# A Decentralized Receiver in Gaussian Interference
## 3 Bounds on Communications Rates
### Achievable Rate through Distributed Compression  
Helper \(n\) for \(n\in[1:N]\) in a realization of the Gaussian distortion system described in Section 3.2, on average and with long codes, will produce an \(r_{n}\)-bit encoding of its observation that can be decoded to produce the helper observation with additional approximately Gaussian noise. Since all the helper's quantizations contain the same signal component (and possibly the same interference), they are correlated and can be compressed before forwarding to allow for less noise to be introduced in quantization. The Slepian-Wolf theorem [PERSON] and [PERSON] (1992) shows that if the LAN is such that helpers can encode at rates \(\boldsymbol{\rho}\triangleq(\rho_{1},\ldots,\rho_{N})\), then the helpers can losslessly convey the encodings to the base at lower rates \(\boldsymbol{r}\preceq\boldsymbol{\rho}\), as long as \(\boldsymbol{r}\) and the encodings \(\boldsymbol{U}\) satisfy the conditions that for all subsets \(S\subseteq[1:N]\), then:  
\[H(\boldsymbol{U}_{S}|\boldsymbol{U}_{S^{C}},Y_{0},\boldsymbol{Q})<\sum_{n\in S }r_{n}. \tag{33}\]  
Note that \(Y_{0}\) is always included in the conditioning because \(Y_{0}\) is available at the base node in full precision. In the Gaussian distortion setting described right before Equation (23), \(H(\boldsymbol{U}_{S}|\boldsymbol{U}_{S^{C}},Y_{0},\boldsymbol{Q})\) can be made close to \(I(Y_{S};\boldsymbol{Z}_{S}(\boldsymbol{\rho})|\boldsymbol{Z}_{S^{C}}( \boldsymbol{\rho}),Y_{0})\) with time expansion. The details of this are given in Lemma 2. Maximizing over the LAN constraint in Equation (9), the following rate is achievable:  
\[R_{DC}(L)\triangleq\max_{\boldsymbol{r}\in\mathcal{R}_{\text{ LAN}}(L)}\,\max_{\boldsymbol{\rho}\in\mathcal{R}_{DC}(\boldsymbol{\rho})}R_{G}( \boldsymbol{\rho}) \tag{34}\]  
where  
\[\mathcal{R}_{DC}(\mathbf{r})=\left\{\mathbf{\rho}:\forall S\subseteq[1:N],I\left(\mathbf{Y}_{S} ;\mathbf{Z}_{S}(\mathbf{\rho})|\mathbf{Z}_{S^{C}}(\mathbf{\rho}),Y_{0}\right)<\sum_{n\in S}r_{n} \right\}. \tag{35}\]  
\(\mathbf{Z}(\cdot)\) in Equation (35) is defined as in Equation (25). We can then state the following:  
Theorem 3.1: _For a distributed receive system as described in Section 2 with noise covariance matrix \(\mathbf{\Sigma}\) and LAN constraint \(L\), \(R_{DC}(L)\) is achievable with the following facilities:_  
* _The transmitter and base coordinate a codebook of appropriate rate and distribution._
* _Each_ \(n\)_-th helper is informed of a quantization rate_ \(\rho_{n}\) _and binning rate_ \(r_{n}\)_, (optimal choice derived from_ \(\mathbf{h},\mathbf{\Sigma},L\)_)._
* _Each helper coordinates a random dither with the base._
* _The base is informed of_ \(\mathbf{\rho},\mathbf{r},\mathbf{h},\mathbf{\Sigma},L\)_._  
The base can unambiguously decompress the helper's compressed encodings with low probability of error if and only if \(\mathbf{\rho}\) is chosen such that Equation (33) is satisfied. However, in analog to Corollary 1 from Reference [1], the rate in 6 can be improved by expanding \(\mathcal{R}_{DC}(\mathbf{r})\) to include some of the \(\mathbf{\rho}\) where the base cannot perform unambiguous decompression. This helps because even if the encoding rates in \(\mathbf{\rho}\) are chosen outside of \(\mathcal{R}_{DC}(\mathbf{r})\) so that helpers cannot convey \(\mathbf{U}\) to the base unambiguously, some extra correlation with \(X\) is retained through this distortion. Theorem 3.2: _For a distributed receive system as described in Section 2 with noise covariance matrix \(\mathbf{\Sigma}\) and LAN constraint \(L\), the following rate is achievable:_  
\[R_{\overline{DC}}(L)\triangleq\max_{ \begin{subarray}{c}\mathbf{r}\in\mathcal{R}_{ \mathrm{LAN}}(L),\\ \lambda>0\end{subarray} }\max_{ \begin{subarray}{c}\mathbf{\rho}\in\mathcal{R}_{DC} ^{\lambda}(\mathbf{r})\end{subarray}