Round Trip Time (RTT) and Doppler Measurements for IoRT Localization by a Single-Satellite

Accurate node localization holds prime significance across a range of future network applications. In this letter, we present a novel scheme for instantaneous localization of static Internet-of-Remote-Things (IoRT) terminals from the satellite/network side, employing a single-channel probing by a single-antenna-equipped single-orbiter. Our method measures both Round Trip Time (RTT) and Doppler shift, which aid the proposed localization scheme. A geometric framework to establish a probabilistic relationship between satellite Transmit (Tx) and Receive (Rx) positions with the target IoRT terminal location is proposed. The key steps of the proposed framework are summarized in an algorithmic form. Precise IoRT node positioning relies on the accurate estimation of RTT, Doppler shift, and satellite position. For instance, at a 200km satellite altitude and precise satellite positioning, an error of 1.25Hz in Doppler and $0.0125~\mu $ sec in RTT estimation translates into a positioning error of 7.081m. Notably, this error increases with an increase in satellite altitude, making the proposed scheme better suited for lower-altitude satellites.


Round Trip Time (RTT) and Doppler Measurements for
IoRT Localization by a Single-Satellite Syed Junaid Nawaz , Ernestina Cianca , Member, IEEE, Tommaso Rossi , and Mauro De Sanctis Abstract-Accurate node localization holds prime significance across a range of future network applications.In this letter, we present a novel scheme for instantaneous localization of static Internet-of-Remote-Things (IoRT) terminals from the satellite/network side, employing a single-channel probing by a single-antenna-equipped single-orbiter.Our method measures both Round Trip Time (RTT) and Doppler shift, which aid the proposed localization scheme.A geometric framework to establish a probabilistic relationship between satellite Transmit (Tx) and Receive (Rx) positions with the target IoRT terminal location is proposed.The key steps of the proposed framework are summarized in an algorithmic form.Precise IoRT node positioning relies on the accurate estimation of RTT, Doppler shift, and satellite position.For instance, at a 200km satellite altitude and precise satellite positioning, an error of 1.25Hz in Doppler and 0.0125µsec in RTT estimation translates into a positioning error of 7.081m.Notably, this error increases with an increase in satellite altitude, making the proposed scheme better suited for lower-altitude satellites.

I. INTRODUCTION
I NTERNET-OF-REMOTE-THINGS (IoRT) is a network of physical objects connected to the internet and widely distributed in the remote regions of the Earth [1].Precise positioning of such extensively interconnected IoRT terminals is critical to advance various critical future applications such as logistics tracking, remote exploration campaigns, disaster management, and environment monitoring -to name a few.Besides, accurate node localization is also critical in assisting various communication operations with the potential to advance the existing 3 rd Generation Partnership Project (3GPP) standards for communication via satellite links in beyond 5 th Generation (5G) integrated terrestrial and non-terrestrial networks [2], [3].
IoRT terminals are required to provide long-range connectivity with low power consumption and low device cost.Syed Junaid Nawaz is with the Department of Electronics Engineering, University of Rome "Tor Vergata," 00175 Rome, Italy, and also with the Department of Electrical and Computer Engineering, COMSATS University Islamabad (CUI), Islamabad 45550, Pakistan (e-mail: junaidnawaz@gmail.com).
Digital Object Identifier 10.1109/LCOMM.2023.3348156aims at meeting the power requirements of Internet-of-Things (IoT) terminals over a long period (years), however, the lack of availability of terrestrial network infrastructures in remote regions restricts the employment of such protocols.In this context, satellite connectivity is seen as a promising resolution.While Global Navigation Satellite Systems (GNSSs) receivers are widely recognized as a fundamental standard for outdoor localization, their employment in IoRT terminals is not suitable due to their comparatively high cost, power consumption, and computational demands, which exceed the capabilities of IoRT nodes.Typically, the node-side has fewer privileges, while the network-side possesses extensive capabilities, including robust computational power.This distinction marks the primary reason for preferring network-side localization over node-side localization of IoRT terminals.In satellite-based localization schemes, the satellites can potentially probe and relay the estimated channel parameters to a ground station equipped with extensive computational capability, thus avoiding concerns about computational complexity associated with the processing of signals.Consequently, satellite-side localization of IoRT terminals has recently gained significant attention of the research community.The satellite-side localization schemes can be categorized into single-and multi-satellite localization schemes that exploit spatio-temporal or instantaneous spatial diversity, respectively.Direct channel measurements such as Doppler, Received Signal Strength Indicator (RSSI), Round Trip Time (RTT), and Angle-of-Arrival (AoA), etc., or differential measurements such as Time-Difference-of-Arrival (TDoA) and Frequency-Difference-of-Arrival (FDoA) are considered in the literature to aid the localization operation.A multisatellite network-side approach using multiple RSSI and Doppler measurements for the localization of IoT terminals is proposed in [4].Another multi-satellite network-side localization scheme is proposed in [5], while it utilizes AoA and Doppler measurements.A single-satellite network-side localization approach using multiple Doppler measurements is proposed in [6].While RTT measurements are commonly used for indoor localization, the pioneering application of RTT measurements for outdoor satellite-side localization has recently been proposed in [1].In this scheme, a single-antennaequipped single-orbiter takes multiple RTT measurements along its mobility orbit to localize the IoRT nodes.Multiple RTT measurements with single LEO satellite for outdoor localization is also motivated in [7].However, performing multiple channel probing imposes cost of reduced algorithm convergence speed and reduced terminal battery life.
To the best of the authors' knowledge, the integration of RTT and Doppler measurements for satellite-side localization of outdoor terrestrial terminals has not been considered in the literature.In this letter, we propose a novel approach utilizing single channel probing at single-antenna-equipped single-satellite to estimate both RTT and Doppler shift for instantaneous IoRT node positioning.Our approach draws inspiration from the principles employed in Synthetic Aperture Radars (SARs) which rely on the intersection of the Doppler cone, ranging sphere, and Earth surface for positioning the target [8].However, our approach involves RTT ellipsoid, Doppler cone, and Earth spheroid to aid the positioning.In contrast to our prior study in [1] that relied solely on RTT measurements obtained at multiple instances along the satellite's mobility orbit, this scheme considers single channel probing of both RTT and Doppler measurements.Notably, this current work also enhances the accuracy of RTT measurements model compared to [1] by eliminating the assumption of equal UpLink (UL) and DownLink (DL) path lengths, which significantly refines the model's accuracy.The major contributions of this letter are listed as follows • A novel RTT and Doppler measurements based geometric framework and algorithm for instantaneous localization of IoRT nodes is proposed.• Likelihood function of the node position is derived considering the parameters estimation error.• Performance analysis of the proposed scheme is conducted and analytical expression for Mean Square Error (MSE) has been derived.The rest of the letter is organized as follows: Sec.II presents the proposed system model for joint RTT and Doppler-based localization of IoRT.Sec.III presents results and analysis.Conclusion is presented in Sec.IV.

A. System Model
The satellite transmits/broadcasts an illuminating signal at time t T at position P s,T = [x s,T , y s,T , z s,T ] T .The target terrestrial static IoRT terminal positioned at P n = [x n , y n , z n ] T reflects (i.e., receives, processes, and retransmits) it back to the satellite after embedding its information (e.g., identification number, sensed data, altitude information, processing time, etc.).The time associated with the processing at IoRT node is represented by t IoRT (i.e., communicated to the satellite in UL transmission packet).The satellite receives the reflected signal along its mobility orbit at position P s,R = [x s,R , y s,R , z s,R ] T at time instant t R .We assume that an appropriate multiplexing mechanism is employed to identify the signal of target IoRT terminal.Moreover, the target IoRT node is positioned comfortably within the satellite's coverage area when probing the channel, ensuring the satellite can receive the reflected signal while within its line-of-sight (LoS).The Round Trip Time (RTT) can be expressed as (1) The proposed system model is illustrated in Fig. 1.In subfigure (a), the geometric configuration of the system model is illustrated.Subfigure (b) demonstrates the geometric layout of the error contours depicted as ellipsoids, spheres, and circular cones.Subfigure (c) presents the RTT-Doppler grid along with definition of important parameters.The Earth origin is taken as the origin of the coordinate system, i.e., P • = [0, 0, 0] T .The z-axis is taken as fixed along the zenith at satellite Transmit (Tx) position (i.e., along the line originated from P • towards P s,T ).The x-axis is taken as fixed along the direction of motion of the satellite.The circular/elliptical plane (in x-z plane) confined by the orbit of satellite is termed as Satellite Orbit Plane (SOP), see Fig. 1 (a).The projection of SOP on Earth's surface is named as subtrack.The localization problem is defined as determining the node position P n given the estimate of satellite Tx position P s,T , satellite Receive (Rx) position P s,R , RTT t RTT , and Doppler shift f D .
Exploiting the unique property of an oblate ellipsoid that the sum of distances from any point on its surface to its two foci points remains constant for all the points on its surface, an oblate ellipsoid ξ RTT can be drawn representing a given RTT value t RTT .The foci points of the ellipsoid ξ RTT are set as fixed at Tx and Rx positions of the satellite P s,T and P s,R , respectively.The direct distance between Tx and Rx positions (i.e., foci points of ξ RTT ) can be obtained as d s = ∥P s,T − P s,R ∥, where ∥.∥ represents the Euclidean norm operation.The angle subtended by the arc on satellite's orbit between Tx and Rx positions can be represents as γ R = arctan(x s,R /z s,R ).The coordinates of the origin of the ellipsoid ξ RTT can be represented by and c τ = b τ , respectively.The surface of ellipsoid ξ RTT can be expressed by where the symbol ← represents labeling a name to a surface.The surface of Earth can be represented as The intersection of ξ RTT and ξ E is a closed-line labeled as ξ c , where all the points on ξ c represent positions on Earth surface that correspond to a given RTT measurement t RTT .
The mobility of satellite imposes Doppler shift f D that is measured at the satellite Rx position P s,R .When f c is the transmit carrier frequency, the frequency of received signal is f R = f c + f D .The Doppler shift can be expressed as where β R and θ R is the azimuth and elevation AoA with reference to the Direction-of-Motion (DoM) of the satellite, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.respectively.The parameter f max = vs c f c represents the maximum Doppler shift, where c is the speed of light and v s is the velocity of satellite.The transmit signal duration typically influences the accuracy of the Doppler estimate as it represents the available number of waveform cycles.In the considered IoRT context, the duration of transmit signals/packets, particularly in the UL, holds critical significance as it directly impacts the battery life of the IoRT nodes.However, modern signal processing techniques such as pulse-Doppler and cyclic autocorrelation can derive reliable Doppler estimates even with short-duration signals.In the IoT context, the Doppler shift is suggested to be estimated from the radial velocity calculating the derivative of the slant distance in [5].The satellite altitude influences its velocity which can be related as v s = GM/R s in m/s, where G = 6.674×10 −11 m 2 /kg/s 2 and mass of Earth M = 5.97219×10 24 kg.To spatially represent the given Doppler shift, a circular cone can be originated from the satellite Rx position along the satellite's DoM.The directions along the cone's surface represent the given Doppler shift f D , see Fig. 1 (b).The cone is labeled with ξ D , where its apex/internal angle can be related to the given Doppler shift as The surface of Doppler cone can be expressed as The above Doppler cone expression is valid for the range 0 < |f D | < f max , while for the scenarios of maximum and zero Doppler shift, the Doppler cone degenerates to a straight line and a plane, respectively.These scenarios can be conveniently expressed mathematically, however, these cases are not relevant under the considered geometric configuration.
By equating (4), (5), and ( 7), the target node position can be computed.The RTT-Doppler grid in unwrapped Cartesian plane is illustrated in Fig. 1 (c), where the ellipses and hyperbolas represent the intersection of Ellipsoids ξ RTT and cones ξ D with the earth surface ξ E , respectively.When there is no error involved, one of the two points of intersection of a constant ellipse and hyperbola represent the target node position.

B. Error and Likelihood
In real-world scenarios, multiple factors such as system noise and modeling assumptions introduce inaccuracies in the estimation of channel parameters.These inaccuracies further lead to errors in the positioning of the target node.The contours of error in Doppler cone, RTT ellipsoid, and Earth radius are illustrated in Fig. 1 (b).The error in RTT and Doppler estimate combined with the uncertainty of Earth terrain/altitude causes the intersection point of these surfaces to spread over a volume around the true position termed as the Volume-of-Interest (VoI).The VoI can be defined as: where the function (•, •, •) represents the region formed by the overlap/intersection of three input volumes and the operator represents the union of multiple surfaces over a given continuous range.The error in the considered channel/geometric measurements/estimates can be modeled using Gaussian distributed random variables with the mean representing the true value, i.e., tRTT ∼ N (t RTT , σ 2 , and RE ∼ N (R E , σ 2 R ).Since, the measurement parameters are independent, their joint Probability Density Function (PDF) can be obtained as the

Compute Node Coordinates
• Compute likelihood function f (P n ) as given in (9) • Search node position that represents maximum likelihood, as given in (17).
product of their marginal PDFs, as, f ( tRTT , fD , RE ) = f ( tRTT )f ( fD )f ( RE ).The IoRT node can autonomously determine its altitude (h n ) using a dedicated atmospheric pressure sensor (barometer) [9].This estimate can be used to determine R e as Re + h n , where Re represents the sea-level radius/altitude of the Earth.Unlike typical satellite communication challenges like Line-of-Sight (LoS) obstructions, this method remains robust, ensuring reliable altitude estimates in urban and mountainous areas (i.e., σ 2 R → 0).The IoRT terminal can transmit this information to the satellite during channel probing stage.Knowing the relationship of considered channel parameters with the spatial coordinates, the joint PDF can be transformed as where J( tRTT , fD , Re ) is the Jacobian transfer function given in (10), as shown at the bottom of the page.The simplification parameters are defined as The position within VoI that constitute maximum likelihood represents the node location, which is expressed as The proposed localization scheme is summarized in Algorithm 1.The MSE of the location estimate is where E[.] represents the expectation operator.

III. SIMULATION RESULTS
The carrier frequency is set as f c = 2 GHz [5] and the radius of earth is set as R E = 6371 km.The satellite altitude is set as h s = 2000km which corresponds to the velocity ω s = 11.335rounds/day.The target terrestrial node's true position is taken as P n = [100, 100, 6369.4]T km.With the satellite's transmit position P s,T = [0, 0, 8371] km, the RTT corresponding to node's position and satellite's altitude is t RTT = 0.0134 sec and satellite Rx position is P s,R = [0.0923049,0, 8370.999999491]km.The likelihood function is plotted in Fig. 2 for unwrapped xy-plane.The x'-y' plane represent the unwrapped earth surface, i.e., the projection of Earth surface on x-y plane.The setting of other important parameters is given in the caption of the corresponding figure.The contours of likelihood function corresponded from error in RTT and Doppler measurements are independently plotted in Fig. 2 (a), while the joint likelihood function is plotted in  Compared to our previous work detailed in [1] that solely relied on RTT measurements but considered multiple instances of measurement, our proposed scheme, for 2 measurement instances and 200km satellite altitude, provides a gain of 3.81m in the positioning accuracy.This gain is mainly achieved due to elimination of equal UL and DL path lengths assumption stated in [1].Notably, when the measurement instances are increased from 2 to 3 in [1], the positioning accuracy of both the schemes become comparable.Besides, the proposed scheme still provides the advantage of employing single-channel probing.

IV. CONCLUSION
A novel network-side localization solution for static IoRT nodes by a single-antenna-equipped single-satellite has been proposed.The proposed scheme involves single channel probing at the satellite-side for measuring both RTT and Doppler shift.Subsequently, the intersection of the RTTellipsoid, Doppler-cone, and Earth-sphere is utilized to determine the node's position.Furthermore, a thorough analysis has been conducted to assess the likelihood function and the positioning accuracy.For instance, at a 200km satellite altitude with precise satellite positioning, it has been observed that even minor errors such as a 1.25Hz discrepancy in Doppler and 0.0125µsec inconsistency in RTT estimation result in an IoRT positioning error of 7.081m.Moreover, increasing satellite altitude has been observed to causes a proportional increase in the positioning error, which makes the proposed scheme more suited for lower-altitude satellites.

Fig. 2 (
Fig.2 (b).One of the two points that correspond the maximum likelihood value represent the node position estimate Pn .The ambiguity in determining the true node position from the two available points representing maximum likelihood value can be resolved by employing several methods including multiple instances of channel probing by a single satellite (multiple satellite passes), multiple antennas, multiple single-antenna satellites, etc. Considering accurate computations of RTT, Doppler, satellite Tx and Rx positions, and quadrant of true node position, the proposed scheme precisely determines the target node position, i.e., |P n − Pn | = 0.This establishes the validity of the derived mathematical framework.Fig.3demonstrateshow the estimation error in Doppler, RTT, and satellite positioning influences the IoRT node positioning accuracy.An increase in the estimation error of RTT and Doppler approximately linearly deviates the intersection point of RTT ellipsoid and Doppler cone on the Earth's surface from the true node position, where the slope of increase is determined by the satellite altitude.The behavior of positioning error also depends on various other factors such as the precision of satellite positioning and the target IoRT node's position relative to the subtrack.For instance, at a 200km satellite altitude with precise satellite positioning, an error of 1.25Hz in Doppler and 0.0125µsec in RTT estimation results in an IoRT positioning error of 7.081m (see Fig.3 (a)).Besides, an increase in the satellite altitude from 200 to 2000km causes a loss of 13.539m in The design of Low Power Wide Area Networks (LPWANs) the radius of earth, and h s is the altitude of satellite.The major, intermediate, and minor axes of ξ RTT can be determined from the RTT measurement t RTT as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Algorithm 1 : RTT and Doppler-based Satellite-Side Geo-Localization of IoRT Nodes Channel Measurement: • Transmit/broadcast illuminating/probe signal -Start timer (mark Tx time reference t T ) -Estimate satellite transmit position P s,T • Receive returned signal, demultiplex, and decode -Stop timer (mark Rx time reference t R ) -Demultiplex and decode parameters (e.g., t IoRT ) -Estimate satellite Rx position P s,R • Compute Channel Parameters -Compute RTT as in (1): t RTT = t R − t T − t IoRT -Compute Doppler shift as in (6):