Cubic Soft-Clipper Spectrum

The cubic soft-clipper, like any polynomial nonlinearity, is defined directly by its series expansion:

$$f(x) = \left\{\begin{array}{ll} -\frac{2}{3}, & x\leq -1 \\ [5pt] x-\frac{x^3}{3}, & -1 \leq x \leq 1 \\ [5pt] \frac{2}{3}, & x\geq 1 \\ \end{array} \right.$$ (7.19)

In the absence of hard-clipping ( $\left\vert x\right\vert\leq1$ ), bandwidth expansion is limited to a factor of three. This is the slowest aliasing rate obtainable for an odd nonlinearity. Note that smoothing the ``corner'' in the clipping nonlinearity can reduce the severe bandwidth expansion associated with hard-clipping.