Recursive Windowed Variational Mode Decomposition

Abstract

The variational mode decomposition (VMD) and its variants aim to decompose a given signal into a set of narrow band modes. The analysis of these modes is usually based on the Fourier analysis. That is, the center frequencies of these modes are found without exploiting the local time varying information of the signal during the iteration in the existing algorithms for performing the VMD. To address this issue, this paper proposes a recursive windowed VMD (RWVMD) approach for performing the signal decomposition. First, the window is sliding across the signal. Then, the variational mode extraction is performed on each frame to obtain the first mode. Then, the difference between the first mode and the signal is computed to obtain the residual signal. The above process is repeated on the residual signal until the algorithm converges. The effectiveness of the RWVMD algorithm is demonstrated through the computer numerical simulations. It is found that the center frequency in the time frequency plane is more accurately matched with the characteristics of the original signal.

Appendix

For the discrete realization, $\omega$ is usually sampled using a finite number of points. Let $M$ be the total number of the sampling points and $\tilde{\omega }_{1} , \ldots ,\tilde{\omega }_{M}$ be these sampling points. Then, we have

$$ {\hat{\mathbf{u}}}_{k,\tau } = \left[ {\begin{array}{*{20}c} {\hat{u}_{k} \left( {\tilde{\omega }_{1} ,\tau } \right)} & {\hat{u}_{k} \left( {\tilde{\omega }_{2} ,\tau } \right)} & \ldots & {\hat{u}_{k} \left( {\tilde{\omega }_{M} ,\tau } \right)} \\ \end{array} } \right]^{T} , $$

(40)

$$ {\hat{\mathbf{f}}}_{\tau } = \left[ {\begin{array}{*{20}c} {\hat{f}\left( {\tilde{\omega }_{1} ,\tau } \right)} & {\hat{f}\left( {\tilde{\omega }_{2} ,\tau } \right)} & \ldots & {\hat{f}\left( {\tilde{\omega }_{M} ,\tau } \right)} \\ \end{array} } \right]^{T} , $$

(41)

$$ {\hat{\mathbf{f}}}_{r,\tau } = \left[ {\begin{array}{*{20}c} {\hat{f}_{r} \left( {\tilde{\omega }_{1} ,\tau } \right)} & {\hat{f}_{r} \left( {\tilde{\omega }_{2} ,\tau } \right)} & \ldots & {\hat{f}_{r} \left( {\tilde{\omega }_{M} ,\tau } \right)} \\ \end{array} } \right]^{T} , $$

(42)

$$ {\hat{\mathbf{\lambda }}} = \left[ {\begin{array}{*{20}c} {\lambda \left( {\tilde{\omega }_{1} ,\tau } \right)} & {\lambda \left( {\tilde{\omega }_{2} ,\tau } \right)} & \ldots & {\lambda \left( {\tilde{\omega }_{M} ,\tau } \right)} \\ \end{array} } \right]^{T} $$

(43)

and

$$ {\mathbf{W}} = \left[ {\begin{array}{*{20}c} {\tilde{\omega }_{1} - \omega_{k} \left( \tau \right)} & {} & {} & {} \\ {} & {\tilde{\omega }_{2} - \omega_{k} \left( \tau \right)} & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & {\tilde{\omega }_{M} - \omega_{k} \left( \tau \right)} \\ \end{array} } \right]. $$

(44)

The discrete realization of the augmented Lagrangian function can be expressed as:

$$ \begin{gathered} L\left( {{\hat{\mathbf{u}}}_{k,\tau } ,\omega_{k} \left( \tau \right),{\hat{\mathbf{f}}}_{r,\tau } ,{\hat{\mathbf{\lambda }}}} \right) = 4\alpha \sum\nolimits_{m = 1}^{M} {\left( {\tilde{\omega }_{m} - \omega_{k} \left( \tau \right)} \right)^{2} \left| {\hat{u}_{k} \left( {\tilde{\omega }_{m} ,\tau } \right)} \right|^{2} } + 2\sum\nolimits_{m = 1}^{M} {\left| {\frac{{\hat{f}_{r} \left( {\tilde{\omega }_{m} ,\tau } \right)}}{{\alpha \left( {\tilde{\omega }_{m} - \omega_{k} \left( \tau \right)} \right)^{2} }}} \right|^{2} } \\ + 2\sum\nolimits_{m = 1}^{M} {\left| {\hat{f}\left( {\tilde{\omega }_{m} ,\tau } \right) - \hat{u}_{k} \left( {\tilde{\omega }_{m} ,\tau } \right) - \hat{f}_{r} \left( {\tilde{\omega }_{m} ,\tau } \right) + \frac{{\hat{\lambda }\left( {\tilde{\omega }_{m} ,\tau } \right)}}{2}} \right|^{2} - \left| {\frac{{\hat{\lambda }\left( {\tilde{\omega }_{m} ,\tau } \right)}}{2}} \right|^{2} } \\ = 4\alpha \left\| {{\mathbf{W\hat{u}}}_{k,\tau } } \right\|_{2}^{2} + \frac{2}{{\alpha^{2} }}\left\| {{\mathbf{W}}^{ - 2} {\hat{\mathbf{f}}}_{r,\tau } } \right\|_{2}^{2} + 2\left( {\left\| {{\hat{\mathbf{f}}}_{\tau } - {\hat{\mathbf{u}}}_{k,\tau } - {\hat{\mathbf{f}}}_{r,\tau } + \frac{{{\hat{\mathbf{\lambda }}}}}{2}} \right\|_{2}^{2} - \left\| {\frac{{{\hat{\mathbf{\lambda }}}}}{2}} \right\|_{2}^{2} } \right). \\ \end{gathered} $$

(45)

Here, the gradients are as follow:

$$ \frac{{\partial L\left( {{\hat{\mathbf{u}}}_{k,\tau } ,\omega_{k} \left( \tau \right),{\hat{\mathbf{f}}}_{r,\tau } ,{\hat{\mathbf{\lambda }}}} \right)}}{{\partial {\hat{\mathbf{u}}}_{k,\tau } }} = 4\alpha {\mathbf{W}}^{2} {\hat{\mathbf{u}}}_{k,\tau } - 2\left( {{\hat{\mathbf{f}}}_{\tau } - {\hat{\mathbf{u}}}_{k,\tau } - {\hat{\mathbf{f}}}_{r,\tau } + \frac{{{\hat{\mathbf{\lambda }}}}}{2}} \right), $$

(46)

$$ \frac{{\partial L\left( {{\hat{\mathbf{u}}}_{k,\tau } ,\omega_{k} \left( \tau \right),{\hat{\mathbf{f}}}_{r,\tau } ,{\hat{\mathbf{\lambda }}}} \right)}}{{\partial \omega_{k} \left( \tau \right)}} = - 8\alpha \sum\nolimits_{m = 1}^{M} {\left( {\tilde{\omega }_{m} - \omega_{k} \left( \tau \right)} \right)\left| {\hat{u}_{k} \left( {\tilde{\omega }_{m} ,\tau } \right)} \right|^{2} } - \frac{8}{{\alpha^{2} }}\sum\nolimits_{m = 1}^{M} {\frac{{\left| {\hat{f}_{r} \left( {\tilde{\omega }_{m} ,\tau } \right)} \right|^{2} }}{{\left( {\tilde{\omega }_{m} - \omega_{k} \left( \tau \right)} \right)^{5} }}} $$

(47)

and

$$ \frac{{\partial L\left( {{\hat{\mathbf{u}}}_{k,\tau } ,\omega_{k} \left( \tau \right),{\hat{\mathbf{f}}}_{r,\tau } ,{\hat{\mathbf{\lambda }}}} \right)}}{{\partial {\hat{\mathbf{f}}}_{r,\tau } }} = \frac{2}{{\alpha^{2} }}{\mathbf{W}}^{ - 4} {\hat{\mathbf{f}}}_{r,\tau } - 2\left( {{\hat{\mathbf{f}}}_{\tau } - {\hat{\mathbf{u}}}_{k,\tau } - {\hat{\mathbf{f}}}_{r,\tau } + \frac{{{\hat{\mathbf{\lambda }}}}}{2}} \right). $$

(48)

As the parameter $\alpha$ is usually large, (A.7) can be approximated as:

$$ \frac{{\partial L\left( {{\hat{\mathbf{u}}}_{k,\tau } ,\omega_{k} \left( \tau \right),{\hat{\mathbf{f}}}_{r,\tau } ,{\hat{\mathbf{\lambda }}}} \right)}}{{\partial \omega_{k} \left( \tau \right)}} \approx - 8\alpha \sum\nolimits_{m = 1}^{M} {\left( {\tilde{\omega }_{m} - \omega_{k} \left( \tau \right)} \right)\left| {\hat{u}_{k} \left( {\tilde{\omega }_{m} ,\tau } \right)} \right|^{2} } . $$

(49)

Set the gradients to zeros, we have:

$$ 4\alpha {\mathbf{W}}^{2} {\hat{\mathbf{u}}}_{k,\tau } - 2\left( {{\hat{\mathbf{f}}}_{\tau } - {\hat{\mathbf{u}}}_{k,\tau } - {\hat{\mathbf{f}}}_{r,\tau } + \frac{{{\hat{\mathbf{\lambda }}}}}{2}} \right) = 0, $$

(50a)

$$ \left( {{\mathbf{I}} + 2\alpha {\mathbf{W}}^{2} } \right){\hat{\mathbf{u}}}_{k,\tau } = \left( {{\hat{\mathbf{f}}}_{\tau } - {\hat{\mathbf{f}}}_{r,\tau } + \frac{{{\hat{\mathbf{\lambda }}}}}{2}} \right), $$

(50b)

$$ {\hat{\mathbf{u}}}_{k,\tau } = \left( {{\mathbf{I}} + 2\alpha {\mathbf{W}}^{2} } \right)^{ - 1} \left( {{\hat{\mathbf{f}}}_{\tau } - {\hat{\mathbf{f}}}_{r,\tau } + \frac{{{\hat{\mathbf{\lambda }}}}}{2}} \right), $$

(50c)

$$ \sum\nolimits_{m = 1}^{M} {\left( {\tilde{\omega }_{m} - \omega_{k} \left( \tau \right)} \right)\left| {\hat{u}_{k} \left( {\tilde{\omega }_{m} ,\tau } \right)} \right|^{2} } = 0, $$

(51)

$$ \left( {\sum\nolimits_{m = 1}^{M} {\left| {\hat{u}_{k} \left( {\tilde{\omega }_{m} ,\tau } \right)} \right|^{2} } } \right)\omega_{k} \left( \tau \right) = \sum\nolimits_{m = 1}^{M} {\tilde{\omega }_{m} \left| {\hat{u}_{k} \left( {\tilde{\omega }_{m} ,\tau } \right)} \right|^{2} } , $$

(52)

$$ \omega_{k} \left( \tau \right) = \frac{{\sum\nolimits_{m = 1}^{M} {\tilde{\omega }_{m} \left| {\hat{u}_{k} \left( {\tilde{\omega }_{m} ,\tau } \right)} \right|^{2} } }}{{\left( {\sum\nolimits_{m = 1}^{M} {\left| {\hat{u}_{k} \left( {\tilde{\omega }_{m} ,\tau } \right)} \right|^{2} } } \right)}}, $$

(53)

$$ \frac{2}{{\alpha^{2} }}{\mathbf{W}}^{ - 4} {\hat{\mathbf{f}}}_{r,\tau } - 2\left( {{\hat{\mathbf{f}}}_{\tau } - {\hat{\mathbf{u}}}_{k,\tau } - {\hat{\mathbf{f}}}_{r,\tau } + \frac{{{\hat{\mathbf{\lambda }}}}}{2}} \right) = 0, $$

(54a)

$$ \left( {{\mathbf{I}} + \frac{1}{{\alpha^{2} }}{\mathbf{W}}^{ - 4} } \right){\hat{\mathbf{f}}}_{r,\tau } = \left( {{\hat{\mathbf{f}}}_{\tau } - {\hat{\mathbf{u}}}_{k,\tau } + \frac{{{\hat{\mathbf{\lambda }}}}}{2}} \right) $$

(54b)

and

$$ {\hat{\mathbf{f}}}_{r,\tau } = \left( {{\mathbf{I}} + \frac{1}{{\alpha^{2} }}{\mathbf{W}}^{ - 4} } \right)^{ - 1} \left( {{\hat{\mathbf{f}}}_{\tau } - {\hat{\mathbf{u}}}_{k,\tau } + \frac{{{\hat{\mathbf{\lambda }}}}}{2}} \right). $$

(54c)

This further implies:

$$ \hat{u}_{k} \left( {\tilde{\omega }_{m} ,\tau } \right) = \frac{{\hat{f}\left( {\tilde{\omega }_{m} ,\tau } \right) - \hat{f}_{r} \left( {\tilde{\omega }_{m} ,\tau } \right) + \frac{{\hat{\lambda }\left( {\tilde{\omega }_{m} ,\tau } \right)}}{2}}}{{1 + 2\alpha \left( {\tilde{\omega }_{m} - \omega_{k} \left( \tau \right)} \right)^{2} }} $$

(55)

and

$$ \begin{gathered} \hat{f}_{r} \left( {\tilde{\omega }_{m} ,\tau } \right) = \frac{{\hat{f}\left( {\tilde{\omega }_{m} ,\tau } \right) - \hat{u}_{k} \left( {\tilde{\omega }_{m} ,\tau } \right) + \frac{{\hat{\lambda }\left( {\tilde{\omega }_{m} ,\tau } \right)}}{2}}}{{1 + \frac{1}{{\alpha^{2} \left( {\tilde{\omega }_{m} - \omega_{k} \left( \tau \right)} \right)^{4} }}}} \\ = \frac{{\alpha^{2} \left( {\tilde{\omega }_{m} - \omega_{k} \left( \tau \right)} \right)^{4} \left( {\hat{f}\left( {\tilde{\omega }_{m} ,\tau } \right) - \hat{u}_{k} \left( {\tilde{\omega }_{m} ,\tau } \right) + \frac{{\hat{\lambda }\left( {\tilde{\omega }_{m} ,\tau } \right)}}{2}} \right)}}{{1 + \alpha^{2} \left( {\tilde{\omega }_{m} - \omega_{k} \left( \tau \right)} \right)^{4} }} \\ \end{gathered} $$

(56)

for $m = 1,...,M$. By representing (53), (55), (56) in the continuous form, we have:

$$ \hat{u}_{k} \left( {\omega ,\tau } \right) = \frac{{\hat{f}\left( {\omega ,\tau } \right) - \hat{f}_{r} \left( {\omega ,\tau } \right) + \frac{{\hat{\lambda }\left( {\omega ,\tau } \right)}}{2}}}{{1 + 2\alpha \left( {\omega - \omega_{k} \left( \tau \right)} \right)^{2} }}, $$

(57)

$$ \omega_{k} \left( \tau \right) = \frac{{\int_{0}^{\infty } {\omega \left| {\hat{u}_{k,m} \left( {\omega ,\tau } \right)} \right|^{2} d\omega } }}{{\int_{0}^{\infty } {\left| {\hat{u}_{k,m} \left( {\omega ,\tau } \right)} \right|^{2} d\omega } }} $$

(58)

and

$$ \hat{f}_{r} \left( {\omega ,\tau } \right) = \frac{{\alpha^{2} \left( {\omega - \omega_{k} \left( \tau \right)} \right)^{4} \left( {\hat{f}\left( {\omega ,\tau } \right) - \hat{u}_{k} \left( {\omega ,\tau } \right) + \frac{{\hat{\lambda }\left( {\omega ,\tau } \right)}}{2}} \right)}}{{1 + \alpha^{2} \left( {\omega - \omega_{k} \left( \tau \right)} \right)^{4} }}. $$

(59)

Then, the ADMM updated rules can be obtained accordingly.