# Sample-Wise Aiding in GPS/INS Ultra-Tight Integration for High-Dynamic, High-Precision Tracking
## 3 Sample-Wise Doppler-Aided PLL  
### Ideal Sample-Wise Doppler Aiding  
In order to reduce the high dynamic stress error, it is effective to increase the aiding rate. Actually, the external aiding data can be interpolated to each sample point of the digital signal, and the control words of the carrier NCO for generating the local reference carrier can be updated at a rate up to the sampling rate \(R_{s}\) without major modification of the receiver architecture, as shown in Figure 4. This is the main idea of the sample-wise Doppler aiding method proposed in this section. Compared with traditionally-aided DPLL in Figure 2, the only additional cost of sample-wisely aided DPLL is the frequency interpolation unit before the NCO phase accumulator.  
Figure 3: Ratio of phase error amplitudes of aided loop to unaided loop.  
In Figure 4 the control word of the carrier NCO is \(f_{sa}\left(0:T_{s}:T_{p}\right)=\text{interp1}\left[f_{a}(0:T_{a}:T_{p})\right]\), where interp1[\(\cdot\) ] is the one-dimensional data interpolation function and \(T_{p}\) is the time length of the data. In contrast to the traditionally aided DPLL, the equivalent input carrier phase in a sample-wisely Doppler aided DPLL becomes:  
\[\theta_{ia}(nT_{s})=\theta_{i}(nT_{s})-\theta_{a}(nT_{s})=\theta_{i}(nT_{s}) -\sum_{q=0}^{n-1}f_{sa}(qT_{s})\cdot T_{s} \tag{13}\]  
If there are still no aiding Doppler error and no time alignment error, ignoring the interpolation error (which will be investigated in the next subsection), we can substitute \(f_{sa}\left(qT_{s}\right)=A\sin\left(qT_{s}\right)\) and \(\theta_{i}(t)=-\frac{A}{\omega}\cos\left(\omega t\right)\) into Equation (13) to obtain:  
\[\theta_{ia}\left(nT_{s}\right)\approx-\frac{A}{\omega}C_{a}\left(\omega T_{s} \right)\cos\left[\omega nt_{s}+\phi_{a}\left(\omega T_{s}\right)\right]+\frac {A}{\omega}C_{a}\left(\omega T_{s}\right)\cos\left[\phi_{a}\left(\omega T_{s }\right)\right] \tag{14}\]  
It is easy to meet that:  
\[C_{a}\left(\omega T_{s}\right)\approx\frac{1}{2}\omega T_{s} \tag{15}\]  
Thereby the equivalent phase error transfer functions of the sample-wisely aided loop is:  
\[H_{e-sa}\left(\omega\right)=H_{e}\left(\omega\right)C_{a}\left(\omega T_{s} \right)\exp\left[\phi_{a}\left(\omega T_{s}\right)\right] \tag{16}\]  
The equations indicate that the aiding performance can be significantly improved by increasing the aiding rate from \(R_{a}\) to \(R_{s}\). Smaller dynamic stress error can be achieved with a higher sampling frequency.  
As stated in [14], a second-order PLL is superior to a third-order PLL when aided. Taking typical third-order and second-order PLL designs in [2] as examples, Figure 5 plots the amplitudes of the phase error transfer functions of the unaided second-order PLL, the unaided third-order PLL, the traditionally-aided second-order PLL, and the sample-wisely-aided second-order PLL, respectively, with the PLL bandwidth \(B_{\text{PLL}}=15\) Hz, the Doppler aiding rate \(R_{a}=1/T_{a}=1\) KHz, and the signal sampling rate \(R_{s}=1/T_{s}=10\) MHz.  
Figure 5 shows that:  
1. When the frequency of the LOS dynamic vibration goes much higher than the natural frequency of the loop, the unaided carrier tracking loop has an amplitude magnification of near 1 with hardly any suppression on the dynamic stress error.
2. Except for the frequency range of 6.7 < \(\omega\) < 28.6 (rad/s), the unaided third-order PLL shows better dynamic performance than the unaided second-order PLL.
3. For the dynamic vibrations with \(\omega>2331\) rad/s, the traditional Doppler aiding cannot improve tracking performance efficiently.
4. Since \(T_{s}\) is normally several orders of magnitude smaller than \(T_{a}\), the phase error can be significantly reduced by sample-wise Doppler aiding.  
Figure 4: Feedback phase of sample-wisely aided DPLL.