# Blind Source Separation of Transformer Acoustic Signal Based on Sparse Component Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}norm decomposition [19,20,21]. Bofill et al. [19] proposed an underdetermined blind source separation method using sparse representations, which successfully separated six source signals from two mixed acoustic signals. Li et al. [21] used density peak clustering and compression sensing model to separate various fault signals from mixed vibration signals, and the separation effect is good.

## 2. Principle of Blind Source Separation for Transformer Acoustic Signal

#### 2.1. Characteristic Analysis of Interference Signals

#### 2.2. Sparse Component Analysis

**X**(t) = [X

_{1}(t),X

_{2}(t),...,X

_{M}(t)]

^{T}are the M mixed acoustic signals at time t,

**A**∈ $\mathbb{R}$

^{M}

^{×N}is the unknown mixing matrix with M < N,

**a**

_{n}= [a

_{1n},a

_{2n}, ..., a

_{mn}]

^{T}is the nth column of the mixing matrix

**A**.

**S**(t) = [S

_{1}(t),S

_{2}(t),...,S

_{N}(t)]

^{T}are the N sound source signals at time t.

_{1}is present at a single source point t, i.e., S

_{1}(t) ≠ 0 and S

_{2}(t) = 0. Equation (1) can be simplified as

_{1}will be distributed on a straight line where the direction is

**a**

_{1}. Therefore, the SSPs dominated by different sound source signals will be distributed on straight lines in different directions. The direction of the lines can be calculated by clustering algorithm and used as the estimation of the column vectors of the mixing matrix so that the transformer acoustic signal can be separated [19].

## 3. Blind Source Separation Method of Transformer Acoustic Signal

#### 3.1. Sparse Enhancement of Mixed Acoustic Signals

**X**(t, f) = [X

_{1}(t, f), X

_{2}(t, f), ..., X

_{M}(t, f)]

^{T}and

**S**(t, f) = [S

_{1}(t, f), S

_{2}(t, f), ..., S

_{N}(t, f)]

^{T}are, respectively, the STFT coefficients of the mixed acoustic signals and sound source signals in the fth frequency bin at time frame t. Although the mixed acoustic signals show approximate sparsity in the TF plane, it does not meet the sparsity requirements of SCA. Therefore, the SSPs are selected from the mixed acoustic signals in the TF plane by the single-source-point identification method based on phase angle. Assume that only one source signal S

_{1}is present at an SSP (t

_{1}, f

_{1}) in the TF plane. Equation (4) can be expressed as

**X**(t

_{1}, f

_{1}), we will get

_{m}(t

_{1}, f

_{1})} and I{X

_{m}(t

_{1}, f

_{1})} can be expressed as

**X**(t, f) will be the same if the sampling point (t, f) is an SSP. However, the probability of obtaining SSPs is very low in practice, hence we relax the determination condition of SSP as follows: the point in the TF plane where the absolute value of the difference β

_{f}between the ratio of the real part to the imaginary part of each component of the mixed acoustic signals is less than $\beta $ is taken as an SSP. The SSP identification formula can be expressed as

**X**

_{SSP}show sufficient sparsity and linear clustering characteristics. In order to ensure that the direction of each line is represented by a unique direction vector, the negative direction vectors of

**X**

_{SSP}are mapped to the positive half unit circle using the normalization method. The transformation process can be expressed as

#### 3.2. Mixing Matrix Estimation Based on Density Space Clustering

**X**(t, f) in the TF plane are stacked into an array, and this array is used as the input for the clustering algorithm. Then, the output of the clustering algorithm is the mixing matrix estimation.

_{1}, P

_{2}, ..., P

_{C}), the definitions of various sample points in the algorithm are shown in Figure 1, and the descriptions are as follows [22,23]:

- (1)
- Eps-neighborhood: for P ∈ C, the set of sample points contained in the hypersphere region with P as the center and Eps as the radius is called the Eps-neighborhood of P, i.e., ${N}_{E}(P)=\text{{}Q\text{}\in \text{}C\mid \mathrm{dist}(P,\text{}Q)\text{}\le \text{}\mathit{Eps}\text{}}$. Where dist(P, Q) is the distance between sample point P and Q in C.
- (2)
- Core point: for P ∈ C, if the number of sample points contained in N
_{E}(P) is more than or equal to mps, i.e., $\mid {N}_{E}(P)\mid \text{}\ge \text{}\mathit{mps}$, P is called the core point of C. - (3)
- Directly density reachable: for Q ∈ C, if $\mid {N}_{E}(P)\mid \text{}\ge \text{}\mathit{mps}$ and Q ∈ N
_{E}(P), Q is directly density reachable from P. - (4)
- Density reachable: for P
_{1}, P_{2}, ..., P_{C}∈ C, if P_{c}_{+1}is directly density reachable from P_{c}, where 1 ≤ c ≤ C−1, P_{C}is density reachable from P_{1}. - (5)
- Boundary point: for Q ∈ C, if N
_{E}(Q) < mps, N_{E}(P) ≥ mps and Q ∈ N_{E}(P), Q is the boundary point of C. - (6)
- Noise point: for Q ∈ C, if Q does not belong to Eps-neighborhood of any core point, Q is the noise point of C.

#### 3.3. Source Signals Recovery Based on Compressed Sensing

#### 3.3.1. Compressed Sensing

**x**= [x(1),x(2),···,x(M1)]

^{T}are the observed signals, $\mathsf{\Phi}\in {\mathbb{R}}^{M1\text{\xd7}N1}$ is the measurement matrix with M

_{1}< N

_{1},

**s**= [s(1),s(2),···,s(N1)]

^{T}are the unknown sparse source signals. Then, the source signals can be recovered by solving the L

_{0}norm optimization problem:

_{0}norm optimization problem is usually solved by the orthogonal matching pursuit (OMP) algorithm [26].

#### 3.3.2. Source Signals Recovery

**X**(f) = [X

_{1}(f), X

_{2}(f), ..., X

_{M}

_{1}(f)]

^{T}

**S**(f) = [S

_{1}(f), S

_{2}(f), ..., S

_{N}

_{1}(f)]

^{T}, f = 1,2,...,L

**X**(f) and S(f) into vectors as follows:

**X**= [X

_{1}(1), ..., X

_{M}

_{1}(1), X

_{1}(2), ..., X

_{M}

_{1}(2), ..., X

_{1}(L), ..., X

_{M}

_{1}(L)]

^{T}

**S**= [S

_{1}(1), ..., S

_{N}

_{1}(1), S

_{1}(2), ..., S

_{N}

_{1}(2), ..., S

_{1}(L), ..., S

_{N}

_{1}(L)]

^{T}

**X**=

**Φ**

**S**

**0**is an M × N matrix of zeros. The measurement matrix estimation is obtained by using the mixing matrix estimation

**B**instead of the mixing matrix

**A**. So far, the compressed sensing model of the mixed acoustic signals has been established.

## 4. Simulation Analysis

#### 4.1. Simulation Signals

_{1}is collected through the microphone under the environment of 10 dB background noise, and two speech signals are randomly selected from the TIMIT as two interference signals S

_{2}and S

_{3}. The sampling frequency of the signals is 16 kHz and the sampling time is 4 s.

**X**(t) = [X

_{1}(t),X

_{2}(t)]

^{T}is generated by Equation (1). Figure 5 shows waveforms and spectrums of the mixed acoustic signals of simulation. From Figure 5 it can be seen that the frequency components of the mixed acoustic signals are relatively complex, and there are low-frequency and high-frequency interferences with large amplitude in the spectrum range of 0–2000 Hz.

#### 4.2. Simulation Experiment and Analysis

**X**(t) is transformed to TF domain by STFT. The parameters of STFT are set as: STFT size 1024, Hanning window as the weighting function and the overlap size of window 512.

_{1}and the y-axis is the amplitude of X

_{2}in the scatter diagram. Figure 6b shows the scatter diagram of the mixed sound signals

**X**(t, f) in the TF domain, the x-axis is the amplitude of R{X

_{1}(t, f)} and the y-axis is the amplitude of R{X

_{2}(t, f)} in scatter diagram.

_{SSP}(t, f) of the mixed acoustic signals in the TF domain are extracted by using the single-source-point identification method and the parameter of the SSP identification method is set to β = 0.02 [19]. Then, the normalized data

**X**

_{nor}(t, f) are obtained by normalizing

**X**

_{SSP}(t, f). Figure 7 shows scatter diagrams of the real part of the SSPs and normalized data, respectively.

_{a}, voice signal S

_{b}, and voice signal S

_{c}. The low-frequency and high-frequency interference signals in the mixed acoustic signals are eliminated, and the frequency characteristic of the transformer acoustic signal is preserved completely by the transformer recovered signal.

#### 4.3. Comparison with Other Methods

_{n}(t) and ${\widehat{S}}_{n}\left(t\right)$ are respectively the nth source signal and its recovered signal; T is the length of the signal. The closer NCC is to 1, the closer the recovered signal is to the source signal.

## 5. Experimental Analysis

#### 5.1. Experimental Setup

#### 5.2. Experimental Results and Analysis

_{a}, voice signal S

_{b}, and voice signal S

_{c}. Figure 13 shows the waveforms and spectrums of the recovered signals. The separation results show that the waveform of the transformer recovery signal S

_{a}is roughly the same as that of the reference signal, and the main frequency peaks in the two signal spectrums correspond to each other one by one.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**Scatter diagram of the mixed acoustic signals in time domain and time-frequency domain; ((

**a**): time domain; (

**b**): time-frequency domain).

**Figure 7.**Scatter diagram of the real part of the SSPs and normalized data; ((

**a**): the real part of the SSPs; (

**b**): the real part of the normalized data).

**Figure 8.**Waveforms and spectrums of three recovered signals of simulation; ((

**a**): waveforms; (

**b**): spectrums).

**Figure 9.**Waveforms and spectrums of the transformer recovered signal using the two methods; ((

**a**): method 1; (

**b**): method 2).

**Figure 10.**Physical diagram and experiment schematic diagram of the transformer; ((

**a**): power transformer, three phases, 50 MVA, 50 Hz; (

**b**): experiment schematic diagram).

**Figure 11.**Waveforms and spectrums of the mixed acoustic signals in the substation; ((

**a**): waveform; (

**b**): spectrum).

**Figure 12.**Waveform and spectrum of the reference signal in the substation; ((

**a**): waveform; (

**b**): spectrum).

**Figure 13.**Waveforms and spectrums of the recovered signals in the substation; ((

**a**): waveform; (

**b**): spectrum).

Interference Source | Interference Factor | Duration/s | Frequency Band/Hz |
---|---|---|---|

Transformer structure | Cooling fan sound | continued | 0–1000 |

OLTC action sound | 0.2 | 0–20,000 | |

power equipment | Reactor sound | continued | 0–2000 |

Capacitor sound | continued | 0–2000 | |

Corona discharge sound | continued | 5000–20,000 | |

operation environment | Speech sound | 0.4 | 150–5000 |

Musical sound | 0.5 | 0–20,000 | |

vehicle sound | 0.5 | 2000–8000 | |

Bird sound | 0.2 | 3000–8000 |

Evaluation Index | Proposed Method | Method 1 | Method 2 |
---|---|---|---|

NCC(S_{1}, S_{a}) | 0.9941 | 0.9648 | 0.8413 |

SNR(S_{1}, S_{a})/dB | 20.1829 | 13.8940 | 4.9459 |

Evaluation Index | (X_{1}, S_{a}) | (X_{2}, S_{a}) | (S_{1}, S_{a}) |
---|---|---|---|

NCC | 0.4927 | 0.6472 | 0.9254 |

SNR/dB | 0.7885 | 1.6663 | 8.8580 |

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**MDPI and ACS Style**

Wang, G.; Wang, Y.; Min, Y.; Lei, W. Blind Source Separation of Transformer Acoustic Signal Based on Sparse Component Analysis. *Energies* **2022**, *15*, 6017.
https://doi.org/10.3390/en15166017

**AMA Style**

Wang G, Wang Y, Min Y, Lei W. Blind Source Separation of Transformer Acoustic Signal Based on Sparse Component Analysis. *Energies*. 2022; 15(16):6017.
https://doi.org/10.3390/en15166017

**Chicago/Turabian Style**

Wang, Guo, Yibin Wang, Yongzhi Min, and Wu Lei. 2022. "Blind Source Separation of Transformer Acoustic Signal Based on Sparse Component Analysis" *Energies* 15, no. 16: 6017.
https://doi.org/10.3390/en15166017