Definition (Companding Quantization)

We will define a few tools here:

• Compressor: $G:\mathbb{R}\to(a,b)$ monotonically increasing and invertible.
• $Q_{\Delta}$: N-level uniform quantizer with support $(a,b)$ and step size $\Delta=\frac{b-a}{N}$.
• Expander: $G^{-1}:(a,b)\to \mathbb{R}$, the inverse of $G$.

Then we define the companding realization of our quantizer $Q$ as $$Q(x)=G^{-1}(Q_{\Delta}(G(x)))$$