## 3 Analysis of Roll Modulation Signals
### SNR Analysis of the Roll Modulation Signals  
The premise for (16) to be established is that the signal-to-noise ratio (SNR) of one millisecond values have to be high enough. And it is essential for a PLL to track the roll modulation signals with high precision, which requires high SNR as well. Thus, the SNR of roll modulation signals is studied in this section. The power of receiver signals can be reduced during antenna rotation and results in attenuation of the power of satellite signals. As shown in Figure 6, it is the normalized average power of received satellite signals in one second at different spin rates with initial relative roll angles \(\alpha_{0}\). Figure 5: The block diagram of a GPS receiver with a roll angle detection module. As we can see from Figure 6, the average power of receiver signals differs when the spin rates and initial angles change. Considering the background of this technology, the spin rate of a spinning vehicle usually exceeds 3 r/s. So, the average power of received satellite signals is approximately equal to the middle value between the maximum and minimum power of the receiver signals. Generally, the maximum and minimum gain of a patch antenna is about 0 dB and \(-12\) dB respectively. While, conversely, the minimum power of received signals is normally 10 times smaller than the maximum power and can be ignored in the average processing. Thus, the average power of receiver signals can be approximately equal to the value of half of the maximum power, which means there is an attenuation of 3 dB in the power compared to the static situation. In addition to the power attenuation received by the antenna, there is an attenuation in the tracking process. According to (16), the power of roll modulation signals depends on the carrier and code tracking errors at the same time. As for the frequency error \(f_{e}\), it can only be influenced by the disappearance of satellite signals when the signals are blocked from satellites during rotation, while the code NCO is hardly affected [12]. The block of signals can induce fluctuations on the carrier NCO, which results in the increase of frequency error \(f_{e}\). The fluctuation induced attenuation can be expressed as:  
\[\Delta\Big{|}\text{sinc}(f_{flu})\Big{|}=\big{|}\text{sinc}(f_{e}T_{car})|- \Big{|}\text{sinc}[(f_{e}+f_{flu})T_{car})\Big{|} \tag{18}\]  
where \(f_{e}\) is the carrier frequency error in static situation, \(f_{flu}\) is the fluctuation induced frequency error during rotation. Figure 7 shows the attenuation caused by fluctuation at different static frequency errors. Figure 6: The diagram of normalized average power in one second. Figure 7: Attenuation caused by fluctuation at different static frequency errors. After the tracking process, there is a square loss \(L_{SQ}\) at the magnitude extraction process, expressed as:  
\[L_{SQ}=\text{SNR}_{corr}-\text{SNR}_{roll} \tag{19}\]  
where \(\text{SNR}_{roll}\) is the SNR of roll modulation signals, which can be defined as [23]:  
\[\text{SNR}_{roll}=\left[E\bigg{(}\sqrt{I_{p}(n)^{2}+Q_{p}(n)^{2}}\bigg{)}-E \bigg{(}\sqrt{n_{I}{}^{2}+n_{Q}{}^{2}}\bigg{)}\right]^{2}/V\bigg{(}\sqrt{n_{I} {}^{2}+n_{Q}{}^{2}}\bigg{)} \tag{20}\]  
where \(E(. )\) and \(V(. )\) represent the expectation and variance respectively. \(\text{SNR}_{corr}\) is the SNR of correlation results, which is defined as \(\text{SNR}_{corr}=\big{|}\
icefrac{{p}}{{p}}(n)\big{|}^{2}/\sigma_{n}^{2}\), \(\sigma_{n}^{2}\) is the variance of \(I_{p}(n)\) or \(Q_{p}(n)\). Figure 8 shows the relationship between squaring loss and SNR of correlation results. As we analyzed above, the power attenuation caused by rotation mainly comes from antenna rotation, frequency error and square loss of magnitude extraction.