Separation of waves governed by the one-dimensional wave equation—a stochastic systems approach

In various applications it is of interest to extract the wave traveling in a certain direction of a medium. As waves that propagate in different directions generally overlap, it is not possible to directly measure the desired wave. This paper presents techniques for separation of waves governed by the one-dimensional wave equation in the presence of measurement noise. The derived estimators are based on stochastic modeling of the waves and measurement noise, state space representations and optimal state estimation. The treatment of measurement noise is of crucial importance since it fundamentally limits the performance of wave separation. The derived estimators are compared and evaluated through analytical results and computer simulations. The efficiency of the wave estimation does not only depend on the magnitudes of the waves and the noise, but also on the structure of the estimator, the number of sensors and the spacing between sensors. It is shown that the wave traveling in a certain direction can be estimated with high accuracy without amplification of the measurement noise.