## 7 Models and Simulations  
In order to testify the proposal algorithm, two sets of simulation data are created. The model of the simulation data is shown as follows:  
\[x[n]=\sigma[n]f[n], \tag{23}\]  
where \(f[n]\) is the white Gaussian noise [5], \(\sigma[n]\) is the standard deviation of \(f[n]\), and \(x[n]\) is simulating data of gyroscopes whose unit is \({}^{\circ}/\text{h}\). The sampling interval of the two sets of simulation data is 0.01 s. The first simulation dataset \(x_{1}[n]\) is stationary white Gaussian noise, whose standard deviation \(\sigma[n]\) is always equal to 1. The number of \(x_{1}[n]\) is \(L_{1}=6\times 10^{3}\) samples. So the time length of \(x_{1}[n]\) is 60 s. The second one \(x_{2}[n]\) is also white Gaussian noise, whose variance \(\sigma[n]\) increases with time linearly as Equation (24) shows. The number of \(x_{2}[n]\) is \(L_{2}=6\times 10^{5}\) samples. Thus the time length of \(x_{2}[n]\) is 6000 s.  
\[\sigma[n]Figure 6: Two sets of simulation data. (**a**) white Gaussian noise \(x_{1}[n]\); (**b**) Allan variance of \(x_{1}[n]\); (**c**) white Gaussian noise with “increase” \(x_{2}[n]\); (**d**) Allan variance of \(x_{2}[n]\). Figure 7: The classical DAVAR and the improved fast DAVAR. (**a**) DAVAR of white Gaussian noise \(x_{1}[n]\); (**b**) The improved fast DAVAR of \(x_{1}[n]\); (**c**) DAVAR of white noise with “increase” \(x_{2}[n]\); (**d**) The improved fast DAVAR of \(x_{2}[n]\). With a rectangular window of \(L=1000\) samples and a step whose width is 30 samples, the analysis results of \(x_{1}[n]\) are obtained. As can be seen in Figure 7a, both DAVAR methods are constant over time and have a typical white noise slope. The value of the improved fast DAVAR is similar to that of the classical DAVAR at fixed \(t\) and \(\tau\). The analysis results of \(x_{2}[n]\) are obtained by a rectangular window of number \(N_{\text{W}}=2000\) samples and a step whose width is 300 samples. In Figure 7b, the improved fast DAVAR and classical DAVAR reveal that noise increases linearly with time. The coefficient of the angle random walk (\(N\)) denotes the magnitude of the white noise [30]. According to the 2D description method of noise terms mentioned in Section 4.2, the coefficient \(N(t)\) of the Angle random walk can be acquired. Figure 8 shows the time-varying parameter \(N(t)\) of the two sets' simulation data. Figure 8a,c shows the results of the classical DAVAR while Figure 8b,d shows the results of the improved fast DAVAR. The coefficient \(N(t)\) of \(x_{1}[n]\) fluctuates around a constant while \(x_{2}[n]\) increases linearly with time. The results prove that the improved fast DAVAR is correct. Both the results are obtained by the Matlab program on an Intel(R) Core(TM) i7-3770 CPU with a 3.4 GHz clock. Table 2 shows a computational comparison of the classical DAVAR and the improved fast DAVAR. In the second column, the number of simulation data points is reported. The third column shows the length of the truncation window and the fourth column shows the step width. The fifth column represents the computational time. The last column indicates the calculation times of the Allan variance in the whole calculation process. Figure 8: Angle random walk of the simulation data. (**a,b**) Angle random walk of \(x_{1}[n]\); (**c,d**) angle random walk of \(x_{2}[n]\). When the length of the time series is short (\(6\times 10^{3}\) samples), the improved fast DAVAR saves 78.93% of computing time. When the length of the time series is long (\(6\times 10^{5}\) samples), the improved fast DAVAR costs only 27.92727 s while the classical DAVAR costs 960.422362 s. The improved fast DAVAR reduces 97.09% of computing time. The last column reports the reason. With the time series increasing, the classical DAVAR needs to calculate the Allan variance more and more times while the improved fast DAVAR just needs to compute it one time (the initial value of the DAVAR). Thus, the improved fast DAVAR could shorten the calculation time.