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Two-Channel Information Fusion Weak Signal Detection Based on Correntropy Method^{ †}

^{1}

^{2}

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## Abstract

**:**

## Featured Application

**The method proposed in the article can be used for weak signal detection and weak fault feature extraction. It is mainly a method proposed to select suitable singular components for singular value decomposition. After that, the cyclic correntropy spectrum is used to replace the demodulation effect of the envelope spectrum.**

## Abstract

## 1. Introduction

## 2. Fundamental Theory

#### 2.1. Principal Components Analysis for Multi-Channel Information Fusion

- Form the original data into an n-by-m data Hankel matrix
**X**; - Each row of matrix
**X**is subtracted from its corresponding mean value, that is, zero mean value; - Calculate the covariance matrix $C=\frac{1}{m}X{X}^{T}$;
- Calculate the eigenvalues and corresponding eigenvectors of the covariance matrix;
- Arrange the eigenvectors into a matrix according to the size of the corresponding eigenvalues from large to small, and take the first k (determined by the cumulative contribution rate of the principal components) columns to form a matrix
**P**; - $Y={P}^{T}X$ is the principal component matrix of the original data
**X**after dimensionality reduction.

#### 2.2. Theory of Singular Value Decomposition

**A**becomes a non-Hankel matrix after SVD, it is decided to obtain each element of ${x}_{i}$ by averaging along the anti-diagonals of ${A}_{i}$.

#### 2.3. Definition of Correntropy

## 3. Presentation of the Proposed Method

_{0}is one of the periods of CE, $\alpha =n/T,\text{}n\in Z$ is taken as the cyclic frequency, T is the period of CE.

## 4. Simulation for Correntropy Induced Metrics and Cyclic Correntropy Spectrum

#### 4.1. The Effect of Correntropy Induced Metrics

_{1}, s

_{2}, s

_{3}, s

_{4}, and s

_{5}are generated. s

_{1}is a sine signal, s

_{2}is a square wave signal, s

_{3}is an amplitude modulated signal, s

_{4}is random noise, where, s

_{1}, s

_{2}, and s

_{3}are expressed as (11)–(13), s

_{4}consists of random numbers from −1 to 1. s

_{5}is the mix of s

_{1}, s

_{2}, s

_{3}, and s

_{4}, expressed in (14). The sampling frequency is 16 kHz, and the number of sampling points is 2048.

_{5}, respectively. From (7), it can be conducted that when $X=Y$, CIM = 0; CIM > 0, for other cases. Additionally, the greater the difference between the variables X and $Y$, the greater the CIM value. Set X in (6) to s

_{5}, set $Y$ in (7) to s

_{1}, s

_{2}, s

_{3}, s

_{4}, respectively. Then four CIM values are displayed in Figure 4, it can be seen that the CIM between s

_{2}and s

_{5}is the smallest which can be explained by the frequency spectrum (Figure 3b).

_{2}contains the most frequency components of s

_{5}, so for a method based on frequency component analysis, it is appropriate to choose s

_{2}with the lowest CIM to represent s

_{5}. Moreover, the largest value of CIM is produced by s

_{4}and s

_{5}, where s

_{4}is the random noise, which also shows that CIM-based selection can effectively reduce noise interference.

#### 4.2. The Role of Cyclic Correntropy Spectrum

## 5. Validation of the Proposed Method on Weak Signal Detection

#### 5.1. Weak Electrical Signal Detection

#### 5.2. Features Extraction of Two-Channel Information Fusion

- (1)
- Construct each signal column into a 3073-by-1024 matrix, and then splice the two matrices into a 3073-by-2048 matrix.
- (2)
- Perform PCA on the newly constructed matrix with each column as one-dimensional, and obtain a new 3073-by-2048 matrix.
- (3)
- To maintain the consistency of the signal length, the first 1024 dimensions are preferred as the principal components.
- (4)
- However, to prove that the selected principal components can represent the main information of the original matrix, the cumulative contribution of the selected matrix should be calculated. The cumulative contribution of the first 352-dimensional vector reaches 95%, and the cumulative contribution of the first 1024-dimensional vector reaches 99.66%. Therefore, it can be considered that the first 1024 dimensional vectors can express most of the information of the original matrix.

## 6. Conclusions

- A method called SVD-CIM is proposed to extract the weak signal features, which includes using CIM to select the SCs from the SVD and estimating CCES for the reconstructed signal.
- CIM is a similarity measure based on frequency components.
- Compared with the frequency spectrum of the reconstructed signals, CCES can express more useful information for the cyclostationary signals.
- Compare with the method of difference spectrum selection, the CIM selection has a better performance on weak signal features.
- For two-channel information, two-SVD-CIM is proposed, and the two-channel information fusion can enhance the fault characteristics.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 8.**The original signal with 93 Hz. (

**a**) the waveform of the original signal. (

**b**) the frequency spectrum.

**Figure 14.**The 108th data of the database. (

**a**) Time-domain waveform of the original signal at the drive end. (

**b**) Frequency spectrum of the original signal at the drive end. (

**c**) Time-domain waveform of the original signal at the fan end. (

**d**) Frequency spectrum of the original signal at the fan end.

**Figure 15.**Difference spectrum and CIM. (

**a**) Difference spectrum of the SVD. (

**b**) CIM between the SCs and original signal.

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**MDPI and ACS Style**

Gong, S.; Lu, J.; Li, S.; Ma, H.; Wang, Y.; Teng, G. Two-Channel Information Fusion Weak Signal Detection Based on Correntropy Method. *Appl. Sci.* **2022**, *12*, 1414.
https://doi.org/10.3390/app12031414

**AMA Style**

Gong S, Lu J, Li S, Ma H, Wang Y, Teng G. Two-Channel Information Fusion Weak Signal Detection Based on Correntropy Method. *Applied Sciences*. 2022; 12(3):1414.
https://doi.org/10.3390/app12031414

**Chicago/Turabian Style**

Gong, Siqi, Jiantao Lu, Shunming Li, Huijie Ma, Yanfeng Wang, and Guangrong Teng. 2022. "Two-Channel Information Fusion Weak Signal Detection Based on Correntropy Method" *Applied Sciences* 12, no. 3: 1414.
https://doi.org/10.3390/app12031414