On-Board Satellite-Based Interference Geolocation Using Time Difference of Arrival Measurements

Transcription

1 On-Board Satellite-Based Interference Geolocation Using Time Difference of Arrival Measurements Luca Canzian, Samuele Fantinato, Giovanni Gamba, Stefano Montagner, Oscar Pozzobon Qascom S.r.l., via O. Marinali 87, Bassano del Grappa, VI, 366 Italy Phone: (+39) , Fax: (+39) s: Abstract Rigas Ioannides, Francisco Amarillo Fernandez, Massimo Crisci ESA-ESTEC, TEC-ETN, Keplerlaan, AZ Noordwijk, The Netherlands Phone: (+3) , Fax: (+3) s: {name.surname}@esa.int This paper studies how satellites can be used to geolocate an interference source. In particular this paper considers on-board satellite geolocation performed by a single satellite. As for the interference source, this paper focuses on a single ground-based static interferer that continuously sends a signal toward the satellite as long as the satellite is visible. The methodology proposed in this paper is based on the aggregation of multiple Time Difference of Arrival (TDoA) measurements collected by the satellite at different time instants. The TDoA measurement at a specific time is obtained by looking at the delay at which the interference signal arrives at the two antennas the satellite is equipped with. The final interferer position estimation is obtained by aggregating all TDoA measurements exploiting the Taylor Series (TS) method. The estimation accuracy of the proposed approach is numerically evaluated. Furthermore, this paper investigates the impact on such accuracy of different system parameters, such as the antenna placement and distance, the accuracy, sampling time, and collection interval of TDoA measurements, and the position of the satellite orbit with respect to the interferer position. This analysis provides many insights on how to configure the parameters of the system and on the strengths and limitations of the proposed approach. Among them: ) the antennas should be placed cross-track and at least meter apart; ) the TDoA collection time window should be at least hours; 3) a reasonable TDoA sampling time is 3 seconds; and 4) the closer the interferer to the projection of the satellite orbit on the Earth the lower the estimation accuracy, though this limitation can be compensated by adding two additional antennas along-track. Introduction Interference represents a major threat for satellite systems and their users. Indeed, a number of incidents have proven the vulnerability of these systems to interference. According to [RD.] nowadays intentional interference (jamming) counts for less than 5% of interference cases but is increasing dramatically. Interference is often unintentional, resulting from faulty equipment or incorrect ground station operation, but interference can also result from deliberate jamming or the illegal use of available bandwidth. In all cases, detecting and geolocate the source of interference is becoming a priority in today s satellite industry. Indeed, although much effort has been placed at the user level to design interference mitigation schemes to increase the system s robustness in the presence of interference, little has been done at the satellite level (i.e., on the uplink communication channel between a ground station and a satellite), whose vulnerability propagates as a vulnerability in the entire system. Also in this context interference mitigation schemes can be useful to improve the performance of the system in the presence of interference, but they cannot deter such type of threats from appearing in the future. Instead geolocation represents a prevention measure for intentional interferers because it allows taking actions such as sending the authorities to the place the interference originates from. This paper proposes and evaluates an on-board satellite geolocation technique for single satellite based on Time Difference of Arrival (TDoA) measurements. On board geolocation is important for satellite systems in which a downlink channel between the satellite and on ground equipment is not always available, and it allows taking immediate interference mitigation actions on the satellite, such as isolating and switching off the interference source. Single satellite geolocation is important because multiple satellite solutions suffer from the drawback that the interference signal may reach other satellites with a very low power, moreover there is not always a good timing synchronization among different satellites and the synergy among the satellites receiving the same interference might not be available since the satellites might belong to different systems.

2 The methodology considered in this paper assumes that the satellite is moving with respect to a fixed point in the Earth and it is equipped with two antennas that are placed cross-track and are exploited to generate TDoA measurements using a Cross Ambiguity Function (CAF) [RD.], [RD.3], [RD.4]. Specifically, the signals received by the two antennas are sampled and correlated, and the time at which the correlation peak occurs represents the TDoA at which the two signals are received by the two antennas. The locus of points associated to a TDoA measurement is a hyperboloid, hence it is not possible to univocally identify the interferer position from a single TDoA measurements. In fact, the proposed approach collects multiple TDoA measurements at different time instants, associated to different satellite positions, and performs a final geolocation by aggregating these TDoA measurements through the Taylor Series (TS) method [RD.5]. The geometric interpretation of this approach consists in finding the point that is the closest, in a least square sense, to all the hyperboloids associated to the collected TDoA measurements. Indeed, since TDoA measurements are affected by estimation errors, these hyperboloids do not intersect exactly at the interference position. The estimation accuracy of the proposed approach is numerically evaluated for a static ground-based interferer and a MEO satellite. An important contribution of this paper consists in investigating the impact on such accuracy of different system parameters, such as the antenna placement and distance, the accuracy, sampling time, and collection interval of TDoA measurements, and the position of the satellite orbit with respect to the interferer position. This analysis provides many insights on how to configure the parameters of the system and on the strengths and limitations of the proposed approach. For example: ) the antennas should be placed cross-track and at least meter apart; ) the TDoA collection time window should be at least hours; 3) a reasonable TDoA sampling time is 3 seconds; and 4) the closer the interferer to the projection of the satellite orbit on the Earth the lower the estimation accuracy, though this limitation can be compensated by adding two additional antennas along-track. Related Work This section briefly describes some works and methods that can be used to geolocate the interference source of a satellite. [RD.5] discusses how to use the Taylor-Series (TS) estimation technique to compute a final geolocation solution from multiple location measurements. [RD.6] presents a method to determine the location of an unknown source transmitting an unknown signal to satellite relays. The receivers receive reference signals via respective relays from a common source. From the unknown signal DTO (Differential Time Offset) and DFO (Differential Frequency Offset) the position of the unknown source is calculated. [RD.7] analyses an emitter location technique that involves space-borne interception of ground-to-satellite communication links. In the proposed scheme a number of interceptor satellites transpond the frequency band of interest to a terrestrial location for processing. [RD.8] resides in a process for locating a transmitter using at least two receiver sites where there is relative motion between at least one of the receiver sites and a stationary transmitter or using at least three receiver sites where there is no relative motion. The location of the transmitter is determined based on Time Difference of Arrival (TDoA) and Frequency Difference of Arrival (FDoA). [RD.9] presents a new positioning algorithm that relies on Kalman filtering of the ARGOS frequency measurements. In [RD.] a multiple target tracking filter incorporating a wide range of capabilities is designed. The primary contribution is a Bayesian formulation for determining the probabilities of alternative data-to-target association hypotheses. Practical aspects in the implementation of Multiple Hypothesis Tracking (MHT) are discussed in [RD.]. [RD.] proposes an extension of the classical particle filter where the stochastic vector of assignment is estimated by a Gibbs sampler. This algorithm is used to estimate the trajectories of multiple targets from their noisy bearings, thus showing its ability to solve the data association problem. A fast particle filter algorithm for accurate initiation and tracking of multiple targets is proposed [RD.3]. The basis of the approach is the Linear Multi-target (LM) method. The above cited works propose new location methods or improvements of existing methods. However, to the best of our knowledge, no work in the literature have applied some of these geolocation approaches to quantify the accuracy of a single satellite interference geolocation solution, investigating also the impact on such accuracy of different system parameters. This represents the main contribution of this paper.

3 Considered Scenario This paper considers a scenario in which a ground-based static interferer is sending a signal to a satellite with a high enough power to interfere with a possible uplink communication between a ground station and the satellite. The interferer is continuously active as long as the satellite is visible, pointing its transmitting antenna toward the satellite position. The satellite is assumed to follow a circular orbits around the Earth, with a constant distance from the center of the earth equal to = + = 937, where = 3 is the altitude of the satellite and = 637 is the radius of the earth. The satellite angular speed is = 7 / =. /, meaning that the satellites covers an entire orbits in about 4 hours. The satellite is equipped with two antennas that are placed such that the straight line connecting them is perpendicular to the direction of travel (cross-track placement). The parameter defines the distance between the antennas. The antennas are exploited to generate Time Difference of Arrivals (TDoA) measurements at regular time intervals. This scenario is schematically represented in Figure, which shows the position of the satellite at different time instants along the circular orbits. Figure : D schematic representation of the considered scenario. The Cartesian coordinate system is selected in the following way: ) it is centered at the center of the earth; ) it is rotated such that the z-axis points toward the interferer; and 3) it is rotated around the z axis such that the x-axis lay in the plane defined by the satellite orbits. With this choice of the reference system, the only degree of freedom that must be set to univocally identify the position of the interferer and of the satellite orbit is the relative orbital inclination, which is defined as the angle between the interferer horizon plane and the orbital plane. Notice that with the selected reference system the interferer horizon plane is parallel to the x-y plane (i.e., the plane spanned by the axes x and y), hence is also equal to the angle between the x-y plane and the orbital plane. These conventions are shown in Figure for = 45. Interferer position and satellite orbit Interferer position and satellite orbit.5.5 Earth Satellite Interferer x-y plane.5.5 z [m] z [m] i x [m] y [m] Figure : Interferer position and satellite orbit: a 3D view (left) and a D view (right) showing the relative orbital inclination. y [m]

4 The TDoA measurements are generated every for a time interval of length during which the satellite reaches its maximum elevation angle (measured either with respect to the x-y plane or with respect to the interferer s horizon plane). For example, Figure 3 shows the interferer position and the satellite orbits limited to the time interval = hour (left side) and = 6 hours (right side), for an orbital inclination = 6. Notice that if > 6 hours the satellite disappears below the interferer horizon at a certain point, for this reason the maximum considered value of will be 6 hours. All the TDoA measurements collected in the considered time interval are aggregated using a Taylor series estimation technique to generate a final interferer position estimate. The approaches used to obtain the TDoA measurements and to aggregate them are described with more details in the next sections. Antennas and jammer trajectories Antennas and jammer trajectories 3 Earth Satellite Jammer 3 Earth Satellite Jammer z [m] z [m] x [m] y [m] x [m] - - y [m] 3 Figure 3: Interferer position and satellite orbit limited to the intervals = hour (left) and = hours (right), for =. Generation of TDoA measurements A TDoA measurement between two antennas l and can be written as: = Δ + where Δ = Δ l / is the real (unknown) TDoA value, Δ l is the differential range between the interference source and the two antennas, is the speed of light, and is the TDoA measurement error. The estimate is assumed to be generated through a Cross-Ambiguity Function (CAF). It can be shown that such estimate is unbiased (i.e., has zero mean) and has a variance that achieves the Cramer-Rao bounds [RD.]. Hence, the accuracy of a TDoA estimate can be quantified through the standard deviation of the TDoA measurement errors To compute a simulative tool has been implemented, in which the signals received by the two antennas have been simulated and a computationally efficient scheme to identify the CAF peak has been exploited to estimate the TDoA. Such scheme is based on the fact that the CAF function has a maximum bandwidth equal to the bandwidth of the receiver band-pass filter. Because the sampling frequency is typically larger than, then the CAF function can be interpolated without errors using for example a sinc filter with bandwidth equal to. However, such an interpolation could be a computationally expensive operation because of the large interpolation factor. For example, if the target TDoA accuracy is = 5 and the sampling time is = 5, then the interpolation factor is. To solve this problem the CAF peak has been identified iteratively using a binary search algorithm, and the CAF function has been reconstructed only in those time instants in which it must be evaluated. With this approach the number of iterations required to achieve the desired accuracy is ( / ) = 4 for = 5 and = 5, and at each iteration the CAF function is interpolated in a single point. Figure 4 shows the achievable with the proposed approach for different interference signal types and for the correlation time windows = (left side) and = (right side). The x axis represents the Interference to Noise plus Signal Ratio (INSR), which is defined as the ratio between the power of the interference signal and the power of the uplink signal plus the power of the noise. Indeed, for interference geolocation purposes the interference signal represents the signal that must be analyzed.

5 To generate the results shown in Figure 4 an interferer signal with unitary power is first generated at MHz (i.e., = 5 ) and a time translated replica of this signal is obtained for a delay Δ generated randomly following a uniform distribution in [, ]. These two signals are then received by the two antennas: each of them is sampled at MHz (i.e., = 5 ), added to a white noise with power spectral density equal to (INSR B), and finally filtered with a band-pass filter with bandwidth B = MHz. The signals that are obtained in this way are then used to estimate the delay Δ with the approach described above, and is obtained by computing the empirical standard deviation of the TDoA estimation error after simulations. The considered interferer signal types are: continuous wave (CW), linear chirp (CHIRP-LFM), non-linear chirp (CHIRP-NLFM), time and frequency hopping (RADAR), spread spectrum (SSS), fast frequency hopping (FH-F), slow frequency hopping (FH-S), broadband noise (NOISE-BB), and narrowband noise (NOISE-NB). As expected different interference signal types produce different results, due to their autocorrelation properties. For example, the accuracy associated to a CW signal is always worst (i.e., larger) than the accuracy associated to a variable signal such as the RADAR. Notice that in the FH-S signal the frequency is changed every, hence for a correlation window = only a single frequency jump occurs and the associated accuracy is even worse than the accuracy of the CW signal. However, for a correlation window = several frequency jumps occur and for this reason the accuracy associated to the FH-S signal becomes similar to the accuracy associated to the FH-F signal and much better than the accuracy associated to a CW signal. Finally, notice that all the accuracies improve with a larger correlation window, but even for = the achievable with the proposed approach is in the order of tenths of for all the jammer signal types for INSR = db, which is much smaller than the sampling time = 5. Std of TDoA estimation [s] 3 x Accuracy (std) of Jammer TDoA estimation with corr window. s CW CHIRP-LFM CHIRP-NLFM RADAR SSS FH-F FH-S NOISE-BB NOISE-NB Std of TDoA estimation [s] 3 x Accuracy (std) of Jammer TDoA estimation with corr window. s CW CHIRP-LFM CHIRP-NLFM RADAR SSS FH-F FH-S NOISE-BB NOISE-NB INSR [db] INSR [db] Figure 4: vs. for different interference signals, with = (left) and = (right). Taylor Series Method The Taylor-Series (TS) estimation [RD.] (or Gauss or Gauss-Newton interpolation) is an iterative scheme for solution of the simultaneous set of algebraic position equations (generally nonlinear), starting with a rough initial guess and improving the guess at each step by determining the local corrections. The TS estimation technique is summarized by the following iterative approach:. Choose an initial position estimate, e.g., the Sub Satellite Point (SSP);. Compute the local corrections by solving the system of equations associated to the first order expansion in the current position estimate; 3. Update the position estimate using; 4. Repeat to 3 until convergence or until failure to convergence is detected; It is important to remark that this iterative approach may not converge (this case can be easily detected) or may converge to a local optimum instead of to the global optimum. To overcome this problem it is possible to repeat the above steps multiple times with different initial position estimates, and keep the best among all the converged solutions. It is also important to remark that the TS method can be used to aggregate different types of location measurements (e.g., TDoA, FDoA, AoA, etc.), however in this case the location measurements must be rescaled with the proper rescaling factors in order to weight the measurement errors properly. The interested reader may find more details about the TS method in [RD.].

6 Results In this section the accuracy of the final interferer position is quantified. To achieve this goal a simulative tool has been developed. Such tool models the movement of the satellite along the orbit with orbital inclination, and generates TDoA measurements every for a time interval of length, each TDoA measurement is generated with a Gaussian error having standard deviation. The final interferer position is obtained by aggregating all the collected TDoA measurements with the TS methods using a least square algorithm to compute the local corrections. For a given combination of the parameters such simulation is performed times and finally the empirical covariance matrix of the errors is computed: = The performance of the scheme are provided in terms of: 3D accuracy: = + + Horizontal accuracy: = + Vertical accuracy: = In the following the impact of the parameters,,,, and on the final interferer position estimate accuracy is evaluated. Table summarizes the adopted default values for these parameters. In turn, each of these parameters will be varied inside a specified range, while maintaining the values of the other parameters to their default values. Parameter Distance between antennas [m] TDoA estimation accuracy [ps] 5 TDoA measurement sampling time [s] 3 Collection time window [hours] 4 Relative orbital inclination [ ] 6 Default Value Table : Default values of the parameters having an impact on the final interferer position estimate. First, the impact of the distance between antennas on the final interferer position estimate is evaluated. Figure 5 shows that the relationships between and - - are hyperbolic. For the considered default values of the parameters, the 3D and horizontal accuracies when the antennas are far apart are 44 and 7, respectively. Figure 5 shows also that, on one hand, the minimum distance at which the antenna should be placed is =, because for smaller values of the accuracies get worse quickly (i.e.,,, and increase rapidly). On the other hand, it is always possible to improve the accuracies by increasing, but such an improvement becomes smaller as increases, hence it seems reasonable to limit to some meters. It is important to remark that these considerations depend on the adopted values for the parameters. For example, if a worse TDoA accuracy is considered, then the plot of Figure 5 shifts to the right, as a consequence larger antenna distances should be adopted to maintain desirable interferer position estimate accuracy. Accuracy of the final interferer position estimate vs. antenna distance D accuracy Horizontal accuracy Vertical accuracy Accuracy [km] Distance between antennas [m] Figure 5: Accuracy of the final interferer position estimate vs. antenna distance.

7 Next, the impact of the TDoA estimation accuracy on the final interferer position estimate is investigated. Figure 6 shows that the relationships between and - - are linear. The considered range for the TDoA estimation accuracy is 6, which covers most of the possible INSR that can occur in practice (see Figure 4 it is implicitly assumed that the jammer power must be at least db larger than the reference signal power to cause reception issues). On one extreme the final accuracies are =.9, =.8, and =.3 for = ; on the other extreme the final accuracies are = 75, = 8, and = 37 for = Accuracy of the final interferer position estimate vs. accuracy of TDoA measurements 3D accuracy Horizontal accuracy Vertical accuracy 4 Accuracy [km] Std TDoA measurements [ps] Figure 6: Accuracy of the final interferer position estimate vs accuracy of TDoA measurements. Figure 7 depicts the relationship between the TDoA measurement sampling time and the accuracy of the final interferer position estimate. As expected, the use of a larger sampling time results in less accurate estimations because fewer measurements are aggregated (indeed the collection time window is fixed: = 4 h ). Figure 7 shows that such a relationship is sublinear: if the sampling time is doubled the accuracy is less than doubled. This behavior can be explained considering that when the sampling time is doubled the collected TDoA measurements are halved, but each measurement carries more information that the measurement with the original sampling rate, because with the new sampling rate the satellite covers a large space before collecting a new measurement. When the sampling time is = 6 the final accuracies are =, =, and = 5 ; as a comparison with the default value = 3 the following accuracies are obtained: = 44, = 7, and = Accuracy of the final interferer position estimate vs. TDoA measurement sampling time 3D accuracy Horizontal accuracy Vertical accuracy Accuracy [km] TDoA measurement sampling time [minutes] Figure 7: Accuracy of the final interferer position estimate vs. TDoA measurement sampling time. The effect of the collection time window is now analyzed. Figure 8 shows that the use of a larger time window results in more accurate estimations, indeed if the collection time windows is increased two positive effects occur: ) more TDoA measurements are collected and aggregated; and ) the satellite covers a large space, resulting in a more effective way of aggregating the measurements (this can be interpreted as a geometric gain ). In particular, the second effect is dominant for very short time windows. This can be seen by comparing the accuracy = 6 achievable with = 4 minutes and = 3 (this values falls out of the range plotted in Figure 8), with the accuracy = 5 achievable with = 4 hours

8 and = 3 minutes (see Figure 7). In both cases a total of 8 TDoA measurements are aggregated, but the performance achievable with the smaller time window = 4 minutes are much worse than those achievable with = 4 hours, because the satellite covers a short angle span in 4 minutes and this results in poor aggregation performance. Figure 8 shows also that, on one hand, the minimum collection time window that should be used is = hours, because for smaller values of the accuracies get worse quickly. On the other hand, it is always possible to improve the accuracies by increasing, but such an improvement becomes smaller as increases, hence it seems reasonable to limit to 4 5 hours. Indeed, the accuracies for = 6 hours ( = 9, =, and = ) are not much better than the accuracies for = 5 hours ( = 34, =, and = 6 ) and for = 4 hours ( = 44, = 7, and = 35 ). Moreover larger than 6 hours should not be considered because the satellite would disappear below the interferer s horizon plane at a certain point (see the right side of Figure 3), in this case the interferer signal is visible anymore. Accuracy of the final interferer position estimate vs. collection time window D accuracy Horizontal accuracy Vertical accuracy Accuracy [km] Collection time window [hours] Figure 8: Accuracy of the final interferer position estimate vs. collection time window. The impact of the relative orbital inclination on the final interferer position estimate is now investigated, for a range of from to 9. Notice that if < then the satellite disappears below the interferer horizon plane, whereas for symmetric reasons the accuracies for = > 9 are equal to the accuracies for = 9 that is lower than 9. Figure 9 shows an interesting behavior of the accuracies with respect to the relative orbital inclination. When approaches 9 the values of,, and increase sharply. This behaviour occurs because when is close to 9 both the antennas and the interferer lay approximatively on the same plane, which is the plane defined by the satellite orbit, and this geometric configuration produces poor aggregation performance (the reason for this behavior will be discussed later in more details). Instead, as soon as the geometric scenario deviates from this unfavorable case the interferer estimates becomes very accurate for a wide range of. For example, for = 7 the accuracies are = 59, = 33, and = 49, and for = 4 the accuracies are = 37, = 5, and = 7. Interestingly, the horizontal accuracy remains stable to = 5 also for < 4, whereas the vertical accuracy (and as a consequence the 3D accuracy as well) gets worse for < 4, reaching the value = 7 for = Accuracy of the final interferer position estimate vs. orbital inclination 3D accuracy Horizontal accuracy Vertical accuracy Accuracy [km] Orbital inclination [ ] Figure 9: Accuracy of the final interferer position estimate vs relative orbital inclination.

9 A different interpretation of the results of Figure 9 can be provided by fixing the Cartesian coordinate system with respect to the satellite orbit, for example considering a coordinate system such that the orbital plane lays in the x-y plane (see Figure ). The locus of points on the Earth associated to constant relative orbital inclination, and hence to a constant estimation accuracy, are circumferences that lays in planes that are parallel to the orbital plane. In particular, the locus of point with = 9 is the circumference that is obtained by projecting the satellite orbit on the Earth surface. This means that the Earth surface can be sectioned into many circumferences using planes that are parallel to the satellite orbital plane, and the interferer position estimate accuracy depends on which of these circumferences the interferer lays on. By defining an estimate accuracy threshold it is possible to identify the zone on the Earth associated to an accuracy worse than. For example, Figure 9 shows that all the points with an associated horizontal accuracy worse than the horizontal accuracy threshold, = must have a relative inclination angle above = 84.55, the associated locus of points on the Earth surface is a strip of width centered in the satellite orbit projection. If the interferer is located inside such strip then its horizontal position estimate accuracy will be worse than, =, whereas if the interferer is located out of this strip then its horizontal position estimate accuracy will be better than, =. If the horizontal accuracy threshold is increased to, = 5 then the width of the strip decreases to 5. The strips associated to, = and, = 5 are represented, with the correct scales, on the left side and on the right side of Figure, respectively. Strip on the Earth associated to an accuracy worse than km Strip on the Earth associated to an accuracy worse than 5 km Earth Satellite orbit Low accuracy strip Earth Satellite orbit Low accuracy strip z [m] z [m] y [m] x [m] y [m] x [m] Figure : Earth strip in which the horizontal accuracy of the estimation is worse than km (left) or 5 km (right). All the above results have been obtained with a cross-track antenna placement. Now the along-track placement is considered as well and compared with the cross-track placement. First, in order to understand the difference between the two approaches, Figure and Figure show the locus of points associated to the cross-track placement and the along-track placement, respectively. Such locus of points are limited to the plane parallel to the x-z plane and passing through the interferer position. It is important to remark that this figures have not been plotted with a proper scale (indeed the numerical labels of the axes are missing), they simply represent a qualitative view of what is going on for the two antenna placements for the orbital inclination = 9 (left side) and < 9 (right side). The 3D locus of points of a TDoA measurement is a hyperboloid, and the intersection between this hyperboloid and the considered plane is a hyperbola. For the cross-track placement and = 9 (left side of Figure ), all the hyperbole degenerate to a line, representing the projection of the satellite orbit in the considered plane. The interferer position, denoted in the figures with the letter J (Jammer), belongs to this line. However the TS method is not able to discriminate in which point of the line the interferer lays. In the 3D space the situation is even worse: the locus of point of each TDoA is the whole y-z plane. Actually, since the TDoA measurements are affected by errors, the locus of points of each location measurement is not exactly the y-z, and the solution at which the TS will converge is determined more by the measurement errors than by the real position of the interferer. This explains the bad performances of the cross-track placement for 9. The right side of Figure shows instead what happens for < 9. In this case the hyperbole are different, have a positive concavity, and they intersect exactly at the interferer position. The smaller, the more the hyperbole diverge from each other after intersecting, as a consequence the final estimation becomes more accurate because it is less affected by measurement errors. A similar behavior

10 occurs for > 9, with the hyperbole having a negative concavity in this case. The above considerations explain the trend observed in Figure 9. Figure : Locus of points of multiple TDoA measurements for the cross-track antenna placement and orbital inclination = 9 (left side) and < 9 (right side). The situation for the along-track placement, shown in Figure, is completely different, with the hyperbole that are now oriented differently compare to the cross-track case. On one hand, for = 9 (left side of Figure ) the hyperbole intersect the considered plane in a unique point, representing the position of the interferer. This suggests that for = 9 the along-track placement should allow to obtain better accuracies than the cross-track placement. On the other hand, for < 9 (right side of Figure ) the hyperbole intersect in two points, one representing the interferer position (J) and the other one representing the reflection of the interferer position with respect to the orbital plane (A). In addition to it, all the hyperbole are very close to the segment connecting J to A, as a consequence TDoA measurement errors may move the convergence of the TS method in any point belonging to this segment (and in the 3D space the situation is even worse because the hyperboloids tend to get even closer before separating). The smaller the farther J from A, meaning that the uncertainty area becomes larger and the estimate accuracy decreases. This suggests that for 9 the cross-track placement should allow to obtain better accuracies than the alongtrack placement. The above considerations explain the trend that will be observed in Figure 3. Figure : Locus of points of multiple TDoA measurements for the along-track antenna placement and orbital inclination = 9 (left side) and < 9 (right side). Finally, the performance obtained with the along-track placement are compared to that obtained with the cross-track placement. A total of simulations are performed, for each simulation the 3D distance and the horizontal distance between the real interferer position and the final estimate are computed, and finally the average 3D and horizontal distances are calculated. Figure 3 shows how such average distances varies with respect to the relative orbital inclination, for both the cross-track and along-track antenna placements. The 3D average and horizontal distances for the cross-track antenna placement are equal to the 3D and horizontal accuracies and of Figure 9. This means that the estimation errors are unbiased, i.e., the averages of the errors in the x, y, and z directions are equal to. Indeed, this is the reason why the previous analyses were focused only on the variances of the estimation errors. Instead, for the along-track antenna placement this is not the case: the resulting errors are biased, and for this reason the average distances between the interferer position and the estimates are now considered in place of the variances of the errors.

11 As expected from the previous considerations, Figure 3 shows that the performance achievable with the along-track placement is worse than the performance achievable with the cross-track placement for most values of the relative orbital inclination, but for close to 9 the estimates with the along-track placement are more accurate than those provided by the cross-track placement. As a final consideration, if only two antennas are available then the cross-track disposition seems the best choice, but if two pairs of antennas are available then by using both placements it may be possible to exploit the strength of both approaches, for example either ) by using the cross-track placement as the default method to generate the final interferer position estimate, and as soon as it is detected that such estimate is highly inaccurate (e.g., because it is located far from the Earth surface) the system automatically switches to the estimate generated by the along-track placement; or ) by aggregating together both the TDoA measurements generated by the cross-track placement and the TDoA measurements generated by the along-track placement, indeed since the locus of points seems to be almost orthogonal (see Figure and Figure ) this approach may be able generate accurate estimates for all values of the orbital inclination. 3 Average distance for cross track and along track antanna placement vs. orbital inclination 5 Average distance [km] 5 3D average distance - cross track Horizontal average distance - cross track 3D average distance - along track Horizontal average distance - along track Orbital inclination [ ] Figure 3: Average distance between interferer position and final estimates, for cross-track and along-track placements. Conclusions and Way Forward This paper proposes and evaluates an on-board satellite geolocation technique for single satellite. The proposed methodology is based on the aggregation, through a Taylor Series (TS) technique, of multiple Time Difference of Arrival (TDoA) measurements collected by the satellite at different time instants. The estimation accuracy of the proposed approach is numerically evaluated, and the impact on it of different system parameters is investigated. This analysis provides many insights on how to configure the parameters of the system and on the strengths and limitations of the proposed approach. For example: ) the antennas should be placed cross-track and at least meter apart; ) the TDoA collection time window should be at least hours; 3) a reasonable TDoA sampling time is 3 seconds; and 4) the closer the interferer to the projection of the satellite orbit on the Earth the lower the estimation accuracy, though this limitation can be compensated by adding two additional antennas along-track. Looking forward, the proposed approach can be improved in a number of different ways. First of all, additional types of location measurements can be collected and aggregated with the TS technique. For example, the Cross Ambiguity Function (CAF) can generate also Frequency Difference of Arrival (FDoA) measurements, and the Amplitude Comparison Monopulse (ACM) scheme implemented with two feeders in one antenna dish can be used to generate Angle of Arrival (AoA) measurements. In this case care must be taken on how to rescale different types of measurements in order to weight the measurement errors properly. Second, the TS method can be used to generate frequent estimations, which can be used by a sequential technique such as the Kalman filter in order to improve the estimation accuracy. This approach may be particularly useful to track a dynamic interference source. Finally, the application of Multiple Hypothesis Tracking (MHT) techniques [RD.] should be explored in order to deal with multiple interferer scenarios.

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