Underwater discharge sound sources are widely utilized. The acoustic signals generated by underwater discharge consist of shock waves and bubble pulsations. The shock wave is a major component of underwater discharge sound sources and is worthy of further study. The minimum-phase cepstrum method is selected to separate the shock wave, and a customized experiment is designed to determine the optimal parameters of this method. The mixed shock wave bubble signal and single shock wave signal are simulated and fitted and are transmitted and received in the anechoic pool, respectively. Different parameters are used to process the signals, and the optimal parameters are selected by comparing the separation results. To verify the validity of the method, the shock waves generated with the same electrode and circuit parameters in the anechoic pool and non-anechoic pool are separated and compared. Experimental results show that this shock wave separation method exhibits significant performance. This method of shock wave separation is of service to the research of underwater discharge acoustic sources.

## I. INTRODUCTION

Underwater discharge based on the electrohydraulic effect^{1} is widely utilized as an effective explosive sound source in many areas, such as ocean exploration,^{2} water sterilization,^{3} lithotripsy,^{4} and wideband acoustic interference.^{5} Extensive research has been carried out to discover the mechanism and characteristics of underwater plasma discharge shock waves. The mechanism^{6} is shown in Fig. 1: The energy storage capacitor *C* is charged with a high voltage *U*. When the gas switch *S* is switched on under the action of an external trigger pulse, the high voltage *U* on the capacitor *C* is suddenly added to the water gap *G*, causing *G* to break down, forming a high temperature, high density, high conductivity discharge channel. This is followed by a rapid discharge of the capacitor *C* through *S* and *G* (with a discharge time of a few nanoseconds to a few milliseconds), generating an extremely strong discharge current (which can reach several thousand A to hundreds of kA magnitude). Due to the huge energy instantly released in the discharge channel of *G*, the water in the channel will quickly vaporize and cause an explosion; the explosion is caused by an impact pressure of up to 1000–10 000 atm, so the discharge channel expands outward at high speed. As the discharge time is very short, this high pressure can be compressed around the discharge channel water medium, which forms a pressure and results in a sudden change in the density of the surface, thus forming a very powerful shock wave and the outward propagation at supersonic speed. Then the plasma channel cools down and starts to oscillate as a bubble several times. The bubble pulsations lag behind shock waves by milliseconds and overlap in the time domain. Figure 2 shows the acoustic signal of underwater discharge, where the shock wave and bubble pulsations can be easily recognized.

Shock waves are the main cause of concern after discharge.^{7} The bubble pulsations are the by-product of underwater discharge shock waves and cannot be eliminated. In the field of underwater communication, bubble pulsations following shock waves seriously interfere with the study of the propagation characteristics of explosive signals in the ocean.^{8} In the area of basic underwater discharge research, it is necessary to study the discharge shock wave characteristics under different dielectric conditions. The adjustment of dielectric properties in free far-field or anechoic environments is often challenging. However, experiments in smaller containers, where the dielectric properties such as hydrostatic pressure and electrical conductivity can be easily adjusted, are also susceptible to reverberation. The signals of shock waves, bubbles, and reverberations overlap, affecting the accuracy of the analysis of the results. Separation of the shock wave component from the measured acoustic signal helps improve the accuracy of the analysis in different test water environments. The traditional methods such as time-domain subtraction or frequency domain division are not effective in removing bubble pulsation components and reverberations.^{8} The essence of this problem is blind source separation. There are two signal sources: shock waves and bubble pulsation. However, there is usually only one signal receiver: a hydrophone. Commonly used blind source separation techniques require the number of receivers to be no less than the number of signal sources. Hence, an approach suitable for these kinds of situations needs to be studied.

The primary aim of this study is to obtain an appropriate method to separate the shock wave component from the originally collected underwater discharge acoustic signal (shown in Fig. 3). The minimum-phase cepstrum method is chosen to extract the shock wave, and an experiment was designed to determine the optimal parameters of the minimum-phase cepstrum method for processing underwater discharge shock waves. Then the shock waves generated with the same electrode and circuit parameters in the anechoic pool and non-anechoic pool are separated and compared to verify the validity of the method. This study provides a practical approach to the analysis and comparison of underwater discharge shock wave characteristics.

## II. METHODS AND THE EXPERIMENTAL SETUP

### A. Shock wave signal separation method

The acoustic signal of underwater spark discharge consists of a shock wave and several subsequent bubble pulsations. In non-anechoic environments, there are also reverberations in the collected signal. There are already some well-established methods of de-reverberation in speech signal processing.^{9,10} Cepstral processing^{11} is one of the more prominent methods. Zhang *et al.*^{8} presented the underwater explosive signal with a convolving model, shown as

where *x*(*t*) is the underwater spark discharge acoustic signal, *s*(*t*) is the shock wave component, *a* · *s*(*t* − *τ*) is the bubble pulsations, *a* is the amplitude variation coefficient of bubble pulsations relative to the shock wave, *τ* is the time delay between the shock wave and bubble pulsations, *δ*(*t*) is the impulse function, and *h*(*t*) = *δ*(*t*) + *δ*(*t* − *τ*). Hence, *x*(*t*) is the output of *s*(*t*) through the system with an impulse response *h*(*t*), that is, *x*(*t*) can be expressed as the convolution of *s*(*t*) and *h*(*t*). When *a* < 1, a homomorphic deconvolution filter can be used to remove the bubble pulsations from the original signal. The traditional complex cepstrum domain filtering method^{12} is used to construct a reversible characteristic system so that *x*(*t*) is mapped to the sum of two separable quantities $s\u0302(t)$ and $h\u0302(t)$ corresponding to *s*(*t*) and *h*(*t*), respectively, in the cepstrum domain [shown in the following equation],

Then $s\u0302(t)$ and $h\u0302(t)$ are separated by the linear filtering technique. After the two quantities are separated, the signal corresponding to *s*(*t*) is reconstructed in the time domain under the inverse transformation of the original system. The detailed separation method is used to add a low time window function to the complex cepstrum domain for filtering^{13} (shown in Fig. 4) or to reduce the influence of reverberation by removing the peak value in the complex cepstrum domain.^{8} In Liu’s study,^{14} the minimum-phase component of the reverberation signal is filtered in the complex cepstrum domain and then combined with the all-pass component to remove the reverberation. This method can easily determine the parameters of the window function and avoid phase ambiguity because it can be calculated by the real cepstrum. Compared with speech signals, underwater discharge signals have a very short duration time, so there is no need of framing. The implementation process of separation is shown in Fig. 5.

$x\u0302r(t)$ is the real cepstrum of *x*(*t*), $x\u0302min(t)$ is the minimum-phase cepstrum of *x*(*t*), and *X*(*ω*) is the FFT spectrum of *x*(*t*). *X*_{min}(*ω*) is the minimum-phase component of *X*(*ω*). *X*_{all}(*ω*) is the all-pass phase component of *X*(*ω*). $x\u0302min(t)$ can be calculated by

$r\u0302(t)$ is a window function whose purpose is to zero the cepstrum for negative frequencies [shown in the following equation],

As shown in Fig. 5, $s\u0302min(t)$ is the minimum-phase complex cepstrum of *x*(*t*) after being filtered by the low time window function $w\u0302low(t)$. Correspondingly, *Y*_{min}(*ω*) is the minimum-phase component of *X*(*ω*). *S*(*ω*) is the FFT spectrum of the reconstructed signal. *s*(*t*) is the reconstructed shock wave signal.

The key to this separation method is to determine the parameter of the cepstrum low time window function $w\u0302low(t)$—the width of the stopband and transition band—as shown in Fig. 6, where *N* is the data length, *M* is the stopband width, *h* is the transition bandwidth, and *h*(*n*) is the curve function of the transition band. Zhang and Chen^{13} revealed that the optimal widths of the stopband and transition band for speech de-reverberation are **1/128** and **1/8**, respectively. The optimal bandwidths for de-reverberation of underwater discharge signals need to be redefined based on the signal waveform characteristics.

### B. Setup of the shock wave separation algorithm’s optimal parameter determination experiment

A customized experiment was implemented to determine the optimal widths of the filter’s stopband and transition band suitable for underwater plasma sound signals. As shown in Fig. 7, a damped oscillation signal similar to the underwater spark discharge signal was constructed, which had a higher first oscillation (corresponding to the shock wave) and more minor subsequent oscillations (corresponding to the bubble pulsations and reverberations), and transmitted by a piezo ceramic transducer (produced by the Hangzhou Institute of Applied Acoustics) in the anechoic tank and then collected by a hydrophone. The transducer and the hydrophone were placed 1 m under water, and the distance between the transducer and hydrophone was 1 m. The collected signal was filtered by low time window filters with different stop and transition bandwidths in the cepstrum domain. The reconstructed signals were compared to find out the optimal stop and transition bandwidths of the cepstrum low time window function for underwater shock wave signals.

### C. Setup of the separation algorithm’s optimal parameter determination experiment

Subsequently, to verify the validity of the method, a validation experiment was implemented. The underwater discharge acoustic signals were collected in two different environments: (1) The anechoic tank was 13.5 m long, 7 m wide, and 7.5 m deep. The hydrophone and the center of the electrode gap were at the same depth, and the distance between them was 0.25 m. The hydrophone and the electrodes were deployed 1 m under water. (2) The non-anechoic pool was 20 m long, 20 m wide, and 10 m deep. The hydrophone and the center of the electrode gap were at the same depth, and the distance between them was 0.25 m. The hydrophone and the electrodes were deployed 0.25 m under water. The distance between the electrode gap and the pool wall was 0.25 m. The test setup is as shown in Fig. 8. A pair of 304 stainless steel cylindrical electrodes, which had a cross section diameter of 5 mm and a length of 150 mm, was fixed on a self-made plastic base for discharging. The gap distance between the electrodes was 1 mm. The electrode tip cross sections were flat and bared; the rest of the parts were wrapped with insulating rubber. The electrode tips were cleaned after each discharge. The underwater plasma discharge system was driven by a high voltage pulse power with a fixed output voltage of 10 kV and an energy storage capacitance of 0.11 *μ*f. The acoustic signals were collected by a hydrophone produced by the Hangzhou Institute of Applied Acoustics with a sensitivity of −205 dB re 1 V/*μ*Pa in the range of 5 Hz–15 MHz. A RIGOL MSO5354 digital storage oscilloscope recorded the collected sound signals. The water conductivity was 0.37 mS/cm, and the water temperature was 17.5.

## III. RESULTS

### A. The optimal parameters for the shock wave separation algorithm

As shown in Fig. 9, the damped oscillation signal was transmitted and collected. Then the signal was processed using the minimum-phase cepstrum method with different parameters of the cepstrum low time window function. The purpose of this method is to separate the shock wave component from the original signal mixed with bubble pulsations and reverberations—the reconstructed waveforms corresponding to the shockwave components.

The index used in the experiment to measure the quality of filter parameters is the wave correlation coefficient between the reconstructed signals and the damped oscillation signal’s first oscillation. It can be concluded from the correlation coefficients (shown in Table I) that the optimal widths of the filter’s stopband and transition band are *N*/64 and *N*/32, respectively, where *N* is the data length.

Stopband . | Transition band . | Correlation coefficient . |
---|---|---|

64 | 4 | 0.5540 |

64 | 8 | 0.5919 |

64 | 16 | 0.5947 |

64 | 32 | 0.8809 |

64 | 64 | 0.3573 |

64 | 128 | 0.2239 |

64 | 256 | 0.2190 |

128 | 4 | 0.5592 |

128 | 8 | 0.5920 |

128 | 16 | 0.5801 |

128 | 32 | 0.4655 |

128 | 64 | 0.2391 |

128 | 128 | 0.1950 |

128 | 256 | 0.2117 |

Stopband . | Transition band . | Correlation coefficient . |
---|---|---|

64 | 4 | 0.5540 |

64 | 8 | 0.5919 |

64 | 16 | 0.5947 |

64 | 32 | 0.8809 |

64 | 64 | 0.3573 |

64 | 128 | 0.2239 |

64 | 256 | 0.2190 |

128 | 4 | 0.5592 |

128 | 8 | 0.5920 |

128 | 16 | 0.5801 |

128 | 32 | 0.4655 |

128 | 64 | 0.2391 |

128 | 128 | 0.1950 |

128 | 256 | 0.2117 |

### B. Result of the validation experiment

To verify the practicality of the method, the acoustic signals generated with the same electrode and circuit parameters were collected in an anechoic pool and a non-anechoic pool, respectively. Then the signals were processed using the method and parameters determined above. The original waveforms, cepstrum waveforms, and reconstructed waveforms of each signal are shown in Fig. 10. The discharge channel was formed at time 0 (electromagnetic noise is not displayed), and the first peaks at about 0.17 ms are the shock wave pressure. The reconstructed signals’ correlation coefficient is **0.8166**. The results show that the shock wave components separated from the original signals collected in different water environments have good consistency.

## IV. CONCLUSIONS

The present study was conducted to design a method to separate the shock wave components from the underwater discharge acoustic signals collected in different water environments. The findings of this study suggest that the optimal widths of the filter’s stopband and transition band suitable for underwater discharge acoustic signals are *N*/64 and *N*/32 (*N* is the data length), respectively, when using the minimum-phase cepstrum method. The method shows good performance in separating shock wave components of acoustic signals collected in an anechoic pool and a non-anechoic pool. This study found that the parameters of the de-reverberation algorithm need to be optimized for the characteristics of the signal to be processed. The results prove that the process can effectively remove the interference of bubble pulsation and reverberation, ensuring the accuracy of the compared results under different conditions.

## ACKNOWLEDGMENTS

The experimental environment and measuring instruments were provided by Chongqing Qianwei Technology Group Co., Ltd.

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to state restrictions.