### 1 Introduction

### 2 SFCIM-based Fault Diagnosis Method

**Step 1**. The time-domain synchronization of the raw signal for training is completed through cross-correlation in the time domain.**Step 2**. Using wavelet packet decomposition (WPD), the normal and faulty signals for training are converted into time-frequency domain spectrograms.**Step 3**. Normal-reference (N-R) and abnormal-reference (A-R) spectrograms for training are obtained by spectral subtraction of a reference spectrogram, which we usually set from one of the normal signals.**Step 4**. Three statistical features are extracted from the coefficients of each frequency level of the N-R and A-R spectrograms. Important frequency ranges for diagnosis are ranked using Fisher’s scores.**Step 5**. Among the user-defined variables, important feature values are selected according to the window selection ratio (WSR). The effective band is the frequency band in which all three features are selected as important areas. Moreover, one band each of higher and lower bands are additionally selected for conservative selection. A further detail WPD analysis in a selected frequency range with a user defined magnification ratio (MR) may be optionally necessary.**Step 6**. Through an optimization process (i.e., training), we determine the OWs in a spectrogram, which areas the areas where the difference between the N-R and A-R spectrograms is consistently observed. We call the spectrogram map with the OWs a Selected Frequency Range Critical Information Map (SFCIM).**Step 7**. A mechanical system is classified as a normal or faulty condition by the number of OWs. If those are closer to the faulty condition, we may determine the system is in the faulty condition.

### 2.1 Time-Domain Data Synchronization

*f*is the signal to be synchronized,

*g*denotes the reference signal for obtaining the time delay, and

*δ*denotes the time delay between the two signals. We define

*δ*

*as the time point that shows the highest cross-correlation value and obtain the synchronized signal by applying the*

_{max}*δ*

*value to the raw signal*

_{max}*f*.

### 2.2 Time Frequency Representation and Spectral Subtraction Process

*j*is a scale factor at the corresponding decomposition level;

*k*is the shifting parameter (i.e., translation along the time axis); and n is a modulation or oscillation parameter. The first and second wavelet packet functions become scaling and mother wavelet functions, respectively, and the general formula is defined as follows:

*h*(

*k*) and

*g*(

*k*) are quadrature mirror filters and orthogonal to each other. Therefore, the wavelet packet coefficients expressed by variables

*j*,

*n*, and

*k*are as follows:

*f*(

*t*) is the signal to be decomposed. The WPD is used to generate the SFCIM. Details of the WPD may be referred in Ref. 17.

*S*

*(*

_{v}*ω*) is a spectrogram of the signal that includes noise,

*S*

*(*

_{n}*ω*) and is a spectrogram of a noise signal without a valid voice. Berouti et al. [18] obtained the noise-removed signal using Fourier inversion of

*S*

*′(*

_{v}*ω*). Denda et al. [19] presented a spectral subtraction method for a wavelet transform-based spectrogram. Our method uses the following equation:

*X̂*(

*b,a*)| is the wavelet spectrogram of the filtered signal, |

*γ*

*(*

_{s}*b,a*)| is the wavelet spectrogram of the raw signal, including noise,

*α*is a reduction factor indicating the degree of noise removal, and

*a*and

*b*are the time-domain position and scale parameters of the wavelet filter, respectively. Bouchikhi et al. [20] applied the noise-reduction concept to bearing fault diagnosis.

### 2.3 Dominant Frequency Range Selection and Additional Detailed Analysis Process

*j*means the class number corresponding to the classification target of the model,

*i*is the sequence number of the feature,

*n*

*is the number of data samples of the*

_{j}*j*th class,

*μ*

*is the overall average value of the*

_{i}*i*th feature,

*μ*

*is the mean value of the*

_{i,j}*j*th class cluster of the

*i*th feature, and

*σ*

*is the internal variance value of the jth class cluster of the ith feature. The larger the Fisher score (*

_{i,j}*f*

*) of each feature, the greater the density of the features in the cluster for each class.*

_{i}*c*

*is the*

_{i}*i*th coefficient value. The frequency level selection process preceding the optimization process selects a range with a big difference between N-R and A-R at each frequency level of the learning data. In other words, frequency levels that show a constant and significant difference that reflects the characteristics of the high Fisher’s score features are selected. In Fig. 2, one frequency level is shown as an example. Three statistical features of the coefficient values within the frequency level indicate the difference between N-R and A-R. In this step, we could select the section where the high-ranked features are intensively distributed.

*frequency levels. The user experimentally sets the WSR, and MR values based on the basic characteristics of the diagnosis object (i.e., AC current frequency or vibration characteristics of the system). We recommend that the users set the WSR value by starting with a value close to one and set the MR value when detailed analysis is required according to the user’s needs.*

^{MR}### 2.4 Optimization Process for SFCIM

*N*

*and*

_{n}*N*

*are training datasets of each normal and abnormal condition, respectively.*

_{a}*D*

*is the number of divisions along the frequency band.*

_{f}*L*

*is the signal length used for model learning (multiplication of time and sampling rate).*

_{s}*N*

*means the number of OWs on the (*

_{n-r,ij}*i*,

*j*) window among training data in a normal condition, in and

*N*

*is the number of OW occurrences on the window among training data in an abnormal condition.*

_{a-r,ij}*x*

*, is the number of divisions in the time band of the spectrogram. As an example, in Fig. 4, as the value of*

_{1}*x*

*increases, the time domain division on the spectrogram increases and create more detailed windows. The second one,*

_{1}*x*

*, is the threshold of the difference value between the wavelet coefficients inside the window divided in the time–frequency band. The third one,*

_{2}*x*

*, is a threshold number for determining whether each window is outstanding based on the number of values that exceed the*

_{3}*x*

*threshold.*

_{2}*x*

*and*

_{2}*x*

*are used as thresholds to determine whether each window is outstanding. For example, in Fig. 4, if*

_{3}*x*

*is 0.4 and*

_{2}*x*

*is 10, the window becomes OW; if*

_{3}*x*

*is 0.4 and*

_{2}*x*

*is 20, it cannot be OW. The*

_{3}*R*

*value is a criterion value for determining the OW; as it decreases, it tends to be more tolerant of noise inside the training data. For a conservative fault diagnosis, the*

_{c}*C*

*value is designated as one or more, to reduce the occurrence of OWs in the data under normal conditions.*

_{p}### 3 Application to Case Studies

### 3.1 Center of Intelligent Maintenance Systems Bearing Data

*x*

*threshold value (0.1972). However, in the case of the A-R subtracted spectrogram shown in Fig. 9(b), more than ten coefficients are greater than the*

_{2}*x*

*threshold. When we apply the optimized x*

_{2}*variable value to the result, we can confirm that the corresponding window is OW, and through this process, all windows in the SFCIM are determined. Through this process, the model can classify the fault using the number defined OW, and we discuss the validation results below.*

_{3}### 3.2 Fault Detection of the Industrial Robot Input Gear

*R*

*) are shown in Figures 14(b) and 14(d).*

_{c}