Definition (High-resolution conditions)

In the context of a companding realization we define high-resolution conditions as the following. Fix the compressor $G$, and let $N\to \infty$. We now state the following assumptions:

1. $N$ is large, $\Delta_{i}=(x_{i}-x_{i-1})$ is small for $i=2,\dots,N-1$
2. $X\sim f$, where $f$ is continuous pdf
3. $G:\mathbb{R}\to(a,b)$ is continuously differentiable.

After this and a lot of computation we come to the result that for MSE and $N$ large $$D(Q_{G,N})\approx \frac{B^{2}}{12N^{2}}\int\limits _{-\infty}^{\infty} \frac{f(x)}{G'(x)^{2}} \, dx$$