Research on Ship-Radiated Noise Denoising Using ... - MDPI

sensors Article

Research on Ship-Radiated Noise Denoising Using Secondary Variational Mode Decomposition and Correlation Coefficient Yuxing Li *, Yaan Li *, Xiao Chen

ID

and Jing Yu

ID

School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710000, China; [email protected] (X.C.); [email protected] (J.Y.) * Correspondence: [email protected] (Y.L.); [email protected] (Y.L.) Received: 27 November 2017; Accepted: 24 December 2017; Published: 26 December 2017

Abstract: As the sound signal of ships obtained by sensors contains other many significant characteristics of ships and called ship-radiated noise (SN), research into a denoising algorithm and its application has obtained great significance. Using the advantage of variational mode decomposition (VMD) combined with the correlation coefficient for denoising, a hybrid secondary denoising algorithm is proposed using secondary VMD combined with a correlation coefficient (CC). First, different kinds of simulation signals are decomposed into several bandwidth-limited intrinsic mode functions (IMFs) using VMD, where the decomposition number by VMD is equal to the number by empirical mode decomposition (EMD); then, the CCs between the IMFs and the simulation signal are calculated respectively. The noise IMFs are identified by the CC threshold and the rest of the IMFs are reconstructed in order to realize the first denoising process. Finally, secondary denoising of the simulation signal can be accomplished by repeating the above steps of decomposition, screening and reconstruction. The final denoising result is determined according to the CC threshold. The denoising effect is compared under the different signal-to-noise ratio and the time of decomposition by VMD. Experimental results show the validity of the proposed denoising algorithm using secondary VMD (2VMD) combined with CC compared to EMD denoising, ensemble EMD (EEMD) denoising, VMD denoising and cubic VMD (3VMD) denoising, as well as two denoising algorithms presented recently. The proposed denoising algorithm is applied to feature extraction and classification for SN signals, which can effectively improve the recognition rate of different kinds of ships. Keywords: variational mode decomposition (VMD); secondary variational mode decomposition (2VMD); correlation coefficient (CC); ship-radiated noise (SN); denoising

1. Introduction In the practical measuring process, measured signals are often mixed with noise and useless signal components which come from the surrounding complex environment and the measurement equipment itself. The time domain waveforms of the polluted signals are often different from those of original signals, and it is not easy to identify the original signal from the polluted signal. In the frequency domain, the bandwidth of the clear signal and the noisy signal also partially or even completely overlap. In this case, traditional spectrum analysis techniques and linear filtering algorithms cannot effectively eliminate noise. Therefore, the question of how to eliminate noise from the polluted signal must be solved in the signal processing field [1,2]. There is a kind of common denoising algorithm, the basic ideas of which is to extract components of signal obtained by one signal decomposition algorithm, identify and remove noise components by screening criteria, then reconstruct the useful components. For this kind of denoising algorithm, Sensors 2018, 18, 48; doi:10.3390/s18010048

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the focus is on selecting signal decomposition algorithm and noise-screening criteria. For instance, the wavelet denoising algorithm [3,4] has been widely used in various fields, and a good effect is gained. However, using different types of wavelet functions and different numbers of decomposition have a great influence on denoising [5]. A kind of self-adaptive signal processing algorithm is empirical mode decomposition (EMD) [6,7], originally proposed by Huang et al. This is an absolutely data-driven and adaptive algorithm that depends on local characteristics of data in the time domain. EMD can decompose complex signal into intrinsic mode functions (IMFs), with each IMF indicating one oscillation mode of the complex signal. However, EMD faces problems of mode-mixing and end effects and also lacks mathematical demonstration. Ensemble EMD (EEMD) [8], as an algorithm that improves on EMD, can reduce the phenomena of modes overlap to some extent. A growing number of researchers are focusing on developing EMD and improved EMD algorithms, and these algorithms are widely employed in various fields, especially in mechanical fault diagnosis [9–11], medical science [12], meteorology [13], oceanography [14–18] and so on. Many denoising algorithms using EMD and improved EMD algorithms have been proposed. For example, high-frequency IMFs are regarded as noise IMFs; the rest of the IMFs are reconstructed for denoising [19]. Nevertheless, this denoising algorithm cannot completely eliminate noise components, and the reconstructed signal lacks some detailed information. Many denoising algorithms have been proposed to solve the problems of this denoising algorithm by the threshold for IMFs [20,21]. As a kind of non-recursive and self-adaptive signal-processing algorithm, variational mode decomposition (VMD) [22–24], originally put forward by Dragomiretskiy et al., can effectively decompose a multi-component signal into several bandwidth-limited IMFs. Every IMF has a corresponding central frequency updated in real-time. Compared with EMD and the improved EMD algorithms, VMD has not only a solid theoretical foundation, but also good robustness to noise. In the field of fault diagnosis, a new diagnosis algorithm based on VMD denoising is proposed in [25], which uses the IMFs obtained by VMD to reconstruct the IMFs according to the correlation coefficients (CCs) between IMFs and the original signal in order to realize denoising, and then extracts the bearing fault characteristics by means of a morphological difference filter to demodulate the signals after denoising, with simulated signal and experimental results showing the validity of the algorithm. In research [26], a new denoising algorithm based on the non-convex framework has been proposed. By comparing with wavelet denoising and 1-order total variation denoising algorithms, the validity is verified by analyzing the simulation signals and the vibration signals. In research [27], an adaptive denoising algorithm for a chaotic signal has been proposed by using independent component analysis (ICA) and EMD. In research [28], an adaptive denoising algorithm using a probability density function and VMD has been proposed, and a small mean-error square and a high signal-to-noise ratio prove the effectiveness of the denoising algorithm. These denoising algorithms also demonstrate the feasibility of EMD and VMD in signal denoising. In this article, we proposed a new denoising algorithm for ship-radiated noise (SN) signals. We used VMD and CC to decompose the original signals into IMFs and identify noise IMFs, respectively; the decomposition number by VMD is equal to the number by EMD. According to the threshold of the CC, noise IMFs and useful IMFs can be distinguished effectively. Then, the first denoising can be realized by reconstructing useful IMFs. Secondary denoising can be accomplished by repeating the above steps of decomposition, mode-identification and reconstruction. The final result of denoising is determined according to the CC between IMFs and the original signal. Simulation results indicate that the proposed denoising algorithm based on the secondary VMD (2VMD) and CC is better than existing denoising ones. The proposed 2VMD denoising algorithm is used to feature extraction and classification for SN signals, which can effectively improve the recognition rate of different kinds of ships. The outline of the article is as follows. Section 2 provides the background to VMD, CC and the evaluation criterion; a review of the proposed 2VMD denoising algorithm is presented in Section 3;

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in Sections 4 and 5, the proposed 2VMD denoising algorithm is used to simulation data and SN signals respectively; finally, the last section is the conclusions. 2. Background 2.1. Variational Mode Decomposition (VMD) In the VMD algorithm, IMFs are defined as amplitude-modulated–frequency-modulated (AM–FM) signals, which are given by: uk (t) = Ak (t) cos(φk (t)),

(1)

where t and Ak (t) represent time and the envelope of IMF; and φk (t) and uk (t) denote the phase and the IMFs. IMFs have center frequencies and limited bandwidths. The decomposition process is the constrained variational problem, which is given by: ( min

{ u k },{ w k }

) 2 K j − jwkt k subject to ∑ uk =s, ∑ k∂t[(δ(t) + πt ) ∗ uk (t)]e 2 k =1 k =1 K

(2)

where s is the original signal; K represents the number of IMFs; and uk and wk are the IMF and the center frequency for each IMF. The constrained variational problems in Equation (2) can be addressed by the penalty factor α and the lagrangian multiplier λ. The augmented lagrangian is expressed as: L({uk }, {wk }, λ)

K

= α ∑ k∂t[(δ(t) + k =1

2 j − jwkt k πt ) ∗ uk ( t )] e 2

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+k f (t) − ∑ uk (t)k + λ(t), f (t) − ∑ uk (t) . k =1

The alternating direction multiplier method (ADMM) is used to obtain the saddle points, then the uk , wk and λ are updated in the frequency domain, which is given by: ˆ n (w) − ∑ ui ˆ n (w) + fˆ(w) − ∑ ui i k

λˆ n (w) 2

R ∞ n+1 2 ˆ dw 0 w uk = R , 2 ∞ n +1 ˆ u dw 0 k

(w) = λ (w) + τ fˆ(w) − ∑ ˆn

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(5) !

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.

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∑ kuˆ nk +1 − uˆ nk k k

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where e represents convergence accuracy. The specific process of VMD is summarized as follows:   • Initialize uˆ 1k , w1k , λˆ 1 and n = 0. n o n o • Update the value of uˆ nk +1 , wkn+1 and λˆ n+1 according to Equations (4)–(6).

•

Judge whether or not uk meets the convergence condition (7). Repeat the steps of updating parameters until the stopped condition is satisfied.

(7)

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The uniformly spaced distribution for initialization of center frequency is expressed as: w0k =

k−1 , 2k

k = 1, · · · , K,

(8)

and the zero initial can be expressed as: w0k = 0 ,

k = 1, · · · , K.

(9)

In addition, K is equal to the decomposition level by EMD. The zero initial is used in this paper. 2.2. Correlation Coefficient (CC) The correlation coefficient (CC), as a parameter of statistical relationships, can measure the degree of dependence and correlation. In this paper, the formula of CC is shown as the following: r=

E(ui. f ) − E(ui ) E( f ) p p , D (ui ) D ( f )

(10)

where f and ui represent the original signal and IMF obtained by mode decomposition; D and E correspond to mathematical expectation and variance; and r represents the CC between the IMF and original signal. High values (close to 1) indicate a strong degree of dependence and correlation. Instead, the closer that this value is to −1, the more inverse the relationship value is. The relationship between CC and correlation is shown in Table 1. If CCs between the original signal and IMFs by VMD are within the range of moderate correlation or strong correlation, IMFs contain useful components. Therefore, the CC threshold is within the range of weak correlation. Table 1. The relationship between correlation coefficient (CC) and correlation.

CC

No Correlation

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Strong Correlation

0~0.1

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A simulation example should make this easier to understand, with the simulation signals as follows:   f 1 (t) = cos(10πt)    f (t) = cos(100πt) 2 ,  f 3 ( t ) = cos(200πt )    f (t) = f 1 (t) + f 2 (t) + f 3 (t)

(11)

where f 1 (t), f 2 (t) and f 3 (t) represent the three components of f (t). Three decomposition algorithms are used to decompose f (t). The original signals are presented in Figure 1. The decomposition result of VMD is shown in Figure 2. As can be seen in Figures 1 and 2, the decomposition result using VMD is similar to the component of the simulation signal. CCs between the simulation signal and corresponding IMFs are shown in Table 2. By comparison with EMD and EEMD algorithms, the CCs between simulation signal and corresponding IMFs by VMD are closer to the true values in Table 2. This shows that the VMD algorithm can better reflect the correlation.

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Figure 2. Decomposition result by variational mode decomposition (VMD). Figure Figure 2. 2. Decomposition Decomposition result result by by variational variational mode mode decomposition decomposition (VMD). (VMD). Table 2. The CCs between simulation signal and corresponding intrinsic mode functions (IMFs). Table intrinsic mode mode functions functions (IMFs). (IMFs). Table 2. 2. The The CCs CCs between between simulation simulation signal signal and and corresponding corresponding intrinsic

Simulation Signals Simulation Signals f (t ) Signals 0.5775 f 33(t ) Simulation 0.5775 f (t ) 0.5775 0.5775 f 22(t ) f 3 (t) 0.5775 f ( t ) 0.5775 2 f (t ) 0.5775 f11(t ) f 1 (t) 0.5775 0.5775

EMD Algorithm EMD Algorithm EMD Algorithm IMF1 0.6040 IMF1 0.6040 IMF2 0.5756 IMF1 0.6040 IMF2 0.5756 IMF2 0.5756 IMF3 0.5774 IMF3 0.5774 IMF3 0.5774

EEMD Algorithm VMD Algorithm EEMD Algorithm VMD Algorithm VMDIMF1 Algorithm 0.5772 IMF5 0.5854 IMF1 0.5772 IMF6 0.6164 IMF1 IMF2 0.5772 0.5775 IMF5 0.5854 IMF6 0.6164 IMF2 0.5775 IMF6 0.6164 IMF8 0.5773 IMF2 IMF3 0.5775 0.5775 IMF8 0.5773 IMF3 0.5775 IMF8 0.5773 IMF3 0.5775

EEMD IMF5Algorithm 0.5854

2.3. Evaluation Criteria for Denoising Algorithm 2.3. 2.3. Evaluation Evaluation Criteria Criteria for for Denoising Denoising Algorithm Algorithm The denoising effects of different decomposition algorithms are compared. Therefore, two The denoisingeffects effects of different decomposition algorithms are compared. two The denoising of different decomposition algorithms are compared. Therefore,Therefore, two evaluation evaluation criteria for denoising algorithms are given as follows: evaluation foralgorithms denoising algorithms given as follows: criteria for criteria denoising are given asare follows:  f  !  SNR = 10 log 10 kf f k ， SNR log10 10  ˆ ， SNR = =1010log  kfˆf fˆ−−−ff fk ,  

(12) (12) (12)

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fˆ −f kf k fˆ − RMSE = ,， RMSE = NN s

(13) (13)

where f is foriginal signal; signal; and Nresult; represents length. Signal-to-noise fˆ is the denoising is original denoising and signal signal length. where N represents fˆ is the result; ratio (SNR) and root mean square the evaluation criteria denoising, respectively. Signal-to-noise ratio (SNR) anderror root(RMSE) mean are square error (RMSE) arefor the evaluation criteria for

denoising, respectively. 3. Denoising Algorithm Using Secondary VMD (2VMD) and CC 3. Denoising Algorithm Using Secondary VMD and CC in Figure 3. The experimental A 2VMD denoising algorithm using VMD and(2VMD) CC is designed procedures are asdenoising follows: algorithm using VMD and CC is designed in Figure 3. The experimental A 2VMD procedures are as follows: Step 1: The target signal is decomposed by EMD, and the decomposition number by VMD is equal to the number by EMD; Step 1: The target signal is decomposed by EMD, and the decomposition number by VMD is equal Step 2: Calculate the CCs between the original signal and IMFs by VMD, screen out the noise IMFs to the number by EMD; according to CC threshold. abundant experiments, theout CCthe threshold is Step 2: Calculate the CCs betweenThrough the original signal simulation and IMFs by VMD, screen noise IMFs fixed at 0.2 to in this according CC paper; threshold. Through abundant simulation experiments, the CC threshold is Step 3: Reconstruct the useful IMFs by removing noise IMFs. After the reconstruction, the first fixed at 0.2 in this paper; denoising is completed; Step 3: Reconstruct the useful IMFs by removing noise IMFs. After the reconstruction, the first denoising is completed; Step 4: Judge the times decomposition satisfies the 2VMD or not; Step times decomposition satisfies the result 2VMDisor not; Step 5: 4: If Judge 2VMDthe is not satisfied, the first reconstructed regarded as the input signal, then repeat Step 5: If 2VMD not satisfied, the first reconstructed is regarded the input signal, Steps 1–3 to is complete the whole denoising process; result if it is satisfied, the as reconstructed signalthen is repeat Steps 1–3 to complete the whole denoising process; if it is satisfied, the reconstructed regarded as the final denoising result. signal is regarded as the final denoising result.

Figure 3. The flow chart of proposed secondary variational mode decomposition (2VMD) denoising Figure 3. The flow chart of proposed secondary variational mode decomposition (2VMD) denoising algorithm. algorithm.

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4. Test with Numerical Simulation Signal Sensors 2018, 18, 48

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Sensors 2018,spectrum 18, 48 of 17 for The line is the important information of the SN signals, and it provides a7 basis ship detection tracking. The periodic 4. Test withand Numerical Simulation Signalsignal can be used as a line spectrum model. Therefore, 4. Test with Numerical Simulation Signal three simulation have been carried out inof Sections the different SNRs The lineexperiments spectrum is the important information the SN 4.1–4.3; signals, and it providesinput a basis for and The line spectrum is the important information of the SN signals, and it provides a basis for the times of decomposition by VMD for the three simulation signals are also discussed in Section ship detection and tracking. The periodic signal can be used as a line spectrum model. Therefore, 4.4. ship and tracking. The signalout can used algorithm, as a linethe spectrum model. Therefore, To further prove the effectiveness ofperiodic the proposed denoising in Section 4.5 we compare it threedetection simulation experiments have been carried inbeSection 4.1–4.3; different input SNRs and three simulation experiments have been carried out in Section 4.1–4.3; the different input SNRs and with two denoising algorithmsbypresented recently using the signals same simulation signals. the times of decomposition VMD for the three simulation are also discussed in Section 4.4.

the times ofprove decomposition by VMD three simulation are also discussed in Section 4.4. To further the effectiveness of for thethe proposed denoisingsignals algorithm, in Section 4.5 we compare it To further prove the effectiveness of the proposed denoising algorithm, in Section 4.5 we compare it 4.1. Simulation 1 with two denoising algorithms presented recently using the same simulation signals. with two denoising algorithms presented recently using the same simulation signals.

The simulation signal s is composed of three different frequency and amplitude cosine signals, 4.1. Simulation 1 4.1.times Simulation 1 and 0.5 standard Gaussian white noise n is added to get the noisy signal y. The simulation The simulation signal s is composed of three different frequency and amplitude cosine signals areThe as follows: simulation signal s is composed of three different frequency and amplitude cosine

signals, and 0.5 times standard Gaussian white noise n is added to get the noisy signal y . The  standard Gaussian white noise n is added to get the noisy signal y . The signals, and 0.5 times simulation signals  are as follows: 0.8 cos(2π f 1 t) + 0.6 cos(2π f 2 t) + 0.3 cos(2π f 3 t) simulation signals  are sas=follows: , (14) n = 0.5randn (t)π f1t ) + 0.6 cos(2π f 2 t ) + 0.3cos(2π f 3t )  s = 0.8 cos(2  cos(2π f1t ) + 0.6 cos(2π f 2 t ) + 0.3cos(2π f 3t )  y = ssn+==n0.8 0.5 randn ( t ) , (14)  randn(t ) , (14) ny == 0.5 s+n  y = s + n where f 1 = 10, f 2 = 50 and f 3 = 100 represent the three frequencies of clear signal s; and y is the noisy where f1 = 10 , f 2 = 50 and f3 = 100 represent the three frequencies of clear signal s ; and y is signalwhere containing andand n. The for clear signal andsignal noisys signal f = 10both f3 =time-domain 100 represent waveform , f 2 =s 50 ; and is is the three frequencies of clear y shown the noisy1 signal containing both s and n . The time-domain waveform for clear signal and noisy in Figure 4. The decomposition result of the EMD, EEMD, VMD and 2VMD for a noisy signal are the noisy signal in containing waveform for VMD clear signal and noisy s and n . The time-domain signal is shown Figure 4.both The decomposition result of the EMD, EEMD, and 2VMD for a presented in Figure 5. signal is shown in Figure 4. decomposition result of the EMD, EEMD, VMD and 2VMD for a noisy signal are presented in The Figure 5. As seen in Figure 5, the number of IMFs by EMD is 9 (containing residue), and the number of noisy signal are presented in Figure 5. As seen in Figure 5, the number of IMFs by EMD is 9 (containing residue), and the number of Asshould seen inbeFigure 5, the number of IMFs by EMD is 9 (containing residue), and number of VMDVMD should be equal to the number of EMD, so we can set K and setthe K= 2VMD. equal to the number of EMD, so we can set=K9 =for for VMD and set 9 VMD K7=for 7 for VMD should be equal to the number of EMD, so we can set for for VMD and set K = 9 K = 7 The CCs between thebetween noisy signal andsignal IMFsand by IMFs VMDby and 2VMD are shown in Table 3. 3. 2VMD. The CCs the noisy VMD and 2VMD are shown in Table 2VMD. The CCs between the noisy signal and IMFs by VMD and 2VMD are shown in Table 3. 2

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(c) Figure 5. The decomposition result of empirical mode decomposition (d) (EMD), ensemble EMD Figure 5. The decomposition result of empirical mode decomposition (EMD), ensemble EMD (EEMD), (EEMD), VMD and 2VMD for the noisy signal. (a) EMD; (b) EEMD; (c) VMD; (d) 2VMD. Figureand 5. 2VMD The decomposition result (a) ofEMD; empirical mode(c) decomposition (EMD), ensemble EMD VMD for the noisy signal. (b) EEMD; VMD; (d) 2VMD. (EEMD), VMD and 2VMD for the noisy signal. (a) EMD; (b) EEMD; (c) VMD; (d) 2VMD. Table 3. The CCs between the noisy signal and IMFs by VMD and 2VMD. Table 3. The CCs between the noisy signal and IMFs by VMD and 2VMD. IMF 1 3.IMF IMF 3 the IMF 4 signal IMF and 5 IMFs IMFby 6 VMD IMFand 7 2VMD. IMF 8 IMF 9 Table The 2CCs between noisy VMD 0.1438 0.1336 0.1592 0.1465 0.1583 0.1479 0.2417 0.4378 0.6721 IMF 1 IMF 2 IMF 3 IMF 4 IMF 5 IMF 6 IMF 7 IMF 8 IMF 9 IMF0.2205 1 IMF 2 IMF 3 IMF 4 IMF 5 0.3935 IMF 6 0.6673 IMF 7 −IMF 8 − IMF 9 2VMD 0.0922 0.0927 0.0913 0.4204 0.1592 0.1465 0.1465 0.1583 0.4378 0.4378 0.6721 0.6721 VMD VMD 0.1438 0.1438 0.13360.13360.1592 0.1583 0.1479 0.1479 0.2417 0.2417 2VMD 0.0927 0.0913 0.6673 to− − As seen in0.2205 Table 3,0.0922 the number of useful IMFs 0.4204 by VMD 0.3935 is three according the CC threshold;

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the 2VMD can further remove noise IMFs (IMF2, IMF3 and IMF4) by the CC threshold. The results of EMD, VMD and VMD 2VMDand for2VMD the noisy are signal shownare in shown Figure 6. SNR denoising resultsEEMD, of EMD, EEMD, for signal the noisy in The Figure 6. and The RMSE for EMD, EEMD, VMD and 2VMD denoising are shown in Table 4. SNR and RMSE for EMD, EEMD, VMD and 2VMD denoising are shown in Table 4. 2

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4. 48 The signal-to-noise ratio (SNR) and root mean square error (RMSE) for EMD, EEMD, VMD9 of 17 SensorsTable 2018, 18, and 2VMD denoising.

Noisy Signal EMD EEMD VMD 2VMD Table 4. The signal-to-noise ratio (SNR) and root mean square error (RMSE) for EMD, EEMD, VMD and SNR/db 2.959 8.133 9.33 15.01 17.34 2VMD denoising. RMSE 0.9797 0.8353 0.8246 0.1508 0.03072 Noisy Signal

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As seen in SNR/db Figure 6, the denoising results of EMD and are obviously 2.959 8.133 9.33 EEMD 15.01 17.34different from the clear signal; the denoising results of VMD and 2VMD are close to the clear signal. RMSE 0.9797 0.8353 0.8246 0.1508 0.03072 To compare the performance of different denoising algorithms, the SNR and RMSE are listed in Table 4. As can be seen in Table 4, EEMD denoising is superior to EMD denoising, and VMD denoising is better than As seen in Figure 6, the denoising results of EMD and EEMD are obviously different from the EEMD denoising; 2VMD denoising is the best denoising algorithm which has high SNR and low clear signal; the denoising results of VMD and 2VMD are close to the clear signal. To compare the RMSE. performance of different denoising algorithms, the SNR and RMSE are listed in Table 4. As can be seen in Table 4, EEMD denoising is superior to EMD denoising, and VMD denoising is better than EEMD 4.2. Simulation 2 denoising; 2VMD denoising is the best denoising algorithm which has high SNR and low RMSE. The simulation signal s is composed of frequency-modulated signal and sine signal, and 0.5 4.2. Simulation times standard2Gaussian white noise n is added to get the noisy signal y . The simulation signals are as follows: The simulation signal s is composed of frequency-modulated signal and sine signal, and 0.5 times standard Gaussian white noise n is to get the noisy signal y. The simulation signals are cos(2 π fadded  s = 0.6 1t + 0.8sin(2π f 2 t )) + 0.4 sin(2π f 3 t ) as follows:  n = 0.5randn(t ) (15)  0.6 cos(2π f t + 0.8 sin(2π f t)) + 0.4 sin(2π f , t)  2 3  s= 1 y = s + n  0.5randn(t) , (15) n=   y = s+n where f = 50 , f = 40 and f = 150 represent the three frequencies of clear signal s ; and y is 1

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where f 1 =signal 50, f 2 = 40 and f 3 both = 150s represent the three frequencies of clear signal s; and y is the noisy the noisy containing time-domain waveform for the clear signal and and n . The signal containing both sinand n. The time-domain for the clear signal and noisy signal is noisy signal is shown Figure 7, with the clearwaveform signal submerged in Gaussian white noise. The shown in Figure theEEMD, clear signal noise. The denoising results7,ofwith EMD, VMD submerged and 2VMD in areGaussian shown inwhite Figure 8, and thedenoising SNR andresults RMSE of shown in and the SNR for EMD, EEMD, forEMD, EMD,EEMD, EEMD,VMD VMDand and2VMD 2VMDare denoising areFigure shown8,in Table 5. As and seenRMSE in Figure 8 and Table 5, VMD and 2VMD denoising are shown Table 5. As Figureis8also and the Table 5, the 2VMD denoising the 2VMD denoising algorithm with in high SNR andseen lowinRMSE most effective denoising algorithm algorithm.with high SNR and low RMSE is also the most effective denoising algorithm. 1

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4.3. Simulation 3

Table 5. The SNR and RMSE for EMD, EEMD, VMD and 2VMD denoising.

Noisy Signal EMD EEMD VMD 2VMD The simulation signal s is composed of an amplitude-modulated signal and sine signal, SNR/db 0.0827 2.5348 3.6687 10.3561 11.4758 and standard Gaussian white noise n is added to get the noisy signal y. The simulation signals RMSE 0.9335 0.7496 0.3384 0.2402 0.0626 are as follows:    s = 2 sin(2π f 1 t) sin(2πt/ f 2 ) + sin(2π f 3 ) 4.3. Simulation 3 , (16) n = randn(t)   y = s+n The simulation signal s is composed of an amplitude-modulated signal and sine signal, and

n is added to get the frequencies noisy signalofyclear . Thesignal simulation as standard Gaussian where f1 = 10, f 2 =white 10 andnoise f 3 = 15 represent the three s; and signals y is the are noisy follows: signal containing both s and n. The time-domain waveform for the clear signal and noisy signal is shown in Figure 9; the clear signals cannot be distinguished from the noisy signal. The denoising results  = 2sin(2π f1t ) sin(2π t / f 2 ) + sin(2π f 3 ) of EMD, EEMD, VMD and 2VMD  for the noisy signal are shown in Figure 10, and the SNR and RMSE , (16) n = randn(t ) for EMD, EEMD, VMD and 2VMD are shown in Table 6. As seen in Figure 10 and Table 6,  y =denoising s n +  the 2VMD denoising algorithm with high SNR and low RMSE, which is smooth and close to the clear signal, is the most effective denoising algorithm.the three frequencies of clear signal s ; and y is the where f = 10 , f = 10 and f = 15 represent 1

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noisy signal containing both s and n . The time-domain waveform for the clear signal and noisy Table 6. The SNR and RMSE for EMD, EEMD, VMD and 2VMD denoising. signal is shown in Figure 9; the clear signal cannot be distinguished from the noisy signal. The denoising results of EMD, EEMD, VMD and EMD 2VMD forEEMD the noisy signal are shown in Figure 10, and Noisy Signal VMD 2VMD the SNR and RMSE for EMD, EEMD, VMD and 2VMD denoising are shown in Table 6. As seen in SNR/db −1.467 4.898 6.335 8.306 9.412 Figure 10 and Table 6, the 2VMD denoising algorithm with high SNR and0.1601 low RMSE, which is RMSE 1.258 0.8652 0.8473 0.3065 smooth and close to the clear signal, is the most effective denoising algorithm.

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Figure Figure 10. 10. The The denoising denoising results results of of EMD, EMD, EEMD, EEMD, VMD VMD and and 2VMD 2VMD for for the the noisy noisy signal. signal. (a) (a) EMD; EMD; (b) (b) Figure 10. The denoising results of EMD, EEMD, VMD and 2VMD for the noisy signal. (a) EMD; EEMD; EEMD; (c) (c) VMD; VMD; (d) (d) 2VMD. 2VMD. (b) EEMD; (c) VMD; (d) 2VMD. Table 6. The SNR and RMSE for EMD, EEMD, VMD and 2VMD denoising.

Table 6. The SNR and RMSE for EMD, EEMD, VMD and 2VMD denoising. 4.4. Different Input Signal-to-Noise Ratio (SNR) and Times of Decomposition by VMD Noisy EMD EEMD VMD 2VMD Noisy Signal Signal EMD2VMD EEMD VMD 2VMD To further prove the suitability of the proposed algorithm, the denoising effect is compared SNR/db −1.467 4.898 6.335 8.306 9.412 SNR/db −1.467 4.898 6.335 8.306 under different input SNRs and times of decomposition by VMD for the 9.412 simulation signals in RMSE 1.258 0.8652 0.8473 0.3065 0.1601 RMSESNRs range 1.258 0.8473 0.3065of decomposition 0.1601 Sections 4.1–4.3. Input from −100.8652 dB to 5 dB, and the times for VMD range from 1 to 3. Figure 11 shows the plots of input SNRs versus output ones for different denoising 4.4. Input Signal-to-Noise Ratio (SNR) and of Decomposition by 4.4. Different Differentand Input Signal-to-Noise Ratio (SNR)each and Times Times Decomposition by VMD VMD algorithms simulation signals, where outputof SNR is calculated by using the mean of To further prove the suitability of the proposed 2VMD algorithm, the denoising effect 100 times. As can be seen Figure 11, SNRs of thealgorithm, 2VMD and VMD (3VMD) To further prove the in suitability of the the output proposed 2VMD thecubic denoising effect is is compared under different input SNRs and times of decomposition by VMD for the simulation denoising algorithms in most cases are higher than the others, especially in the case of low-input SNRs, compared under different input SNRs and times of decomposition by VMD for the simulation signals in Sections 4.1–4.3. SNRs range −10 55 dB, and of for which suitable forInput SN signal denoising. However, 2VMD has the advantage of low signalsare in more Sections 4.1–4.3. Input SNRs range from from −10 dB dB to to dB, denoising and the the times times of decomposition decomposition for VMD range from 1 to 3. Figure 11 shows the plots of input SNRs versus output ones for different computational cost1over VMD range from to 3.3VMD Figuredenoising. 11 shows the plots of input SNRs versus output ones for different denoising denoising algorithms algorithms and and simulation simulation signals, signals, where where each each output output SNR SNR is is calculated calculated by by using using the the mean of 100 times. As can be seen in Figure 11, the output SNRs of the 2VMD and cubic mean of 100 times. As can be seen in Figure 11, the output SNRs of the 2VMD and cubic VMD VMD

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(3VMD) denoising algorithms in most cases are higher than the others, especially in the case of low-input SNRs, which are more suitable for SN signal denoising. However, 2VMD denoising Sensors 2018, 18, 48 12 ofhas 17 the advantage of low computational cost over 3VMD denoising. 19

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4.5. Comparison with Denoising Algorithms Presented Recently 4.5. Comparison with Denoising Algorithms Presented Recently In recent research [26,29], two denoising algorithms have been proposed for the vibration signal In recent [26,29], denoising algorithms have proposed and SN signal,research respectively. Thetwo same simulation signals in [26]been are as follows:for the vibration signal and SN signal, respectively. The same simulation signals in [26] are as follows: ( = xx11++x2x2 xx = (17) x = sin(30πt + cos(60πt)) (17) x  11 = sin(30π t + cos(60π t ))

where x is a typical modulating signal; x I s Gaussian white noise, whose mean value and variance where x11 is a typical modulating signal; x2 2is Gaussian white noise, whose mean value and variance are 0 and 0.5, respectively. The clear signal x and noisy signal x2 are shown in Figure 12. The SNRs are 0 and 0.5, respectively. The clear signal x1 1and noisy signal x2 are shown in Figure 12. The SNRs for for different variances of Gaussian white noise different denoising algorithms shown Table different variances of Gaussian white noise andand different denoising algorithms areare shown in in Table 7. 7. As be seen in Table the proposed algorithm high for variances different variances of As cancan be seen in Table 7, the 7, proposed algorithm has highhas SNR for SNR different of Gaussian Gaussian white noise compared with thedenoising wavelet denoising and the denoising algorithms white noise compared with the wavelet and the denoising algorithms presentedpresented recently recently in [26,29]. in [26,29]. 2

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5. Application Application in in Feature Feature Extraction Extraction for for Ship-Radiated Ship-Radiated Noise Noise (SN) (SN) 5. First, three three kinds kinds of of SN SN signals signals are are performed performed by by the the proposed proposed 2VMD 2VMD denoising denoising algorithm; algorithm; First, then, the features of SN signals are extracted by the feature extraction algorithm in [16]; finally, the then, the features of SN signals are extracted by the feature extraction algorithm in [16]; finally, classification results before and after denoising are compared. the classification results before and after denoising are compared. 5.1. 5.1. Denoising of of SN SN Three SNSN signals, which are the as the as signals in [16], were recorded calibrated Threekinds kindsofof signals, which aresame the same the signals in [16], were using recorded using omnidirectional hydrophoneshydrophones at a depth of 29 the South there were calibrated omnidirectional at ma in depth of 29China m inSea. theDuring Southrecording, China Sea. During no observed disturbances from biological or man-made sources.or The distance sources. between The the ship and recording, there were no observed disturbances from biological man-made distance hydrophone 1 km. The sampling frequency and sampling sampling frequency points wereand set sampling as 44.1 kHz and between thewas shipabout and hydrophone was about 1 km. The points 5000, Theand samples normalized get the time-domain waveform kinds of were respectively. set as 44.1 kHz 5000,were respectively. Thetosamples were normalized to get for thethree time-domain SN signals shown in Figure The denoising for three kinds of denoising SN signals results by the proposed waveform for three kinds 13a,c,e. of SN signals shown results in Figure 13a,c,e. The for three denoising algorithm arethe shown in Figure 13b,d,f. kinds of SN signals by proposed denoising algorithm are shown in Figure 13b,d,f.

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Figure 13. Three Figure 13. Three kinds kinds of of ship-radiated ship-radiated noise noise (SN). (SN). (a) (a) Ship Ship 11 without without denoising; denoising; (b) (b) Ship Ship 11 after after denoising; (c) Ship 2 without denoising; (d) Ship 2 after denoising; (e) Ship 3 without denoising; (f) denoising; (c) Ship 2 without denoising; (d) Ship 2 after denoising; (e) Ship 3 without denoising; (f) Ship Ship 3 after denoising. 3 after denoising.

5.2. Feature Extraction of SN 5.2. Feature Extraction of SN According to the research in [16], three kinds of SN signals after denoising are decomposed by According to the research in [16], three kinds of SN signals after denoising are decomposed by VMD. It is then easy to obtain the IMF with the highest energy (EIMF) by calculation, and the center VMD. It is then easy to obtain the IMF with the highest energy (EIMF) by calculation, and the center frequency of EIMF is regarded as characteristic parameter in this paper. Forty samples for each kind frequency of EIMF is regarded as characteristic parameter in this paper. Forty samples for each kind of of SN were selected to calculate the center frequency of the EIMF. The center frequency distribution SN were selected to calculate the center frequency of the EIMF. The center frequency distribution of of EIMF before and after denoising is shown in Figure 14. The proposed denoising algorithm is EIMF before and after denoising is shown in Figure 14. The proposed denoising algorithm is useful for useful for distinguishing the first and second kinds of ships. distinguishing the first and second kinds of ships.

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5.3. Classification 5.3. Classification To further prove the effectiveness of the proposed 2VMD denoising algorithm, the center To further prove the effectiveness of the proposed 2VMD denoising algorithm, the center frequencies of EIMF are classified by a support vector machine (SVM), and the polynomial kernel frequencies of EIMF are classified by a support vector machine (SVM), and the polynomial kernel function is used for training and identifying. The classification results of train and test samples are function is used for training and identifying. The classification results of train and test samples are shown in Tables 8 and 9. As shown in Tables 8 and 9, the accuracy of ship 3 is 100%. However, the shown in Tables 8 and 9. As shown in Tables 8 and 9, the accuracy of ship 3 is 100%. However, recognition rates of Ship 1 and Ship 2 have been improved significantly. The accuracy after the recognition rates of Ship 1 and Ship 2 have been improved significantly. The accuracy after denoising is 95.67%, which is obviously superior to that before denoising. denoising is 95.67%, which is obviously superior to that before denoising. Table 8. The center frequency classification results without denoising. Table 8. The center frequency classification results without denoising.

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6. Conclusions 6. Conclusions In order to achieve denoising of SN signals, a hybrid secondary denoising algorithm is proposed in to this article.denoising The proposed denoising algorithm employs 2VMDalgorithm and CC. isThe target In order achieve of SN signals, a hybrid secondary denoising proposed signal decomposed using VMD, and the CC threshold used toand determine useful IMFs. in this isarticle. The proposed denoising algorithm employsis 2VMD CC. Thethe target signal is Through abundant simulation experiments and analytical comparisons, the proposed denoising decomposed using VMD, and the CC threshold is used to determine the useful IMFs. Through algorithm its superiority the following contributions: abundant demonstrated simulation experiments and and analytical comparisons, the proposed denoising algorithm demonstrated its superiority and the (1) A secondary VMD algorithm forfollowing denoisingcontributions: is put forward for the first time in this paper. (2) A denoising using 2VMD andfor CC forfirst thetime SN signal the field of (1) A novel secondary VMD algorithm algorithm is forproposed denoising is put forward the in this in paper. underwater acoustic signal processing. (2) A novel denoising algorithm is proposed using 2VMD and CC for the SN signal in the field of (3) Compared with EMD and EEMD, the CCs between the simulation signal and its IMFs using underwater acoustic signal processing. VMD are closer to the true values. This shows that the VMD algorithm can better reflect the correlation.

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Compared with EMD and EEMD, the CCs between the simulation signal and its IMFs using VMD are closer to the true values. This shows that the VMD algorithm can better reflect the correlation. Compared with EMD, EEMD and VMD denoising, the proposed denoising algorithm is a better denoising algorithm which has a high SNR and low RMSE by numerical simulations. Compared with the different input SNRs and the times of decomposition by VMD, the proposed 2VMD denoising algorithm has high SNRs for different simulation signals, especially in the case of low-input SNRs. In addition, the proposed 2VMD denoising algorithm is superior to the two denoising algorithms presented recently in [26,29]. Using the proposed 2VMD denoising algorithm and the feature extraction method in [16], the dominant frequency information is extracted. Compared with the feature extraction algorithm without denoising, the experimental results indicate that the proposed 2VMD denoising algorithm can effectively improve the recognition rate of different kinds of ships.

Acknowledgments: The authors gratefully acknowledge the support of the National Natural Science Foundation of China (No. 51179157, No. 51409214, No. 11574250 and No. 51709228). Author Contributions: Yuxing Li and Yaan Li conceived and designed the research, Yuxing Li analyzed the data and wrote the manuscript, Xiao Chen and Jing Yu collected the data. All authors have read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.

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