High-Order Domain Feature Extraction Technology for Ocean Acoustic Observation Signals: A Review

For the acquisition of ocean observation information, the underwater acoustic signal is the only known medium that can propagate over long distances in water. However, the underwater acoustic field environment is highly time-varying and spatially variable, and ocean acoustic observation signals are mixed Gaussian, non-cooperative, and nonlinear. Therefore, feature extraction of underwater target signals has been a challenging research direction. The high-order domain feature extraction method of ocean acoustic observation signal is widely used because of its advantages of anti-Gaussian background noise and preservation of signal phase information. Firstly, this paper analyzes three aspects, including the characteristics of ocean acoustic observation signals, the limitations of second-order domain processing methods, and advantages of high-order domain processing methods. Then, this paper focuses on theoretical advantages and application areas of signal high-order domain feature extraction technology. The technical development and application performance of common feature extraction technology in the field of ocean acoustic observation signal processing by higher-order domain methods in recent years are analyzed in the paper, and the challenges and future trends of higher-order domain feature extraction technology in marine acoustic observation signal processing are discussed in the context of recent studies.


I. INTRODUCTION
With the development of human society and technology, the need for resource development and Marine environmental protection is becoming more and more prominent. In the field of marine research, underwater observation has been a key topic of ocean related research in recent years because underwater observation technology plays an extremely important role in most marine underwater engineering fields such as marine resource exploration, marine environment detection, The associate editor coordinating the review of this manuscript and approving it for publication was Donato Impedovo . and marine biological monitoring. While the development of underwater electromagnetic and optical observation is limited due to the high energy attenuation rate of electromagnetic and optical signals propagating underwater [1], the good propagation characteristics of acoustic waves underwater have attracted more and more attention in underwater observation [2]. At the same time, low visibility of underwater observation makes underwater signal sensing mainly relies on hydrophones, and acoustic waves have become the main medium of underwater observation. Although lots of studies have shown the great potential of acoustic waves for underwater observation, they are limited by complex underwater environments, and obtained detection results include target signals and substantial background noises [3], [4]. At the same time, generally acquired oceanic acoustic observation signals are nonstationary nonlinear signals, and the conventional frequency domain analysis method and second-order domain method cannot accurately distinguish the target signal from background noise. In addition, marine acoustic observation technology started late, and underwater acoustic signals have complex characteristics such as sparsity, diversity, and non-stationarity, and there is a lack of standardized largescale underwater acoustic signal data sets. Therefore, the research on underwater targets using underwater acoustic observation mainly focuses on the problem of how to reduce the effect of background noise and extract the target signal from nonstationary nonlinear ocean acoustic observation signals submerged in noise in the absence of prior knowledge of the signal [5].
General nonstationary nonlinear observation signal processing flow is signal preprocessing, feature extraction, target recognition, and target classification. The research on underwater targets mainly focuses on detection, recognition, and feature extraction of underwater target signals, where feature extraction of signals determines the final recognition and classification results. A large number of studies have been carried out for feature extraction of nonstationary nonlinear signals. Common feature extraction methods of nonstationary signals mainly include matched filtering, short-time Fourier transform, wavelet analysis [6], Hilbert-Huang transform, adaptive filtering, stochastic resonance [7], etc. In addition, since conventional power spectrums cannot characterize the time-varying behavior of nonlinear signals [8], high-order domain feature extraction technology has been developed. What advantages of higher-order cumulants are that Gaussian noise and symmetrically distributed noise are not sensitive, and phase information can be preserved by extracting features [9], [10]. Based on these advantages, the high-order domain feature extraction technology is widely used in the processing of nonlinear non-Gaussian signals, such as analyzing electrocardiogram signals based on the fourth-order cumulant and trispectrum correlation features [11], and blind extracting nonstationary mixed signals based on the statistical properties of correlation peaks of the fourth-order cumulant [12].
In the field of ocean acoustic observation signal processing, based on characteristics of nonlinear non-Gaussian of the target signal and Gaussian of the pure ocean background noise, high-order domain feature extraction technology is widely used in ocean acoustic observation signal processing. However, there are few related reviews to systematically describe the application and performance of high-order domain feature extraction technology in the field of ocean acoustic observation signal processing [13]. Therefore, the purpose of this paper is to research, analyze, and synthesize high-order domain feature extraction technology for ocean acoustic observation signals. According to the purpose of this paper, 75 highly relevant papers are selected to be included in the scope of review, among which the numbers  of papers related to different feature extraction technology in the higher-order domain methods are shown in Fig. 1.
Among high-order domain feature extraction technology, the third-order cumulant and its extensions such as bispectrum, and bispectrum diagonal slice are mostly used as the tools for signal feature extraction and analysis based on the advantages of anti-Gaussian noise of the third-order cumulant early. The third-order cumulant and bispectrum methods are the main methods in the third-order domain method. Later, the advantages of the fourth-order domain method, which can be used in a low signal-to-noise ratio (SNR) environment by retaining symmetric distribution information, have led to the development of fourth-order domain technologies such as fourth-order cumulant, trispectrum, and the diagonal slice of trispectrum. In addition to conventional methods, a small number of papers also use other extended methods in the higher-order domain for signal feature extraction. At the application level, the percentages of different observation applications in the papers related to high-order domain methods are shown in Fig. 2.
Where the high-order domain feature extraction technology is mainly applied to the non-Gaussian detection of target signals for ocean acoustic observations, accounting for more than 60%, while the proportions of high-order feature classification of target signals and arrival estimation of vector hydrophone gradually decrease.
On the other hand, with the development of neural networks and feature fusion concepts, the higher-order domain method has a far-reaching development prospect in the subsequent field of ocean acoustic observation signal processing based on its superior characteristic performance.
To deeply study and analyze the high-order domain feature extraction technologies for ocean acoustic observation signals, and to explore its development and future trends, this paper is organized as follows. Section II briefly introduces characteristics of ocean acoustic observation signals and concepts and advantages of higher-order domain feature extraction technology. In Section III, the development of higher-order feature extraction technology for ocean acoustic observation signals is studied in detail. In Section IV, a comprehensive analysis of the application of higher-order feature extraction technology for ocean acoustic observation signals is presented. The future research trends of higher-order feature extraction technology for ocean acoustic observation signals are briefly introduced in Section V. The conclusion is given in Section VI.

II. ANALYSIS OF PROCESSING METHODS FOR OCEAN ACOUSTIC OBSERVATION SIGNALS A. CHARACTERISTICS OF OCEAN ACOUSTIC OBSERVATION SIGNALS
Although acoustic signal has a better underwater propagation effect than electromagnetic signal and optical signal, acoustic wave belongs to mechanical waves, and their underwater propagation is susceptible to interference, which makes collected underwater acoustic signal data often contain complex and substantial background noise. The background noise greatly hinders the results of signal processing applications such as target identification, target classification, and target localization. In order to ensure the effectiveness and accuracy of operations such as identification, classification, and localization, noise reduction processing for signal data is of great importance. Based on the principle that sound waves are mechanical waves, acoustic observation signals can be classified according to transmission media. Among them, this paper focuses on the processing and application of ocean acoustic observation signals, so the ocean acoustic observation signals are classified according to the application fields. The related acoustic observation signals in this paper are classified as shown in Fig. 3. Common ocean acoustic observation signal areas include ship and vessel noise identification and detection, marine biological vocalization investigation, submarine oil, and gas detection, underwater locator beacon search, underwater explosion acoustic monitoring, etc.
Although the characteristics of these signal areas are different, they have the same observation characteristics: 1) Signal generation mechanism and complex environmental impact [14], [15] result in the oceanic acoustic observation signals are often nonlinear Gaussian mixture signals with complex background noises; 2) Statistical characteristics of underwater background noises included in the observation results are usually Gaussian distribution-dominated; 3) The triggering of signals is mostly random, sparse, and nonstationary in the observation time domain.
The common flow of ocean acoustic observation signal processing is shown in Fig. 4. The signal data is first observed and acquired by receiving transducer, hydrophone, or vector hydrophone, and then preprocessed by prefiltering, analogto-digital conversion, or interception to obtain signal data segments containing the target signal and background noise. If background noise contained in the signal data segment is pure ocean noise, it is linear and Gaussian. The signal data segments are processed by feature extraction to separate the background noise and obtain the feature representation of the target signal.
Aiming at the nonstationary characteristics of oceanic acoustic observation signals, the time-frequency analysis combined with time-frequency domain feature representation is developed and applied to oceanic acoustic observation signals, but they do not take advantage of inherent non-Gaussian and nonlinear characteristics of underwater sources. Based on nonlinear and non-Gaussian characteristics of the target signal observed by oceanic acoustics, in order to correctly filter noise and characterize the target signal, it is required that the Gaussian feature extraction algorithm is insensitive and capable of characterizing the nonlinear features of the signal. To solve this problem, a higher-order domain feature extraction method is constructed from the perspective of the statistical domain by increasing the number of orders, based on the advantages and shortcomings of the second-order domain method.

B. LIMITATIONS OF THE SECOND-ORDER DOMAIN METHOD
Among statistical feature detection methods, the secondorder statistical method power spectral density functions (PSD) [13] are commonly used for feature extraction, which can completely represent feature information of Gaussian processes with known mean values. Suppose there is a wide smooth random signal x(t), then its autocorrelation function is (1) VOLUME 11, 2023 where E is the expectation function. Assuming that the signal x(t) satisfies the absolute integration and has a finite number of intermittent points, the signal has a Fourier transform The energy conservation in the time domain and frequency domain can be known from Parseval's theorem when the Fourier transform of the signal exists. Thus the energy spectral density of the signal can be defined as However, the Fourier transform of the random signal does not converge, so the spectrum cannot be used to describe the random signal. The PSD function can reflect the distribution of the average power of the signal in the frequency domain along with the frequency change, so it can be used to characterize random signals. If there is a limit to the energy spectral density of the signal, the PSD of the signal is The autocorrelation function and PSD of the signal are known as a pair of Fourier transform pairs by the Wiener-Sinchin theorem. For a stationary stochastic process, the power spectral function is deterministic, so the power spectral function can be used for feature extraction of stationary random signals. The correlation function and power spectrum can fully characterize characteristics of the random normally distributed signals, but there are limitations as follows.
1) Phase information is suppressed in the correlation function and power spectrum methods, so phase information is not included in feature expression, and non-minimum phase systems cannot be distinguished.
2) Correlation function and power spectrum are sensitive to additive noise, and can only handle observations with ideal additive white noise. For non-Gaussian signals, the first and second order cannot fully represent its statistical characteristics information.
3) Correlation function and power spectrum are applied assuming that the signal is generalized stationary, but actual signals are mostly nonstationary signals.
Given the nonstationary nature of actual signals such as oceanic acoustic observation signals, second-order domain methods such as the time-frequency analysis method which simultaneously represents time-frequency characteristics have been gradually developed on the basis of traditional frequency domain analysis methods [16]. However, the limitations of the second-order domain method in the detection of phase information and non-Gaussian properties still exist. With the expansion of the application field of ocean acoustics, requirements for the processing of observation signals in practical engineering are constantly improving, and the limitations of the second-order processing technology are gradually emerging, which also makes researchers continuously explore higher-order feature extraction methods.

C. ADVANTAGES OF THE HIGH-ORDER DOMAIN METHOD
In response to non-Gaussian nonlinear characteristics of signals, the research on non-Gaussian signal processing theories and methods has been gradually developed and matured. Among them, Higher order statistics (HOS), which is based on the knowledge of higher order statistics, is widely used in the processing of non-Gaussian signals because of its advantages of resisting Gaussian noise and retaining phase information [9]. Higher-order statistics refer to statistics with an order greater than the second order, such as higher-order moments, higher-order cumulants, higher-order spectra, and other tools. Based on the research foundation of HOS by scholars in multiple fields, Nikias and Mendel [10] published a monograph related to HOS. Since then, HOS has been widely used and developed. People usually use higherorder cumulants and higher-order spectra in signal processing to suppress additive Gaussian color noises in signals with unknown power spectra, identify non-minimum phase systems or reconstruct non-minimum phase signals, extract information deviating from Gaussian properties, detect and characterize nonlinear characteristics in signals, and identify nonlinear systems.
Assuming that there is a random variable x whose probability density function is f (x), its moment generating function (first characteristic function) is defined as Taking the logarithm of the first characteristic function of x to obtain its second characteristic function (the cumulant generating function) The value of the k-order derivative of the cumulant generating function at the origin is defined as the k-order cumulant of the random variable x When k is greater than 2, c k is called the higher-order cumulant. Compared with the second-order domain methods, higher-order cumulants mainly have following advantages: 1) For Gaussian random variables, the cumulant generating function can be deduced from the probability density function as follows According to the cumulant generating function, the cumulants of each order of Gaussian random variables can be obtained as Since the high-order (k>2) cumulants of Gaussian noise are always zero, high-order cumulants can automatically suppress the effect of additive Gaussian-colored noise. Similarly, the derivation of higher-order (k>3) cumulants can also inhibit the influence of symmetrically distributed noises.
2) For non-minimum phase signals, higher-order cumulants not only contain amplitude information but also retain real phase information of the signals, thus they can be used to reconstruct non-minimum phase signals.
3) Since the non-Gaussian signal has a non-zero high-order spectrum, more useful information deviating from the Gaussian property can be extracted from the high-order cumulants, and different classification characteristics can be extracted in pattern recognition, signal detection, and classification.
4) The higher-order cumulants can detect and characterize nonlinear characteristics of the signal while preserving the phase information, and can also deal with the quadratic cubic phase coupling problem caused by a nonlinear system.
Due to the advantages of the higher-order domain method, it can identify the non-Gaussian [17] and nonlinear inherent in the acoustic signal data of oceanic observation targets. Therefore, the use of higher-order spectral analysis in oceanic acoustic observation signals can improve the conventional feature representation and provide important additional information for analysts [18].

III. ANALYSIS OF THE DEVELOPMENT OF HIGH-ORDER DOMAIN TECHNOLOGY
Based on the definition of cumulant given in (7), to better observe the performance of different order feature extraction technology, a simulation signal is set and processed by noise, and the noise signal is processed by different order methods, and the processing results are shown in Fig. 5.
Contact (9), Fig. 5 illustrates the defects of the secondorder domain method in processing signals containing Gaussian noise and the anti-Gaussian advantages of higher-order cumulants. Based on the defects of conventional second-order domain feature extraction methods in ocean acoustic observation signal processing, third-order domain methods have been proposed first. The third-order domain method is usually converted into non-Gaussian detection of the signal, which suppresses the Gaussian noise and the non-zero skewness signal, which would reduce the original information of signal and therefore require a high SNR of the original signal. When the SNR of the signal decreases, the processing effect of the third-order domain method is affected. Since then fourthorder domain methods have been proposed to suppress Gaussian noise while preserving the symmetric part of the retained signal, which can be applied to environments with lower SNR. In a few papers, higher-order statistics such as fifthorder cumulants and three-dimensional half-spectra are used, but the complexity of calculation limits the development of higher-order methods. In conclusion, conventional highorder domain analysis methods are mainly third-order and fourth-order methods. Common high-order feature extraction methods in the field of ocean acoustic observation are shown in Fig. 6.

A. THIRD-ORDER DOMAIN FEATURE EXTRACTION TECHNOLOGY
The third-order domain method mainly utilizes the advantages of the third-order cumulants in suppressing Gaussian VOLUME 11, 2023 noise, preserving phase information, extracting non-Gaussian information, identifying nonlinear signals, and applying the third-order domain method to the fields of target recognition and signal classification of ocean acoustic observation signals. It mainly contains tools such as bispectrum and the diagonal slice of bispectrum based on third-order cumulants.

1) THIRD-ORDER CUMULANTS AND BISPECTRUM
The third-order cumulant is subjected to vectorization processing [19] to obtain (10) The obtained third-order accumulation matrix can be used for direction of arrival (DOA) estimation for vector hydrophone beamforming.
The power spectrum is defined as the Fourier transform of the autocorrelation function. Similarly, when higher-order cumulants are absolutely summable, the higher-order cumulative spectrum is defined as the k −1 order Fourier transform of the k-order cumulants, so the third-order spectrum (bispectrum) [18], [20] is defined as As the result of the second Fourier transform of the thirdorder cumulant, the bispectrum retains the advantages of the third-order cumulant in suppressing Gaussian noise, retaining phase information, extracting non-Gaussian information, and identifying nonlinear signals. The underwater background noise is usually a composite signal close to the Gaussian random field [21], and its bispectrum is basically close to zero according to the high-order cumulant property [22], [23]. For target signals, the generation mechanism of actual signals is mostly nonlinear, which leads to most of the target signals having obvious non-Gaussian structures, and exhibiting nonzero representation in bispectrum [24], [25]. Therefore, the results obtained when bispectrum is applied to detection and classification such as small and medium-sized vessel monitoring tend to have a higher SNR than the original data [26], [27], [28], [29], [30], [31], [32], [33], [34]. Moreover, since higher-order spectroscopy has the advantage of revealing the phase coupling mode of the signal [35], [36], bispectrum can be used to determine the phase coupling between the signals that generate certain spectral peaks, thus enabling the spectral peaks to be associated with different physical generation mechanisms.
Bispectrum estimation processing is performed in signal noise recognition by dividing the signal into k segments, estimating the bispectrum values of the k segmentsB (k) x [24] respectively, and calculating the bispectrum average value of dual-frequency smoothing as a consistent estimator [37], [38] Although the third-order cumulants have many advantages, a large amount of calculation and large output caused by bispectrum computation hinder their practical application. The bispectrum is represented as 12 symmetrical regions. According to the symmetrical characteristics of the bispectrum, limiting the feature extraction of the bispectrum to a triangular region of ω 1 ≥ 0, ω 2 ≥ ω 1 , ω 1 + ω 2 ≤ π [39] can greatly reduce the computational complexity while preserving the signal features.
In addition, in order to optimize the detection of weak signals in the presence of additive independent stationary non-Gaussian noise, [40], [41], [42] set skewness to help quantify the deviation from symmetry According to the definition and principle of cepstrum, the multispectrum similar to the cepstrum can be obtained by taking the logarithm of the corresponding multispectrum and inverse transforming the logarithmic spectrum to obtain a multispectrum similar to the cepstrum, and the cepstrum of the retrieval source wavelet can be analyzed [43]. The bispectrum cepstrum is defined as a two-dimensional z-inverse transform of a logarithmic bispectrum When the reflectivity does not contain a column of peaks, the multispectral-derived cepstrum will also retain phase information because the bispectrum and the trispectrum retain the phase characteristics of the wavelet.
In order to better identify acoustic noise sources, the concept of bicoherence was proposed [36] bic(ω 1 , where P represents signal energy. As a normalized form of bispectrum, it is independent of the signal energy or signal amplitude and can be used as a convenient test statistic for detecting non-Gaussian, nonlinear, and coupled processes [36], [44], [45], [46]. Aiming at the high computational cost and over-fitting possibility caused by directly using the bispectrum matrix or bicoherence matrix as a feature vector, the integrated bispectrum with low dimension, constant scaling, and translation is introduced in [47], including Radially Integrated Bispectrum (RIB), Cyclicly Integrated Bispectrum (CIB), Axially Integrated Bispectrum (AIB), and Bispectral-MFCC (BMFCC)

2) DIAGONAL SLICE OF BISPECTRUM
In order to compress the amount of data and reduce the amount of calculation, the diagonal slice of the bispectrum is defined on the basis of the bispectrum. The diagonal slice of the bispectrum is obtained by taking the diagonal region of the bispectrum spectrogram where the two frequency components are equal (i.e., the Fourier transform of the diagonal slice of the third-order cumulant [48]). Let ω 1 = ω 2 = ω in formula (11) to get Due to the symmetric nature of the bispectrum, diagonal slices of the bispectrum retain the non-Gaussian and nonlinear characteristics of the bispectrum, but some information in the bispectrum plane is missing. Based on this, [49] introduced a correction q in the original diagonal slice, which represents the distance between the original diagonal slice and other slices parallel to it in the dual-frequency plane, and proposed a generalized diagonal slice of third-order cumulant and its spectrum The generalized diagonal slice of the third-order cumulative spectrum provides more information than the original slice.
For the diagonal slice property of the bispectrum, peak distribution can be extracted in addition to directly using it as the feature [50], and identifiable features can be comprehensively extracted by combining PSD estimation.
Technical development of third-order domain feature extraction methods in the field of ocean acoustic observation is shown in Table 1.

B. FOURTH-ORDER DOMAIN FEATURE EXTRACTION TECHNOLOGY
In practical applications, bispectrum detection of the target is usually converted to non-Gaussian detection of the signal. Bispectrum detection is feasible when a non-Gaussian property of the signal is strong. However, the non-Gaussian property of the signal becomes weak under the condition of low SNR, the bispectrum detection effect becomes poor. In contrast, the fourth-order cumulative spectrum method is unaffected by additive Gaussian noise and suppresses symmetric distribution noise in addition to conventional advantages, which can be used in low SNR environments [17].

1) FOURTH-ORDER CUMULANT
The fourth-order cumulant [51], [52], [53] is defined as The fourth-order cumulant [53] which preserves that symmetric part of the signal while suppressing gaussian noise can be used to realize the target detection [54] in a low SNR environment, and it is also possible to determine whether to use the fourth-order or the third-order cumulant for estimation [55] according to the symmetry of the PDF. Overall performance of the fourth-order detector is superior to that of the second-order (energy) and third-order detectors [53], [56], [57], [58], and it can effectively detect transient energy with unknown form and duration [59], [60], [61], [62], [63].
In order to realize blind channel estimation in multipath fading channels, [64], [65] define the normalized fourth-order cumulant of signals as The classification error problem caused by the uncertainty factor β in blind communication can be eliminated by the theoretical derivation of the normalized fourth-order cumulants.
Meanwhile, for the difficult problem of large computation of fourth-order cumulative estimation, [66] proposed a simplified calculation method of fourth-order cumulant based on the direct estimation formula of fourth-order cumulant and the relationship between recursive estimation and computational complexity of fourth-order cumulant According to the expansion characteristics of the fourthorder cumulants, a fourth-order cumulant matrix [67], [68], [69], [70], [71], [72] can be constructed The expansion of beamforming array aperture is achieved by weighted Bartlett beamforming through the fourthorder cumulants accumulation of a uniformly spaced linear array, and the acoustic source can be located using the algorithm.

2) TRISPECTRUM AND THE DIAGONAL SLICE OF THE TRISPECTRUM
The fourth-order cumulative spectrum (trispectrum) [73], [74] is defined as In contrast to bispectrum, trispectrum achieves the simultaneous suppression of Gaussian noise and symmetrically distributed noise by increasing computational cost, therefore outperforms the third-order accumulation of the second-order domain method in signal detection, ship noise, and other fields [75], [76], [77], [78].
Similar to the third-order cumulants, the fourth-order cumulants can also define the concept of slices [79]. Non-Gaussian coefficients will give a larger value of fourth-order statistical cumulants (kurtosis) to the signal, while conventional noise coefficients with Gaussian PDF have small kurtosis values [23]. Thus kurtosis can distinguish the target signal part from the noise part of data for feature extraction purposes. On this basis, the estimated value of diagonal slices of the fourth-order cumulant [17] iŝ Based on the retention of advantages of suppressing Gaussian noise while preserving symmetrical parts of the signal, the diagonal slices of trispectrum can also achieve target detection in a low SNR environment [80]. According to the relationship between the fourth-order cumulant definition and delay amount, the slice spectrum [81]  of the fourth-order cumulant can be defined as Compared with the power spectrum, the slice spectrum of the fourth-order cumulant can be used in the detection of both effective extractions of harmonic components of the signal and detection of secondary phase coupling. In order to overcome the problem that the diagonal slice of the fourth-order cumulant of the random sinusoidal signal with variable frequency or non-uniform distribution of the original phase is not equivalent to the non-diagonal slice of the fourth-order cumulant of the time-varying signal, the diagonal slice and the non-diagonal slice of the fourthorder cumulant of the time-varying signal are integrated into a weighted fourth-order cumulant slice and updated by the leakage iterative algorithm, and a weighted fourth-order cumulant slice with adaptive linear enhancement is proposed in [82] C(k, l) = aC 1 (k, l) + bC 2 (k, l) =Ē{ax(k)e(k, l) + bx(k + l)e(k)} (26) where a and b are constants. By adjusting the ratio of cumulant weighting coefficients a and b, the performance of the algorithm can be greatly improved.
Characteristics of different slices of fourth-order cumulant of the quasistationary stochastic process are analyzed, and the off-diagonal slice of fourth-order cumulant is proposed in [83] C 4x (k, l) = µC 4x (k − 1, l) + (1 − µ)x(k + 1)e 2 (k, 0) (27) where µ < 1 is the smoothing factor. The dynamic line spectrum feature of the signal can be enhanced by an offdiagonal slice of the fourth-order cumulant. According to the polarization characteristics between the target signal and noise in ocean, the diagonal slice of the fourth-order mixed cumulants is used in feature extraction. In order to improve the estimation performance of the fourthorder mixed cumulants slice spectrum and avoid spectrum leakage, the diagonal slice of the fourth-order mixed cumulants is windowed in [84] z where f is frequency and w is window function. Based on polarization characteristics between the target signal and noise in ocean, the diagonal slice of the fourth-order mixed cumulant is used to calculate the polarization filter coefficient in feature extraction, and the noise spectrum can be filtered to extract the line spectrum. Technical development of the fourth-order domain feature extraction methods in the field of ocean acoustic observation is shown in Table 2.

C. OTHER RELATED TECHNOLOGIES IN THE HIGH-ORDER DOMAIN
According to the definition of a higher-order spectrum, the feature larger than cubic coupling can be extracted with the cumulant larger than the fourth order. At the same time, the cumulant larger than the fourth order can completely suppress the symmetrically distributed noise and frequencyindependent components, but its computational complexity is extremely high. To address the problem of high computational complexity for higher-order spectrum, [85] proposed an optimization algorithm for fifth-order cumulant estimation and developed diagonal slices of the fifth-order cumulant that can be used for adaptive linear enhancement In general, expected characteristics can be obtained from conventional underwater acoustic target signals after the third-order or fourth-order processing, so the cumulants larger than the fourth-order are rarely applied to the field of signal processing for ocean acoustic observation. Only in the case of complex nonlinear vibration systems, in order to explore its higher-order frequency coupling characteristics, higher-order cumulants are used for calculation. For example, the three and one half dimension spectrum is obtained by using different definitions of the fifth-order cumulants of a complex signal, which can extract the fourth-order or secondorder to third-order frequency coupling components involved in the coupling or generate frequency coupling components, respectively. In addition, the three and one half dimension spectrum of the actual signal can also synchronously extract the coupled original frequency component and the derivative frequency component.
In order to explore the extraction method of high-order frequency coupling characteristics for underwater motion radiation noise, the three and one half dimension spectrum of complex real signals [86], [87] is To increase the number of observable sources, it is proved by experiments that the vector hydrophone data with fourthorder statistics can resolve up to eight sources [63]. By further processing HOS on the order of 2q (when q≥2), the number of sources is where each source has a p-dimensional direction vector, the autocorrelation function of the zero-mean signal received by M hydrophone arrays from the D sources is A(θ)R S A H (θ) + σ 2 I, and ρ is the rank of A(θ )R S A H (θ).
In addition to higher-order cumulants, higher-order domain analysis also includes higher-order moment analysis. Such as kurtosis detection [88] based on the second-order moment and fourth-order moment where m 2 is the second-order moment and m 4 is the fourthorder moment of the signal. Kurtosis detection can overcome the problem that the order of windowed transform is difficult to obtain accurately, so as to improve the accuracy and reliability of the algorithm. However, higher-order moments have few applications because their overall performance is weaker than higher-order cumulants.
In the aspect of ship radiation noise recognition, the diagonal slice of the Wigner trispectrum [89] can be used to extract low-frequency line spectrum components m kx (t, τ 1 , τ 2 , . . . , τ k ) Although Wigner's higher-order spectrum can effectively extract the low-frequency line spectrum component, and without generating pseudo line spectrum components, there is a limitation that line spectral amplitude cannot be too low.
In addition, conventional high-order spectral characteristics of the signal cannot be extracted in the α-stable distributed noise model of the underwater acoustic channel, so generalized cyclic cumulants are usually applied as characteristics of the underwater communication signal [90].
Other high-order domain feature extraction technologies in the field of ocean acoustic observation are shown in Table 3.
In general, the goal of high-order domain feature extraction is to suppress background noise and enhance signal components. In practical situations, the overall performance of the fourth-order method is higher than that of the third-order method, while the computational complexity of higher-order methods is too complicated. As the fourth-order cumulants that retain characteristics of the third-order and can adapt to low SNR environments have a wider prospect of subsequent applications, their computational complexity can also be solved by finding diagonal slices. Although lots of papers have demonstrated the effectiveness of high-order domain feature extraction technology in the processing of ocean acoustic observation signals, complex acoustic data make the application of various methods in the high-order domain limited, and no feature extraction algorithm that can be adapted to any situation. The selection of algorithms in practical applications needs to be integrated with the target signal characteristics and background noise performance.

IV. OBSERVATION APPLICATION OF HIGH-ORDER DOMAIN METHOD
In the process of ocean acoustic observation signal processing, the feature extraction step is mainly for the preparation of later engineering applications. Based on extracted features, the target signal part of the ocean acoustic observation data can be targeted, and subsequent signal detection recognition, signal classification, beamforming, and other work can be performed.

A. NON-GAUSSIAN DETECTION OF TARGET SIGNAL
For typical underwater acoustic signal target detection application scenarios such as the difference between target signal and background noise in ship noise recognition, the most commonly used detection algorithm is based on the statistical representation of background noise and target signal, which is estimated to determine the Gaussianity of the process, and subsequent comparison between the data and the PDF model can be performed by modeling the probability density function. The process of applying high-order domain feature extraction technology for signal detection in the field of ocean acoustic observation signals is shown in Fig. 7. Among them, the Gaussian test (non-Gaussian detection) mainly judges the Gaussian property of the signal by processing the signal containing Gaussian noise with high-order domain technology, and detects the target signal part which is non-zero in the high-order domain. The schematic diagram of the non-Gaussian detection of signals is shown in Fig. 8.
In that field of third-order cumulant correlation estimation, bispectral features are commonly used for the detection and identification of ship radiate underwater acoustic noise and underwater acoustic transient signals according to characteristics of the non-Gaussian test of the third-order detector [18], [22], [24], [25], [26], [27], [28], [29], [30], [38]. [20], [31], [32] simultaneously carried out PDF modeling and analysis on the ship radiation underwater acoustic noise and the underwater acoustic transient signal by using frequency spectrum and bispectrum. The third-order characteristic skewness is also used in the construction of underwater noise PDF models in [40] and [41]. Based on the advantage that third-order spectra can reveal signal phase coupling patterns, [35], [42], [44] applied bispectrum analysis technology to accelerometer and hydrophone data recorded in shallow water experiments in the Baltic Sea. In the field of the diagonal slice of bispectrum technology, based on the advantages of the diagonal slice of bispectrum in reducing Gaussian white noise, enhancing fundamental frequency, and eliminating uncoupled phase harmonics, [48] used the Radon transform to transform the straight line in the diagonal slice of bispectrum into the peak in the Radon domain, so as to calculate frequency and frequency modulation and realize estimation of the underwater target radiation noise line spectrum. In addition to the peaks of the diagonal slice of the bispectrum, [50] also extracted the peak distribution and PSD estimation of the non-diagonal slices of the bispectrum for feature extraction.
In terms of the fourth-order domain feature extraction technology, the signal detection purpose can be achieved [58], [59], [60], [61], [62], [63] by virtue of the high fourth-order cumulant (kurtosis) value of non-Gaussian coefficient and small kurtosis of the Gaussian noise coefficient [23], [51]. For example, [63], [64] used the fourth-order cumulant kurtosis to achieve the detection of low-frequency underwater acoustic signals and ship-radiated noise, while [88] used high-order moment kurtosis to statistically characterize the ship radiation noise propagated from the source through ocean medium to receiver, and [40] further extended the designed non-Gaussian noise PDF model by inserting the fourth-order kurtosis. In [53], asymmetric generalized Gaussian PDF construction of low-frequency underwater noise was achieved based on the asymmetric Gaussian PDF using parameters of the third and fourth orders to control the sharpness, standard deviation, and symmetry of the model. Based on the fact that the overall performance of the fourth-order detector is better than the second-order (energy) and thirdorder detectors but more computationally complex [56], [60], [53] used both third-order cumulants and fourth-order cumulants for shallow-water ship noise detection, while [55] decided to use fourth-order or third-order cumulants for the estimation process based on whether the input reflectance PDF is symmetric. Reference [66] designed an adaptive linear enhancer with low computational load and suppression of Gaussian noise for underwater targets using a simplified calculation of fourth-order cumulants.
In the aspect of diagonal slice technology with lower computational complexity, [17], [79], [80] realized the detection of underwater target noise in the low SNR environment using the diagonal slice of fourth-order cumulant. Reference [81] modeled the calculation of the sound field, the line spectral component, and a continuous spectral component of radiated noise respectively by analyzing characteristics of the radiated noise from ships based on the slice spectrum of fourth-order cumulant. In order to overcome the problem that the diagonal slice of the fourth-order cumulant of the random sinusoidal signal with variable frequency or non-uniform distribution of VOLUME 11, 2023  the original phase is not equivalent to the non-diagonal slice of the fourth-order cumulant, [82] designed an adaptive linear enhancer to identify underwater moving target radiation line spectrum signal in Gaussian noise using a weighted slice of fourth-order cumulant, and [83] designed an active dynamic line enhancer using different slices of fourth-order cumulant, which output SNR is greater than that of the adaptive linear enhancer. Based on the mixed diagonal slices of the fourth-order cumulant, [84] also constructed the underwater signal polarization parameters according to the polarization difference between the target and the noise to filter the noise spectrum to extract the signal line spectrum.
In other applications of high-order domain feature extraction technologies, underwater target radiated noise is also detected using an enhancer. Reference [85] developed an adaptive line enhancer based on the diagonal slices of the fifth-order cumulant, [86], [87] explored the higher-order frequency coupling features of underwater motion radiation noise using the three-and-one-half dimension spectrum, [89] used the diagonal slice of Wigner trispectrum for ship radiation noise identification, and [90] applied generalized cyclic cumulant as a feature for underwater communication signal detection, etc. However, the computational complexity and redundancy of the higher-order methods make their applications much less than the third-order and fourth-order methods.
The performance and complexity of different order feature extraction technologies in non-Gaussian detection of target signals are shown in Table 4. As shown in Table 4, the anti-noise ability of the fourth order is stronger than that of the third order, but the computational complexity is higher. The improvement of computer operation performance and the appearance of the diagonal slicing method of the spectrum reduce the computational complexity, so the technology in the field of signal detection development in the direction of the fourth-order slice.

B. HIGH-ORDER CLASSIFICATION OF SIGNALS
In signal classification, high-order domain feature extraction technology has been applied to clutter classification [46], mine-like target parallel detection and classification of autonomous underwater vehicles [78], automatic modulation classification of blind channel estimation and pattern recognition [65], underwater acoustic noise point source separation classification [62], underwater acoustic signal modulation classification [54], underwater acoustic signal feature fusion analysis [33], etc. In most signal classification applications, high-order domain feature extraction technology is often combined with a support vector machine (SVM) and neural network to achieve the purpose of classification. The schematic diagram of the high-order classification of signals is shown in Fig. 9.
Traditional classifier in the field of feature extraction in the high-order domain of ocean acoustic observation signals is mainly SVM. In the third-order domain, [39] proposed a bispectrum feature extraction scheme based on a short sample length of 512 points according to the symmetry property of bispectrum, which sums the amplitudes on each column or row in the triangular region of one of the 12 symmetry regions of the bispectrum as feature vectors. Based on the one-to-one (One-against-One, OAO) method of SVM multiclassification, the classification performance is improved by using the per-row summation feature vector as supplementary features, but the classification computation is increased. In [43], a radial basis function kernel (RBF) SVM was trained using bispectrum analysis and bispectrum cepstrum analysis  to extract the target features to achieve the classification of artificial noise sources in ocean. In view of the high computational cost and overfitting possibility associated with the direct use of the bispectrum matrix as the feature vector, [47] introduced integrated bispectrum, bispectrum MFCC coefficients, and self-coupling frequencies and derived a set of robust features for underwater noise classification based on bispectrum analysis, and evaluated the classification performance using RBFSVM. To improve the classification accuracy of underwater target radiated noise, [49] proposed generalized diagonal slices of third-order cumulant and their spectrum, and composed feature vectors from the sum or maximization of each generalized slice to substantially extract the target features in the dual frequency domain, and confirmed the classification accuracy higher than the previous algorithms using the multi-classification OAO of SVM method. After developing the fourth-order domain technology, [61] used the fourth-order cumulants to construct characteristic parameters of the underwater acoustic modulated signals and used a multiclassification SVM for interclass and intra-class recognition.
Traditional classification methods such as SVM rely on an accumulation of prior knowledge and experience of scholars on data, which cannot meet the requirements of the amount of underwater acoustic sample data. And the relatively powerful performance of deep learning has led a large number of scholars to apply it to underwater acoustic signal recognition and classification and gradually replace the traditional machine learning algorithm at present. In the higher-order domain, for feature extraction of underwater target radiation VOLUME 11, 2023  noise, [37] extracted features after bispectrum estimation of the target signal to obtain a low-dimensional feature vector and input an RBF neural network classifier as a way to suppress Gaussian noise and reduce the dimensionality of the feature space. References [36] and [45] extract the specific characteristics of the source using the dual bicoherence spectrum and feed these characteristics back to the neural network classifier to achieve the classification of ocean noise. Based on the effectiveness of bispectrum features being used for neural network classification, [52] combined higherorder cumulants and Hilbert transform for feature extraction to extract the ratio of instantaneous frequency, relative standard deviation, central frequency, average intensity, higherorder moment, and higher-order cumulants between adjacent eigenmode functions, which are extracted and input into a back propagation neural network to identify and classify two types of ship targets. Based on traditional classification processing, [34] used STFT amplitude spectrum, phase spectrum, and bispectrum feature input to the integrated neural network to improve the classification effect of separate higher-order domain analysis based on the concept of feature fusion. The performance and complexity of different-order feature extraction technologies in the high-order classification of signals s are shown in Table 5. As shown in Table 5, the higher-order domain techniques are mainly third-order and fourth-order in traditional classification, and after the development of neural network classifiers, the higher-order domain techniques gradually develop towards the fusion of higher-order accumulation and other methods.

C. DOA ESTIMATION OF VECTOR HYDROPHONE
With the needs of research, the vector hydrophone which can locate the target more accurately has been developed among traditional omnidirectional hydrophones [93], while in the array design and signal processing of sonar systems, a beam formation method with high resolution and good stability is needed to achieve a robust estimation of target arrival direction [73]. Therefore, in the research field of high-order domain feature extraction technology for ocean acoustic observation signals, the third-order, and fourth-order domain DOA estimation methods based on single vector hydrophone were first developed because of the advantages of vector hydrophone and high-order cumulant, and then the fourth-order domain DOA estimation methods based on multi-vector hydrophone and vector hydrophone arrays were subsequently developed. The schematic diagram of the DOA estimation of vector hydrophone is shown in Fig. 10.
A single acoustic vector hydrophone is small in size, has a clear direction of arrival, and is insensitive to the range, frequency, and bandwidth of the impinging acoustic signal source. In the aspect of single vector hydrophone, based on characteristics of aperture expansion of higherorder cumulant array, [19] used third-order cumulants to estimate the two-dimensional direction of arrival of multiple non-Gaussian sources, and [57], [69] used fourth-order cumulants to provide efficient and accurate beamforming and DOA estimation methods for vector hydrophones processing non-Gaussian sources in the presence of generalized Gaussian noise or interference sources. Reference [67] extended the fourth-order cumulant beam formation method combined with fractional-order Fourier transform (FRFT) to extend the algorithm to DOA estimation of broadband LFM signals and apply the algorithm to active sonar target detection. Reference [68] obtained multiple invariance matrices by defining a series of fourth-order cumulant matrices, and then extracted the localization information of the near-field source from these invariance matrices. Complex three-dimensional search is avoided by closed solutions of azimuth, elevation, and distance. Reference [72] used fourth-order cumulant for two-dimensional direction-of-arrival estimation of multiple non-Gaussian signals and improves the performance of the conventional fourth-order domain algorithm [73] for DOA estimation based on a compressed sensing approach.
However, the feature that a single vector hydrophone uniquely identifies two sources of the same frequency in any direction likewise limits the utility of a single vector hydrophone in practical systems. To increase the number of observable sources, [63], [91] demonstrated that eight sources with no more than four in a single azimuth/elevation plane can be uniquely resolved when processing vector hydrophone data with fourth-order statistics, and [63], [91], [92] proved that the number of sources can be increased by further processing HOS of order 2q (when q ≥ 2). To break the limitations of single-vector hydrophones, [70] implemented direct extraction of source arrival angles from the estimated single vector hydrophone steering vectors using means of 2 fourth-order cumulant matrices in a doublevector hydrophone, thus avoiding the two-dimensional search problem involved in many other Fourth-order cumulant algorithms, and demonstrated that up to 9 sources can be identified. In the vector hydrophone array, [71] defined 3 fourth-order cumulant matrices to form a third-order tensor, obtains the direction cosine of the signal by using parallel factor (PARAFAC) fitting, and extended the PARAFAC model to multiple cumulant matrices and a fourth-order tensor frame, which also avoids the open-form two-dimensional search and parameter matching problems. The performance and complexity of different order feature extraction technologies in DOA estimation of vector hydrophone are shown in Table 6. As shown in Table 6, for DOA estimation of vector hydrophones, the high-order domain technology is dominated by the fourth order and DOA estimation of multiple sources is achieved by increasing the number of matrices and by increasing the order by 2.

V. FUTURE TRENDS
Existing research articles show that high-order domain feature extraction technology has gained a lot of attention in the field of ocean acoustic observation signal processing in the past decades due to its advantages of anti-Gaussianity and nonlinearity. According to the development performance and application performance of existing technology, the research problems that need to be paid attention to or deeply explored in future research are: • Based on the background that the fourth-order domain method is superior to the third-order and secondorder domain methods, and the calculation complexity of the higher-order domain method is too high while the efficiency is low, the subsequent development of the high-order domain feature extraction technology of the ocean acoustic observation signals will mainly focus on the fourth-order cumulants, the fourth-order cumulative diagonal slices and spectra thereof in the fourth-order domain technology.
• Traditional detection and recognition methods of ocean acoustic observation signals need to extract target features manually. With the development of deep learning theory, deep learning which has been widely used in computer vision, signal processing, and other fields [94], [95], [96], [97], [98], [99], [100] is also introduced into this field. In terms of higherorder cumulants, the feasible performance of using higher-order spectral features to implement the ocean noise classifier based on an artificial neural network VOLUME 11, 2023 [45] indicates that the application of higher-order domain feature extraction technology in the field of ocean acoustic observations is about to evolve from traditional methods such as vector machines to neural networks.
• The non-stationarity and noise complexity of ocean acoustic observation signals lead to the inability of traditional single-feature extraction methods to fully exploit the comprehensive information in the signals.
To address this phenomenon, the concept of feature fusion [101], [102], [103], [104] has been developed to improve classification recognition accuracy and noise robustness. As a feature extraction technology that highly conforms to the performance of oceanic acoustic observation signals, the successful application of cumulants and higher-order spectra in feature fusion [33], [52] suggests that high-order domain method has a wide application prospect in the field of feature fusion in ocean acoustic observation signal processing.
• Deep learning and neural networks are also applied to feature fusion [105], [106], [107], [108]. In terms of high-order spectrum, [34] fused the amplitude spectrum, phase spectrum, and bispectral features of STFT by integrating neural networks for the purpose of underwater acoustic target recognition, demonstrating that higher-order accumulations have a wide scope of application in the field of neural network superimposed feature fusion.
• The introduction of the neural network reduces the problem caused by the lack of prior knowledge of ocean acoustic observation signals. However, since ocean acoustic observation signals are usually burst signals with short duration, the training data available are much less than the test data, and the training of reliable networks is still a major problem.

VI. CONCLUSION
In this paper, the technology of feature extraction in the highorder domain of oceanic acoustic observation signals is summarized. Technical advantages, technical development, and application in this field are discussed respectively. Technical principles and observation applications of the most highly relevant articles are analyzed, and the same characteristics of different ocean acoustic observation signals are synthesized in this paper. Due to the advantages of anti-Gaussian noise and retaining phase information, the high-order domain method is a common feature extraction technology in the field of non-Gaussian nonlinear oceanic acoustic observation signal processing. On the technical development of highorder domain feature extraction of ocean acoustic observation signals, this paper summarizes the development process of theoretical innovation and performance improvement of high-order domain technology in the field of ocean acoustic observation signals in a linear vein from the third-order domain, the fourth-order domain to other related technology.
Among them, high-order domain feature extraction technology is mainly based on the third-order domain method and the fourth-order domain method, and the overall theoretical performance of the fourth-order domain method is superior to the third-order do-main method and others-order domain method. Based on the current status of technology development, high-order domain methods are widely used in target detection, and signal classification. Therefore, this paper reviews the performance of higher-order domain methods in oceanic acoustic observation from three major observation applications: non-Gaussian detection of target signals, highorder spectral feature classification of signals, and the estimation of the direction of arrival of vector hydrophones. In the end, the paper discusses the future development trend of the higher-order domain method based on its overall performance and recent research.