Kalman filtering with noncoherent integrations for Galileo E6-B tracking

2 SIGNAL MODEL

A GNSS receiver collects signals from several satellites and processes them in an independent way leveraging the properties of Direct Sequence Spread Spectrum (DSSS) modulations. Independent processing of different signals is based on the quasi-orthogonality of the codes used to modulate the different signal components. For this reason, signals from a single satellite are considered in the following. The continuous signal received from a single satellite can be modelled as the sum of L useful components and can be expressed as (Kaplan & Hegarty, 2005):

1

where L is the total number of signals transmitted by the satellite. The terms, e_j(t), represent the different GNSS baseband signal components modulated by a spreading code and carrying the navigation message. For each signal component identified by the subscript j, C_j is the received power and f_RF,j is the central frequency, whereas τ_j, ϕ_j, and f_j are the code delay, the carrier phase, and the Doppler shift introduced by the propagation channel. Finally, n(t) is a zero mean Additive White Gaussian Noise (AWGN) introduced by the propagation channel and with spectral density N₀. The ratio between the signal power, C_j, and N₀, defines the Carrier-to-Noise Power Spectral Density Ratio (C/N₀), one of the main signal quality indicators used in GNSS signal processing. When only the Galileo E6-B/C signals are considered, signal (1) can be expressed as (European Union, 2016):

2

This signal is given by the combination of a pilot and data component, which are characterized by the same power so that

3

Both data and pilot components are Binary Phase Shift Keying (BPSK) modulated. The data component can be expressed as:

4

where d_E6-B(t) is the navigation message with a data rate of 1 ksymbol/s and c_E6-B(t) is the spreading code with a 5.115 Mchip/s rate. The pilot component can be expressed as:

5

where c_E6-B(t) is the primary spreading code with a 5.115 Mchip/s rate anda1msduration while s_E6-C(t) is the secondary spreading code leading to a tiered spreading code of length equal to 100 ms. The E6-B/C spreading codes can be found in European Union (2019).

Continuous signal (2) is amplified, filtered, down-converted to the baseband, and sampled by the receiver front end, which provides the following digital complex sequence:

6

where is the sampling interval used by the front end to digitize the signal and n is the discrete time index. The symbol ̃⋅ is used to denote the impact of the receiver front end on the useful signal components. When only the data component is considered, Equation (6) finally becomes:

7

After digitalization, the signal is processed by the acquisition stage that has the goal to provide coarse estimates of the signal code delay and Doppler shift, τ₀ and f₀. These estimates are then refined by the tracking stage that also determines the signal carrier phase. In addition to this, the receiver tracking stage has the purpose to provide a dynamic estimate of the above parameters, following their temporal variations. In this paper, a standard acquisition approach (Kaplan & Hegarty, 2005) is adopted. For this reason, acquisition is not further detailed. The selected tracking stage is detailed in the next section.

3 TRACKING STAGE

In this paper, a software Matlab receiver has been implemented. The receiver implements different processing strategies including KF tracking with noncoherent integrations. A general representation of the tracking loops implemented for the processing of the Galileo E6-B signal is shown in Figure 1.

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FIGURE 1

Schematic representation of the tracking schemes implemented for the processing of the Galileo E6-B signal. A single Kalman filter replaces the standard code and tracking loop filters, which are represented with dashed lines

Independently from the strategy adopted, input signal (7) is correlated with local carrier and code replicas. This is a standard process that leads to the Prompt, Early, and Late correlator outputs (Kaplan & Hegarty, 2005):

8

where , f_d, and are the current estimates of the code delay, Doppler frequency, and carrier phase. d_s is the Early-minus-Late chip spacing. In Equation (8), N samples are combined together to form a correlator output. The integration over N samples effectively down-samples the original signal, y[n]. In particular, the index k in Equation (8) is used to denote the different correlators produced over time. The sampling interval of the correlators is equal to T_c = NT_s and defines the coherent integration time. It is possible to show (Del Peral-Rosado et al., 2010; Van Dierendonck, 1996) that the correlator outputs can be effectively modelled as

9

where R(.) is the BPSK correlation function and

10

are the residual delay and phase errors. In Equation (9), the impact of residual frequency errors is neglected in order to simplify the notation. A sinc term depending on the residual frequency error should be introduced in order to account for the effect of the residual Doppler frequency (Del Peral-Rosado et al., 2010; Van Dierendonck, 1996).

In Equation (9), the three correlator outputs are modulated by the E6-B navigation message defined in Equation (4) and assumed constant over the coherent integration time:

11

Moreover, three noise terms are present. These terms are obtained from the processing of the input noise, η(nT_s), in Equation (6). They will be better analyzed in Section 5.

The Galileo E6-B navigation message limits the coherent integration time, and a nonlinear operation is required to remove its impact. This operation is performed through squaring as shown in Figure 1. After squaring, the integration process can continue and extended correlators are obtained (Borio & Lachapelle, 2009):

12

In order to simplify the notation, the temporal index for the extended correlators has been omitted in Equation (12). At each loop update, a new set of correlators is, however, produced. The extended correlators are obtained by combining signal samples over a total duration equal to

13

which also defines the update rate of the loop.

The extended correlators are passed to the code and carrier discriminators. In this paper, we used the following discriminators:

14

and

15

Carrier discriminator (14) is an extension of the standard Costa loop discriminator (Borio & Lachapelle, 2009; Kaplan & Hegarty, 2005; Van Dierendonck, 1996), and the factor 1/2 considers the fact that phase errors are doubled through squaring. Equation (18) is a generalization of the normalized early-minus-late envelope discriminator. The square roots present in Equation (18) are required to compensate for the effect of squaring in Equation (12).

In standard tracking loops (Kaplan & Hegarty, 2005; Van Dierendonck, 1996), the discriminator outputs are fed to two independent loop filters. The output of such filters is then used to control the Numerically Controlled Oscillators (NCOs) that generate the new local code and carrier replicas and close the processing loop. This process is shown in Figure 1 with dashed lines and dashed boxes. This convention has been used to indicate that, in this work, standard tracking loop filters have been replaced by a single KF, which is better detailed in the next section.

The full software receiver used for the analysis implements several processing options and transitions between different processing stages depending on the output of a Frequency Lock Indicator (FLI) and a Phase Lock Indicator (PLI) (Kaplan & Hegarty, 2005; Van Dierendonck, 1996). These processing stages are shown in Figure 2a): after acquisition the receiver uses a standard DLL and a Frequency Lock Loop (FLL). When the FLI indicates that frequency lock has been achieved, the FLL is replaced by a standard PLL. Finally, when the PLI indicates that also phase lock has been achieved, KF tracking is started. Only at this stage, standard loop filters are replaced by the KF. Figure 2b) shows the normalized code and carrier Doppler produced by the standard loop filters and the KF: the transition between the two stages can be clearly distinguished. The KF implicitly performs carrier smoothing with a single Doppler state variable. This leads to very smooth code Doppler estimates. More details are provided in the next section.

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FIGURE 2

Different tracking stages and example of operations implemented in the KF Matlab software receiver. a) Different tracking stages including the decision logic to transit between the different stages. b) Normalized code and carrier Doppler highlighting the transition to KF tracking

4 KALMAN FILTER TRACKING

Being an adaptive filter, the KF is particularly indicated under harsh and variable working conditions. In addition to this, the KF is an optimum filter in the Minimum Mean Square Error (MMSE) under the assumption of additive Gaussian noise (Kalman, 1963). This assumption is approximately true for the architecture considered here and under high/medium C/N₀ conditions. In particular, the carrier and code discriminator outputs can be approximated as Gaussian random variables for high/medium C/N₀ conditions. When the C/N₀ drops, the Gaussian approximation is no longer valid and the KF does not operate under optimal conditions.

As a consequence, the filter can be subjected to poor performance or divergence. To overcome this issue, a possible solution is to use an Extended Kalman Filter (EKF) (e.g., Del Peral-Rosado et al., 2010; Vila-Valls et al., 2017; Vila-Valls, Brossier, & Ros, 2010; Vila-Valls, Ros, & Brossier, 2012). The latter has the advantage to directly cope with nonlinear Gaussian observation models by implementing a linearization process. Specifically, this approach allows dealing with measurements that are not linearly dependent on the state vector and with nonlinear dynamics (Vila-Valls et al., 2017). Therefore, the complex signal samples can be directly input to the EKF without the necessity of a discriminator (Vila-Valls et al., 2017). In this way, it is possible to avoid the saturation and noise amplification effects, which can be produced by classical discriminators operating in the presence of very weak signals (Del Peral-Rosado et al., 2010), as shown in Figure 3a.

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FIGURE 3

Empirical pdfs of the carrier discriminator outputs at low C/N₀ conditions. a) Costas loop discriminator without noncoherent integrations and pure pilot PLL. b) Extended Costas loop with 20 noncoherent integrations

The latter figure compares the empirical probability density functions (pdfs) estimated for the output of a Costas arctangent discriminator without noncoherent integrations and for a pure four-quadrant arctangent discriminator used for pilot processing. The pdfs have been estimated using real data collected under low C/N₀ conditions. In the Costas case, the data carrier discriminator output does not follow a Gaussian distribution, and the optimality conditions for KF processing are violated. On the contrary, given the larger phase discrimination range of a pure four-quadrant arctangent, the pilot carrier discriminator output is able to maintain a pdf that is approximately Gaussian at lower C/N₀ conditions.

In order to be able to use the classical KF, the validity range of the Gaussian approximation can be extended by increasing the equivalent SNR at the discriminator input. This operation is performed using noncoherent integrations that are effective in restoring the optimality working conditions of the KF. This fact is highlighted in Figure 3b) that shows the empirical pdf obtained when 20 noncoherent integrations are used along with the discriminator in Equation (14). In this case, the data carrier discriminator output approximately follows a Gaussian distribution. This result is the main justification for combining together noncoherent integrations and KF.

The KF is based on the design of a dynamic system model, which is used to predict and estimate the time evolution of the parameters of interest. This prediction is then corrected by using actual measurements.

In this research work, the KF states are a four-dimensional vector composed of the code delay error, the carrier phase error, the carrier Doppler frequency, and the Doppler rate errors as in Susi et al. (2017). This model has been widely used in the literature. The state vector can be expressed as:

16

where Δτ and Δϕ are the code delay and carrier phase errors defined in Equation (10). Δf is the residual Doppler frequency shift, and Δa is the Doppler frequency rate correction.

The system dynamic model, which defines the state evolution by providing the state vector at the instant (k + 1), is defined as:

17

where A_k is the following transition matrix

18

with β indicating the factor converting cycles in unit of code chips and T_u indicating the loop update rate.

The term w_k in Equation (18) indicates an additive zero mean and uncorrelated Gaussian noise process, which can be expressed as:

19

where Q_k is the discrete transition noise covariance matrix modelling the noise of the state elements. Assuming that the noise affecting the elements of the state vector is due to independent sources, Q_k can be defined as a diagonal matrix where the diagonal elements are the spectral densities of the process noises. In this case, Q_k has been defined as in Macchi et al. (2010).

As previously mentioned, the KF corrects the prediction performed using the system model by exploiting the noise of the actual measurements, which are, in this case, the output of the code and carrier discriminators.

The measurement model can be expressed as

20

where is the vector including the code and carrier discriminator outputs, which represent the estimates of the code and phase errors over the time of integration. H_k is the observation matrix mapping the filter states into the measurements, and it is defined as follows

21

v_k ∼ N(0,R_k) is the zero mean and uncorrelated Gaussian noise affecting the measurements.

Specifically, considering the code and carrier measurements as independent variables, R_k can be represented as a diagonal matrix having as diagonal elements the carrier and code noise variance:

22

The variances of the code and carrier discriminators, which are used to fill the measurement noise matrix, are derived in Section 5 and in Appendix A where it is also shown their dependence on the signal C/N₀. The latter is computed from the correlator output by implementing the Narrow-Wide Band Power Ratio (NWPR) estimator (Kaplan & Hegarty, 2005) at the loop update rate, T_u. As detailed in Appendix A, the use of an extended integration time allows the reduction of the carrier and code noise by averaging the term on a longer temporal interval.

The code and carrier discriminator outputs are related to the state vector by a linear relationship justifying the use of a linear KF. Therefore, after implementing the linear dynamic and measurement models, the KF recursive and predictions steps can be applied as described in Kalman (1963). As shown in Equation (18) and in Equation (21), the extended time of integration is included in both transition and observation matrices, and it implicates that the KF measurements will be updated at a lower rate. Specifically, extending the integration interval K times the measurements input rate will be K times lower (Del Peral-Rosado et al., 2010).

5 DISCRIMINATOR OUTPUT VARIANCES

In order to derive the variances of the discriminator outputs, it is necessary to propagate the signal model in Equation (6) through the correlation process and the nonlinear discriminators shown in Figure 1. In this respect, it is possible to show (Van Dierendonck, 1996) that the noise terms in Equation (9) are complex circularly symmetric Gaussian random variables. Their real and imaginary parts have variance:

23

The three random variables in Equation (9) are correlated, and their covariance is determined by the Early-minus-Late correlator spacing, d_s. In particular,

24

These properties are used in this section to evaluate the discriminator output variances, and . In particular, using the results derived in Borio (2011), it is possible to show that the variance of Equation (14) can be effectively approximated as

25

where SNR_P is the SNR after noncoherent integrations:

26

After squaring and assuming perfect phase and code recovery (Δφ = 0,Δτ = 0), each prompt correlator assumes the following form:

27

and

28

Thus,

29

and

30

The results in Equation (30) have been obtained by exploiting the properties of Gaussian random variables. In particular, the covariance term in the second line of Equation (30) is zero since it leads to odd terms in zero mean Gaussian random variables. The variance of is obtained by exploiting the properties of moments of order four of zero mean Gaussian variables.

The SNR after noncoherent integrations can be finally computed from Equations (29) and (30):

31

From this result, it is finally possible to determine the variance of :

32

The derivation of is more complex and requires approximations based on the Delta method (Ver Hoef, 2012). For this reason, the derivations are reported in Appendix A, where it is shown that:

33

The validity of Equations (32) and (33) has been empirically investigated using the data collected during the hardware simulator experiment described in Section 6, where E6-B signals were collected for different C/N₀ values. Several instances of the code and carrier discriminator outputs were evaluated and used to empirically estimate the variances, and . Finally, the results were plotted as a function of the input C/N₀ and compared with the theoretical formulas derived in this section.

The results are shown in Figure 4 for three different values of K, the number of noncoherent integrations, and for d_s = 1. A good agreement between theoretical and experimental results can be observed, most of all for high/medium C/N₀ values. For low C/N₀ values, nonlinear effects start playing a significant role, and the approximations made to derive Equations (32) and (33) are no longer valid. Differences can be clearly observed for the code discriminator output variances analyzed in Figure 4b): for C/N₀ values below 35 dB-Hz, the code discriminator output variance starts saturating and converges to a constant value. While this effect is difficult to quantify from a theoretical point of view, it has been accounted for in the KF implementation. In particular, has been fixed to a constant value for C/N₀ levels below 35 dB-Hz. These values have been empirically determined for different values of K and reflect the results shown in Figure 4b).

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FIGURE 4

Carrier and code discriminator output variances as a function of the input C/N₀. The empirical variances were estimated using the data collected during the hardware simulator test described in Section 6

6 TRACKING STRATEGIES AND EXPERIMENTAL SETUP

In addition to KF tracking, different data and pilot processing strategies have been implemented into the SDR receiver and used for comparison purposes. They are:

• Standard tracking on the data channel with integration time limited to 1 ms, fixed bandwidth of 10 Hz, and a Costas loop as carrier discriminator (Kaplan & Hegarty, 2005).

• Data KF tracking on the data channel with integration time limited to 1 ms and a Costas loop as carrier discriminator.

• Pure pilot KF tracking with long coherent integration and a four-quadrant arctangent as carrier discriminator. This discriminator is characterized by a wider linearity region in comparison to other discriminators, such as the Costas, and consequently, it allows enhanced sensitivity performance (Curran, 2015). Secondary code synchronization is at first performed and the secondary code symbols are removed from the correlator outputs. This process allows the extension of the coherent integration time.

• Data KF with noncoherent integrations: the data bit effect is removed through squaring before implementing extended integrations. Moreover, a modified four-quadrant arctangent is adopted as carrier discriminator (Borio & Lachapelle, 2009).

While a detailed description of pure pilot KF tracking is out of the scope of this paper, the architecture adopted for this type of processing is shown in Figure 5. The figure has been organized in order to highlight the commonalities and differences with respect to noncoherent data KF processing analyzed in Figure 1. In this case, the pilot component in Equation (6) is processed, and the local codes match those of the E6-C signal. As for the data component, the primary code is at first removed, and coherent integration is performed on the primary code duration. In this way, correlators similar to those in Equation (9) are obtained. For the pilot signal, the data navigation message is replaced by the E6-C secondary code, s_E6-C [k] = s_E6-C(nT_s), that is known at the receiver side and constant over the primary code coherent integration process.

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FIGURE 5

Schematic representation of the tracking schemes implemented for the processing of the Galileo E6-C pilot signal. A single Kalman filter replaces the standard code and tracking loop filters, which are represented with dashed lines

In Figure 5, the block labeled as “Sec. code synchro” represents the secondary code synchronizer that is used to detect the start of the secondary code and to generate a local version of this component. In this paper, we used a simple secondary code synchronizer, which maximizes the correlation of the incoming prompt correlators with the known secondary code sequence. After secondary code wipe-off, additional coherent integrations can be performed extending the integration time on several pilot primary code periods. In this case, no nonlinear operations are performed, and no noise amplification occurs due to squaring that is replaced by secondary code removal. The correlators are then passed to the code and carrier discriminators. We used a standard normalized early-minus-late envelope discriminator for the code component and a pure four-quadrant arctangent discriminator (Kaplan & Hegarty, 2005) for the carrier component:

34

where P_p indicates the prompt correlator obtained after extended coherent integrations on the secondary code. The output of the pure four-quadrant arctangent discriminator was analyzed in Figure 3a): this type of discriminator has a larger phase discrimination range and provides improved performance with respect to data processing for a fixed integration time.

Code and carrier discriminator outputs are then passed to the KF that has the same structure used for data processing. As for data processing, the software receiver implements several states that include standard processing, which is used for initialization before KF tracking.

Several tests have been carried out using both simulated and real data, which were then used to assess the tracking schemes described above. Specifically, a first test was performed using the Spirent GSS9000 simulator shown in Figure 6. The simulator was set to generate E6-B/C signals with C/N0 levels progressively decreasing at steps of 1 dB-Hz per minute. This test was performed in order to assess the sensitivity of the different tracking schemes.

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FIGURE 6

Details of the simulated and live static tests:(upper left side) USRP X310 and the external OCXO used to collect the baseband data;(right side) Spirent GSS9000 simulator used to simulate E6 signals with decreasing C/N0 levels;(lower left side) High geodetic antenna placed in proximity of buildings producing signal multipath

The simulator was connected to a Universal Software Radio Peripheral (USRP) X310 front end by Ettus, which was driven by a 10 MHz external low phase noise Oven Controlled Crystal Oscillator (OCXO), as shown in Figure 6 (upper left side). The external oscillator was used since the clock embedded in the front end is not stable and can introduce significant drifts.

The data collected with the USPR X310 was then processed using the different tracking strategies implemented in the SDR receiver. Further tests were performed under real scenarios in both static and dynamic conditions. For the static test, the above front end, again driven by the external OCXO, was connected to a high-grade geodetic antenna placed in the proximity of buildings obstructing the signal reception, as shown in the lower left side of Figure 6. The aim of the test was to assess the performance of the tracking algorithms under real multipath conditions.

Finally, a dynamic test was carried out inside the Joint Research Centre (JRC) campus in Ispra, Italy. This location was selected because of the presence of buildings and foliage, which make the signal reception very difficult. For the dynamic test, a USRP2 front end was used. The front end, in this case driven by an external rubidium clock, was placed inside a van and connected to an antenna mounted on the roof of the van.

Details related to the dynamic experimental setup are shown in Figure 7. The upper right side of Figure 7 provides an external view of the van used for the test. The interior of the van with the USRP2, the rubidium oscillator, and the COTS receiver are shown in the upper left side of Figure 7. Also in this case, a 10 MHz external oscillator was used since the USRP2 houses a poor internal clock, which could introduce apparent dynamics, as discussed in Borio and Susi (2018).

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FIGURE 7

Views of the experimental setup used for the dynamic test: (upper left side) Internal view of the van with the USRP 2, connected to an external rubidium clock, and the COTS receiver; (upper right side) External view of the van with the high geodetic antenna mounted on the roof;(lower left and right sides) Views of the environment selected for the dynamic test. The path is characterized by the presence of tall trees with rich foliage and sporadic buildings

The lower left and right sides of Figure 7 show two sections of the test environment characterized by the presence of tall trees with rich foliage and sporadic buildings. As mentioned above, for all the tests, an E6-capable COTS receiver was connected to the same high-end geodetic antenna used by the SDR front end. Tracking results obtained using the COTS receiver were used as a reference term for SDR processing, as detailed in the next section.

7 EXPERIMENTAL RESULTS

7.1 Static simulated data

The results obtained processing the simulated data are shown in Figure 8 where the C/N₀ time series, as estimated by the different tracking strategies, are compared.

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FIGURE 8

C/N₀ values obtained using the simulated data collected with the USRP X310 front end and processed using different tracking strategies

The experimental analysis clearly demonstrates the benefits of combining KF with noncoherent integrations for the processing of E6-B signals. When considering the simulated scenario, the synergy between KF and noncoherent integration provides a significant performance gain as compared to standard processing.

Standard data-only processing with a fixed loop bandwidth equal to 10 Hz and 1 ms integration time achieves a tracking threshold of around 29 dB-Hz. Using the data-only KF with integration time equal to 1 ms provides only a slight improvement in tracking sensitivity with a tracking threshold around 28 dB-Hz.

However, when the data-only KF is combined with noncoherent integrations (in this case K = 20 and T_u = 20 ms), the loop never loses lock for the whole duration of the simulation, and indeed, the signal is tracked down to 24 dB-Hz. When pilot processing is adopted, the KF is always able to maintain signal lock even when the coherent integration time is equal to 1 ms. Therefore, the combination of KF and noncoherent extended integrations allows achieving results comparable to the ones obtained using the pilot processing.

Figure 9 shows the normalized carrier and code Doppler time series for:

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FIGURE 9

Normalized carrier and code Doppler for data-only standard tracking (upper left side), data-only KF tracking with 1 ms integration time (lower left side), pilot-only KF with 1 ms integration (upper right side), and data-only KF with 20 ms integration time (lower right side)

• standard data-only processing (upper left side),

• data-only processing with KF and 1 ms integration time (lower left side),

• data-only processing with KF and 20 noncoherent extended integration (upper left side)

• and pilot processing with 1 ms integration time (lower right side).

From these figures, we can clearly observe that for standard tracking and KF tracking with 1 ms integration time, the carrier and code Doppler values diverge when loss of lock occurs. On the contrary, in the case of KF data-only processing with extended integration time, the Doppler is correctly estimated, and indeed, it follows a linear trend as for the pilot-only processing case.

7.2 Static live data under multipath conditions

As described in Section 6, a static test with live signals has been carried out by placing a high-grade geodetic antenna in proximity of buildings to test the tracking performance under real multipath conditions.

Figure 10a) reports the C/N₀ values recorded by the COTS receiver for all the four Galileo satellites tracked during the data collection, which lasted around four hours. From Figure 10, it can be observed that the signal from one satellite, PRN 30, was very powerful, and it is characterized by C/N₀ values quite stable, above 50 dB-Hz, for the entire duration of the test.

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FIGURE 10

a) C/N₀ time series extracted from the COTS receiver for the static test under multipath. Frequent loss of lock is observed on signal from PRN 2. b) Page reception map indicating the number of errors per E6-B page as function of time. White areas indicate the corresponding E6-B pages were not provided

On the contrary, the signals from the other satellites show different levels of C/N₀ with fluctuations that can be attributed to multipath. In particular, the signal from PRN 2 is the one with the largest variations in the C/N₀ values, and thus, it was selected to challenge the different algorithms. Figure 10b) shows the page reception maps that provides, as a function of time and for the different signals, the number of errors recorded in the different E6-B pages. Galileo E6-B signals currently broadcast a dummy message that contains a known bit sequence. For this reason, it was possible to count the number errors occurring in the received E6-B pages. The white areas in Figure 10b) indicate that for that portion of the test the receiver was unable to decode the E6-B navigation message. This figure confirms the findings of Figure 10a): the scenario selected was affected by severe multipath with discontinuous tracking and page decoding.

The time span between 40 and 1,800 seconds in Figure 10a) has been selected for the analysis with the SDR receiver.

It is worth underlining that the different tracking strategies have been applied without enabling any reacquisition technique in order to be able to better analyze the different tracking behaviors. Figure 11 shows the C/N₀ time series obtained for the different tracking strategies. For data-only processing, both standard processing and the KF with 1 ms integration lose the lock after about 180 seconds in correspondence of a deep signal fade.

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FIGURE 11

C/N₀ time series estimated using different tracking strategies for the static test under multipath. Signal from PRN 2

On the contrary, data-only KF processing with 20 noncoherent integrations is able to track the signal until the end of the experiment, achieving similar performance of the pilot-only case. As for the simulator test, KF with 20 noncoherent integrations achieves a tracking threshold of about 24 dB-Hz, a value that occurs in correspondence of a deep signal fade. These results agree with the ones obtained using simulated data and confirm the effectiveness of KF combined with extended integration in enhancing the sensitivity of the data-only processing.

As already mentioned, the E6-B signal is currently transmitting a dummy navigation message with a known sequence of bits. Thus, it was possible to evaluate the demodulation capabilities of the different approaches. More specifically, the Bit Error Rate (BER) and the Symbol Error Rate (SER) have been evaluated from the data correlator output. The SER has been estimated before decoding the convolutional code of the E6-B signal while the BER has been estimated after applying a soft Viterbi decoding (Proakis, 2018).

In the COTS receiver case, the same evaluation was performed by exploiting the demodulated navigation data that the receiver provides as output. It is worth underlining that the COTS receiver implements pilot-only processing that is used to aid the demodulation of the data channel. The data channel is then demodulated and decoded using soft Viterbi decoding. The BER evaluated from the COTS receiver are reported in Figure 12 for signal PRN 2 and PRN 27 cases, namely the satellite signals with lower C/N₀ values.

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FIGURE 12

BER obtained from the COTS receiver for PRN 2 and 27

The curves show a good agreement with the BER theoretically expected for a Gaussian channel with soft Viterbi decoding. In both cases, the BER starts significantly increasing for C/N₀ values lower than 28 dB-Hz. Specifically, the BER approaches the random guess probability, 0.5, in correspondence of 28 dB-Hz. This means that the bits are randomly guessed and correctly modulated only half of the time. So, even with pilot-only processing and even if we are able to track weak signals, the demodulation performance is already degraded below 28 dB-Hz.

These results agree with the ones presented in Borio and Susi (2018, 2019). The same analysis has been performed using the SDR receiver and implementing data-only tracking with 20 noncoherent integrations and soft Viterbi decoding. In this case, also, the SER has been evaluated. BER and SER results for the SDR receiver are presented in Figure 13. As for the COTS receiver, a good agreement between experimental and theoretical curves is found. As for pilot-only processing, the demodulation performance strongly decreases below 28 dB-Hz. The demodulation performance for data-only and pilot-only processing is comparable.

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FIGURE 13

BER and SER (PRN 2) estimated with the SDR receiver and data-only KF tracking with 20 noncoherent integrations

7.3 Dynamic live data

As described in Section 6, the dynamic test has been carried out in a very challenging scenario with rich foliage strongly attenuating the signals and buildings blocking signal reception. Consequently, the tracking approaches needed to cope with both weak signals and signal dynamics. In addition to this, frequent signal blockage was observed.

The results of the dynamic tests are reported in Figure 14, which compares the C/N₀ time series obtained with the COTS receiver and the ones obtained with the SDR receiver using the data-only KF with 20 noncoherent integrations. Specifically, the C/N₀ time series obtained by the COTS and the SDR receiver are reported as dashed and continuous lines, respectively. For this test, contrarily to what is done for the static test under multipath, reacquisition was enabled. This was necessary in order to cope with signal blockage.

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FIGURE 14

C/N₀ time series obtained from the COTS receiver (dashed red lines) and data-only KF tracking with 20 ms integration time (continuous line) for the dynamic test

Also in this case, four Galileo signals were tracked during the whole data collections. In particular, the signal arriving from satellite PRN 9 was very strong with C/N₀ values quite stable around 55 dB-Hz. The other three satellites showed C/N₀ values characterized by several fluctuations due to the very challenging scenario.

For all signals, there is a very good agreement between the C/N₀ obtained with the proposed data-only processing scheme and the one provided by the COTS receiver that is implementing pilot processing. Also for this challenging scenario, the lack of the pilot component is compensated by the use of the proposed data-only KF with extended integration time.