# Sample-Wise Aiding in GPS/INS Ultra-Tight Integration for High-Dynamic, High-Precision Tracking
## 4 Test Results and Discussion
### Aiding Effects on Carrier Phase Errors with no Thermal Noise in the Signal  
To exclude the effects of thermal noises and cross-correlation interferences, a single-SV scenario is necessary to achieve high C/N\({}_{0}\) (carrier-to-noise ratio) and, thus, high precision dedicated for the dynamic stress error test [30]. Although no noise is added to the output signal of the digital IF signal simulator, the C/N\({}_{0}\) estimated by the receiver is around 128 dB-Hz owing to both the inevitable quantization noise and the limited range of the wideband-narrowband power ratio method (PRM) [31]. This is near the best PRM-estimated C/N\({}_{0}\) achieved by our test platform so far. The carrier phase tracking errors in a 5 g (\(\approx\)50 m/s\({}^{2}\)) dynamic condition are shown in Figure 8.  
In the first 30 s, the receiver is in unaided state as the epoch time has not been solved. The top plot in Figure 8 shows that the carrier phase error is greatly reduced after external Doppler aiding being turned on. The bottom plot discards the first 30s data and examines the aided period. It can be seen that the carrier phase error remains a sinusoidal wave with the same angular frequency of 1 rad/s, as predicted by Section 2. The tests have been conducted using six sets of signals with the dynamics of 5 g, 20 g, 50 g, 100 g, 200 g, and 500 g, respectively. The carrier phase tracking errors using traditional Doppler aiding are shown in Figure 9a. Figure 9b calculates their amplitudes by combining Equations (4), (10) and (12), and the experimental results shown in the red triangle symbols are well in accordance with the theoretical expectation in the blue line. Figure 8: Carrier phase error using traditional aiding with 5 g dynamics and no thermal noise. (**a**) From unaided (0–30s) to aided (30–200s) status; (**b**) Aided period (30–200s). Table 2 summarizes the test results of the error amplitudes for the unaided third-order PLL and the aided second-order PLL and their ratios. The theoretical calculation of this ratio follows:  
\[r=\frac{|H_{e3}(j\omega)|}{C_{a}\left(\omega T_{a}\right)|H_{e2}(j\omega)|}\approx 2 30.04 \tag{18}\]  
The test results demonstrate the correctness of the equations of carrier tracking errors in Section 2. The small discrepancies between the test results and the theoretical value of the ratio are due to the \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Dynamics** **(Acceleration)**} & **Error Amplitude (m) for** & **Error Amplitude (m) for** & **Ratio of Unaided to** \\  & **Unaided 3 rd-Order PLL,** & **Aided 2 nd-Order PLL,** & **Aided Error** \\  & **B = 15 Hz** & **B = 15 Hz** & **Amplitudes** \\ \hline
5 g & 0.00345 & \(1.50\times 10^{-5}\) & 230.00 \\
20 g & 0.0138 & \(6.03\times 10^{-5}\) & 228.86 \\
50 g & 0.0345 & \(1.49\times 10^{-4}\) & 231.54 \\
100 g & N/A & \(3.00\times 10^{-4}\) & N/A \\
200 g & N/A & \(6.00\times 10^{-4}\) & N/A \\
500 g & N/A & \(1.50\times 10^{-3}\) & N/A \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Test results of carrier phase errors using traditional Doppler aiding. Figure 10: Carrier phase errors using sample-wise aiding with different dynamics and no thermal noise. Figure 9: Carrier phase errors using traditional aiding with different dynamics and no thermal noise. (**a**) Test results; and (**b**) Comparison with theoretical error amplitude. impacts of the time alignment errors and the quantization errors of the test platform, as well as the round-off errors of the error amplitudes. Corresponding to Figure 9a, Figure 10 shows the tracking results using the sample-wise aided PLL with 5 g, 50 g, and 500 g dynamics. It is seen that the dynamic errors have been further eliminated. It should be noted that the tracking error achieved by the test platform even in a static user scenario is at the 0.8 \(\times\) 10\({}^{-6}\) m level in terms of the standard deviation of the carrier phase error.