The Effects of Finite Resolution on Radar Images of Fractal Profiles

In this paper a first step toward a complete model of the fractal imaging process is taken: for the sake of simplicity the mathematical details are here provided for a fractal profile with topological dimension equal to one. In particular, we show how the signal backscattered from a fractal profile modeled as a fractional Brownian motion (fBm) stochastic process is strictly linked to an associated fractional Gaussian noise (fGn) process. We compute in closed form the power density spectrum of the received signal in the simplified hypothesis of a linear dependence of the backscattered signal on the profile derivative process. Our results apply to physical fBm processes, as dictated by the low-pass filtering introduced by both the incident electromagnetic field wavelength and the finite sensor resolution. In the last section a numerical study of the abovementioned is also provided.