非对称复合信号的干扰对齐-解决Host-Madsen-Nosratinia猜想
Interference Alignment with Asymmetric Complex Signaling - Settling the Host-Madsen-Nosratinia Conjecture
摘要：
Host-Madsen和Nosratinia已经证明，具有恒定信道系数的复高斯干扰信道只有一个自由度，与用户的数量无关。虽然已知有几个例子的恒定通道，实现超过1个自由度，这些特殊情况下，只有跨越一个子集的测量零。换句话说，对于几乎所有的信道系数值，不知道是否可以实现多于1个自由度。本文对Host-Madsen-Nosratinia猜想作了否定的解决。我们表明，至少1.2度的自由度是可实现的复信道系数的所有值，除了测量零的子集。对于本文所考虑的线性波束形成和干扰对准方案的类，还示出了对于几乎所有的信道系数值，1.2是在具有恒定信道系数的复高斯3用户干扰信道上可实现的最大自由度。为了建立1.2自由度的可重复性，我们引入了不对称复杂信号的新思想-即，输入被选择为是复数的但不是循环对称的。它示出，不像高斯点到点，多址接入和广播信道，其中循环对称的复高斯输入是最佳的，干扰信道的最佳输入一般是不对称的。对于非对称复信令，我们还证明了具有恒定信道系数的2用户复高斯X信道达到4/3自由度的外界，即，不需要在先前工作中用于建立相同结果的时间变化/频率选择性的假设。
英文摘要：
It has been conjectured by Host-Madsen and Nosratinia that complex Gaussian interference channels with constant channel coefficients have only one degree-of-freedom regardless of the number of users. While several examples are known of constant channels that achieve more than 1 degree of freedom, these special cases only span a subset of measure zero. In other words, for almost all channel coefficient values, it is not known if more than 1 degree-of-freedom is achievable. In this paper, we settle the Host-Madsen-Nosratinia conjecture in the negative. We show that at least 1.2 degrees-of-freedom are achievable for all values of complex channel coefficients except for a subset of measure zero. For the class of linear beamforming and interference alignment schemes considered in this paper, it is also shown that 1.2 is the maximum number of degrees of freedom achievable on the complex Gaussian 3 user interference channel with constant channel coefficients, for almost all values of channel coefficients. To establish the achievability of 1.2 degrees of freedom we introduce the novel idea of asymmetric complex signaling - i.e., the inputs are chosen to be complex but not circularly symmetric. It is shown that unlike Gaussian point-to-point, multiple-access and broadcast channels where circularly symmetric complex Gaussian inputs are optimal, for interference channels optimal inputs are in general asymmetric. With asymmetric complex signaling, we also show that the 2 user complex Gaussian X channel with constant channel coefficients achieves the outer bound of 4/3 degrees-of-freedom, i.e., the assumption of time-variations/frequency-selectivity used in prior work to establish the same result, is not needed.
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