I am hoping to build a function $f_{A,B,\alpha}(x \in \mathbf{R} ) \rightarrow y \in \mathbf{R}$ that serves as a positive signal compressor. The function acts on an input signal $x\left(t\right)$ one value at a time.
Note the behavior:
- For $\alpha$=0.5, we leave the input $x\left(t\right)$ xuntouched
- For $\alpha$=1, we stretch the input $x\left(t\right)$ linearly as much as possible
- For $\alpha$=0, we compress the input $x\left(t\right)$ to a flat nominal value (e.g. 1).
- For all other $\alpha$ we interpolate linearly.
How can I approach this problem? What analytical expression for $f_{A,B,\alpha}(x)$ would satisfy this?
