Variable-step multi-scale fractal dimension and its application to ship radiated noise

Fractal dimension (FD) is a common tool for detecting dynamic changes in signals, however, it can only reflect the signal information at a single scale and has certain limitations. Currently, several scholars have proposed a few indexes based on multi-scale coarse-grained processing (MCP) and refined composite MCP (RCMCP). Nevertheless, the coarse-grained characteristics of MCP and RCMCP lead to their inadequate extraction of information from time series in the calculation process. To address this issue, variable-step multi-scale FD (VSMFD) was proposed, which not only reflects signal information from multiple scales, but also is more accurate and comprehensive when extracting signal information. The results of three sets of simulation experiments indicate that VSMFD is the most sensitive to the dynamic transform of chirped signals, and has the best separability for simulated noise signals and chaotic signals. In addition, both sets of ship radiated noise experiments indicate that VSMFD performs better in distinguishing ship radiated noise than indexes based on MCP and RCMCP, and has broad application prospects in the field of underwater acoustic signal processing.