Aligned Image Sets Under Channel Uncertainty: Settling Conjectures on the Collapse of Degrees of Freedom Under Finite Precision CSIT
Aligned Image Sets Under Channel Uncertainty: Settling Conjectures on the Collapse of Degrees of Freedom Under Finite Precision CSIT
复制标题
通道不确定性下的对齐图像集：解决有限精度 CSIT 下自由度崩溃的猜想
DOI：
10.1109/tit.2016.2586918
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发表时间：
2016
影响因子：
2.5
通讯作者：
S. Jafar
中科院分区：
文献类型：
--
作者：
Arash Gholami Davoodi;S. Jafar
Lapidoth等人在Allerton 2005年提出的猜想（也是ITA 2006年提出的开放问题）指出，在发射机配备有两个天线并且每个用户配备有一个天线的情况下，两个用户广播信道的自由度（DoF）必须在发射机处的有限精度信道状态信息（CSIT）下崩溃。这一猜想早于干扰对齐，但仍未得到解决，这表明人们普遍缺乏对无线网络DoF的理解，包括在发射机信道不确定性下的干扰和X网络。在本文中，我们证明了该猜想在所有非退化设置（例如，其中未知信道系数的概率密度函数存在并且有界）。即使当发射机可获得一个用户的完美信道知识时，DoF也会崩溃。这也解决了Tandon等人最近提出的一个相关猜想。我们证明的关键是码字数量的限制，这些码字可以在信道受到有限精度CSIT影响的非期望接收机上投射相同的图像（噪声失真内），同时在信道被发射机精确获知的期望接收机上保持可分辨。我们还可以将结果沿着两个方向推广。首先，如果概率密度函数的峰值被不允许缩放为O（（α P）α），表示概率密度的集中（改善CSIT），例如，如果量化反馈的速率为（α/2）log（P），则DoF的上界为1+α，这在量化反馈下也是可实现的。其次，我们推广的结果，任意数量的天线在发射机，任意数量的单天线用户，和复杂的信道。推广直接意味着在非退化信道不确定性下，对于一般的K用户干扰和M × N用户X网络，DoF塌陷为1。
A conjecture made by Lapidoth et al. at Allerton 2005 (also an open problem presented at ITA 2006) states that the degrees of freedom (DoF) of a two user broadcast channel, where the transmitter is equipped with two antennas and each user is equipped with one antenna, must collapse under finite precision channel state information at the transmitter (CSIT). That this conjecture, which predates interference alignment, has remained unresolved, is emblematic of a pervasive lack of understanding of the DoF of wireless networks-including interference and X networks-under channel uncertainty at the transmitter(s). In this paper, we prove that the conjecture is true in all non-degenerate settings (e.g., where the probability density function of unknown channel coefficients exists and is bounded). The DoF collapse even when perfect channel knowledge for one user is available to the transmitter. This also settles a related recent conjecture by Tandon et al. The key to our proof is a bound on the number of codewords that can cast the same image (within noise distortion) at the undesired receiver whose channel is subject to finite precision CSIT, while remaining resolvable at the desired receiver whose channel is precisely known by the transmitter. We are also able to generalize the result along two directions. First, if the peak of the probability density function is √ allowed to scale as O(( √P)α), representing the concentration of probability density (improving CSIT) due to, e.g., quantized feedback at rate (α/2) log(P), then the DoF is bounded above by 1+α, which is also achievable under quantized feedback. Second, we generalize the result to arbitrary number of antennas at the transmitter, arbitrary number of single-antenna users, and complex channels. The generalization directly implies a collapse of DoF to unity under non-degenerate channel uncertainty for the general K-user interference and M × N user X networks as well.