## 3 Allan's Method of Variance
The value of the quantization noise coefficient is obtained by reading AV over a slope of \(-1\) dB/dec at \(\tau=\sqrt{3}\) and substituting with Equation (19):  
\[\sigma_{QN}(\tau)^{2}=\frac{3Q^{2}}{\tau^{2}} \tag{19}\]  
\[\sigma(\tau)^{2}=\frac{1}{2}\langle{(\overline{y}_{i+1}-\overline{y}_{i})}^{2}\rangle \tag{11}\]0  
The value of the coefficient \(N\), related to ARW and VRW noise, is read from the section of the curve with a decrease of \(-0.5\) dB/dec for \(\tau=1\). The Allan variance of ARW and VRW noise is given by  
\[\sigma(\tau)^{2}=\frac{1}{2}\langle{(\overline{y}_{i+1}-\overline{y}_{i})}^{2}\rangle \tag{11}\]1  
where \(N\)--noise coefficient of the gyroscope. In the case of sensor drift, the Allan variance is given by the formula  
\[\sigma(\tau)^{2}=\frac{1}{2}\langle{(\overline{y}_{i+1}-\overline{y}_{i})}^{2}\rangle \tag{11}\]2  
where \(x=2\pi f\tau_{0}\) and \(Ci(x)=-\int_{x}^{\infty}\frac{\cos(t)}{t}dt\) and  
where \(B\)--drift factor (coefficient) and \(Ci\)--integral cosine. The drift coefficient B is interpreted as the smallest value on the Allan variance curve, as the function \(\sigma(\tau)\) sums up over a section with a drop of \(-1\) dB, considering both ARW/VRW noise and quantization noise [40]. Relationship (22) is often reduced to the form  
\[\sigma(\tau)^{2}=\frac{1}{2}\langle{(\overline{y}_{i+1}-\overline{y}_{i})}^{2}\rangle \tag{11}\]3  
Allan variance for gyroscope or accelerometer noise (RRW/RW) is given by Relation (24). The value of the \(K\) factor is read from the AV curve at the point \(\tau=3\) (on the curve with a slope of \(0.5\) dB/dec):  
\[\sigma(\tau)^{2}=\frac{1}{2}\langle{(\overline{y}_{i+1}-\overline{y}_{i})}^{2}\rangle \tag{11}\]4  
The trend coefficient \(R\) is determined from a section with a slope of \(1\) dB/dec at the point \(\tau=2\). The Allan variance for harmonic signals is given by  
\[\sigma(\tau)^{2}=\frac{1}{2}\langle{(\overline{y}_{i+1}-\overline{y}_{i})}^{2}\rangle \tag{11}\]5  
In the examples analyzed in the following sections, devices powered only from DC power sources were used, both during laboratory tests and during practical, in-flight tests. As a consequence, the cumulative AV function for interference is given by  
\[\sigma(\tau)^{2}=\frac{1}{2}\langle{(\overline{y}_{i+1}-\overline{y}_{i})}^{2}\rangle \tag{11}\]6  
The course of Allan variance for a hypothetical inertial sensor presented in Figure 15 was drawn on the basis of the documentation normalizing the process of testing inertial sensors [23]. Examples of AV graphs obtained for the gyroscopes of the PCIM-01 IMU [42] are shown in \[\sigma(\tau)^{2}=\frac{1}{2}\langle{(\overline{y}_{i+1}-\overline{y}_{i})}^{2}\rangle \tag{11}\]7 AV was determined for the data recorded during the test illustratedin \[\sigma(\tau)^{2}=\frac{1}{2}\langle{(\overline{y}_{i+1}-\overline{y}_{i})}^{2}\rangle \tag{11}\]8 Thus, the AV graphs (Figure 16) can be compared with the PSD graphs presented in Figure 8 (roll angular velocity \(p\)) and with Figure 9 (pitch angular velocity q and yaw angular velocity \(r\)). When interpreting the graphs, one should pay attention to the fact that the average time in the AV graphs is related to the inversely proportional relationship with the frequency value in the PSD graphs. In the case of the tested sensors, both methods failed to identify the quantization noise characterized by a \(-1\) dB/dec slope in the AV curve (on the left side of the graph) and a \(+2\) dB/dec slope in the PSD graphs (right side of the graph). ARW noise with a decrease of \(-0.5\) dB/dec in the Allan variance plot was identified for all angular velocities. The values of white noise coefficients \(N\) determined for gyroscopes using the PSD and AV methods are presented in Table 1 (in the case of this coefficient, the differences reach 25%). For the drift coefficient B, the differences in the results obtained, depending on the method, reach 42%.