## 4 Error Analysis
### Error Analysis of Tracking Loop  
Measurement error of tracking loop is from both third-order PLL and second-order FLL. For a general PLL, the error resource mainly comes from dynamic stress error and phase jitter including thermal noise \(\sigma_{\text{tPLL}}\), oscillator jitter \(\sigma_{v}\), and Allan deviation \(\sigma_{\text{A}}\) related with the frequency of input signal [29].  
For a third-order PLL, the dynamic stress error is given by  
\[\varphi_{\text{d}}^{\text{P}}=1/\omega_{\text{np}}^{3}\left(\text{d}^{3}\varphi ^{\text{P}}/\text{d}t^{3}\right) \tag{22}\]  
where, \(\varphi^{\text{P}}\) is the phase of \(B_{Z}\) and in the circumstance of the projectile rotating in the form of acceleration, \(\text{d}^{2}\varphi^{\text{P}}/\text{d}t^{2}\) is the phase acceleration of \(B_{Z}\), and \(\text{d}^{3}\varphi^{\text{P}}/\text{d}t^{3}=0\), corresponding to the analysis of steady-state error \(e_{\text{ss}}(\infty)\) in chapter 3.  
Owing to frequency of \(B_{Z}\) being no more than 400 Hz, phase error caused by \(\sigma_{v}\) and \(\sigma_{\text{A}}\) is too small so that they can be ignored. The thermal noise \(\sigma_{\text{tPLL}}\) can be expressed as  
\[\sigma_{\text{tPLL}}=\frac{180^{\circ}}{\pi}\sqrt{\frac{B_{\text{PLL}}}{C/N_{0}}(1+\frac{1}{2T_{\text{roll}}C/N_{0}})} \tag{23}\]  
where, \(B_{\text{PLL}}\) is the noise bandwidth, \(C/N_{0}\) is carrier-to-noise ratio (Hz), and \(T_{\text{roll}}\) is integration time. According to (23), effects from \(\sigma_{\text{tPLL}}\) can be reduced by decreasing noise bandwidth \(B_{\text{PLL}}\) and increasing \(C/N_{0}\) as well as \(T_{\text{roll}}\).  
Thus, tracking error of PLL depends on the phase jitter caused by thermal noise \(\sigma_{\text{tPLL}}\) to a large degree. According to the rule of thumb for tracking threshold [29], only in the condition that 3-sigma jitter is no more than one fourth of the pull-in range of PLL discriminator could PLL track input signal constantly and stably. For example, for a four-quadrant arctangent discriminator, the pull-in range of phase discriminator is \(360^{\circ}\) and the tracking threshold of PLL should satisfy  
\[3\sigma_{\text{tPLL}}\leq 90^{\circ} \tag{24}\]  
Similarly, for a pure FLL, tracking error in this designed system mainly consists of frequency jitter caused by thermal noise \(\sigma_{\text{tFLL}}\) expressed as  
\[\sigma_{\text{tFLL}}=\frac{1}{2\pi T_{\text{roll}}}\sqrt{\frac{4 FB_{\text{PLL}}}{C/N_{0}}(1+\frac{1}{T_{\text{roll}}C/N_{0}})} \tag{25}\]  
where, \(F=1\) when carrier-to-noise ratio \(C/N_{0}\) reaches a high level and \(F=2\) when the \(C/N_{0}\) is at a low level and loop tracking works near to threshold. Based on the rule of thumb for tracking threshold mentioned above, \(\sigma_{\text{tFLL}}\) should satisfy  
\[3\sigma_{\text{tFLL}}\leq\frac{1}{4T_{\text{roll}}} \tag{26}\]  
In conclusion, the measurement error caused by PLL and FLL in this designed tracking loop mainly originates from phase jitter and frequency jitter in which thermal noise \(\sigma_{\text{tPLL}}\) and \(\sigma_{\text{tFLL}}\) account for the most. In order to eliminate this kind of error, the selection of parameters of noise bandwidth and integration time should meet the requirement that 3-sigma jitter is no more than one fourth of the pull-in range of discriminator.