The model of fractal signal having a phase portrait in a form of two-scale Cantor set provides a possibility to describe many real signals generating by dynamical systems at the onset of chaos and to treat them in a unified way. The points in the parameter plane of the fractal signal are outlined, which correspond to these real types of dynamical behavior. Simple electronic circuit admitting experimental realization is suggested, that generates the fractal signal with tunable parameters. Renormalization group analysis is developed for the case of period-doubling system forced by the fractal signal. It is shown that a bifurcation takes place in the RG equation, and the behavior at the onset of chaos may be described by either Feigenbaum or non-Feigenbaum fixed point solutions. The results of numerical simulations are presented to illustrate the scaling properties of the dynamics forced by the fractal signal.