In this age of information technology, signal processing plays a pivotal role as a key technology. By processing the collected signals, we can extract valuable information from complex data and address various practical problems. The development of signal processing not only drives the advancement of science and technology but also profoundly influences our daily lives, spanning multiple fields such as industrial engineering [1], economics [2], and hydroacoustic [3]. The core of signal processing lies in feature extraction [4]. However, influenced by various factors such as complex environments and equipment self-noise, collected signals often exhibit nonlinear characteristics [5], making traditional feature extraction methods less advantageous in processing real measured signals. Therefore, it is imperative to apply new features that can effectively characterize nonlinear signals.
Nonlinear dynamical features are used to describe the characteristics of nonlinear behavior within dynamical systems, and many scholars have applied them in the processing of real measured signals [6 ~ 9]. Commonly employed nonlinear dynamical features include Lempel-Ziv complexity (LZC) [10], Lyapunov exponent [11], and entropy [12]. Specifically, LZC can characterize the rate at which new patterns emerge in a signal, whereas entropy quantifies the uncertainty of the signal. In contrast to the high computational complexity associated with Lyapunov exponent calculation, LZC and entropy not only have lower computational complexity but also vary with the complexity of the signal. Specifically, as the signal becomes more complex, both entropy and LZC values tend to increase [13].
Complex networks are another commonly used method in signal processing. They define nodes and links in a specific way, where nodes represent individuals or elements in the system, and links in the network represent connections or relationships between nodes [14 ~ 16]. Unlike metrics such as LZC and entropy, which directly act on signals, complex networks calculate various relationships between nodes and links, and the obtained features can be used to describe the complex structures and dynamic behaviors of various systems. Therefore, complex networks are widely applied in fields such as mechanical engineering, biomedicine, and hydroacoustics [17–19].
Currently, commonly used complex networks include the visibility graph (VG) [20], recurrence network (RN) [21], and transition network (TN) [22]. In VG, each sampling point in the signal data is regarded as a node, and they are connected based on visibility, offering the advantage of not requiring parameter selection [23]. On the other hand, both RN and TN use the embedding dimension to segment signal data as different nodes. However, RN establishes links through similarity calculations, whereas TN considers adjacent nodes as connected links, thereby preserving temporal causality [24].
However, the aforementioned complex networks also face certain issues. For instance, VG encounters potential data loss and exhibits limited applicability to large datasets [24]. Incorrect threshold selection in RN can significantly impact the network, and during network transformation [25], TN may lead to the loss of effective information in the original signal due to the settings of embedding dimension and time delay. To illustrate, consider the pattern \({\pi }_{123}\), based on the temporal sequence, the next pattern can only be \({\pi }_{23x}\). While setting a larger time delay can mitigate this phenomenon, it may also affect subsequent patterns.
In addition, existing complex network metrics, such as the global clustering coefficient [26], network transitivity [27], and average path length [28], although capable of providing a profound understanding of the overall structure, properties, and functionality of networks [29], still have certain limitations in quantifying network irregularities. This is also a challenge faced by the application of complex network theory in underwater acoustic signal processing.
In response to the existing issues, we propose a novel solution that combines the concepts of complex networks and information entropy, leveraging the advantages of both to characterize the complexity of nonlinear signals from the perspective of network information distribution. Specifically, inspired by the dispersion pattern [30], we initially use the values of the cumulative distribution function after scaling as network nodes. Following the principles of Markov chains, we treat the relationships between connected nodes as links, thus introducing the concept of dispersion complex network (DCN). Subsequently, we calculate the distribution probabilities of each node and link in the DCN and use the Shannon entropy formula to obtain the DCN-transition entropy (DCN-TE). While constructing DCN, we do not employ embedding dimensions for pattern segmentation of nodes to reduce information redundancy. In the computation of TE, we also simultaneously consider the importance of both network nodes and links. Therefore, compared with other metrics, DCN-TE exhibits more stable and accurate characteristics in signal processing. Subsequently, experimental validations are conducted on simulated chaotic models and two types of real measured hydroacoustic signal datasets, further confirming the outstanding performance of DCN-TE in nonlinear signal processing.
The main contribution of this paper is the introduction of a new complexity metric, DCN-TE, which demonstrates excellent performance in nonlinear signal processing. Furthermore, the structure of this paper is as follows: Section 2 progressively introduces the theoretical steps of DCN-TE and discusses its parameters. Section 3 uses simulation experiments to compare DCN-TE's ability to detect dynamic changes and differentiate chaotic models, while also evaluating its computational cost. In Section 4, the practical application capabilities of DCN-TE are validated using two types of real measured hydroacoustic datasets. Section 5 summarizes the entire paper.