# Performance Limits for Signals of Opportunity-Based Navigation

Francesco Zanirato1 Francesco Ardizzon1 Laura Crosara1 Alessio Curzio2
Luca Canzian2 Stefano Tomasin1,3 and Nicola Laurenti1,3
1 Department of Information Engineering, University of Padova, Padova, Italy
2 Qascom S.r.l, Bassano del Grappa, Vicenza, Italy
3 Consorzio Nazionale Interuniversitario per le Telecomunicazioni (CNIT),
Parma, Italy

## biography

Francesco ZaniratoFrancesco Zanirato received the B.Sc. degree in electronic
engineering from the University of Padova in 2022. He is currently a M.Sc.
student in telecommunications engineering and he is conducting his research
training activity for the master thesis in the field of signals of
opportunity.

Francesco Ardizzon Francesco Ardizzon (Member, IEEE) received the B.Sc.
degree, the M.Sc. degree, and the Ph.D. degree in information engineering from
the University of Padova, Italy, in 2016, 2019, and 2023, respectively. In
2022, he has been a Visiting Scientist with the ESA European Space Research
and Technology Centre. He is currently an Assistant Professor at the
University of Padova. His current research interests include authentication
for global navigation satellite systems, physical layer security, and
underwater acoustic communications.

Laura CrosaraLaura Crosara received the B.Sc. degree in information
engineering and the M.Sc. degree in telecommunications engineering from the
University of Padova, Italy, in 2019 and 2021 respectively, where she is
currently pursuing the Ph.D. degree in Information Engineering. She is
currently a Visiting Expert at the European Space Research and Technology
Centre in Noordwijk, The Netherlands. Her current research interests include
authentication techniques for global navigation satellite systems, physical
layer security, and wireless communications.

Alessio CurzioAlessio Curzio is a GNSS Receiver and Test Engineer at Qascom
with one year of experience in GNSS, PNT, and signal processing. Since 2023,
he has been working on the development of receivers for LEO PNT systems and
using signals of opportunity. He is also involved in testing user receivers
with Galileo 1st generation signals. Alessio holds a Master’s degree in
Telecommunications Engineering from the University of Padua (Italy), focusing
his thesis on prototyping a Starlink receiver for opportunistic positioning.

Luca CanzianLuca Canzian joined Qascom in 2015 and he is currently leading the
R&D domain area. He has worked in several projects with ESA, ASI, NASA, the
European Commission and Industry. His main expertise involves ground-based and
space-based location systems, detection and location of interference signals,
navigation using signals of opportunity, inertial navigation systems, orbit
determination techniques, GNSS authentication and anti-spoofing techniques. He
holds a MSc degree and a PhD in Electrical Engineering from University of
Padova (Italy)

Stefano TomasinStefano Tomasin received the Ph.D. degree from the University
of Padova, Italy, in 2003. He joined the University of Padova where he has
been Assistant Professor (2005-2015), Associate Professor (2016-2022), and
Full Professor (since 2022). His research interests include physical layer
security, security of GNSS, and signal processing for wireless communications.
He is a senior member of IEEE since 2011. He is Editor of the EURASIP JWCN
(since 2011) and Deputy Editor-in-Chief of the IEEE TIFS.

Nicola LaurentiNicola Laurenti is an Associate Professor at the Department of
Information Engineering, University of Padua, Italy. His research interests
mainly focus on wireless network security at lower layers (physical, data
link, and network), satellite navigation systems, information theoretic
security, and quantum cryptography. He has led several activities related to
the information security of satellite navigation systems, funded by the
European Space Agency.

## Abstract

This paper investigates the potential of non-terrestrial and terrestrial
signals of opportunity (SOOP) for navigation applications. Non-terrestrial
SOOP analysis employs modified Cramér-Rao lower bound (MCRLB) to establish a
relationship between SOOP characteristics and the accuracy of ranging
information. This approach evaluates hybrid navigation module performance
without direct signal simulation. The MCRLB is computed for ranging accuracy,
considering factors like propagation delay, frequency offset, phase offset,
and angle-of-arrival (AOA), across diverse non-terrestrial SOOP candidates.
Additionally, Geometric Dilution of Precision (GDOP) and low earth orbit (LEO)
SOOP availability are assessed. Validation involves comparing MCRLB
predictions with actual ranging measurements obtained in a realistic simulated
scenario. Furthermore, a qualitative evaluation examines terrestrial SOOP,
considering signal availability, accuracy attainability, and infrastructure
demands.

##  1 Introduction

Signals of opportunity (SOOP) are signals that, despite being designed for
communications, are opportunistically used by the receiver for navigation
purposes as well.

SOOP signals encompass both satellite and terrestrial sources. Among
terrestrial sources are cellular, i.e., LTE and 5G/6G networks, Wi-Fi, and
Bluetooth, while satellite SOOP include low earth orbit (LEO) satellite
constellations like Starlink and Orbcomm. Indeed, SOOP may be exploited to
extract additional ranging estimations when global navigation satellite
systems (GNSS) signals are compromised or unavailable. In particular,
terrestrial SOOP offer dense deployment and bandwidth advantages, particularly
beneficial in indoor settings and urban areas, where the number of visible
GNSS satellites is limited. Moreover, LEO satellite constellations are of
interest concerning the satellite SOOP as they provide a good decorrelation of
measurement errors, potentially assisting GNSS when a hybridized position,
velocity, and time (PVT) solution is computed. Also due to their limited cost
with respect to medium Earth orbit (MEO) GNSS satellites, thousands of LEO
satellites orbit around the Earth, and each of these may represent a source of
ranging.

Still, employing SOOP for positioning presents challenges, as these signals
are not designed for navigation but serve various purposes. For instance,
retrieving pseudoranges requires information such as exact transmission time
and transmitter location, which may not be trivial to obtain. For instance,
while knowledge of the location of stationary transmitters is typically easy
to retrieve and often already known to the receiver (e.g., the base station
location in 5G/6G) when transmitters are moving, such information is often
inaccurate and hard to retrieve. Moreover, most communication systems used as
SOOP are not time-synchronized to a nanosecond accuracy (as GNSS are) and a
time correction may not be available to the receiver. Other relevant issues
are non-line-of-sight (NLOS) and multipath effects.

Various measurements can be drawn from SOOP, such as carrier phase, time-of-
arrival, angle-of-arrival (AOA), or Doppler frequency. These measurements are
later processed to extract ranging information such as the pseudorange or
pseudorange rate, and may be integrated with GNSS-ranging information in a
hybrid navigation module. Indeed mixing signals from diverse systems/networks,
even if not originally intended for positioning, may represent a huge
opportunity for navigation.

For instance, the use of terrestrial SOOP from 5G/6G cellular or Wi-Fi
networks may aid navigation in urban canyon environments. Moreover, SOOP
represent a valid resource in contexts where the signal navigation cannot be
trusted, e.g., after the detection of jamming or spoofing attacks. Thus,
approaches relying on hybrid navigation solutions enhance PVT resilience,
availability, and accuracy.

This paper explores the usage of non-terrestrial and terrestrial SOOP for
navigation purposes. For non-terrestrial SOOP, we employ the modified Cramér-
Rao lower bound (MCRLB) analysis to establish a connection between the
characteristics of SOOP and the accuracy of the ranging information estimated
from each signal. This approach allows us to assess the performance potential
of a hybrid navigation module without direct signal simulation. Specifically,
we calculate the MCRLB for ranging accuracy, considering various measurements
such as propagation delay, frequency offset, phase offset, and AOA, across
different non-terrestrial SOOP candidates. We then evaluate geometric dilution
of precision (GDOP) and the availability of various LEO SOOP. Finally, to
validate the derived bounds rigorously, we compare the MCRLB with actual
ranging measurements obtained in a realistic simulated scenario.

Lastly, we provide a qualitative assessment of the positioning performance
achievable by terrestrial SOOP, taking into account signal availability,
achievable accuracy, and infrastructure requirements.

##  2 Methodology

To identify the best SOOP, either terrestrial or non-terrestrial, we consider
two criteria: the positioning accuracy and the SOOP availability.

We analyze the achievable positioning accuracy of the PVT when using different
systems. This can be characterized by considering

$\sigma_{\mathrm{pos}}^{2}\triangleq\mathbb{E}\left[\left(\bm{p}-\hat{\bm{p}}\right)^{2}\right]\,,$
(1)

where $\bm{p}$ and $\hat{\bm{p}}$ are respectively the true and measured PVT
results.

Moreover, as in Reid et al., (2016), positioning accuracy can be approximated
as

$\sigma_{\mathrm{pos}}=\sigma_{\mathrm{UERE}}\,\mathrm{GDOP}\,,$ (2)

thus it is related to both the ranging accuracy, expressed by
$\sigma_{\mathrm{UERE}}$, that is the user equivalent ranging error (UERE)
variance, and the GDOP. The former term pertains to the estimation accuracy of
the distance between the transmitter and receiver, considering both signal
processing aspects and signal characteristics. Conversely, the latter term,
$\mathrm{GDOP}$, addresses the accuracy influenced by the spatial arrangement
of satellites relative to the receiver, e.g., the number of visible satellites
and their distribution.

Thus, given (2), the best SOOP will be ones minimizing
$\sigma_{\mathrm{UERE}}$ and $\mathrm{GDOP}$.

##  3 Limits on the Accuracy of the Estimation of Non/Terrestrial SOOP
Observables

We consider a receiver acquiring signals from different LEO systems, and from
each of them, it estimates the propagation delay $\tau$, the phase offset
$\theta$, and the frequency offset $\upsilon$. The objective of this analysis
is to determine the best non-terrestrial LEO candidate by considering the
ranging and ranging rate accuracy via MCRLBs, considering various
measurements. In other terms, MCRLBs will provide the minimum variance of the
estimator associated with each measurement and LEO system under consideration.

In detail, we will consider Iridium NEXT, OrbComm, OneWeb, and Starlink as LEO
candidates.

###  3.1 Channel and Signal Model

The signal at the receiver in baseband can be decomposed into useful signal,
$s(t)$, and additive white Gaussian noise (AWGN) noise $w(t)$ at the receiver,
having zero mean and variance $N_{0}$, i.e.,

$r(t)=s(t)+w(t)\>,$ (3)

where $s(t)$ is defined as

$s(t)=e^{j(2\pi\upsilon t+\theta)}\sum_{i=1}{N}c_{i}g_{i}(t-iT-\tau)\>.$ (4)

In detail, $g_{i}$ describes the waveform of the $i$th pulse, $T$ is the
symbol duration, and $c_{i}$ denotes the message symbols. In particular, we
consider the information to be modulated in (eventually complex) amplitude
(e.g., QAM) or in phase (PSK).

We remark that the phase and frequency offsets are defined with respect to the
receiver local oscillator, having instantaneous phase $\Phi_{\rm L}$ and
frequency $f_{\rm cL}$. Thus the frequency offset is

$\upsilon=f_{\mathrm{c}}-f_{\rm cL}\,,$ (5)

modeling, for instance, also the shift induced by the Doppler effect, while
the phase shift term $\theta$ is instead

$\theta=\Phi_{\rm L}-2\pi f_{\mathrm{c}}\tau\>.$ (6)

###  3.2 Modified Cramer Rao Lower Bounds for LEO Observables

The traditional Cramér-Rao lower bound (CRLB) Mengali and D’Andrea, (1997)
provides the estimator variance when all the parameters are known. However, we
aim at deriving estimates from SOOP data signals, where the modulated data is
not known by the receiver. Thus, we resort to the MCRLB of D’Andrea et al.,
(1994), where for a measurement $\lambda$ and a signal in baseband

$\mathrm{MCRLB}(\lambda)\triangleq\frac{N_{0}/2}{\mathbb{E}_{\bm{u}}\left[\int_{0}^{T_{0}}\left|\frac{\partial
s(t,\lambda,\textit{{u}})}{\partial\lambda}\right|^{2}\,dt\right]}\>,$ (7)

where $\mathbb{E}_{\bm{u}}$ represents the expectation taken with respect to
the random components of the signal (e.g., data) and $N_{0}$ the noise power
spectral density (PSD). Indeed,

$\mathrm{CRLB}(\lambda)\geq\mathrm{MCRLB}(\lambda)\,,$ (8)

so the CRLB represents a more conservative version of the lower bound on
variance, and the two values coincide if the unwanted parameters are known at
the receiver or no unwanted parameters exist at all.

Considering the signal pattern in (4), we compute the MCRLB for the variance
of $\tau$, $\upsilon$ and $\theta$ estimators

$\mathrm{MCRLB}(\tau)=\frac{T^{2}}{T_{0}\xi\frac{C}{N_{0}}}{8\pi^{2}}\,,$ (9)
$\mathrm{MCRLB(\theta)}=\frac{1}{{T_{0}}\frac{C}{N_{0}}}2$ (10)
$\mathrm{MCRLB}(\upsilon)=\frac{3}{{T_{0}}^{3}\frac{C}{N_{0}}}{2\pi^{2}}$ (11)

which include the carrier to noise ratio ($C/N_{0}$) (in dB-Hz), the
observation time $T_{0}$. $\xi$ denotes the normalized mean square bandwith
(NMSB), defined, for a signal with PSD $G(f)$, as

$\xi=T^{2}\frac{\int_{-\infty}^{+\infty}f^{2}\left|G(f)\right|^{2}df}{\int_{-\infty}^{+\infty}\left|G(f)\right|^{2}df}\,.$
(12)

A fourth method could involve the estimation of the AOA, $\beta$, of one of
the tones using an array of $M$ (synchronized) equally spaced antennas, with a
spacing $d$. The maximum accuracy with which the angle of arrival can be
estimated as Kay, (1997)

$\mathrm{MCRLB}(\beta)\geq\frac{12}{(2\pi)^{2}M\frac{C}{N_{0}}T_{0}\frac{M+1}{M-1}\left(\frac{Lf_{\mathrm{c}}}{c}\right)^{2}\mathrm{sin}^{2}\beta}\,,$
(13)

which, with respect to the previously described MCRLBs also depends on $M$,
the overall array length $L=d(M-1)$, and the AOA itself.

All the MCRLBs show that as the $C/N_{0}$ and signal observation time
increase, the estimator accuracy increases as well. Specifically, concerning
the $\tau$ estimation of (9), the lower bound also depends on the actual
signal bandwidth. Thus, to compute the bounds, we need the exact signal
parameters.

An important issue to be tacked is the estimation of the $C/N_{0}$. In Ferre
and Lohan, (2021), the authors report for systems LEO between 20 dB-Hz and 50
dB-Hz of $C/N_{0}$. The maximum $C/N_{0}$ can be estimated via the link budget
formula reported in Dey et al., (2023) here written in its linear form

$C/N_{0}=\mathrm{EIRP}\frac{G}{T_{\rm temp}}L_{P}\frac{1}{k}\,,$ (14)

where $\mathrm{EIRP}$ stands for the effective isotropic radiated power, which
takes into account the amount of power radiated by the transmitter antenna and
its radiation pattern, $G$ is the gain of the receiver antenna,
$T_{\mathrm{temp}}$ represents the system temperature, $L_{\rm P}P$ is the
free-space path loss, which in turn is function of the altitude of the
satellite and the carrier frequency of the signal, and $k$ is the Boltzmann
constant.

In particular for the path loss we considered the minimum distance over the
ground receiver position, i.e., the altitude of the satellite (e.g., for
Starlink $550\text{\,}\mathrm{km}$). In the following, we describe each LEO
signal parameter.

#### Starlink

Concerning Starlink, the structure of the signal has been reverse-engineered
in Humphreys et al., (2023). Starlink uses an orthogonal frequency division
multiplexing (OFDM) of which we know the overall symbol duration, cyclic
prefix duration, number of subcarriers as well as the number and amplitude of
each channel. Analyses on the $C/N_{0}$ have been provided in Osoro and
Oughton, (2021); Aguilar et al., (2019); del Portillo Barrios et al., (2019);
Xia et al., (2019); Ferre et al., (2022) provides an estimate of $42.6$ dB-Hz
for outdoor. On the other hand, this quantity turns out to be much lower than
the one we derived through (14) and Osoro and Oughton, (2021); Aguilar et al.,
(2019); del Portillo Barrios et al., (2019); Xia et al., (2019). We remark
that according to Humphreys et al., (2023), consecutive frames from the same
satellite and the same beam may be transmitted with different (not specified)
powers. The duration of a frame is about 1.33 ms of which the last 5.33 $\mu$s
of silence, however, frames are not transmitted with perfect regularity (the
frequency depends on the demand for data traffic) but at least one frame is
always detected about every 40 ms (1 in 30) Humphreys et al., (2023).

The calculation of $\xi$ for the Starlink system was performed by considering
the entire channel and assuming a trapezoidal impulse shape, so (12) becomes

$\begin{split}\xi=T^{2}\frac{\sum_{i=-N/2+1}^{N/2}\int_{-\infty}^{+\infty}f^{2}|G_{i}(f)|^{2}\,df}{\sum_{i=-N/2+1}^{N/2}\int_{-\infty}^{+\infty}|G_{i}(f)|^{2}\,df}&=T^{2}\frac{\sum_{i=-N/2+1}^{N/2}\int_{-\infty}^{+\infty}f^{2}|G_{0}(f-iF)|^{2}\,df}{\sum_{i=-N/2+1}^{N/2}\int_{-\infty}^{+\infty}|G_{0}(f-iF)|^{2}\,df}\\\
&=T^{2}\left(-\frac{1}{(2\pi)^{2}}\frac{2}{T_{C}}\frac{1}{T_{\rm
sym}-\frac{1}{3}T_{C}}+\frac{{FN}^{2}}{12}+\frac{F^{2}}{6}\right)\end{split}$
(15)

where $F$ is the width of a single sub-band and $N$ is the number of
subcarriers, $T_{\rm sym}$ the OFDM symbol period, $T_{\rm C}$ the OFDM chip
period.

#### OneWeb

Only partial information on OneWeb signals is available. In Kozhaya et al.,
(2023) the authors claim that, as for Starlink, its spectrum is divided into 8
channels of 250 MHz placed between 10.7 GHz and 12.7 GHz. Each of them
implements an OFDM structure. The data used to estimate the link budget were
obtained from del Portillo Barrios et al., (2019), Xia et al., (2019) and
Osoro and Oughton, (2021).

#### Iridium

According to ICAO, (2007), the signal transmitted by the Iridium satellites
consists of $240$ channels of $31.5\text{\,}\mathrm{kHz}$ each. These use a
QPSK modulation whose symbols last for $40\text{\,}\mathrm{\SIUnitSymbolMicro
s}$ and their waveforms have a $40\%$ roll-off factor. The link budget was
computed by using the data reported in Dey, (2014).

#### Orbcomm

The Orbcomm signal is the one using the least amount of bandwidth: the
spectrum is only $4.8\text{\,}\mathrm{kHz}$ wide. The modulation is a
symmetric differential PSK whose pulse shape has a $40\%$ roll-off. The data
for the link budget were obtained from ORBCOMM, (2001).

Finally, the parameters for the LEO systems under consideration have been
summarized in Table 1.

Table 1: Summary of the LEO SOOP parameters derived from the literature, where N/A stands for not currently available. |  Starlink | OneWeb | Iridium | Orbcomm
---|---|---|---|---
Modulation |  OFDM $N=1024$ $T_{\text{sym}}=4.4\,\mu\text{s}$ $T_{C}=4.167\,\text{ns}$ $F=234375$ Hz |  OFDM $N=$ N/A $T_{\text{sym}}=$ N/A $T_{C}=$ N/A $F=$ N/A |  QPSK 40% roll-off $T_{\text{sym}}=40\mu\text{s}$ |  SD-QPSK 40% roll-off $T_{\text{sym}}=208.33\mu\text{s}$
Bandwidth |  240 MHz (8 channels) |  250 MHz (8 channels) |  31.5 kHz (240 channels) |  4.8 kHz
Carrier frequency |  11.57 GHz ($4$th channel) | 11.075 GHz |  1.621 GHz ($120$th channel) | 137.5 MHz
Altitude and relative p.l. | 550 km; 168.5 dB | 1200 km; 174.9 dB | 780 km; 154.5 dB | 750 km; 132.7 dB
Estimation of the maximum achievable $C/N_{0}$ |  109.3 dB-Hz |  105.53 dB-Hz |  80.6 dB-Hz |  79.6 dB-Hz
Value of $C/N_{0}$ found in the literature | 42.6 dB-Hz | 31.9 dB-Hz |  N/A |  N/A
Beacon Length |  1.33 ms (max $D=99.7\%$) | 10 ms |  90 ms (max $D=36.8\%$ ) |  1 s (max $D=50$ % commonly $D$ varies from $6$ % to $10$ %)
Notes |  Presence of unmodulated carriers around 11.325 GHz spaced by 44 kHz. C/N0 varies between 24 dB-Hz and 36 dB-Hz |  Unknown inner structure of the OFDM. The computation of $\mathrm{MCRLB}(\tau)$ is estimated via flat spectrum model. |  $-$ |  $-$

###  3.3 Numerical Results: MCRLBs

The following results have been converted from the time domain to the space
domain by considering $c=3\times{10^{8}}$\mathrm{m}\mathrm{/}\mathrm{s}$$. In
order to evaluate the dependency of ranging accuracy on both $C/N_{0}$ and
observation time, we first consider the Starlink signal. Next, we will report
the results of the comparative analysis. Finally, to validate the use of the
MCRLBs we report the results of a simulation of an acquisition from Starlink,
comparing the statistic of the estimated measurements with the MCRLB.

In Fig. 1 are represented the MCRLB of the propagation time, frequency offset,
phase shift, and angle of arrival for Starlink signal, as computed in (9),
(10), (11) and (13), respectively. In Figs. LABEL:fig:mcrb_tau \-
LABEL:fig:mcrb_theta, the observation time is equal to the duration of a
frame, while in Fig. LABEL:fig:mcrb_aoa it is fixed to
$5\text{\,}\mathrm{ms}$. In particular, for the latter, we consider an angle
of $50\text{\,}\mathrm{\SIUnitSymbolDegree}$, $M=2$ antennas, and array length
of $L=$0.5\text{\,}\mathrm{m}$$. The conversions on the right axes were made
using $c$ as propagation speed and $f_{\mathrm{c}}=11.57$ GHz as carrier
frequency.

(a)

(b)

(c)

(d)

Figure 1: MCRLB vs $C/N_{0}$ for the measurement under consideration
considering Starlink as SOOP.

Fig. 2 reports the MCRLBs for the same parameter keeping $C/N_{0}$ fixed to 60
dB-Hz and varying the observation time $T_{0}$. As in the first batch, we see
that the MCRLB decreases inversely with the parameters under observation.

(a)

(b)

(c)

Figure 2: MCRLB vs the observation time $T_{0}$ for (a) $\tau$, (b)
$\upsilon$, and (c) $\theta$.

Finally Fig. 3 reports the MCRLBs as a function of the $C/N_{0}$ for a fixed
$T_{0}=$1.33\text{\,}\mathrm{ms}$$, corresponding to the frame of a Starlink
frame. In particular, we consider a single antenna receiver, thus the AOA is
not a viable measurement. While the results are comparable for phase and
frequency shift, the best performance for the propagation delay is achieved by
Starlink and OneWeb. Indeed, the propagation delay is a valid candidate, due
to the easier ambiguity resolution.

(a)

(b)

(c)

Figure 3: MCRLB vs the observation time $C/N_{0}$ for (a) $\tau$, (b)
$\upsilon$, and (c) $\theta$ considering OrbComm, Iridium, Starlink and
OneWeb, with $T_{0}=$1.33\text{\,}\mathrm{ms}$$.

###  3.4 MCRLB validation under realistic conditions

To task the validity of the proposed model we run a simulation using the OFDM
Starlink signal simulator published in Komodromos et al., (2023). In
particular, we compare the accuracy that characterizes the estimated
propagation delay $\tau$ and its MCRLB as a function of $C/N_{0}$. For the
acquisition process, we used the whole channel bandwidth of
$240\text{\,}\mathrm{MHz}$. The procedure is similar to the one used for GNSS.
However, since no pseudo-random noise (PRN) sequence is available, cross-
correlation was performed making use of the two known synchronization
sequences, the primary synchronization sequence (PSS) and secondary
synchronization sequence (SSS), that are transmitted by Starlink satellites at
the beginning of each frame. The total duration of these two OFDM symbols is
$8.8\text{\,}\mathrm{\SIUnitSymbolMicro s}$.

We performed 300 simulations for each $C/N_{0}$ value, with a time interval of
$20.83\text{\,}\mathrm{\SIUnitSymbolMicro s}$. The plot in Fig. 4 shows that
for values of $C/N_{0}$ above $66$ dB-Hz, sub-meter accuracy can be achieved
while below this threshold the results deviate from the MCRLB and tend to have
a uniform distribution within the search interval.

Figure 4: Acquisition results of Starlink OFDM and Modified Cramér-Rao Lower
Bound as function of $C/N_{0}$

##  4 Geometry and Availability Considerations

In this Section, we investigate the performance limits of the SOOP-aided
navigation due to availability and geometry, i.e., the placement of the
satellites with respect to the receiver position.

To do so we consider two metrics. First, we consider the complementary
cumulative distribution function (CCDF) of the number of satellites in view:
indeed, while we are interested in having at least 4 satellites in view for
most of the time, having a high number of satellites typically leads to a more
accurate PVT solution. Next, we consider the cumulative distribution function
(CDF) of the GDOP, as the GDOP also reports information about how spread out
the satellites are with respect to the receiver, with sparser constellation
having a lower GDOP. Then, the derivation of the GDOP can be used to derive
the overall positioning accuracy by using (2).

We consider two scenarios, which we call Scenario 1 and 2, and run a
Montecarlo simulation for each, where the satellites’ positions are obtained
via an orbital propagator with used real-world broadcast ephemeris. In both
scenarios, we considered as representative locations Padova, the Svalbard
islands, La Reunion island, and ESTEC. In Scenario 1 we neglected the
visibility effect induced by the beamwidth and masking angle, thus considering
$\varphi=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and
$\theta=$10\text{\,}\mathrm{\SIUnitSymbolDegree}$$, as typically chosen for
MEO-GNSS navigation. On the other hand, in Scenario 2, we consider a
$\varphi=[$30\text{\,}\mathrm{\SIUnitSymbolDegree}$,\ldots,$80.92\text{\,}\mathrm{\SIUnitSymbolDegree}$]$
and $\theta=$40\text{\,}\mathrm{\SIUnitSymbolDegree}$$.

#### Analysis for Scenario 1

The parameters of Scenario 1 are reported in Tab. 2. For this first set of
simulations, we neglected the visibility effect induced by the beamwidth by
considering $\varphi=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$. The impact of
beamwidth will be analyzed instead in Scenario 2.

Table 2: List of simulation parameters for Scenario 1. Constellations | Orbcomm, Iridium, Starlink, OneWeb
---|---
Receiver Locations | Padova, Italy [$45.409\text{\,}\mathrm{\SIUnitSymbolDegree}$ N, $11.894\text{\,}\mathrm{\SIUnitSymbolDegree}$ E]
Longyearbyen, Svalbard [$78.224\text{\,}\mathrm{\SIUnitSymbolDegree}$ N,
$15.637\text{\,}\mathrm{\SIUnitSymbolDegree}$ E]
ESTEC, the Netherlands [$52.219\text{\,}\mathrm{\SIUnitSymbolDegree}$ N,
$4.419\text{\,}\mathrm{\SIUnitSymbolDegree}$ E ]
Saint Denis, La Reunion [$20.883\text{\,}\mathrm{\SIUnitSymbolDegree}$ S,
$55.450\text{\,}\mathrm{\SIUnitSymbolDegree}$ E]
Time | April $19$, $2024$, $00:00$ \- $23:59$
Beamwidth, $\varphi$ | $90\text{\,}\mathrm{\SIUnitSymbolDegree}$
Masking Angle, $\theta$ | $10\text{\,}\mathrm{\SIUnitSymbolDegree}$

Fig. 5 reports the CCDFs of the number of satellites in view during the
observation period computed for Scenario 1. As expected, due to the larger
size of the constellation the best performance is achieved by Starlink and
OneWeb which both always guarantee at least $20$ satellites in view from all
the considered receiver locations. Interestingly, we note that, due to higher
orbit inclination differently from Starlink, OneWeb achieves better
performance at higher-latitude positions.

(a)

(b)

(c)

(d)

Figure 5: CCDFs of the number of satellites in view, considering the scenario
described in Tab. 2.

Fig. 6 reports the CDF of the GDOPs computed for Scenario 1 compared with the
one achieved instead by using Galileo. The results mirror the one achieved in
Fig. 5, with Starlink and OneWeb achieving the best performance due to their
larger constellation size and even outperforming Galileo. Additionally, we
notice again that OneWeb achieves better performance than Starlink from high-
latitude locations.

(a)

(b)

(c)

(d)

Figure 6: GDOPs of the number of satellites in view, considering the scenario
described in Tab. 2, and comparison with Galileo dashed).

The results concerning the GDOP are summarized in Tab. 3 where we reported the
mean and standard deviation (STD) of the measured GDOP.

Table 3: Average and STD of the GDOP for the different constellations, computed over the parameters reported in Tab. 2. In red the systems which outperform Galileo. With ’/’ we indicated that no PVT could be computed and thus no GDOP can be reported. | Receiver Locations
---|---
Constellation | Svalbard | Padova | La Reunion | ESTEC
Iridium NEXT | $3.608\pm 2.557$ | / | / | /
OrbComm | / | / | / | /
OneWeb | $\color[rgb]{0.609375,0,0.078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.609375,0,0.078125}0.692\pm 0.035$ | $\color[rgb]{0.609375,0,0.078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.609375,0,0.078125}1.162\pm 0.074$ | $\color[rgb]{0.609375,0,0.078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.609375,0,0.078125}1.353\pm 0.085$ | $\color[rgb]{0.609375,0,0.078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.609375,0,0.078125}1.079\pm 0.058$
Starlink | $\color[rgb]{0.609375,0,0.078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.609375,0,0.078125}1.500\pm 0.227$ | $\color[rgb]{0.609375,0,0.078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.609375,0,0.078125}0.482\pm 0.086$ | $\color[rgb]{0.609375,0,0.078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.609375,0,0.078125}0.614\pm 0.108$ | $\color[rgb]{0.609375,0,0.078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.609375,0,0.078125}0.523\pm 0.064$
Galileo | $\bm{2.465\pm 0.523}$ | $\bm{2.380\pm 0.634}$ | $\bm{2.589\pm 0.640}$ | $\bm{2.661\pm 0.635}$

#### Analysis for Scenario 2

In order to compare the performance of OneWeb and Starlink, we now consider
the stricter Scenario 2, whose parameters are reported in 4, where the
parameters for beamwidth $\varphi$ the masking angle $\theta$ follow the
descriptions reported in the proposals SpaceX, (2016); Eutelsat OneWeb,
(2017). In particular, a more narrow masking angle
$\theta=$40\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and beamwidth
$\varphi=$30\text{\,}\mathrm{\SIUnitSymbolDegree}$,$40\text{\,}\mathrm{\SIUnitSymbolDegree}$,\ldots,$80.92\text{\,}\mathrm{\SIUnitSymbolDegree}$$
are considered, where
$\varphi_{\mathrm{max}}=$80.92\text{\,}\mathrm{\SIUnitSymbolDegree}$$ is the
maximum beamwidth used by Starlink SpaceX, (2016). Indeed, since the GDOP
increases with the number of satellites in view, for the sake of conciseness,
we will only report the CCDF of the number of satellites in view.

We remark that OneWeb satellites vehicles (SVs) are designed to have a dynamic
beam able to limit overlap between spots of different SVs Eutelsat OneWeb,
(2017), therefore the analysis can considered to be a best case for OneWeb.

Table 4: List of simulation parameters for Scenario 2. Constellations | Starlink, OneWeb
---|---
Time | April 19, 2024, 00:00 - 23:59
Receiver Locations | Padova, Italy [$45.409\text{\,}\mathrm{\SIUnitSymbolDegree}$ N, $11.894\text{\,}\mathrm{\SIUnitSymbolDegree}$ E]
Longyearbyen, Svalbard [$78.224\text{\,}\mathrm{\SIUnitSymbolDegree}$ N,
$15.637\text{\,}\mathrm{\SIUnitSymbolDegree}$ E]
ESTEC, The Netherlands [$52.219\text{\,}\mathrm{\SIUnitSymbolDegree}$ N,
$4.419\text{\,}\mathrm{\SIUnitSymbolDegree}$ E ]
Saint Denis, La Reunion [$20.883\text{\,}\mathrm{\SIUnitSymbolDegree}$ S,
$55.450\text{\,}\mathrm{\SIUnitSymbolDegree}$ E]
Beamwidth, $\varphi$ |  $30\text{\,}\mathrm{\SIUnitSymbolDegree}$, $40\text{\,}\mathrm{\SIUnitSymbolDegree}$, $50\text{\,}\mathrm{\SIUnitSymbolDegree}$, $60\text{\,}\mathrm{\SIUnitSymbolDegree}$, $70\text{\,}\mathrm{\SIUnitSymbolDegree}$, $80.92\text{\,}\mathrm{\SIUnitSymbolDegree}$
Masking Angle, $\phi$ | $40\text{\,}\mathrm{\SIUnitSymbolDegree}$

The results for the Montecarlo simulations for OneWeb and Starlink concerning
the availability are reported in Figs. 7 and 8, respectively. For Starlink,
only the results from Padova are reported since, as shown from the previous
batch of results, no relevant difference is observed from one location to the
other. Concerning OneWeb, following the trend shown in the previous results,
due to the high orbit inclination, the best results are measured from the
Svalbard islands, from which it is also possible to outperform the performance
obtained instead by Starlink. On the other hand, when considering lower
latitude locations, Starlink achieves much better performance than OneWeb, for
all the considered beamwidths. More in detail, numerical results show that
except from the Svalbard, even considering $\theta_{\mathrm{max}}$ a PVT can
be computed using only OneWeb satellites at most the $76\%$ of the time (Fig.
LABEL:fig:oneWebBeamPadova). On the other hand, when using Starlink, the same
percentage is achieved using a much narrower beamwidth of
$\theta=$60\text{\,}\mathrm{\SIUnitSymbolDegree}$$.

$0$$10$$20$$30$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$N, # of SV
in
view$P(n>N)$$\varphi=$30\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$40\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$50\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$60\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$70\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$80.92\text{\,}\mathrm{\SIUnitSymbolDegree}$$
(a)

$0$$10$$20$$30$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$N, # of SV
in
view$P(n>N)$$\varphi=$30\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$40\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$50\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$60\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$70\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$80.92\text{\,}\mathrm{\SIUnitSymbolDegree}$$
(b)

$0$$10$$20$$30$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$N, # of SV
in
view$P(n>N)$$\varphi=$30\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$40\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$50\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$60\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$70\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$80.92\text{\,}\mathrm{\SIUnitSymbolDegree}$$
(c)

$0$$10$$20$$30$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$N, # of SV
in
view$P(n>N)$$\varphi=$30\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$40\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$50\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$60\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$70\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$80.92\text{\,}\mathrm{\SIUnitSymbolDegree}$$
(d)

Figure 7: CCDF of the number of OneWeb SVs in view, for different $\varphi$
and $\theta=$40\text{\,}\mathrm{\SIUnitSymbolDegree}$$, with scenario
described in Tab. 4.
$0$$10$$20$$30$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$N, # of SV
in
view$P(n>N)$$\varphi=$30\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$40\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$50\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$60\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$70\text{\,}\mathrm{\SIUnitSymbolDegree}$$$\varphi=$80.92\text{\,}\mathrm{\SIUnitSymbolDegree}$$
Figure 8: CCDF of the number of Starlink SVs in view, for different beam width
angles $\varphi$, with scenario described in Tab. 4 (Padova only).

##  5 Discussion on Terrestrial SOOP

Suitable candidates as terrestrial SOOP are i) Bluetooth 4.0 and Bluetooth Low
Energy (BLE), ii) Wi-Fi, and iii) 5G cellular network.

#### Bluetooth

Bluetooth 4.0 and BLE have several drawbacks when used for positioning. First,
their range is typically limited to around $10\text{\,}\mathrm{m}$,
restricting their availability and effectiveness in larger spaces.
Additionally, the achievable position accuracy is limited since positioning
algorithms mainly rely on received signal strength indicator (RSSI)
measurements. Lastly, a significant limitation of using Bluetooth 4.0 and BLE
as a SOOP is the lack of infrastructure as the positions of Bluetooth beacons
are typically not known at the receiver.

#### Wi-Fi

Similar drawbacks can be associated with Wi-Fi positioning, which can
guarantee good performance in terms of accuracy only in an indoor environment.
Wi-Fi networks are typically designed for indoor use, resulting in limited
coverage and signal strength in outdoor environments. In an outdoor scenario,
a solution adopted for positioning through Wi-Fi is to use fingerprinting
Bisio et al., (2019), thus creating a database of Wi-Fi signal characteristics
at various locations. Each location’s unique fingerprint is recorded,
capturing signal strength and other parameters from nearby Wi-Fi access
points. When positioning is needed, the current WiFi signals are compared to
the database to estimate the device’s location. This method can provide
improved accuracy compared to simple RSSI-based methods but faces challenges
such as environmental variability, signal interference, and the need for
extensive initial data collection and maintenance of the fingerprint database.
Finally, effective Wi-Fi positioning requires a dense network of access points
with known locations, which may not be available or feasible to deploy in many
outdoor areas.

#### Cellular Networks

On the other hand, 5G measurements can offer improved reliability, and
accuracy in urban scenarios, especially considering the soon-to-be massive
deployment of base stations (BSs), and thus anchors for positioning. For
instance, they may serve as a backup for GNSS when unavailable and enhance
overall performance by integrating measurements from both systems Gonzalez-
Garrido et al., (2022). Differently from both Bluetooth and Wi-Fi, the
information about the 5G infrastructure, such as the positions of 5G BSs, is
publicly available. Moreover, a new positioning method relying on 5G has been
proposed, using the 5G positioning reference signals (PRSs), defined in TS
38.211 3GPP, (2020). The PRSs are transmitted to the user equipment (UE) by
multiple BSs. Next, the UE performs measurements for positioning that are then
reported back to the network, which calculate the UE’s position by using
trilateration or triangulation methods.

The allocation of time-frequency resources for PRSs is adaptable Wei et al.,
(2023), enabling the multiplexing of different downlink PRS signals from
multiple BSs across different subcarriers in a comb-like pattern. Moreover,
the comb structure of PRS manages the interference caused by PRS signals
transmitted by multiple BSs. Thus, depending on the application, the
requirements may vary and the 5G flexibility will help optimize the resources.
Moreover, apart from PRS, other 5G synchronization signals (SS), such as the
primary SS (PSS), the secondary SS (SSS), and the demodulation reference
signals (DMRS), can be used for positioning Tedeschini et al., (2023),
achieving good performance.

To summarize, the qualitative comparison between all the terrestrial SOOP is
provided in Table 5.

Table 5: Qualitative comparison of terrestrial SOOP. |  Accuracy |  Availability |  Supporting Data
---|---|---|---
BT |  Only based on RSSI. |  BLE and BT 4.0 have limited range. |  No infrastructure. The positions of BT beacons are not known.
5G |  High PVT accuracy with PRS. |  Extensive coverage in urban environments, good coverage in rural areas. |  Data on BSs positions are available and collected in a dataset.
Wi-Fi |  Based on RSSI or fingerprinting. |  Mainly Indoor. Outdoor-only fingerprinting. |  In fingerprinting WiFi, the AP position is known only within an area of 100m.

##  6 Conclusions

In this paper, we have investigated the potential of non-terrestrial and
terrestrial SOOP for navigation applications. Our analysis of non-terrestrial
SOOP has employed MCRLB to establish a relationship between SOOP
characteristics and the accuracy of ranging information. We have computed a
lower bound on the ranging accuracy using MCRLB, considering the estimation of
propagation delay, frequency offset, phase offset, and AOA across various non-
terrestrial SOOP candidates. Additionally, we assessed GDOP and the
availability of LEO SOOP. Our validation process involved comparing MCRLB
predictions with actual ranging measurements obtained in realistic simulated
scenarios. Furthermore, we qualitatively evaluated various terrestrial SOOP,
considering aspects such as signal availability, achievable accuracy, and
infrastructure requirements.

Based on the comprehensive investigation into non-terrestrial and terrestrial
SOOP for navigation applications, this study concludes that Starlink and the
5G cellular network emerge as the optimal choices. The Starlink LEO
constellation demonstrates significant advantages in accuracy of ranging
information. The terrestrial SOOP evaluation highlights the suitability of the
5G cellular network, emphasizing its widespread signal availability,
achievable accuracy, and existing infrastructure support. These qualities
position 5G as a strong contender for terrestrial navigation applications.

## Acknowledgements

This work was partially funded by the European Space Agency under contract n.
4000143575/24/NL/WC/kg: “Navigation Using Machine lEaRning applied to Signals
of Opportunity (NUMEROSO)”. This work was also partially supported by the
European Union under the Italian National Recovery and Resilience Plan (PNRR)
of NextGenerationEU, partnership on “Telecommunications of the Future”
(PE0000001 - program “RESTART”).

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