From: Michael_RW on

Averaging several time-correlated observation vectors is a fundamental way of eliminating interfering noise. In your case, averaging 128 realizations (each with 1,300 samples) should drop the noise-level considerably.

With respect to your Kalman filter work, how did you apply Kleder's KALMANF Matlab function script-file?

The example included by Kleder's script-file considers a 1-dimensional state-vector or simple scalar case. This seems to be a solid way to introduce Kalman filter principals; see the article by Roger M. du Plessis, "Poor Man's Explanation of Kalman Filtering or How I Stopped Worrying and Learned to Love Matrix Inversion."

With respect to your case, the state transition matrix would not be unity or 1, this is because the echo-signal received by the sensor is not constant. It would be some time-dependent discrete function characterizing this behavior; I do not know what this would look like.

You should also note that there are important assumptions with respect to application of Kalman filtering. In basic cases, likely yours, the underlying noise processes (both process and measurement) are Gaussian in nature and the state-transition and measurement matrices are linear. With both of these assumptions met and having correct state transition and measurement matrices in place, the Kalman filter will operate as the optimal minimum mean-square estimator.

In more advanced cases, the associated noise processes are not Gaussian and system dynamics are not linear. Non-linear dynamics require linearization, thus the presence of the extended Kalman filter.