# Validation of Acoustic Emission Waveform Entropy as a Damage Identification Feature
## Appendix A  
Entropy of a variable is the average number of bits needed to represent the variable. According to [PERSON]'s theory, the average of variable {x\({}_{1}\), x\({}_{2}\), x\({}_{3}\),..., x\({}_{n}\)}, is represented using Equation (A1).  
\[Mean=g^{-1}{\sum_{i=1}^{n}}[w(x_{i})g(x_{i})]\] (A1)  
where:\(w(x_{i})\) represent the weight of the variable, such that \(\sum\limits_{i=1}^{n}w(x_{i})=1\);  
\(g(x)\) is a strictly continuous monotonic function and \(g^{-1}(x)\) is its inverse function. For arithmetic mean \(g(x)=x\) and \(g^{-1}(x)=x\).  
When [PERSON]'s theory is used to determine the average or the arithmetic mean number of bit needed to represent a variable, Equation (A1) takes the form as shown in Equation (A2).  
\[Mean=g^{-1}\sum\limits_{k=1}^{m}\left[P(x_{k})g(\log(\frac{1}{P(x_{k})})\right]\] (A2)  
where:  
\(P(x_{k})\) represents the discrete probability distribution of the voltage values with the k\({}^{\text{th}}\) number of bin; \(m\) represents the total number of bins;  
\(\log(\frac{1}{P(x_{k})})\) represents the number of bits need to represent the variable \(x_{k}\).  
Since, \(g(x)\) (i.e., \(g(-\log P(x_{k}))\) in Equation (A2)) is a strictly continuous monotonic function, [PERSON] proved that it has two solutions. The first solution is \(g(x)=ax\) (i.e., \(g(-\log P(x_{k}))=-a\log P(x_{k})\) for Equation (A2)), with \(a\
eq 0\). Substituting the first solution in Equation (A2) yields [PERSON]'s entropy. [PERSON]'s entropy is the average number of bits required to represent a variable; in other words, it is the arithmetic mean of the total number of bits. The second solution is \(g_{a}(x)=2^{(a-1)x}\) (or \(g_{a}(-\log P(x_{k}))=2^{(a-1)-\log P(x_{k})}\) for Equation (A2). Substituting the second solution in Equation (4) results in [PERSON]'s entropy. [PERSON]'s entropy is considered to be an exponential mean of the total number of bits. Both [PERSON]'s and [PERSON]'s entropy can be defined with five sets of postulates. The first four postulates are common between both the entropy measures. However, they differ in their fifth postulates as a result of the choice of \(g(x)\). In contrast to [PERSON]'s entropy, [PERSON]'s entropy has an additional term 'a' in the equation. The term 'a' in [PERSON]'s entropy appears as a power of the discrete probability distribution. As a result, the choice of 'a' makes [PERSON]'s entropy unique.  
In order to investigate the performance of [PERSON]'s and quadratic [PERSON]'s entropy in terms of their sensitivity to disorderness, a decaying sinusoidal waveform (shown in Figure A1) and its corresponding signal-to-noise ratio (SNR) of 10, 20, 30 and 40 were analysed. The waveform in Figure A1 is considered to be highly ordered compared to its corresponding SNR. Therefore, the difference in the respective entropy measurement techniques (e.g., [PERSON]'s and quadratic [PERSON]'s) between the waveform in Figure A1 and its corresponding SNR should provide a true measure of performance against capturing the disorderness. Figure A2 shows the delta entropy against SNR for both [PERSON]'s and quadratic [PERSON]'s entropy. The delta entropy corresponds to the difference in the entropy between waveform A1 and its corresponding SNR.  
It can be observed in this graph that for each of the SNR, the delta entropy for quadratic [PERSON]'s entropy is noticeably higher than that of [PERSON]'s entropy. In other words, the spread in data between the ordered and disordered waveform will be much higher for quadratic [PERSON]'s entropy compared to that of [PERSON]'s entropy. Therefore, it can be concluded that quadratic [PERSON]'s entropy is much more efficient in capturing the disorderness, which is an indication of disorderness.