Real scalar Gaussian multicast relay network and power constraints

A real scalar Gaussian relay network has node set $[N]$ for an integer $N\ge 1$, source set $\mathcal{S}\subseteq [N]$, destination set $\mathcal{D}\subseteq [N]$, one real input $X_i$ and output $Y_i$ at each node, and channel equations $Y_j={\sum }_{i=1}^N{g}_{ji}{X}_i+Z_j,j\in [N],$ where the gains $g_{ji}\in \mathbb{R}$ are fixed and the noises $Z_j$ are independent $N(0,1)$ random variables, independent across channel uses and of all source messages. At each use the current noise vector is independent of all past observations and the current inputs; future inputs may depend on past noises through feedback. Each node has a nonnegative power budget $P_i\ge 0$ and average power constraint $\frac{1}{n}{\sum }_{t=1}^n\mathbb{E}[X_{i,t}^2]\le P_i$. Each source $i\in \mathcal{S}$ has an independent uniform message $M_i$ in a finite nonempty set ${\mathcal{M}}_i$. For every non-source node $i\mathrm{\notin }\mathcal{S}$, set ${\mathcal{M}}_i=\{\ast \}$ and $M_i=\ast$, so its encoder has no message information. A length-$n$ causal code uses measurable encoders $X_{i,t}=f_{i,t}(M_i,Y_i^{t-1})$ and destination decoders based on $(Y_d^n,M_d)$, where $M_d$ is its own source message when $d\in \mathcal{S}$ and a singleton otherwise; multicast means every destination must decode every other source message, while its own message is already known. The average error probability is the probability that any required destination estimate is wrong. Write $G_{B,A}$ for the matrix with rows indexed by $B$ and columns by $A$ whose entries are $g_{ji}$.

Rates in this page's Gaussian formulas are measured in bits per use, $R_i=n^{-1}{\log }_2\mathrm{\mid }{\mathcal{M}}_i\mathrm{\mid }$.