Acoustic feature extraction of radiation pressure signal induced by bubble oscillation under dual-frequency acoustic excitation based on CEEMDAN and bubble entropy

Under the influence of sound waves, oscillations of a cavitation bubble will emit a radiation pressure signal (RPS) to the surroundings. To characterize the properties of underwater targets such as submarines,^1,2 humpback whales,³ and hydro-turbines,^4,5 the RPS can be measured and employed for further analysis.^6–8 In these cases, however, the RPS is usually subject to interference from various kinds of background noise, and it is difficult to effectively identify the required acoustic feature information.⁹ Especially under multifrequency acoustic excitation, the oscillations of a cavitation bubble will exhibit more complex characteristics, and the difficulty of acoustic feature extraction will be significantly increased. Despite the importance of this topic, however, there remains a lack of relevant research. Therefore, there is an urgent need to develop methods based on advanced signal processing techniques that will provide accurate acoustic feature extraction and reduce the impact of noise on target identification.¹⁰ In this article, complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and bubble entropy (BE) are combined to extract effective acoustic feature information from the RPS induced by the oscillations of a bubble under dual-frequency acoustic excitation.

Several signal processing methods have recently been utilized to identify the oscillating characteristics in an RPS. Zhang et al.¹¹ used the fast Fourier transform (FFT) to predict the acoustic frequencies of an RPS. Si et al.¹² analyzed the FFT spectrum of radiation noise from cavitation produced by a hydrofoil. Although the FFT spectrum is very useful for acquiring the frequency characteristics of a signal, its disadvantage is that it cannot capture the time-varying characteristics of the signal.¹³ Several other methods are available that can concurrently obtain the time–frequency distribution of a signal, but they still have many shortcomings (e.g., the existence of cross-items in the Wigner–Ville distribution¹⁴ and the fixed resolution of the short-time Fourier transform¹⁵). Among other signal extraction methods employed in this field, Bao et al.¹⁶ used singular value decomposition (SVD) to extract a better modulated mode of cavitation noise radiated from a ship. Herbertson et al.¹⁷ used a wavelet transform to remove background noise from the acoustic pressure signal measured from a heart valve. The disadvantage of these two methods is that there are many input parameters that need to be manually selected (e.g., the selected threshold for extraction and the wavelet basis function¹⁸).

Huang and co-workers proposed the method of empirical mode decomposition (EMD)¹⁹ and its improved version, ensemble empirical mode decomposition (EEMD),²⁰ which can adaptively decompose a signal into several modes. Gong et al.²¹ found that when a tip leakage vortex cavitation cloud appears in a pump, the root-mean-square value of the first-order component in the EMD analysis of the vibration signal will increase rapidly. Dao et al.²² combined the wavelet threshold and EEMD to extract the effective components in the acoustic vibration signals of a model hydro-turbine. EMD is highly adaptable to changes in the amplitude and frequency of a signal, but it is easily affected by noise contained in the original signal and also suffers from the problem of mode aliasing.²³ EEMD is an improvement of EMD in which white noise is added to the signal multiple times before decomposition, and the average of the analysis results is taken when obtaining the modes. The disadvantage of EEMD is that it cannot adaptively adjust the level of added noise. CEEMDAN is a further improvement that eliminates the problem of mode aliasing during decomposition by applying an adaptive noise algorithm to EEMD. CEEMDAN is more effective at suppressing the influence of noise and improves the accuracy of decomposition by adaptively adjusting the noise level.²³ Therefore, in the present work, the more reliable CEEMDAN algorithm is employed to extract the effective acoustic components of an RPS.

In this article, CEEMDAN and BE algorithms are combined to extract the effective acoustic feature information from the RPS induced by the oscillations of a bubble under dual-frequency acoustic excitation. The signal extraction performances of SVD, EEMD, and CEEMDAN are analyzed by comparing the time-domain plots of the extracted RPSs and several evaluation indices. In addition, the signal extraction performances under different conditions are also discussed. The remainder of this article is organized as follows. In Sec. II, the fundamental formulas of bubble dynamics and the induced RPS are presented. In Sec. III, the principles of the proposed signal extraction scheme are introduced in detail. In Sec. IV, the performance of the scheme is demonstrated. In Sec. V, signal extraction capabilities of different schemes under different conditions are discussed and compared. In Sec. VI, the main conclusions of this article are presented.

In this section, the fundamental formulas for bubble oscillations are presented, and the fundamental formula for the RPS induced by these oscillations is introduced.

The fourth-order Runge–Kutta method is adopted to solve Eqs. (1)–(3) and calculate R, $Ṙ$⁠, and $R̈$⁠. In the calculation, the parameters adopted are c_l = 1486 m/s, μ_l = 1.0 mPa·s, μ_th = 0, ρ_l = 998.2071 kg/m³, κ = 1.33, σ = 0.0728 N/m, and P_a = 101 300 Pa.

CEEMDAN adopts the idea from EEMD of superimposing Gaussian white noise several times and canceling the noise through an overall average calculation. The difference between the two approaches is that after obtaining the first mode, CEEMDAN performs the overall average calculation to obtain the final first mode. After the first mode has been determined, rather than white noise being added directly to the signal, the noise component that is added is the mode that contains auxiliary noise after the previous EMD analysis. This can effectively solve the problem that the white noise during the EEMD analysis cannot be completely eliminated after overall averaging, resulting in residual white noise in the final modes. The detailed implementation of the CEEMDAN method is as follows:²³ 

1. The number of times that noise is added, Q, is set.

2. Q groups of positive and negative pairs of Gaussian white noise with a standard normal distribution are superimposed on the original signal S(t) to obtain N new signals, which can be expressed as follows:

   $Si(t)=S(t)+ε0ni(t),i=1,2,…,Q,$

   (5)

   where S_i(t) is the new signal generated after the ith superposition of Gaussian white noise, ɛ₀ is the standard deviation of the noise, and n_i(t) is the Gaussian white noise added at the ith time.
3. The new signal S_i(t) is decomposed by EMD, and Eⁱ(·) is set to be the ith signal through EMD analysis. The first component obtained by the ith decomposition can be expressed as follows:

   $Ei[Si(t)]=c1i(t)+Ri(t),i=1,2,…,Q,$

   (6)

   where $c1i(t)$ is the c₁ obtained by the ith decomposition, and Rⁱ(t) is the residual obtained by the ith decomposition.
4. The overall average of N components of c₁ is calculated, and the final c₁ component is obtained as follows:

   $c1(t)=1Q∑i=1Qc1i(t).$

   (7)
5. The residual after the first iteration, r₁, is calculated as follows:

   $r1(t)=S(t)−c1(t).$

   (8)
6. The mode containing auxiliary noise after the last EMD analysis is superimposed on r₁(t) as the original signal of the next iteration. The expression for the original signal $Sij(t)$ of the jth iteration is thus as follows:

   $Sij(t)=rj−1(t)+ε0Ej−1[ni(t)], j≥2,i=1,2,…,Q,$

   (9)

   where j is the iteration number, i is the signal number decomposed in the iteration, r is the residual generated by the previous iteration, and E is the mode that contains auxiliary noise after EMD analysis in the previous iteration.
7. Finally, steps 3–6 are repeated until the residual r becomes a monotonic function and cannot be further decomposed by EMD. A total of K components are obtained when the decomposition has been completed, and the original signal S(t) after CEEMDAN analysis can be expressed as follows:

   $S(t)=∑k=1Kck(t)+r(t).$

   (10)

The BE algorithm is based on permutation entropy, which creates a coarser-grained distribution through a bubbling sorting process, thus greatly reducing the influence of parameter selection. The calculation procedure of the BE algorithm can be summarized as follows:²⁸ 

1. A signal of length N (>10) is reconstructed in phase space and mapped to e-dimensional space with time delay τ:

   $X1={x1,x1+τ,…,x1+(e−1)τ},⋮Xh={xh,xh+τ,…,xh+(e−1)τ},⋮XN−(e−1)τ={xN−(e−1)τ,xN−(e−1)τ,…,xN}.$

   (11)
2. The elements x_h of each vector X_h in Eq. (11) sorted in ascending order using the bubbling sorting method, the number n_h of exchanges required for sorting in each vector is counted, and the probability p_h of the distribution of the number of exchanges is calculated based on the number of vectors reconstructed in phase space:

   $ph=nhN−(e−1)τ.$

   (12)
3. The conditional Rényi entropy $Hswapse$ of the distribution of the number of exchanges is calculated as follows:

   $Hswapse=−ln∑h=1nph2.$

   (13)
4. Finally, the reconstruction dimension of the phase space is increased to e + 1, and the above steps are repeated to calculate $Hswapse+1$⁠. The BE is given by the following formula:

   $BE=Hswapse+1−Hswapselne+1e−1.$

   (14)

In the scheme proposed in this article, to accurately extract the acoustic feature components from the RPS, CEEMDAN and BE are combined to perform signal extraction. First, the contaminated RPS is obtained through summing the background noise and the original simulated RPS induced by the oscillation of a bubble under dual-frequency acoustic excitation, as described by Eqs. (1)–(4). Second, on the basis of Eqs. (5)–(10), the contaminated RPS is decomposed into several modes by CEEMDAN. Then, the BE of each mode is calculated using Eqs. (11)–(14). The noise components are discarded and the useful signal components are retained on the basis of the preset threshold. After signal reconstruction, the extracted RPS is obtained. Finally, the signal extraction performance of CEEMDAN is compared with those of SVD and EEMD under a variety of conditions. Figure 1 shows the flow diagram of the proposed signal extraction scheme.

In this section, the signal extraction scheme presented above is illustrated in detail. First, the original simulated RPS is presented in the time and frequency domains. Then, the simulated RPS are processed by the proposed signal extraction scheme to extract effective acoustic feature information. Finally, the signal extraction performances of different schemes are evaluated in terms of several quantitative indices.

The frequency ω₁ of the first externally excited sound wave is 0.03 times the intrinsic frequency ω₀ of the bubble. The nondimensionalized pressure of the sound waves, ε = ε₁ = ε₂, is 0.06. The equilibrium radius during the bubble oscillation is R₀ = 10 μm. During the oscillation process, R, $Ṙ$⁠, and $R̈$ are numerically calculated using Eqs. (1)–(3). The original simulated RPS P₀ is calculated using Eq. (4). It is assumed that the distance between the acquisition point of the RPS and the center of the bubble is d = 100R₀. The initial conditions for the numerical calculation are set as R = R₀ and $Ṙ=0$⁠. The sampling frequency of the simulated RPS is 8.09 × 10⁵ Hz, and each RPS contains M = 2000 data points.

In this subsection, the typical simulated RPS under dual-frequency acoustic excitation are presented. Figure 2 shows the time-domain plot and the FFT spectrum of the original simulated RPS P₀. In Fig. 2(a), it can be seen that the waveform of RPS is relatively stable, indicating that the bubble oscillation enters a stable state under dual-frequency acoustic excitation. The FFT spectrum in Fig. 2(b) shows that there are six main frequencies in the RPS, including the frequencies of two acoustic waves (ω₁ and ω₂), their harmonics (2ω₁ and 2ω₂), and their sum (ω₁ + ω₂) and difference (ω₂ − ω₁). These characteristic frequencies are mainly concentrated in the low-frequency band of the FFT spectrum.

In this subsection, the extraction scheme for the RPS is presented. First, the contaminated RPS P_con is decomposed into 11 independent components c₁, …, c₁₁ and a residual r using CEEMDAN. The corresponding time-domain plots of these independent components are shown in Fig. 4. When the order of a component increases, its time-domain waveform gradually becomes stable and the corresponding frequency gradually decreases. The first few components are noise components, and the later components are useful signal components. Table I shows the specific BE values of all components based on CEEMDAN. During the signal extraction, it is important to determine the critical component that can distinguish these two types of components. The first four components c₁, …, c₄ exhibit strong randomness and typical noise characteristics. On the basis of the preset BE threshold (0.2) and the specific BE values in Table I, the first four components c₁, …, c₄ are considered noise and should be discarded. The remaining components are mainly useful signal components. The extracted RPS P_ext is obtained by summing these components c₅, …, c₁₁ and r. In addition, the amplitudes of c₅, …, c₈ are much higher than those of c₉, …, c₁₁ and r, indicating that the acoustic feature information of the RPS is mainly contained in these components c₅, …, c₈.

Figures 5(a)–5(c) show waveform plots of the extracted RPS P_ext obtained from SVD, EEMD, and CEEMDAN, respectively. Both SVD and EEMD are mainstream signal extraction methods. In SVD, the mean value of the singular difference spectrum is adopted to extract the acoustic feature information from the RPS. In EEMD, the extraction process is similar to that of CEEMDAN. The critical BE value distinguishing between noise and useful signal components is also taken as 0.2. The noise components are again discarded, and the remaining components are summed to obtain the extracted RPS. In Fig. 5(a), there are still many noise components in the RPS extracted using SVD, indicating that the effective acoustic feature information has not been separated from background noise. In Fig. 5(b), although many noise components have been removed in the RPS extracted using EEMD, some high-frequency details are missing compared with the original RPS P₀, as shown by the red circles. In Fig. 5(c), the time-domain plot of the RPS extracted using CEEMDAN contains fewer noise components than that obtained with SVD and retains most of the high-frequency details. Also, the time-domain waveform is much closer to the original RPS P₀ in Fig. 2(a). Thus, CEEMDAN is able to identify the precise period and amplitude of the RPS.

Table II shows the calculated evaluation indices for extraction using SVD, EEMD and CEEMDAN. The RPS extracted using CEEMDAN has a larger OSNR, smaller RMSE, and larger CI, indicating that CEEMDAN has better extraction ability than SVD and EEMD.

In this section, contaminated RPSs are simulated under different conditions, with the nondimensionalized pressure ε of the sound waves, the frequency ω₂ of the second sound wave, and the ISNR being varied. An extraction process similar to that in Sec. IV is performed on the contaminated RPS under each condition, and the extraction performances of SVD, EEMD, and CEEMDAN are compared.

Figure 6 shows time-domain plots and FFT spectra of the original RPS P₀, together with time-domain plots of the contaminated RPS P_con for ε = 0.01, 0.03, and 0.1. The ISNR remains at 10 dB in each contaminated RPS. The settings of other parameters (ω₁, ω₂, R₀, ISNR, and d) are the same as in Sec. IV. It can be seen from Figs. 6(b), 6(e), and 6(h) that as ε increases, the acoustic frequency information in the RPS becomes richer, which also means that the complexity of the RPS increases.

Figure 7 shows the evaluation indices OSNR, RMSE, and CI for different nondimensionalized pressures of the sound wave (ε = 0.01, 0.03, 0.06, 0.08, and 0.1) obtained using SVD, EEMD and CEEMDAN. It can be seen that for each ε, the RPS extracted by CEEMDAN has a larger OSNR, smaller RMSE, and larger CI, indicating that CEEMDAN has better signal extraction capability than SVD and EEMD. It is worth noting that when the contaminated RPS becomes more complex (i.e., when ε = 0.08 and 0.1), the signal extraction performance of EEMD drops dramatically (as can be seen from the trend of change of the red lines), but CEEMDAN retains an efficient extraction performance.

Figure 8 shows the time-domain plots and FFT spectra of the original RPS P₀, together with time-domain plots of the contaminated RPSs P_con for ω₂ = ω₁, 1.8ω₁, and 2ω₁. The settings of other parameters (ISNR, ε, ω₁, R₀, and d) are the same as in Sec. IV. It can be seen from Figs. 8(b), 8(e), and 8(h) that as ω₂ increases, the acoustic frequencies in the RPS become more complex.

Figure 9 compares the three evaluation indices OSNR, RMSE, and CI for different angular frequencies ω₂ of the second sound wave obtained using SVD, EEMD, and CEEMDAN. For each ω₂, the RPS extracted by CEEMDAN has larger OSNR, smaller RMSE, and larger CI, further illustrating that CEEMDAN has better signal extraction capability than SVD and EEMD.

Figure 10 shows time-domain plots of the contaminated RPS P_con for ISNR = −10, −2, 0, 5, 10, and 20 dB. The settings of other parameters (ε, d, ω₁, ω₂, and R₀) are the same as in Sec. IV. When the ISNR decreases, the contaminated RPS becomes more complex because more noise and impulse components appear in the signal.

Figure 11 shows the evaluation indices OSNR, RMSE and CI for different ISNRs obtained using SVD, EEMD, and CEEMDAN. It can be seen that for each ISNR, the RPS extracted by CEEMDAN has larger OSNR, smaller RMSE and larger CI, demonstrating that CEEMDAN has better signal extraction capability than SVD and EEMD. Figures 11(b) and 11(c) reveal that when the ISNR decreases, the signal extraction performance of SVD drops dramatically (as can be seen from the trend of change of the black lines), but CEEMDAN is still able to efficiently extract the acoustic features, as is EEMD, albeit with a slightly poorer performance.

In this article, the CEEMDAN and BE algorithms have been combined to improve the accuracy of extracting acoustic feature information from the RPS induced by bubble oscillations. Following the signal extraction process, accurate acoustic characteristics of the RPS under dual-frequency acoustic excitation can be captured, facilitating identification of the real-time status of underwater targets.

The main conclusions of this article are as follows:

1. Using the signal extraction scheme proposed here, it is possible to accurately extract acoustic feature information from an RPS to allow better identification of underwater targets. More noise components are removed from time-domain plots of the RPS extracted by CEEMDAN than when SVD is used for extraction, and more high-frequency details are retained than when EEMD is used, which facilitates observation of the oscillation characteristics of the RPS.

2. Quantitative analysis results show that the RPS extracted by CEEMDAN has larger OSNR, smaller RMSE, and larger CI than those extracted by SVD and EEMD, providing further confirmation of the superior signal extraction ability of CEEMDAN.

3. Under a variety of conditions, CEEMDAN shows better signal extraction performance than SVD and EEMD. When a contaminated RPS becomes more complex, CEEMDAN retains its efficient extraction performance, while the extraction capabilities of SVD and EEMD drop sharply.

In the future work, measurements of the RPS induced by bubble oscillations under acoustic excitation will be utilized to verify the effectiveness of the proposed signal extraction scheme.

This work was financially supported by the Scientific Research Starting Foundation of Fuzhou University (Project No. 511429).

The authors have no conflicts to disclose.

Xianghao Zheng: Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Writing – original draft (equal). Yuning Zhang: Conceptualization (equal); Data curation (equal); Supervision (equal); Writing – review & editing (equal). Yiming Li: Investigation (equal); Validation (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.