## 3 Allan Variance  
FOGs sense the angle velocity. Their output signal is the angle velocity \(\omega(t)\). Allan variance [26] has commonly been used to evaluate the stability of FOG signal \(\omega(t)\). The standard concept of the Allan variance is:  
\[\sigma_{\omega}^{2}(\tau)=\frac{1}{2}\left\langle(\overline{\omega}(t+\tau)- \overline{\omega}(t))^{2}\right\rangle\!, \tag{1}\]  
where \(\tau\) is the observation interval and \(\left\langle\ \right\rangle\) indicates a time averaging. The average of \(\omega(t)\) is given by  
\[\overline{\omega}(t)=\frac{1}{\tau}\!\int_{t}^{t+\tau}\omega(u)du, \tag{2}\]  
where \(u\) is the integral variable. Equation (1) is the definition of Allan variance for continuous data. If the Allan variance is to be estimated on discrete samples \(\omega\left[n\right]\) whose total number is \(N\), \(t\) is sampled as  
\[t=n\tau_{0},(n\ =\ 1,2,\cdots N), \tag{3}\]  
where \(\tau_{0}\) is the sampling interval. Consequently, the observation interval \(\tau\) is discretized is  
\[\tau=k\tau_{0},(k=1,2,\cdots\frac{N}{2}). \tag{4}\]  
For the discrete data, the Allan variance becomes  
\[\sigma_{\omega}^{2}(k)=\frac{1}{2k^{2}\tau_{0}^{2}}\frac{1}{N-2k}\overset{N-2 k-1}{\underset{n=0}{\sum}}\overset{2k-1}{\underset{n=0}{\sum}}(\overline{\omega}[n+k] -\overline{\omega}[n])^{2}). \tag{5}\]  
The average of \(\omega[n]\) is given by  
\[\overline{\omega}[n]=\frac{1}{k}\overset{n+k-1}{\underset{i=n}{\sum}}\omega[ i]. \tag{6}\]  
Therefore, according to the different observation interval \(k\), the corresponding Allan variance can be obtained. Ng [26] shows that a unique relationship existing between \(\sigma_{\omega}^{2}(\tau)\) and the power spectral density (PSD) of the intrinsic random processes. This relationship is  
\[\sigma_{\omega}^{2}(\tau)=4\!\int_{0}^{\infty}\mathrm{S}_{\Omega}(f)\frac{ \sin^{4}{(\pi f\tau)}}{{(\pi f\tau)}^{2}}du, \tag{7}\]  
where \(\mathrm{S}_{\Omega}(f)\) is the PSD of the random process \(\Omega\left(t\right)\), namely the instantaneous output rate of the gyro. Equation (7) is the focal point of the Allan variance technique. The PSD of any physically meaningful random process can be substituted in the integral and an expression for the Allan variance \(\sigma_{\omega}^{2}(\tau)\) as a function of cluster length \(\tau\) is identified [26; 27]. Equation (7) states that the Allan variance is proportional to the total power output of the random process when passed through a filter with the transfer function \(\dfrac{\sin^{4}{(\pi f\tau)}}{{(\pi f\tau)}^{2}}\). This particular transfer function is the result of the method used to create and operate on the cluster. It is seen from Equation (7) that the filter bandpass depends on \(\tau\). This suggests that different types of random processes can be examined by adjusting the filter bandwidth, namely by varying \(\tau\)[28]. Consequently, since \(\sigma_{\omega}^{2}(\tau)\) is a measurable quantity, a plot of \(\sigma_{\omega}^{2}(\tau)\) versus \(\tau\) provides a direct indication of the types of random processes that exist in the gyro data. A log-log plot of the square root of the Allan variance \(\sigma_{\omega}(\tau)-\tau\) provides a means of identifying and quantifying various noise termsthat exist in the gyro sensor data, as Figure 3 shows. Table 1 gives a representation of noise terms using Allan variance. In general, any number of the random processes discussed above can be present in the data. Thus, a typical Allan variance plot looks like the one shown in Figure 3. Different noise terms appear in different regions of \(\tau\). This allows for easy identification of the various random processes that exist in the gyro data. If it can be assumed that the existing random processes are all statistically independent, then it can be shown that the Allan variance at any given \(\tau\) is the sum of the Allan variances due to the individual random processes at the same \(\tau\)[27]. In other words,  
\[