*2.4. Signal Processing with Weighted Root Mean Square*

The direct analysis of the excited guided wave field usually does not give useful information about the shape, position and size of defects occurring in an examined structure. The utility of the results can be enhanced by further signal processing. The simple and widely used approach is based on the calculation of the effective value, also called the root mean square (RMS). For damage identification and imaging, it is advantageous to use the weighted variant of RMS (WRMS) that decreases the impact of the incident wave on the results obtained. The WRMS for a discrete time domain signal *sr* = *s*(*tr*) consisted of *N* values sampled with a constant Δ*t* interval can be calculated with respect to the formula [31,50–53]:

$$\text{WRMS} = \sqrt{\frac{1}{N} \sum\_{r=1}^{N} w\_r s\_r^2} \tag{1}$$

in which *wr* is a so-called weighting factor, defined as the function of the number of consecutive sample *r* as:

$$w\_{\mathcal{I}} = r^m, \; m \ge 0, \; r = 1, 2, \dots, N \tag{2}$$

where *m* is a non-negative power of the weighting factor. For *m* = 0 the above formulas describe the simple root mean square (without weighting). According to the literature cited above, the value of *m* was usually set arbitrary, however, it can be concluded that efficient results can be obtained for the higher values of *m*, e.g., *m* = 1 (linear weighting factor) or *m* = 2 (square variant). Further increasing of power *m* does not enhance the results in a significant manner.
