PATIENT REPORT

6.4.2.1 Maximum differential tracking errors entailed by second order distortions

The following plots represent the worst differential tracking error for all considered reference configurations. Results are presented for Galileo E1C in Figure 6-15, for Galileo E5a and GPS L5 signals in Figure 6-16 and for GPS L1 C/A in Figure 6-17.

Figure 6-15. Worst differential tracking error for different signal distortion parameters. On the right, only the 1 m limit is shown. Blue limits give the remaining conservative TM. Red limit underlines that

the TM cannot be bounded for high 𝜎 values. Galileo E1C.

Figure 6-16. Worst differential tracking error for different signal distortion parameters. On the right, only the 1 m limit is shown. Blue limits give the remaining conservative TM. Red limit underlines that

the TM cannot be bounded for high 𝜎 values. Galileo E5a and GPS L5.

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Figure 6-17. Worst differential tracking error for different signal distortion parameters. On the right, only the 1 m limit is shown. Blue limits give the remaining conservative TM. Red limit underline that

the TM cannot be bounded for high 𝜎 values. GPS L1 C/A.

As a reminder, the withheld TS is the parameters range leading to a worst case differential error higher than ∆𝑒𝑟𝑟_𝑚𝑎𝑥 = 1 m (dark colored area on right plots).

To have a simple TM definition, it is decided to adopt a rectangular TS. Table 6-5 represents the TM limits (also representing with blue and red lines in the above figures) for 𝑓𝑑 and low 𝜎 that can be chosen from differential tracking error considerations or that have been fixed in the previous section 6.4.1. This rectangle is referred to as TM-B “area 1”.

Galileo E1C Galileo E5a and GPS L5 GPS L1 C/A

𝑓_𝑑 (𝑀𝐻𝑧) 1 𝑡𝑜 19 3 𝑡𝑜 19 1 𝑡𝑜 19

𝜎 (𝑀𝑛𝑒𝑝𝑒𝑟𝑠/𝑠) 0 𝑡𝑜 26 0 𝑡𝑜 24 0 𝑡𝑜 28

Table 6-5. Analog parameters proposed range for different signals on area 1.

It is however noticeable that the rectangular area does not include large differential tracking errors generated by high 𝜎 distortions when 𝑓𝑑 is low. There is thus a need to define a complementary area to the area 1 to define the TS.

To define this second area, another representation to observe the impact of high 𝜎 on the tracking error is used. It consists in plotting the tracking error for signal distortions with ( ^𝜎

(𝑓_𝑑)²; 𝑓_𝑑) axis as shown in Figure 6-18. As an example, this figure gives the differential tracking error after applying filter 3 (150 ns differential group delay resonator) on the user receiver (and taking the worst case among the two different correlator spacing values and the seven different bandwidths) and filter 1 (6^th-order Butterworth) on the reference for the four signals of interest. It is noteworthy to mention that differential errors can also be large (i.e. larger than the considered ∆𝑒𝑟𝑟_𝑚𝑎𝑥= 1 m limit) for high 𝜎 values. Galileo E1C results are presented on the left, Galileo E5a and GPS L5 results on the middle and GPS L1 C/A results on the right.

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Figure 6-18. Differential tracking errors in meter generated by a TM-B distortion, function of ^𝝈

(𝒇𝒅)^𝟐

and 𝒇_𝒅 for signals of interest.

This new ( ^𝜎

(𝑓𝑑)²; 𝑓_𝑑) representation has a lot of interest because it illustrates that:

- even strongly attenuated distortions can lead to high differential tracking errors. This phenomenon can be explained by the fact that the correlation function is strongly rounded and distorted in an asymmetric way for high 𝜎 values (as illustrated in Figure 6-19) for an unfiltered GPS L1 C/A signal. In this figure, the ratio 𝜎 (𝑓⁄ 𝑑)² is set to 3 and four different 𝑓_𝑑 values are tested. It is noticeable that curves obtained for 𝑓𝑑= 6 MHz (𝜎 = 36 Mnepers/s), 𝑓_𝑑= 11 MHz (𝜎 = 121 Mnepers/s) and 𝑓_𝑑 = 16 MHz (𝜎 = 256 Mnepers/s) are superimposed.

- the tracking error is (almost) constant for a given 𝜎 (𝑓⁄ 𝑑)² and high frequencies. More details about this property is given in appendix G.

Figure 6-19. Impact of highly attenuated TM-B distortions on the correlation function.

Figure 6-20 represents the tracking errors observed by a reference station with the considered configurations for a Galileo E1C (left), a Galileo E5a and GPS L5 (middle), and a GPS L1 C/A signal (right).

An advantage of the 𝜎 (𝑓⁄ 𝑑)² representation is that high values of 𝜎 (𝑓⁄ 𝑑)² can be easily bounded using the condition related to the reference capability to detect large absolute tracking bias. Consequently,

Wors t di fferentia l tra cki ng error, Ga l i l eo E1C

Wors t di fferentia l tra cki ng error, Ga lileo E5 a nd GPS L5

Wors t di fferentia l tra cki ng error, GPS L1 C/A

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it is decided to establish a TM-B “area 2” upper limit in the 𝜎 (𝑓⁄ 𝑑)² representation based on this reference capability. It is reminded that in this document the reference minimum detectable bias is assumed equal to 20 m.

Figure 6-20. Tracking errors affecting the reference in meter generated by TM-B distortions, function of ^𝜎

(𝑓𝑑)² and 𝑓𝑑. Blue rectangles represent area 2 limits, black lines area 1 upper limits.

The “area 2” lower limit is based on its complementarity with “area 1”. To be conservative, the lower limit for

- Galileo E1C is given by:

( 𝜎 (𝑓_𝑑)²)

𝑚𝑖𝑛

= 26

19² ≈ 0.07 𝑛𝑒𝑝𝑒𝑟𝑠/𝑠/𝐻𝑧/𝑀𝐻𝑧 - Galileo E5a and GPS L5 is given by:

( 𝜎 (𝑓_𝑑)²)

𝑚𝑖𝑛

= 24

19² ≈ 0.06 𝑛𝑒𝑝𝑒𝑟𝑠/𝑠/𝐻𝑧/𝑀𝐻𝑧 - GPS L1 C/A is given:

( 𝜎 (𝑓_𝑑)²)

𝑚𝑖𝑛

= 28

19² ≈ 0.07 𝑛𝑒𝑝𝑒𝑟𝑠/𝑠/𝐻𝑧/𝑀𝐻𝑧

One important point is that distortions with 𝜎 (𝑓⁄ 𝑑)² value higher than (𝜎 (𝑓⁄ 𝑑)²)_𝑚𝑖𝑛, can be studied in the 𝜎 (𝑓⁄ _𝑑)² representation. Indeed, from this (𝜎 (𝑓⁄ _𝑑)²)_𝑚𝑖𝑛, the new representation is able to take into account most of the different threatening distortions even for high 𝑓𝑑 where less 𝜎 are tested.

This is supported by the fact that above this limit, distortions vary slowly as intuited on Figure 6-20 and as it will be demonstrated in the next part.

TM-B “area 2” could be reduced to the area between the 20 m absolute tracking error upper limit and the black line representing the “area 1” upper limit in the 𝜎 (𝑓⁄ 𝑑)² representation (Figure 6-20).

Nonetheless, to be conservative and simplify the TMs definition, it is decided to limit “area 2” to the blue rectangles. Finally, the proposed “area 2” limits are given in Table 6-6 for all considered signals.

Tra cki ng error Galileo E1C Tra cki ng error Galileo E5a and GPS L5 Tra cki ng error GPS L1 C/A

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Galileo E1C Galileo E5a and

GPS L5 GPS L1 C/A

Table 6-6. Analog parameters proposed range for different signals on area 2.