Scaling Region of Weierstrass-Mandelbrot Function: Improvement Strategies for Fractal Ideality and Signal Simulation

Abstract

Fractal dimension (D) is widely utilized in various fields to quantify the complexity of signals and other features. However, the fractal nature is limited to a certain scope of concerned scales, i.e., scaling region, even for a theoretically fractal profile generated through the Weierstrass-Mandelbrot (W-M) function. In this study, the scaling characteristics curves of profiles were calculated by using the roughness scaling extraction (RSE) algorithm, and an interception method was proposed to locate the two ends of the scaling region, which were named corner and drop phenomena, respectively. The results indicated that two factors, sampling length and flattening order, in the RSE algorithm could influence the scaling region length significantly. Based on the scaling region interception method and the above findings, the RSE algorithm was optimized to improve the accuracy of the D calculation, and the influence of sampling length was discussed by comparing the lower critical condition of the W-M function. To improve the ideality of fractal curves generated through the W-M function, the strategy of reducing the fundamental frequency was proposed to enlarge the scaling region. Moreover, the strategy of opposite operation was also proposed to improve the consistency of generated curves with actual signals, which could be conducive to practical simulations.