## 3 Structure and Phase Error Analysis of Dual Antenna Tracking Loop
### Clock Phase Noise  
The clock phase noise in a receiver can be modeled as colored phase noise. The wave generated by the oscillator can be represented as [1]:  
\[ \begin{split} Asin\left[(\omega_{0}t+\phi_{0})+\left(\phi_{l}+ \omega_{l}t+2\pi d\frac{t^{2}}{2}\right)+x(t)\right]\end{split}  \tag{11}\]  
Here the amplitude is as assumed to be a constant. In the phase item, \((\omega_{0}t+\phi_{0})\) represents the theoretical phase, and \(\left(\phi_{l}+\omega_{l}t+2\pi d\frac{t^{2}}{2}\right)\) is the causal phase error, which is mainly induced by the temperature, magnetism, acceleration, and atmospheric pressure, _etc._, the last item \(x(t)\) represents the stochastic error. In the causal phase error, the frequency and frequency change rate have the same effect as the dynamic, which can be eliminated by the master loop assistance in the slave loop. Thus, only the effect of the stochastic phase error \(x(t)\) is considered in this paper. For the clock phase noise \(\theta_{c}(s)\) in an analogous loop system, it can be easily known from Figure 2 that the output phase noise of the master loop includes two items: one is the \(\theta_{c}(s)\) directly from the NCO, the other is the clock phase noise fed back from the loop output. Hence, the equation below can be derived:  
\[ \begin{split}\theta_{o1}(s)=\theta_{c}(s)-\frac{\theta_{o1}(s)KF _{1}(s)}{s}\end{split}  \tag{12}\]  
Here \(K=K_{d1}K_{o}\). A more concise formula could be expressed as follows:  
\[ \begin{split}\theta_{o1}(s)&=\frac{s}{s+KF_{1}(s)} \,\theta_{c}(s)\\ &=H_{e1}(s)\theta_{c}(s)\end{split}  \tag{13}\]  
This equation is the clock phase noise of an independent loop, from which the clock phase noise variance could be accumulated. Assuming the Power Spectral Density (PSD) of the input clock phase noise is \(G_{\phi}(\omega)\), the phase jitter induced by clock phase noise can be derived by accumulation, which will be in accord with Equation (10). The phase error induced by clock phase noise of the slave loop can be shown as:  
\[ \begin{split}\theta_{o2}(s)=\theta_{c}(s)-\theta_{o2}(s)\frac{ KF_{2}(s)}{s}-\theta_{o1}(s)\frac{KF_{1}(s)}{s}\end{split}  \tag{14}\]  
\[\theta_{\omega 2}(s) = \theta_{i2}(s)\,\frac{KF_{2}(s)}{s+KF_{2}(s)}+\theta_{i1}(s)\,\frac{KF _{1}(s)}{s+KF_{1}(s)}\cdot\frac{s}{s+KF_{2}(s)}\] \[= \theta_{i2}(s)H_{2}(s)+\theta_{i1}(s)H_{1}(s)H_{e2}(s)\] \[= \theta_{i2}(s)H_{2}(s)+\theta_{i1}(s)H_{1}(s)(1-H_{2}(s))\]  
In the equation above, \(H_{1}(s)\) and \(H_{2}(s)\) are the loop transfer functions of the master and slave loops respectively. \(H_{e2}(s)\) is the error transfer function of the slave loop. From the equation, an equivalent double input single output system can be obtained:  
From Equation (18) and Figure 3, besides the thermal noise from the slave antenna, the noise along with the assistance from the master loop should also be considered. The noise from the Doppler aiding is equivalent to the noise from the master antenna going through a new sub system, whose transfer function is:  
\[H_{d}(s)=H_{1}(s)\big{(}1-H_{2}(s)\big{)}=\frac{KF_{1}(s)}{s+KF_{1}(s)}\cdot \frac{s}{s+KF_{2}(s)} \tag{19}\]  
The frequency response of the sub-system are compared with those of the master and slave loops in Figure 4. In this figure, the bandwidth of the master loop \(B_{\mathrm{L1}}\) is 15 Hz, \(\omega_{\mathrm{n1}}=12.5\,rad/s\), and the bandwidth of the slave loop is \(B_{\mathrm{L2}}\) is 0.5 Hz, \(\omega_{\mathrm{n2}}=0.4167\,rad/s\). From Figures 3 and 4a, the new sub system is a cascaded by a low pass filter with a higher cutoff frequency and a high pass filter with a lower cutoff frequency. Thus, it is equivalent to a band pass filter with the cutoff frequencies as the two cutoff frequencies from the two cascaded filters. Its bandwidth approximately is equal to \(B_{\mathrm{L1}}-B_{\mathrm{L2}}\). Mathematically, the equivalent noise bandwidth of the master loop, slave loop, as well as the sub system can all be computed by Equation (4). A numerical integration method is used to get the actual bandwidth of each filter.