PSI - Issue 54

Claudia Barile et al. / Procedia Structural Integrity 54 (2024) 225–232 C. Barile et al. / Structural Integrity Procedia 00 (2023) 000–000

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3

Fig. 2. An Example Signal, its Envelope, Absolute Level and Characteristic Form

After this, the signal is segmented into k segments, each having equal length. The Yule-Walker autoregressive (AR) model of order M is fit to each segment. The coe ffi cients of the AR model are given by a i m , where m = 1 , 2 , 3 , .., M . The AR model is given in Equation (3) [Carpinteri et al. (2012)].

M  m = 1

a i

m x t − m + e i t

x n =

(3)

e i t is the Gaussian noise with zero mean and a standard deviation of σ i , which is used to create the AR model. Now the goal is to divide the signal into a deterministic part of intervals 1 , ..., k and a non-deterministic part of intervals k + 1 , ..., K (where K is the total number of segments) and to identify the segment where the ToA may appear. The approximate likelihood function that divides the deterministic and non-deterministic parts is given by

exp    −

2   

n i  j = p i    x j −

m x j − m   

2  i = 1    1 σ 2

i 2 π    2

M  m = 1

1 2 σ 2 i

a i

G ( x ) =

(4)

The maximum likelihood function of Equation (4) is given by

n i  j = p i    x j −

m x j − m    2

M  m = 1

1 n i

σ 2

a i

(5)

i , max =

For splitting the AR models of the segments k into non-deterministic (before ToA) and deterministic (post ToA) parts, i = 1 , 2. For i = 1 , 2, Equation (5) can be solved by assigning p 1 = 1, p 2 = k + 1, n 1 = k and n 2 = n − k . The entropy of σ 2 i , max , k is calculated for k = 1 , 2 , 3 .., K by considering that the maximum likelihood solution is a series of data for k . The entropy used in this study is Renyi’s entropy, which is given by H ( k ) = − log    K  k = 1 P ( k ) 2    , P ( k ) = σ 2 i , max , k (6) By minimizing H ( k ), the segment where ToA may appear can be identified. ToA appears in the segment where H ( k ) is minimum. Now, in order to find the precise ToA, Akaike Information Criterion (AIC) is used [Kitagawa and Akaike (1978)]. AIC ( w ) = wlog [ var ( x { 1 , w } )] + ( N − w − 1) log [ var ( x { w + 1 , N } )] (7)