## 2 Power Spectral Density of Selected Disturbances  
The process of quantization of the signal causes a measurement error that cannot be removed. Assuming an even distribution of the probability of the occurrence of a value in the quantization interval \(q\), the dependence on the covariance of the quantization noise is obtained from (3) [22]:  
\[\sigma^{2}=\frac{q^{2}}{12} \tag{3}\]  
Figure 4: Simplified block diagram of an integrated triaxial MEMS accelerometer, gyroscope, and magnetometer. The spectral density is determined by the relationship (4) [22; 39; 40]:  
\[S(f)=\left\{ \begin{array}{c}\frac{4Q^{2}}{\tau_{0}}\sin^{2}(\pi f\tau_{0})\\ (2\pi f)^{2}\tau_{0}Q^{2}\quad dla\ f<\frac{1}{2\tau_{0}}\end{array} \right. \tag{4}\]  
where \(\tau_{0}\) is the sampling period and \(Q\) is the quantization noise coefficient (factor). Angle random walk (ARW) gyroscope white noise or velocity random walk (VRW) accelerometer white noise can be represented as power spectral density:  
\[S(f)=N^{2} \tag{5}\]  
where \(N\) is the noise coefficient \(\left({}^{\circ}/\sqrt{s}\right)\) from the gyroscope and \(\left(\left(\frac{m}{s}\right)/\sqrt{s}\right)\) from the accelerometer. Power spectral density (PSD) in the case of drift of inertial sensors is determined by the following formula:  
\[S(f)=\left\{ \begin{array}{c}\left(\frac{B}{2\pi}\right)^{2}\frac{1}{f},\ f\leq f_{c}\\ 0,\ f>f_{c}\end{array} \right. \tag{6}\]  
where \(B\) is the drift coefficient and \(f_{c}\) is the crossover frequency. The power spectral density of the Brownian noise (red noise) of the gyroscope (rate random walk--RRW) is represented by the formula  
\[S(f)=\left(\frac{K}{2\pi}\right)^{2}\frac{1}{f^{2}} \tag{7}\]  
where \(K\) is the RRW/RW noise coefficient. The PSD of the accelerometer's Brownian noise (random walk--RW) is similarly described. The power spectral density of the trend function, which expresses a uniform change in the signal value, is determined from the formula  
\[S(f)=\frac{R^{2}}{\left(2\pi f\right)^{3}} \tag{8}\]  
where \(R\) is the trend coefficient. Harmonic interference may occur when equipment is powered from the 115 VAC/400 Hz onboard network or during laboratory testing of equipment powered by 230 VAC/50 Hz. The power spectral density of this type of interference is expressed by the following formula:  
\[S(f)=\frac{1}{2}A[\delta(f-f_{0})+\delta(f+f_{0})] \tag{9}\]  
where \(A\)--the amplitude of the disturbance, \(f_{0}\)--the frequency of the jamming signal, and \(\delta\)--the Dirac delta. The hypothetical course of the PSD characteristic for the inertial sensor is presented in Figure 5[23]. In practice, in order to determine this characteristic, it is necessary to register a set of test data in static conditions, including at least \(10^{6}\) samples [41]. The test data should be corrected for the initial offset value [23]. Figure 6 shows time courses of roll rate (\(p\)), longitudinal acceleration (\(a_{x}\)), and sensor module temperature (\(T\)) obtained from the PCIM-01 IMU module [42] cooperating with the PRP-W2 recorder (Figure 7). The signal sampling frequency was 1 kHz. Figure 8 shows the power spectral density for two data sets of roll rate selected from one test, which lasted about an hour in total. For comparative purposes, the PSD of the signal was generated for two cases. First, PSD was generated for \(10^{6}\) samples recorded just after switching on the device. In this case, the temperature rose from 27.1 \({}^{\circ}\)C to 36 \({}^{\circ}\)C. A second series of \(10^{6}\) samples began to be recorded 2490 s after switching on, when the temperature of the IMU assembly (Figure 6)reached a relatively constant temperature (in practice it changed between 2490 and 3490 s in the range of 37.3-37.5 \({}^{\circ}\)C). Figure 6 shows the temperature course of the sensor during the entire experiment, as well as the time course of the angular velocity \(p\) around the \(x\)-axis. The recorded angular velocity oscillated during the static test in the range of \(-\)0.4-0.4 deg/s.