To complete the above narrowband analysis, a wideband study was performed particularly addressing the GNSS case. The goal is to quantify the influence of the environment resolution on the receiver performance when estimating the pseudo-range to a satellite. First, the simplified GNSS receiver model used is described and then the wideband channel model used to generate the channel response is presented. Finally, a qualitative analysis of the pseudo-range estimation error is shown followed by a statistical analysis of the influence of environment resolution.

### 4.1. Simplified GNSS receiver model

The simplified GNSS receiver used in this study was developed by the French Space Agency, CNES. It aims at reproducing the various internal processes of a typical receiver in a simplified way in order to estimate pseudo-range errors. The simplifications are mainly based on the fact that the receiver directly operates on post-correlation synthetic data instead of operating on baseband synthetic data. Hence, the standard C/A GPS code [18], a spreading *Pseudo Random Noise* (PRN) sequence, is here modelled as a simple triangular function representing the autocorrelation of the PRN code. Then, in the presence of multipath, the overall correlation function is constructed by combining the complex delayed correlation functions due to each individual MP path. The DLL is implemented as a dot-product discriminator followed by a first-order loop filter [17]. The "early" and "late" replicas of the local code are spaced 0.25 chips from the "prompt", and the DLL noise bandwidth is set to 10 Hz. The output of the *Phase Lock Loop* (PLL) is emulated assuming noise-free conditions, and then, the phase noise is added to the "prompt" signal using randomly generated samples whose standard deviation corresponds to that of a Costas PLL with 1 Hz noise bandwidth. The tracking time is set to 20 ms.

Two input files are required: the first contains the receiver parameters and the second is devoted to the channel characteristics. The channel file contains several parameters such as the time of week, the navigation data, the ionospheric delay or the number of MP paths. Each path, including the direct signal, is defined considering its power, phase, delay and Doppler. Power and phase parameters are given relative to the direct signal. The output file gathers the pseudo-range error, the tracked phase and the demodulated navigation data. Note that an acquisition step is performed by the receiver but all the simulations presented below have been realised assuming that the receiver was in the "locked" state. The thermal noise level was assumed to be -202 dBW and the direct signal power -153 dBW.

### 4.2. Wideband representation of the channel

As for the narrowband analysis, the MoM-based tool ELSEM3D was used to determine the EM field along linear trajectories. The urban canyon under consideration here has been built using the same principles as in Figure 4. Contrary to the narrowband analysis where the contributions of canonical buildings were coherently summed all together, each building contribution is now represented by one ray (delta function) discriminated in terms of power, phase, delay and Doppler. The power of each ray is assumed to be equal to the narrowband power radiated by each canonical building and depends on the environment resolution as presented in Figure 10. Figure 10 permits a better understanding of how the chosen environment resolution could affect the GNSS receiver since the evolution of one specific echo has been highlighted for all four resolutions to observe its power variation along the receiver route. As visible in Figure 10, the maximum power of the red ray slightly decreases when adding details to the facade. Also visible a back scattering zone around 50 m where power is increasing when adding details. The phase, delay and Doppler shift are determined using a geometrical approach where rays originate from the centre of each canonical building. This construction implies that phase, delay and Doppler do not depend on the resolution employed. The power and delay of each ray are considered relative to the LOS signal. The phase and Doppler are considered in their absolute values. Note that the navigation satellite is supposed to be stationary and the observed Doppler shifts are only due to the receiver's motion. Figure 11 illustrates the evolution of the channel parameters, power, delay and Doppler, while the receiver is moving through the Canyon1. As predictable, Figure 11 reveals that the strongest echoes are the nearest. The evolution of Doppler and delay of each echo are also noticeable. Note that the considered red echo is the same as in Figure 10.

In order to model the wideband channel, the major assumption made in this simplified approach is the fact that each canonical building gives rise to one single ray. This approach induces discontinuities since buildings are reduced to one scattering point. This assumption comes from the MoM tool ELSEM3D we used, whose outputs are only the EM fields. A solution would have been to divide each façade into smaller pieces and record their EM fields. Using the same reradiating principle as in Section 3.1, we could have obtained 64 rays per façade instead of 1, leading to more than 1,500 rays for Canyon1 in the high-resolution case. However, the GNSS receiver is a software model and is not adapted to process such amount of data in a reasonable time. Even more, a statistical analysis based on numerous cases would not be possible. Furthermore, this assumption is a reasonable one considering the spatial resolution of a standard 20 MHz bandwidth system is 15 m. This assumption is conservative since typical GNSS receivers use bandwidths between 10 and 2 MHz. In our simulated scenarios, the maximum distance between two points from the same canonical building is 7 m, smaller than system's spatial resolution.

### 4.3. Influence of environment on navigation devices

Here, the influence of the employed environment resolution is analysed from a GNSS receiver point of view, considering the pseudo-range error as the key parameter and using the simplified wideband channel model previously described. The pseudo-range to a satellite represents the estimated distance between receiver and satellite. The pseudo-range error represents the bias induced by multipath and is the difference between the estimated and the real distance. In the following, the pseudo-range error parameter will be analysed as a function of the environment resolution for 81 different configurations (9 incidences, 3 speeds, 3 urban canyons). We introduced one further parameter called the relative error, which represents the offset between the pseudo-range error for the simplified resolutions with respect to the high resolution.

#### 4.3.1. Influence of resolution and speed

The first part of the analysis presents the influence of the environment resolution on navigation devices when the direct LOS signal is not attenuated. One example is presented in Figure 12 to illustrate a typical LMS configuration with 40° elevation and 40° azimuth. This figure presents, from top to bottom, the narrowband MP scattered power relative to LOS, the absolute pseudo-range error and the relative pseudo-range error with respect to that for the high-resolution representation.

A first remark concerns the configuration in Figure 12 and others not presented in the figure. It has been observed that the receiver can be sensitive to a particular wave combination resulting in punctual errors and small discontinuities visible around *y* = 25 m for the high-resolution case and *y* = 40 m for the low-resolution case on the pseudo-range error diagram. This phenomenon does not seem dependent on the chosen resolution. The consequence of this is a slight shift in the relative error, bottom diagram, while other resolutions are not affected in the same way. However, these irregularities are negligible when extracting statistical parameters.

From the narrowband analysis, it has been shown how, for typical LMS configurations, the total MP scattered power is usually overestimated. This phenomenon was particularly pronounced for null resolution. With a wideband approach, the same observation has been made. For typical LMS incidences, strong echoes are present for the null resolution resulting in high and punctual biases visible from 40 to 60 m in Figure 12. Contrary to the medium and low resolutions, which show good agreement with respect to the high-resolution case, the null resolution is not suitable for typical LMS configurations.

To get a better overview and generalise the observations made in Figure 12, statistical results are next presented. For all the 81 scenarios, the mean and standard deviation error of the relative pseudo-range error have been extracted in the canyon's reflection zone, from 0 to 85 m in the Canyon1 case. Figure 13 presents the cdf of both error parameters. As visible here, the mean and standard deviation error for the medium and the low resolution are good approximations with respect to the high resolution. The null resolution has critical limitations with strong mean errors.

Figure 14 presents a parametric plot considering the {Mean;Std} error couple, showing the influence of two parameters: environment resolution and receiver speed. The speed parameter is here taken into account since it has an important influence on the integration process taking place in the receiver. From the analysis of Figure 9, it has been shown that slow motion was the most critical case. From the analysis of Figure 12, it has been shown that the null resolution is not suitable to represent the environment in the GNSS context. Removing both cases from Figure 14, the null resolution (pink) and the slow motion (circles), all other cases are in the dashed line black square. This square represents a mean error in the interval of ± 1 m and standard deviation error inferior to 1 m. We can consider all these configurations as acceptable simplifications for the analysis of GNSS systems when LOS is not attenuated.

#### 4.3.2. Influence of incidence

In a second step, the influence of the incidence angle is here analysed. As a remainder of Section 2.3, incidences are grouped into two categories: typical LMS configurations which represent all incidences with elevation ≤ 40° and azimuth ≤ 40° and extreme LMS configurations which represent all other incidences with elevation > 40° or azimuth > 40°. Figure 15 illustrates one extreme LMS incidence case with 80° elevation and 80° azimuth using the same representation as Figure 12. Figure 15 is representative of the fact that for extreme incidences pseudo-range errors are very small (± 0.5 m) and are noise-like. Furthermore, relative errors are almost null in the range of ± 0.1 m. In general, for all configurations, it has been noted that pseudo-range errors occur when the MP scattered power is larger than -15 dB relative to the LOS signal. This explains the low impact of MP for the extreme incidences. In this case, the environment resolution has no impact compared to the typical geometrical LMS configurations.

To confirm this statement, statistics are presented in Figure 16. They have been extracted considering the null resolution. This case is the most conservative one since the null resolution has the largest spread in the mean error as shown in Figure 13. From Figure 16, it can be observed how all extreme cases have a mean error almost equal to zero and standard deviation below 1 m, in agreement with Figure 15. This confirms that for extreme LMS incidences even the null resolution produces acceptable results.

#### 4.3.3. Influence of LOS attenuation

In the previous scenarios, the direct signal was not attenuated. To extend the domain of validity of this analysis and better take into account real cases, attenuation on the direct signal has been investigated. Two attenuations are considered: -5 and -20 dB corresponding to tree shadowing and building blockage. It is assumed that the direct signal is the only component affected by this attenuation. The consequence is that the margin between the MP echoes and the direct signal is reduced. It may even occur that some echoes are stronger than the direct signal.

Simulations realised for -5 dB show similar results to those for no direct signal attenuation. Thus, for this case, the same conclusions are applicable. However, results differ when considering a -20-dB attenuation as presented in Figure 17. Here, the MP scattered power is about 20 dB higher than the attenuated direct signal. When a long blockage event occurs, the receiver tends to track the strongest echo. This results in all resolutions being biased in the same way. As visible in Figure 17, the pseudo-range error is about 6 m, representative of the distance between the receiver trajectory and the first row of buildings, situated at 4 m from the receiver's route. We can also notice fluctuations around *y* = 20 m due to the second row of buildings, situated 10 m from the receiver. A similar behaviour is particularly visible from *y* = 70 m to the end of receiver's route. From point *y* = 70 m onwards specular reflections are not possible. While the receiver continues down its route, the DLL keeps tracking the strongest diffuse contribution coming from last building corner. The result is that pseudo-range error increases until the diffuse contribution falls down at around *y* = 90 m. Then, the attenuated direct signal becomes predominant over the MP, and the receiver recovers track of the LOS signal. Finally, looking at the first half of the pseudo-range error diagram in the figure, for high and medium resolutions, the GNSS receiver seems to be affected by MP earlier than for low and null resolutions. The explanation is that when the LOS signal is attenuated, any small multipath contribution becomes noticeable by the receiver. Since scattering is almost absent in the low and null resolutions, both resolutions are affected later by MP, and only in the reflection zone. The medium and high resolutions show a longer impact, before and after. Not presented in the figure, the above observations are even more visible for extreme azimuths where the corner effect from the details does not remain negligible when comparing the null and low resolutions to the medium and high ones.

Figure 18 shows the statistics on the influence of the resolution when the direct signal is attenuated by 20 dB. Contrary to unblocked case, the environment resolution has a significant impact on the receiver's performance.