Theorem (Gaussian Rate Distortion Function)

Let $X\sim\mathcal{N}(0,\sigma^{2})$ and MSE distortion then the rate distortion function is defined as $$R(D)=\begin{cases} \frac{1}{2}\log \frac{\sigma^{2}}{D}&0<D\le \sigma^{2} \\ 0&D>\sigma^{2} \end{cases}$$and the distortion rate function is $$D(R)=\sigma^{2}2^{-2R}$$