On the Degrees of Freedom of Finite State Compound Wireless Networks - Settling a Conjecture by Weingarten et. al

TL;DR

This paper investigates the degrees of freedom in finite state compound wireless networks, resolving conjectures about bounds and demonstrating when channel uncertainty affects network capacity.

Contribution

It provides a complete characterization of DoF for various finite state compound networks, confirming the tightness of outer bounds and revealing when DoF loss occurs due to channel uncertainty.

Findings

Outer bounds are tight in complex and real MISO BC cases.

Scalar interference and X networks do not lose DoF due to channel uncertainty.

Finite state MISO BC loses DoF compared to perfect CSIT, but scalar networks do not.

Abstract

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.