Z干扰信道的容量问题

Interference channel is an important multiuser channel model in network information theory. As a special case of the interference channel, Z-interference channel is the simplest model for interference in wireless communciations. However, the capacity region of most interference channels or Z-interference channels remain open. One of the main difficulties in characterizing the capacity region of the Z-interference channel stems from the difficulty in defining auxiliary random variables in the proof of converse. Traditional tools for defining auxiliary random variables work well in networks with one transmitter, i.e., centralized coding. Direct application of these tools to distributed-coding networks such as the interference channel will result in auxiliary random variables being correlated with all other variables in the network, causing a gap between the lower and upper bounds of the capacity region. In this proposal, we would like to investigate how to appropriate apply converse tools for defining auxiliary random variable traditionally used in centralized-coding networks to the Z-interference channel (distributed coding). Along this line, we have already made some progress: for Z-interference channels that satisfy the shift-invariant condition and the maximum entropy condition, we have successfully defined auxiliary random variables that are independent to some of the transmitted symbols, and gave the first capacity region of the interference channel that contains an auxiliary random variable. As part of future work, we would like to investigate deeper into the problem and find ways to relax or remove the shift-invariant conditon and/or the maximum entropy condition. As a result, we expect to derive converses and capacity regions for more general Z-interference channels, and contribute towards eventually solving the capacity region of the Gaussian Z-interference channel and the general Z-interference channel.

干扰信道是网络信息论中的重要模型。Z干扰信道作为干扰信道的特例，是对干扰现象的最简单建模。然而求解干扰信道或Z干扰信道的容量区域非常困难。容量区域难以求出的重要原因在于容量上界中辅助随机变量的设置。已有的设置方法多适用于集中式编码的网络，直接应用此方法在干扰信道（分布式编码）中会导致设置出的辅助变量不满足独立性，致使容量上下界不一致。为此，本项目研究如何将传统用于集中式编码网络辅助变量的设置方法灵活的运用到Z干扰信道中。前期工作已取得一定进展：当Z干扰信道满足移不变条件和最大熵条件时，申请人成功的设置了满足独立性的辅助变量，给出了第一个包含辅助随机变量的干扰信道容量区域。从已有的工作基础展望，我们拟对移不变条件和最大熵条件进行推敲，尝试将这两个条件进一步放松或者完全去掉，将结论拓宽到更广泛的Z干扰信道，增强我们对干扰的理解，向完全解决高斯Z干扰信道甚至最广泛Z干扰信道的容量问题迈进。

本课题主要研究Z干扰信道的容量上界问题，探索如何灵活将现有辅助变量设置方法（适用于集中式编码网络）应用到Z干扰信道（分布式编码）中。在已有的工作基础上，针对更广泛的Z干扰信道推导容量上界，设置满足独立条件的辅助随机变量，给出更广泛Z干扰信道的容量区域，证明叠加编码部分解码干扰方案在Z干扰信道的最优性。 在Z干扰信道中，我们进行了两方面的工作。第一方面，是找到了一种特殊Z干扰信道的容量区域。此类信道的特殊性在于不受干扰的信道是一个完美信道。通过分析，我们发现此信道的容量问题等同于接收者拥有编码信道状态信息的单用户信道。我们拓展了已知结果，在更广泛的信道条件下，给出了一个紧的容量上界，并证明了信道状态信息用Wyner-Ziv压缩编码传给接收者的可行性方案的最优性。另一方面，我们给出了满足移不变条件和独立同分布最优条件的Z干扰信道的新颖容量上界。在某些情况下，我们证明了此容量上界是紧的，即我们找到了一些以前未知容量的Z干扰信道的容量区域。在其他干扰信道模型中，我们着重研究了降阶消息集的Z信道。在某些信道条件下，我们找到它的整个容量区域。我们解决问题所用的方法，可以应用到各种包含分布式编码元素的网络中，为分布式编码网络提供了一种新的寻求容量上界的方法。