# Performance Evaluation of the New Compound-Carrier-Modulated Signal for Future Navigation Signals
## 3 Evaluation Criteria
### Multipath Error Envelope  
The multipath error envelope (MPEE) commonly relies on the assumption of a multiple ray signal model to quantify the worst possible multipath-induced bias. The envelope graphs, as a significant criteria, directly illustrate the differences of the multipath performance for modulation schemes [34].  
The two-ray signal model allowing for the direct line of sight (DLOS) signal and one single multipath signal, visualizes the common multipath environment. In this dedicated model, the output S-curve of the Early-Minus-Late (EML) discriminator can be expressed as [35]:  
\[\begin{array}{ll}D_{EML}\left(\varepsilon\right)&=\left[R\left(\varepsilon+ d/2\right)-R\left(\varepsilon-d/2\right)\right]\\ &+a\text{cos}\left(\Delta\varphi\right)\left[R\left(\varepsilon-\Delta\tau_{1 }+d/2\right)-R\left(\varepsilon-\Delta\tau_{1}-d/2\right)\right]\end{array} \tag{14}\]  
where \(\varepsilon\) is the code delay estimation error, \(d\) is the correlator spacing between the early and late reference signal, \(R\) is the correlator function of the received signal and reference signal, \(a\) is the multipath signal to DLOS amplitude ratio (MDR), \(\Delta\varphi\) is the phase difference between the multipath and the DLOS, \(\Delta\tau_{1}\) is the multipath signal time delay refer to DLOS. Note that the multipath error envelops are computed using the two extreme cases, where \(\Delta\varphi=180^{\circ}\) and \(0^{\circ}\).  
Expanding the \(D_{EML}(\varepsilon)\) with the first-order Taylor's series approximation about 0, Equation (14) becomes:  
\[D_{EML}\left(\varepsilon\right)\approx D_{EML}\left(0\right)+D_{EML}^{\prime} \left(0\right)\times\varepsilon+o\left(\varepsilon^{2}\right) \tag{15}\]  
where \(o(\varepsilon^{2})\) represents higher-order errors. Neglecting the influence of higher-order terms, Equation (15) can be expressed as:  
\[\varepsilon\approx-D_{EML}\left(0\right)/D_{EML}^{\prime}\left(0\right) \tag{16}\]  
According to the relationship between the ACF and PSD of a signal, the MPE can be obtained as:  
\[\varepsilon\approx\frac{\pm a\int\limits_{-B_{r}/2}^{B_{r}/2}G\left(f\right) \sin\left(2\pi f\tau\right)\sin\left(\pi f\,d\right)df}{2\pi\int\limits_{-B_{r}/ 2}^{B_{r}/2}fG\left(f\right)\sin\left(\pi f\,d\right)\left[1\pm a\text{cos} \left(2\pi f\tau\right)\right]df} \tag{17}\]where, the operation symbols \"+\" and \"\(-\)\" correspond to the aforementioned two extreme cases, respectively. From Equation (17), we can obtain that the MPEE lobes strictly depend on the characteristics of the signal ACF and PSD. The sharper peak of a signal ACF means more high-frequency components a signal PSD processes, then, the corresponding MPEE will be smaller and decrease faster towards zero.  
The related average multipath error envelop (AMPEE) expressions are given by:  
\[\varepsilon_{av}=\frac{1}{\tau}\int\limits_{0}^{\tau}\left[\frac{abs\left( \varepsilon\mid_{\Delta\varphi=0}\right)+abs\left(\varepsilon\mid_{\Delta \varphi=180}\right)}{2}\right]d\tau \tag{18}\]  
The AMPEE reflects the susceptibility to multipath interference at any delay.