GNSS C/N₀ degradation model in presence of continuous wave and pulsed interference

2.1 Generic airborne civil aviation GNSS receiver

In order to understand the C/N₀ degradation analytical model, a generic airborne civil aviation GNSS receiver structure, as well as the behavior and effect of its components on the received signals, is described. In Figure 1, the receiver structure is presented.

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FIGURE 1

Generic civil aviation GNSS receiver block scheme

First, the antenna is the element responsible for capturing the incoming signal: at the antenna port (point A), there is a mix of all incoming signals; useful signals, GNSS and SBAS signals; and RFI signals such as DME/TACAN, JTIDS/MIDS, etc. Once the signals have been captured by the antenna, they are passed to the RFFE block. This block amplifies the received signals, shifts them from their received signal frequency carrier to the intermediate frequency, and filters them (removing the image frequency, removing the spurious frequencies, and removing the signal outside the interest frequency bandwidth). The filtered signals are modeled in Figure 1 at the RF (Radio-Frequency) and IF (Intermediate Frequency) filters output at point B. RTCA DO-292 (RTCA, 2004) defines the joint effect of these two filters plus the antenna filtering effect with an equivalent filter transfer function; the equivalent transfer function, H_RF(f), for a 20 MHz filter bandwidth is provided in Figure 2.

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FIGURE 2

RFFE plus antenna equivalent transfer function defined in DO292 customized for a BW = 20 MHz

The RFFE block is also responsible for gain control and digitizing the filtered signals with the application first of the AGC circuit followed by ADC. In the proposed airborne civil aviation L5/E5a GNSS receiver, the digital pulse blanker is introduced after the RFFE block. As explained in the introduction, the blanker is a device, which is going to blank (put to zeroes) the time and/or frequency samples of the incoming signal (mix of signals), which exceed a set threshold; the digitized and blanked signal is found at point C. In RTCA DO-292 (RTCA, 2004), the defined blanker is a temporal blanker called instantaneous blanker.

This blanking mechanism removes all the incoming signal time samples, which have a power over a given threshold (issues concerning its actual description and physical implementation are addressed in Garcia-Pena et al. (2019)); see Figure 3. For an optimal functioning, the blanker should also be coupled with the ADC/AGC blocks: to ensure that high-power pulses are not saturating the ADC/AGC and the blanked signal spans the ADC quantization range. The effect of the AGC/ADC and its coupling with the blanker are out of the scope of this paper. Finally, digitized and blanked signals are fed to the correlator, and it is at its output (point D) where the impact of the RFI signals and the blanking method is measured. Finally, the RFI signals at the correlator output (point D of Figure 1) are where the demodulation, acquisition, and tracking capabilities of the receiver can be impacted. It is at this point that these impacts are predicted and simulated within the analysis in this paper.

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FIGURE 3

Example of the behavior of the DO-292 instantaneous blanker over the signal complex envelope

2.2 General analytical model

The key figure of merit to analyze the RFI signals and the blanking method impact is the signal C/N₀, or more specifically, the difference between the C/N₀ when only the useful signal is present at the receiver antenna port (no RFI signals) and the C/N₀ when the useful signal and RFI signals are present at the receiver antenna port (with blanker activation), also called effective C/N₀ or (C/N₀)_eff. The difference between these two C/N₀ values is also called the C/N₀ degradation introduced by the RFI signals and the blanking method.

Although the blanking method is going to reduce the average power of the useful signal (part of the information signal is removed as well as part of the noise), RTCA DO-292 (RTCA, 2004) proposes to model the (C/N₀)_eff by defining an equivalent N_0,eff while keeping the original useful power C. Note that N_0,eff represents the effective noise power spectrum density that a receiver will observe at the correlator output if the receiver captures a useful signal with power C at the correlator output. This assumes that subsequent RFFE elements are considered as ideal (RF filter, IF filter, AGC/ADC), the correlator is also considered ideal, there are no RFI signals present, and the blanker is not activated. In other words, in section 2.6.2.3, RTCA DO-292 (RTCA, 2004) recommended a generic formula to compute the degradation of the C/N₀ through the increase of the background noise due to pulsed and continuous RFI, based on rigorous evaluation within the RTCA Special Committee 159. Note that from this definition, the (C/N₀)_eff and C/N₀ when only the useful signal is present values, and the difference between the two are expressed at point A of Figure 1 while they are calculated at point D (correlator output) when considering the impact of all the elements between the two points.

In order to mathematically model (C/N₀)_eff, the following considerations about the blanking and the incoming signals must be considered. First, although all received GNSS and SBAS signals are by definition useful signals, since the receiver has to isolate the signals one-by-one to exploit them, the GNSS and SBAS signals, which are not tackled by a specific channel (correlator), are also considered RFI signals: if the receiver is trying to isolate GNSS (or SBAS) signal i in one correlator block, all the other GNSS (or SBAS) signals j, j ≠ i, falling in the L5/E5a band are considered RFI signals. These signals are continuous (non-pulsed) signals, and its contribution is modeled within the term I_0,WB. Note that the blanking method will not target these signals since they are continuous, and thus, the blanking method settings will be determined by the pulsed RFI signals, such as DME/TACAN and JTIDS/MIDS. Moreover, note that this term, I_0,WB, also includes the contribution of all continuous aeronautical signals from other systems as well as continuous non-aeronautical signals, which are assumed to not trigger the blanker.

Second, pulsed RFI signals impact in two different ways (C/N₀)_eff:

• Part of the signal is removed due to the blanking and since the impact on the removed useful signal power, (1 − bdc)², is higher than the impact on the power of the noise, (1 − bdc), the equivalent N_0,eff can be seen to be increased by a factor of 1∕(1 − bdc). bdc is the blanker duty cycle, or in other words, the percentage of time the incoming signal is blanked (bdc ∈ [0, 1]).

• Not all the RFI signal samples have a power above the threshold; therefore, there is a part of the RFI signal that is not removed, and its influence must be added to the thermal noise; R_I is the below-threshold interfering-signal-to-thermal-noise ratio.

From these considerations, (C/N₀)_eff general mathematical model at the correlator output can be modeled as is:

1

where:

• P_u is the received useful signal power at the antenna input,

• G_u is the receiver antenna gain affecting the received useful GNSS signal,

• bdc is the blanker duty cycle if a blanker is used,

• L_u corresponds to losses on the useful signal due to the RF front-end filter, quantization, etc.,

• I_0,l represents the equivalent noise PSD level created by each individual source of interference, the interference source l = 0 being the equivalent thermal noise PSD,

• L is the total number of interference sources.

The equivalent thermal noise PSD term can be expressed as

2

3

where:

• (1 − bdc)𝛽₀ represents the combined effect of the blanker (1 − bdc) and the RFFE filter plus correlator, 𝛽₀, on the noise power degradation,

• N₀ is the thermal noise power spectrum density level generated by the RFFE,

• is the normalized PSD of the local replica of the m^th PRN code used by the correlator,

• H_RF(f) is the baseband transfer function of the equivalent RFFE plus antenna filter.

Expression (1) can thus be rewritten as

4

5

where:

• L_impu is defined as the receiver implementation losses on the useful GNSS signal.

Denoting C as the received signal power at the correlator output, with C = P_u ⋅ L_imp ⋅ G_u, the mathematical model of N_0,eff can be finally provided from Equation (4). In this new expression, the RFI signal impact is separated into continuous RFI interferences, I_0,cont, and pulsed RFI signals, I_0,pulse:

6

Note that K represents the number of continuous RFI interference signals and that I represents the number of pulsed RFI interference signals. This equation is usually expressed in a more compact form:

7

8

where R_I,i is the i^th source below-blanker interfering-signal-to-thermal-noise ratio. It is possible to identify:

9

10

Usually, the part of the term I_0,WB, which models the intra/inter-system RFI signals impact, is called the I_GNSS.

3.1 Generic structure of a correlator

A generic scheme of a correlator is given in Figure 4. Note that in this scheme the carrier wipe-off has been included and that for simplification purposes the correlation is assumed to be conducted in the continuous time instead of the discrete domain (digital samples at the RFFE output).

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FIGURE 4

Generic civil aviation GNSS receiver block scheme

The model of the useful signal inside the interval at the correlator input (RFFE output) assuming that no blanker mechanism is implemented is given below:

11

where:

• 𝜏 and are the signal code delay and the signal code delay estimation,

• f_d and are the signal doppler frequency and the signal doppler frequency estimation,

• 𝜙₀ and are the signal initial carrier phase and the signal initial carrier phase estimation,

• d(t) is the data signal (contains the transmitted information),

• c(t) is the PRN code signal (materialization or pulse shaping of the PRN code sequence).

Note that the carrier wipe-off introduces an amplitude term to have at the correlator output the useful signal power equal to C^′ where C^′ = P_u G_uL_u. Moreover, note that the RFFE losses, L_u, are already introduced in C^′, and thus, the correlator is expected to be ideal from the useful signal processing point of view (no additional losses). The final missing term to obtain C from C^′, 𝛽₀, is brought by the passing of the thermal noise through the correlator.

3.2 General large band interference power formula derivation with spectral separation coefficient

For simplification purposes, in this section, the influence of the blanker is omitted. To correctly derive I_0,l, some definitions must be given about this term. I_0,l represents the increase of the noise PSD, I_0,0, resulting from the reception of interference l. However, at the correlator output, the thermal noise impact, as well as of the interference signal l impact, is just a single value per interference source characterized by its distribution and power. Therefore, assuming that the interference signal correlator output can be approximated by a Gaussian distribution, I_0,l can be determined by comparing the thermal noise and interference powers at the correlator output and by inspecting the contribution of the noise PSD, I_0,0 = N₀ 𝛽₀, on the power of the noise term. Note that 𝛽₀ is the noise power degradation due to the RFFE filter plus correlator; therefore, modeling I_0,0 as N₀𝛽₀ implies to decrease the original thermal noise PSD, N₀, and to assume the RFFE and correlator elements are ideal for the noise.

The power of the l^th interference signal i(t) at the correlator output, P_l,post, when assuming a large bandwidth in comparison to the inverse of the integration time, 1/T_I, can be approximated by the following expression:

12

where:

• S_lpre(f) is the PSD of the l^th interfering signal before the integration,

• F(f) is the transfer function of the filter equivalent to the application of the integration (accumulation); the time impulse response of this filter is square pulse of length T_I centred at .

Therefore, since the integration term in Equation (12) is the same irrespective of the interfering signal i(t), in order to determine I_0,l, S_lpre (0) must be determined for the noise and for each interfering signal. The general expressions for the noise, S_npre (0), and for a large-bandwidth-compared-to-1/T_I interfering signal, S_lpre (0), are given below. Note that in the S_lpre (0) expression, the Spectral Separation Coefficient (SSC), SSC_l(Δf), between the l^th incoming signal after RFFE filtering and the PRN code signal, is introduced:

13

14

15

16

17

where:

• P_wf,l is the l^th interfering signal waveform power at the RFFE input,

• are the receiver quantization losses for the l^th interfering signal,

• G_l is the receiver antenna gain for the l^th interfering signal,

• 𝛽_l(Δf) is the RFFE filter frequency dependent rejection (FDR) to the received RFI signal,

• is the baseband normalized l^th interfering signal PSD,

• is the baseband filtered normalized l^th interfering signal PSD,

• Δf is approximated as the frequency difference between the l^lh interfering signal central frequency and the receiver central frequency, in this case, the L5 frequency.

Moreover, P_wf,l can be decomposed into two terms to facilitate the blanker mechanism integration in the next section:

18

where:

• P_peak,l is the peak power of the l^th interfering source at the antenna input,

• dc_l is the duty cycle of l^th interference source; dc_l ∈ [0, 1] represents the duration in percentage of time of an equivalent baseband square pulse interference with peak power P_peak,l to the analyzed interference; for example, if dc_l = 10⁻³ with respect to 1s, the equivalent squared pulse interference duration is 1 ms.

Therefore, I_0,l can be expressed as shown below, where is the implementation losses of the l^th interfering signal:

19

3.4 Signal power and signal power spectrum density considerations

Equations (20) and (21) depend on the RFI signal power and on the RFI signal PSD before blanking. Therefore, a key point for the correct application of these two equations is to correctly define the RFI signal mathematical model being analyzed (as input). Some considerations can be made depending on the nature of this signal.

First, in order to be modeled as an increase of the effective N₀ at the correlator output, the RFI signal characteristics (or some of them) must be random; in other words, the receiver must perceive the RFI signal as a random signal. If this is not the case, the effect at the correlator output is just a constant contribution for a long period of time: the constant value is derived from a probability density function, but it does not evolve in time, contrary to the noise that does (magnitude evolution determined by N₀).

Second, in the strict sense, the RFI signal to be considered for which its power and PSD must be mathematically modeled is the complete RFI signal. This means that in the case of pulsed RFI signals, the PSD mathematical model must include the contribution of all the pulses. Nevertheless, if the RFI signal is observed by the receiver as a random signal, the PSD representation could potentially be simplified. For example, in the pulsed RFI signal case, the mathematical model could be reduced to the PSD modeling of an individual pulse. Therefore, in this potential case, the interfering analysis of one pulsed RFI interfering source with 𝜆 pulses per seconds with a complex PSD mathematical model could be transformed into the analysis of 𝜆T_I pulsed RFI interfering sources with one pulse per T_I seconds with a simple PSD mathematical model.

These two considerations will be used to validate the given proposed mathematical Equations (20) and (21) for DME/TACAN signals in Section 5.

3.5 Blanker analysis constraint

As stated in the previous section, the RFI signal impact analysis conducted from the post-blanker signal PSD can be simplified to the RFI signal impact analysis conducted from the PSD of only one signal pulse. However, although this simplification is possible, to obtain a very accurate PSD model of even one pulse remains a challenge in scenarios with a high number of RFI sources: the bdc value created by the RFI sources is very significant, and the signal pulses collisions are not a rare event. In fact, since the blanking mechanism is not an LTI system, its impact cannot be directly modeled by its impulse response, h(t). Therefore, the post-blanker signal PSD cannot be directly modeled from the blanker impulse response if an equivalent h(t) is not previously derived, where the equivalent h(t) modeling increases its complexity along the increase of the number of interference sources. This modeling problem is not tackled in this paper.

4.1 RTCA DO-292 customization of general C/N₀ degradation analytical model

The effective C/N₀ ratio expression described in RTCA DO-292 (RTCA, 2004) equation 2–1 can be found by customizing the general equivalent C/N₀ analytical formulas with the following assumptions of and bdcⁱ:

1. The blanking mechanism has an effect of uniformly spreading the interfering signal PSD along the RFFE plus antenna equivalent filter bandwidth, BW. This means that can be approximated as the normalized square of length BW.

2. The useful signal, as well as all the interfering signal sources, is affected by the same applied blanking duty cycle: bdc ≈ bdcⁱ, i ∈ I.

From these two assumptions, RTCA DO-292 (RTCA, 2004) customizes the pulsed interfering signals impact on the C/N₀ degradation, R_I,i,as

23

Note that following RTCA DO-292 (RTCA, 2004), refers to the peak power at the antenna output: this term contains (although are the ADC/AGC quantization losses). An additional set of assumptions in RTCA DO-292 (RTCA, 2004) is about the bdc value observed/applied to the useful signal and interfering signals. The RTCA DO-292 formula of bdc is given below:

24

where bdc_i is the blanking duty cycle triggered by the i^th interfering source and bdc_i is calculated without considering other RFI sources. Therefore, bdc is computed by assuming:

1. No below-to-below threshold pulses collisions or, more generally, no true pulse collision,

2. Uniform distribution for time pulse arrivals,

3. No consideration on pulse duration for above-to-above threshold and for above-to-below threshold pulse collisions (Hegarty et al., 2000,).

Nevertheless, note that for DME/TACAN signals a bdc expression that englobes all DME/TACAN pulse collisions except for below-to-below threshold ones is proposed in RTCA DO-292 (Kim & Grabowski, 2003; RTCA, 2004).

4.2 Limitation of RTCA DO-292 assumptions

The limitation of the three previously defined assumptions applied in the RTCA DO-292 (RTCA, 2004) is described below:

1. The application of the temporal blanking over a signal has as consequence the spreading of the signal PSD. This is because the abrupt nulling of some signal time-domain samples introduces fast variations of the signal amplitude, and thus, high-frequency components appear on the post-blanker signal PSD. Nevertheless, the assumption of a uniformly spread PSD of the i^th post-blanker interfering signal appears to be too conservative. In fact, as it will be shown in Section 6.3, the spreading of the signal is far from transforming the signal into a uniformly spread PSD signal. A priori, this assumption seems to be a worst-case scenario since all interference signals, irrespective of Δf, are processed equally except for the 𝛽_i term, multiplication between and .

2. This assumption is fulfilled for all the continuous interfering signals, which are assumed to not trigger the blanker (always below the threshold). However, this assumption may not be fulfilled for the pulsed interfering signals. For pulsed interference, it is assumed that the blanking activation (above-threshold RFI pulses arrivals) can happen at any moment in time and that the RFI pulses below-threshold can also be received at any moment in time; this implies a uniform distribution of the pulse arrivals in time. Therefore, if due to the interfering scenario, either the blanker activation or the below-threshold RFI pulses arrivals do not follow a uniform distribution, this assumption will lead to an inaccurate R_I prediction: it might be that the bdc term applied to an individual interfering signal is not the same for all pulsed interfering sources. Moreover, the pulse duration is not considered, or in other words, the interval of time during which the blanker is activated after its initial activation. This implies that each instant of time is analyzed independently (no time correlation considerations). Therefore, the final effect of this assumption on R_I,i is difficult to predict since it will depend on the analyzed interfering scenario and interfering sources. Garcia-Pena et al. (2020) illustrates this effect in the case of JTIDS/MIDS signals. In the case of JTIDS/MIDS signals, although there is a random component of the signal structure, the starting point of a time slot cannot be considered as a uniform variable. Therefore, the impact of the temporal blanker activation cannot be seen as uniformly distributed over the signal. In fact, in Garcia-Pena et al. (2020), it was shown that while the general bdc in interfering scenario Case 8 was equal to 0.0929, the exact bdcⁱ of interference sources RG1 and RG2 were 0.18 and 0.14, respectively.

3. The inspection of the last set of assumptions (a), b), c)) shows that an increase of the predicted bdc value accuracy could be possible. Assumption a) implies that one type of collisions, below-threshold pulses with below-threshold pulses collisions, is not considered during the calculation of bdc since each bdc_i term is calculated only considering the i^th interfering signal source: bdc_i is calculated assuming that no other signal is present at the temporal blanker input; therefore, the potential event of constructive combination of below-threshold pulses to generate a resulting above-threshold pulse is not considered since the actual resulting signal obtained from the addition of all received signals is never truly compared to the temporal blanker threshold. In fact, note that no actual resulting signal is ever compared to the threshold, and constructive/destructive combination of the RFI signals could modify whether the blanker is activated as well as its activation time. bdc calculation could be refined from this perspective.

Assumptions b) and c) also imply that the final bdc calculated value could be more accurate when considering the true (or worst) scenario statistics. In fact, this set of assumptions implies a uniform time distribution of the pulses arrival and considers each instant of time as independent from the other instants of time (no consideration about the duration of the blanker activation). On one hand, the uniform time distribution of the pulse arrivals implies that at any moment an RFI could be received or not, which is not true for RFI signals with partially random signal structures such as JTIDS/MIDS. Therefore, the influence of the RFI signal on the temporal blanker activation is concentrated in specific moments in time, which means that pulse blanker collision activations cannot be statistically computed as a random variable (as it is done in Equation (24)). On the other hand, to consider each instant of time independent from the other instant of time overlooks the fact that once a RFI pulse arrives at the receiver, this pulse activates the blanker for a given period of time (pulse duration over the threshold). Therefore, for this duration, the blanker is activated regardless of the other sources (neglecting for simplification purposes destructive pulse collision). In fact, note that this limitation was already identified in RTCA DO-292 (RTCA, 2004) for DME/TACAN signals, and instead of modeling them separately in Equation (24), a joint bdc_DME/TACAN value is proposed, which takes into account true blanker activation duration and true pulse arrival distribution (Kim & Grabowski, 2003; RTCA, 2004).

4.3 RTCA DO-292 analytical model advantage

The R_I,i formula presented in this paper determines more accurately the impact of RFI signals below-threshold than the formula proposed in RTCA DO-292. However, this new formula requires the knowledge of the PSD of the nonlinearly distorted RFI signals at the temporal blanker output. This means that time-domain simulations must be conducted to determine the desired PSD, which should be geographically dependent (DME/TACAN induced bdc depends on the DME/TACAN beacons in view of the airplane). Therefore, to reduce the final complexity of the proposed formula application and to guarantee an upper bound of C/N₀ degradation, a possible solution is to use only one PSD representing all the received DME/TACAN signals and representing the worst-case scenario (wide PSD). Note that this is the methodology chosen in Section 6.3 and in Figure 8. In opposition to this situation, the RTCA DO-292 formula does not require conducting any time-domain simulations since the interfering signals PSD at the temporal blanker output was already approximated as totally spread on the RFFE filter bandwidth. Therefore, although the RTCA DO-292 formula is less accurate, it does not require time-domain simulations, and thus, it has a lower complexity to be applied. Note that this feature was probably highly valued during RTCA DO-292 development back in the early 2000s.

5.1 Systems and signals description

DME and its military equivalent, TACAN, are two systems used by aircrafts to know their distance to a ground station, which position is known. The systems operate as follows (Borden et al., 1951): the aircraft DME equipment (called interrogator) sends pulses to ground stations. Once the interrogation is detected, the station transponder replies to the interrogator. The distance is then determined by measuring the time elapsed between each pulse transmitted by the interrogator and the reception of its corresponding reply pulse from the transponder. This time corresponds to twice the distance between the aircraft and the station, plus fixed processing time inside the ground station.

According to RTCA (2004), only the signals emitted in the band of interest of the study disturb GNSS receivers’ operations. Indeed, the band of interest is the E5a/L5 one and equals [1164 MHz; 1191 MHz], and the aircraft’s DME interrogators emitting their signals between 1025 and 1150 MHz are ignored herein. The study focuses on DME ground stations/repeaters, as they emit their signals between 962 and 1213 MHz, which includes the above defined band of interest.

The emitted signal of an individual DME station is composed of a pair of Gaussian pulses modulated by a cosine, which can be modeled as

25

where:

• P_peak is the interference beacon transmitting peak power (dBW),

• f_DME is the carrier frequency of the DME/TACAN signal (Hz),

• 𝛼 = 4.5 ⋅ 10⁻¹¹ s⁻²,

• Δt = 12𝜇s is the inter pulse time separation,

• t_k is the emission time of the k^th pulse pair,

• 𝜃_DME is the DME/TACAN signal initial carrier phase shift.

Figure 5 (left part) represents a normalized DME/TACAN pulse pair, modulated at 14 MHz. Figure 5 (right part) represents the normalized pulse complex envelope of a DME/TACAN signal.

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FIGURE 5

DME signal pattern

5.2 US hot spot scenario

RTCA DO-292 (RTCA, 2004) defines three US hot spots from the theoretical C/N₀ degradations (equivalently, N_0,eff degradation) due to RFI DME/TACAN signals. In this paper, out of the three identified hot spots, the worst one is selected: near Harrisburg, PA at 40^◦ North latitude and 76^◦ West longitude. Using RTCA DO-292 (RTCA, 2004) table E-8 information, DME/TACAN beacons significantly degrading the N_0,eff are presented in Figure 6.

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FIGURE 6

DME/TACAN beacons around Harrisburg, PA (RTCA 2004)

5.3 Power spectrum density for pair of pulses

Equation (26) shows the normalized PSD of a pair of pulses of an equivalent baseband DME/TACAN signal, S_pp,DME (f). Figure 7 shows the graphical representation of this PSD.

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FIGURE 7

Normalized PSD of a pair of pulses of an equivalent baseband DME/TACAN signal

26

6.1 Signal power and signal power spectrum density considerations for DME/TACAN signal

In the case of DME/TACAN signals, where each DME repeater transmits 𝜆 (𝜆_DME or 𝜆_TACAN) pair of pulses per second, its contribution to the effective N₀ (in terms of R_I) can be analyzed as the contribution of 𝜆T_I independent DME repeaters with one pair of pulses per T_I seconds. The justification for this assumption is that the time of arrival between two pairs of pulses follows a Poisson distribution and that the arrival time of a pair of pulses inside a given interval of time, , for example, follows a uniform distribution. Therefore, each pair of pulses inside can be seen as an independent RFI source.

The signal power of a pulsed RFI source can be calculated by just dividing the signal energy by the time: P = E/T. Therefore, knowing that one pair of pulses per T_I seconds is transmitted by an interference source, the energy can be calculated as the area below the instantaneous energy |s(t)|² (Kim & Grabowski, 2003). There are two pulses, and the energy of the equivalent baseband signal must be divided by two in order to take into account the signal carrier frequency:

27

where represents the equivalent DME/TACAN pulse width. P_wf,l is given in Equation (28) considering the 𝜆_lT_I DME repeaters, and from expression (28), it can be observed that d c_l = 𝜆_l T_eq:

28

The baseband before-blanker PSD signal of a DME/TACAN pair of pulses is provided in Equation (26); note that this expression must be normalized to be used as in Equation (21). The computation of terms and (1 − bdc^l) is not straightforward in a scenario with more than one original DME/TCAN repeater (before decomposition in independent DME/TACAN sources with one pair of pulses per T_I seconds) due to pulse collisions.

6.2 C/N₀ degradation results for simplified DME signal

As commented in the previous section, the computation of and (1 − bdc^l) is not straightforward due to the collision between pulses. Therefore, in order to validate Equations (20) and (21), the original DME/TACAN signal and scenario are simplified in order to allow the exact computation of these two terms. The implemented simplifications are:

• Scenario: Only one DME (or TACAN) repeater is considered (L = 1).

• Signal: Only one pulse instead of a pair of pulses is transmitted. A fixed number of pulses per T_I seconds are considered; this number is called RF_TI, and 𝜆 is calculated as 𝜆 = RF_TI/T_I. No collisions are allowed; this means bd c^l = 0. All the pulses have the same peak power per inspected case. Note that this problem is equivalent to having 𝜆 T_I = RF_TI RFI sources transmitting one pulse per T_I seconds.

Due to these simplifications, Table 1 expressions for S_l,BB(f), S_l,PBf(f) (for simplifications, theRFFE filter is assumed to not modify the PSD), dc_l, and ldc_l are used depending on whether the signal is partially blanked or not (P_peak above or below threshold, T_h). Note that for this case, the blanker can just be modeled as a nulling of the signal when the signal is above the threshold (represented by a square pulse of width 2w_l); since there are no collisions, no other parts of the signal will be additionally set to zero.

Where w_l represents half of the width of the pulse above the threshold,

29

Finally, since no collisions are allowed, the total bdc can be computed as

30

Tables 2, 3, and 4 present the C/N₀ degradation comparison between theoretical values computed using Equations (7), (8, 10, 20) and (21) and a simulated simplified DME signal. The comparison is made for different peak power, P_peak, threshold, T_h and RF_TI values and for two DME carrier frequencies. Moreover, all the simulations are generated with the same random seed, which means that all the random parameters (initial phase, delays, noise relative amplitude, etc) for the DME signals are the same for all the simulations. Additionally, note that for simplification purposes, the implementation losses and antenna gain are not considered, G_l L_impl = 1. The C/N₀ degradation simulation is conducted as follows. First, a GNSS L5 signal is generated (in this case a GPS L5 pilot signal without secondary code). Second, AWGN is added and its power adjusted to obtain a C/N₀ between the useful GNSS signal and the noise PSD equal to 60 dB/Hz. Third, the simplified DME signal is added with the desired peak power to the GNSS plus AWGN signal. Fourth, the resulting signal is filtered by the RFFE plus antenna block filter presented in Figure 2. Fifth, the signal is tracked, carrier phase and code delay, by a simplified GNSS receiver; the receiver assumes to have already achieved PRN code synchronization and (although not implemented) secondary code synchronization to allow for 20 ms of coherent integration. Sixth, the C/N₀ is estimated by applying a C/N₀ estimator (Parkinson & Spilker, 1996). Finally, this value is subtracted to the C/N₀ value estimated when the simplified DME signal is not included in order to obtain the targeted C/N₀ degradation.

From Tables 2, 3, and 4, it can be observed that the theoretical and the simulated C/N₀ degradation match very well; only a maximum difference of about 0.1∼0.15 dB is observed. Moreover, this small difference remains quite constant regardless of the absolute C/N₀ degradation; this means that either a finer tuning of the simplified GNSS receiver/C/N₀ estimator is required, or that the single simulated signal generates a punctual C/N₀ degradation having this difference with respect to the average theoretical C/N₀ degradation (deviation between the mean and one simulation running, or RFFE no distorting the signal simplification); remember that all signals/scenarios are simulated with just one seed.