## 3 The Preliminaries
### The Proposed Localized Adaptive Denoising Technique  
Generally, a vibration signal obtained from a gearbox contains fault-related vibration signatures and noise. Denoising of the signal is required for the extraction of fault-related vibration signatures. Suppose the observed signal is \(s\) and the informative signal is \(x\); then \(\mathrm{s}=x+\partial\), where \(\partial\) represents the noise. The denoising technique tries to filter out noise for obtaining the estimation signal \(\hat{\mathrm{x}}\) in a tendency to approximate the useful signal x as much as possible. The adaptive denoising technique uses the concept of destructive interference for denoising of an impaired signal. This technique utilizes the noise-simulated reference signal to access frequency segments in a frequency domain of the observed impaired signal in order to remove noise. The adaptive noise-reducer-based Gaussian reference signal (ANR-GRS), which has been proposed and verified in [22; 23], has achieved great performance in reducing noise and avoiding distortion of the fault-related ingredients. In this method, the noise \(\partial\) in a gearbox vibration signal is analyzed and divided into two types of noise: white noise (\(\alpha\)) and band noise (\(\beta\)), \(\partial=\alpha+\beta\). Then, the reference signal is created by combining two noise-simulated signals, which are analogous with two existing sources of noise in the observed signal, a white noise signal and a Gaussian signal. Moreover, the parameters of the reference signal are adjustable by adaptive algorithm regarding the varying input values of rotation shaft speeds. The Gaussian signal is responsible for building the simulated noise reference signal. The parameters of the Gaussian signal (a mean value and a standard deviation value) are adaptively adjusted so as to reduce the noise between two consecutive sideband frequencies (the sideband frequency is the gear frequency in this study). The process for generating a reference signal is depicted in Figure 2, and the Gaussian signal is formulated as follows:  
\[\mathrm{G_{ns}}(k)\ =\ \sum_{k=1}^{K}\mathrm{e}^{-\frac{(k-F_{m})^{2}}{2 \sigma^{2}}} \tag{3}\]  
where \(K\) is the number of sideband segments, and the mean value \(F_{m}\) and the standard deviation value \(\sigma\) are the function of the shaft rotation frequency. Those parameters are adjusted by an optimization process to select the optimized vibration sub-band as an output of the ANR-GRS module [22]. From each parameter set (\(F_{m}\), \(\sigma\)), which is randomly selected from the specific required range defined in [22], a noise-simulated signal is generated. This reference signal is provided as an input to the adaptive filter along with the impaired observed signal. The adaptive noise filter contains a digital filter, which employs an L-tap FIR type digital filter and weight vector as \(\mathrm{w}(n)\equiv[\mathrm{w_{0}},\ \mathrm{w_{1}},\ \dots\,\ \mathrm{w_{L-1}}]^{\mathrm{T}}\), and a least mean square adaptive algorithm. The adaptive filter works as follows: The noise-simulated signal is provided as an input to the digital filter, then the filtered output signal is summed with the vibration signal (impaired observed signal) to compute the error signal. This output error signal is provided as a feedback input to the adaptive algorithm to measure its mean square value. Next, the adaptive algorithm tunes the weights of the digital filter according to the converging criterion of least mean square (LMS) error to obtain the optimal weight vector (\(\mathrm{w_{0}}\)) and then expose the optimal vibration sub-band corresponding to the particular parameter set. The schematic diagram of the ANR-GRS is provided in Figure 2.  
From Figure 2, it can be observed that the ANR-GRS method tries to look for the general optimal parameters of the Gaussian reference signal applied to the whole frequency range of input vibration signals (0-10 kHz).