1. Introduction
With the development of science and technology, nonstationary signal processing and its applications in engineering are gaining more and more attention. During recent decades, scholars have developed many approaches to process single-channel nonstationary signals, or even multi-channel ones, which are not discussed in this paper. Short-Time Fourier Transform (STFT) [
1] and Wavelet Transform (WT) [
2] are two of the most popular algorithms used to perform time–frequency (TF) transform on nonstationary signals. These transform methods exhibit limited TF resolutions [
3], and cannot separate a multi-component signal into mono-components. These sometimes suffer from the consequences of the Heisenberg uncertainty principle. However, data-driven signal decomposition methods can decompose a multi-component signal into several modes—for example, Empirical Mode Decomposition (EMD) [
4], and Variational Mode Decomposition (VMD) [
5]. We develop a new signal decomposition method here.
Variational Mode Decomposition (VMD) [
5] and Variational Nonlinear Chirp Mode Decomposition (VNCMD) [
6] are proposed to adaptively extract a set of modes, which are called Intrinsic Mode Functions (IMFs). VMD is a non-recursive algorithm method to decompose a signal into several modes with quasi-orthogonality, intrinsics, and adaptivity [
7]. VMD can concurrently look for the IMFs and their respective center frequencies. Each IMF is compact at a particular band. Unlike the EMD-based methods, VMD is built on well-founded mathematical theories.
Several other VMD-based algorithms have emerged. Due to the difficulty of selecting the mode number, successive VMDs (SVMD) [
8] need not predefine the mode number
. The adaptive chirp mode pursuit (ACMP) [
9] is proposed to recursively extract the nonlinear chirp modes. However, VNCMD and ACMP require high-limited instantaneous frequency (IF) initialization [
6,
9], and VMD and SVMD suffer from the narrowband assumption of IMFs.
The VMD was proposed as a one-dimensional algorithm [
5], and a two-dimensional algorithm was later published [
10,
11]. Then, multivariate VMD (MVMD) [
12] was developed to achieve a better performance than the direct use of univariate VMD in a channel-by-channel method. However, MVMD still suffered from the limited narrowband assumption, and the VMD-based developed algorithms could not decompose signals composed of wideband multivariate IMFs (MIMFs). A multivariate nonlinear chirp mode decomposition (MNCMD) and its improved version, multivariate intrinsic chirp mode decomposition (MICMD) [
13], were developed. These two algorithms could process multichannel signals involving wideband MIMFs.
The VMD has attracted a broad variety of time–frequency analysis applications, such as signal decomposition in multivariate time–frequency analysis [
3], speech signal processing [
14,
15], emotional speech classification [
7,
16], system identification [
17], medicine [
18], fault diagnosis [
19], seismic signal analysis [
20], and so on.
VMD suffers from the narrow band-limited mode, which has a center frequency, and VMD cannot decompose a complex signal with harmonics, in theory [
21]. In this paper, we further develop a more adaptive variation method by augmenting the concept of flattest response in the mathematical model with extra adaptive bandwidth, and we also consider the high-order harmonics of the decomposed mode.
The rest of this paper is organized as follows:
Section 2 reviews VMD primarily on the definition of the mode and the model of VMD;
Section 3 introduces our idea for improving VMD mainly on the concept of the flattest response and bandwidth;
Section 4 presents our improved model and its solution;
Section 5 contains our rich experiments and results; and
Section 6 concludes the discussion on iVMD.
3. Ideas for Improving VMD
In this section, we briefly propose a few ideas for improving VMD. These ideas constitute the building blocks of our improved VMD, which is simply abbreviated to iVMD.
3.1. The Flattest Response
VMD can recover an AM–FM mode with a low-pass, narrow-band selection of the input signal. The form in (6) was taken as a Wiener filter, and thus the recovered mode had a lowpass power spectrum. Based on the heuristic method of the filtering concept in (5) and (6), we rewrite the differential part
of the model in (3) as a time differential equation to solve the model in (3), and generalize it as
Here, is the -th derivative operator with , the highest derivative order, and . We have noted that is the coefficient of and .
Therefore, we can obtain the corresponding frequency domain form of (7),
We set the ratio of
as the filter system; therefore,
When
, their amplitude spectra are, respectively,
and thus,
Table 1 provides the coefficients of the different lowpass filters, and
Figure 1 shows the squared amplitude frequency characteristic,
. The system is a lowpass filter expressed by
, with its coefficients carefully selected via many methods of filter designing from Butterworth, Chebyshev, etc. Here, we design the filter as a Butterworth filter [
25], which has the flattest response in the frequency as depicted in
Figure 1. The parameters of the Butterworth filter are calculated in the following equations:
Here, is meant to take the maximum integer and add 1, while are the band pass and stop attenuations, respectively, and are the responding frequencies. Certainly, other filter-type designs can also be applied here.
Figure 1.
Amplitude Spectra of different order with normalization.
Figure 1.
Amplitude Spectra of different order with normalization.
Table 1.
Coefficient of the lowpass Butterworth filter in the model (7).
Table 1.
Coefficient of the lowpass Butterworth filter in the model (7).
| | | | | | |
---|
1 | 1 | | | | | |
2 | | 1 | | | | |
3 | 2 | 2 | 1 | | | |
4 | 2.61312593 | 3.41421356 | 2.61312593 | 1 | | |
5 | 3.23606798 | 5.23606798 | 5.23606798 | 3.23606798 | | |
6 | 3.86370331 | 7.46410162 | 9.14162017 | 7.46410162 | 3.86370331 | 1 |
3.2. To Set the Bandwidth
In the design of the lowpass Butterworth filter, we can adjust the bandwidth by normalizing the frequency. We set the normalized frequency as
, and thus we can set the lowpass bandwidth as
.
Figure 1 shows the bandwidth is normalized by dividing with
, where the cutoff frequency is 1 kHz.
From (9), we rewrite the system function as
Based on the property of the Fourier transform, if denormalization means
is divided by
in the frequency domain, then the time domain response is
, where
is the inverse Fourier transform of
. We obtain the denormalized version of the filter as
3.3. Harmonics
Continuous periodic signal (mode),
, may have multiple harmonic components with its base frequency of
, each of which has a gradually attenuated amplitude
with the harmonic frequency
. We find that
is the highest order of harmonic frequency. In theoretical application,
. That is,
Therefore, the composite signal may consist of one harmonic mode with maximum harmonic order at , and the center frequencies of the harmonic mode are .
4. Improved VMD
4.1. Improved Optimal Problem
In this section, we introduce our improved mathematical model for the variational mode decomposition based on the VMD idea [
5] and the previous section.
The new model is similar to the model found in (3), except in a few aspects. The sparsity in each mode is chosen to be its bandwidth,, in the spectral domain. Each mode without the harmonical frequencies, , is compact around a center pulsation, , which is to be determined among the decomposition. Each mode with the harmonical frequencies is compact around the harmonical frequencies, . Here, the sparsity also indicates full quasi-orthogonality.
We propose the following improved idea to decompose the signal : (1) for each mode, , that has an adaptive bandwidth of , we design the flattest response lowpass filter which permits the mode to pass through; (2) for each mode, , we shift the mode’s harmonic frequencies spectrum with the baseband, by multiplying it with an exponential, , which is tuned to the respective estimated center frequency, .
We set the analytical signal of
as,
Here,
is the convolution operator. The resulting constrained variational problem is
where
is the mode to be decomposed, where
is the given number of the modes, where
is the basic bandwidth of the mode
, and where
is the center frequency corresponding with the mode
.
4.2. Solution to the Problem
The constraint optimal problem (17) can be solved via the augmented Lagrangian method. Lagrangian multipliers are set with a quadratic penalty term to render the problem unconstrained. The weight of the penalty term is set as the factor of each mode .
First, we project the minimization problem (17) into solving the extreme point of the augmented Lagrangian equation [
26], which is
The augmented Lagrangian (18) is in a sequence of alternate direction methods of multipliers (ADMM) [
27]. Next, we detail how the respective sub-problems can be solved.
4.3. Minimization w.r.t
To update the modes
, the problem (18) is rewritten as the following unconstraint goal function for
:
This was achieved via Parseval–Plancherel Fourier isometry [
28], and we take
in the first term; then,
By exploiting the Hermitian symmetry of the real signals,
Letting the first variation vanish, i.e.,
for the positive frequencies. Thus,
When
,
,
, and
which is taken from
Table 1, then the above equation is simplified as,
If we set
,
,
, and
in Equation (22), then,
When
,
, and
which is taken from
Table 1, then Equation (22) is,
When and , Equation (25) is clearly identified as a Butterworth filtering of the current residual.
4.4. Minimization w.r.t
The center frequency
is solved via the optimization of the following goal function,
As described previously, the minimization of (26) can work in the Fourier domain; that is,
We also take the derivative of
to
, and set it to be zero; then,
Applying the binomial theorem, we get,
Here, we find that
,
, and
. Equation (29) is a polynomial
-power equation about
. We rewrite (29) as
Here,
is the n-power coefficient, and
Solving the above equation in (30), we can obtain the solution of
via the Newton–Raphson method, or others. Since Equation (28) is complex, it is not easy to obtain the solution. In fact, we find that
are not large, so we provide the different possible values of
, and obtain the corresponding solutions.
Table 2 shows the different solutions of
under
, and shows that
should be selected via the conditions, (1)
; (2)
being a real number. Additionally, the solution exists in practice, which can be clearly proven since the power order is odd.
4.5. Minimization w.r.t
The Bandwidth,
, is solved via optimization of the following goal function:
The minimization of (32) can be completed in the Fourier domain; that is,
We also take the derivative of
to
, and set it to be zero; that is,
, then, via the binomial theorem, we get,
Here, we still rewrite (34) as
Here,
is the
-power coefficient, and
As in the previous section, when solving the above equation in (35), we can obtain the solution of via the Newton–Raphson method, or others.
4.6. Complete Algorithm
The Lagrangian multiplier
is updated with the following equation [
5]:
As well as in the frequency domain,
Here, is the iterative number, and is the update parameter.
We directly optimize in the Fourier domain, and then we obtain the complete algorithm for iVMD in Algorithm 1.
Algorithm 1: Complete optimization of iVMD |
Initialize , , , , |
Repeat |
|
For do |
Update for all : |
|
Update : |
in Table 2 |
|
End for |
Update Lagrangian multiplier for all : |
|
Until convergence |
|
4.7. Reconstruction versus Denoising
The role of the Lagrangian multiplier [
5]
is the same in iVMD as in VMD, which serves to enforce the constraint, while the quadratic penalty
improves convergence.
The iVMD algorithm adds the extra bandwidth , and it acts as a penalty factor, as detailed in Equation (24). Both the penalty factor and the bandwidth improve convergence, and we can initially set the factor and leave the bandwidth adaptively undated. If we set the bandwidth as , the penalty factor of iVMD acts as the VMD.
6. Conclusions and Outlook
We further developed the algorithm of VMD as iVMD from three points: (1) flattest response, (2) harmonic, and (3) bandwidth. The flattest response is applied in iVMD and thus, we can set the higher differential order with respect to time, which results in the added weighting coefficient which can be obtained via Butterworth filter designing. As the harmonics may exist in the input signal, the mathematical model of VMD is further studied and modified via the harmonic order , and the improved version can support -order harmonical center frequency, . Each mode may have its adaptive bandwidth, and we set it in the model in (13) and (17). Through the above three points, we developed the algorithm iVMD.
In our experiments, iVMD works effectively with the same abilities as VMD and achieves a better performance than VMD.
The assumption of iVMD is the same as VMD, except that we can set the differential order and harmonic order with adjustable bandwidth, . We explain the reasons behind decomposing the two sawtooth composite signals, and it is due to setting the M-order harmonics in the mathematical model.
The algorithm iVMD is now being further extended with two-dimension decomposition, and we expect further challenges to decompose more complex composite signals.