Knigge, C., & Steeghs, D. 2010, MNRAS, 409, 237, doi: 10.1111/j.1365-2966.2010.17046.x
  * Vacca et al. (2003) Vacca, W. D., Cushing, M. C., & Rayner, J. T. 2003, PASP, 115, 389, doi: 10.1086/346193
  * Valenti et al. (2019) Valenti, S., Sand, D. J., Wyatt, S., et al. 2019, Transient Name Server Discovery Report, 2019-1915, 1
  * Valenti et al. (2014) Valenti, S., Sand, D., Pastorello, A., et al. 2014, MNRAS, 438, L101, doi: 10.1093/mnrasl/slt171
  * Valenti et al. (2016) Valenti, S., Howell, D. A., Stritzinger, M. D., et al. 2016, MNRAS, 459, 3939, doi: 10.1093/mnras/stw870
  * van den Bergh & Younger (1987) van den Bergh, S., & Younger, P. F. 1987, A&AS, 70, 125
  * Vanderplas (2015) Vanderplas, J. 2015, Gatspy: General Tools For Astronomical Time Series In Python, v0.1.1, Zenodo, doi: 10.5281/zenodo.14833
  * VanderPlas & Ivezić (2015) VanderPlas, J. T., & Ivezić, Ž. 2015, ApJ, 812, 18, doi: 10.1088/0004-637X/812/1/18
  * Werner et al. (2004) Werner, M. W., Roellig, T. L., Low, F. J., et al. 2004, ApJS, 154, 1, doi: 10.1086/422992
  * Whitelock (1987) Whitelock, P. A. 1987, PASP, 99, 573, doi: 10.1086/132019
  * Whitelock et al. (2017) Whitelock, P. A., Kasliwal, M., & Boyer, M. 2017, in European Physical Journal Web of Conferences, Vol. 152, European Physical Journal Web of Conferences, 01009, doi: 10.1051/epjconf/201715201009
  * Whitelock et al. (2009) Whitelock, P. A., Menzies, J. W., Feast, M. W., et al. 2009, MNRAS, 394, 795, doi: 10.1111/j.1365-2966.2008.14365.x
  * Williams (2012) Williams, R. 2012, AJ, 144, 98, doi: 10.1088/0004-6256/144/4/98
  * Williams (2013) —. 2013, AJ, 146, 55, doi: 10.1088/0004-6256/146/3/55
  * Williams & Mason (2010) Williams, R., & Mason, E. 2010, Ap&SS, 327, 207, doi: 10.1007/s10509-010-0318-x
  * Williams et al. (2008) Williams, R., Mason, E., Della Valle, M., & Ederoclite, A. 2008, ApJ, 685, 451, doi: 10.1086/590056
  * Williams et al. (1991) Williams, R. E., Hamuy, M., Phillips, M. M., et al. 1991, ApJ, 376, 721, doi: 10.1086/170319
  * Wolf et al. (2018) Wolf, C., Onken, C. A., Luvaul, L. C., et al. 2018, PASA, 35, e010, doi: 10.1017/pasa.2018.5
  * Wolf et al. (2013) Wolf, W. M., Bildsten, L., Brooks, J., & Paxton, B. 2013, ApJ, 777, 136, doi: 10.1088/0004-637X/777/2/136
  * Yaron et al. (2005) Yaron, O., Prialnik, D., Shara, M. M., & Kovetz, A. 2005, ApJ, 623, 398, doi: 10.1086/428435