# Sample-Wise Aiding in GPS/INS Ultra-Tight Integration for High-Dynamic, High-Precision Tracking
## 3 Sample-Wise Doppler-Aided PLL
### Effects of Interpolation Techniques  
In practice, we cannot obtain the exact value of the signal Doppler at each sample point by interpolation. This section explores the effects of different interpolation techniques on the dynamic stress error, still ignoring the errors in the INS estimates. Two typical interpolation approaches are investigated: (1) the linear interpolation; and (2) the cubic spline interpolation. A linear interpolation is generally the simplest that yields a good result when up-sampling. Figure 6 illustrates the block diagram of a second-order PLL sample-wisely-aided by the Doppler data linearly-interpolated from the INS data rate \(R_{a}\) to the sampling rate \(R_{s}\). Figure 6 follows the phase model of Figure 1 in the time domain and omits the input noise for simplicity. The components shown in purple are operated at the sampling rate \(R_{s}\) and often implemented in hardware for real-time processing; whereas the components shown in white and green are operated at the loop filter update rate \(R\) (usually the reciprocal of the pre-detection integration time \(T\)) and the INS data update rate \(R_{a}\) respectively. One exception is the carrier phase discriminator, which is usually composed of the carrier lookup table, the multipliers for carrier and code wipe-offs, and the integration and dump accumulator. The accumulator performs low-pass filtering and down-samples the data from \(R_{s}\) to \(R=1/T\). The cost of replacing traditional aiding with sample-wise aiding is the  
Figure 5: Amplitudes of phase error transfer functions. Figure 6: Block diagram of second-order PLL aided by linearly-interpolated Doppler. linear interpolation unit shown in the top block of Figure 6, which can also be regarded as an extra stage of NCO accumulator. In high dynamic situations, the simple linear interpolation may not meet the requirements of curve smoothness and approximation. More complicated up-sampling should be used to construct a smoother velocity and approximate complex trajectory. Among the interpolation techniques including polynomials, splines, Gaussian/low-pass processes, wavelets, _etc._, the cubic spline is very popular for its tradeoff between accuracy and computational cost. The trajectory constructed by cubic splines makes the velocity, acceleration, and jerk (VAJ) continuous, and the VAJ of the initial and final positions can be flexibly configured [28; 29]. Simulations are carried out for the two interpolation techniques from a wide range of INS data rates to three sampling rates. Figure 7 compared their effects on the ratios of phase error amplitudes of aided PLLs to unaided loops, with (a) \(\omega=1\) rad/s, (b) \(\omega=50\pi\) rad/s, respectively. The simulation results of traditional aiding are also included and compared with the theoretical prediction in Equation (10). It can be seen that:  
1. Equation (10) is accurate to quantify the effectiveness of traditional Doppler aiding, regardless of the aiding data error and the time alignment error. 2. When \(\omega T_{a}>2.33\) rad/s, any aiding technique cannot suppress the dynamic stress error efficiently and, thus, is not worth the bother. 3. When \(T_{a}\) begins to decrease from its upper threshold \(T_{u}\left(\omega\right)\approx 2.33/\omega\), sample-wise aiding with linear interpolation shows better tracking performance than traditional aiding, yet its improvement cannot catch up with the spline interpolation, until that \(T_{a}\) reaches a certain lower threshold, denoted by \(T_{l}\left(\omega,R_{s}\right)\). When \(T_{a}<T_{l}\left(\omega,R_{s}\right)\), the ratio of the phase error amplitudes of sample-wisely aided PLL to unaided PLL will be only determined by the theoretical limit \(C_{a}\left(\omega T_{s}\right)\approx 0.5\,\omega T_{s}\), independent of \(T_{a}\), and cannot be improved by employing other more advanced interpolation techniques.