Theorem

The KLT transform for a transform coder is optimal under High-Resolution Conditions, i.e. we assume - $\mathbf{X}$ is a Gaussian random vector with $E[\mathbf{X}]=0$ and $\mathbf{R_{X}}$ is nonsingular - We maintain the high-resolution approximation to the optimal scalar quantizer is valid with equality - Each $Q_{i}$ is the optimal $b_{i}$-bit quantizer for $Y_{i}$ - The $b_{i}$ are optimally allocated in accordance with the constraint $\sum_{i=1}^{k}b_{i}\le B$ We also observe that each $Y_{i}$ is Gaussian since $\mathbf{X}$ is Gaussian meaning each $Y_{i}$ has the same shape coefficient: $h_{i}=h_{g}$ in the high resolution formula $$E[(Y_{i}-\hat{Y}_{i})^{2}]=\sigma_{i}^{2}h_{g}2^{-2b_{i}}$$where we know from KL Transform Decorrelates X that $E[Y_{i}^{2}]=\sigma_{i}^{2}$ and $$h_{g}=\frac{1}{12}\left( \int\limits _{-\infty}^{\infty}\left( \frac{1}{\sqrt{ 2\pi }}e^{-x^{2}/2} \right)^{1/3} \, dx \right)^{3}$$ Then for any $k\times k$ orthogonal matrix $\mathbf{A}$, we have the following $$D_{\text{tc},\mathbf{A}}\ge D_{\text{tc},\mathbf{T}}=kh_{g}2^{-2\bar{b}}\det(\mathbf{R_{X}})^{1/k}$$