# A Decentralized Receiver in Gaussian Interference
## Appendix B Achievability of Bounds
By the extension of Theorem 1 in Appendix A of Reference [29], \[=\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+th(W_ {A})+I(\mathbf{Y}_{A};\mathbf{Y}_{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A})]\]9 (101) Further, by Theorem 6 in Reference [20], \[\geq\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+ th(W_{A})+tI(Y_{A};Y_{A}+W_{A})]\]0 (102) where \(Y_{n}\) has the distribution of any individual component of \(\mathbf{Y}_{n}\) and \(W_{Q,n}\sim\mathcal{CN}(0,G^{t}(\mathcal{L}_{n}^{t},\mathcal{P}_{n}^{t}))\) (in agreement with notation in Section 3.2). Thus, the \(t\) encoded messages produced by helper \(n\), \((U_{n}^{\ell})_{\ell}\), are within the LAN constraint for large enough block length \(T\). Helper \(n\in[1:N]\) forwards \((U_{n}^{\ell})_{\ell}\) to the base. * **Decoding** At the base, receive all of the lattice quantizations, \[\geq\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+ th(W_{A})+tI(Y_{A};Y_{A}+W_{A})]\]1 (103) Take \(A_{\varepsilon}^{T}(\mathbf{X},\mathbf{U}|\mathbf{W}_{Q})\) to be the collection of \[\geq\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+ th(W_{A})+tI(Y_{A};Y_{A}+W_{A})]\]2 (104) which are jointly \(\varepsilon\)-weakly-typical with respect to the joint distribution of \((\mathbf{X},\mathbf{U})\), conditional on all the dithers \((\tilde{\mathbf{W}}_{Q,n})_{n\in[0:N]}\). Weak- and joint-typicality are defined in Reference [14]. At the base, find a message estimate \(\hat{m}\in[1:2^{TR}]\) where \((\phi(\hat{m}),\mathbf{U})\in A_{\varepsilon}^{T}(\mathbf{X},\mathbf{U}|\mathbf{W}_{Q})\). Declare error events \(\mathcal{E}_{0}\) if \(M\) is not found to be typical with \(\mathbf{U}\), and \(\mathcal{E}_{1}\) if there is some \(\hat{m}\
eq M\) where \((\phi(\hat{m}),\mathbf{U})\in A_{\varepsilon}^{T}(\mathbf{X},\mathbf{U}|\mathbf{W}_{Q})\). * **Error analysis** By typicality and the law of large numbers, \(P(\mathcal{E}_{0})\to 0\) as \(t\to\infty\). In addition, \[\geq\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+ th(W_{A})+tI(Y_{A};Y_{A}+W_{A})]\]3 (43) \[\geq\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+ th(W_{A})+tI(Y_{A};Y_{A}+W_{A})]\]4 (44) \[\geq\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+ th(W_{A})+tI(Y_{A};Y_{A}+W_{A})]\]5 (45) \[\geq\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+ th(W_{A})+tI(Y_{A};Y_{A}+W_{A})]\]6 (46) \[\geq\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+ th(W_{A})+tI(Y_{A};Y_{A}+W_{A})]\]7 (47) \[\geq\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+ th(W_{A})+tI(Y_{A};Y_{A}+W_{A})]\]8 (48)  
\(\boldsymbol{Z}(\cdot)\) is defined as in Equation (25). Equation (47) follows by the data processing inequality. Equation (48) follows by combining the entropies into a mutual information, then using the worst additive noise lemma [27]. Thus, if \(R\) is chosen less than \(I(X;\boldsymbol{Z}(\boldsymbol{r}-\boldsymbol{1}\cdot\varepsilon))-4\varepsilon\) then \(P(\mathcal{E}_{0}\cup\mathcal{E}_{1})\!\to\!\!0\) as \(T\to\infty\). Mutual information is lower semi-continuous so \(I(X;\boldsymbol{Z}(\boldsymbol{r}-\boldsymbol{1}\cdot\varepsilon))-4\varepsilon\) can be made arbitrarily close to \(R_{G}(\boldsymbol{r})\). Theorem 4.2: _For a distributed receive system as described in Section 2 with noise covariance matrix \(\boldsymbol{\Sigma}\) and LAN constraint \(L\), a rate \(R_{DC}(L)\) is achievable._  
Proof. : Apply Theorem 4.2 with \(\lambda=0\) (proof shown below). Theorem 4.3: _For a distributed receive system as described in Section 2 with noise covariance matrix \(\boldsymbol{\Sigma}\) and LAN constraint \(L\) then \(R_{\overline{DC}}(L)\) is achievable._  
Proof. : It is enough to show that any component in the maximization from Equation (36) is achievable. Fix \(\lambda>0\), a helper rate vector \(\boldsymbol{r}\in\mathcal{R}_{\text{LAN}}(L)\), and a compression rate vector \(\boldsymbol{\rho}\in\mathcal{R}_{\text{DC}}^{\lambda}(\boldsymbol{r})\).