## 3 Obtaining the True Degradation Distribution via Deconvolution  
In this section, it is considered that \(Z_{i}(t_{j})\) represents the observed degradation measurement of the \(i\)th unit at time \(t_{j}\), and that the observed degradation is contaminated with some measurement error \(\varepsilon_{ij}\). Thus, \(\varepsilon_{ij}\) is also observed at \(t_{j}\) for each \(i\)th unit. Which means that for each observed degradation a measurement error is observed. Such that \(\varepsilon_{ij}\) is a random variable that follows a Gaussian distribution as \(G(\mu,\sigma)\) with a PDF defined as,  
\[f\big{(}\varepsilon_{ij}|\mu,\sigma\big{)}=\frac{1}{\sqrt{2\pi}\sigma}exp \Bigg{\{}-\frac{\big{(}\varepsilon_{ij}-\mu\big{)}^{2}}{2\sigma^{2}}\Bigg{\}}. \tag{5}\]  
Based on this measurement error, an additive function of the observed degradation can be considered as \(Z_{i}(t_{j})=S_{i}(t_{j})+\varepsilon_{ij}\), where \(S_{i}(t_{j})\) denotes a hidden true degradation measurement. Indeed, the observed degradation and measurement error are considered to be known as (4) and (5), respectively. Then, the true degradation may be obtained via deconvolution [30]. This operation consists in obtaining the subtraction of random variables, for example consider the function \(H=E+G\), where \(H\) represents an observed measurement, \(E\) represents an unknown variable and \(G\) represents a measurement error. The PDFs of \(H\) and \(G\) are known to be \(f_{H}\) and \(f_{G}\), respectively, and the characteristic functions (CF) of such PDFs are defined as \(\varphi_{H}\) and \(\varphi_{G}\). The CFs are also known as Fourier transforms (FT). In the first instance, the deconvolution operation consists of determining the CF of \(E\), which is defined as \(\varphi_{E}=\varphi_{H}/\varphi_{G}\). In the second instance, the function of the deconvoluted true measurement \(f_{E}\) is obtained by considering the inverse Fourier transform (IFT) of \(\varphi_{E}\). Consider this approach for \(Z_{i}\big{(}t_{j}\big{)}=S_{i}\big{(}t_{j}\big{)}+\epsilon_{ij}\), where a gamma distribution is defined for \(Z_{i}\big{(}t_{j}\big{)}\) as \(f\big{(}Z_{i}\big{(}t_{j}\big{)}\big{)}\) and a Gaussian distribution is defined for \(\epsilon_{ij}\) as \(f\big{(}\epsilon_{ij}\big{)}\). Firstly, the CF of \(S_{i}\big{(}t_{j}\big{)}\) can be obtained by considering the CF of the gamma and normal distributions in (6) and (7), respectively. \[\varphi_{Z}(\zeta)=(1-ui\zeta)^{-\varphi}. \tag{6}\]  
\[\varphi_{\epsilon}(\zeta)=exp\Big{\{}i\mu\zeta-\sigma^{2}\zeta^{2}/2\Big{\}}. \tag{7}\]  
Thus, \(\varphi_{S}(t)\) is obtained as  
\[\varphi_{S}(\zeta)=\frac{(1-ui\zeta)^{-\varphi}}{exp\{i\mu\zeta-\sigma^{2} \zeta^{2}/2\}}. \tag{8}\]  
The PDF of \(S_{i}\big{(}t_{j}\big{)}\) is obtained via the IFT of (8) as,  
\[f\big{(}S_{i}\big{(}t_{j}\big{)}\big{)}=\int_{-\infty}^{\infty}\varphi_{S}( \zeta)exp\{-i\zeta s\}d\zeta. \tag{9}\]  
It can be noted that the IFT represented by the integral in (9) does not have a closed analytical expression. For this, the discrete Fourier transform (DFT) is considered to obtain an approximation of \(f\big{(}S_{i}\big{(}t_{j}\big{)}\big{)}\). The DFT considers a discrete version of the IFT in (9) as a Riemann sum approximation which can solved via the fast Fourier transform (FFT) algorithm [46; 47]. The FFT is known to reduce the complexity of the DFT of a function sampled on a regular grid of \(2^{p}\) points [48; 49]. By considering that any integral, such as (9), can be viewed as the sum of infinitely many small rectangles, then for the sampled regular grid, \(p\) equally spaced sub-intervals with range \([-L_{0},L_{0}]\) are considered, where \(L_{0}=\mu+5\sigma+q_{g}\), \(q_{g}\) is the 0.9999 quantile of the gamma distribution, \(-L_{0}=0\) and \((\mu,\sigma)\) are the parameters of the measurement error PDF.