Definition (Gain of Transform Coding)

Given the TC distortion, $D_{\text{tc},\mathbf{T}}$. We compare this to the distortion of a scalar quantizer without transform coding where we assume equal variance $E[X_{i}^{2}]=\sigma_{X}^{2}$ for all $i$, and each $X_{i}$ independently and optimally scalar quantized with optimal bit allocation $b_{i}=\bar{b}=\frac{B}{k}, i=1,\dots,k$ and MSE $$D_{\text{PCM}}=\sum_{i=1}^{k}E[(X_{i}-Q_{i}(X_{i}))^{2}]=kh_{g}\sigma_{X}^{2}2^{-2\bar{b}}$$ we have the gain defined as follows $$\frac{D_{\text{PCM}}}{D_{\text{tc}}}=\frac{\sigma_{X}^{2}}{\left( \prod_{i=1}^{k}\lambda_{i} \right)^{1/k}}=\frac{ \frac{1}{k}\sum_{i=1}^{k}\lambda_{i}}{\left( \prod_{i=1}^{k}\lambda_{i} \right)^{\frac{1}{k}}}\ge1$$