Noise reduction of ship-radiated noise based on noise-assisted bivariate empirical mode decomposition

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1 Indian Journal of Geo-Marine Sciences Vol. 45(4), April 26, pp Noise reduction of ship-radiated noise based on noise-assisted bivariate empirical mode decomposition Guohui Li,2,*, Yaan Li 2 & Hong Yang,2 School of Electronic and Engineering, Xi an University of Posts and Telecommunications, Xi an, 72, China 2 College of Marin, Northwestern Polytechnical University, Xi an, 772, China *[ lghcd@63.com] Received 26 March 25; revised 27 April 25 Underwater acoustic signal has the non-linear and non-stationary characteristics. Aiming at the issue on noise reduction of underwater acoustic signal, an adaptive noise reduction method of ship-radiated noise based on noiseassisted bivariate empirical mode decomposition is proposed. Firstly, a two-dimensional complex data is built by using one-dimensional real signal and adding Gaussian white noise as the real and imaginary parts, respectively. Secondly, each order intrinsic mode function is obtained by noise-assisted bivariate empirical mode decomposition. Thirdly, the noise components and the signal components can be adaptively determined by estimating the noise level of each order intrinsic mode function. Lastly, the noise reduction is done by reconstruction of the signal components. The proposed method is used in not only noisy Lorenz signal, but also real ship-radiated noise. Simulation and experiment results show that i) the time-frequency distribution of the original signal can be got accurately by the noise-assisted bivariate empirical mode decomposition, and ii) by the proposed noise reduction method, the clear chaotic attractor can be recovered from noisy signal. So the proposed method is an effective method of noise reduction for underwater acoustic signal. [Keywords: bivariate empirical mode decomposition, noise assistant, ship-radiated noise, noise reduction] Introduction Ship-radiated noise is target signal source for passive sonar. It can be used by sonar equipment for target detection, target identification, information extraction such as target azimuth, distance and depth. However, the real shipradiated noise often contains noise caused by measurement and environment. Some research results show that underwater acoustic signal not only has the non-linear and non-stationary characteristics, but also has the characteristics of chaos and fractal 2. In addition, because the ocean channel is very complex, one now little about the chaotic dynamics mechanism of ship-radiated noise. Compared with the other chaotic signal, the noise reduction of ship-radiated signal is very difficult. Therefore, it is very necessary to find the effective noise reduction method for ship-radiated noise. Chaotic signal has the characteristics of broadband and pseudo random in the timefrequency domain. The frequency band of chaotic signal and the frequency band of noise often appear partially or completely overlap. Therefore, the traditional filtering method is not suitable for noise reduction of chaotic signal. In recent years, people propose many inds of noise reduction method for chaotic signal, such as phase space reconstruction method 3, local proection method 4, wavelet method 5, shadow theorem method 6, singular spectrum analysis method 7, and so on. However, these methods have some disadvantages. Noise reduction method based on shadow theorem needs to foresee the mapping function, so it is very difficult in practice. Wavelet analysis has been well applied to the field of chaotic signal noise reduction, but it is difficult to select suitable wavelet threshold value. Local

2 47 INDIAN J. MAR. SCI., VOL. 45, NO. 4 APRIL 26 proection does not need to foresee dynamics characteristics and model of the system, so it has been widely applied. However, too high or too low noise level can lead to poor noise reduction effect. For singular spectrum analysis method, the number of principal component has great impact on noise reduction effect. As an adaptive time-frequency analysis method, the empirical mode decomposition (EMD) has been proved quite versatile in a broad range of applications for extracting signals from data generated in noisy non-linear and non-stationary processes 8. EMD can adaptively decompose data into several intrinsic mode functions (IMFs). In order to achieve the signal after noise reduction, noisy IMFs are removed by using some criterions, and the rest of the IMFs are reconstructed. As effective as the method of noise reduction based on EMD proved to be, it still leaves some annoying difficulties which have not solved because EMD is not complete. One of the maor drawbacs of the EMD is the frequent appearance of mode mixing. The mode mixing could not only cause serious aliasing in the time frequency distribution, but also mae the physical meaning of individual IMF unclear. To overcome the problem, Wu et al. proposed a new noise-assisted data analysis method in 29, the ensemble EMD (EEMD) 9. Compared with the EMD, the EEMD effectively alleviates the phenomenon of mode mixing, but it is difficult to accurately determine the magnitude of the white noise amplitude and the superposition times. Based on the bivariate EMD (BEMD) and the noise-assisted analysis idea, we propose the noiseassisted BEMD method (NABEMD). Twodimensional complex data is constructed by adding Gaussian white noise in the original signal. Then the decomposition is performed by using relevant information between the real and imaginary parts of the complex data. The proposed method can effectively reduce mode mixing phenomenon. In addition, white noise only is added one time in the proposed method. Compared with EEMD, the complexity and the running time of the algorithm are greatly reduced. Materials and Methods A.Noise-assisted bivariate empirical mode decomposition EMD method proposed by Huang et al. 8 is an adaptive time-frequency analysis method, but this method is only limited to use in one-dimensional real signal. In order to solve the problem of the decomposition of two-dimensional series, Tanaa et al. cleverly uses the original EMD to separately decompose real and imaginary channels of bivariate time series, based on properties of the complex fields. However, one disadvantage of the method is unable to ensure that the same number of IMFs across data channels, which is a maor requirement in real-world applications. Altaf et al. proposed rotation-invariant EMD (RI-EMD) to extend the original EMD, which operates by taing proections of the bivariate input signal along two directions in the complex plane to compute the local mean. The bivariate signal envelopes are calculated by interpolating the envelopes of those univariate (real-valued) proections, and the local mean of a complex signal is determined by taing the mean of the envelopes. Although the RI-EMD gives the same number of IMFs for both the signal components, it is not well-equipped to deal with fast signal dynamics due to a low number of signal proections, which limits its practical usefulness. To alleviate these problems, Rilling et al. 2 developed the BEMD, which taes multiple (univariate) proections of a complex (bivariate) signal to determine the local mean. Unlie RI- EMD, the multiple proection directions within BEMD can sample the whole complex (bivariate) plane, enhance accuracy and mae it much better suit for signals with fast changing dynamics than RI-EMD. Therefore, in recent years, BEMD has been widely used in the empirical mode decomposition of complex data. The steps required for calculating the local mean of a bivariate signal in BEMD is described as follows:

3 LI et al.: NOISE REDUCTION OF SHIP-RADIATED NOISE 47. Given a set of directions, N. 2. Proect the complex-valued signal x(t) on direction : i p ( t) Re[ e x( t)]. ( ) 3. Extract the maxima of p ( t) at the moment of ( ) ( ) t :{ t, p }. ( ) ( ) i ( ) 4. Interpolate the set {[ t, e p ]} to obtain the ' partial envelope curve in direction : e ( t). 5. Compute the mean of all tangents: N 2 ' m( t) e ( t). (2 N ) 6. Subtract the mean to obtain h ' ( t) x( t) m( t) ( 3) Its subsequent decomposition processes is the same as the original EMD. If h ' ( t ) satisfies the IMF conditions, h ' ( t) is defined as a twodimensional IMF. x( t) h ( t) is defined as a new ' signal, and BEMD sifting process is restarted. ' Otherwise, let x( t) h ( t) as a new signal, and repeat the steps to 6. In 29, Wu et al. 9 proposed a noise-assisted data analysis method, the ensemble EMD. The noise-assisted idea mainly is used to overcome the mode mixing phenomenon which appears in the EMD. The noise-assisted idea is applied to BEMD for the decomposition of one-dimensional real signal. Two-dimensional complex data is constructed by using the one-dimensional real signal and adding white Gaussian noise as the real part and imaginary part, respectively. The noiseassisted BEMD is used in constructed data. Compared with EEMD, because the NABEMD only needs to add a white noise in the whole process of decomposition, the complexity and the running time of the algorithm are greatly reduced. The NABEMD method for one-dimensional real signal is described as follows.. y( t) is the observed one-dimensional real signal. n( t) is finite amplitude Gaussian white noise. Two- dimensional complex data signal is constructed: x( t) y( t) in( t). ( 4) 2. Give a set of directions ( ), proect the complex data x(t) along the directions : i i p ( t) Re[ e x( t)] Re{ e [ y( t) in( t)]} Re{[cos( ) isin( )][ y( t) in( t)]} ( y( t)cos( ) n( t)sin( ). 5) Then repeat the steps of BEMD, the realization of one- dimensional time series based on BEMD is achieved. Eq.(5) shows that p ( t) is equivalent to add the finite amplitude Gaussian white noise to the scaled y( t ). In the BEMD, the extreme value points are selected based on p ( t). Therefore, instead of the original way of extreme point selection by y( t ), the NABEMD chooses the extreme points by referring the IMF of the noise n( t) proection, so different scales of the signal y( t) are automatically proected onto proper scales according to its own time-frequency characteristics. Thus, the mode mixing phenomenon is reduced. Noise reduction idea of NABEMD is described as follows. Firstly, a group of IMFs from high frequency to low frequency can be obtained by NABEMD. Secondly, the noise components and the signal components can be adaptively determined by estimating the noise level of each order IMF. Lastly, the purpose of noise reduction is achieved by the recombination of the signal components. The IMF evaluation criteria usually include correlation analysis, mutual information value, adacent signal standard deviation, continuous variance, etc. Noise level estimation method is applied to noise reduction based on NABEMD. The signal is decomposed into a set of IMFs by NABEMD. The noise components and the signal components can be adaptively determined by estimating the noise level of each order IMF and comparing with the setted noise threshold.

4 472 INDIAN J. MAR. SCI., VOL. 45, NO. 4 APRIL 26 si ( t ) is the ith IMF component by the NABEMD. ˆi is noise level estimation value of si ( t ). MADi denotes the absolute median deviation MADi Median{ si ( t) Median{ si ( t)}}. ( 6) Then, noise level estimation value ˆi is given by ˆ i MADi / ( 7) When the noise level estimation ˆi is greater than the noise threshold, the IMF is considered as noise component. Otherwise, it is considered as signal component. Signal components in the noise threshold range are restructured in order to realize noise reduction. Because the IMF can be adaptively udged by estimating the noise level, the undesirable noise reduction problem caused by the noise component selection too much or too little can be effectively avoided. Results and discussions When parameters, r and b are, 28 and 8/3, respectively, we use the Runge-Kutta method to calculate the integral step length which is., and get x component. After the initial unstable transition process is removed, x component signal whose data length is 2 points is adopted. These signals are respectively denoised by EMD and NABEMD. The attractor traectory before and after noise reduction for Lorenz signal when signal-to-noise ratio (SNR) is db are shown in Fig.. It can be seen from Fig. (b) that after noise reduction by EMD, the Lorenz attractor is part clearly shown irregularity and has a large deviation with the original orbit. Thus, the self similarity of the attractor traectory is low. It can be seen from Fig.(c) that after noise reduction by NABEMD, the Lorenz attractor is smooth, natural transition and a small deviation from the original orbit. Thus, the self similarity of the attractor traectory is high. Therefore, comparing Fig. (b) and Fig. (c), it can be seen that the noise reduction effect of NABEMD is obviously better than that of EMD (a) The attractor traectory before reduction (b) The attractor traectory after noise reduction by EMD (c) The attractor traectory after noise reduction by NABEMD Fig. Lorenz signal when SNR is db

5 LI et al.: NOISE REDUCTION OF SHIP-RADIATED NOISE 473 In order to accurately evaluate the effect of noise reduction, some standards are usually used. These standards include signal-to-noise ratio (SNR), the root mean square error (RMSE) and correlation coefficient (R) 3. Each noise reduction experiment is repeated 2 times. Then the average is calculated. Before and after noise reduction, SNR, RMSE and R are calculated as shown in Table. It can be seen from Table that after noise reduction, SNR and r have greatly improved, and RMSE has significantly reduced. Compared with the EMD method, the noise reduction effect of the NABEMD has a certain degree of improvement. Table SNR, RMSE and R values before and after noise reduction by EMD and the method for the noisy Lorenz signal before noise reduction after noise reduction by EMD after noise reduction by NABEMD SNB/dB RMSE R SNB/dB RMSE R SNB/dB RMSE R B. Noise reduction for real ship-radiated noise Three different types of real ship-radiated noise are chosen as sample data. Each type of signal includes a lot of samples. Each sample length is 248 points, and the sampling interval is.5ms. The original data will be normalized in advance. For three different types of real ship-radiated noise, their time-domain waveforms are respectively shown in Fig. 2(a), Fig. 3(a) and Fig. 4(a). Three types of real ship-radiated noise are respectively decomposed by EMD and NABEMD, and denoised. The attractor traectories before noise reduction for ship-radiated noise are respectively shown in Fig. 2(b), Fig. 3(b) and Fig. 4(b). The attractor traectories after noise reduction by EMD for ship-radiated noise are respectively shown in Fig. 2(c), Fig. 3(c) and Fig. 4(c). The attractor traectories after noise reduction by the proposed method for ship-radiated noise are respectively shown in Fig. 2(d), Fig. 3(d) and Fig. 4(d). It can be seen from Fig. 2, Fig. 3 and Fig. 4 that above two methods can reduce the noise and obtain the strong regular chaotic attractor. The attractor traectory by the NABEMD is more smooth and clear than that of EMD n (a) The time-domain waveform (b) The attractor traectory before noise reduction

6 474 INDIAN J. MAR. SCI., VOL. 45, NO. 4 APRIL (c) The attractor traectory after noise reduction by EMD (b) The attractor traectory before noise reduction (d) The attractor traectory after noise by NABEMD Fig.2 The first type of real ship-radiated noise n (a) The time-domain waveform (c) The attractor traectory after noise reduction by EMD (d) The attractor traectory after noise by NABEMD Fig.3 The second type of real ship-radiated noise

7 LI et al.: NOISE REDUCTION OF SHIP-RADIATED NOISE n (a) The time-domain waveform (b) The attractor traectory before noise reduction (c) The attractor traectory after noise reduction by EMD (d) The attractor traectory after noise by NABEMD Fig.4 The third type of real ship-radiated noise In order to quantitatively analyze noise reduction effect by the EMD and the NABEMD for real ship-radiated noise, we calculate feature parameters such as correlation dimension, Kolmogorov entropy, Lyapunov exponent and noise intensity shown in Table 2. Table 2 shows that after noise reduction by EMD and the NABEMD for ship-radiated noise, the above feature parameters were improved obviously. Noise reduction of ship-radiated signal is the foundation of the underwater acoustic signal processing. The noise reduction effect will have a direct impact on subsequent processing result. Aiming at the nonlinear, non-stationary and non- Gaussian characteristics of underwater acoustic signal and some shortcomings of the existing noise reduction methods of chaotic signal, the adaptive noise reduction method of ship-radiated noise based on NABEMD is proposed. The noisy Lorenz signal and real ship signal are denoised by EMD and the proposed method. The results showed that compared with the existing EMD noise reduction method, the proposed method can further reduce the noise, and more clearly restore chaotic attractor.

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