2.2. Principle of SSD

Singular spectral decomposition (SSD) is a new adaptive signal processing method recently proposed. It can decompose the non-linear and non-stationary signals from a high frequency to a low frequency into the sum of several singular spectral components (SSC) and residual terms. The specific process is as follows:

First, a new trajectory matrix is constructed. For a time series x(n), its data length and embedding dimensions are N and M, respectively. It is constructed as a matrix of X of N columns and M rows, and the i-th row of matrix X is Xi=(x(i),⋯x(N),x(1),⋯x(i−1)), and i=1,⋯,M; that is, matrix X=[x1T,x2T,⋯,xMT]T. Selecting K=N−M+1, the lower-right corner of matrix X is moved to the upper-left position of matrix X, and the improved trajectory matrix X(N×K) is obtained. The improved trajectory matrix can enhance the vibration component of the original signal and make the residual component decrease after iteration.

The embedding dimension M is selected adaptively. Considering the defect, SSA chooses the embedding dimension according to experience, and the adaptive rule is used to select the embedding dimension M used in the j-th iteration. Firstly, the power spectral density (PSD) of the residual component vj(n) at the j-th iteration is calculated, where the residual component vj(n) is

The frequency fmax corresponding to the maximum peak value in the PSD is then estimated. In the first iteration, if the normalized frequency fmax/Fs is less than a given threshold of 10⁻³, the residual is considered as a large trend term, and M is set to N/3, where Fs is the sampling frequency. Otherwise, when the number of iterations J > 1, the embedding dimension is set to M=1.2Fs/fmax, which improves the analysis effect of SSA.

The j-th component signal is reconstructed in the order of high frequency to low frequency. In the first iteration, if a large trend item is detected, only the first or so feature cards are used to obtain g(1)(n), so that X1=σ1μ1v1T and g(1)(n) can be obtained from the diagonal average of X1. Otherwise, when the number of iterations J > 1, a sequence of components g(1)(n) must be obtained to describe a time scale with a clear physical meaning. In this sense, its frequency components are mainly concentrated in the frequency band [fmax−δf,fmax+δf], where δf represents the half bandwidth of the main peak in the residual power spectral density. Therefore, a subset Ij(Ij={i1,⋯,ip}) is created according to all the characteristic groups of the left eigenvector with prominent principal frequencies in the spectrum [fmax−δf,fmax+δf] range and one of the characteristic groups with the greatest contribution to the principal peak energy of the selected modal components. Then the corresponding component signals are reconstructed by the diagonal averaging method of the matrix XIj=Xi1+⋯+Xip.

Setting the stop condition of an iteration. The iteratively estimated component sequence g˜(j)(n) is separated from the original signal and a residual term v(j+1)(n)=v(j)(n)−g˜(j)(n) is obtained. The normalized mean square deviation between the residual term and the original signal is calculated; that is, the normalized mean square deviation between the residual term and the original signal:

When the normalized mean square deviation is less than the given threshold th = 1%, the whole decomposition process terminates. Otherwise, the residual term is used as the original signal to repeat the above iteration process until the iteration stopping condition is satisfied, and the final decomposition result is obtained:

where m is the number of component sequences obtained. It is noteworthy that after each iteration, the energy of the residual v(j+1)(n) decreases.

The SSD method has a higher decomposition accuracy and can better suppress the generation of modal aliasing and pseudo components. In order to compare the decomposition performance of SSD and EEMD, we construct a simulation signal, such as Equation (14), for comparison:

where f1=30Hz,fn1=20Hz,fz=150Hz. The simulation signal consists of a sinusoidal signal and a modulation signal with a modulation source. The waveform of the simulation signal is shown in Figure 6.

Time domain waveform of the simulation signal.

Figure 7 shows the time–frequency graph of each component of the simulation signal decomposed by EEMD, and Figure 8 shows the time–frequency graph of each component of the simulation signal decomposed by SSD. It can be seen intuitively that the decomposition performance of SSD is more excellent. The decomposed components are almost identical with the simulation signals, and there is no modal aliasing and no false components. However, mode aliasing occurs in IMF, IMF2, IMF3 and IMF4 after EEMD decomposition, which shows that the decomposition performance of SSD is more reliable. Therefore, this paper chooses the SSD method to stratify the original vibration signal.

The spectrum of intrinsic mode functions (IMFs) after ensemble empirical mode decomposition (EEMD) and its corresponding spectrum.

The spectrum of IMFs after singular spectral decomposition (SSD) and its corresponding spectrum.