Robust Interference Mitigation in GNSS Snapshot Receivers

2.1 Jamming Signals

The received jamming interference can assume different waveforms. According to Morales Ferre et al. (2019), jamming signals can be classified into five categories: Class I jammers are CW-modulated signals with a bandwidth of up to 100 kHz; Class II and III jammers include single-chirp and multi-chirp signals; Class IV are chirp signals with frequency bursts that aim to expand the band of disrupted frequencies; and Class V jammers include pulsed and DME-like signals. Class I jammers, also known as amplitude-modulated jammers, generate single- or multi-tone sinusoidal signals. These jammers represent the simplest and most widely studied waveform type. Chirp jammers from Classes II–IV are characterized by an instantaneous frequency f_J(t) that changes over time, often showing a sawtooth-like waveform with rapidly changing frequency or phase characteristics. In this paper, we consider a single-chirp jammer with frequencies periodically sweeping from f_min to f_max every T_sweep seconds. Sweep periods are commonly around 10 μs, and sweep ranges (Δf = f_max – f_min) can vary within the range of 10–40 MHz (Mitch et al., 2011). Regarding Class V, signals transmitted by DME and TACAN systems are examples of legitimate pulsed jamming waveforms that can potentially disrupt GNSS performance. Aircraft interrogators in DME systems transmit paired high-power pulses, with each pulse potentially modeled by a Gaussian function. TACAN combines pulse pairs with amplitude modulation, where the amplitude of the pulses is modulated by a rotating antenna to provide both distance and azimuth information. Gao (2007) described the DME/TACAN pulse structure, including DME pulse pairs as well as TACAN pulse amplitude modulation and reference pulse group patterns.

Several metrics have been adopted in the literature to characterize the relationship between jamming and noise signals. The jammer-to-noise ratio (J/N) is defined as the ratio between the received jamming power J and the noise power N, whereas the jammer-to-noise PSD ratio (J/N₀) is defined as the ratio between J and N₀. These two ratios are related as follows:

$\frac{J}{N}=\frac{J}{N_0}\frac{1}{B_{R_x}}=\frac{JT}{N_0}\frac{1}{N_{\text{snap}}}=\frac{A_J^2}{2{\sigma }^2}$ 3

where T is the snapshot duration in seconds, A_J is the amplitude of the jamming signal, and N_snap corresponds to the number of snapshot samples or chunk length. One of the main signal quality indicators in GNSS is the carrier-to-noise PSD ratio (C/N₀), defined as the ratio between the carrier power C and C/N₀ depends solely on the received signal and is continuously estimated by the receiver and determined post-correlation, usually in logarithmic units, i.e., dB-Hz. GNSS receivers first estimate the signal-to-noise ratio (SNR) and then use a model that relates the SNR to C/N₀ (Borio et al., 2012). The C/N₀ estimated by the receiver may be referred to as an effective C/N₀, given that SNR estimation does not usually account for the degradation resulting from, for instance, jamming interferences (Betz, 2000, 2001). Considering this, it is possible to quantify the effect of a jammer as the degradation in the effective C/N₀.

2.2 Signal Quantization

RFI detection and mitigation techniques usually assume that input data are quantized with a significant number of bits B that is sufficient to fully represent the jamming signal, i.e., at least 8 (Díez-García & Camps, 2019). Nevertheless, this is not the case with most GNSS receivers, as 2-bit ADCs are very common in the market. Moreover, in the absence of jamming interferences and under suitable conditions, GNSS receivers provide reasonable results for B = 1. To maximize the effectiveness of state-of-the-art jamming detection and mitigation techniques, it is beneficial to have enough quantization levels available so that the full jamming signal is represented. A sufficient number of quantization bits is required to ensure that the receiver is sensitive to the power differences between non-interfered and interfered samples.

The number of quantization bits and the quantization dynamic range can be dynamically altered according to the characteristics of the received signal. The quantization dynamic range is defined as the difference between the largest and smallest values that a quantized signal can assume. A fixed dynamic range guarantees a proper representation of the expected noise signal. However, samples affected by jamming suffer from clipping because, as the name implies, the range is fixed and is not susceptible to changes in the received power. This clipping effect occurs when the number of quantization bits is not sufficient to represent the signal, in cases such as when the signal is overpowered by a jammer. Díez-García and Camps (2019) proposed an adaptive dynamic range system based on the AGC, where the quantization thresholds and the dynamic range are determined by the variance of the input signal.

The quantized signal can assume the amplitude values in the set B = {−(2^B−1), …, −3, −1, 1, 3, …, 2^B−1}, which corresponds to odd numbers ${\{2i+1\}}_{i=-2^{B-1},-2^{B-1}+1,\dots ,2^{B-1}-1}$ (Borio, 2008, §6.2.3). The quantization thresholds can be calculated as Q₀ = {−(2^B – 1) + 2, …, −3, −1, 0, 1, 3, …, (2^B – 1) – 2}. Additionally, the input signal must be multiplied by an optimal AGC gain factor. This factor depends on the standard deviation of the snapshot noise before the quantization block, denoted as σ. This term may be found by minimizing the quantization loss expression proposed by Borio (2008, §6.2.4), which is expressed as follows:

$L\left(A_g\right)=\frac{2}{\pi }\frac{{\left(1+2{\sum }_{i=1}^{2^{B-1}-1}{e}^{-{\alpha }_i^2}\right)}^2}{1+8{\sum }_{i=1}^{2^{B-1}-1}i\cdot \text{erfc}\left({\alpha }_i\right)}$ 4

where ${\alpha }_i=-i/\left(\sqrt{2}{A}_g\sigma \right)$ and consequently $A_{\text{opt}}={\min }_{A_g}L\left(A_g\right)$ . Figure 2 shows quantization loss values as a function of the normalized AGC gain, denoted as A_gσ, for the values of B considered in the experimental analysis. The figure suggests that the requirement for an optimal AGC gain can be relaxed for higher values of B because these cases exhibit a larger range of normalized AGC gain values at which the quantization loss remains almost constant. The optimal AGC gain factor values and their corresponding quantization losses are shown in Table 1. The loss introduced by signal quantization for B = 8 is almost negligible for an increasing noise standard deviation up to a normalized AGC gain of 70. In contrast, the minimum loss for B = {2, 4} is higher, with a very limited range of normalized AGC gain. The loss in 1-bit quantization systems cannot be improved by adjusting the AGC gain, as it remains constant at 1.96 dB.

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FIGURE 2

Quantization loss as a function of the normalized AGC gain for the values of B considered in the experimental analysis

The quantization loss expression in Equation (4) was proposed by Borio (2008, §6.2.4).

GNSS positioning can still be reliable with 1-bit quantization owing to the robustness of pseudorandom noise codes. This feature makes low-bit quantization receivers especially attractive for low-power applications requiring simple hardware. However, higher quantization levels are often necessary in scenarios involving interference, multipath, or high-precision needs. As a result, most studies on RIM assume a sufficient bit depth to fully represent jamming signals, typically B > 8. Recognizing that such high bit depths may not suit low-power applications in which snapshot architectures are more practical, our work evaluates the performance of snapshot RIM at lower ADC quantization levels. This approach allows us to investigate the limits of the well-established RIM strategy.

3.1 Huber’s ZMNL

Following the work by Borio et al. (2018), we use Huber’s nonlinearity to reduce the impact of interfered samples in the TD. This ZMNL is the derivative of Huber’s loss function and allows us to identify outliers by comparing the amplitude of the TD samples with a decision threshold. Huber’s ZMNL is defined as follows (Wang & Poor, 1999):

${\psi }_H\left(\Upsilon \right[k\left]\right)=\{\begin{array}{cc}\Upsilon \left[k\right]& \text{for}\left|\Upsilon \right[k\left]\right|\le T_h\\ T_h\mathrm{csign}\left(\Upsilon \right[k\left]\right)& \text{for}\left|\Upsilon \right[k\left]\right|>T_h\end{array}$ 6

where |·| is the absolute value operator and T_h is the nonlinearity decision threshold, with an optimal value of T_opt = 1.345σ for the real-valued signal case (Fox, 2002). The complex signum operator preserves the phase of the samples ϒ[k] and can be defined as follows:

$\mathrm{csign}\left(\Upsilon \right[k\left]\right)=\{\begin{array}{cc}\frac{\Upsilon \left[k\right]}{\left|\Upsilon \right[k\left]\right|}& \text{for}\Upsilon \left[k\right]\ne 0\\ 0& \text{for}\Upsilon \left[k\right]=0\end{array}$ 7

The standard deviation of the snapshot noise must be known to calculate the Huber nonlinearity decision threshold T_h. In the presence of interference, we need a robust measure of statistical dispersion such as the median absolute deviation (MAD) to estimate σ. The MAD can be expressed as follows:

$\mathrm{MAD}\left(x\right)=\beta \mathrm{Med}\left(\right|x-\mathrm{Med}\left(x\right)\left|\right)$ 8

with Med(x) denoting the median of x and β being a constant scale factor. This method is widely used for obtaining an asymptotically consistent estimator of the standard deviation σ due to its simplicity and its breakdown point of 0.5 (Rousseeuw & Croux, 1993). The value of the scale factor β is distribution-dependent, being set to β = 1.4815 under a Gaussian assumption. The variance must be estimated after the first linear transformation has been applied, i.e., with the samples in ϒ[k].

Borio et al. (2018) interpreted Huber’s nonlinearity as a switch between the Gaussian and Laplace regimes, based on the fact that the nonlinearity in Equation (6) when a TD sample is recognized as an outlier, i.e., when |ϒ[k]| > T_h, is equivalent to the nonlinearity that results from the Laplace assumption (Borio & Closas, 2018). The Gaussian regime is applied when interference is weak, effectively modeling noise under nominal conditions but remaining sensitive to outliers. In contrast, the Laplace regime is more robust to outliers owing to its heavy-tailed distribution, although it sacrifices some precision under nominal conditions. The impact of this trade-off in performance is quantified through an LoE analysis, as detailed in the next subsection. The combination of these two regimes, as depicted in Figure 3, allows the framework to dynamically adapt to varying interference environments without requiring specific detection or estimation of the interference waveform.

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FIGURE 3

This overview of the RIM framework includes the interpretation of Huber’s nonlinearity, denoted as Ψ_H(·), as a switch between Gaussian and Laplace regimes, as proposed by Borio et al. (2018). The RIM algorithm is implemented in three processing steps: a first linear transformation to the TD denoted as T₁, a ZMNL such as Huber’s nonlinearity, and a second linear transformation subject to T₁ o T₂ = I. To compute the Huber ZMNL threshold T_h, we estimate σ² with the MAD measure of statistical dispersion. When applied in the frequency domain, the MAD is recursively calculated by dividing the snapshot into batches, as shown in Figure 4.

3.2 Loss of Efficiency

The LoE caused by Huber’s nonlinearity has been evaluated by Borio et al. (2018). The LoE is defined as the performance degradation caused by the ZMNL in the absence of interference and is given by the following ratio:

$L_0\left(T_h\right)=\frac{{\text{SNR}}_{\text{out}}^{\psi }}{{\text{SNR}}_{\text{out}}}$ 9

where SNR_out corresponds to the post-correlation SNR that measures the quality of C(τ, f_d). This ratio may be defined as follows (Betz, 2000, 2001; Borio et al., 2018):

${\text{SNR}}_{\text{out}}=\underset{\tau ,f_d}{\max }\frac{\left|E\right\{C(\tau ,f_d)\}{|}^2}{\frac{1}{2}\text{Var}\left\{C\right(\tau ,f_d\left)\right\}}$ 10

${\text{SNR}}_{\text{out}}^{\psi }$ corresponds to the post-correlation SNR that measures the quality of C_ψ(τ, f_d). Borio et al. (2018) demonstrated that the LoEs obtained under time- and frequency-domain processing are comparable, with a maximum value of 1.05 dB. As mentioned in Section 1, the DD-RIM framework used by Li et al. (2019) exploits the sparsity of the jamming signals in the time and frequency domains simultaneously, with the cost of a duplicated LoE. This approach results in a maximum LoE of 2.1 dB.

4.1 Snapshot RIM

In this section, we describe how frequency-domain RIM has been adapted for snapshot processing in the presence of interferences with time-varying instantaneous frequency. Chirp interferences commonly sweep a significant portion of the receiver bandwidth over time periods that are significantly shorter than the snapshot duration. Under these conditions, the snapshot samples no longer admit a sparse representation of the jamming component in the frequency domain. To address this issue, we recursively update the snapshot MAD every Δt_MAD seconds. Figure 4 provides an overview of how the snapshot MAD is robustly estimated by using recursion and how this connects to the regime selector from the RIM framework in Figure 3. The number of snapshot samples in the time domain is denoted as N_snap = M × N_M, where N_M = f_s Δt_MAD represents the number of samples per batch and M is the number of batches into which the snapshot is divided. Here, m = [1, …, M] represents the batch index, and Ψ_m denotes the application of the ZMNL in Equation (6) applied to the samples of batch m. The snapshot MAD is calculated recursively as follows:

${\overline{\sigma }}_m^2=\frac{{\sigma }_m^2}{m}+\frac{m-1}{m}{\overline{\sigma }}_{m-1}^2$ 11

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FIGURE 4

Overview of the snapshot RIM framework, where the outlier detection module is adapted to preserve the sparse nature of chirp jammers in the frequency domain

The snapshot samples are divided into M batches of Δt_MAD seconds, and Huber’s nonlinearity, denoted as Ψ_m, is applied to the N_M samples of each batch independently. The MAD measure of statistical dispersion is calculated recursively.

where $\overline{.}$ is defined as the recursive operator. Considering this, ${\overline{\sigma }}_m^2$ corresponds to the recursive estimate at update m and is dependent on the recursive estimation at update m – 1, denoted as ${\overline{\sigma }}_{m-1}^2$ . The non-recursive estimate ${\sigma }_m^2$ is computed as the MAD of the N_M samples of batch m in the frequency domain, i.e., after the first linear transformation T₁ has been applied as an FFT. Once we have obtained the robust estimate ${\overline{\sigma }}_M^2$ , we calculate the corresponding optimal value of the decision threshold for this snapshot, i.e., $T_h=1.345{\overline{\sigma }}_M^2$ . The ZMNL is also applied to every N_M samples, with the difference that no recursion is used for this step and, consequently, each batch is treated independently. Moreover, batches from the same snapshot share the same decision threshold value after its estimation via the recursive MAD. In the conducted experiments, frequency-domain RIM is applied every Δ t_MAD = 1 ms, and the robust MAD measure is computed recursively, considering snapshot batches of the same duration. In contrast, time-domain RIM is applied to the whole snapshot, and the MAD measure is computed by considering all snapshot samples N_snap.

Several aspects are worth discussing regarding the implementation of RIM. In conventional receivers, an FFT can be used in fast parallel algorithms to accelerate CAF computation during acquisition. These same FFTs can serve as the outputs of the first linear transformation T₁ when RIM is applied in the frequency domain. Based on findings in the literature and our results presented in Section 5.1.1, frequency-domain RIM is typically sufficient to mitigate most interferences, owing to the periodic nature of common jamming signals (Borio & Closas, 2019). Consequently, no additional computational load is required for FFT calculation with RIM when FFT-based algorithms for rapid acquisition are used. Furthermore, because RIM is applied pre-correlation, this approach only requires that the ZMNL be applied once, rather than repeating the processing for each correlator or satellite (Borio & Closas, 2019). In a conventional receiver, RIM is applied to the samples within the coherent integration segment, which is usually shorter than a snapshot. This shorter duration does not allow partitioning into multiple batches, as is done in snapshot RIM. In contrast, the practical implementation of RIM in snapshot receivers benefits from the availability of multiple batches within a snapshot, allowing for recursive estimation of signal variance across these batches. As previously mentioned, batches are selected to have a duration of 1 ms, matching the duration of a coherent integration segment. This choice ensures consistency with the implementation of RIM in a conventional architecture. Ultimately, if snapshots are shortened, fewer batches are available for refining variance estimates; however, RIM still mitigates interference with 1-ms chunks, as demonstrated in the literature, when applied to conventional receivers (Borio et al., 2018). Overall, the main challenge for the implementation of RIM in snapshot receivers does not arise from the lack of tracking module or snapshot duration constraints, as coherent integration segments in conventional receivers are typically shorter than snapshots. Rather, the primary challenge arises from the need for low-bit quantization to meet communication and power constraints in low-power applications.

4.2 RFI Simulation

In the first experiment, we use Python to generate 120-ms snapshots with GNSS and jamming data under various configurations. Global Positioning System (GPS) L1, Galileo L1, and BeiDou B1 signals are generated with the same sampling frequency as the two-sided bandwidth of the baseline receiver, i.e., $f_s=B_{R_x}=4.5\text{MHz}$ . The baseband signal models used for interference simulation are presented in Table 2, which considers the taxonomy proposed by Morales Ferre et al. (2019). Jammer 1 is a single-tone CW with f_J set to −1 kHz with respect to the baseband frequency, Jammer 2 is a chirp interference sweeping the whole receiver bandwidth ($f_{\min }=-\frac{f_s}{2}$ and $f_{\max }=-\frac{f_s}{2}$ ), and Jammer 3 is a wideband jammer with a sweep range of 20 MHz (from −10 to 10 MHz). Both Jammers 2 and 3 have a sweep period of duration T_sweep = 10 μs. The behavior of CW, chirp, and pulsed chirp jammers in the frequency domain over time can be understood from their spectrogram plots (Morales Ferre et al., 2019, Figure 1).

Wideband interferences with a sweep range greater than the receiver bandwidth can be interpreted as pulsed interferences, as the receiver is only affected by the jamming attack when the instantaneous frequency is within the receiver bandwidth. This is the case for Jammer 3, whose bandwidth (20 MHz) exceeds the receiver bandwidth (4.5 MHz), causing the interference to appear as pulsed interference in the time domain. Because of this pulsed effect, Jammer 3 does not contaminate all samples within the snapshot, as there are periods during which its instantaneous frequency lies outside the receiver bandwidth. In contrast, Jammer 2 consistently targets frequencies within the receiver bandwidth over the entire snapshot duration, t ∈ [0, 120] ms. The three jammers show a sparse nature in the frequency domain, whereas only Jammer 3 shows a sparse nature in the time domain, given its pulsed effect. Jammer 3 is categorized as both a Class II single-chirp jammer and a Class V DME-like jammer; thus, we are evaluating the snapshot RIM performance against signals similar to those from DME/TACAN systems.

As a preliminary step to proceed with the experimental section, we determined the appropriate ranges of J/N ratio values to consider for evaluation. Characterizations of mass-market jammers in previous literature (Mitch et al., 2011) have shown that the maximum jamming transmission peak power typically varies from −10 dBm to 30 dBm. This power corresponds to a J/N ratio of approximately 95-130 dB, respectively, in the case of our receiver. To convert from J/N to J/N₀ (see Equation (3)), we consider the two-sided receiver bandwidth $B_{R_x}$ to be equal to the sampling frequency f_s = 4.5 MHz, as we are performing complex sampling. To consider the worst-case scenario, in which the jammer suffers from the least possible attenuation, the free-space path loss (FSPL) factor is computed as $\text{FSPL}=\frac{4\pi d^2}{\lambda }$ , where d corresponds to the jammer-receiver distance and λ corresponds to the signal wavelength, which depends on the frequency of transmission. As L-band wavelengths range between 15 and 30 cm, the worst-case scenario (i.e., least path-loss attenuation for the jammer) is given by λ = 30 cm, resulting in 60 dB for d = 100 m. Considering that the maximum jammer transmitted peak power can reach 130 dB, the maximum J/N ratio that a receiver will experience is 70 dB for d = 100 m. Consequently, the J/N ratio values that we considered for assessing the performance of the RIM algorithm in the experimental section vary from 0 to 95 dB, with an additional 25 dB considered as a safety margin.

A diagram of the system model for this experiment is presented in Figure 5. In this diagram, r[n] corresponds to the clean snapshot samples generated by the GNSS signal simulator, and i[n] represents the jamming interference generated by the jamming signal simulator. The anti-jamming module is placed between the simulation tool, which is equivalent to the system front-end, and the baseband processing block. The parameter J controls the option to simulate and add a jamming signal to the clean GNSS snapshot, and the parameter M controls the option to enable RIM. If J = 1, the output of the simulation tool is polluted by the jammer as y[n] = r[n] + i[n], whereas y[n] = r[n] if J = 0. When M = 1, RIM is applied to the snapshot samples y[n]. In this case, the input of the baseband processing block is $\tilde{y}\left[n\right]$ , where $\widetilde{.}$ represents the impact of RIM. If the anti-jamming module is successful, $\tilde{y}\left[n\right]\approx r\left[n\right]$ . If M = 0, the input signal of the baseband processing block corresponds to the simulated snapshot samples y[n]. The LoE analysis is conducted under the conditions of J = 0 and M = 1.

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FIGURE 5

Diagram of the system setup in the first experiment with simulated RFI data

The parameter J controls the simulation and addition of a jamming signal to the clean GNSS signal, whereas the parameter M determines whether the RIM algorithm is enabled.

The quantization block in Figure 5 reduces the resolution of the snapshot samples y[n] from B = 16 to the desired value of B. The performance of RIM is assessed for B = {2, 4, 8} and an adaptive quantization dynamic range that is proportional to the snapshot variance, which increases with the presence of jamming. For large dynamic ranges, a higher resolution is needed to perceive the contribution of useful GNSS signal, which can be achieved by increasing the number of quantization bits. The optimal AGC gain factor is calculated according to the expressions provided in Table 1. It is relevant to highlight that the variance used to calculate the AGC gain factor is not calculated robustly and, consequently, contains the effect of the jamming component. Each case of study within the first experiment is defined by a combination of values for J and M and a specific jammer type. For each case of study, ten snapshots with a duration of 120 ms (i.e., 1200 ms of recording) are processed. For the LoE assessment, two 120-ms snapshots are processed.

4.3 ESA Test-Bed

In the second experiment, the performance of a COTS GNSS receiver is compared with that of the baseline receiver in the processing of realistic GNSS jammed signal recordings provided by ESA. A diagram of the setup used to generate and record realistic GPS L1 and Galileo L1 signals under three different scenarios is shown in Figure 6. A different jamming interference is simulated for each scenario, including a single-tone CW centered at −1 kHz with respect to the baseband frequency, a chirp with T_sweep = 10 ms and Δf = 9 MHz, and another chirp with T_sweep = 10 ms and Δf = 20 MHz. These interferences resemble those simulated in the first study for consistency, but with experimental data. As in Experiment 1, Jammer 3 resembles the pulsed signals emitted by DME/TACAN systems. This feature is a result of the pulsed effect described in Section 4.2, which arises when the interference bandwidth is significantly greater than the receiver bandwidth.

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FIGURE 6

Setup used by ESA to generate and record realistic GPS L1 and Galileo L1 signals in the presence of three RFI signals with I/S values from 0 to 100 dB

Benchmark PVT solutions are derived based on RINEX observation files recorded by receiver B and are compared with the baseline positioning results. Receiver A is included only to confirm results during testing.

To generate RFI data, three configurations are defined as the input of the radio frequency constellation simulator (RFCS), each consisting of several files describing the characteristics of the signals to be generated, including the RFI features (e.g., sweep period, transmit power). Two GNSS benchmark receivers are used. The first one (receiver A) is connected to the RFCS, and the second one (receiver B) is connected to the signal recorder. The output of receiver A is not used for further evaluation; its purpose is to provide a confirmation of the results while the test-bed is being used. The signal recorder provides a binary file that can be processed by the snapshot baseline receiver, providing a baseline PVT solution. Benchmark PVT solutions are derived based on the receiver-independent exchange (RINEX) observation file recorded by receiver B. In Experiment 1, the J/N ratio was considered in the definition of the interference features; in contrast, in this experiment, the interference-to-signal ratio (I/S) is used instead. These two ratios are proportional to the interference power, with I/S ratio values typically being lower because they correspond to the ratio between the interference power and the useful signal power, not just the noise. For this experiment, I/S values range from 0 to 100 dB, with power transitions of 1 dB every 2 s of recording (i.e., approximately 16 120-ms snapshots).

5.1 Experiment 1

In this section, we assess the performance of the snapshot RIM framework with the GNSS and jamming data described in Section 4.2. We first compare results obtained with the frequency- and time-domain RIM algorithms in Section 5.1.1, investigating which is the most suitable option to ensure that the baseline receiver is resilient against typical interferences while avoiding an unnecessary increase in LoE. In Section 5.1.2, we discuss the effect of low-bit quantization on the baseline receiver in the presence of interference and in terms of the effective J/N, waveform distortion, and quantization gain. The impact of quantization on robust variance estimation is also studied. We provide a general performance assessment and LoE results in Section 5.1.3.

5.1.1 Frequency- vs. Time-Domain Snapshot RIM

It is of interest to compare the results obtained with the snapshot RIM approach in the time and frequency domains. The three implemented jamming signals assume a sparse representation in the frequency domain, with Jammer 3 also assuming a sparse representation in the time domain owing to its wide sweep range, which causes a narrowband pulsed effect. Figures 7(c) and 7(d) show the performance of the frequency- and time-domain snapshot RIM for B = 8 in the presence of the three jammers implemented in the first experiment (see Table 2). The number of satellite vehicles (SVs) detected by the receiver is plotted as a function of the simulated interference J/N. When the receiver is jeopardized by a jammer and RIM is not enabled, this metric drops dramatically. Nevertheless, when RIM is enabled in a domain where the interfered samples can be identified as outliers, we observe an increase in the range of J/N values at which the receiver can detect a considerable number of SVs.

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FIGURE 7

Results on the effect of signal quantization and snapshot RIM performance: (a) evolution of the effective J/N value at the receiver for B = {1, 2, 4, 8} as a function of the actual J/N, (b) performance of snapshot RIM under Jammer 1 attack for B = {2, 4, 8}, with the number of detected SVs used as a metric, (c)–(d) comparison of frequency- and time-domain RIM frameworks for 8-bit quantization under three types of jamming, with detected SVs versus J/N used as a performance metric

The results show that Jammer 3 can be successfully mitigated by both the time- and frequency-domain snapshot RIM, with the performance of the frequency-domain RIM being slightly better. Frequency-domain RIM can reduce the impact of Jammers 1 and 2, whereas time-domain RIM is not successful, as the interfered samples from these two jammers cannot be identified as outliers in the time domain. For instance, in the presence of Jammer 1, the gain offered by RIM is 35 and 5 dB in the frequency and time domains, respectively. Morales Ferre et al. (2020) suggested that most interferences found in typical environments assume a sparse representation at least in the frequency domain. This trend suggests that applying the frequency-domain RIM is sufficient to mitigate jamming interferences in typical environments. A solution would be to use DD-RIM, in which the RIM algorithm is sequentially applied in the time and frequency domains. However, this approach would lead to a duplicated LoE. Considering this, the RIM algorithm is only applied in the frequency domain throughout the remainder of our study.

5.1.2 Effect of Signal Quantization

Effective J/N

Snapshot samples affected by jamming suffer from a clipping effect when the number of quantization bits is low. This phenomenon occurs because the number of available quantization values is not sufficient to allow for full signal representation. Consequently, the J/N observed at the receiver, which we refer to as the effective J/N, differs from its actual value, similar to how the effective C/N₀ differs from its actual value, as explained in Section 2.1. In Figure 7(a), the value of effective J/N observed at the receiver is studied as a function of the actual value of J/N used to simulate the jamming signal, which is a known input parameter of the interference simulator. This figure shows that the quantization dynamic range increases proportionally with B and that a saturation point occurs when the whole dynamic range is used. Once this saturation point is reached, an increase in jamming power cannot be further perceived by the receiver. The saturation point occurs at J/N values of 0, 12, 17, and 35 dB with effective J/N values of approximately 0, 5, 12, and 33 dB for 1, 2, 4, and 8 quantization bits. Thus, for instance, in the case of B = 4, the value of J/N measured at the receiver cannot be greater than 12 dB, and this value is reached when the J/N of the simulated interference is 17 dB.

For a low B, the magnitude of J/N that the receiver can perceive is also low. For B = 2, as quantization values are limited and signal power is more spread owing to waveform distortion, the effective J/N at the receiver is very low. Although one might think that this low value would diminish the effect of powerful interference without the need for mitigation, this is not the case. The results in Section 5.1.3 suggest that when RIM is not enabled, the receiver observes a protection gain in terms of detected SVs and PVT availability only by lowering the number of quantization bits B from 8 to 2. However, the C/N₀ plots show a strong performance decrease for B = 2 when RIM is both enabled and disabled. The pseudo-resilience offered by low resolution does not translate in terms of actual signal quality, i.e., the number of detected SVs or PVT availability can be high while the actual C/N₀ at the receiver is very low. This finding suggests that, in the presence of interference, it is indeed necessary to use high signal resolution and to enable RIM to achieve true resilience.

Waveform Distortion

Figure 7(a) shows how, for B = 8 and before saturation, effective J/N values coincide with the actual J/N used for interference simulation. However, this is not the case for lower values of B. We believe that this phenomenon is due to the waveform distortion introduced by a low number of quantization bits, which causes artifacts in the form of multiple frequency harmonics in the signal spectrum. This phenomenon is evidenced by the results presented in Figure 8, where the PSD and spectrogram of Jammer 1 are shown for B = {2, 8}. The interference waveform strongly differs from the baseband signal model, as shown in Table 2, when there is a low number of quantization bits. Additionally, in the case of B = 2, the power is spread across multiple frequencies, differing from the single tone obtained for B = 8.

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FIGURE 8

PSD and spectrograms of 120-ms simulated GPS L1, Galileo E1, and BeiDou B1 signals with interference from Jammer 1 (see Table 2) for B = {2, 8}, C/N₀ = 45 dB, and J/N = 30 dB (a) PSD for B = 2 (b) PSD for B = 8 (c) Spectrogram for B = 2 (d) Spectrogram for B = 8

Quantization Gain

In Figure 7(b), the performance of snapshot RIM is assessed in the presence of Jammer 1 as a function of the J/N ratio for B = {2, 4, 8}. In this figure, the performance metric is the number of detected SVs. Results are presented for when RIM is both enabled and disabled. A gain of 10, 20, and 35 dB is observed when the RIM algorithm is applied for 2, 4, and 8 quantization bits, suggesting that the gain obtained when the RIM algorithm is applied is proportional to the number of quantization bits.

When RIM is not applied, the baseline performance is better for B = 2 in terms of detected SVs. Nevertheless, this improvement is due to the pseudo-resilience discussed earlier.

Impact on Robust Variance Estimation

In Table 4, snapshot variance estimation results with the MAD measure of statistical dispersion are shown with both clean and interfered signal for B = {2, 4, 8}. These results have been normalized based on the quantization gain values from Table 1. The simulated snapshot contains GPS L1, Galileo E1, and BeiDou B1 signals with a C/N₀ of 45 dB. Three different values of quantization dynamic range (DR) are considered, namely, DR₁, DR₂ and DR₃. These dynamic ranges correspond to the variances of snapshots interfered with Jammers 1, 2, and 3, respectively, for J/N = 30 dB (i.e., DR_l is proportional to the variance of a snapshot interfered by Jammer l). In each row of the Interfered Signal subtable, we present MAD results obtained when estimating the variance of the GNSS signal subject to interference from the jammer used to compute the dynamic range for that particular row. For example, the row corresponding to DR_l contains results for the signal with interference from Jammer l. Given the robustness of the MAD estimation method, it is expected that when the variance of an interfered snapshot is estimated, only the contribution of useful GNSS signal has an impact on the result. In other words, we expect to obtain the same variance estimation results for both a clean and an interfered snapshot because the MAD measure should be able to cancel the contribution of the jamming component.

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TABLE 3

RFI Features for the Three RFCS Input Scenarios in the Second Experiment

These interferences resemble those used in the first experiment (see Table 2). Values of I/S under study vary from 0 to 100 dB.

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TABLE 4

Snapshot Variance Estimation Results with the MAD as a Robust Measure of Statistical Dispersion for y[n] = r[n] (Clean Signal, Left Subtable) and y[n] = r[n] + i[n] (Jammed Signal, Right Subtable), with C/N₀ = 45 dB and J/N = 30 dB

Signal quantization is performed with different numbers of bits B = {2, 4, 8} over three different dynamic ranges. DR₁, DR₂, and DR₃ correspond to the dynamic ranges calculated from the variance of a snapshot with interference from Jammers 1, 2, and 3, respectively (see Table 2). After the ADC, both the signal and its variance are represented in unitless digital counts.

The results suggest that MAD offers robust results if the following two conditions apply: (1) the number of quantization bits is high enough to allow for full signal representation (i.e., B ≥ 8) and (2) the MAD is applied in the domain where the interference exhibits a sparse representation. Considering this, results for B = 8 in the frequency domain match for all dynamic ranges, as the number of quantization bits is sufficiently high and the three jammers assume sparse representation in the frequency domain. Meanwhile, in the time domain and for B = 8, only results with Jammer 3 match between the clean and interfered signals, as this is the only jammer assuming a sparse representation in the time domain.

For B = {2, 4} and in the time domain, all results in the Clean Signal subtable are equal to 0, indicating that the MAD measure discards all of the useful signal contribution as a consequence of the waveform distortion caused by low-bit quantization. For B = {2, 4} and in the frequency domain, the MAD method intercepts the contribution of useful signal when applied to the clean snapshots. Better results are obtained in the frequency domain because samples are mapped to a wider range of values, allowing for more quantization levels than those defined by B. In the Interfered Signal column for the same configuration, the variance estimation values are lower, suggesting that the interference causes the MAD measure to mistake clean samples for interfered samples. Overall, high resolutions are necessary to discard the jamming component when estimating the snapshot variance using the robust MAD measure.

5.1.3 Overview of Snapshot RIM Performance

An assessment of the snapshot RIM performance in the frequency domain for B = {2, 8} is presented in Figure 9. The key performance indicators (KPIs) under study are the average C/N₀, number of detected SVs, and PVT availability. C/N₀ values are plotted only when the receiver passes the integrity test. Overall, the results suggest that in the presence of interference, it is necessary to use high signal resolution to experience the resilience offered by the snapshot RIM framework, with the gain being proportional to B. In the analysis with Jammer 3, we observe that reducing the number of quantization bits can be slightly beneficial when no anti-jamming technique is available, but only in the presence of interferences that are sparse in the time domain. Nevertheless, the best performance is achieved with the highest signal resolution and with snapshot RIM enabled.

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FIGURE 9

Assessment of the frequency-domain snapshot RIM algorithm for B = {2, 8} (a) B = 2 (b) B = 8

The performance metrics under study are the number of detected SVs, PVT availability, and average C/N₀. C/N₀ values are plotted only when the receiver passes the integrity test.

In the case of Jammer 1 without RIM, the number of detected SVs and PVT availability fall to 0 at J/N = 30 dB for B = 8 and at J/N = 40 dB for B = 2. The 10-dB gain provided by reducing the signal resolution is attributed to the pseudo-resilience experienced by the receiver, as discussed earlier. The fact that this resilience is not genuine becomes apparent when observing the average C/N₀ plots, which show equally poor performance when RIM is not enabled for both signal resolutions. When snapshot RIM is enabled, a gain of 30 dB in terms of average C/N₀ is observed for B = 8 with respect to when RIM is disabled, while the improvement for B = 2 is only 5 dB.

In the case of Jammer 2 without RIM, the number of detected SVs and PVT availability fall to 0 at J/N = 20 dB under both signal resolutions, although the metrics start degrading earlier for B = 8. Average C/N₀ values are similar under these two configurations for disabled RIM. When snapshot RIM is enabled, the receiver acquires all of the satellites with 100% PVT availability for B = {2, 8} for the full range of tested jamming powers. This result is due to an actual improvement in signal quality only for the case of high signal resolution, i.e., B = 8, for which we observe a gain of 30 dB in terms of average C/N₀. This gain is only 5 dB for B = 2. For J/N > 40 dB and B = 8, the high values of detected SVs and PVT metrics are not meaningful, as the average C/N₀ drops dramatically.

The clipping effect caused by a low number of quantization bits, as described in Section 2.2, slightly mitigates the impact of the interference component in the presence of the pulsed chirp. Jammer 3 assumes a sparse representation not only in the frequency domain but also in the time domain. Consequently, the clipping resembles the use of a ZMNL by setting a limit for the magnitude of the samples overpowering the receiver in the domain in which the interference shows a sparse nature, i.e., the time domain. Consequently, for B = 2 and disabled RIM, all satellites are acquired with 100% PVT availability and stable average C/N₀ values. For B = 8, the receiver performance begins to decrease for J/N > 10 dB when RIM is disabled. Nevertheless, in the presence of Jammer 3, the best overall performance is obtained for B = 8 when snapshot RIM is enabled, resulting in the acquisition of all satellites with 100% PVT availability and the maximum average C/N₀ for J/N > 35 dB.

Loss of Efficiency

Results have been obtained by averaging each satellite LoE and by setting Huber’s nonlinearity threshold to T_h = 1.345σ as in the remainder of the experiments. The measured LoE introduced by the frequency-domain snapshot RIM with MAD applied every 1 ms is 0.45, 0.48, and 0.49 dB for 2, 4, and 8 quantization bits, respectively. These values differ from the theoretical loss of 0.29 dB. The difference of approximately 0.15 dB between the theoretical value and the empirical loss is considered to be small and is not a potential issue for satellite acquisition.

5.2 Experiment 2

In this section, we report on a benchmark study between the baseline receiver and a COTS GNSS receiver in the presence of three different interference types (see Table 3). The setup used in this experiment is detailed in Section 4.3. The KPIs under study are the average C/N₀ and the number of detected SVs as a function of I/S values from 0 to 100 dB. The results show a significant performance improvement for the baseline receiver when snapshot RIM is used to process realistic RFI recordings, even when compared against the benchmark GNSS receiver.

5.2.1 Scenario 1: CW

In Figure 10(a), we assess the performance of the baseline and benchmark receivers when processing the recording from scenario 1. An adaptation curve in terms of detected SVs is observed for the benchmark because it needs to obtain the navigation message, unlike the baseline snapshot receiver. The KPIs begin to decline at I/S = 55 dB for the benchmark. When the anti-jamming module is disabled, the baseline shows a modest improvement of approximately 2 dB compared with the benchmark results in terms of detected SVs. However, with the anti-jamming module activated, this improvement increases to approximately 20 dB for the baseline. Overall, the results indicate that the performance of the snapshot-based receiver surpasses that of the benchmark receiver in terms of average C/N₀ and the number of detected SVs.

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FIGURE 10

Results obtained in Experiment 2 when processing 200 ms of realistic recordings provided by ESA containing GNSS data with interference from the jammers described in Table 3 (a) Jammer 1 (b) Jammer 2 (c) Jammer 3

The baseline receiver performance is assessed with disabled and enabled snapshot RIM and compared with the performance of the benchmark receiver for different values of I/S. The metrics under study are the number of detected SVs and average C/N₀.

5.3 24-h Specifications

The performance of the proposed mitigation technique was validated by processing a 24-h RFI-free recording of L1/G1 and L5 bands in open-sky conditions, with the goal of computing KPIs such as the horizontal position accuracy, velocity accuracy, and TTFF. These KPIs are defined in Table 5. The 24-h test was conducted under three different configurations: one in which the anti-jamming module is disabled and two in which the anti-jamming module is enabled, with the application of frequency-domain RIM and DD-RIM, respectively. Results are presented in Table 6. In the case of the L1/G1 band, the horizontal position accuracy is 2.298 m both without RIM and with frequency-domain snapshot RIM. This value increases to 2.377 m with DD-RIM. In the L5 band, the worst accuracy is also obtained with DD-RIM, whereas the best accuracy is obtained with frequency-domain RIM. We do not consider the error introduced by DD-RIM to be a concern because this error is small and our previous experiments show that applying RIM in the frequency domain is sufficient to counter the jamming signal component. The velocity accuracy obtained in the L1/G1 band is 0.290 m/s with disabled RIM, 0.299 m/s with frequency-domain RIM, and 0.305 m/s with DD-RIM. These errors are considered to be sufficiently small. Moreover, no error is introduced by RIM in terms of velocity accuracy in the L5 band.

The collected results show that the use of the FFT and IFFT functions as the first and second linear transformations in the frequency-domain RIM accounts for the most important increase in TTFF. When frequency-domain RIM is applied in the L1 band, the TTFF increases by 0.25 s with respect to the case in which RIM is disabled. However, only 0.01 s is added to this value when the time-domain RIM algorithm is also enabled with the DD-RIM. The same trend is observed in the L5 band, where the TTFF increases by 0.31 s when frequency-domain RIM is applied and only 0.03 s is added for DD-RIM. Although this could be a motivation to use time-domain RIM, frequency-domain RIM offers a stronger protection against interferences and better position and velocity accuracies. Furthermore, we consider these TTFF increases of 0.25 and 0.31 s to be sufficiently small. Together, the present findings suggest that the implemented anti-jamming module does not negatively affect the product performance.