Uncorrelated case

Consider $K$ independent AWGN channels in parallel with a common power constraint $P$ i.e.

• $Y_{i}=X_{i}+Z_{i}, \ i=1,\ldots,K$ where $X_{i}\perp\!\!\!\perp Z_{j}, \ \forall i,j$
• $Z_{i}\sim\mathcal{N}(0,\sigma_{i}^{2}), \ i=1,\ldots,K$
• $Z_{1}\perp\!\!\!\perp\cdots\perp\!\!\!\perp Z_{K}$ and $$\sum\limits_{i=1}^{k}E[X_{i}^{2}]\le P$$ Note that the network of $K$ parallel channels is equivalent to a vector channel with input $X^{K}=(X_{1},\ldots,X_{K})$ and output $Y^{K}=(Y_{1},\ldots,Y_{K})$ where overall capacity is $$\begin{align*} C(P)&=\sup_{F_{X^{K}}:\sum\limits_{i=1}^{K}E[X_{i}^{2}]\le P}I(X^{K};Y^{K})\\ &\le \sum\limits_{i=1}^{K} \frac{1}{2}\log(2\pi e\mbox{Var}(Y_{i}))-\sum\limits_{i=1}^{K} \frac{1}{2}\log(2\pi e\sigma_{i}^{2}) \end{align*}$$but $Y_{i}=X_{i}+Z_{i}$ and $X_{i}\perp\!\!\!\perp Z_{i}$ so $$\begin{align*} \mbox{Var}(Y_{i})&=\mbox{Var}(X_{i})+\mbox{Var}(Y_{i})\\ &\le E[X_{i}^{2}]+\sigma_{i}^{2}\\ &= P_{i}+\sigma_{i}^{2} \end{align*}$$so we get that $$I(X^{K};Y^{K})\le\sum\limits_{i=1}^{K}\left[ \frac{1}{2}\log\left(1+ \frac{P_{i}}{\sigma_{i}^{2}}\right)\right]$$with equality iff $X_{i}\perp\!\!\!\perp X_{j}, \ \forall i,j$ and $X_{i}\sim\mathcal{N}(0,P_{i}), \ \forall i$. Hence $$\tag{$*$}C(P)=\sum\limits_{i=1}^{K} \frac{1}{2}\log\left(1+ \frac{P_{i}}{\sigma_{i}^{2}}\right)$$where $P_{i}=E[X_{i}^{2}]$ and $\sum\limits_{i=1}^{K}P_{i}\le P$.

Correlated case

Now let’s look to a similar case to Capacity of Parallel Gaussian Channels but now we deal with $K$ correlated AWGN Channels, with a common power constraint $P$. i.e.

• $Y_{i}=X_{i}+Z_{i}, \ i=1,\ldots,K$ where $X_{i}\perp\!\!\!\perp Z_{j}, \ \forall i,j$.
• $Z^{K}=(Z_{1},\ldots,Z_{K})=\underline Z^{T}\sim\mathcal{N}(\underline0,K_{\underline Z})$ where $K_{\underline Z}$ invertible. and $$\sum\limits_{i=1}^{k}E[X_{i}^{2}]=tr(K_{\underline X})\le P$$ and similar to how we applied the methodology with Optimal Power Allotment we find that network capacity is given by $$C(P)=\sum\limits_{i=1}^{K} \frac{1}{2}\log_{2}\left(1+ \frac{b_{ii}}{\lambda_{i}}\right)$$where $b_{ii}=\max\{0, \Theta-\lambda_{i}\}, \ i=1,\ldots,K$ with $\Theta$ chosen such that $\sum\limits_{i=1}^{K}b_{ii}=P$ and $$\lambda_{1},\ldots,\lambda_{K}$$are eigenvalues of $K_{\underline Z}$.