# Comparison of L1 and L5 Bands GNSS Signals Acquisition
## 2 GNSS Signals and Acquisition  
### GNSS Signals  
GNSS signals contain three essential components, namely navigation data, a pseudorandom code (also called primary code with the advent of modern signals) and a carrier. Modern signals may also contain other optional elements, such as a pilot channel where there is no random data; a secondary code, helping the data synchronization, reducing cross-correlation, and mitigatinginterference; and a sub-carrier, for multipath mitigation [6; 7]. Table 1 summarizes the characteristics of the GNSS signals considered. The signal after the front-end of a receiver is a combination of several GNSS signals and noise:  
\[s_{\text{b}}(nT_{\text{S}})=\sum_{u=1}^{U}s_{\text{b}}^{u}(nT_{\text{S}})+\eta_{ \text{b}}(nT_{\text{S}}), \tag{1}\]  
where \(s_{\text{b}}^{u}\) is the discrete baseband signal from satellite \(u\), \(n\) is the discrete time index, \(T_{\text{S}}\) is the sampling period equal to \(1/f_{\text{S}}\) with \(f_{\text{S}}\) the sampling frequency, and \(\eta_{\text{b}}\) is a white Gaussian noise. The signal from one satellite can be expressed in a generic way as  
\[ \begin{array}{c}s_{\text{b}}^{u}(nT_{\text{S}})=a_{\text{d}}^{u}c_{2,\text{ d}}^{u}(nT_{\text{S}}-\tau^{u})c_{1,\text{d}}^{u}(nT_{\text{S}}\\ -\tau^{u})sc_{\text{d}}^{u}(nT_{\text{S}}-\tau^{u})d^{u}(nT_{\text{S}}-\tau^{ u})\cos(2\pi\big{(}f_{\text{IF}}+f_{\text{D}}^{u}+f_{\text{S}\text{C}})nT_{\text{S}}+ \varphi_{\text{d}}^{u}\big{)}\\ +a_{\text{P}}^{u}c_{2,\text{P}}^{u}(nT_{\text{S}}-\tau^{u})c_{1,\text{P}}^{u} (nT_{\text{S}}-\tau^{u})sc_{\text{P}}^{u}(nT_{\text{S}}-\tau^{u})\sin\Bigl{(} 2\pi\big{(}f_{\text{IF}}+f_{\text{D}}^{u}+f_{\text{O}\text{S}\text{C}}\big{)} nT_{\text{S}}+\varphi_{\text{P}}^{u}\Bigr{)},\end{array}  \tag{2}\]  
where the subscript \(\text{d}\) and \(\text{p}\) denotes the data and pilot channels respectively, \(a^{u}\) is the amplitude, \(c_{1}^{u}\) is the primary code, \(c_{2}^{u}\) is the secondary code, \(sc^{u}\) is the subcarrier, \(d^{u}\) is the navigation data, \(\tau^{u}\) is the unknown code delay, \(f_{\text{D}}^{u}\) is the unknown Doppler frequency, \(f_{\text{O}\text{S}\text{C}}\) is the oscillator offset (not known in a cold start, but may be known or have a good estimation in warm and hot starts [8]), \(f_{\text{IF}}\) is the intermediate frequency, and \(\varphi^{u}\) is the carrier phase. Note that in Equation (2), the data channel is denoted with a \(\cos\) and the pilot channel is denoted with a \(\sin\), however it does not mean that they are necessary in quadrature, as shown in Table 1; it depends on the value of \(\varphi_{\text{d}}\) and \(\varphi_{\text{P}}\)[9]. Furthermore, the code Doppler is not considered in this model to keep it simple. Next, it is assumed that the code Doppler is compensated in the receiver. Since it is proportional to the carrier Doppler, it is relatively simple to adapt the local code chipping rate or to shift the correlation results during their accumulation. Next, only the pilot channels are considered for the acquisition (except for the GPS L1 C/A signal that has only a data channel), because they allow longer coherent integration time, and it is known that the longer the coherent integration time the better the sensitivity [8; 10]. However, it is not always possible to extend the coherent integration time to very long duration, due to the instability of the local oscillator [11; 12; 13] or the unknown dynamics of the receiver [8; 14].