1. Introduction
The data-driven method is an effective research method in both scientific research and practical applications [
1,
2,
3]. The complexity of the ocean background makes it hard to obtain the features of ship radiated noise (S-RN) [
4,
5,
6]. The existing feature extraction methods based on Fourier analysis are not suitable for underwater acoustic signals [
7]. The empirical mode decomposition (EMD) is a kind of data-driven and self-adaptive signal decomposition method for non-linear and non-stationary signal [
8,
9,
10]. Due to the mode mixing of EMD, ensemble EMD [
11] was put forward after EMD, which inhibits this phenomenon to some extent by adding noise. Still, EEMD and other improved EMD methods are all empirical ones without mathematical derivation [
12].
With the development of data-driven methods, some more effective methods have been proposed. In 2014, to avoid the aforementioned problems of EMD, variational mode decomposition (VMD) was employed as a novel kind of data-driven signal analysis tool, which has better decomposition performance and robustness to noise than improved EMD methods [
13]. In reference [
14], several different data-driven methods are compared, including empirical wavelet transform, VMD, Vold–Kalman filter order tracking, EMD and its four kinds of improved methods. In reference [
15], three underwater acoustic signals denoising methods are compared based on EMD, EEMD and VMD respectively, the results show that the VMD-based denoising methods are superior to other EMD-based and EEMD-based denoising methods. Meanwhile, the VMD-based feature extraction methods for underwater acoustic signal also have better performance [
16].
In recent years, the two kinds of data-driven decomposition methods have developed rapidly. On the one hand, the existing theoretical problems of themselves have been addressed; on the other hand, their application areas have also been expanded. Some modified EMD methods have been proposed to inhibit mode mixing; unfortunately, they are all empirical methods. To overcome the influence of the parameter selection problem in VMD, some modified VMD methods are put forward, which are suitable for analyzing different kinds of signals, such as magnetocardiography (MCG) [
17] and bearing signals [
18]. The application of VMD in underwater acoustic field is very limited. To solve the parameter selection problem of VMD, the existing methods depend on engineering experience or are based on EMD assistant selection. However, we have not found a modified VMD method for underwater acoustic signals.
Sample entropy (SE) [
19], like permutation entropy (PE), has the ability to explain the complexity of time series. Unlike PE [
20,
21], SE represents complexity by measuring the probability of new patterns, which have high reliability and consistency for data of different lengths. The method based on mode decomposition and entropy has been applied in many aspects, such as fault diagnosis [
22,
23,
24], medical science [
25] and other fields [
26,
27,
28]. In reference [
22], EEMD and PE are used in bearing fault diagnosis. In reference [
28], EEMD and SE are employed in feature extraction of partial discharge. In underwater acoustic field, PE is first used to extract S-RN feature by EMD [
26]. However, only three types of S-RN are applied, and SE is seldom used to analyze the complexity of underwater acoustic signals.
In this paper, IVMD, as a new data-driven method, is first put forward to solve the problem of choosing parameters for VMD by using frequency-aided method. Then, we proposed a novel feature extraction method for S-RN based on IVMD and SE. For underwater acoustic signal processing, feature extraction using IVMD and SE is seldom discussed. In
Section 2, IVMD and the novel feature extraction methods are described; the SE and center frequency of all IMFs are compared and analyzed by IVMD, EMD and EEMD in
Section 3; the SE of the maximum energy IMF (EIMF) is seen as a novel feature and compared with other methods in
Section 4; the last Section is the conclusion.
2. Theoretical Framework
2.1. VMD
As a data-driven signal decomposition method, VMD can decompose the target signal into a group of IMFs. Unlike the IMF definition of EMD, VMD defines IMF as Amplitude Modulation Frequency Modulation (AM-AF) signal, which has frequency center and limited bandwidth by solving the non-constrained variational model, as follows:
Table 1 lists the mathematical symbols of the non-constrained variational model. Alternating direction multiplier method is used to obtain the saddle points, then we can update
,
and
according to Equation (2).
where
represents frequency domain. The process of VMD is as follows:
- (1)
Make , and and equal to 0.
- (2)
. Update , and by using Equation (2).
- (3)
Repeat step 2 until the end condition is met as follows:
More detailed instructions are available in reference [
13].
The differences between IVMD, VMD, EMD and EEMD are as follows:
- (1)
EMD, EEMD and other EMD-based improved algorithms are all empirical data-driven decomposition methods, however, VMD and IVMD are not empirical algorithms and based on a foundation of mature mathematical theories and methods, which are wiener filtering, Hilbert transform, analytic signal and heterodyne demodulation.
- (2)
The sensitivity of VMD and IVMD to noise is lower than that of EMD, EEMD and other improved EMD algorithms.
- (3)
Based on the above two points, VMD and IVMD have better decomposition performance than EMD and EEMD. In addition, IVMD, as an improved method of VMD, solved the problem of choosing decomposition layers for VMD by using the frequency-aided method.
2.2. IVMD
In order to better analyze S-RN signals, a frequency-aided VMD method, called IVMD, is proposed. IVMD mainly solves the problem of choosing decomposition layers for VMD. The processing of VMD has been expressed in many studies. Therefore, here we focus on the IVMD approach as follows:
- (1)
Because of the complexity of S-RN, we initialize K = 5 according to EMD results.
- (2)
Decompose S-RN to obtain K IMFs and corresponding center frequency by VMD.
- (3)
Make K = K + 1.
- (4)
We can also obtain K IMFs and corresponding center frequency by VMD.
- (5)
Determine the new center frequency. When K increases by 1, we get a new center frequency. For example, when K = n + 1, compare the n + 1 center frequencies with the center frequencies obtained under K = n, find the nearest center frequencies, calculate the frequency difference separately, and regard the maximum frequency difference as the new center frequency.
- (6)
Judge whether or not the decomposition is excessive. We make the new center frequency equal to A (K = n + 1), the center frequency nearest to A is B (K = n). When , the decomposition is excessive, we make K = n − 1. When , we repeat steps from (3) to (6) until .
- (7)
Process S-RN by VMD with the decomposition layer K.
2.3. SE
Data of length N constitutes time series
, the steps of SE are as follows [
11]:
- (1)
is reconstructed into a set of vector sequences with dimension .
- (2)
Define the distance between
and
as follows:
- (3)
Set a tolerance threshold to
, the number of
is
.
can be expressed as follows:
- (4)
The mean value of
can be expressed as follows:
- (5)
can be obtained by increasing dimension to .
- (6)
Finally, SE can be defined as follows:
More detailed instructions are available in reference [
19].
2.4. Feature Extraction Based on IVMD and SE
IVMD has a strong ability of analysis in the time-frequency domain. Combined with the property of the SE, a new feature extraction approach can be designed as shown in
Figure 1. The main steps are as follows:
Step 1: The four types of S-RN signal are sampled and then normalized. Also, some parameters of IVMD were compared and selected. Then, we can obtain all the IMFs components by EMD, EEMD and IVMD.
Step 2: The order of IMF was reordered by center frequency and energy descending respectively, and the center frequency and SE of each arranged IMF was calculated. After analysis and comparison, the optimal IMF was chosen to represent the original signal.
Step 3: Calculate the center frequency and SE of 20 optimal IMFs for each type, and then analysis the statistic characteristic parameters of S-RN signal. Compare with feature extraction methods by IVMD, EMD and EEMD, we can select one optimal feature which is easy to use to distinguish the four types of S-RN signals.
4. Comparison of Feature Extraction Methods
In order to verify the universality of the difference between the four types of S-RN signal, 20 samples were randomly selected for each type to calculate the center frequency and SE of EIMF by EMD, EEMD and IVMD respectively.
Figure 15 shows the center frequency distribution of the EIMF by the three decomposition methods with 20 samples for each type. Unlike
Figure 15,
Figure 16 shows the SE distribution. It is discovered that the center frequency of EIMF are at the same level for similar ships, however, there are differences for different types of ships. The center frequency of EIMF obtained by the IVMD has better robustness. The three decomposition methods can distinguish the third and the fourth types of S-RN signals, however, due to the similar center frequency of the EIMF for the first and second type, the method based on the center frequency of EIMF cannot effectively distinguish the four types of S-RN signals.
SE of EIMF by EMD and EEMD can distinguish the third and fourth types, while SE of EIMF for the first and second types too similar to distinguish from each other in
Figure 16. Otherwise, the SE of EIMF by IVMD for four types of S-RN signal is obviously different.
In order to verify the validity of the method based on IVMD and SE of EIMF.
Table 7 shows the average, standard deviation (SD) and range of the SE of EIMF for the four types of S-RN signals under the three decomposition methods with 20 samples each type. As can be seen in
Table 7, the mean value of SE of EIMF for the four types by IVMD is obviously different, and their fluctuation range is not overlapping. However, for the first and second types, the mean value is so similar, and their range is overlapping by the EMD and EEMD method; the SD obtained by the EEMD method is less than that obtained by EMD, and the SD obtained by IVMD method is less than EEMD for the third and fourth types. Compared with EMD and EEMD methods, the proposed method can better distinguish the four types of S-RN signals.
5. Conclusions
IVMD as a novel algorithm combining SE is first proposed for feature extraction of S-RN signals. The selection of parameters is the key problem in IVMD. At first, according to the priori characteristics of S-RN signal and the analysis of experimental data, the main parameters are fixed in an orderly manner. Then four types of S-RN signals are decomposed by EMD, EEMD and IVMD, by analyzing and comparing the center frequency and the SE of IMF, it is found that the center frequency and the SE of IMF are similar for the same type of S-RN signal, while showing difference for different ones. Simulation results show the SE of EIMF by IVMD is more different than other methods for four types of S-RN signals.
Based on the above results, the SE of EIMF by IVMD is selected as a novel feature for the S-RN signal. Compared with the center frequency and SE of the EIMF by EMD and EEMD, the proposed method can represent the complexity of the EIMF of S-RN signals, and has a better distinguishing ability than the existing ones. Consequently, the novel method also can be used as the basis of classification and recognition.