Aligned Image Sets Under Channel Uncertainty: Settling Conjectures on the Collapse of Degrees of Freedom Under Finite Precision CSIT

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 $O((\sqrt {P})^\alpha )$ , representing the concentration of probability density (improving CSIT) due to, e.g., quantized feedback at rate $({\alpha }/{2})\log (P)$ , then the DoF is bounded above by $1+\alpha $ , 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 $M\times N$ user $X$ networks as well.