## 5 Open Loop Processing Architecture  
The key concept of GNSS open loop processing, which is depicted in Figure 4, is that, as long as the receiver is static, its reference position and time are known, and a network connection providing A-GNSS messages is available; some of the components of the signal can be considered as known, or properly predicted. The knowledge of such components can then be exploited to remove them from the incoming signal, leaving only the random contributions that are due to the thermal noise and other disturbances, among which are the scintillation effects. Scintillation contributions can then be separated from the GNSS signal, which can be considered a sort of deterministic, periodic contribution in the overall received signal. Figure 3: Example of Loss of Lock (LoL) under strong amplitude scintillation in a closed loop receiver; (**a**) second order Phase Lock Loop (PLL) implementation; (**b**) experimental third order PLL. Let  
\[r[n]=\sum_{i=1}^{N_{\text{sat}}}\left\{\sqrt{2P_{RX}\left[n\right]}d_{i}\left(nT_ {s}-\tau_{i}\right)c_{i}\left(nT_{s}-\tau_{i}\right)\cos\left(2\pi f_{i}nT_{s}+ \varphi_{i}\right)\right\}+\eta[n] \tag{1}\]  
be the discrete version of the received signal at the output of the analog-to-digital converter of the front-end which operates at a sampling frequency \(f_{s}=1/T_{s}\) on the GNSS signal at IF. \(N_{\text{sat}}\) is the number of satellites in view, each of one contributing to the overall received signal; the pedix \(i\) refers to the \(i\)-th satellite; \(d_{i}\) is the navigation message; \(\tau_{i}\) is the code delay; \(c_{i}\) is the spreading code; \(f_{i}=f_{IF}+f_{D}\) is the carrier frequency, which includes the nominal intermediate frequency \(f_{IF}\) and the Doppler frequency shift \(f_{D}\); \(\varphi_{i}\) is a generic phase contribution; and \(\eta[n]\) is thermal noise. The architecture does not include either a PLL or a Frequency Lock Loop (FLL) demodulating the signal to baseband by removing the residual carrier at frequency \(f_{i}\). The residual carrier is wiped-off using the Doppler information from assistance. Such assistance information is a good estimate of \(f_{D}\), but it does not include the spurious effect that can be due to the instability of the front-end clock. For such a reason, a good quality clock with negligible frequency drift has to be used to drive both the sampling stage and the frequency references in the front-end. The effect of the non-perfect carrier removal and the issues related to the Doppler shift estimation will be discussed in Section 5.2. Let us assume, for sake of simplicity at this stage, that the estimation of the Doppler frequency is perfect and then that the carrier the wipe-off demodulates the signal to baseband. The mixing generates images of the spectrum at a harmonics frequency, which are removed by the accumulation process, acts as a low pass filter. The signal \(x\left[n\right]\), after carrier wipe-off, and assuming perfect time synchronization and ideal filtering, can then be expressed as:  
\[x[n]=\sqrt{2P_{RX}\left[n\right]}d_{i}\left(nT_{s}-\tau_{i}\right)c_{i}\left( nT_{s}-\tau_{i}\right)+\eta[n]\, \tag{2}\]  
and is, in general, a complex number, due to the multiplication times a complex exponential. As a following step, if a data channel is considered, the navigation message has to be removed, (e.g., exploiting assistance message and the reference time at the receiver). Small errors in this process can be tolerated, as they introduce negligible losses, as it will be shown later. The signal \(z\left[n\right]\), after navigation message wipe-off, and assuming perfect time synchronization, can then be expressed as:  
\[z[n]=\sqrt{2P_{RX}\left[n\right]}c_{i}\left(nT_{s}-\tau_{i}\right)+\eta[n]. \tag{3}\]  
Because of the presence of thermal noise \(\eta[n]\), the Pseudo Random Noise (PRN) spreading code \(c_{i}\) is not visible and therefore it is not yet possible to estimate and remove it.