# GNSS Performance Modelling and Augmentation for Urban Air Mobility +
## 3 GNSS Performance Characterization
### Error Modelling
#### 3.1.8 Receiver Model  
In addition to geometrical effects, pseudorange error also has a dependency on the receiver architecture and the specific type of discriminator function used to extract a pseudorange measurement from a satellite. The primary focus of this paper is the effect of multipath on the code-phase. Therefore, the receiver model implementation is limited to the Delay Lock Loop (DLL) aspects viz the employed discriminator function. In order to estimate the pseudorange error, the estimated power delay profile must be fed through the In-phase (I) and Quadrature-phase (Q) arms in order to form a discriminator function as demonstrated in [37]. In order to simplify the analysis in this paper, a weighted average is computed over all the echoes for a given receiver-satellite-reflector configuration at a given altitude in the urban canyon. This yields a single characteristic multipath ray. This allows us to employ a simplified single-ray multipath error envelope which maps the multipath delay and relative amplitude to the pseudorange error. The error is dependent on the two geometry-dependent parameters discussed so far, i.e., the multipath delay and the relative amplitude, and one receiver-dependent parameter, viz. the correlator spacing (specified in units of chips). Multipath error envelopes for Binary Phase Shift Keying (BPSK) (1) modulation for GPS are estimated by the following inequalities [38]:  
\[\tau=\tfrac{a\delta}{a+1};\,\text{for}\,\,0\leq\,\delta\leq\tfrac{d}{2}(\alpha+1) \tag{24}\] \[\tau=\tfrac{a\delta}{2};\,\text{for}\,\,\tfrac{d}{2}(\alpha+1) \leq\,\,\delta\leq\left(1+\tfrac{d}{2}\right)(\alpha-1)\] \[\tau=\tfrac{\left(d-\tfrac{\delta}{2}-1\right)\alpha}{a-2};\, \text{for}\,\left[\left(1+\tfrac{d}{2}\right)\right](\alpha-1)\leq\,\,\delta \leq 1+\tfrac{d}{2}\]  
Standard correlator spacings of one chip (d=1 chip) have the largest potential pseudorange error. However, most modern receivers employ narrow correlators, where the spacing is 0.1 chip. Equation 16 assumes infinite bandwidth of the autocorrelation function. Modeling finite bandwidth would be more accurate in terms of the estimated pseudorange error. However, peak error is reduced when accounting for bandwidth in the model and therefore the infinite bandwidth assumption yields more conservative estimates [38]. The receiver model is used to set an upper bound on the pseudorange error based on the conservative estimates of delays and echo strengths for the ground and airframe reflections. For reflections from building faces, the error estimation is based on the more severe NLOS condition. An urban canyon test domain of (L) 100m \(\times\) (W) 50m is defined to model the described electromagnetic interactions with building facades. The following parameters are tunable in the implemented urban canyon model and are set to:  
* Street width: 30m (common in the U.S, Europe and Australia)
* Number of buildings: 8
* Building widths: 50m
* Building spacing: 5m
* Building height: (\(\mu_{h}:65\text{m}\), \(\sigma_{h}:5\text{m}\))  
A grid of receiver points is defined over the model domain with a horizontal resolution of 1m and vertical resolution of 5m up to a ceiling of 10m above the maximum building height. Satellite elevation is varied in increments of 10 \({}^{\circ}\) from a masking angle of 10\({}^{\circ}\) to 90\({}^{\circ}\). The previously described steps to check for NLOS/multipath conditions are applied for each angular increment. Figure 10 is a plot of histograms of delays for three altitudes and three satellite elevation angles. As expected, the likelihood of tracking an NLOS signal is strongly dependent on the altitude and satellite elevation. In general, NLOS conditions were met more frequently at low operating altitudes and low elevation angles.