# A Decentralized Receiver in Gaussian Interference
## 3 Bounds on Communications Rates
### Achievable Rate through Gaussian Distortion  
Rate-distortion theory shows that to encode a Gaussian source \(Y\sim\mathcal{N}(0,\sigma^{2})\) with minimum rate such that distortion does not exceed some maximum allowable mean squared error \(\mathcal{D}>0\), the rate must be at least  
\[\mathcal{R}(\mathcal{D})=\log_{2}(\sigma^{2})-\log_{2}(\mathcal{D}). \tag{21}\]  
To achieve this rate, the encoding operation must emulate the following test channel (Reference [14]):  
(22)  
where \(\alpha=\sqrt{1-\mathcal{D}/\sigma^{2}}\) and \(\beta=\sqrt{\mathcal{D}}\). Dithered lattice quantization [20] is one method that can be used to realize such a quantizer in limit with block length in addition to having other nice analytic properties. Using this method, in limit with block length, the base's estimates for a helper's observation at any single timepoint can be made to approximate  
\[Z_{n}(r_{n})\triangleq Y_{n}+W_{Q,n}(r_{n}) \tag{23}\]  
where \(W_{Q,n}(r_{n})\) is independent Gaussian distortion with variance some function of the helper's rate \(r_{n}\). We will now derive this function. Define  
\[W_{Q}(\mathbf{r})\triangleq(W_{Q,1}(r_{1}),\ldots,W_{Q,N}(r_{n}))\in\mathbb{C}^{N \times 1}, \tag{24}\]  
and the vector of \(Z_{n}\)s corresponding to this choice of \(W_{Q,n}\) as:  
\[\mathbf{Z}(\mathbf{r})\triangleq\mathbf{Y}+W_{Q}(\mathbf{r}). \tag{25}\]  
The subscript \(Q\) here is a decoration to distinguish quantization distortion terms \(W_{Q},W_{Q,n}\) from environment noise, \(\mathbf{W},W_{n}\). If the \(n\)th helper is to forward information to the base at rate \(r_{n}\), then the amount of distortion in the helper's encoding under this strategy can be determined by setting Equation (21) equal to \(r_{n}\) and solving for \(\mathcal{D}\), with \(\sigma^{2}\) equal to the helper observation's variance, \(\|h_{n}\|^{2}+\mathbf{\Sigma}_{n,n}\). Scaling the output of the test channel the helper emulates (Equation (22)) by \(1/\alpha\) causes it to equal the helper's observation plus independent Gaussian noise with variance \((\beta/\alpha)^{2}=\mathcal{D}/(1-\mathcal{D}/\sigma^{2})\). This means that, in such a system, the distortion from the helpers is equivalent to adding independent additive Gaussian noise with variance:  
\[\mathrm{Var}(W_{\mathcal{Q},n}(r_{n}))=\frac{\|h_{n}\|^{2}+\mathbf{\Sigma}_{n,n}}{2 ^{r_{n}}-1}. \tag{26}\]  
By Equations (1) and (26), all the noise and distortion on the signal present in the helpers' messages to the base are summarized by the vector \(\mathbf{W}+W_{\mathcal{Q}}\) with covariance matrix  
\[\mathbf{D}(\mathbf{r})\triangleq\mathbf{\Sigma}+\mathrm{diag}\left(0,\frac{\|h_{1}\|^{ 2}+\mathbf{\Sigma}_{1,1}}{2^{r_{1}}-1},\ldots,\frac{\|h_{N}\|^{2}+\mathbf{\Sigma}_{N, N}}{2^{r_{N}}-1}\right). \tag{27}\]  
We refer to this system as a _Gaussian distortion system_. A block diagram of this system is shown in Figure 3. This system is equivalent to a multi-antenna Gaussian receiver and its achievable rate has a closed form:  
\[R_{G}(\mathbf{r})\triangleq I(X;\mathbf{Z}(\mathbf{r}))=\log_{2}\left[\frac{|\mathbf{D}(\bm {r})+\mathbf{h}\mathbf{h}^{\dagger}|}{|\mathbf{D}(\mathbf{r})|}\right]. \tag{28}\]  
**Theorem 2**. : _For a distributed receive system as described in Section 2 with noise covariance matrix \(\mathbf{\Sigma}\), LAN constraint \(L\) and fixed average helper quantization rates \(\mathbf{r}\in\mathcal{R}_{\mathrm{LAN}}(L)\), then a rate \(R_{G}(\mathbf{r})\) 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 an encoding 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_ \(\mathbf{r},\mathbf{h},\mathbf{\Sigma},L\)_._  
All the details of this derivation are formalized in Appendix B.