Fractal renewal processes generate 1/fnoise

1/${\mathit{f}}^{\mathit{D}}$ noise occurs in an impressive variety of physical systems, and numerous complex theories have been proposed to explain it. We construct two relatively simple renewal processes whose power spectral densities vary as 1/${\mathit{f}}^{\mathit{D}}$: (i) a standard renewal point process, with 0<D<1, and (ii) a finite-valued alternating renewal process, with 0<D<2. The resulting event number statistics, coincidence rates, minimal coverings, and autocorrelation functions are shown also to follow power-law forms. These fractal characteristics derive from interevent-time probability density functions which themselves decay in a power-law fashion. A number of applications are considered: trapping in amorphous semiconductors, electronic burst noise, movement in systems with fractal boundaries, the digital generation of 1/${\mathit{f}}^{\mathit{D}}$ noise, and ionic currents in cell membranes.