# Validation of Acoustic Emission Waveform Entropy as a Damage Identification Feature
## 3 Materials and Methods  
### Disorderness as an AE Feature  
The disorderness of a waveform is a measure of how randomly distributed its discrete voltage values are. There are several theorems in the literature to successfully measure the disorderness of a waveform. The initial theorem was proposed by [PERSON] in 1928 [PERSON] (1928), which was later known as'max entropy'. [PERSON] in 1948 [PERSON] (1948) used a linear weighing parameter in [PERSON]'s disorderness measure. [PERSON]'s theorem of disorderness measure was later known as '[PERSON]'s entropy'. [PERSON] in 1960 [PERSON] (1960) proposed a more flexible form of disorderness measure by introducing an exponential weighing parameter in the computation. [PERSON]'s theorem was termed as '[PERSON]'s entropy'. Given a waveform with a discrete amplitude distribution {X\({}_{1}\), X\({}_{2}\), X\({}_{3}\),..., X\({}_{n}\)}, [PERSON]'s entropy \(H_{a}(x)\) is calculated using Equation (1):  
\[H_{a}(x)=\frac{1}{1-a}\log\left(\sum_{k=1}^{m}\left(P(x_{k})\right)^{a}\right) \tag{1}\]  
The term \(P(x_{k})\) in Equation (1) is the probability of mass with the K\({}^{\text{th}}\) number of bin. Figure 3 shows a typical probability distribution of the voltage values against the voltage distribution range (i.e., the range of voltage over which the probability distribution is calculated) of an AE waveform. The term 'a' in Equation (1) is called the order of the entropy. With a higher value of 'a', more weight is provided to the larger probability of mass [PERSON] (1948); [PERSON] (1948). When 'a' is 1, the weight provided to all the probabilities of mass are the same. Depending on the choice of 'a', [PERSON]'s entropy generalizes other notions of disorderness measures such as 'Hartley's entropy', 'Shannon's Entropy', 'collision entropy' (quadratic entropy) and the'min Entropy'.  
Figure 1: Working principle of the state-of-the-art acoustic emission (AE) monitoring technique.  
Figure 2: Traditional AE features.  
Shannon's entropy is a widely used measure of disorderness. However, in this study, 'quadratic Renyi's entropy' was chosen as a measure of disorderness in the AE waveform by using the value of 'a' as 2. There are two reasons for choosing the 'quadratic Renyi's entropy'. Firstly, as it is evident from Figure 3 that the probability of mass for the voltage values close to zero are higher compared to the ones that are further away from zero. Therefore, it is reasonable to provide more weight to the dominant probability of mass for the voltage values, as they have occurred with higher frequency and can disclose the true nature of disorderness in the waveform. Choosing a very high 'a' would significantly reduce the contribution from a lower probability of mass. Therefore, a = 2 provides a trade-off such that the computation provides more weight to the larger probability of mass and also considers the lower probability of mass, as 'a' is close to 1 (i.e., close to [PERSON]'s entropy). The first advantage of quadratic Renyi's entropy over Shannon's entropy as a measure of disorderness is provided in Appendix A. Secondly, quadratic Renyi's entropy is easy to calculate. In fact, it can be estimated directly from the discrete voltage distribution if 'Prazen window' [35] and Kernel is implemented in the AE data acquisition system. The bin width used for the probability distribution in this research is 0.00305 mV, which is equal to the resolution of the AE data acquisition system. The 'quadratic Renyi's entropy' of a transient AE waveform will be referred to as AE entropy in this paper. In order to calculate AE entropy, the transient AE waveform was converted into a spreadsheet containing the discrete voltage values. A probability distribution of the discrete voltage values was then calculated. Using the probability distribution, the calculation of AE entropy was performed by using Equation (2).  
\[H_{2}(x)=-\text{log}\!\left(\sum_{k=1}^{n}\left(P(x_{k})\right)^{2}\right) \tag{2}\]