*2.1. Singular Value Decomposition of Signals*

SVD is an orthogonal transformation. For a real matrix, **<sup>A</sup>** <sup>∈</sup> **<sup>R</sup>***m*×*n*, there exist two orthogonal matrices, **<sup>U</sup>** <sup>∈</sup> **<sup>R</sup>***m*×*<sup>m</sup>* and **<sup>V</sup>** <sup>∈</sup> **<sup>R</sup>***n*×*n*, that satisfy the equation given below [14]

$$\mathbf{A} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^T = \sum\_{i=1}^{q} u\_i \sigma\_i \boldsymbol{\upsilon}\_i^T \tag{1}$$

where the diagonal matrix, **Σ**, is [**diag**(σ1, σ2, ··· , σ*q*), 0] or its transposition. The elements, σ*i*(σ<sup>1</sup> > σ<sup>2</sup> > ··· > σ*q*), are the singular values of the matrix **A**, and *q* = min(*m*, *n*). **U** and **V** are the unitary matrices of **A**, and their column vectors *ui* and *vi* are the eigenvectors of the covariance matrices, **AA***<sup>T</sup>* and **A***T***A**, respectively.

The singular values correspond to the feature components of the decomposition matrix. Apart from their high stability, they also have the characteristics of proportional and rotational invariance. Therefore, SVD can ensure the robustness of the signal features represented by different singular values, in compliance with the properties required by the feature vectors in pattern recognition. In the SVD-based process of signals, the Hankel matrix is typically accepted as the trajectory matrix because of its characteristic of zero phase shift [11]. A signal containing a noise is indicated as a vector form, **s** = [*s*(1),*s*(2), ··· ,*s*(*N*)], and its corresponding *m* × *n* dimensional Hankel matrix form is expressed as

$$\mathbf{A} = \begin{bmatrix} a\_{\bar{i}\cdot\bar{j}} \end{bmatrix}\_{m \times n} = \begin{bmatrix} s(1) & s(2) & \cdots & s(n) \\ s(2) & s(3) & \cdots & s(n+1) \\ \vdots & \vdots & & \vdots \\ s(m) & s(m+1) & \cdots & s(N) \end{bmatrix} \tag{2}$$

where *m* = *N* − *n* + 1 and 1 < *n* < *N*.

The sampling signal can be expressed by Equation (2) as

$$\mathbf{s} = \begin{bmatrix} \mathbf{A}(1, \ : \end{bmatrix} \mathbf{A}(2:m, n) \end{bmatrix} \tag{3}$$

Defining **A***<sup>i</sup>* = *ui*σ*ivi <sup>T</sup>*, the signal component, **Pi**, can be expressed as [28]

$$\mathbf{P}\_{\mathbf{l}} = [\mathbf{A}\_{\mathbf{l}}(1, :), \mathbf{A}\_{\mathbf{l}}(2:m, n)]. \tag{4}$$

Based on Equations (1), (3), and (4), the original signal can be written as

$$\mathbf{s} = \sum\_{i=1}^{q} \mathbf{p}\_i. \tag{5}$$

Based on Equation (5), by SVD using the Hankel matrix, the polluted signal can be decomposed into a simple linear superposition of a series of component signals [27]. For an additive noise signal, **s** = **x** + **wnoise**, an advantage of this decomposition is that the clean signal can be solved by the order of the effective singular values.

$$\mathbf{x} = \sum\_{i=1}^{k} \mathbf{P}\_{\mathbf{i},i} \tag{6}$$

where **x** is the clean signal, and *k* is the order of the effective singular values.

### *2.2. Order Determination of Akaike Information Criterion*

The AIC is an estimated measure of the fitting goodness of statistical models [34], and is currently used in the estimation of the source number. The decision functions of the AIC are as follows [35]:

$$AIC(d) = -2N(n-d)\log\_{10}(L\_d) + 2d(2n-d) \tag{7}$$

and

$$L\_d = \frac{\prod\_{i=d+1}^n \lambda\_i^{\frac{1}{n-d}}}{\frac{1}{n-d} \sum\_{i=d+1}^n \lambda\_i} \tag{8}$$

where λ*<sup>i</sup>* = σ*<sup>i</sup>* <sup>2</sup> denotes the eigenvalues of the unitary matrices, *Ld* is the maximum likelihood estimation of the eigenvalues, and *d* = 1, 2, ··· , *n* − 1 denotes the number of sources.

The AIC function consists of two parts. The former term is the maximum likelihood estimation of the model parameters, which reflects the parameter fitness of the principal components. The second term is the bias correction term inserted to convert the AIC into an unbiased estimator. The former term decreases with the increase in the number of sources, whereas the second term is contrary to the former. When the sum of the two terms is minimum, the best estimate of the effective order is obtained by balancing both the terms as

$$k = \underset{d}{\text{argmin}} (AIC(d)). \tag{9}$$

### **3. Signal Denoising of Akaike Information Criterion–Singular Value Decomposition**

#### *3.1. Selection of Hankel Matrix Rows and Columns*

To select the number of rows and columns of the Hankel matrix, the energy characteristics of the singular values are considered. The energy of the singular values indirectly reflects the information richness of the trajectory matrix [36], which is defined as

$$E(n) = \sum\_{i=1}^{q} \sigma\_i^2. \tag{10}$$

The relationship between the energy of the singular values and elements of the Hankle matrix can be derived from Equation (11).

$$\mathbf{A}\mathbf{A}^T = [\boldsymbol{u}\_1 \boldsymbol{u}\_2 \cdots \boldsymbol{u}\_q] \cdot \begin{bmatrix} \lambda\_1 \\ & \lambda\_2 \\ & & \ddots \\ & & & \lambda\_q \end{bmatrix} \cdot [\boldsymbol{u}\_1 \boldsymbol{u}\_2 \cdots \boldsymbol{u}\_q]^T \tag{11}$$

$$E(n) = \lambda\_1 + \lambda\_2 + \dots + \lambda\_q = \sum\_{j=1}^n \sum\_{i=1}^m a\_{ij} \, ^2. \tag{12}$$

The difference in the number of rows and columns will modify the singular value energy. To easily distinguish the singular components and avoid feature coupling, the optimal number of matrix columns is selected based on the maximum energy of the singular values, i.e.,

$$\mathfrak{H} := \operatorname\*{argmax}\_{n} \left( E(n) \right) = \operatorname\*{argmax}\_{n} \left( \sum\_{j=1}^{n} \sum\_{i=1}^{N-n+1} a\_{ij} \,^2 \right). \tag{13}$$

According to Equation (13), the energy of the singular values is equal to the sum of the squares of all the matrix elements. When the structure of the Hankel matrix is a square or an approximate square, the corresponding energy of the singular values is maximum. Specifically, if *N* is even, the energy of the singular values is maximum at *n* = *N*/2 and *m* = *N*/2 + 1. If *N* is odd, the energy of the singular values is maximum at *n* = *m* = (*N* + 1)/2. As the basis for selecting the optimal structure of Hankel matrix, the maximum criterion of singular value energy makes it convenient to identify the effective singular components.

### *3.2. Verification and Improvement of Order Determination*

To verify the validity of the order determination based on the AIC, the different types of signals are designed. The expressions of the periodic, attenuation and sweep signal are given as

$$\begin{cases} x\_1 = \sin(40\pi t) + 1.8\sin(100\pi t) + 0.5\sin(200\pi t) \\ x\_2 = \exp^{-2t}[\sin(40\pi t) + 0.5\sin(200\pi t)] \\ x\_3 = \operatorname{clirp}(t, 10, 1, 100) \end{cases}.\tag{14}$$

Mixed with Gaussian white noise of different signal-to-noise ratios (SNRs), the initial signals turn into a series of polluted signals *s*1(*SNRs*), *s*2(*SNRs*) and *s*3(*SNRs*), respectively. At a sampling rate of 1 kHz and sampling time of 1 s, the polluted signals are constructed as 501 × 500 Hankel matrices to calculate by SVD. For simulation signals of different SNRs, the AIC is used to determine orders in comparison with cumulative contribution rate (CCR) and singular value curvature spectrum (CSM), as shown in Figure 1.

**Figure 1.** Comparison of the different methods in order determination: (**a**) signals *s*1(*SNRs*); (**b**) signals *s*2(*SNRs*); (**c**) signals *s*3(*SNRs*). CCR: cumulative contribution rate; CSM: singular value curvature spectrum; AIC: Akaike information criterion; SNR: signal-to-noise ratio.

Based on the main frequency analysis method, the effective orders of *s*1(*SNRs*) and *s*2(*SNRs*) can be rapidly determined as 6 and 4. In Figure 1a,b, the results calculated by the AIC are consistent with those by main frequency analysis method, remaining constant irrespective of the change in the SNR. Concurrently, violent jumps occur in the curves of both the CCR and CSM. As can be observed in Figure 1c, the effective orders by the AIC are more stable than the compared methods for sweep signals *s*3(*SNRs*). Therefore, the AIC improves the accuracy and robustness of the order determination, yielding results better than those obtained with other methods at different SNRs. The AIC can achieve viable noise separation, which is beneficial for reasonable noise reduction and feature extraction.

Apart from a white noise of uniform power, the actual vibration signals are also mixed with an uneven colored noise. To smooth the interference components in the background of the colored noise, the eigenvalues are modified by the diagonal loading technique [37] as follows:

$$
\mu\_i = \sigma\_i^2 + \sqrt{\sum\_{i=1}^n \sigma\_i^2}. \tag{15}
$$

Substituting the modified eigenvalues into the maximum likelihood estimation of the signals, the improved AIC function becomes as expressed in Equation (16). Therefore, the adaptive determination of the singular components can be achieved by minimizing the AIC objective function for the signals containing the colored noise.

$$AIC(d) = -2N(n-d)\log\_{10}\left(\frac{\prod\_{i=d+1}^{n}\mu\_i\frac{1}{n-d}}{\frac{1}{n-d}\sum\_{i=d+1}^{n}\mu\_i}\right) + 2d(2n-d).\tag{16}$$

### *3.3. Denoising of Akaike Information Criterion–Singular Value Decomposition*

Combining the energy characteristics and AIC-based order determination of the singular values, a signal denoising method called AIC–SVD is proposed, as shown in Figure 2. The detailed steps of the method can be described as follows:


$$\mathfrak{X}(i) = \frac{1}{h-l+1} \sum\_{j=l}^{h} \hat{A}(i-j+1, j), \tag{17}$$

where *i* = 1, 2, ... , *N*, *l* = max(1, *i* − *n* + 1), and *h* = min(*n*, *i*).

**Figure 2.** Flow chart of the signal denoising method using AIC–SVD. SVD: singular value.decomposition.

### **4. Simulation of Akaike Information Criterion–Singular Value Decomposition**

#### *4.1. Numerical Simulation*

To verify the effectiveness of AIC–SVD in signal denoising, simulation experiments are performed with signal *s*1, *s*<sup>2</sup> and *s*<sup>3</sup> mixed with a Gaussian white noise of 5 dB. At a sampling rate of 1 kHz and sampling time of 1 s, the corresponding waveform diagrams of the clean and polluted signals are shown in Figure 3. After selecting 501 × 500 Hankel matrices to construct the trajectory matrix of the signals, the singular values and the AIC values are calculated, as shown in Figure 4. Concurrently, the relevant parameters are extracted and listed in Table 1.

**Figure 3.** Waveform diagrams of simulation signals: (**a**) clean signals; (**b**) polluted signals with Gaussian white noise of 5 dB.

**Figure 4.** Calculation of simulation signals based on AIC–SVD: (**a**) singular values; (**b**) AIC values.


<sup>1</sup> Energy ratio is the energy ratio of clean signals to polluted signals.

As listed in Table 1, the minimum AIC value indices of the above-mentioned three signals are 6, 4, and 46, respectively. Concurrently, the corresponding effective singular spectral values are 89.19%, 54.82%, and 66.75%. The values of the valid singular spectrum are extremely close to the energy ratio of the initial pure signals, and the maximum error is 8.36%. This illustrates that the AIC exhibits a high performance for the order determination of singular values. To determine the reliability of the method, AIC–SVD is compared to WTD and EMD–SG by reconstructing the signals. The processed signals are shown in Figures 5–7.

**Figure 5.** Comparison of denoising effects by different methods for signal *s*1. WTD: wavelet threshold denoising; EMD-SG: empirical mode decomposition with Savitzky–Golay filter.

**Figure 6.** Comparison of denoising effects by different methods for signal *s*2.

**Figure 7.** Comparison of denoising effects by different methods for signal *s*3.

The comparison reveals that the signals processed by AIC–SVD are well restored by the morphology of the pure signal without a phase shift. For a periodic signal, the denoising effects of WTD and EMD–SG are similar overall to that of AIC–SVD. However, the attenuated signal and swept frequency signal have a notable issue. Specifically, the reconstructed signals exhibit a major waveform distortion, which is not conducive to the subsequent extraction and analysis of the features. The denoising method of AIC–SVD can prevent signal distortion while effectively removing noise. With zero phase shift characteristics, the method of AIC–SVD is suitable in the denoising of different types of signals.

### *4.2. Denoising Performance Evaluation*

To describe the performance of denoising more intuitively and accurately, the simulation signals are further quantitatively analyzed by combining the SNR, root mean square error (RMSE), and waveform correlation coefficient (NCC). These evaluation indicators are defined as follows [38]:

$$SNR = 10\log\_{10}\frac{\sum\_{i=1}^{N} \mathbf{x}(i)^2}{\sum\_{i=1}^{N} \left[\mathbf{x}(i) - \hat{\mathbf{x}}(i)\right]^2},\tag{18}$$

$$RMSE = \sqrt{\frac{1}{N} \sum\_{i=1}^{N} \left[ \mathbf{x}(i) - \hat{\mathbf{x}}(i) \right]^2},\tag{19}$$

$$\text{NCC} = \frac{\sum\_{i=1}^{N} \text{x}(i) \cdot \text{x}(i)}{\sqrt{\left(\sum\_{i=1}^{N} \text{x}(i)^2\right) \cdot \left(\sum\_{i=1}^{N} \text{x}(i)^2\right)}} \cdot \tag{20}$$

The SNR and RMSE reflect the global characteristics of the denoising performance, whereas the NCC describes the local characteristics of the signals. To avoid the limitations of a single evaluation index, a comprehensive evaluation index (CID) of denoising is introduced by integrating the SNR, RMSE and NCC. It can be defined as

$$\text{CID} = \frac{\text{SNR} \cdot \text{NCC}}{\text{RMSE}}.\tag{21}$$

According to Equation (21), a large value of CID corresponds to a good performance in signal denoising. For the simulation signals, the denoising performance parameters of different methods are calculated and listed in Table 2. Subsequently, Gaussian white noise with different SNRs (2 dB, 5 dB, and 10 dB) is added to the pure signals. The CID values of the denoising at the different SNRs are shown in Figure 8.


**Table 2.** Denoising performance parameters at SNR of 5 dB.

RMSE: root mean square error; NCC: waveform correlation coefficient; CID: comprehensive evaluation index.

**Figure 8.** Comparison of the CID for the different denoising methods.

The data in Table 2 prove that the SNRs of the signals are improved after denoising by both the methods, of which AIC–SVD leads to the largest increase. The minimum NCC value of AIC–SVD is 0.954, which can preserve the local waveform characteristics of the initial signal well, avoiding signal distortion. In Figure 8, the CID values of the different denoising methods increase with the improvement in the SNR of the initial signal, and the overall denoising performance of AIC–SVD is significantly better than those of the compared methods. Specifically, for the attenuated signal, the corresponding CID value of AIC–SVD at a 5 dB SNR is 657, which is much larger than those of the other methods. The powerful denoising performance of AIC–SVD for the attenuated signal shows that it is an effective pre-processing tool for vibration signals with pulse characteristics.
