# Sample-Wise Aiding in GPS/INS Ultra-Tight Integration for High-Dynamic, High-Precision Tracking
## 2 Performance of Traditional Doppler Aiding
Ideally, if the INS-estimated Doppler is exactly the derivative of the input phase, there will be \(\theta_{ia}\left(t\right)=0\) and \(\theta_{\varepsilon}\left(s\right)=H\left(s\right)n\left(s\right)\), which means that the LOS dynamics can be completely eliminated by Doppler aiding. As a matter of fact, the signal frequency shift is also contributed by the receiver clock drift, the satellite clock drift, and even induced by the ionosphere/troposphere propagation, (which are beside the point of the paper). In this case, the design of the loop filter only needs to account for the phase noise instead of tracking the signal dynamics. The analog domain analysis, however,  
Figure 1: Linear phase model of Doppler-aided PLL. cannot cover all aspects of a digital realization. (1) In reality, a PLL in a modern receiver is implemented in its digital version--the DPLL where, traditionally, the aiding Doppler value remains constant over an external data update interval \(T_{a}\) (which may be equal to, smaller than, or larger than the pre-detection integration time \(T\); nevertheless most reported systems keep \(T_{a}\geqslant T\) as considered in [19]); whereas the Doppler variation within this interval cannot be ignored in high dynamic situations; and (2) the carrier NCO control word actually comes from the Doppler estimate of the previous interval rather than the current interval. Taking these two factors into account, we replace the analog NCO integrator in Figure 1 with a digital accumulator operated at the sampling rate \(R_{s}\), as shown in Figure 2. Note that another input of the adder before the integrator in Figure 1, namely the loop filter output, has been omitted in Figure 2, being dedicated to investigating the feedback phase of the aiding data. Assuming that the aiding Doppler is provided at an update rate \(R_{a}=1/T_{a}\), and the Doppler data update interval is the integral multiple of the digital signal sample interval \(T_{s}\): \(m=\frac{T_{a}}{T_{s}}=\frac{R_{a}}{R_{a}}\). At the \(n\)th sample time \(nT_{s}=(km+l)\,T_{s}\)\(=kT_{a}+lT_{s}\), \(l=0,1,\cdots,m-1\), the feedback phase can be expressed as \(\theta_{a}(nT_{s})=\sum\limits_{p=0}^{k-1}f_{a}(pT_{a})\cdot T_{a}+f_{a}(kT_{ a})\cdot lT_{s}\). The equivalent input phase of the loop at each aiding Doppler update epoch (\(n=km\), \(l=0\)) is:  
\[\theta_{\varepsilon}\left(s\right)=H_{\varepsilon}\left(s\right)\theta_{i} \left(s\right)-H\left(s\right)n\left(s\right) \tag{1}\]0  
Assuming that the INS can provide exact Doppler estimates synchronized with the LOS signal Doppler, we have \(f_{a}\left(pT_{a}\right)=A\sin\left(\omega pT_{a}\right)\) for the sinusoidal input carrier phase in Equation (3). Substituting them into Equation (8) and neglecting the constant item (because it contributes nothing to the steady state tracking error), we obtain:  
\[\theta_{\varepsilon}\left(s\right)=H_{\varepsilon}\left(s\right)\theta_{i} \left(s\right)-H\left(s\right)n\left(s\right) \tag{1}\]1 \[\theta_{\varepsilon}\left(s\right)=H_{\varepsilon}\left(s\right)\theta_{i} \left(s\right)-H\left(s\right)n\left(s\right) \tag{1}\]2 \[\theta_{\varepsilon}\left(s\right)=H_{\varepsilon}\left(s\right)\theta_{i} \left(s\right)-H\left(s\right)n\left(s\right) \tag{1}\]3  
\[\theta_{\varepsilon}\left(s\right)=H_{\varepsilon}\left(s\right)\theta_{i} \left(s\right)-H\left(s\right)n\left(s\right) \tag{1}\]4 \[\theta_{\varepsilon}\left(s\right)=H_{\varepsilon}\left(s\right)\theta_{i} \left(s\right)-H\left(s\right)n\left(s\right) \tag{1}\]5  
Consequently, the phase error of the aided loop will follow the sinusoidal shape with the same angular frequency of the input signal phase as shown in Equation (4), yet with an amplitude of \(C_{a}\left(\omega T_{a}\right)\) times that of the unaided PLL.