关于有限状态复合无线网络的自由度-解决Weingarten等人的一个猜想。al
On the Degrees of Freedom of Finite State Compound Wireless Networks - Settling a Conjecture by Weingarten et. al
摘要：
本文研究了三类有限状态复合无线网络的自由度。首先，我们研究了多输入单输出（MISO）有限状态复合广播信道（BC）与任意数量的用户和天线在发送端。在先前的工作中，Weingarten et.等人已经在具有2个用户的DoF上找到了内边界和外边界。边界具有不同的特性。当状态数增加时，内边界坍缩为1，而当状态数超过阈值时，外边界不会随着状态数的增加而减小。它已被证明，外界是松散的，内界代表实际的自由度。在复杂的设置（所有的信号，噪声和信道系数是复变量），我们解决了一些情况下，发现外边界-而不是内边界-的Weingarten等。艾尔很紧。对于真正的设置（所有的信号，噪声和信道系数是实变量），我们完全特征的自由度，再次证明，外边界的Weingarten等。艾尔很紧。我们还将结果扩展到任意数量的用户。其次，我们描述了有限状态标量（单天线节点）复合X网络的自由度与任意数量的用户在现实环境中。第三，我们描述了有限状态标量复合干扰网络的自由度与任意数量的用户在实数和复数设置。关键的发现是，标量干扰网络和（实）X网络不会失去任何自由度，由于在有限状态复合设置的发射机的信道不确定性。有限状态复合MISO BC相对于完美CSIT场景确实失去了DoF。然而，丢失的只是在发射天线处的联合处理的DoF益处，没有它，MISO BC减少到X网络。
英文摘要：
We explore the degrees of freedom (DoF) of three classes of finite state compound wireless networks in this paper. First, we study the multiple-input single-output (MISO) finite state compound broadcast channel (BC) with arbitrary number of users and antennas at the transmitter. In prior work, Weingarten et. al. have found inner and outer bounds on the DoF with 2 users. The bounds have a different character. While the inner bound collapses to unity as the number of states increases, the outer bound does not diminish with the increasing number of states beyond a threshold value. It has been conjectured that the outer bound is loose and the inner bound represents the actual DoF. In the complex setting (all signals, noise, and channel coefficients are complex variables) we solve a few cases to find that the outer bound -and not the inner bound- of Weingarten et. al. is tight. For the real setting (all signals, noise and channel coefficients are real variables) we completely characterize the DoF, once again proving that the outer bound of Weingarten et. al. is tight. We also extend the results to arbitrary number of users. Second, we characterize the DoF of finite state scalar (single antenna nodes) compound X networks with arbitrary number of users in the real setting. Third, we characterize the DoF of finite state scalar compound interference networks with arbitrary number of users in both the real and complex setting. The key finding is that scalar interference networks and (real) X networks do not lose any DoF due to channel uncertainty at the transmitter in the finite state compound setting. The finite state compound MISO BC does lose DoF relative to the perfect CSIT scenario. However, what is lost is only the DoF benefit of joint processing at transmit antennas, without which the MISO BC reduces to an X network.
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