Adaptive WVD Cross-Term Removal Method Based on Multidimensional Property Differences

Zou Yifei; Li Xiukun; Yu Ge

doi:10.1007/s11804-024-00469-4

Adaptive WVD Cross-Term Removal Method Based on Multidimensional Property Differences

https://doi.org/10.1007/s11804-024-00469-4

Yifei Zou^{1, 2, 3},
Xiukun Li^{1, 2, 3},
Ge Yu^{1, 2, 3, ,}

1
National Key Laboratory of Underwater Acoustic Technology, Harbin Engineering University, Harbin, 150001, China
2
Key Laboratory of Marine Information Acquisition and Security (Harbin Engineering University), Ministry of Industry and Information Technology, Harbin, 150001, China
3
College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin, 150001, China

Funds:

the National Natural Science Foundation of China 62201171

Corresponding author:
Ge Yu yuge221@hrbeu.edu.cn

Received: 17 June 2024

Accepted: 13 August 2024

Abstract

Abstract
Wigner–Ville distribution (WVD) is widely used in the field of signal processing due to its excellent time–frequency (TF) concentration. However, WVD is severely limited by the cross-term when working with multicomponent signals. In this paper, we analyze the property differences between auto-term and cross-term in the one-dimensional sequence and the two-dimensional plane and approximate entropy and Rényi entropy are employed to describe them, respectively. Based on this information, we propose a new method to achieve adaptive cross-term removal by combining seeded region growing. Compared to other methods, the new method can achieve cross-term removal without decreasing the TF concentration of the auto-term. Simulation and experimental data processing results show that the method is adaptive and is not constrained by the type or distribution of signals. And it performs well in low signal-to-noise ratio environments.
- Cross-term removal ·
- Multidimensional property ·
- Approximate entropy ·
- Rényi entropy ·
- Seeded region growing
Article Highlights
● The method can remove cross-term WVD without reducing the time–frequency concentration of the auto-term.

● The paper examines the differences in properties between WVD auto-term and cross-term in one-dimensional sequences and two-dimensional planes. Approximate entropy, and Rényi entropy are used to individually characterize these properties.

● The proposed method is versatile and can handle signals of any type or distribution, working effectively in conditions with low signal-to-noise ratios.

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1 Introduction

Time–frequency distribution (TFD), which is one of the useful tools for analyzing and processing nonstationary signals, can simultaneously provide signal time and frequency information. Most TFD methods, such as the shorttime Fourier transform and wavelet transform (WT), can offer effective time or frequency resolution but fail to achieve both simultaneously. Thus, a type of TFD method with superior performance, namely Wigner–Ville distribution (WVD), is required (Wigner, 1932; Cohen, 1995). WVD is recognized as a potent method for representing time – frequency (TF) information. This method does not use the window function; therefore, WVD is unaffected by the relationship of time and frequency resolutions in linear representations. The WVD method surpasses other TFD methods in terms of TF concentration, clustering, and edge features, making it one of the most accessible and superior TFD methods.

However, as a bilinear transformation, the main drawback of WVD lies in its introduction of intrinsic crossterm interference when handling signals with more than one component. This limitation restricts its capability to effectively analyze signals and extract characteristics such as time and frequency. Research has shown that the crossterms of WVD are unavoidable (Zou et al., 2002). Efficiently suppressing or removing the cross-terms while preserving the dominance of the WVD algorithm is a popular subject. In recent years, several approaches have been presented for this goal. The current methods to deal with crossterms can be broadly categorized into the following groups: kernel function methods, signal decomposition, multispectral superposition, and image processing methods.

First, the kernel function method is the commonly employed method for suppressing cross-terms, with pseudoWVD (PWVD) and smooth pseudo-WVD (SPWVD) as typical representatives (Choi and Williams, 1989). The kernel function approach effectively mitigates the cross-term by modifying the kernel or window function in the time domain. This approach is governed by the characteristics and durations of the window function and must satisfy specific requirements in terms of time and frequency domains as well as boundary constraints. Moreover, this approach necessitates achieving a balance between eliminating crossterms and maintaining a high level of TF resolution. The removal of cross-terms is not only incomplete but also diminishes TF concentration. Baraniuk and Jones (1994) and Jones and Baraniuk (1995) introduced the adaptive kernel function to address the shortcomings of the earlier kernel function approach. This adaptive optimal kernel TFD (AOK-TFD) can suppress cross-terms while maintaining better TF resolution and concentration than previous ones. Sang and Williams (1995) then proposed to use Rényi entropy to generate minimum uncertainty product kernels, which are highly effective at suppressing cross-terms and maintaining high resolutions. Guo and Wang (2008) also proposed an optimal procedure for a bi-directional Gaussian kernel based on a third-order entropy measure with normalized volume. However, these methods require considerable computer processing resources. Second, signal decomposition is also one of the key areas of cross-term suppression. The signal is decomposed into several single-component signals based on a certain pattern. WVD is then computed for each of these signals. All the individual results are combined using linear summation to create the new result. Thus, the cross-terms are further weakened. For instance, WT uses the decomposition coefficients to reconstruct the TFD by extracting the signal features of each frequency component of the lowest layer, progressively ranging from low to high frequencies (Narasimhan and Kumar, 2004; Pachori and Nishad, 2016). The empirical mode decomposition (EMD) breaks down signals into separate intrinsic mode functions that are independent (Wang et al., 2014). EMD reconstructs the WVD of the original signal using the WVD of the true intrinsic mode obtained from EMD. Additionally, the category of signal decomposition of cross-term suppression techniques includes the filter bank (Kumar et al., 2015) and Gabor transform (Sattar and Salomonsson, 1999). However, these signal decomposition methods are highly restricted by signal type and have limited applicability. The third group is multispectral superposition. Jeong and Williams (1992) superimposed multiple spectrograms with different TF resolutions and then limited the superimposed results to determine the region of support of the WVD auto-term in the TF plane. Multiplying the WVD by the region yields the new TFD. However, some of the mathematical properties of WVD, including edge distribution, can only be downgraded due to the nonlinear nature of thresholding. Finally, a few methods are based on image processing. The morphological filtering transforms the TFD into a binary image and suppresses cross-terms by utilizing the length difference between the auto-terms and cross-terms (Li et al., 2015; Yang and Li, 2016). However, retaining all the information is difficult when the auto-term energy is inconsistent or noise interference occurs. The Hough transform is used as a line detection method for auto-term detection (Barbarossa, 1995), and the Radon transform rotates the signal components (Wood and Barry, 1992). However, both are notably limited by the signal type and cannot directly provide the TFD of the signal.

In recent years, scholars have also proposed new solutions to the problem from different angles. Wu and Li (2016) used linear frequency modulation (LFM) signal auto-term and cross-term frequency difference filtering. Moghadasian and Gazor (2020) utilized sparse computing techniques to effectively suppress cross-terms. Wacker and Witte (2011) proposed the novel TF analysis method that combines matching pursuit (MP) and WVD. They calculated the WVD of each matched atom generated by MP and modified the result by the original WVD. Hao et al. (2019) introduced an MP-based double WVD to suppress cross-term interference and used synthetic signals to demonstrate the high TF concentration of the proposed TF analysis method. Fu et al. (2022) combined MP and SPWVD to analyze the problem of cross-term interference elimination in measured blasting vibration signal frequency spectra. Khan and Sandsten (2016) adjusted the direction of the smoothing kernel locally at each TF point to maintain the alignment of the smoothing kernel with the ridges of the auto-terms. Ren et al. (2016) approached WVD through the improvement of PWVD refinement but failed to achieve the same level of TF concentration as WVD. Singh and Pachori (2021) focused on sliding eigenvalue decomposition and the WVD and decomposed the multicomponent signal into a set of mono-component signals. The WVD of analytic mono-components is then added to obtain a crossterm-free TF representation. Liu et al. (2023) proposed WVD Net, which is a semi-supervised learning model for TF analysis based on the WVD. This approach is based on the mean–teacher model and the existing cross-term in WVD, thereby relaxing the requirements of the training dataset. The cross-term also has stronger energy than the auto-term. One of the future research directions involves combining the idea of weak signal detection with crossterm suppression (Qiao and Shu, 2021; Qiao et al., 2023). Therefore, researchers continuously explore and attempt other research directions. However, simultaneously balancing the cross-term suppression effect and TF concentration is still difficult.

This study presents a new method to address the shortcomings of current methods. The new method effectively removes cross-terms while preserving excellent TF concentrations of WVD. This method is also adaptable to signals with multiple types and distributions. The primary contribution of this method can be briefly summarized as follows: on the one hand, the method utilizes the similarity between TF and traditional images to employ the seeded region growing (SRG) algorithm as the basic framework. On the other hand, the method extracts multidimensional property differences in auto-terms and cross-terms by processing WVD results in several dimensions. Approximate entropy and Rényi entropy were selected as one- and two-dimensional metrics, respectively, to improve and enhance the critical stages of the SRG method.

The paper is organized as follows: Section 2 explores the multidimensional property. We introduce the fundamental theory of WVD, comprehensively investigate autoterm and cross-term from a multidimensional perspective, and choose approximate entropy and Rényi entropy to describe them, respectively. Section 3 presents a comprehensive explanation of the principle and process of the cross-term removal method proposed in this paper, combined with the content of information from Section 2 and the SRG. Section 4 validates the efficacy of the proposed method based on simulation and real experiment data. Section 5 presents the conclusion.

2 Multidimensional property differences of WVD auto-term and cross-term

The WVD and its several other variants have been widely used for TF analysis of nonstationary signals in a wide range of application domains (Liu et al., 2013; Wu et al., 2018). The WVD provides the instantaneous TF coefficients of a signal and can be expressed as

$$ W(t, \omega)=\frac{1}{2 \pi} \int s^*\left(t-\frac{1}{2} \tau\right) s\left(t+\frac{1}{2} \tau\right) \mathrm{e}^{-\mathrm{j} \tau \omega} \mathrm{~d} \tau $$

(1)

where ∗ denotes the complex conjugation, t is the time, ω is the frequency, and s(t) is the analytic signal defined by

$$ s(t)=x(t)+\mathrm{j} H[x(t)] $$

(2)

where H [ x(t) ] is the Hilbert transform of the actual signal x(t). Obtaining the analytic form of the signal is essential to prevent interference between positive and negative frequency components. The WVD almost perfectly localizes the signal with a single component signal in the TF plane. However, when the signal is multicomponent,

$$ s(t)=\sum\limits_{i=1}^{\mathrm{NC}} s_i(t) $$

(3)

where NC represents the number of components, and its WVD is shown as follows:

$$ W(t, \omega)=\underbrace{\sum\limits_{i=1}^{\mathrm{NC}} W_i(t, \omega)}_{\text {auto - terms }}+\underbrace{2 \sum\limits_{i=1}^{\mathrm{NC}} \sum\limits_{j=1, j \neq i}^{\mathrm{NC}} \operatorname{Re}\left\{W_{i, j}(t, \omega)\right\}}_{\text {cross - terms }} $$

(4)

The WVD of the multicomponent signal comprises two parts. The first item on the right is the WVD of each component and is called auto-terms. The second item of WVD between each pair of the components is called cross-terms. Auto-terms and cross-terms differ from one another. Thus, the key to dealing with cross-terms lies in utilizing this difference. The paper combines specific signal types to conduct a detailed analysis. Common types of signals include LFM, hyperbolic frequency modulation (HFM), and continuous wave (CW) signals. LFM is chosen as the representative and has received considerable attention due to its extensive application in various fields, such as radar, sonar, communication, and seismic analyses. In the single LFM $S_{\mathrm{LFM} 1}(t)=\exp \left(\mathrm{j} \omega_0 t\right), \omega_0=2 \pi\left(f_0+0.5 \times k t\right)$ amplitude is 1, f₀ is the initial frequency, k is the frequency modulation ratio, and its WVD is a pulse function as shown in Eq. (5):

$$ W_{\mathrm{LFM} 1}(t, \omega)=2 \pi \delta\left(\omega-\omega_0\right) $$

(5)

There exists a multi-component LFM signal, which is defined as $S_{\mathrm{LFM} 2}(t)=\exp \left(\mathrm{j} \omega_1 t_1\right)+\exp \left(\mathrm{j} \omega_2 t_2\right)$ω₁ and ω₂ are the frequency of the two components, and t₁ and t₂ indicate the time of their appearance. The WVD of s_LFM2 (t) is

$$ \begin{aligned} & W_{\text {LFM2 }}(t, \omega)=\underbrace{2 \pi \delta\left(\omega-\omega_1\right)}_{\text {auto - term1 }}+\underbrace{2 \pi \delta\left(\omega-\omega_2\right)}_{\text {auto - term2 }} \\ & \quad+\underbrace{4 \pi \cos \left(\omega_1 t_1-\omega_2 t_2\right) \delta\left(\omega-\left(\omega_1+\omega_2\right) / 2\right)}_{\text {cross - term }} \end{aligned} $$

(6)

As shown in Figure 1, this paper obtains and drafts the three-dimensional spatial distribution of single auto-term and cross-term of the LFM.

Figure 1 Three-dimensional spatial representation of auto-term and cross-term of LFM signal

Download: Full-Size Img

Figure 1 visually illustrates the distinctions between the auto-term and cross-term in the three-dimensional spatial distribution. The amplitude of the LFM auto-term exhibits a smooth variation, and the energy distribution reveals strong continuity. However, the LFM cross-term shows notable oscillation, and its amplitude value can reach twice that of the auto-term. The cross-term peak alternates in the positive and negative domains, and the energy value of these connected domains dramatically change. Essentially, the auto-term remains constant and changes at a steady pace, but the cross-term fluctuates and changes rapidly.

The above content is intuitive but insufficiently comprehensive, and obtaining usable deterministic information to describe the auto-terms and cross-terms is difficult. Thus, additional reliable information is necessary to achieve the purpose of the article and describe the differences between auto-terms and cross-terms. Two critical issues are encountered in cross-term removal: the determination of the positions of the auto-terms and cross-terms and the discrimination between the two.

Therefore, this section proposes to discuss the auto-term and cross-term property differences in the other two dimensions (one-dimension and two-dimension). The plan is to obtain and study the auto-term and cross-term properties in the two-dimensional TF plane and one-dimensional sequences and then select the appropriate parameter to describe the differences.

Taking the multicomponent LFM signal as an example, Figure 2 shows the proposed idea for this paper. Figures 2(b)(c)(d) represent the distribution WVD of the entire signal, auto-term, and cross-term on the two-dimensional TF plane, respectively. This distribution is investigated in Section 2.1. Meanwhile, Figures 2(e)(f)(g) are their one-dimensional sequences. These sequences are indirectly available in the WVD results and require special processing, which will be extensively explained in Section 2.2.

Figure 2 Multidimensional representation of WVD results

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2.1 Property of auto-terms and cross-terms based on the two-dimensional TF plane

As shown in Figures 2(c)(d), the differences between auto-terms and cross-terms on the two-dimensional TF plane are highly intuitive. The auto-term is continuous, and energy is concentrated, while the cross-term is discontinuous, and energy is dispersed. However, the two exhibit completely different information properties. Using this point to distinguish between the terms is essentially a judgment of information content. Rényi entropy is chosen as the feature quantity to describe the information content of the TFD in the TF plane. Rényi entropy originated in the field of thermodynamics and is used to define the degree of order or the amount of information contained in a system. Moreover, Rényi entropy represents the uncertainty of a situation or problem in mathematics and the capability to describe information in an information system in information theory (Campbell, 1965).

For Rényi entropy, a finite number of random variables with an incomplete probability density distribution is ζ =$V$, satisfying $\sum \varsigma_i \leqslant 1$. Rényi entropy is defined as:

$$ R^\alpha(\varsigma)=\frac{1}{1-\alpha} \log _2 \frac{\sum\limits_i \varsigma_i^\alpha}{\sum\limits_i \varsigma_i} $$

(7)

where α is the order of Rényi entropy. When α= 1, Rényi entropy of time degenerates into Shannon entropy. Rényi entropy of a continuous two-dimensional probability density distribution ξ(x, y) is defined as

$$ R^\alpha(\xi)=\frac{1}{1-\alpha} \log _2 \frac{\iint \xi^\alpha(x, y) \mathrm{d} x \mathrm{~d} y}{\iint \xi(x, y) \mathrm{d} x \mathrm{~d} y} $$

(8)

Williams et al. (1991) proposed the concept of Rényi entropy in TFD, which obtains the essential information characteristics of a signal by statistically analyzing the Rényi entropy of the signal on the TF plane. This approach has later been widely applied (Baraniuk et al., 2001; Sucic et al., 2014; ANS et al., 2016; Vedran, 2024).

In this approach, if the TFD result TF (t, f) of a signal satisfies the characteristics of time edge, frequency edge, and energy retention, then TF (t, f) and ξ(x, y) have similar properties. The Rényi entropy of TFD is defined as follows:

$$ R^\alpha=\frac{1}{1-\alpha} \log _2 \iint\left(\frac{\mathrm{TF}(t, f)}{\iint \mathrm{TF}(x, y) \mathrm{d} x \mathrm{~d} y}\right)^\alpha \mathrm{d} t \mathrm{~d} f $$

(9)

A key point must be emphasized. That is, WVD is the distribution of signal energy over time and frequency. Using LFM as an example, the auto-term is theoretically expected to be consistently positive, whereas the cross-term can have positive and negative values. In fact, the auto-term, which is the Fourier transform of the autocorrelation function of the signal, is always a real value but may not be positive. Therefore, Rényi entropy must permit the existence of negative information.

Based on existing research content, TFDs of signal components with high temporal frequency concentration and low dispersion have reduced information richness and correspond to a low Rényi entropy value. However, if the TFD comprises clutter-scattered signal components, then its TFD contains additional information content and a low TF concentration, and its corresponding entropy value is large.

As shown in Figure 3, the Rényi entropy distribution of some common signal types is simulated to comprehensively explore the correlation between the Rényi entropy and signal characteristics, such as type, auto-term, cross-term, and α. Taking LFM, HFM, and CW signals as examples, the Rényi entropy of a single auto-term and cross-term TFD is discussed. Three signals also have the same duration and sampling rate. The initial frequency and range of LFM and HFM agree, and the frequency of CW matches the central frequency of LFM. The cross-term in Figure 3 is produced by two signal auto-terms with a substantial frequency difference and strong distinguishability. However, the crossterm is actually influenced by parameters such as the time and frequency intervals of the auto-term, rendering its Rényi entropy highly difficult to predict.

Figure 3 Different orders of Rényi entropy values of the cross-term and auto-term

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In Figures 3(a)(b)(c), for each signal type, the auto-term Rényi entropy exhibits a gradual decline as the α increases, revealing minimal overall variation. The cross-term values have a discontinuous distribution at low α and rough increases with the rise of α, leveling out at a high α. HFM exhibits the maximum entropy value among several signal types. The auto-term entropy value of the LFM signal is approximately equal to that of the CW signal. However, the cross-term entropy value is higher than that of the CW signal. Thus, Rényi entropy possesses the necessary attributes to function as a distinctive signal characteristic.

The situation of same-type signal auto-terms and crossterms differ. For HFM, the entropy value of the cross-term is substantially larger than that of the auto-term when α ≥ 3. Meanwhile, the difference is largest when α = 3. However, the entropy value of the cross-term in CW is substantially higher than that of the auto-term. Regarding LFM, the situation is highly intricate. The entropy value of the crossterm is elevated for even α, while the inverse is true for odd α. When α > 14, the entropy value of the cross-term grows, and the disparity between the two diminishes.

α is typically selected as an odd number. According to Figure 3, selecting α = 3 is the most effective way to differentiate between auto-terms and cross-terms. Considering HFM and CW, α has a greater range of options.

Section 2.1 shows an intuitive property difference between the auto-term and the cross-term in the two-dimensional TF plane. Thus, choosing the appropriate order of Rényi entropy can describe and distinguish the two and adapt to changes in signal types.

2.2 Property of auto-terms and cross-terms on one-dimensional sequences

Figures 2(e)(f)(g) illustrates the intention of Section 2.2, which is to examine the characteristics of auto-terms and cross-terms in one-dimension. The objective is to utilize this property to efficiently determine the position of autoterms and cross-terms. First, we obtain the one-dimensional sequence of the auto-term and cross-term by utilizing the signal characteristics to reconstruct the WVD results.

Assuming the presence of a multicomponent signal S (t), which comprises various deformations of the basic signal s₀(t)

$$ S(t)=\sum s_i(t), s_i(t)=s_0\left(t+\tau_i\right) \text { or } s_i(w)=s_0\left(w+w_i\right) $$

(10)

We set the pulse width of s₀(t) as T and the bandwidth as B. Instantaneous frequency is represented by F(t). When s₀(t) belongs to LFM, F(t) satisfies the following equation:

$$ F_{\mathrm{LFM}}(t)=k t+f_0 $$

(11)

where k= B/T represents the frequency modulation slope, and f₀ is the starting frequency of s₀(t). When s₀(t) belongs to the HFM signal,

$$ F_{\mathrm{HFM}}(t)=f_c^{\prime} /\left[1-\left(v / f_c^{\prime}\right) t\right] $$

(12)

where $f_c=2 f_L f_H /\left(f_L+f_H\right), v=f_c^2\left(f_H-f_L\right) /\left(T f_H f_L\right)$, f_L is the lowest frequency, and f_H= f_L+ B is the highest frequency. When s₀(t) belongs to the CW signal, the frequency of the signal does not vary with time.

$$ F_{\mathrm{CW}}(t)=f_0 $$

(13)

Taking the three signal types above as examples, the WVD of in the TF plane is shown in Figure 4 (omitting cross-term). Notably, distinct auto-terms of the multicomponent signal are essentially identical except for their different positions on the TF plane. Moving one auto-term can result in other auto-terms.

Figure 4 Schematic of two-dimensional WVD of multicomponent signals in the TF plane (omitting cross-terms)

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We use a line to simplify the auto-term, and the line type is defined by F(t) of the signal. There is only the "difference in position" between the different lines in one plane, which is defined as the "intercept" and is denoted by "⋅" A finite number of "lines" make up the TF plane, and each one is a one-dimensional sequence W_{b_i}. In the three-dimensional schematic Figure 2(e), W_{b_i} is equivalent to the value of WVD on the F(t) + b_i curve section.

Therefore, this paper aims to obtain W_{b_i} from the WVD result (WVD_S) of S(t). W_{b_i} is set as a one-dimensional sequence:

$$ \boldsymbol{W}_{b_i}=\left\{\boldsymbol{W}_{b_i}\left(t_1, f_1\right), \cdots, \boldsymbol{W}_{b_i}(t, f), \cdots\right\} $$

(14)

Every point in the sequence satisfies the following:

$$ \boldsymbol{W}_{b_i}(t, f)=\mathbf{W V D}_S\left(t, F(t)+b_i\right) $$

(15)

Thus, W_{b_i} is the set of points taken along "the line F (t) + b_i" on WVD_S, and WVD_S= ∪_{b_i}W_{b_i}. We reconstruct WVD_S as matrix S. Therefore, each column of Scorresponds to b_i

$$ \boldsymbol{S}=\left[\boldsymbol{W}_{b_1}, \boldsymbol{W}_{b_2}, \cdots, \boldsymbol{W}_{b_1}, \cdots, \boldsymbol{W}_{b_t}\right] $$

(16)

In practical signal processing, S(t) is the discrete signal, and WVD_S is a matrix of M×N with coordinate origin (1, 1). The signal sampling rate is f_s, and the number of points of its FFT is N_fft. The quantities in Eqs. (14)–(16) are discretized and expressed as Eq. (17).

$$ \begin{cases}t=n \times \Delta t=n / f_s & t=1, 2, \cdots, N \\ f=m=F\left(n / f_s\right)+b_i, & b=b_1, b_2, \cdots, b_i, \cdots, b_L \\ \boldsymbol{W}_{b_i}(n, m)=\mathbf{W V D}_s\left(n, F\left(n / f_s\right)+b_i\right), & i=1, 2, \cdots, L\end{cases} $$

(17)

Meanwhile, S is denoted as

$$ \boldsymbol{S}=\left[\boldsymbol{W}_{b_1}, \boldsymbol{W}_{b_2}, \cdots, \boldsymbol{W}_{b_x}\right]=\left[\begin{array}{ccc} \mathbf{W V D}_s\left(1, F\left(1 / f_s\right)+b_1\right) & \cdots & \mathbf{W V D}_s\left(1, F\left(1 / f_s\right)+b_L\right) \\ \vdots & \ddots & \vdots \\ \mathbf{W V D}_s\left(N, F\left(N / f_s\right)+b_1\right) & \cdots & \mathbf{W V D}_s\left(N, F\left(N / f_s\right)+b_L\right) \end{array}\right] $$

(18)

We take the LFM signal as an example and combine Eq. (11) to obtain S, which corresponds to Figure 2(e).

$$ \begin{aligned} & \boldsymbol{S}=\left[\begin{array}{ccc} \mathbf{W V D}_S\left(1, k+b_1\right) & \cdots & \mathbf{W V D}_S\left(1, k+b_L\right) \\ \vdots & \ddots & \vdots \\ \mathbf{W V D}_s\left(N, k \times N+b_1\right) & \cdots & \mathbf{W V D}_S\left(N, k \times N+b_L\right) \end{array}\right] \\ & k=\mathrm{d} f / \mathrm{d} t=2 N_{\mathrm{fft}} \times B /\left(T \times f_s^2\right) \\ & b_i \in[-N \times k, M] \end{aligned} $$

(19)

As shown in Figure 2(e), the W_{b_i} of auto-term, cross-term, and other positions demonstrates considerable differences. For unknown signals, b_i of the auto-term or cross-term cannot be directly obtained. However, W_{b_auto} and W_{b_cross} can be obtained using the property differences of W_{b_i} and choosing the appropriate feature for measurement and screening b_auto and b_cross corresponding to the positions of cross-term and auto-terms, respectively, can then be obtained. As shown in Figures 2(e)(f)(g), W_{b_i} mainly differs in the distribution and volatility. The paper chooses approximate entropy as the feature.

Approximate entropy is a nonlinear dynamical parameter used to quantify the regularity and unpredictability of fluctuations in time sequences (Vaillancourt, 2010; Sampaio and Nicoletti, 2016; Yang et al., 2022). This parameter reflects the likelihood of new information occurring in the time sequence, with the highly complex time sequence corresponding to a high value. The definition of approximate entropy is as follows:

There is a set of sequences x(1), x(2), …, x(P), with P data points. We choose q as the embedding dimension. According to Eq. (20), the data points comprise q dimensional vectors X(i)

$$ \begin{aligned} \boldsymbol{X}(i) & =[x(i), x(i+1), \cdots, x(p+q-1)], \\ i & =1, \cdots, P-q+1 \end{aligned} $$

(20)

where d[X(i), X(j)] is the largest distance between the corresponding elements of vectors X(i) and X(j):

$$ d[\boldsymbol{X}(i), \boldsymbol{X}(j)]=\max _{c=0 \sim q-1}[|x(i+c)-x(j+c)|] $$

(21)

There is similarity tolerance r. We count the number of d[X(i), X(j)], for each iwhen d[X(i), X(j)] < r. The ratio of the number to the total number is defined as

$$ \begin{aligned} c_i^q(r) & =\frac{1}{P-q+1} \times\{\text { number of } d[X(i), X(j)] \text { if } d[X(i), X(j)]<r\} \\ i & =1,2, \cdots, P-q \end{aligned} $$

(22)

The above equation is then logarithmized, and the outcomes are combined to obtain the average.

$$ \phi^q(r)=\frac{1}{P-q+1} \sum\limits_{i=1}^{P-q+1} \ln C_i^q(r) $$

(23)

The above content is repeated to obtain ϕ^{q + 1}(r). The approximate entropy of the sequence is defined as

$$ \operatorname{ApEn}(q, r)=\lim _{P \rightarrow \infty}\left[\phi^q(r)-\phi^{q+1}(r)\right] $$

(24)

When P is finite, the estimate of ApEn(q, r) is

$$ \operatorname{ApEn}(q, r, P)=\phi^q(r)-\phi^{q+1}(r) $$

(25)

In general, q = 2, r = (0.10~0.25) × SD (SD is the standard deviation of sequence). Approximate entropy measures the likelihood that the patterns of P observations will remain similar in the next incremental comparison. The value is smaller when the likelihood of maintaining proximity is higher and vice versa.

Based on the above properties, the paper solves for each column W_{b_i} of the S

$$ \mathbf{A p E n}_s=\left[\operatorname{ApEn}\left(b_1\right), \operatorname{ApEn}\left(b_2\right), \cdots, \operatorname{ApEn}\left(b_L\right)\right] $$

(26)

Taking the multicomponent LFM in Figure 2 as an example, the paper reconstructs S and obtains its ApEn_S, as shown in Table 1. The position of one auto-term and two cross-terms corresponds to the b_auto = b_{7, 11}, b_cross = b₉.

Table 1 ApEn_S of multicomponent LFM signal

i	1	2	3	4	5	6
ApEn(b_i)	0.782	0.926	0.650	0.911	0.726	0.922

i	7	8	0	10	11	12
ApEn (b_i)	0.362	0.836	0.556	0.849	0.309	0.926

i	13	14	15	16	17	18
ApEn(b_i)	0.734	0.934	0.634	1.01	0.739	0.945

Combining Figure 2(e) and Table 1, the sequence of cross-terms violently fluctuates, revealing high values. The signal auto-terms fluctuate smoothly, demonstrating low values. Despite their small overall amplitude, the change is still highly complex. Therefore, the approximate entropy value of cross-term should be greater than the auto-term.

The approximate entropy of the signal auto-term, crossterm, and other locations under interference-free conditions satisfies the following relationship:

$$ \mathrm{ApEn}_{\text {auto }}<\mathrm{ApEn}_{\text {cross }} \ll \mathrm{ApEn}_{\text {else }} $$

(27)

Eq. (27) helps obtain ApEn_auto and ApEn_cross from ApEn_S, then W_{b_auto}, W_{b_cross} and b_auto, b_cross can then be recognized. Section 2.2 utilizes one-dimensional sequences of autoterms and cross-terms to determine the position, which notably reduces the difficulty and workload of cross-term removal.

3 Adaptive WVD cross-term removal method based on multidimensional property differences

Based on the property differences between auto-terms and cross-terms in Section 2, this paper proposes a new method for cross-term removal. The flowchart of the proposed method is shown in Figure 5.

Figure 5 Flowchart of the proposed method

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Figure 6 Schematic of the SRG algorithm

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This method is based on SRG. SRG is a simple and effective segmentation method for segmenting sections in an image that have similar characteristics (Adams and Bishof, 1994; Carvalho et al., 2010). This method can preserve clear boundary contour information and produce accurate segmentation results. SRG is extensively utilized in medical imaging and various other disciplines.

SRG can combine individual pixels or small regions into large regions using predesigned growth criteria. Figure 5 illustrates the basic steps in the algorithm. First, the algorithm identifies a point within the target region that acts as an initial point for expansion, referred to as a "seed." Subsequently, SRG finds a pixel within the vicinity that has the same or similar properties to the "seed" based on the similarity criterion. This pixel is integrated into the region where the seed is situated. The algorithm repeats the above process using the newly selected pixel as the updated "seed" until no such pixels remain. Figure 5 successfully extracts the core yellow area.

The SRG algorithm has a simple process and clear extraction results. WVD and images are matrices; thus, the difference in continuity concentration of auto-terms and crossterms determines that the SRG algorithm can be utilized to extract auto-terms while removing cross-terms. However, employing the SRG directly will introduce several problems. For example, seeds are typically subjected to artificial selection, which can result in notable biases and expansion in non-signal areas. In addition, the similarity rules for growth and the prerequisites for the completion of the growing process remain unclear. Thus, controlling growth is difficult, and outcomes include an excessive number of interfering elements. Based on the auto-term and crossterm properties in Section 2, the proposed method improves the SRG algorithm and solves the above problems. The specific steps of the method are as follows, referring to the implementation in Figure 7.

Figure 7 Schematic of the proposed method

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The proposed method is divided into nine steps:

Step 1: Graying. We convert the WVD_S to grayscale matrix Gray_S, maintaining the TF information and laws.

Step 2: Seeds are selected, and the "initial seed space" is created. According to Section 2.2, we calculate S and ApEn_S of WVD_S And filter out some W_{b_auto} and b_auto are filtered out based on Eq. (28). The point in every W_{b_auto} with the highest energy is referred to as "the initial seed, " and its coordinates are represented as (m, n). They constitute "the initial seed space."

Step 3: One point is selected from "the initial seed space, " which is denoted as "Seed1, " represented by the o₁. The coordinate of this point is (m₁, n₁), and the grayscale value is Gray_S(o₁) = Gray_S(m₁, n₁). A zero matrix I₁ of the same size is created as Gray_S.

Step 4: The collection of pixels around o₁ is referred to as o'₁. The points from o'₁ that satisfy the similarity criterion are chosen and then placed in "the processing space." o₁ is recorded as completed and produces I₁ (m₁, n₁) = 1. The step is illustrated in Figures 7(a)(b).

The study selects the gray value difference as a criterion for evaluating the similarity rules for growth.

This point satisfies the similarity rules if $\left.\left\lvert\, \operatorname{Gray}_S(\text { point })-\frac{\sum \operatorname{Gray}_S(\text { the completed space })}{\text { number of the completed space }}\right. \right\rvert\, < T_h .$ where T_h. is the growth threshold and varies in accordance with the complexity of the signal environment.

Step 5: o₂ is selected from the "the processing space, " and Step 4 is repeated. When all the points in "the processing space" have been processed and no new pixels have been added, the "Seed1" growth process is completed. At this point, I₁ recorded the positions of all growth points. All the selected points in I₁ are 1, and others are designed as 0.

Step 6: Determination of cessation of growth. The operation WVD₁ = WVD_S⋅ I₁ is performed. Considering Section 2.1, this paper calculates the specified order Rényi entropy value as a prerequisite for the completion of the growing process. For example, the multicomponent LFM signal is shown in Figure 7, and the Rényi entropy with α= 3 is chosen. The Rényi entropy of WVD₁ is calculated. If the Rényi entropy is close to the Rényi entropy value of the auto-term, then WVD₁ and I₁ are retained.

Otherwise, the value of T_h is modified based on the level of Rényi entropy, and steps 4–6 are iterated until the specified condition is satisfied.

Step 8: The second seed point is obtained from "the initial seed space, " noted as "Seed2, " and Steps 3–6 are repeated until "the initial seed space" is exhausted.

Step 9: We merge the above outputs to obtain the result of the new cross-term removal. WVD_R = WVD_S⋅(I₁ ∪ I₂ ∪ …)

Steps 2–7 in the aforementioned process correspond to the regional growth of a single seed. Steps 8–9 pertain to the regional growth of a plurality of seeds. Thus, the method described herein is not constrained by the quantity of signal components.

This approach presents some improvements over the conventional way.

1) Step 2 improves the seed selection process. Utilizing the signal instantaneous frequency as a priori knowledge helps determine the positions of the auto-terms, cross-terms, and the rest of the positions based on their one-dimensional sequence properties. This approach ensures a sensible seed selection.

2) Step 7 introduces the prerequisites for the completion of the growing process. Rényi entropy quantifies the distinct distribution properties of auto-terms and cross-terms on the two-dimensional TF plane. Combined with the signal type, this method adaptively selects the proper order of Rényi entropy as a judgment criterion for the result. A constant adjustment is made until the entropy value of the growth result is as expected.

3) Steps 1 and 4 convert the WVD_S values to Gray_S as a similarity criterion.

The new method utilizes the auto-term and cross-term properties: continuity, one-dimensional sequence properties, and two-dimensional TF plane distribution properties. This method achieves the elimination of the cross-term and the complete preservation of the auto-term.

4 Emulation signals and real experimental data verification

This section evaluates the effectiveness of the proposed method by analyzing synthetic emulation signals and real experimental data. The objective is to examine the performance of the proposed method in comparison with the WVD, PWVD, SPWVD, and AOK-TFD.

The PWVD is one of the simplest and most direct crossterm removal methods, which achieves the reduction of cross-terms by adding a window function h(τ) to the variable τ.

$$ \operatorname{PW}(t, \omega)=\frac{1}{2 \pi} \int s^*\left(t-\frac{1}{2} \tau\right) s\left(t+\frac{1}{2} \tau\right) \times h(\tau) \times \mathrm{e}^{-\mathrm{j} \tau \omega} \mathrm{~d} \tau $$

(28)

The SPWVD is windowed simultaneously for u and τ using the window functions g(u) and h(τ) respectively, which are real even window functions satisfying h(0) = G(0) = 1.

$$ \begin{aligned} \operatorname{SPW}(t, \omega) & =\frac{1}{2 \pi} \iint s^*\left(t-u-\frac{1}{2} \tau\right) s\left(t-u+\frac{1}{2} \tau\right) \\ & \times h(u) \times g(\tau) \mathrm{e}^{-\mathrm{j} \tau \omega} \mathrm{~d} u \mathrm{~d} \tau \end{aligned} $$

(29)

The AOK-TFD adapts the spread of the radially Gaussian kernel along each direction based on the characteristics of a signal at each time-instant. Φ_opt(t; θ, τ) is the optimal kernel.

$$ \begin{aligned} \operatorname{TFD}_{\text {AOK }}(t, \omega) & =\frac{1}{4 \pi^2} \iint A(t ; \theta, \tau) \Phi_{\text {opt }}(t ; \theta, \tau) \mathrm{e}^{-\mathrm{j} \omega \tau-\mathrm{j} \theta t} \mathrm{~d} \theta \mathrm{~d} \tau \\ & =\frac{1}{4 \pi^2} \iint s^*\left(v+t-\frac{\tau}{2}\right) w^*\left(v-\frac{\tau}{2}\right) s\left(v+t+\frac{\tau}{2}\right) w\left(v+\frac{\tau}{2}\right) \mathrm{e}^{\mathrm{j} \theta v} \mathrm{~d} v \times \Phi_{\text {opt }}(t ; \theta, \tau) \mathrm{e}^{-\mathrm{j} \omega \tau} \mathrm{~d} \theta \mathrm{~d} \tau \end{aligned} $$

(30)

The following energy concentration measure is used to compare the performance of selected TFDs (Staniović, 2001).

$$ \mathrm{EC}=\left.\left.\left|\sum\limits_{n=0}^{N-1} \sum\limits_{k=0}^{N-1}\right| \rho[n, k]\right|^{1 / \sigma}\right|^\sigma $$

(31)

where ρ[n, k] is a normalized TFD. $\sum_{n=0}^{N-1} \sum_{k=0}^{N-1} \rho[n, k]=1$ and σ is any integer constant. σ = 2 is selected in this study. Unlike entropy-based measures, this measure was selected because it does not compromise the energy concentration of weak components over strong components. The energy concentration measure has a higher value if the signal energy is spread in the TF domain and a lower value if the energy is concentrated.

4.1 Emulation signals

4.1.1 Multitypes of emulated signals

The above content shows that the method proposed in the paper is adaptive and applicable to multiple types of signals and is not limited by signal type. Cross-term removal can be achieved when the instantaneous frequency of the signal base of the multicomponent signal is known. According to Figure 8, the paper deal with three types of signals as example of removing cross-terms in signals.

Figure 8 Performance comparison of multitype signals (LFM, HFM, and CW) in different TFDs

(ⅰ), (ⅱ), (ⅲ), (ⅳ), and (ⅴ) are the results of WVD, PWVD, SPWVD, AOK-TFD, and the proposed method, respectively

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Figure 8 comprises multiple columns, each representing a specific signal type. These signal types include singlecomponent LFM, multicomponent LFM, single-component HFM, multicomponent HFM, single-component CW, and multicomponent CW. Multicomponent signals exhibit variations in frequency among their different components. Each row in Figure 8 corresponds to an algorithm. Rows (ⅰ), (ⅱ), (ⅲ), (ⅳ), and (ⅴ) represent the WVD, PWVD, SPWVD, AOK-TFD, and the method described in this study, respectively. These signals are defined as follows:

$$ \begin{aligned} & S_a(t)=\exp (20000 \pi \mathrm{j} t(1+75 t)) \\ & \begin{aligned} S_b(t) & =\exp (20000 \pi \mathrm{j} t(1+75 t)) \\ & +\exp (20000 \pi \mathrm{j} t(2+75 t)) \\ S_c(t) & =\exp \left(-106 \pi \mathrm{j} \ln \left(1-300 t^{\prime}\right)\right) \\ S_d(t) & =\exp \left(-106 \pi \mathrm{j} \ln \left(1-300 t^{\prime}\right)\right) \\ & +\exp \left(-266 \pi \mathrm{j} \ln \left(1-214.3 t^{\prime}\right)\right) \end{aligned} \\ & \begin{aligned} S_e(t) & =\exp (60000 \pi \mathrm{j} t) \\ S_f(t) & =\exp (60000 \pi \mathrm{j} t)+\exp (40 000 \pi \mathrm{j} t) \\ & ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 \leqslant t \leqslant 0.002 \mathrm{~s} \\ & ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 \leqslant t^{\prime} \leqslant 0.004 \mathrm{~s} \end{aligned} \end{aligned} $$

(32)

The energy concentration EC of the cross-term suppression results in Figure 8 is calculated, as shown in Table 2.

Table 2 Energy concentration comparison of multitype signals in different TFDs

EC	PWVD	SPWVD	AOK-TFD	The proposed method
S_a	6.76×10⁴	4.09×10⁴	5.14×10⁴	3.77×10²
S_b	1.33×10⁵	7.91×10⁴	7.93×10⁴	6.49×10³
S_c	2.16×10⁵	1.38×10⁵	8.39×10⁴	1.15×10⁴
S_d	4.28×10⁵	2.14×10⁵	1.39×10⁵	3.70×10⁴
S_e	6.90×10⁴	4.65×10⁴	4.03×10⁴	5.09×10³
S_f	1.31×10⁵	6.03×10⁴	7.56×10⁵	7.80×10³

Figure 8(b)(ⅰ) display the WVD result of the multicomponent LFM signal. There are two auto-terms and one cross-term. Figure 8(b)(ⅱ) show the PWVD and do not completely eliminate cross-terms. SPWVD and AOK-TFD eliminate cross-terms, but the TF concentration is lower in Figures 8(b)(ⅲ) and (ⅳ). The proposed method is displayed in Figure 8(b)(ⅴ). Cross-term removal is achieved, but the results still maintain a high TF concentration. Herein, clear and slim auto-terms are observed, preserving the advantages of the WVD algorithm. The contents of Figures 8(d) and 8(f) are similar to Figure 8(b).

Figure 8(a) (ⅰ) displays the WVD of the single-component LFM signal without the interference of cross-term. Notably, the PWVD (Figure 8(a)(ⅱ)), SPWVD (Figure 8(a) (ⅲ)), and AOK-TFD (Figure 8(a)(ⅳ)) both directly lead to a decrease in the TF concentration, and the signal becomes coarse in the TF plane. The proposed method is shown in Figure 8(a)(ⅴ), and its TF concentration is higher than other TFDs with the lowest EC value. The contents of Figures 8(c) and 8(e) are similar to Figure 8(a).

Combining Figures 8(a)(c)(e) or Figures 8(b)(d)(f), the proposed method in this paper is also applicable to different types of signals.

4.1.2 Multidistributed emulated signals

The emulated signals serve the goal of thoroughly assessing the performance of the method described in this research. In addition to the types of signals in Section 4.1.1, considering other distributions of signals is also necessary. The paper obtains some multidistributed emulated signals, as depicted in Figure 9. Six signal distributions are commonly found and are defined as follows:

$$ \begin{aligned} S_a^{\prime}(t) & =\exp \left(20000 \pi \mathrm{j} t_1\left(1+75 t_1\right)\right) \\ & +\exp \left(20000 \pi \mathrm{j} t_2\left(1+75 t_2\right)\right) \\ S_b^{\prime}(t) & =\exp \left(20000 \pi \mathrm{j} t_1\left(1+75 t_1\right)\right) \\ & +0.8 \exp \left(20000 \pi \mathrm{j} t_2\left(1+75 t_2\right)\right) \\ S_c^{\prime}(t) & =\exp \left(20000 \pi \mathrm{j} t_1\left(1+75 t_1\right)\right) \\ & +0.8 \exp \left(20000 \pi \mathrm{j} t_2\left(1+75 t_2\right)\right) \\ & +0.6 \exp \left(20000 \pi \mathrm{j} t_3\left(1+75 t_3\right)\right) \\ S_d^{\prime}(t) & =\exp \left(20000 \pi \mathrm{j} t_1\left(1+75 t_1\right)\right) \\ & +\exp \left(20000 \pi \mathrm{j} t_2\left(1+75 t_2\right)\right)+\text { noise }_{-3 \mathrm{~dB}} \\ S_e^{\prime}(t) & =\exp \left(20000 \pi \mathrm{j} t_1\left(1+75 t_1\right)\right) \\ & +\exp \left(20000 \pi \mathrm{j} t_2\left(1-75 t_2\right)\right) \\ S_f^{\prime}(t) & =\exp \left(20000 \pi \mathrm{j} t_1\right)+\exp \left(22000 \pi \mathrm{j} t_1\right) \\ & +\exp \left(26000 \pi \mathrm{j} t_1\right) \\ &\quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 \leqslant t_1 \leqslant 0.002 \mathrm{~s}, \\ & \quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0.001 \mathrm{~s} \leqslant t_2 \leqslant 0.003 \mathrm{~s}, \\ & ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0.0025 \mathrm{~s} \leqslant t_3 \leqslant 0.0045 \mathrm{~s} \end{aligned} $$

(33)

Figure 9 Performance comparison of multidistributed signals in different TFDs

(ⅰ), (ⅱ), (ⅲ), (ⅳ), and (ⅴ) are the results of WVD, PWVD, SPWVD, AOK-TFD, and the proposed method, respectively

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Figure 9(a) shows the different TFD results of the twocomponent LFM signal, with different signal components exhibiting only time delay intervals and changing the strength of the signal, which comprises two signals (one strong and one weak in Figure 9(b)). Changing the number of components, a three-component LFM signal emerges in Figure 9(c). Figure 9(d) adds Gaussian white noise to Figure 9(a) with a signal-to-noise ratio of -3 dB. Figure 9(e) shows a special case of signal crossover. Figure 9(f) shows the three closest CW signals. The energy concentration EC of the cross-term suppression results is calculated, as shown in Table 3.

Table 3 Energy concentration comparison of multidistributed signals in different TFDs

EC	PWVD	SPWVD	AOK-TFD	The proposed method
S'_a	2.08×10⁵	1.21×10⁵	8.25×10⁴	1.53×10²
S'_b	2.10×10⁵	1.22×10⁵	8.32×10⁴	1.21×10⁴
S'_c	4.80×10⁵	2.65×10⁵	1.22×10⁵	2.32×10⁴
S'_d	1.92×10⁵	1.13×10⁵	8.71×10⁴	1.02×10⁴
S'_e	9.50×10⁵	2.17×10⁵	1.16×10⁵	8.67×10³
S'_f	1.21×10⁵	7.32×10⁵	9.05×10⁴	9.13×10³

Section 4.1 considers several factors, such as the number of signals, time delay, frequency delay, intensity, and noise. In most cases, the method in this paper is better than other TFDs. For PWVD, SPWVD, and AOK-TFD, signal auto-terms are fuzzy and unclear. Moreover, PWVD does not even remove the cross-terms completely. SPWVD and AOK-TFD methods eliminate the cross-terms with fuzzy auto-terms but compromise between preserving the autoterms and removing the cross-terms. Combining Tables 2 and 3, the proposed method has the minimum EC value when processing the same signal. The method achieves the removal of cross-terms while maintaining clear auto-terms and high TF concentrations, as shown in Figure 9(iv). Overall, the proposed method outperforms other TFDs.

4.2 Real experimental data verification

Next, we verify the effectiveness of the proposed method using hydroacoustic real experimental data, as shown in Figure 10 and Table 4. These data include the underwater target acoustic scattering echo data collected when the transmit signal is an LFM signal. The WVD of the signal is shown in Figure 10(a), where determining the number of echoes and analyzing the echo information is difficult due to noise and cross-terms. Figure 10(b) shows the results of PWVD, Figure 10(c) represents the results of SPWVD, and Figure 10(d) represents the results of AOK-TFD. PWVD cannot completely eliminate the cross-terms, while SPWVD eliminates the cross-terms and interference but notably reduces the TF concentration. The processing effect of AOKTFD is better than the first two but is still not ideal.

Figure 10 Performance comparison of experimental data in different TFDs

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Table 4 Energy concentration comparison of experimental data in different TFDs

EC	PWVD	SPWVD	AOK-TFD	The proposed method
data	5.49×10⁵	2.59×10⁵	1.29×10⁵	2.54×10⁴

Figure 10(f) uses the algorithm of this paper to process the experimental data. The effect shows that the algorithm achieves cross-term removal and auto-term retention and does not degrade the TF concentration. Overall, the performance of the proposed algorithm is notably better than other methods.

5 Conclusions

The cross-term removal method proposed in this paper targets known signal type but is not limited by the type or number of signals. The results of the processing can balance the dealing effect and TF concentration. The key to the method lies in the utilization of the multidimensional property difference between auto-terms and cross-terms. First, based on the instantaneous frequency distribution of signals, this method extracts a one-dimensional sequence in the TF plane and reconstructs the WVD matrix. The different amplitude fluctuations of one-dimensional sequences of cross-terms and auto-terms can lead to differences in approximate entropy so as to determine the position of the auto term and achieve initial seed selection for SRG. Second, the method selects energy difference as the similarity criterion for regional growth. Finally, Rényi entropy can measure the different distribution properties of auto-terms and cross-terms on the two-dimensional TF plane, which is used as a condition for the completion of the growing process in this paper. Feedback regulation is applied to the growth results and similarity criteria by utilizing the sensitivity of Rényi entropy to signal types, auto-terms, and cross-terms, ultimately achieving the goal. Simulation and experimental data processing results show that the method not only removes the cross-terms and maintains the excellent TF concentration of WVD but is also adaptive to multiple types and multi-distribution of signal with noise immunity. The overall performance is also substantially better than that of the classical cross-term suppression method.

However, the proposed method in this paper has some limitations when dealing with overlapping signals, which will be further investigated.

Competing interest

Xiukun Li is an editorial board member for the Journal of Marine Science and Application and was not involved in the editorial review, or the decision to publish this article. All authors declare that there are no other competing interests.

Figure 1 Three-dimensional spatial representation of auto-term and cross-term of LFM signal

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Figure 2 Multidimensional representation of WVD results

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Figure 3 Different orders of Rényi entropy values of the cross-term and auto-term

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Figure 4 Schematic of two-dimensional WVD of multicomponent signals in the TF plane (omitting cross-terms)

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Figure 5 Flowchart of the proposed method

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Figure 6 Schematic of the SRG algorithm

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Figure 7 Schematic of the proposed method

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Figure 8 Performance comparison of multitype signals (LFM, HFM, and CW) in different TFDs

(ⅰ), (ⅱ), (ⅲ), (ⅳ), and (ⅴ) are the results of WVD, PWVD, SPWVD, AOK-TFD, and the proposed method, respectively

Download: Full-Size Img

Figure 9 Performance comparison of multidistributed signals in different TFDs

(ⅰ), (ⅱ), (ⅲ), (ⅳ), and (ⅴ) are the results of WVD, PWVD, SPWVD, AOK-TFD, and the proposed method, respectively

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Figure 10 Performance comparison of experimental data in different TFDs

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Table 1 ApEn_S of multicomponent LFM signal

i	1	2	3	4	5	6
ApEn(b_i)	0.782	0.926	0.650	0.911	0.726	0.922

i	7	8	0	10	11	12
ApEn (b_i)	0.362	0.836	0.556	0.849	0.309	0.926

i	13	14	15	16	17	18
ApEn(b_i)	0.734	0.934	0.634	1.01	0.739	0.945

Table 2 Energy concentration comparison of multitype signals in different TFDs

EC	PWVD	SPWVD	AOK-TFD	The proposed method
S_a	6.76×10⁴	4.09×10⁴	5.14×10⁴	3.77×10²
S_b	1.33×10⁵	7.91×10⁴	7.93×10⁴	6.49×10³
S_c	2.16×10⁵	1.38×10⁵	8.39×10⁴	1.15×10⁴
S_d	4.28×10⁵	2.14×10⁵	1.39×10⁵	3.70×10⁴
S_e	6.90×10⁴	4.65×10⁴	4.03×10⁴	5.09×10³
S_f	1.31×10⁵	6.03×10⁴	7.56×10⁵	7.80×10³

Table 3 Energy concentration comparison of multidistributed signals in different TFDs

EC	PWVD	SPWVD	AOK-TFD	The proposed method
S'_a	2.08×10⁵	1.21×10⁵	8.25×10⁴	1.53×10²
S'_b	2.10×10⁵	1.22×10⁵	8.32×10⁴	1.21×10⁴
S'_c	4.80×10⁵	2.65×10⁵	1.22×10⁵	2.32×10⁴
S'_d	1.92×10⁵	1.13×10⁵	8.71×10⁴	1.02×10⁴
S'_e	9.50×10⁵	2.17×10⁵	1.16×10⁵	8.67×10³
S'_f	1.21×10⁵	7.32×10⁵	9.05×10⁴	9.13×10³

Table 4 Energy concentration comparison of experimental data in different TFDs

EC	PWVD	SPWVD	AOK-TFD	The proposed method
data	5.49×10⁵	2.59×10⁵	1.29×10⁵	2.54×10⁴

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Article Contents

Adaptive WVD Cross-Term Removal Method Based on Multidimensional Property Differences

https://doi.org/10.1007/s11804-024-00469-4

Corresponding author: Ge Yu yuge221@hrbeu.edu.cn

1 Introduction

2 Multidimensional property differences of WVD auto-term and cross-term

2.1 Property of auto-terms and cross-terms based on the two-dimensional TF plane

2.2 Property of auto-terms and cross-terms on one-dimensional sequences

3 Adaptive WVD cross-term removal method based on multidimensional property differences

4 Emulation signals and real experimental data verification

4.1 Emulation signals

4.1.1 Multitypes of emulated signals

4.1.2 Multidistributed emulated signals

4.2 Real experimental data verification

5 Conclusions

Publishing history

目录

Corresponding author:
Ge Yu yuge221@hrbeu.edu.cn