###### Abstract
For the minimalization of errors, both sensors should be used. Figure 2 presents a scheme with a complementary filter. Gyroscopic data, after integration, are filtered by a high-pass filter, which eliminates low-frequency errors. A low-pass filter allows the elimination of noises from accelerometers. A sum of both filtered signals is a satisfactory approximation of the real value. Time constant \(T\) is the designing parameter. For the proper choice of the \(T\) parameter, information about the sensors' stochastic parameters is essential. The presented analysis shows that quantitative analysis of the stochastic properties of accelerometers and gyroscopes is a significant element of the measurement system design. The aim of this article is to compare methods used for the analysis of stochastic properties, and to present possibilities of application in engineering. The comparative analysis of stochastic properties is performed with the PSD method, the Allan variance method (the application of these two methods is determined by appropriate standards [22; 23]), and the generalized method of wavelet moments [24; 25; 26]. The GMWM is a relatively new, and hence less known, but potentially the most interesting method. Studies in which the GMWM method is used for the analysis of stochastic errors in AHRS systems [27] and inertial navigation units [28], including smartphones [29], are scarce. They are also relatively  
Figure 2: Scheme of the complementary filter. new, as the method was made public in 2013 [24]. The GMWM method is usually compared with the AV method [30], or the GMWM alone is used to determine the stochastic errors [28]. The analyses in papers [27; 28; 29; 30] focus on sampling frequency from 10 to max 200 Hz. This article presents the stochastic properties of MEMS sensors, especially for high sampling frequency (1 kHz). Analyses of the stochastic properties of MEMS sensors with such a high frequency can rarely be found in research papers. With high sampling frequency, it is possible to realize an AHRS synthesis that enables high precision control of high-maneuverability objects. With the oversampling of sensor data, it is also possible to apply additional methods of signal processing due to which a lower frequency AHRS can be fed with more accurate and reliable input data. Application of high-sampling-frequency accelerometers and gyroscopes enables precise analysis of flight parameters, including, e.g., short-period oscillations, with frequencies as high as hundreds of Hz, of mini-UAVs [31; 32]. The properties relating to the construction of MEMS sensors, their dimensions, and their weight are a separate problem linked with the technical possibilities of their application. In the data sheets of most MEMS-class products, the potential user will find information about the sensitivity of the measurement system, noise level (mean square value, power spectral density), and natural oscillator frequencies. The parameters of the output filters are usually configurable (programmable). The programmable parameters are usually the sampling rates, gain factors, and offsets [33; 34]. In the case of some measurement systems, manufacturers also provide a calibration certificate, containing programmed values of gains and offsets and the results of tests performed for a specific unit. The mass sample contained in a MEMS accelerometer is typically a few tenths of a microgram (Figure 3). The smallest detectable capacity change is currently several attofarads, and the distances between the plates are of the order of one micrometer [35]. The operating frequency of the oscillator must be several orders of magnitude higher than the desired acceleration measurement frequency since the capacitance reading process using the demodulator requires a large number of cycles to determine the output value. The measurement noise resulting from this process is inversely proportional to the root of frequency and can be modeled as quantization noise. For static measurements near zero, the value of this noise may exceed the measured acceleration.