Analysis of the influence of sample rates on the Allan variance

: pp. 53-61
Lviv Polytechnic National University

The paper considers the problem of interpreting the Allan deviation plot for signals
from sensors polled more frequently than data are refreshed. The Allan variance is a standard
tool for analysis of noise terms inevitably present in signals of inertial sensors. There exists a
well-defined algorithm for its calculation both for time domain and frequency domain. Having
calculated the Allan variance as a function of time (or frequency) one fetches its square root,
called Allan deviation, and builds its plot in a logarithmic format. Each region of the Allan
deviation plot characterizes a specific noise kind (white noise, flicker noise, random walk, etc).
The plot is expected to have a well recognizable, predefined shape. However, in practice it may
be that a plot obtained for real time series does not follow its textbook pattern. In this case it is
unobvious how to interpret the plot and whether it is applicable or not. We observed quite
untypical Allan deviation plots for signals of a magnetometer sampled too frequently, which
suggested that the sample rate can be responsible for the unusual shape of the plot. Our work
is aimed at analyzing the influence of the sample rate on the Allan deviation plot and
evaluating the applicability of such a plot obtained for signals sampled too frequently. We
reproduced experimental results by simulation and detected that the sample rate for
synthesized white noise signals impacts the shape of the Allan deviation plot. The same idea
was corroborated by filtering out repeated measurement points from experimentally obtained
magnetometer signals. The simulation results are backed up by analytical calculations.
Therefore, all the applied approaches such as simulation, filtering reading of a real sensor and
analytical considerations confirmed that the shape of the Allan deviation plot depends on the
signal sample rate. Moreover, we have shown that the Allan deviation plot built under these
conditions is completely inapplicable unless all repeated measurement points are filtered out.
Our analytical explanation of this fact is confirmed by a set of experiments. We provide a
detailed description of a procedure for evaluation of the applicability of the Allan deviation
plot using a magnetometer.

1. Lee, J., Mellifont, R., & Burkett, B. (2010). The use of a single inertial sensor to identify stride, step, and stance durations of running gait. Journal of Science and Medicine in Sport, 13(2), 270–273. doi:10.1016/j.jsams.2009.01.005

2. Tang, Z., Sekine, M., Tamura, T., Tanaka, N., Yoshida, M., & Chen, W. (2015). Measurement and Estimation of 3D Orientation using Magnetic and Inertial Sensors. Advanced Biomedical Engineering, 4, 135–143. doi:10.14326/abe.4.135

3. Mao, A., Ma, X., He, Y., & Luo, J. (2017). Highly Portable, Sensor-Based System for Human Fall Monitoring. Sensors, 17(9), 2096. doi:10.3390/s17092096

4. Marina, H., Pereda, F., Giron-Sierra, J., & Espinosa, F. (2012). UAV attitude estimation using unscented Kalman filter and TRIAD. IEEE Transactions on Industrial Electronics, 59(11), 4465–4474. Retrieved from

5. Faragher, R. (2012). Understanding the Basis of the Kalman Filter Via a Simple and Intuitive Derivation. IEEE Signal Processing Magazine, 29(5), 128–132. doi: 10.1109/MSP.2012.2203621.

6. Mahony, R., Hamel, T., & Pimlin, J.-M. (2008). Nonlinear complementary filters on the special orthogonal group. IEEE Transactions on Automatic Control, 53(5), 1203–1218.

7. El-Sheimy, N., Hou, H., & Niu, X. (2008). Analysis and Modeling of Inertial Sensors Using Allan Variance. IEEE Transactions on Instrumentation and Measurement, 57(1), 140–149. doi: 10.1109/TIM.2007.9086354.

8. Vukmirica, V., Trajkovski, I., & Asanović, N. (2010). Two Methods for the Determination of Inertial Sensor Parameters. Scientific Technical Review, 60(3–4), 27–33.

9. U.S. Army Research, Development and Engineering Center. (2015). Allan variance calculation for nonuniformly spaced input data (Publication No. ARWSE-TR-14011). Retrieved from

10. Friederichs, T. (2019). Analysis of geodetic time series using Allan variances. Retrieved from

11. University of Cambridge. Computer Laboratory. (2007). An introduction to inertial navigation (Publication No. UCAM-CL-TR-696). Retrieved from

12. Riley, W. J. (2008). Handbook of frequency stability analysis. Washington: U. S. Government Printing Office.

13. Barrett J. M. (2014). Analyzing and modeling low-cost MEMS IMUs for use in an inertial navigation system (Master's thesis). Retrieved from