Decomposition of one-dimensional waveform using iterative Gaussian diffusive filtering methods

The Gaussian smoothing method is shown to have a wide transition zone around the cut-off frequency selected to filter a given dataset. We proposed two iterative Gaussian smoothing methods to tighten the transition zone: one being approximately diffusive and the other being strictly diffusive. The first version smoothes repeatedly the remaining high-frequency parts and the second version requires an additional step to further smooth the resulting smoothed response in each of the smoothing operation. Based on the choice of the criterion for accuracy, the smoothing factor and the number of iterations are derived for an infinite data length in both methods. By contrast, for a finite-length data string, results of the interior points (sufficiently away from the two endpoints) obtained by both methods can be shown to exhibit an approximate diffusive property. The upper bound of the distance affected by the error propagation inward due to the lack of data beyond the two ends is numerically estimated. Numerical experiments also show that results of employing the iterative Gaussian smoothing method are almost the same as those obtained by the strict diffusive version, except that the error propagation distance induced by the latter is slightly deeper than that of the former. The proposed method has been successfully applied to decompose the wave formation of a number of test cases including two sets of real experimental data.