## 2 Materials and Methods
### The Noise-Reduction Algorithm  
Concerning the processing of signals for noise reduction, different works have been carried out. These works can be classified according to various criteria. However, according to the number of signals at the input of the noise-reduction system, they can be classified into methods that use two or more input signals, and methods that use only one input signal [22]. A classification proposal can then be provided; it tries to group different techniques according to their characteristics and requirements:  
* Classical FIR (Finite Impulse Response) filtering methods [35; 36; 37].
* Adaptive noise reduction methods [38; 39; 40].
* Artificial intelligence methods. Neural Networks [41; 42].
* Wavelet-based methods [43; 44; 45].
* Statistical signal processing methods [46; 47].  
From previous works, classical filtering techniques limit their use to reduce noise only in the frequency band to which the filter is limited. On the other hand, although adaptive filtering algorithms have the ability to adapt, most of them need a reference sample for obtaining tangible results. Similarly, artificial intelligence and wavelet-based methods need a reference to adjust the desired output and a comparison threshold, which depends on the noise power, respectively. The proposed noise reduction algorithm is intended to simplify the signal and keep only the information relative to the motor behavior that leads to identifying the broken bars. It distinguishes and separates the spectral components of the acoustics that do not inform about them. These components are considered as random interferences from the environment. We consider them part of a Gaussian noise. We split the propose method into the following steps:  
1. We set a filter from the convolution of the signal with its autocovariance. More details about these operations will be shown below and in the recent survey paper of the authors [34].
2. The result of the convolution is rescaled in amplitude by a non-linear factor: \[A^{\prime}_{k}=4(A_{k})^{1/3},\ \ \text{with}\ k\ \in\ \mathbb{N}_{0}\] (3) where \(A^{\prime}_{k}\) is the signal amplitude obtained from the convolution-autocovariance calculation and \(A_{k}\) is the amplitude of the original signal.
3. We apply an envelope detector to the outcome of the IFT (inverse Fourier transform) from the previous step. This demodulates the signal thanks to the loss of symmetry after the amplitude and phase rescaling.
4. Finally, the filtered signal is recovered after a division of the result of the envelope detector.  
To summarize, we represent the whole in Figure 2, where the input signal is the acoustic signal obtained directly from the induction motor.  
For the sake of completeness, we give more details about using second-order statistics to explain the result of the convolution of a signal with its autocovariance. The autocovariance measures the scattering of the signal around the mean value [33; 34; 48; 49]. If \(y=\left\{y(t)\right\}_{t\geq 0}\) is the signal, then:  
\[C_{2}^{y}(t_{1},t_{2})=E[(y(t_{1})-u_{y}(t_{1}))(y(t_{2})-u_{y}(t_{2}))]=R_{2}^ {y}(t_{1},t_{2})\ -\ u_{y}(t_{1})u_{y}(t_{2}). \tag{4}\]  
where \(u_{y}(t)\) is the mean and \(R_{2}^{y}\) is the autocorrelation of \(y(t)\). If \(u_{y}(t)\) is zero then the autocorrelation matches the autocovariance. For ergodic stationary data, we may assume \(u_{y}(t)\) to be constant for all \(t\geq 0\). Then, the autocovariance function of (3) becomes:  
\[C_{2}^{y}(t_{1},t_{2})=R_{2}^{y}(t_{1},t_{2})\ -\ u_{y}(t)^{2} \tag{5}\]