Characterization of the Multipath Situation Under Meaconing

2.1 Model of Correlator Outputs

GNSS implements direct-sequence spread spectrum. The received signal is correlated with a local replica controlled by the code and carrier numerically controlled oscillators over the integration time T_i (Hofmann-Wellenhof et al., 1997). Typically, GNSS receivers encompass at least three correlators: the early (Λ_E), prompt (Λ_P), and late (Λ_L) correlators, where the local replicas are shifted in code by –c_τ / 2,0, and c_τ /2, respectively (c_τ is the chip spacing). The correlator output in the presence of meaconing interference has been derived as a function of the tracking errors ε_η = [ε_τ, ε_θ, ε_f]^⊤ and relative parameters Δv by Hussong et al. (2023). The correlator output at epoch k is expressed as the following linear combination:

$\Lambda ({\epsilon }_{\eta },\Delta \nu )={\Lambda }_a\left({\epsilon }_{\eta }\right)+{\Lambda }_s({\epsilon }_{\eta },\Delta \nu )+{\Delta }_n$1

with the nominal and spoofing contributions, Λ_a and Λ_s, expressed as follows:

$\begin{array}{c}{\Lambda }_a\left({\epsilon }_{\eta }\right)=\sqrt{C_a}{d}_k{\zeta }_{\tau }\left({\epsilon }_{\tau }\right){\zeta }_f\left({\epsilon }_f\right)e^{j{\epsilon }_{\theta }}\\ and\hspace{0.33em}{\Lambda }_S({\epsilon }_{\eta },\Delta \nu )=\sqrt{\Delta g{C}_a}{d}_k{\zeta }_{\tau }({\epsilon }_{\tau }+\Delta \tau ){\zeta }_f({\epsilon }_f+\Delta f)e^{j({\epsilon }_{\theta }+\Delta \theta )}\end{array}$2

Here, ζ_τ and ζ_f are the code and frequency synchronization mismatch functions defined by Ghizzo et al. (2024, Equation (7)). C_a is the received signal power, and d_k is the navigation message bit (considered constant over the integration time). The noise contribution, Λ_n, can be defined as Gaussian noise with power P_n, expressed by Ghizzo et al. (2024) as follows:

$P_n=\frac{N_0}{T_i}(1+\Delta N){\zeta }_{\tau }\left(0\right)$3

2.2 Model of Tracking Loops

The dynamic behavior of the tracking loops has been modeled by Ghizzo et al. (2025) as a nonlinear system of two difference equations, depending on the relative parameters Δη. This paper focuses on the system’s dynamic value at lock (i.e., where the loop has successfully established and maintains synchronization with the incoming signal dynamics). The tracking errors at lock have been shown to be equivalent to the system’s stable equilibrium (SE) values (ε_τ, ε_θ, and ε_f), being the solutions of this system (without stress error):

$\{\begin{array}{ccc}{D}_{\tau }({\epsilon }_{\eta },\Delta \nu )=0,& D_{\theta }({\epsilon }_{\eta },\Delta \nu )=0,& {\chi }_f({\epsilon }_{\eta },\Delta \nu )=0\\ \frac{\partial D_{\tau }({\epsilon }_{\eta },\Delta \nu )}{\partial {\epsilon }_{\tau }}>0,& \frac{\partial D_{\theta }({\epsilon }_{\eta },\Delta \nu )}{\partial {\epsilon }_{\theta }}>0,& \frac{\partial {\chi }_f({\epsilon }_{\eta },\Delta \nu )}{\partial {\epsilon }_f}>0\end{array}$4

D_τ and D_θ are the code and phase discriminator outputs, and χ_f is the phase discriminator difference. Details about these discriminators are provided in Appendix B. These terms can be expressed as follows:

$\begin{array}{c}{D}_{\tau }({\epsilon }_{\eta },\Delta \nu )=\frac{C_a}{2P_d}\left({\zeta }_f{\left({\epsilon }_f\right)}^{2}Z\right({\epsilon }_{\tau },0)+\Delta g{\zeta }_f{({\epsilon }_f+\Delta f)}^{2}Z({\epsilon }_{\tau }+\Delta \tau ,0)\\ +2\sqrt{\mathrm{\Delta g}\hspace{0.17em}}\cos \left(\Delta \theta \right){\zeta }_f\left({\epsilon }_f\right){\zeta }_f({\epsilon }_f+\Delta f)Z({\epsilon }_{\tau },\Delta \tau ))\\ D_{\theta }({\epsilon }_{\eta },\Delta \nu )={\epsilon }_{\theta }+\frac{\Delta \theta }{2}-\mathrm{atan}\left({\gamma }_{\Delta }\right({\epsilon }_{\tau },{\epsilon }_f\left)\tan \right(\frac{\Delta \theta }{2}\left)\right)-P\pi \\ {\chi }_f({\epsilon }_{\eta },\Delta \nu )={\epsilon }_f+\Delta f-\frac{1}{2\pi T_{\dot{1}}}\left[\mathrm{atan}\right({\gamma }_{\Delta }({\epsilon }_{\tau },{\epsilon }_f)\tan \left(\frac{\Delta \theta }{2}\right))\\ -\mathrm{atan}\left({\gamma }_{\Delta }\right({\epsilon }_{\tau },{\epsilon }_f\left)\tan \right(\pi \Delta f{T}_{\dot{1}}+\frac{\Delta \theta }{2}\left)\right)]-\frac{p^{\text{'}}}{2T_{\dot{1}}}\end{array}$5

The integers p and p' represent the phase and frequency ambiguity, respectively, and we have the following:

$Z({\tau }_1,{\tau }_2)\triangleq {\zeta }_{\tau }({\tau }_1+\frac{c_{\tau }}{2}){\zeta }_{\tau }({\tau }_1+{\tau }_2+\frac{c_{\tau }}{2})-{\zeta }_{\tau }({\tau }_1-\frac{c_{\tau }}{2}){\zeta }_{\tau }({\tau }_1+{\tau }_2-\frac{c_{\tau }}{2})$6

${\gamma }_{\Delta }({\epsilon }_{\tau },{\epsilon }_f)\triangleq \frac{{\zeta }_{\tau }\left({\epsilon }_{\tau }\right){\zeta }_f\left({\epsilon }_f\right)-\sqrt{\mathrm{\Delta g}\hspace{0.17em}}{\zeta }_{\tau }({\epsilon }_{\tau }+\Delta \tau ){\zeta }_f({\epsilon }_f+\Delta f)}{{\zeta }_{\tau }\left({\epsilon }_{\tau }\right){\zeta }_f\left({\epsilon }_f\right)+\sqrt{\mathrm{\Delta g}\hspace{0.17em}}{\zeta }_{\tau }({\epsilon }_{\tau }+\Delta \tau ){\zeta }_f({\epsilon }_f+\Delta f)}$7

The code tracking error at lock ε_τ can be found by solving Equation (4) numerically and is presented in Figure 3 (explained hereafter) for GPS L1 C/A signals; the structure is detailed in Appendix B. The tracking error at lock (i.e., the SE) does not represent the actual errors, but rather the tracking error to which the system tries to converge. In slow-dynamic scenarios (i.e., static, pedestrian, car), the SE can be assimilated to the DLL output error in the code pseudorange.

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FIGURE 3

Maximum and mean values of the DLL errors at lock with respect to Δf, Δτ, and Δg, considering $\Delta \theta \sim u[0,2\pi ]$ for GPS L1 (a) $\left|\underset{\Delta \theta \in [0,2\pi ]}{mean}\left[{\epsilon }_{\tau }\right]\right|\mathrm{wrt}\hspace{0.17em}\mathrm{\Delta f}$ and Δτ, (b) $\underset{\Delta \theta \in [0,2\pi ]}{\max }\left[{\epsilon }_{\tau }\right]\mathrm{wrt}\hspace{0.17em}\Delta f$ and $\Delta \tau ,\left(c\right)\left|\underset{\Delta \theta \in [0,2\pi ]}{mean}\left[{\epsilon }_{\tau }\right]\right|\mathrm{wrt}\hspace{0.17em}\Delta f$ and Δg.

C/A; wrt: with respect to

Figure 3(a) presents the absolute value of the tracking mean error at lock, when averaged over Δθ uniformly distributed between 0 and 2π, as a function of Δf and Δτ. The results are given for GPS L1 C/A signals. This mean error is a good approximation of the DLL output error found in the code pseudorange estimation, in the situations when Δθ varies more than 2π during T_i (as evidenced in Section 5). Figure 3(a) highlights that the code pseudoranges can be corrupted, with an error of ±11 m for many relative parameter values (mainly when Δτ < 1 chip and when $\left|\Delta f\right|<25\mathrm{HZ}$). Figure 3(b) reports the maximum absolute value of the tracking error at lock. The DLL errors can potentially reach ±15 m when Δτ < 1 chip and $\left|\Delta f\right|<25\mathrm{Hz}$ and for specific values of Δθ. Particularly when Δf ∈ [1; 10] Hz, large DLL errors are observed, even for a very small relative delay Δτ. Figure 3(c) shows the influence of the relative power Δg on the maximum tracking errors at lock, demonstrating that the largest errors are almost proportional to Δg (on a linear scale).

Overall, Figure 3 shows that the DLL outputs may be corrupted by huge deterministic errors of up to ±15 m under meaconing multipath, especially when Δτ < 1 chip and $\left|\Delta f\right|<25\mathrm{Hz}$ . When the relative delay Δτ exceeds 1 chip, the deterministic part of the DLL output errors decreases to zero. When the absolute relative Doppler $\left|\Delta f\right|$ becomes larger than 25 Hz, the deterministic part of the DLL output errors decreases, but might still be significative depending on the error tolerance of the user’s application.

2.3 Impact of Tracking Loop Distortions on C/N₀

The expected value of the C/N₀ estimate in the multipath situation (determined by the moment method) has been modeled by Ghizzo et al. (2024) as a function of the relative parameters Δη:

$C_{\Delta \eta }=E\left[\frac{\hat{C}}{N_0}\right]=\frac{1}{T_{\dot{1}}}\frac{\sqrt{{\overline{P}}_d^2-C_a^2{\sigma }_d^2}}{P_n+{\overline{P}}_d-\sqrt{{\overline{P}}_d^2-C_a^2{\sigma }_d^2}}$8

with:

$\begin{array}{c}{\overline{P}}_d=(\zeta {\left({\epsilon }_{\eta }\right)}^2+\Delta g\zeta {({\epsilon }_{\eta }+\Delta \eta )}^2\\ +2\sqrt{\Delta g}\zeta \left({\epsilon }_{\eta }\right)\zeta ({\epsilon }_{\eta }+\Delta \eta )\frac{\sin \left(\pi M\Delta f{T}_i\right)}{M\sin \left(\pi \Delta f{T}_i\right)}\cos \left(\overline{\Delta \theta }\right))C_a\end{array}$9

$\begin{array}{c}{\sigma }_d^2=2\Delta g\zeta {\left({\epsilon }_{\eta }\right)}^2\zeta {({\epsilon }_{\eta }+\Delta \eta )}^2[1+\frac{\sin (2\pi M\Delta f{T}_i)}{M\sin (2\pi \Delta f{T}_i)}\cos (2\overline{\Delta \theta })\\ -2\frac{{\sin }^2(\pi M\Delta f{T}_i)}{M^2{\sin }^2(\pi \Delta f{T}_i)}{\cos }^2\left(\overline{\Delta \theta }\right)]\end{array}$10

where T_e is the C/N₀ estimation time, $\overline{\Delta \theta }=\pi M\Delta f{T}_i+\Delta \theta$ is the mean relative phase within T_e, and M = T_e / T_i is the number of integration periods within T_e.

Figure 4 shows the value of Equation (8) as a function of Δf, Δτ, and Δg, with the approximation ε_η = 0, with a nominal C/N₀ of 40 dB‧Hz, T_e = 1s, T_i = 20 ms, and still considering GPS L1 C/A signals. Figure 4(a) shows the mean value of the theoretical C/N₀, averaged over Δθ uniformly distributed between 0 and 2π (i.e., $\Delta \theta \sim u[0,2\pi ]$). The results are plotted for Δg = ΔN = –3 dB, as a function of Δf and Δτ. This figure demonstrates that when Δτ < 1 chip, C/N₀ is significantly degraded (as for the DLL output errors), even for a low received meaconer power. The degradations are mainly observed for $\left|\Delta f\right|<25\hspace{0.17em}\mathrm{Hz}$, except when Δf is close to zero ($\left|\Delta f\right|<0.1\hspace{0.17em}\mathrm{Hz}$), where both authentic and meaconer signals can combine with constructive interference.

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FIGURE 4

Mean and minimum values of the C/N₀ estimates (moment method) with respect to Δf, Δτ, and Δg, considering $\Delta \theta \sim u[0,2\pi ]$ for GPS L1 C/A (a) $\underset{\Delta \theta \in [0,2\pi ]}{mean}\left[C_{\Delta \eta }\right]Wrt\Delta f$ and Δτ, (b) $\underset{\Delta \theta \in [0,2\pi ]}{mean}\left[C_{\Delta \eta }\right]Wrt\Delta f$ and Δτ, (c) $\underset{\Delta \theta \in [0,2\pi ]}{mean}\left[C_{\Delta \eta }\right]Wrt\Delta f$ and Δg.

Figure 4(b) presents the minimum theoretical C/N₀value for Δθ varying between 0 and 2π. These results are similar to the mean values shown in Figure 4(a). This similarity indicates that Δθ does not significantly affect the C/N₀ estimations, except for small relative Doppler values Δf, when $\left|\Delta f\right|<T_e^{-1}=1\hspace{0.17em}\mathrm{Hz}$. Additionally, when $\left|\Delta f\right|<0.1\hspace{0.17em}\mathrm{Hz}$, the mean C/N₀ is marginally affected by meaconing interference, whereas the lowest values show degradations of approximately 10 dB.

Figure 4(c) depicts the mean C/N₀ values, averaged over $\Delta \theta \sim u[0,2\pi ]$, and for Δτ = 0 chip (as this relative delay produces large C/N₀ degradations). C/N₀ is plotted against Δf and Δg. Even for a small relative power of Δg = –30 dB, C/N₀ is degraded by approximately 3 dB. The degradations exceed 15 dB when Δg>–10 dB and $\left|\Delta f\right|\in [0.5,25]\hspace{0.17em}\mathrm{Hz}$. In all plots, the C/N₀ values are symmetrical with respect to Δf.

Overall, Figure 4 shows that the estimated C/N₀ can be drastically reduced under meaconing multipath, particularly when Δτ < 1 chip and $\left|\Delta f\right|<25\hspace{0.17em}\mathrm{Hz}$. This decrease highlights the lack of robustness of the moment method C/N₀ estimation under meaconing interference, potentially causing a loss of lock of the GNSS signals.

3.1 Conditions for Observing the Multipath Situation

The multipath situation is dependent on two factors (as illustrated by Figures 3 and 4): the delay condition and the Doppler criterion, defined as follows:

• Delay condition - To observe the multipath situation, both the authentic and meaconer peaks must affect the correlators. For GPS L1 C/A, the signal peaks after correlation form a triangle with a width of 2 chips. The early and late correlators are computed at a delay difference of c_τ/2 with respect to the prompt correlator. The meaconer peak can only be observed with a delay greater than the authentic peak (i.e., Δτ ≥ 0 s) because the meaconer signal detours before reaching the user GNSS antenna, resulting in a larger propagation time. Assuming that the receiver tracks the authentic peak, the meaconer peak distorts the late correlator if the delay difference between the two peaks is smaller than $T_c+\frac{c_{\tau }}{2}$, where T_c is the chip duration. Conversely, if the receiver tracks the meaconer peak, the authentic peak distorts the early correlator if $\Delta \tau \le T_c+\frac{c_{\tau }}{2}$. Figure 5 displays the correlator outputs in the presence of authentic and meaconer signals. The sum of the authentic and meaconer peaks modifies the correlator values only when $\Delta \tau <T_c+\frac{c_{\tau }}{2}$, resulting in a new equilibrium different from the nominal equilibrium (as also evidenced by Figure 3). In Figure 5, c_τ = T_c for pedagogic reasons (more visible in the illustration), but the remainder of this paper uses c_τ = T_c /10 because this value is commonly used in GNSS receivers, as it reduces the tracking estimation error (Van Dierendonck et al., 1992). Larger choices of c_τ would cause the same impacts but with larger amplitudes.

  

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  FIGURE 5

  Illustration of the delay condition for observing the multipath situation for GPS L1 C/A signals

• Doppler criterion - The C/N₀ and DLL outputs strongly depend on the relative Doppler Δf as evidenced by Figures 3 and 4. To observe large multipath errors and C/N₀ degradations, the authentic signal and the meaconer signal should have a relative Doppler shift of less than 25 Hz at the user GNSS antenna. Otherwise, the multipath situation causes smaller degradations of the GNSS observables. Whereas the delay condition is mandatory for a signal to be in the multipath situation, the Doppler criterion only indicates the most affected signals among all signals respecting the delay condition and therefore helps to identify the satellite signals experiencing high degradations caused by meaconing multipath.

3.2 Definition of the Geometry in This Study

In this paper, the meaconer impact is computed according to the position of the user relative to the meaconer. This approach allows for better visualization and understanding of the meaconing effects on GNSS receivers. The meaconer impact also depends on the velocities of both the meaconer and the user, as well as the positions of the satellites and the meaconer characteristics. The term geometry refers to the relative position and velocities of the user, the meaconer, and the satellites, as well as the meaconer gain, intrinsic delay, frequency, and phase offsets.

3.3 Mapping of the Delay Condition to the Geometry

The delay condition 𝕄 is defined with respect to the relative parameters as follows:

$M\iff \Delta \tau <T_c(1+\frac{c_{\tau }}{2})=T_{\max }$11

The relative delay Δτ can be expressed as the difference between the meaconer signal propagation time τ_s and the authentic signal propagation time τ_a. From Figure 1, the relative delay can be expressed as follows:

$\begin{array}{c}\Delta \tau ={\tau }_s-{\tau }_a=(\frac{d_{SM}}{c}+{\tau }_m+\frac{d_{MU}}{c}+{\tau }_{ant,s})-(\frac{d_{SU}}{c}+{\tau }_{ant,a})\\ \frac{d_{SM}+d_{MU}-d_{SU}}{c}+{\tau }_m+\Delta {\tau }_{ant}\end{array}$12

where c represents the speed of light, τ_m is the meaconer intrinsic delay, and Δτ_ant accounts for the antenna group delay difference between the authentic τ_{ant, a} and meaconer τ_{ant, s} signal delays at the user’s receiver antenna. The expression of At in Equation (33) can be rearranged using the parameters defined in Figure 6 to obtain a geometric mapping of the delay condition. In Figure 6, M' is the orthogonal projection of M onto the line (SU), and O is the orthogonal projection of S onto the line (MU). β₁ represents the non-oriented angle between the meaconer and the satellite as seen by the user, and β₂ represents the non-oriented angle between the imaginary point O and the satellite as seen by the meaconer. γ₁ represents the non-oriented angle between the user and the meaconer as seen by the satellite, and γ₂ represents the non-oriented angle between the meaconer and the imaginary point O as seen by the satellite.

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FIGURE 6

Illustration of the satellite (S), meaconer (M), and user (U) in the (SMU) hyperplane, with the authentic signal (green) and the meaconer signal (red)

The objective is to obtain an expression of Δτ that depends only on terms known in the geometry. For this, Δτ is derived in terms of d_SU, d_MU, and the angle β₁. Using the parameters from Figure 6, the relative delay in Equation (33) becomes the following:

$\begin{array}{c}\Delta \tau =\frac{d_{SM}+d_{MU}-d_{S{M}^{\text{'}}}-d_{M^{\text{'}}U}}{c}+{\tau }_m+\Delta {\tau }_{ant}\\ \frac{d_{SM}-d_{S{M}^{\text{'}}}}{c}+\frac{d_{MU}}{c}(1-\frac{d_{M^{\text{'}}U}}{d_{MU}})+{\tau }_m+\Delta {\tau }_{ant}\\ \frac{d_{SM}-d_{S{M}^{\text{'}}}}{c}+\frac{d_{MU}}{c}(1-\cos ({\beta }_1\left)\right)+{\tau }_m+\Delta {\tau }_{ant}\end{array}$13

The computation of (d_SM – d_SM') is presented in Appendix A, proving that d_SM – d_SM', ≈ 0 m is an excellent approximation of the difference. Thus, we have the following:

$\Delta \tau \approx \frac{d_{MU}}{c}(1-\cos ({\beta }_1\left)\right)+{\tau }_m+\Delta {\tau }_{ant}$14

By inserting Equation (14) in Equation (11), the delay condition can now be expressed as follows:

$M\iff d_{MU}(1-\cos ({\beta }_1\left)\right)<c(T_{\max }-{\tau }_m-\Delta {\tau }_{ant})$15

In particular, if T_max < τ_m + Δτ_ant, the delay condition is never satisfied, and the multipath situation is never observed. Furthermore, if $d_{MU}<c(T_{\max }-{\tau }_m-\Delta {\tau }_{ant})/2$ the delay condition is always satisfied, and the multipath situation is observed for all visible satellites. Otherwise, for a given geometry, the satellites in the multipath situation 𝐌 are those seen inside a cone with summit U, symmetry axis (UM), and aperture angle β_max given by the following:

${\beta }_{\max }={\cos }^{-1}(1-c\frac{T_{\max }-{\tau }_m-\Delta {\tau }_{ant}}{d_{MU}})$16

A representation of the multipath cone 𝕄 depicted by Equation (15) is shown in Figure 7, where the user U is in the air and the meaconer M is on the ground. In this configuration, only some signals are in the multipath situation. These satellite signals are received in a direction close to the direction of the meaconer from the user’s perspective. As the meaconer is often seen with an elevation around 0˚, the signals in the multipath situation are more likely to have low elevations. Moreover, when the user is closer to the meaconer, the multipath cone aperture β_max becomes larger, as illustrated by the two examples on the skyplot in Figure 7(b).

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FIGURE 7

Illustrations of the multipath cone 𝕄 (a) In the (UMS) hyperplane, (b) In a skyplot from the user’s perspective, with two meaconers: τ_m = 0 s, d_M1U = 500m, and d_M2U = 6500m.

3.4 Mapping of the Doppler Criterion to the Geometry

As reported by Petovello (2015), the relative Doppler Δf can be expressed as follows:

$\Delta f=f_s-f_a=\frac{{(v_S-v_M)}^T\cdot u_{SM}+{(v_M-v_U)}^T\cdot u_{MU}}{\lambda }-\frac{{(v_S-v_U)}^T\cdot u_{SU}}{\lambda }$17

where f_s (f_a) is the received frequency at the user’s GNSS antenna of the mea-coner (authentic) signal. v represents the velocity vectors, and u denotes the unit direction vectors. λ is the wavelength of the GNSS signal. Note that the receiver clock drift has no impact on the relative Doppler Δf, as it is equally present in f_s and f_a. As a satellite is seen in almost the same direction from the user’s and from the meaconer’s point of view, u_SM ≈ u_SU. This approximation leads to the following:

$\Delta f\approx \frac{{(v_U-v_M)}^T\cdot (u_{SU}-u_{MU})}{\lambda }$18

The C/N₀ degradation and the DLL output distortions in the multipath situation are strongly affected by the absolute value of the Doppler shift $\left|\Delta f\right|$ . The relative Doppler can be mapped to the geometry to identify the regions 𝔽 of small absolute relative Doppler $\left|\Delta f\right|<f_{\max }$, where the impact of meaconing multipath is the strongest. The value of f_max can be chosen depending on the tolerated degradations of the DLL outputs and of C/N₀. For instance, if an application tolerates 3 m of DLL output error and a 6-dB decrease in the estimated C/N₀ caused by meaconing multipath, Figures 3 and 4 show that these degradations are only observed when $\left|\Delta f\right|<25Hz$. In this case, choosing f_max = 25 Hz for region 𝔽 would identify the satellite signals associated with larger potential degradations:

$\begin{array}{c}𝔽\iff \left|\Delta f\right|<f_{\max }\iff \left|\frac{{(v_U-v_M)}^T\cdot (u_{SU}-u_{MU})}{\lambda }\right|<f_{\max }\\ \iff \left|\frac{v_{MU}}{\lambda }\left(\cos \right({\alpha }_U)-\cos ({\alpha }_M\left)\right)\right|<f_{\max }\end{array}$19

$\iff 2\left|\frac{v_{MU}}{\lambda }\sin \left(\frac{{\alpha }_U+{\alpha }_M}{2}\right)\sin \left(\frac{{\alpha }_U-{\alpha }_M}{2}\right)\right|<f_{\max }$20

where $v_{MU}=\Vert V_{MU}\Vert =\Vert V_U-V_M\Vert$ is the relative velocity between the user and the meaconer, α_U is the non-oriented angle between v_MU and u_SU, and α_M is the non-oriented angle between v_MU and u_MU, defined as follows:

${\alpha }_U={\cos }^{-1}\left(\frac{v_{MU}\cdot u_{SU}}{{\nu }_{MU}}\right)\hspace{1em}and\hspace{1em}{\alpha }_M={\cos }^{-1}\left(\frac{v_{MU}\cdot u_{MU}}{{\nu }_{MU}}\right)$21

If the relative direction u_MU and velocities v_MU between the user and the meaconer are fixed, the relative Doppler Δf depends only on the direction of the satellite with respect to the user u_SU. Consequently, it is possible to represent the received relative Doppler on a skyplot as a function of the elevation and azimuth angles of u_SU. When mapped on a skyplot, the region 𝔽 corresponding to $\left|\Delta f\right|<f_{\max }$ forms a ring perpendicular to the vector v_MU (and containing the meaconer location). Three examples of the regions with f_m = 0 Hz are shown in Figure 8, for three different positions of the meaconer (shown in red on the sky-plots). These plots highlight the Doppler rings 𝔽 as a function of the elevation and azimuth angles of the received satellite signal. The relative velocity v_MU is indicated by a black arrow and is directed eastward (0˚ elevation and 90˚ azimuth) with a velocity norm of v_MU = 10 m/s. The relative Doppler does not depend on the distance between the user and the meaconer d_MU, but only on the direction vector u_MU with respect to the velocity vector v_MU. The absolute value of the relative Doppler Δf can reach (in Hz) up to 10 times the value of the relative velocity (in m/s) for GPS L1. The satellites observed far from the meaconer direction and its Doppler ring have a high absolute relative Doppler $\left|\Delta f\right|$ and are therefore less impacted by multipath distortions (as the magnitudes of the C/N₀ degradations and the DLL errors decrease as $\left|\Delta f\right|$ increases).

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FIGURE 8

Three examples of relative Doppler values Δf as a function of the geometry (v_MU = 10 m/s) (a) Meaconer perpendicular to the relative velocity, (b) Meaconer at 135˚ from the relative velocity, (c) Meaconer at 220˚ azimuth and 37˚ elevation.

4.1 Mapping of the Multipath Error Envelope on Skyplots

Figure 9 presents the meaconer multipath error envelope (MMEE), which represents the maximum code tracking errors at lock, obtained by solving Equation (4), as a function of the satellite elevation and azimuth angles. The MMEEs have been computed for Δg = ΔN = 0 dB and for three different relative velocities v_MU = [0, 5, 15] m/s between the meaconer and the user. Even if the received satellite signal power changes as a function of satellite elevation, the ratios AΔg and ΔN remain unchanged. Indeed, both the meaconer and the user view the satellite with the same elevation (as they are close one to each other); thus, they both receive the same power at their antenna input.

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FIGURE 9

Maximum tracking error at lock $\underset{\Delta \theta \in [0,2\pi ]}{\max }\left|{\epsilon }_{\tau }\right|$ as a function of the geometry, with Δg = ΔN = 0 dB

The figure highlights significant MMEE values (up to ±15 m) when the emitting satellite is observed in the multipath cone. Only those satellites that are viewed extremely close to the meaconer direction do not show large DLL errors; the remainder of the satellite signals are distorted by the meaconing interference depending on the relative velocity. In the static case, MMEE values are almost constant at ±15 m over the entire skyplot. When the relative velocity v_MU increases, the large MMEE values at lock agglomerate around a ring perpendicular to the velocity vector and containing the meaconer direction (shaping the Doppler rings 𝔽 evidenced in Section 3). The errors above 10 m are contained inside the Doppler ring corresponding to f_max = 25 Hz. The remainder of the sky-plot exhibits reduced but still significant DLL errors. In all cases, the errors outside of the multipath situation equal zero, because the receiver noise is neglected in these skyplots.

4.2 Mapping of C/N₀ Degradations on Skyplots

Figure 10 shows the minimum C/N₀ as a function of the geometry. The minimum value is computed by evaluating C/N₀ for all Δθ ∈ [0; 2π] (knowing the satellite elevation and azimuth angles) and taking the smallest result. While Δθ significantly influences the C/N₀ estimations for small values of v_MU (below 0.1 m/s, corresponding to $\left|\Delta f\right|<1Hz$), it plays a marginal role for larger values of v_MU. Therefore, the results displayed in Figure 10 provide a excellent approximation for dynamic cases of the received C/N₀. For the static case, the results show the largest degradation eventually observed.

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FIGURE 10

Minimum $C/N_0=\underset{\Delta \theta \in [0,2\pi ]}{\min }\left|C_{\Delta v}\right|$ as a function of the geometry, with Δg = ΔN = –3 dB

The C/N₀ values in Figure 10 are heavily degraded by the presence of the mea-coner in the multipath situation. In this figure, the nominal C/N₀ is set at 40 dB‧Hz. Outside of the multipath cone (in the northern part of the skyplots), the nominal/ jamming situation is observed, and C/N₀ decreases to approximately 38 dB‧Hz when Δg = ΔN = –3 dB. These values align with the results of Hussong et al. (2023). For the satellites in the multipath situation, the C/N₀ degradation exceeds 15 dB‧Hz when Δg = ΔN = –3 dB for almost all satellites in the southern hemisphere when v_MU = 5 m/s. When v_MU = 15 m/s, these degradations occur for the satellite in the Doppler ring 𝔽 corresponding to f_max = 25 Hz, particularly for satellites that are viewed close to the meaconer direction.

5.1 Definition of the Scenarios Under Scrutiny

Three scenarios were designed to represent a static scenario, a pedestrian trajectory, and a car trajectory. The pedestrian and car both pass close to a fixed meaconer on the ground. In each scenario, the same GPS satellite is under scrutiny, and its DLL outputs and estimated C/N₀ are monitored.

Figure 11 illustrates the pedestrian and car scenarios. The trajectories are each 1200 m long, with 200 m of the nominal/jamming situation, followed by 800 m where the GPS satellite under scrutiny is inside the multipath cone, and finally 200 m back in the nominal/jamming situation. The meaconer is located 100 m away from the trajectory, with a gain of 76 dB chosen to compensate the free-space losses to ensure that Δg = ΔN = 0 dB when the user is closest to the meaconer. Both the user and the meaconer antennas are omnidirectional, with a gain of 0 dBi in all directions. The meaconer intrinsic delay is set to τ_m = 0 s, and its phase offset is equal to θ_m = 0 rad. The only difference between the pedestrian and car scenarios is that the pedestrian travels the 1200 m at 1.4 m/s (in approximately 857 s), whereas the car travels the same distance at 14 m/s (in approximately 86 s).

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FIGURE 11

Illustration of the pedestrian and car trajectories

The relative parameters of both scenarios are displayed in Figure 12. Outside the multipath cone, the relative power and noise PSD remain below -10 dB, and the satellite is in the nominal or jamming situation, producing negligible C/N₀ and DLL output distortions, as previously investigated by Hussong et al. (2023). The relative Doppler is positive in the first half of the trajectory because the user is moving toward the meaconer, perceiving the meaconer signal with a higher frequency. Conversely, the relative Doppler is negative in the second half of the trajectory, with the zero-crossing of Δf occurring when the user is closest to the meaconer. The Doppler values are 10 times larger in the car scenario because the car is moving 10 times faster than the pedestrian. Finally, all of the relative parameter curves are almost symmetrical because the user trajectory is symmetric with respect to the meaconer, and the satellite does not move a significant distance during the trajectory duration.

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FIGURE 12

Relative parameters for the pedestrian scenario (top) and car scenario (bottom) (a) Pedestrian scenario, (b) Car scenario.

The static scenario represents a 120-s static user located 100 m from the meaconer (at the “closest point to the meaconer” in Figure 11). The meaconer is activated after 20 s; thus, the first 20 s of the static scenario are in the nominal situation. Similarly, the meaconer is shut down for the last 20 s of the scenario. Between these two time points, the meaconer gain is set to 76 dB to ensure that Δg = ΔN = 0 dB, and the satellite under scrutiny is in the multipath situation for 80 s.

The relative parameters for the static scenario are displayed in Figure 13. The relative distance cΔτ is almost constant but increases slowly (by approximately 1 m, or 5 GPS L1 wavelengths, during the meaconer activation) owing to the satellite’s motion. Because v_MU = 0 m/s in this scenario, Equation (18) indicates that Δf ≈ 0 Hz. The exact relative Doppler is indeed extremely close to zero but varies slightly owing to the satellite’s different observation angles with respect to the user and the meaconer (u_SM ≈ u_SU).

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FIGURE 13

Relative parameters for the static scenario

5.2 Degradation of DLL outputs and C/N₀ in the Scenarios Under Scrutiny

The three scenarios were run N times (N = 8000 for the static and car scenarios, N = 800 for the pedestrian scenario) with a Monte Carlo method. Only the random receiver-generated noise changed between the different runs. For each second of the scenarios, the mean value of C/N₀ and the mean value and standard deviation of the DLL outputs were computed with the N values available. The results are shown in Figures 14–16. Note that in the third subplots of the figures, the DLL outputs (shown in red on the figures) are the actual DLL errors comprised in the code pseudorange estimations, not the DLL errors at lock as in the previous sections (also plotted in the figures, but in black). Indeed, the DLL errors at lock assume that the loop manages to converge toward its SE, whereas the actual DLL error considers the transient response and the smoothing effect of the loop.

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FIGURE 14

C/N₀, DLL standard deviation, and DLL mean values for the static scenario

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FIGURE 15

C/N₀, DLL standard deviation, and DLL mean values for the pedestrian scenario

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FIGURE 16

C/N₀, DLL standard deviation, and DLL mean values for the car scenario