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Successive multivariate variational mode decomposition

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Abstract

This paper presents an extension of variational mode decomposition (VMD) for successively extracting the modes of multi-sensor data sets. First, we achieve the multi-channel extension of the univariate mode by introducing the multivariate modulated oscillation model, which can take the correlation between multiple data channels into account. Then the successive scheme is accomplished by adding some new criteria to VMD: the current extracting mode has no or less spectral overlap with the previously obtained modes and the residual signal. Finally, we employ the alternate direction method of the multiplier algorithm (ADMM) to solve it. Compared with other multivariate extending methods whose performances will be degraded if the number of modes is not precisely known, this extension can recursively extract modes and does not need to know the number of modes. Therefore, it achieves better performance on convergence and computation requirements. Moreover, it is more robust to the initial center frequency and possesses the mode-alignment property. We also investigate the relationships between the regularization parameter \(\alpha \) and the spectrum property of modes. Some suggestions for selecting proper solution parameters are provided. Finally, we show promising practical decomposition results on a series of simulating and real-life multi-channel data.

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Acknowledgements

The helpful and constructive comments from the referees have led to the improvements of this paper; the authors gratefully acknowledge this assistance.

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Correspondence to Kaiping Yu.

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Appendices

Appendix A

In this appendix, we show that Eq. (28) is tenable around the center frequency \(\omega _k\) and has the same effective bandwidth as Eq. (6). The filter in Eq. (20) can be expressed as:

$$\begin{aligned} \begin{aligned} {H_k}\left( {\omega ,{\omega _k},\left\{ {{\omega _i}} \right\} _{i = 1}^{k - 1}} \right) = \frac{1}{{1 + 2\alpha {{\left( {\omega - {\omega _k}} \right) }^2} + \sum \limits _{i = 1}^{k - 1} {\frac{1}{{{\alpha ^2}{{\left( {\omega - {\omega _i}} \right) }^4}}}} }} \end{aligned} \end{aligned}$$
(A.1)

Assume that the algorithm of SMVMD is converged to the true modes and also, the modes are spectrally distinct. When \(\omega \) is close to \(\omega _k\), then \(\left| {\omega \mathrm{{ - }}{\omega _i}} \right| \ge \rho > 0\) and the following relation is given:

$$\begin{aligned} 0< \sum \limits _{i = 1}^{k - 1} {\frac{1}{{{\alpha ^2}{{\left( {\omega - {\omega _k}} \right) }^4}}}} < \sum \limits _{i = 1}^{k - 1} {\frac{1}{{{\alpha ^2}{\rho ^4}}}} = \frac{{k - 1}}{{{\alpha ^2}{\rho ^4}}} \approx 0 \end{aligned}$$
(A.2)

where the last step results from that \(\alpha \) is usually very big and so \({{\alpha ^2}{\rho ^4}} \gg 0\). Therefore, the third term in denominator can be neglected, which result in Eq. (28). Finally, the bandwidth of it can be expressed as Eq. (6).

Appendix B

For multi-dimensional signals coming from multi-sensor measurement, such as the radar array, MIMF \({\varvec{{{\hat{u}}}}}_k(\omega )\) decomposed by MVMD or SMVMD can be expressed as:

$$\begin{aligned} {\varvec{{{\hat{u}}}}}_k(\omega )={{\varvec{a}}_k} {{{\hat{s}}}}_k{(\omega )} \end{aligned}$$
(B.1)

where \({{\varvec{a}}_k}\) with the same size as \({\varvec{{{\hat{u}}}}}_k(\omega )\) is the k-th receiver steering vector and \({{{\hat{s}}}}_k{(\omega )}\) denotes the corresponding source. The following operation can be conducted:

$$\begin{aligned} \begin{aligned} \int _{ - \infty }^{ + \infty } {{\varvec{{{\hat{u}}}}}_k(\omega )\cdot {{{\hat{u}}}}_{k,c}^{H}(\omega ) \mathrm{d} \omega }&=\int _{ - \infty }^{ + \infty } {{{{\varvec{a}}}_k} {{{\hat{s}}}}_k{(\omega )}\cdot {a_{k,c}} {{{\hat{s}}}}_k^{H}{(\omega )} \mathrm{d} \omega }\\&={{{\varvec{a}}}_k}\int _{ - \infty }^{ + \infty } {{a_{k,c}} \left| {{{{\hat{s}}}}_k{(\omega )}} \right| ^2 \mathrm{d} \omega }\\ \end{aligned} \end{aligned}$$
(B.2)

where \((*)^{H}\) is the conjugate operator, \({{{\hat{u}}}}_{k,c}(\omega )\) is the c-th channel of MIMF \({\varvec{{{\hat{u}}}}}_k(\omega )\), and \(a_{k,c}\) is the c-th channel of \({{\varvec{a}}}_k\). The above equation can be rewritten as:

$$\begin{aligned} \begin{aligned} {{{\varvec{a}}}_k} = \frac{\int _{ - \infty }^{ + \infty } {{\varvec{{{\hat{u}}}}}_k(\omega )\cdot {{{\hat{u}}}}_{k,c}^{H}(\omega ) \mathrm{d} \omega }}{\int _{ - \infty }^{ + \infty } {{a_{k,c}} \left| {{{{\hat{s}}}}_k{(\omega )}} \right| ^2 \mathrm{d} \omega }} \end{aligned} \end{aligned}$$
(B.3)

One can easily note that although the denominator of Eq. (B.3) is unknown, it is a scalar and makes no difference on the direction of \({{\varvec{a}}}_k\). Therefore, it is easy to obtain the following \({{\varvec{a}}}_k\) with the first element normalized:

$$\begin{aligned} \begin{aligned} {{{\varvec{a}}}_k} = \frac{\int _{ - \infty }^{ + \infty } {{{{\hat{\varvec{u}}}}}_k(\omega )\cdot {{{\hat{u}}}}_{k,c}^{H}(\omega ) \mathrm{d} \omega }}{\int _{ - \infty }^{ + \infty } {{{{\hat{u}}}}}_{k,1}(\omega )\cdot {{{\hat{u}}}}_{k,c}^{H}(\omega ) \mathrm{d} \omega } \end{aligned} \end{aligned}$$
(B.4)

Then \({{{\hat{s}}}}_k{(\omega )}\) can be recovered based on Eq. (B.1):

$$\begin{aligned} \begin{aligned} {{{\hat{s}}}}_k{(\omega )} = \frac{{{\varvec{a}}}_k^H{{{\hat{\varvec{u}}}}}_k(\omega )}{{{\varvec{a}}}_k^H{{\varvec{a}}}_k} \end{aligned} \end{aligned}$$
(B.5)

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Liu, S., Yu, K. Successive multivariate variational mode decomposition. Multidim Syst Sign Process 33, 917–943 (2022). https://doi.org/10.1007/s11045-022-00828-w

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