Comparison of chip domain and correlation function domain observables

Advantages and drawbacks of both techniques are detailed in [Thevenon et al., 2014]. A summary is presented in the following.

The advantages of the CDO are:

- Inputs of the CDO (IF signal samples) are given directly by the RF front-end while multi-correlator outputs have to be computed specifically for a given code delay.

- The noise affecting the CDO is an uncorrelated white noise (or weakly correlated by the RF front-end filter), while the noise affecting a correlator output is correlated through the multiplication with the local replica.

- The resolution of the CDO can be increased beyond the sampling frequency of the signal based on a principle called dithered sampling. [Pini and Akos, 2007]

- The CDO permits to observe independently different types of signal sections. An important consequence is that falling and rising edges can be visualized separately whereas it is not possible on the correlation function.

However, correlation function observables have also advantages compared to the CDO because of the place of the correlation operation in the tracking processing. Then:

𝐵𝑃𝑆𝐾(1) 𝐶𝐵𝑂𝐶(6,1,1/11)

𝐵𝑃𝑆𝐾(10)

4.5 Visualization of GNSS signal distortions

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- The tracking is directly dependent upon the correlation function. As a consequence, the distortion visible on the correlation function is directly related to the pseudorange error. In that sense, the distortion on the correlation function appears more representative of the potential problems on the pseudoranges. A corollary of this is that some of the distortions visible on the CDO could be filtered/transformed by the corre lation process which is based on the entire PRN code. Consequently, some signal distortions visible on the CDO could not influence the correlation function and by consequence the tracking process.

- Correlation processing is already available in conventional receivers, although multi-correlator outputs are not yet widely available.

- Correlator outputs are much less noisy than IF samples.

The last point is illustrated by the comparison of estimated standard deviation for the two observables.

From (4-4) and (4-10), the ratio of the two standard deviations can be estimated:

(4-12) is a general equation that can be used on all signals with different observable parameters. As an example a particular GPS L1 C/A case is considered. Only rising transitions are superimposed to estimate the CDO. The observed part is chosen with a 𝑇_𝑐 length and bins uniformly distributed along this time interval. It entails that:

- 𝑁𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑_𝑝𝑎𝑟𝑡_𝑐𝑜𝑑𝑒= 𝑁_{𝑡𝑟𝑎𝑛𝑠} where 𝑁_{𝑡𝑟𝑎𝑛𝑠}≈ 250 is the number of rising or falling transitions in one GPS L1 C/A spreading code period.

- Δ_𝑏𝑖𝑛 = 𝑇_𝑐/𝑁_{𝑏𝑖𝑛𝑠} where 𝑁𝑏𝑖𝑛𝑠 is the number of bins in the observed section.

In this particular case, (4-12) becomes:

And the ratio between the two standard deviations is only dependent upon the number of bins 𝑁_𝑏𝑖𝑛 computed for the CDO.

It is demonstrated in [Pagot et al., 2015] that the correlation and the chip domain observables, estimated from the average of all chips of the spreading code and convolved by a rectangular shape, have the same expression assuming that:

- the correlation function is null outside the peak (for delay smaller than −𝑇_𝑐 and higher than 𝑇_𝑐 around the prompt),

- all chips are used to estimate the CDO (positive and negative chips, after a transition or not).

In any case, for GPS L1 C/A, a triangular shape (code correlation function-like) can be obtained by the convolution of the CDO on one chip with an ideal rectangle. Some receivers use this property and derive “correlation functions-like” from the CDO [NovAtel Inc., 2012] to estimate if GNSS signals are distorted from CDO observables.

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4.6 Conclusions

The aim of this chapter is to present the different problematics linked to GNSS signal distortions and more precisely GNSS signal distortions generated by the payload that are a problem for all GNSS users and particularly for SBAS users.

In section 4.1 nominal and non-nominal distortions that can affect a GNSS signal are described. As nominal distortions affect continuously GNSS signals, they can be observed precisely using particular setups to collect signals at every time. Results from previous studies about nominal distortions are introduced. A ringing effect after each transition and a lead/lag between zero-crossings of rising and falling edges of code transitions is observed even on signals generated by healthy satellites. It has been seen that nominal distortions generated at payload level are challenging to characterize especially because they are time varying and they are difficult to dissociate from distortions induced by the receiver. The concept of signal distortion also appears in faulty conditions (these distortions are also called EWF) and is different from the study about nominal distortions. Indeed, due to the lack of examples about signal distortions generated in faulty conditions, it is difficult to characterize the kind of distortion that could appear in a case of a satellite failure.

In section 4.2 an example of signal distortion is considered on a 𝐵𝑃𝑆𝐾(1)-modulated signal (GPS L1 C/A), a 𝐵𝑃𝑆𝐾(10)-modulated signal (GPS L5 and Galileo E5a) and a 𝐶𝐵𝑂𝐶(6,1, 1 11⁄ , −)-modulated signal (Galileo E1C). This signal distortion is used in section 4.3 and 4.5 to illustrate the impact of signal distortions at different levels of the GNSS receiver signal processing.

The main issue with GNSS signal distortions is that their consequences on different user’s receivers are dependent upon several characteristics of receivers presented in section 4.3: the technology and the bandwidth of the antenna and RF front-end filter, the discriminator and the correlator spacing used for the tracking. In particular, in part 4.3.2 the impact of the signal distortion on tracking loops is tackled.

In section 4.4, the issue entailed by non-nominal deformation is exposed. Even if EWF are not frequent, GNSS users with stringent performance, as civil aviation users, have to be protected from such threats.

To deal with the EWF issue, the Most EWF concept was introduced in a previous study but the preferred solution was the definition of a TM. Nowadays a TM is adopted by ICAO to represent distortions expected on the GPS L1 C/A signal.

In section 4.5, the impact of the distortion is looked at in the chip and in the correlation function domains. The technique to generate the CDO is detailed and wi ll be reused in the following.

From this chapter, it can be seen that the study of GNSS signal distortions is made difficult because the characterization of these distortions is complicated and because the impact of GNSS distortions are dependent upon several features of the receiver. This chapter introduces all important notions linked to GNSS signal distortions and can be viewed as an introduction to the following chapters (chapter 5 and chapter 6), where a deepened study about nominal and non-nominal distortions is undertaken.

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5 Nominal distortions

The first GNSS signal distortions that are tackled in details are distortions that aff ect the signal in fault-free conditions. Even if these distortions have a limited impact on GNSS receivers, they can be a problem for users with high requirements. Real data were collected to observe nominal distortions on real GNSS signals. Two types of data collections were performed:

- using high-gain dish antennas and - using omnidirectional antenna.

Nominal distortions that affect GNSS signals recorded with the two types of antennas are estimated and results are presented and compared to the state-of-the-art (chapter 4). The aim of this chapter is to confirm results already available in the state-of-the-art relatively to the study and the characterization of nominal distortions and to present new results on Galileo E1C signal. More precisely, in this chapter different results are provided:

- A characterization of GPS L1 C/A nominal deformations that affect high-gain dish antennas collected signals. This characterization is based on [Phelts et al., 2009].

- A visualization of nominal distortions that affect Galileo E1C signals collected with high-gain dish antennas.

- A visualization of nominal distortions that affect GPS L1 C/A and Galileo E1C correlation functions.

- A description of advantages and drawbacks between the study of nominal distortions using a high-gain dish antenna and using an omnidirectional antenna.

- A characterization of GPS L1 C/A tracking biases that affect signals collected with an omnidirectional antenna. This characterization is based on [Wong, 2014].

Even if one of the purposes of the nominal distortions study is to establish a limit between the nominal case and the non-nominal case, because of the lack of measurements and the difficulties to characterize nominal distortions in an absolute way, this task is not developed in this chapter.

In section 5.1, the setup that was used to collect GNSS signals with high-gain dish antennas is presented. Indeed, the antenna, the digitizer and the software used to process signals have an influence on the observed nominal distortions and have to be defined.

In sections 5.2 and 5.3, results obtained from high-gain dish antennas are presented. The first section introduces nominal distortions visualized on the Chip domain Observable (CDO), and the second section nominal distortions visualized on the correlation function domain and the S-curve zero-crossing plot. It is seen in 5.4 that some distortions generated by the receiver cannot be distinguished from nominal distortions when the setup is not calibrated.

Distortions generated by the receiver have the particularity to affect all received signals. Therefore, to remove the main part of the distortion induced by the receiver, omnidirectional measurements were collected and processed. Results are provided in section 5.5. The common bias that affects all PRNs measurements collected at the same time are subtracted from pseudorange measurements error to estimate inter-PRN biases which effectively affect a GNSS user.

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In section 5.6, inter-PRN biases estimated from the omnidirectional antenna data collection and from one calibrated high-gain antenna data collection are compared to the state-of-the-art.

Finally, section 5.7 makes a conclusion about all results provided in this chapter.