# A Decentralized Receiver in Gaussian Interference
## 2 Problem Setup  
Throughout the paper, we use the notation in Table 1. A broadcaster seeks to convey a message to a base at some rate \(R>0\) bits per time period. The message \(M\) is uniformly distributed along \([1:2^{TR}]\) over some \(T\in\mathbb{N}\) time periods. Our goal is to determine the greatest rate \(R\) at which the base can recover \(M\) with low probability of error. Before communicating, the broadcaster and base agree on a modulation scheme \(\phi:[1:2^{TR}]\rightarrow\mathbb{C}^{T}\), where each possible message \(m\in[1:2^{TR}]\) is mapped to a \(T\)-length signal through \(\phi\). Each of these signals is denoted \(\mathbf{x}_{m}=(x_{m,1},\ldots,x_{m,T})=\phi(m)\), and it is chosen within the broadcaster's power constraint: \(\frac{1}{T}\sum_{\ell=1}^{T}|x_{m,\ell}|^{2}\leq 1\). To send message \(M\), the broadcaster transmits \(\mathbf{x}_{M}\), which we denote \(\mathbf{X}=(X_{1},\ldots,X_{T})=\mathbf{x}_{M}\). \begin{table}
\begin{tabular}{r l} \hline \(\mathcal{CN}(0,\mathbf{\Sigma})\) & Complex circularly symmetric normal distribution with zero mean and Hermitian covariance matrix \(\mathbf{\Sigma}\) \\ \(I(\cdot\cdot),H(\cdot),h(\cdot)\) & Mutual information, Shannon entropy and differential entropy \\ \(\mathbf{A}^{\dagger},\mathbf{a}^{\dagger}\) & Conjugate transpose of matrix \(\mathbf{A}\), vector \(\mathbf{a}\) \\ \(\mathbf{a}_{S}\); \(\mathbf{S}\subseteq T\) & Vector with components indexed by \(S\) \\ \(|\mathbf{A}|\) & Determinant of a matrix \(\mathbf{A}\) \\ \(|a:b|\) & Integers from \(a\) to \(b\), inclusive \\ \(\mathbf{a}\succeq 0\) & Vector \(\mathbf{a}\) with non-negative elements \\ \(\mathbf{A}\succeq 0\) & Positive semidefinite matrix \(\mathbf{A}\) \\ \(\mathbf{1}_{N}\) & \(N\)-row vector of \(1\) s \\ \(\mathbf{A}_{j,k}\) & Element on row \(j\), column \(k\) of a matrix \(\mathbf{A}\) \\ \(\mathrm{diag}(\mathbf{a})\) & Diagonal matrix where the \((i,i)\)th diagonal element is the \(i\)th element of vector \(\mathbf{a}\) \\ \(\mathbf{1}_{n\times n}\) & \(n\times n\) identity matrix \\ \(\|x\|\) & Vector magnitude/2-norm \\ \(\mathbb{R}_{+}\) & Nonnegative real numbers \\ \hline \end{tabular}
\end{table} 
Table 1: Notation and terminology. The signal is observed by \(N+1\) single-antenna receive nodes through a static flat fading channel and additive white Gaussian noise. Enumerating these receivers \(0\)-\(N\), we identify the base as the \"0 th\" receiver, and call the rest _helpers_. For \(n\in[0:N]\), at each time \(t\), the \(n\)th receiver observes  
\[Y_{n,t}=h_{n}X_{t}+W_{n,t}, \tag{1}\]  
where  
* the channel \(\mathbf{h}\triangleq\left[ \begin{smallmatrix}h_{0}\\ \vdots\\ h_{N}\end{smallmatrix} \right]\in\mathbb{C}^{(N+1)\times 1}\) is constant over time \(t\), and
* the noise \((W_{0,t},\ldots,W_{N,t})\sim\mathcal{CN}(0,\mathbf{\Sigma})\) has covariance \(\mathbf{\Sigma}\) across receivers, and is independent over \(t\). The receiver observations are written as vectors:  
\[\mathbf{Y}_{n} \triangleq(Y_{n,1},\ldots,Y_{n,T})\in\mathbb{C}^{T}, \tag{2}\] \[\mathbf{Y} \triangleq(\mathbf{Y}_{0},\mathbf{Y}_{1},\ldots,\mathbf{Y}_{N})\in(\mathbb{C} ^{T})^{N+1}. \tag{3}\]  
Helper \(n\) for \(n\in[1:N]\) employs a (possibly randomly chosen) vector quantizer \(Q_{n}\) on its received sequence \(\mathbf{Y}_{n}\) to produce a coarse \(r_{n}\)-bitrate summary of its observations, \(U_{n}\triangleq Q_{n}(\mathbf{Y}_{n})\). The vector of helper messages is denoted:  
\[\mathbf{U} \triangleq(U_{1},\ldots,U_{N}) \tag{4}\] \[=(Q_{1}(\mathbf{Y}_{1}),\ldots,Q_{N}(\mathbf{Y}_{N})) \tag{5}\]  
and the quantizers \((Q_{1},\ldots,Q_{N})=\mathbf{Q}\) satisfy two properties:  
* Every node in the system is informed of each quantizer's behavior, and quantizers produce their compression using only local information.