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

Arash Gholami Davoodi, Syed Jafar

Abstract:

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.