Simplifications…

Before we state the theorem we must state some high-resolution approximations that are made. 1. High-Resolution Conditions: Each $X_{i}$ has pdf $f_{i}$ and $$W_{i}(b)=\frac{1}{12}\lVert f_{i} \rVert_{\frac{1}{3}}2^{-2b}, \ \ i=1,\dots,k$$ 2. Let $\sigma_{i}^{2}=\text{Var}(X_{i})$ and $\tilde{X}_{i}=\frac{X_{i}}{\sigma_{i}}$. Then $\tilde{X}_{i}$ has unit variance (i.e. $\text{Var}(\tilde{X}_{i})=1$) and its pdf $\tilde{f}_{i}(x)=\sigma_{i}f_{i}(\sigma_{i}x)$ satisfies $$\lVert f_{i} \rVert _{\frac{1}{3}}=\lVert \tilde{f}_{i} \rVert _{\frac{1}{3}}\sigma_{i}^{2}$$ 3. Hence $$\begin{align*} W_{i}(b)&=\underbrace{ \frac{1}{12}\lVert \tilde{f}_{i} \rVert _{\frac{1}{3}} }_{ h_{i} }\sigma_{i}^{2}2^{-2b}\\ &=h_{i}\sigma_{i}^{2}2^{-2b} \end{align*}$$ ## Note $h_{i}$ is invariant to scaling and hence, is the same for all $X_{i}$ so we treat it as a constant. # Theorem Our optimal distortion for the bit allocation problem is defined as $$D(\mathbf{b})=\sum^{k}_{i=1}h_{i}\sigma_{i}^{2}2^{-2b_{i}}$$and is minimized subject to our bit constraint $\sum_{i=1}^{k}b_{i}\le B$ if and only if $$b_{i}=\bar{b}+\frac{1}{2}\log_{2}\frac{\sigma_{i}^{2}}{\rho^{2}}+\frac{1}{2}\log_{2} \frac{h_{i}}{H}, \ \ i=1,\dots,k$$where $$\bar{b}=\frac{B}{k}\quad\rho^{2}=\left( \prod_{i=1}^{k}\sigma_{i}^{2} \right)^{\frac{1}{k}\quad}H=\left( \prod_{i=1}^{k}h_{i} \right)^{1/k}$$i.e. $\rho^{2}$ is the geometric mean of variance and $H$ is the geometric mean of our shape.

Remark

We see that the optimal $\mathbf{b}$ must satisfy $$\sum_{i=1}^{k}b_{i}=B$$