## 3 Analysis of Roll Modulation Signals
### The Extraction Method of Roll Modulation Signals  
Figure 5 shows the block diagram of a GPS receiver with a roll angle detection module. In a receiver, the power of the \(i\)-th satellite's signal is centralized in the correlation results \(lp(n)\) and \(Qp(n)\) after processed by RF (Radio Frequency) front-end and baseband processing [22]. The correlation results of the \(i\)-th satellite's signals can be represented by:  
\[I_{p}(n)=a^{(i)}(\alpha,\beta)R^{(i)}(\delta)D^{(i)}(n)\text{sinc}\bigg{(} \frac{1}{2}\omega_{e}T_{carr}\bigg{)}\cos\bigg{(}\omega_{e}\bigg{(}t_{0}+\frac {T_{carr}}{2}\bigg{)}+\theta_{e}\bigg{)}+n_{I} \tag{14}\]  
\[Q_{p}(n)=a^{(i)}(\alpha,\beta)R^{(i)}(\delta)D^{(i)}(n)\text{sinc}\bigg{(} \frac{1}{2}\omega_{e}T_{carr}\bigg{)}\sin\bigg{(}\omega_{e}\bigg{(}t_{0}+\frac {T_{carr}}{2}\bigg{)}+\theta_{e}\bigg{)}+n_{Q} \tag{15}\]  
where \(a^{(i)}(\alpha,\beta)\) is the amplitude of receiver signals received by a rotating antenna at the vehicle attitude of \((\alpha,\beta)\), \(\omega_{e}\) is the angular frequency difference, \(\theta_{e}\) is the phase difference between local carrier signal produced by carrier NCO (Numerically Controlled Oscillator) and the input satellite signal, \(t_{0}\) is the initial time point of the coherent integration, and \(T_{carr}\) is the integration time in the carrier tracking loop of receiver, \(n_{I}\) and \(n_{Q}\) are the noises of each path. \(R^{(i)}(\delta)\) is the self-correlation function of C/A codes. When the code errors do not exist, \(R^{(i)}(\delta)\) reaches the highest value, where \(R^{(i)}(0)=1\).  
The magnitude extraction equation of roll modulation signals can be expressed as follows when the noises \(n_{I}\) and \(n_{Q}\) are ignored:  
\[\begin{array}{rl}\left|\hat{r}_{p}(n)\right|=\sqrt{I_{p}(n)^{2}+Q_{p}(n)^{2}}=& \left|a^{(i)}(\alpha,\beta)R^{(i)}(\delta)D^{(i)}(n)\text{sinc}(f_{e}T_{carr})e ^{\left|2\pi f_{e}(t_{0}+\frac{T_{carr}}{2})+\theta_{e}\right|}\right|\\ &\hskip 113.811024 pt=a^{(i)}(\alpha,\beta)R^{(i)}(\delta)|\text{sinc}(f_{e}T_{ carr})|\end{array} \tag{16}\]  
where \(f_{e}=\omega_{e}/2\pi\), sinc(\(f_{e}T_{carr}\)) is the correlation loss caused by frequency error, data code \(D(n)=\pm 1\) is omitted in the absolute value. Integration time is usually one millisecond in the receiver, \(I_{p}\) and \(Q_{p}\) are also called the one-millisecond accumulation values.  
Equation (16) cannot represent the relationship between the roll modulation signals and the relative roll angle \(\alpha_{i}\) directly. To better understand their connections, \(\left|\hat{r}_{p}(n)\right|\) can be rewritten as:  
\[\left|\hat{r}_{p}(n)\right|=A_{r}\cos(\omega_{r}t+\alpha_{r})+D \tag{17}\]  
where \(\omega_{r}=\dot{\alpha}_{i}\), is the angular frequency of antenna rotation; \(\omega_{r}t+\alpha_{r}=\alpha_{i}\) is the relative roll angle; \(A_{r}=\left(a_{\text{max}}^{(i)}-a_{\text{min}}^{(i)}\right)R^{(i)}(\delta)| \text{sinc}(f_{e}T_{carr})|/2\) is the amplitude of the roll modulation signals, and \(D=\int_{0}^{2\pi}a^{(i)}(\alpha,\beta)R^{(i)}(\delta)|\text{sinc}(f_{e}T_{carr })|d\alpha/2\pi\) is the direct current (DC) component.  
According to (10), it is obvious that the code error \(\delta\) and the frequency error \(f_{e}\) can influence the amplitude of roll modulation signals directly. If the carrier and code tracking loops in GPS receiver are stable during rotation, \(R^{(i)}(\delta)\) and \(\mid\text{sinc}(f_{e}T_{carr})\mid\) can be considered as constants, and the variation of roll modulation signals only depends on the amplitude \(a^{(i)}(\alpha,\beta)\), which is directly related to the power of receiver signals.