If the in equation 9 are time dependent, then we have:

which is an under-determined linear system. There are many methods for solving under-determined linear system, such as Tikhonov method (Tikhonov, 1963). In this paper, we apply shaping regularization (Fomel, 2009,2007) to regularize the under-determined linear system, and obtain (for details see Appendix):

where is a vector composed of , the elements of vector are , where , stands for the complex conjugate of and is the shaping operator. The elements of matrix are:

where is the regularization parameter. Solving equation 11, we obtain the coefficients vector and form a polynomial below:

(13) |

For the roots computation of the above polynomial, we use the method proposed by Toh and Trefethen (1994). The instantaneous frequency of each different component is derived from the following equation:

(14) |

From the instantaneous frequency, we compute the local phase according to the following equation:

(15) |

Solving the following equation using regularized non-stationary regression method (Fomel, 2013):

Finally the narrow-band intrinsic mode functions are computed based on equation 16

2020-07-18