## 2 Materials and Methods
### Universal GNSS Channel Design Decisions
#### 2.1.6 Discriminator and Filter  
In a multi-signal receiver, the phase relationship from one signal to another may not be cancelled out as part of a common timing error and must thus be specifically accounted for. Similarly, dual-component signals are bound by their phase relationship. With a standard definition where the quadra-phase component leads the in-phase one, we have:  
\[s=\sin{(x)}+j\cdot\cos{(x)} \tag{9}\]  
That is to say, an in-phase (e.g., sin) signal (such as GPS L1 CI) may use the \(I\) and \(Q\) correlator values, while a signal in phase quadrature (e.g., cos) with its RF carrier (such as GPS L1 C/A) should use \(-Q\) and \(I\). In the current implementation, the discriminators are programmed into the embedded MicroBlaze controller, thus allowing for great flexibility. Basically, any coherent and/or non-coherent discriminator could be used based on the signal characteristics; this is simpler than generating sinusoidal waveforms with different phases. More precisely, considering the infinite bandwidth signal auto-correlation function, Figure 6 shows that the BOC main peak has a slope of \(\pm 1.5n\) and a correlation main peak width of \(\pm\frac{1}{n}\) chip, with the BOC modulation ratio \(n=2\cdot\frac{f_{5}}{f_{5}}=2\cdot\frac{p}{q}\). However, the squaring involved in non-coherentcorrelation steepens the peak slopes: which can be approximated by \(\pm\,2n\) between the correlation peak and the zero-amplitude level, separated by approximately \(\pm\,\frac{1}{2n}\) chip. The coherent correlator spacing should not extend beyond \(\pm\,\frac{1}{n}\) chip, above which an inversion of the EML discriminator S-curve in Figure 7 could compromise the DLL behavior (_i.e._, it would amplify the error) [39]. Each one of the \(2\,(n-1)\) side peaks in the squared BOC correlation function leads to a potential false-lock (_i.e._, a biased discriminator output) as a result of as many side S-curves. Also, a rule of thumb imposes, neglecting dynamic stress error ([8] (Chapter 5):  
\[3\cdot\sigma_{\tau_{\text{EMLP}}}<\delta \tag{10}\]  
The normalized correlation function \(R\left(\tau\pm\delta\right)\) is estimated by its main peak positive and negative slopes: \(\{1+m\left(\tau-\delta\right)\}\) and \(\{1-m\left(\tau-\delta\right)\}\) with EML correlators spaced by \(\pm\,\delta\) chip and a chip code delay error \(\left|\tau\right|<\frac{1}{2n}-\delta\). The EML tracking architectures for BOC, should offer a code tracking improvement of \(m\) over BPSK. Figure 6: Infinite Bandwidth BPSK (1) and BOC(1,1) \(-\) n \(=\) 2 \(-\) Coherent and Non-Coherent Normalized Correlation Functions. Figure 7: Effect of Correlator Spacing ((chip)) on a BOC(1,1) Coherent EML Discriminator (assuming an infinite front-end bandwidth). In non-coherent discriminators, \(C/N_{0}\) squaring losses are due to doubled random noise, while the \(\pm 1\) data is wiped off. Non-coherent processing would typically be 3 dB less sensitive than coherent processing for a given duration, although it allows for much longer integration periods, thus achieving a better overall sensitivity. This squaring loss was isolated in square brackets in the code noise jitter equations below. Hence, in non-coherent discriminators, the associated code noise may have a larger variance while preserving the same null mean. It is well known that code phase jitter performances depend on the slope of the discriminator curve (_i.e._, better performances for steeper slopes). In fact, the code phase \(1-\sigma\) error (m) derived from the non-coherent Early Minus Late Power (EMLP) code discriminator closed loop noise variance (squared chip periods) are defined as (extended from [12; 40] to BOC derived modulations):  
\[\sigma_{\tau}\cong c\cdot T_{c}\cdot\sqrt{\sigma_{\tau_{\text{EMLP}}}^{2}} \tag{11}\]  
\[\sigma_{\tau_{\text{EMLP}}}^{2}\cong\left\{