Sensors 2019, 19, 1594 5 of 23
of the matrices in
I
m
can be denoted as
A
I
m
=
∑
i∈I
m
A
i
. Adding all
A
I
m
together will reconstruct the
original Hankel matrix A.
Step 4: Diagonal averaging
A common way to recover the signal
X
is called the diagonal averaging method, which uses the
average of the diagonal elements of reconstructed Hankel matrix as the element of X.
3. Proposed Denoising Method
3.1. Adaptive VMD
As discussed before, the performance of VMD is heavily dependent on the choice of its parameters,
especially for the number of modes
K
. Some related studies have discussed the parameter selection
problem of VMD, which mainly focus on optimizing by heuristic optimization algorithms. For example,
a searching method for quadratic penalty term
α
and mode number
K
is presented in [
24
] based on
the artificial fish swarm (AFS) algorithm. Besides, the particle swarm optimization (PSO) algorithm
has also been applied to VMD in [
8
]. Although these methods can obtain appropriate parameters to
some extent, they may need to go through many iterations before convergence. Moreover, one may
have to set initial parameters of heuristic optimization algorithms manually, making the VMD method
more complicated.
In this paper, the quadratic penalty term a and time-step of the dual ascent
τ
are respectively
set to 2000 and 0, and we focus on the optimization method of the mode number K. The basic idea
behind this adaptive VMD (AVMD) method is simple and straight: increase the value of K (starting
from 2) step by step and obtain the BLIMFs by VMD during each step. When there is no mode mixing
happened in all BLIMFs for a certain K, then this value is considered as optimal. Based on this concept,
the schematic diagram of the proposed method is demonstrated in Figure 1.
Sensors 2019, 19 FOR PEER REVIEW 5
of the matrices in
m
together will reconstruct the
original Hankel matrix
.
Step 4: Diagonal averaging
A common way to recover the signal
is called the diagonal averaging method, which uses
the average of the diagonal elements of reconstructed Hankel matrix as the element of
.
3. Proposed Denoising Method
3.1. Adaptive VMD
As discussed before, the performance of VMD is heavily dependent on the choice of its
parameters, especially for the number of modes
. Some related studies have discussed the
parameter selection problem of VMD, which mainly focus on optimizing by heuristic optimization
algorithms. For example, a searching method for quadratic penalty term
α
and mode number
is presented in [24] based on the artificial fish swarm (AFS) algorithm. Besides, the particle swarm
optimization (PSO) algorithm has also been applied to VMD in [8]. Although these methods can
obtain appropriate parameters to some extent, they may need to go through many iterations before
convergence. Moreover, one may have to set initial parameters of heuristic optimization algorithms
manually, making the VMD method more complicated.
In this paper, the quadratic penalty term a and time-step of the dual ascent τ are respectively set
to 2000 and 0, and we focus on the optimization method of the mode number K. The basic idea behind
this adaptive VMD (AVMD) method is simple and straight: increase the value of K (starting from 2)
step by step and obtain the BLIMFs by VMD during each step. When there is no mode mixing
happened in all BLIMFs for a certain K, then this value is considered as optimal. Based on this concept,
the schematic diagram of the proposed method is demonstrated in Figure 1.
Figure 1. Flow chart of the proposed AVMD.
As can be seen from Figure 1, the key of this method is to determine whether there is mode
mixing occurred in decomposed BLIMFs. For this purpose, the first important task is to find out the
local maximum points (LMPs) of BLIMF spectrum, which are considered to belong to the potential
effective components. This can be achieved by the following two steps:
(i) Preliminary screening
To begin with, the frequency spectrum is divided into several consecutive segments, and each
segment has the same length
1
L
(the last segment may not). Then, the maximal point in each
segment will be picked out to form a new sequence
max max max
{ 1 ,..., ,...., }segmax seg seg n seg N
−− −
=
,
where
max
eg n
−
denotes the maximal point of the nth segment, and
max
N
is the number of segments.
Figure 1. Flow chart of the proposed AVMD.
As can be seen from Figure 1, the key of this method is to determine whether there is mode mixing
occurred in decomposed BLIMFs. For this purpose, the first important task is to find out the local
maximum points (LMPs) of BLIMF spectrum, which are considered to belong to the potential effective
components. This can be achieved by the following two steps:
(i) Preliminary screening
To begin with, the frequency spectrum is divided into several consecutive segments, and each
segment has the same length L
1
(the last segment may not). Then, the maximal point in each segment