Time-Varying Weighted Optimal Empirical Mode Decomposition Using Multiple Sets of Basis Functions

Abstract

Empirical mode decomposition (EMD) is a favorite tool for analyzing nonlinear and non-stationary signals. It decomposes any signal into a finite set of oscillation modes consisting of intrinsic mode functions and a residual function. Superimposing all these modes reconstructs the signal without any information loss. In addition to satisfying the perfect reconstruction property, however, there is no implication about the reconstruction optimality of the EMD. The lack of optimality restricts the signal recovery capability of the EMD in the presence of disturbances. Only a few attempts are made to meet this deficiency. In this paper, we propose a new algorithm named as time-varying weighted EMD. By this algorithm, original signal is reconstructed in the minimum mean-square error sense through the EMD followed by time-varying weightings of the oscillation modes. Determining the time-varying weights for the oscillation modes constitutes the backbone of the algorithm. Aiming to determine the time-varying weights of the oscillation modes; we use multiple sets of basis functions. The effectiveness of the proposed algorithm is demonstrated by computer simulations involving real biomedical signals. Simulation results show that the proposed algorithm exhibits better performance than that of its existing counterparts in terms of lower mean-square error and higher signal-to-error ratio.

Appendix: Description of the Sub-blocks in Matrix \({\varvec{\Delta }}\)

Sub-blocks in the matrix (23) are described as follows:

$$\begin{aligned} {\varvec{\Delta }}^{\alpha \alpha }= & {} \left[ {{\begin{array}{lll} {{\varvec{\Delta }}_{1,1}^{\alpha \alpha } }&{} \quad \cdots &{} \quad {{\varvec{\Delta }}_{1,N}^{\alpha \alpha } } \\ \vdots &{} \quad \ddots &{} \quad \vdots \\ {{\varvec{\Delta }}_{N,1}^{\alpha \alpha } }&{} \quad \cdots &{} \quad {{\varvec{\Delta }}_{N,N}^{\alpha \alpha } } \\ \end{array} }} \right] , \end{aligned}$$

(24)

$$\begin{aligned} {\varvec{\Delta }}^{\alpha \beta }= & {} \left[ {{\begin{array}{lll} {{\varvec{\Delta }}_{1,N+1}^{\alpha \beta } }&{} \quad \cdots &{} \quad {{\varvec{\Delta }}_{1,M+1}^{\alpha \beta } } \\ \vdots &{} \quad \ddots &{} \quad \vdots \\ {{\varvec{\Delta }}_{N,N+1}^{\alpha \beta } }&{} \quad \cdots &{} \quad {{\varvec{\Delta }}_{N,M+1}^{\alpha \beta } } \\ \end{array} }} \right] , \end{aligned}$$

(25)

$$\begin{aligned} {\varvec{\Delta }}^{\beta \alpha }= & {} \left[ {{\begin{array}{lll} {{\varvec{\Delta }}_{N+1,1}^{\beta \alpha } }&{} \quad \cdots &{} \quad {{\varvec{\Delta }}_{N+1,N}^{\beta \alpha } } \\ \vdots &{} \quad \ddots &{} \quad \vdots \\ {{\varvec{\Delta }}_{M+1,1}^{\beta \alpha } }&{} \quad \cdots &{} \quad {{\varvec{\Delta }}_{M+1,N}^{\beta \alpha } } \\ \end{array} }} \right] , \end{aligned}$$

(26)

$$\begin{aligned} {\varvec{\Delta }}^{\beta \beta }= & {} \left[ {{\begin{array}{lll} {{\varvec{\Delta }}_{N+1,N+1}^{\beta \beta } }&{} \quad \cdots &{} \quad {{\varvec{\Delta }}_{N+1,M+1}^{\beta \beta } } \\ \vdots &{} \quad \ddots &{} \quad \vdots \\ {{\varvec{\Delta }}_{M+1,N+1}^{\beta \beta } }&{} \quad \cdots &{} \quad {{\varvec{\Delta }}_{M+1,M+1}^{\beta \beta } } \\ \end{array} }} \right] , \end{aligned}$$

(27)

each of which is $N\left( {M-N+1} \right) \times N\left( {M-N+1} \right)$ matrix.

From (12) to (14) and (16) to (18), the entries of matrices defined by (24)–(27) are defined as in the following forms:

$$\begin{aligned} {\varvec{\Delta }}_{l,m}^{\alpha \alpha }= & {} \left[ {{\begin{array}{lll} {\varDelta _{l,0,m,0}^{\alpha \alpha } }&{} \quad \cdots &{} \quad {\varDelta _{l,0,m,M-N}^{\alpha \alpha } } \\ \vdots &{} \quad \ddots &{} \quad \vdots \\ {\varDelta _{l,M-N,m,0}^{\alpha \alpha } }&{} \quad \cdots &{} \quad {\varDelta _{l,M-N,m,M-N}^{\alpha \alpha } } \\ \end{array} }} \right] , \end{aligned}$$

(28)

$$\begin{aligned} {\varvec{\Delta }}_{l,z}^{\alpha \beta }= & {} \left[ {{\begin{array}{lll} {\varDelta _{l,0,z,0}^{\alpha \beta } }&{} \quad \cdots &{} \quad {\Delta _{l,0,z,N-1}^{\alpha \beta } } \\ \vdots &{} \ddots &{} \vdots \\ {\varDelta _{l,M-N,z,0}^{\alpha \beta } }&{} \quad \cdots &{} \quad {\varDelta _{l,M-N,z,N-1}^{\alpha \beta } } \\ \end{array} }} \right] , \end{aligned}$$

(29)

$$\begin{aligned} {\varvec{\Delta }}_{y,m}^{\beta \alpha }= & {} \left[ {{\begin{array}{lll} {\varDelta _{y,0,m,0}^{\beta \alpha } }&{} \quad \cdots &{} \quad {\varDelta _{y,0,m,M-N}^{\beta \alpha } } \\ \vdots &{} \quad \ddots &{} \quad \vdots \\ {\varDelta _{y,N-1,m,0}^{\beta \alpha } }&{} \quad \cdots &{} \quad {\varDelta _{y,N-1,m,M-N}^{\beta \alpha } } \\ \end{array} }} \right] , \end{aligned}$$

(30)

$$\begin{aligned} {\varvec{\Delta }}_{y,z}^{\beta \beta }= & {} \left[ {{\begin{array}{lll} {\varDelta _{y,0,z,0}^{\beta \beta } }&{} \quad \cdots &{} \quad {\varDelta _{y,0,z,N-1}^{\beta \beta } } \\ \vdots &{} \ddots &{} \vdots \\ {\varDelta _{y,N-1,z,0}^{\beta \beta } }&{} \quad \cdots &{} \quad {\varDelta _{y,N-1,z,N-1}^{\beta \beta } } \\ \end{array} }} \right] , \end{aligned}$$

(31)

where $1\le \left( {l,m} \right) \le N$ and $N+1\le \left( {y,z} \right) \le M+1$. Matrices in (28)–(31) have a size of $\left( {M-N+1} \right) \times \left( {M-N+1} \right)$, $\left( {M-N+1} \right) \times N$, $N\times \left( {M-N+1} \right)$, and $N\times N$, respectively.