# A Decentralized Receiver in Gaussian Interference
## 1 Introduction
### Background  
The problem of communicating with the help of an array of distributed receivers has been studied in a variety of contexts, most generally as an instance of communications over a single-input multiple-output (SIMO) channel. A variety of SIMO channels have been extensively analyzed, but most analyses are subtly different than the problem considered here. Our work is done in the context of information theory and bounds achievable communications rates, while in contrast most existing studies work towards minimizing distortion metrics or bit error rates.  
Results presented here are a significant extension and generalization of the work in [3], where there is no treatment of the system's performance in the presence of an interferer. A tighter inner bound than all those in [3] is included in Theorem 7, and full proofs are provided for all the results.  
Achievable communications rates for this network topology were studied in References [1; 2], although their bounds do not directly apply to channels with a Gaussian interferer. The studies provide an example demonstrating the sub-optimality of a Gaussian broadcaster when no interference is present, and suggest that non-Gaussian signaling techniques are needed for this sort of network. In contrast, we demonstrate in Section 4 that an achievable rate using Gaussian signaling typically comes quite close to the system's upper bound in many practical regimes.  
Figure 1: A broadcast node speaks to many helper nodes, which forward limited amounts of information to a base node. Additive interference from neighboring systems is present in the link from broadcaster to helpers.  
The helpers in the system we have described seem like relay nodes but there is an important distinction. Here, the ensemble of links from each helper to the base are like those in a graphical network, available for design within distributional and rate constraints. This is not usual in the context of relay networks, including those studied in very general setting such as Reference [4]. Studies such as References [5; 6; 7] detailed using a collection of receivers as beam forming relays to a destination node in a network with structure similar to the system considered here. In our situation, each helper-to-base link is orthogonal to the others, so beam forming studies are not directly applicable.  
Performance of specific coding schemes for this system were studied in References [8; 9; 10]. In particular, [10] was able to identify a scheme that can perform to within 1.5 dB of allowing the base and receivers to communicate without constraints. Results in our study further work towards characterizing achievable rates of the system rather than designing and analyzing the performance of specific modulation schemes.  
The topology we consider is similar to many-help-one problems such as the \"Chief Executive Officer (CEO) problem\" posed by [PERSON] in Reference [11]. In this scenario, some node called the CEO seeks to estimate a source by listening to \"agent\" nodes which communicate to the CEO at fixed total rates. Many variations of this problem have been studied, for instance in Reference [12] in limit with number of agents. The focus of the CEO problem and most of its derivatives are to estimate a source to within some distortion (often mean squared error), whereas the focus here is on finding achievable rates for lossless communications.  
The noisy Slepian-Wolf problem [13] may be interpreted as communications in the opposite direction of our system: distributed, cooperative helpers have a message for a base, but their cooperation ability is limited and their modulations must be sent over a noisy channel.