# Addressing the Potential of L5/E5a Signals for Road ITS Applications in GNSS-Harsh Environments +
## 2 Methods and Materials  
### Methods  
Considering that the smartphone pseudorange measurements are characterized from high noise levels compared to high-grade geodetic GNSS receivers, at the pre-processing stage, they are smoothed using the Hatch filter [7; 8]. \[\overline{p}_{r,k}^{s,S}=wp_{r,k}^{s,S}+(1-w)\Big{(}\overline{p}_{r,k-1}^{s,S }+\varphi_{r,k}^{s,S}-\varphi_{r,k-1}^{s,S}\Big{)}\overline{p}_{r,0}^{s,S}=p_{ r,0}^{s,S} \tag{1}\]  
where \(k\), \(k-1\) and \(0\) refer to the current, previous and initialization epochs accordingly, \(p_{r,k}^{s}\) represents the pseudorange measurement, \(S\) indicates the GNSS system with \(S\in(G:GPS,\ E: Galileo,\ C:BeiDou)\), \(s\) represents the satellite used, \(\widetilde{p}_{r}^{s}\) is the smoothed pseudorange, \(\varphi_{r}^{s}\) is the precise but ambiguous carrier phase measurement and \(w\) is a weight factor that is inversely proportional to the number of consecutive epochs. To obtain and adapt a weighting value for \(w\), a similar approach to [8] was adopted. For computing the GNSS PVT solution, the Single Point Positioning (SPP) technique is implemented for the case of single-frequency, multi-constellation measurements [7]. The linearized equations for the observed minus computed single-frequency, multi-constellation pseudoranges is given as follows (the time notation \(t\) is omitted for simplicity): [7]:  
\[E\left( \begin{bmatrix}\Delta\mathbf{r}_{r,j}^{s,G}\\ \Delta\mathbf{r}_{r,j}^{c}\\ \Delta\mathbf{p}_{r,j}^{c}\end{bmatrix} \right)= \begin{bmatrix}\mathbf{G}_{r}^{G}&\bm {u}_{m_{G}}&0&0\\ \mathbf{G}_{r}^{E}&\mathbf{u}_{m_{E}}&\mathbf{u}_{m_{E}}&0\\ \mathbf{G}_{r}^{C}&\mathbf{u}_{m_{C}}&0&\mathbf{u}_{m_{C}}\end{bmatrix}\begin{bmatrix} \Delta\mathbf{r}_{r}\\ dt_{r,j}^{G}\\ \left|ISB_{r,j}^{GC}\right|\\ \end{bmatrix}  \tag{2}\]  
where \(E(\cdot)\) denotes the expected operation, \(G\), \(E\) and \(C\) are notations for the GPS, Galileo and BeiDou, respectively, \(r\) is the subscript for the receiver, \(j\) denotes the frequency used, \(m_{S}\) is the corresponding number of observations for each constellation and \(\widetilde{\Delta\mathbf{r}_{r,j}^{S}}\) denotes the observed minus computed pseudoranges for each constellation \(S\in(G,E,C)\). Also, \(\mathbf{G}_{r}^{S}\) and \(\mathbf{u}_{m_{S}}\) are the matrix containing the LOS (Line of Sight) unit directions vectors and the unit vector, respectively, for each constellation \(S\), \(\Delta\mathbf{r}_{r}\) is the receiver position vector, \(dt_{r,j}^{G}\) is the receiver clock error with respect to GPS lumped with the receiver hardware delays, and \(ISB_{r,j}^{GE}\) and \(ISB_{r,j}^{GC}\) are the GPS-Galileo and GPS-BeiDou inter-system biases (ISB), respectively. The notation \(\widetilde{\cdot}\) denotes that the observed pseudoranges are corrected for the orbit, clock, atmospheric delays, etc. Another bias that should be handled appropriately refers to hardware delays that relate to the signals that the broadcasted satellite clock corrections are based on [7]. For example, the broadcasted satellite clocks of GPS satellites are based on the ionosphere-free (IF) linear combination (LC) of L1 + L2 frequencies, while for Galileo, they are based either on the IF LC of E1 + E5a or E1 + E5b frequencies. Instead, for the BeiDou system, the broadcasted satellite clock corrections refer to the B3 signal. Therefore, for the broadcasted satellite clocks (\(dt^{s,S}\)) to refer to the signals utilized in this study, the Timing Group Delay (TGD)and Broadcast Group Delay (BGD) should be applied for GPS/BeiDou and Galileo systems, respectively, as follows [7; 9]:  
\[dt^{s,G}-\frac{\mu_{j}^{G}}{\mu_{1}^{G}}TGD^{s},\qquad j\in(1,5) \tag{3}\]  
\[dt^{s,E}-\frac{\mu_{1}^{E}}{\mu_{j}^{E}}BGD_{Esq-E1}^{s},\qquad j\in(1,5) \tag{4}\]  
\[dt^{s,C}-TGD_{1},\ for\ the\ B1\ signals \tag{5}\]  
\[dt^{s,C}-TGD_{B2 ap},\ for\ the\ B2a\ signals \tag{6}\]  
where \(\mu_{j}^{S}\) is the frequency dependent coefficient, with \(S\in(G:GPS,E: Galileo,C:BeiDou)\).