The uncertainty in estimating the position using a noisy accelerometer increases with time.
Let us model the acceleration as a uniformly random real number . Suppose you have a random walk where . This is the velocity. Then, we would like to investigate , which is the position.
To do this, we conduct a numerical experiment. We repeat the simulation times and then plot the RMS value of the trials at each time step.
By inspection, we see that and where and are constants.
Number of trials: 100
Noise type: 0 dB/octave