Equation 7 can be written as:
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):
is a vector composed of
the elements of vector
, stands for the complex conjugate of
is the shaping operator.
The elements of matrix
where is the regularization parameter.
Solving equation 11, we obtain the
and form a polynomial below:
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:
From the instantaneous frequency, we compute the local phase according to the following equation:
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