# A Decentralized Receiver in Gaussian Interference
## 2 Problem Setup
Formally, the probability distribution of \((\mathbf{Q},\mathbf{X},\mathbf{Y},\mathbf{U})\) factors: \[P_{(\mathbf{Q},\mathbf{X},\mathbf{Y},\mathbf{U})}=P_{\mathbf{Q}}\cdot P_{\mathbf{X}|\mathbf{Q}}\cdot P_{ \mathbf{Y}|\mathbf{X}}\cdot\prod_{n=1}^{N}P_{U_{n}|\mathbf{Y}_{n},\mathbf{Q}}.\] (6)
* Quantizations are within the LAN constraint: \[\frac{1}{T}H(U_{n}|\mathbf{Q})\leq r_{n},\ n\in[1:N].\] (7) The vector of the helper's quantization rates is denoted: \[\mathbf{r}\triangleq(r_{1},\ldots,r_{N})\in\mathbb{C}^{(N+1)\times 1}.\] (8)  
Each \(U_{n}\) is conveyed precisely to the base over a Local Array Network (LAN). We assume the LAN only supports a limited amount of communication between helpers and the base. For instance, the helpers must all share a small frequency band. We model this by asserting that feasible rate vectors \(\mathbf{r}\) belong to a bounded set \(\mathcal{R}_{\text{LAN}}(L)\) with a sum-capacity \(L>0\):  
\[\mathcal{R}_{\text{LAN}}(L)\triangleq\left\{\mathbf{r}\left|\sum_{n=1}^{N}r_{n} \leq L,\ r_{n}\geq 0\right.\right\}. \tag{9}\]  
We refer to the condition that \(\mathbf{r}\in\mathcal{R}_{\text{LAN}}(L)\) as the _LAN constraint_. The base reconstructs an estimate \(\mathbf{\hat{X}}\) of \(\mathbf{X}\) using its own full-precision observation \(\mathbf{Y}_{0}\) and side information \(\mathbf{U}\) from the helpers, then recovers an estimate \(\hat{M}\) of message \(M\) from \(\mathbf{\hat{X}}\). A block diagram of the system is shown in Figure 2.  
Channel \(h\) and noise covariance \(\Sigma\) are assumed to be static throughout the transmission of each message and are known to all the receive nodes. Both the transmitter and the receivers are assumed to have knowledge of the set of feasible rates, \(\mathcal{R}_{\text{LAN}}(L)\). The environment determines:  
* channel fades \(h\),
* noise covariance \(\Sigma\) and
* maximum LAN throughput \(L\). Available for design are:  
* quantizers \(Q\),
* message modulation \(\{\mathbf{x}_{m}\}_{m\in[1:2^{TR}]}\),
* rates each helper should send to the base \(\mathbf{r}\) and
* fusion and decoding methods at the base to produce \(\hat{M}\). We say a communication rate \(R\) is _achievable_ if for any \(\varepsilon>0\) then for large enough \(T\) there is some distribution of quantizers \(Q\) satisfying \(\frac{1}{T}H(Q_{k}(Y_{k})|\mathbf{Q})\leq r_{k}\) for each \(k\), encoders \(\phi_{\mathbf{Q}}:[1:2^{TR}]\to\mathcal{X}^{T}\) and decoders \(f_{\mathbf{Q}}\) where:  
\[P(f_{\mathbf{Q}}(\mathbf{U})\
eq M)<\varepsilon. \tag{10}\]  
Theorem 3.1: _The maximum communications rate achievable with arbitrarily low probability of error is:_  
\[C=\sup_{T}\max_{ \begin{subarray}{c}(\mathbf{X},\mathbf{Y},\mathbf{Q},\mathbf{U})\sim P\in \mathcal{S}_{T}:\\ \mathbb{E}|X|^{2}\leq 1\end{subarray} }\frac{1}{T}I(\mathbf{X};\mathbf{Y}_{0},\mathbf{U}| \mathbf{Q}) \tag{11}\]  
_where \(\mathcal{S}_{T}\) is the collection of joint distributions with block length \(T\) that satisfy conditions in Equations (6) and (7) for some \(\mathbf{r}\in\mathcal{R}_{\text{LAN}}(L)\)._  
Proof: \(C\) is achievable through the usual noisy channel coding argument [14]. For any achievable rate \(R\), applying [PERSON]'s inequality to \(H(M|\mathbf{Q})\) gives \(R\leq C\). In the sequel, we bound \(C\) from Theorem 3.1 with computable expressions. Figure 2: The system in consideration. A message \(M\) is broadcast as \(\mathbf{X}\) with average power \(1\). The signal is received by a base and \(N\) helper nodes, occluded by correlated AWGN \(W\). Helper \(n\) for \(n\in[1:N]\) quantizes \(\mathbf{Y}_{n}\) through \(Q_{n}\) to produce an \(r_{n}\)-bit summary \(U_{n}\). The quantized observations \(\mathbf{U}\) and the base’s full-precision reception \(\mathbf{Y}_{0}\) are combined to produce an estimate \(\hat{\mathbf{X}}\) of \(\mathbf{X}\), then an estimate \(\hat{M}\) of \(M\).