A novel noise reduction technique for underwater acoustic signals based on complete ensemble empirical mode decomposition with adaptive noise, minimum mean square variance criterion and least mean square adaptive filter

Yu-xing Li , Long Wang

a Faculty of Information Technology and Equipment Engineering, Xián University of Technology, Xián 710048, Shaanxi, China

b School of Marine Science and Technology, Northwestern Polytechnical University, Xián 710072, Shaanxi, China

Keywords:Underwater acoustic signal Noise reduction Empirical mode decomposition (EMD)Ensemble EMD (EEMD)Complete EEMD with adaptive noise(CEEMDAN)Minimum mean square variance criterion(MMSVC)Least mean square adaptive filter (LMSAF)Ship-radiated noise

ABSTRACT Underwater acoustic signal processing is one of the research hotspots in underwater acoustics. Noise reduction of underwater acoustic signals is the key to underwater acoustic signal processing. Owing to the complexity of marine environment and the particularity of underwater acoustic channel, noise reduction of underwater acoustic signals has always been a difficult challenge in the field of underwater acoustic signal processing.In order to solve the dilemma,we proposed a novel noise reduction technique for underwater acoustic signals based on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), minimum mean square variance criterion (MMSVC) and least mean square adaptive filter (LMSAF). This noise reduction technique, named CEEMDAN-MMSVC-LMSAF, has three main advantages: (i) as an improved algorithm of empirical mode decomposition (EMD) and ensemble EMD (EEMD), CEEMDAN can better suppress mode mixing, and can avoid selecting the number of decomposition in variational mode decomposition(VMD); (ii)MMSVC can identify noisy intrinsic mode function (IMF), and can avoid selecting thresholds of different permutation entropies; (iii) for noise reduction of noisy IMFs,LMSAF overcomes the selection of decomposition number and basis function for wavelet noise reduction. Firstly, CEEMDAN decomposes the original signal into IMFs, which can be divided into noisy IMFs and real IMFs. Then, MMSVC and LMSAF are used to detect identify noisy IMFs and remove noise components from noisy IMFs. Finally, both denoised noisy IMFs and real IMFs are reconstructed and the final denoised signal is obtained. Compared with other noise reduction techniques,the validity of CEEMDAN-MMSVC-LMSAF can be proved by the analysis of simulation signals and real underwater acoustic signals,which has the better noise reduction effect and has practical application value. CEEMDAN-MMSVC-LMSAF also provides a reliable basis for the detection, feature extraction,classification and recognition of underwater acoustic signals.

1. Introduction

With the development of science and technology, people pay more and more attention to the utilization, exploitation and protection of the oceans. Due to the complexity of the marine environment and the continuous innovation of underwater target noise reduction technique, the detection, feature extraction and recognition of underwater acoustic signals by modern sonar systems are facing severe challenges [1-3]. With the continuous optimization of stealth technology and quiet submarines, the radiated noise of submarines is decreasing year by year, which makes the identification of submarines more and more difficult.Therefore,there is an urgent need to carry out research on underwater acoustic signal processing methods and their applications, especially on noise reduction of underwater acoustic signals, so as to lay a foundation for further research on detection, feature extraction, classification and recognition of underwater acoustic signals [4].

Underwater acoustic signals have typical non-linear, non-Gaussian and non-stationary characteristics. Traditional signal analysis and processing techniques are based on Fourier analysis,which cannot express the time-frequency local performance of the signal. Wavelet transform can multi-scale refine the signal by calculating of flex and transition,which solves the problem that the size of Fourier transform window cannot change with frequency.However, wavelet transform is still based on Fourier analysis, and limited by the selection of wavelet basis function and decomposition layer.For underwater acoustic signal processing,we hope that we can not only get the frequency information of the signal,but also get the law of the frequency changing with time. As an empirical signal analysis method, empirical mode decomposition (EMD)overcomes the limitation of Fourier transform fundamentally and can theoretically decompose any signal into IMFs [5-7]. Some experts have put forward improved EMD algorithms to solve mode mixing of EMD. Among them, the more universal and effective algorithms are ensemble EMD(EEMD)[8,9]and complete EEMD with adaptive noise (CEEMDAN) [10,11].

Noise reduction techniques using EMD, improved EMD algorithms and variational mode decomposition (VMD) have been applied in many fields [12-14]. In Ref. [15], a noise reduction technique for high-g micro-electromechanical system accelerometer were proposed based on EMD, continuous mean square error criterion and wavelet threshold,which can remove 96%of the noise in the original signal. In Ref. [16], a two-stage noise reduction scheme for electrocardiogram signals were proposed based on grey spectral noise estimation,EMD and EEMD,the results show that the proposed noise reduction scheme is superior to the traditional noise reduction methods using EMD and EEMD.In Ref.[17],a noise reduction technique for acoustic-based system were proposed based on EMD and improved fruit fly optimization algorithm(IFFOA), IFFOA was used to determine the threshold of IMF, the validity of the noise reduction technique was verified by simulation and actual acoustic-based diagnosis system. In Ref. [18], a hybrid noise reduction method for the gear transmission system were proposed, time-frequency peak filtering with different window lengths was used to filter noisy IMF according to the permutation entropy of IMF by CEEMDAN,the noise reduction results show that the proposed method was superior to the other methods. In Refs. [19,20], VMD was used to decompose mold level signal and fiber bragg grating deformation spectrum signal, the noisy IMFs were denoised by wavelet denoising to obtain the denoised signals.The results show that the two proposed denoising methods have better effect than traditional ones.

Wavelet denoising is an effective denoising technique, which includes three basic steps:(i)wavelet transform of noisy signal;(ii)wavelet coefficients thresholding;(iii)inverse wavelet transform to obtain the denoised signal.However,it is difficult to choose the best wavelet basis function,decomposition level and thresholding rule.Least mean square adaptive filter is also an effective denoising technique [21]. The criterion of LMSAF is to minimize the mean square error, that is, to minimize the square of mathematical expectations of the difference between the expected signal and the actual output of the filter, and to modify the tap-weight vector according to this criterion. It has the characteristics of low computational complexity, strong stability and wide range of applications. However, the noise reduction effect is limited by only using LMSAF.

In recent years, some underwater acoustic signal noise reduction techniques have been proposed based on different kinds of mode decomposition algorithms.CEEMDAN and VMD[22-24]are applied to decompose underwater acoustic signals into IMFs; correlation coefficient [25], mutual information [26] and different kinds of permutation entropies[27]are used to identify noisy IMFs;wavelet threshold denoising is usually applied to process noisy IMFs [28]. However, the above noise reduction techniques have some limitations: (i) VMD requires pre-set decomposition quantities and balancing parameter; (ii) a threshold needs to be set to identify noisy IMFs by using correlation coefficient, mutual information and different kinds of permutation entropies; (iii) wavelet threshold denoising also needs to choose the appropriate decomposition level and basis function.

In order to overcome the above-mentioned difficulty, a novel noise reduction technique for underwater acoustic signals is proposed in this study by taking advantage of CEEMDAN, minimum mean square variance criterion (MMSVC) and least mean square adaptive filter (LMSAF). Firstly, difficulties in parameter selection can be overcome by using CEEMDAN. Secondly, using threshold to identify noisy IMFs is avoided by MMSVC. Thirdly, threshold selections for noisy IMFs denoising are solved by using LMSAF.Lastly,the proposed denoising technique presents better performance than other similar techniques.

The structure of this paper is organized as follows: Section 2 is the basic theories of CEEMDAN,MMSVC and LMSAF;the proposed noise reduction technique is presented in Section 3;Section 4 and 5 give the simulation and actual experiments, Section 6 is the conclusion.

2. Basic theories

2.1. CEEMDAN

In this paper, we use the advantages of CEEMDAN with better decomposition performance and without preset parameters to process underwater acoustic signals.The purpose of CEEMDAN is to decompose underwater acoustic signals into IMFs with different oscillation modes one by one [10]. Therefore, the decomposition process of CEEMDAN consists of three steps.

Step 1: Obtain the first IMF.

(1) The mixed signal xi（t） can be expressed as:

where x（t） and Ni（t） are underwater acoustic signal and the i-th standard Gaussian white noise sequences.represented as:

(2) Decompose all xi（t） into ci1（t） and resi（t） as follows:

where ci1（t）and resi（t）are the first IMF and residual item by EMD. EMD algorithm can be referred to in Ref. [29].

(3) Calculate the mean of ci1（t）:

where C1（t） is the first IMF of x（t）, named IMF1.

Step 2: Obtain the other IMFs.

(1) Calculate the residual item Res1（t）:

(2) Decompose all Ni（t） as follows:

(4) Construct xnew1（t） and decompose it by EMD as follows:

(5) Calculate the mean of cRes1Ni1（t）to obtain the C2（t）of x（t）and its residual item Res2（t） as follows:

(6) The other IMFs can be obtained according to the following formulas:

Step 3.The original signal x（t） can be expressed as:

where N and Res（t）represent the number of Cj（t）and the residual item of x（t）.

2.2. MMSVC

MMSVC is used to identify noisy IMFs by CEEMDAN in this paper. The specific steps of MMSVC are summarized as follows:

(1) Define New（n） as the residual signal of removing the first n

IMFs from original signal.

where x（t）and Ci（t）represent the original signal and the i-th IMF of x（t） by CEEMDAN.

(2) Mean square variance of New（n） and New（n+1） can be expressed as:

where N and cn+1（ti）represent the length of original signal and the（n+1）-th IMF of x（t） by CEEMDAN.

Fig.1. The schematic diagram of LMSAF.

(3) Mean square variances of two adjacent IMFs are calculated and minimum mean square variance is obtained.

MMSVC takes minimum mean square variance as critical point.When mean square variance of New（n）and New（n+1）is reaches its minimum value, the first n-1 IMFs are regarded as noisy IMFs[30].

2.3. LMSAF

LMSAF has the advantages of stable performance, simple structure and easy implementation. Its basic principle is noise cancellation for the noisy signal and the reference noise signal,so as to eliminate the noise in the noisy signal [31]. Reference noise signal is correlated with noisy signal, but not with real signal in LMSAF.

Fig. 1 is the schematic diagram of LMSAF. Input of LMSAF includes noisy signal s+n1 and reference noise signal n2,n1 and n2 are correlated,and the correlation between noise and real signal is small, the output y of LMSAF can be expressed as:

where n′is the estimation of n2 through LMSAF. The specific processes of LMSAF are as follows:

(1) Initialize the number of taps of filter M, step factor μ, tapweight vector W（0）.

(2) Calculate the estimated output of the current filter n′.

where WT（n） represents the current tap-weight vector.

(3) Calculate estimation error e（n）.

where e（n） is equal to y.

(4) Update tap-weight vector W（n + 1）.

Fig. 2. The flow chart of CEEMDAN-MMSVC-LMSAF.

(5) Repeat steps (2), (3) and (4) until complete output is obtained.

3. Noise reduction technique for underwater acoustic signals

This paper puts forward a novel noise reduction technique for underwater acoustic signals based on CEEMDAN, MMSVC and LMSAF, named CEEMDAN-MMSVC-LMSAF. The flow chart of CEEMDAN-MMSVC-LMSAF is shown in Fig. 2. The experimental steps are as follows:

(1) Underwater acoustic signals are decomposed into a set of IMFs by CEEMDAN.

(2) Calculate mean square variances of two adjacent IMFs, and identify noisy IMFs and real IMFs according to MMSVC.

(3) LMSAF is carried out on noisy IMFs,denoised noisy IMFs are obtained.

(4) Reconstruct denoised noisy IMFs and real IMFs, denoised underwater acoustic signals can be obtained.

4. Noise reduction of simulation signals

4.1. CEEMDAN of simulation signals

We apply CEEMDAN-MMSVC-LMSAF to noisy bumps signal.Fig.3 is bump signal and noisy bump signal with 5 dB.The sampling frequency and the number of sampling point are 1 KHz and 4096,respectively. CEEMDAN result of noisy bumps signal is shown in Fig.4.As shown in Figs.3 and 4,bump signal is submerged in noise,and the CEEMDAN result is arranged in descending order of frequency.

Fig. 3. Bumps and noisy bumps signals.

Fig. 4. CEEMDAN result of noisy bumps signal.

Table 1 Mean square variances of two adjacent IMFs.

4.2. Identification of noisy IMFs

We calculate mean square variances of two adjacent IMFs and get the results in Table 1. As shown in Table 1, M（5） is minimum mean square variance. Therefore, we can know that the first five IMFs and the last five IMFs are noisy IMFs and real IMFs according to MMSVC, respectively.

4.3. Noise reduction of noisy IMFs and reconstruction

Fig. 5. Noise reduction results of noisy signals.

Table 2 Results of different noise reduction techniques (a) Bump (b) Block (c) Doppler (d) Heavysine.

Fig. 6. Normalized time-domain waveforms for the four ships.

We use LMSAF to process the first five IMFs,five denoised noisy IMFs are obtained. Then, noise reduction result of noisy bumps signal can be obtained by reconstructing denoised noisy IMFs and real IMFs. Three noise reduction experiments are conducted on doppler signal, blocks signal and heavysine signal, which can be obtained by the MATLAB software.Noise reduction results of noisy signals with 5 dB are shown in Fig. 5. As shown in Fig. 5, after comparing with clear signals, most of noise components are removed,the denoised signals are close to the original ones.

4.4. Comparison of noise reduction techniques

To prove the validity of CEEMDAN-MMSVC-LMSAF,experiments of different noise reduction techniques for different signals under different input signal-to-noise ratios (SNRs) are added. Results of different noise reduction techniques are shown in Table 2. Simulation signals are bump, block, doppler and heavysine. Input SNRs are -10 dB, -5 dB, 0 dB and 5 dB. Noise reduction techniques include EMD combined with MMSVC (EMD-MMSVC), CEEMDAN combined with MMSVC (CEEMDAN-MMSVC), CEEMDAN-MMSVCLMSAF and LMSAF.

As shown in Table 2, maximum SNRs and minimum RMSEs under different input SNRs are marked in bold.The noise reduction techniques based CEEMDAN are better than EMD-MMSVC and LMSAF, CEEMDAN-MMSVC-LMSAF has lower root mean square error(RMSE)and higher SNR than the other three noise reduction techniques.

5. Noise reduction of underwater acoustic signals

5.1. CEEMDAN of underwater acoustic signals

To further prove the effectiveness of CEEMDAN-MMSVC-LMSAF for underwater acoustic signals, CEEMDAN-MMSVC-LMSAF is carried out on ship-A, ship-B, ship-C and ship-D. Four types of ships have different degrees of ocean background noise, which received by hydrophones, including biological noise, seismic noise, rain noise and man-made noise. The sampling frequency and the number of sampling point are 44.1 KHz and 2048, respectively.Normalized time-domain waveforms for the four ships are shown in Fig. 6. As shown in Fig. 6, there are different degrees of ocean background noise in four types of ship signals, which pollute the original ship signals.CEEMDAN results of the four ships are shown in Fig.7.As shown in Fig.7,each ship signal is decomposed and the IMFs are arranged in descending order of frequency.

Fig. 7. CEEMDAN results of the four ships.

Fig. 7. (continued).

CEEMDAN is the key step of underwater acoustic signal denoising. Ocean background noise is typically found in highfrequency IMFs. The decomposition result directly affects the next steps of denoising.

5.2. CEEMDAN-MMSVC-LMSAF for underwater acoustic signals

Firstly,we calculate mean square variances of two adjacent IMFs for the four ships.Then,we can get a result that the first four IMFs are noisy IMFs for the four ships according to MMSVC. Finally,denoised noisy IMFs by LMSAF and real IMFs are reconstructed.Noise reduction results of the four ships by CEEMDAN-MMSVCLMSAF are shown in Fig. 8.

As shown in Fig. 8, time-domain waveforms of the four ships after CEEMDAN-MMSVC-LMSAF are smoother than before,most of the high-frequency noise is removed.

5.3. Noise reduction effect

Fig. 8. Noise reduction results of the four ships by CEEMDAN-MMSVC-LMSAF.

Chaos theory is an important theory for studying non-linear random signals, and the underwater acoustic signal has chaotic characteristics. Therefore, we can evaluate the noise reduction effect of underwater acoustic signal according to attractor trajectory[32]. The attractor trajectory is the trajectory depicted by the solution of the dynamic equation in the phase diagram.Generally,the attractor trajectory of clear signal is very smooth and regular, and the attractor trajectory of noisy signal is very rough and irregular.The attractor trajectories before and after noise reduction for the four ships are shown in Fig. 9. For better observation, we chose a time lag of 2 to avoid similar values between adjacent amplitudes[26]. n is the sampling point, the abscissa of attractor trajectories represent the amplitudes of x(n), the ordinate represent the amplitudes of x(n+2), and the range of n is from 1 to 2046.

As shown in Fig. 9, attractor trajectories after CEEMDANMMSVC-LMSAF are more regular and smooth than before. Similar results can be obtained by comparing a large number of denoising results for four types of ships. It can be concluded from the experiments that the proposed CEEMDAN-MMSVC-LMSAF noise reduction technique in this paper can effectively eliminate the noise component.Thus,CEEMDAN-MMSVC-LMSAF noise reduction technique also can lay a good foundation for further engineering application.

6. Conclusions

In this paper,a novel noise reduction technique for underwater acoustic signals is proposed based on CEEMDAN, MMSVC and LMSAF. The main advantages of CEEMDAN-MMSVC-LMSAF are as follows:

(1) CEEMDAN-MMSVC-LMSAF combines the advantages of three popular methods.CEEMDAN has a better performance on suppression of mode mixing, and can avoid selecting parameters of VMD.MMSVC is easier to identify noisy IMFs,and can avoid selecting thresholds of different permutation entropies. LMSAF can avoid the selection of decomposition number and basis function for wavelet noise reduction.

(2) CEEMDAN-MMSVC-LMSAF has better noise reduction performance than EMD-MMSVC,CEEMDAN-MMSVC and LMSAF for simulation signals, which has the highest SNR and the lowest RMSE under different input SNR, and SNR and RMSE are averagely increased by about 0.5 dB and decreased by about 0.02, respectively.

(3) CEEMDAN-MMSVC-LMSAF can realize the noise reduction of underwater acoustic signals. The attractor trajectory after noise reduction are more smoother and regular than before,which proves the validity of the proposed noise reduction technique.Moreover,it is beneficial to the further processing of underwater acoustic signals, such as detection, feature extraction, classification and recognition.

In further studies, we should study an improved CEEMDAN algorithm to further suppress mode mixing. In addition, we should analyze the noise reduction effect of underwater acoustic signal under different sampling rates and samples.

Fig. 9. The attractor trajectories before and after noise reduction for the four ships.

Conflicts of interest

The authors declare no conflict of interest.

Acknowledgement

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (No.11574250).