## 2 Materials and Methods
### Universal GNSS Channel Design Decisions
#### 2.1.6 Discriminator and Filter
\begin{array}{l}\frac{B_{t}(1-0.5 B_{t}T_{p})}{2\cdot\frac{C}{N_{0}}}\cdot\left[1+\ \frac{4}{\frac{\kappa_{0}}{T_{p}\cdot(2-m\cdot\Delta)}}\right]\cdot\left\{ \frac{\Delta}{m}\right\},\ \pi\leq\Delta\cdot b\\ \frac{B_{t}(1-0.5B_{t}T_{p})}{2\cdot\frac{C}{N_{0}}}\cdot\left[1+\ \frac{4}{\frac{\kappa_{0}}{T_{p}\cdot(2-m\cdot\Delta)}}\right]\cdot\left\{ \frac{1}{b}+\frac{b}{\pi-1}\cdot\left(\frac{\Delta}{m}-\frac{1}{b}\right)^{2} \right\},\ 1<\Delta\cdot b<\pi\\ \frac{B_{t}(1-0.5B_{t}T_{p})}{2\cdot\frac{C}{N_{0}}}\cdot\left[1+\ \frac{2}{\frac{\kappa_{0}}{T_{p}}}\right]\cdot\left\{ \frac{1}{b}\right\},\ \Delta\cdot b\leq 1\end{array} \right. \tag{12}\]  
where: \begin{tabular}{l l} \(c\) & is the speed of light (m/s); \\ \(T_{c}\) & is the chip period, the inverse of the chipping rate \(f_{c}\); \\ \(B_{L}\) & is the unilateral noise equivalent bandwidth of the code tracking loop, a.k.a. one-sided equivalent rectangular bandwidth, with the time frame of interest \(\epsilon\ \left[1/B_{L},T_{obs}\right]\); \\ \(T_{p}\) & is the pre-integration time (s); \\ \(\tau\) & is the signal _vs._ replica misalignment (chip); \\ \(\Delta\) & is the early-late correlator spacing (chip), _i.e_, \(2\cdot\delta\); \\ \(\delta\) & is the early to prompt and prompt to late correlator spacing (chip); \\ \(\frac{\zeta}{N_{0}}\) & is the Carrier power to Noise density ratio (dB-Hz); \\ \(m\) & is the slope of the correlation function; \\ \(b\) & is the normalized receiver front-end complex bandwidth (\(\beta_{r}\cdot T_{c}/n\)); \\ \(\beta_{r}\) & is the ideal front-end complex bandwidth (with a brick-wall filter (Hz)). \\ \end{tabular}   
In Equation (12), the term in square brackets reflects the squaring losses attributed to the non-coherent discriminator computations, while the term in braces results from approximations depending on the value of \(\Delta\cdot\ b\). Given a fix front-end bandwidth and an equivalent chip spacing during tracking, the approximation mainly involves the signal modulation represented by \(T_{c}/n\). The Cramer-Rao Lower band is reported by [PERSON] _et al._[40] to be:  
\[\sigma_{\tau_{\text{LB}}}^{2}\cong\left\{ \begin{array}{l}\frac{B_{t}(1-0.5 B_{t}T_{p})}{2\cdot\frac{C}{N_{0}}b^{2}},b\leq 1\\ \frac{B_{t}(1-0.5B_{t}T_{p})}{2\cdot\frac{C}{N_{0}}b},b>1\end{array} \right. \tag{13}\]  
It thus becomes interesting to determine what DLL noise variance can be expected for each GNSS signal when tracked with the proposed channel. Analysis in [40] reports that for limited front-end bandwidths, the discriminator gain diminishes as the early-late correlator spacing \(\Delta\) decreases, while increasing the loop bandwidth and thus the loop variance. Three discriminator regions are identified as: Spacing-Limited, Transition and Bandwidth-Limited, in accordance with Equation (12). Looking at MBOC, while assuming \(\beta_{r}=22.3\) MHz and \(T_{c}/n=1/\left(12\cdot 1.023\times 10^{6}\right)\), \(b\approx\ 2\) for the BOC(6,1) signal component, which rapidly falls under the Bandwidth-Limited during tracking area with \(\Delta\leq 0.5\) chip. One should bear in mind that the relative power ratio of BOC(6,1) is one tenth that of BOC(1,1), for which \(b\approx 12\), well within the Spacing-Limiting function. Looking at other GNSS signals, it appears that in the presented configuration, \(b\) ranges from 2 to 47, as depicted in Figure 8. For signals where \(b\) is high, it still is beneficial to reduce \(\delta\), also mitigating multipath errors. Nevertheless, unless dedicated RAM blocks are available for all the code phases used, a 60 MHz sampling frequency poses a 1 sample limit on \(\delta\) based on delayed code phases based on the shift registers approach described above, the impact of which will vary with the GNSS signal chipping rates.