The distortion of an optimal $N$-level quantizer is defined as $$D^{*}(N)=\min_{Q\in\mathcal{Q}_{N}}E[d(X,Q(X))]$$Let $X\sim f$, let our distortion measure be MSE then $$D^{*}(N)=\min_{y_{1}<\dots<y_{N}}\underbrace{ \sum_{i=1}^{N}\int\limits _{x_{i-1}}^{x_{i}}(x-y_{i})^{2}f(x) \, dx }_{ g(y_{1},\dots,y_{N}) }$$where $x_{i}=\frac{1}{2}(y_{i}+y_{i+1}), \ i=1,\dots,N-1$.

So we see that determining the optimal distortion, $D^{*}(N)$ and the optimal quantizer $Q^{*}$ involves the minimization of a real function of $N$ variables. This is computationally very complex hence we need a separate approach!

Companding Quantization

Now we fix $(a,b)$ and let $Q_{G,N}$ be the $N$-level companding realization of our quantizer. By the proposition $$D^{*}(N)=\min_{Q\in\mathcal{Q}_{N}}E[(X-Q(X))^{2}]=\min_{G}E[(X-Q_{G,N})^{2}]$$ Now we would like to solve for $\min_{G}E[(X-Q_{G,N})^{2}]$. We do this under the assumption that $N$ is large (i.e. “high-resolution conditions”).