Robust Sum-GDoF of Symmetric 2 × 2 × 2 Weak Interference Channel With Heterogeneous Hops

Abstract

The symmetric 2 × 2 × 2 weak interference channel setting with heterogeneous hops is explored from a Generalized Degrees of Freedom (GDoF) perspective, especially under the robust assumption that limits the channel state information at the transmitters (CSIT) to finite precision. Specifically, in the ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>th</i></sup> hop, ℓ ∈ {1, 2}, both direct channels have strength α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[ℓ]</sub> , both cross channels have strength β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[ℓ]</sub> (in logarithmic scale), and β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[ℓ]</sub> ≤ 0.5α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[ℓ]</sub> . Thus, while assuming symmetry within each hop, the model allows heterogeneity across hops (α <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</sup> , β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</sub> ) ≠ (α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[2]</sub> , β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[2]</sub> ). Because β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[ℓ]</sub> ≤ 0.5α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[ℓ]</sub> , each hop corresponds to an interference channel in the weak interference regime where power control and treating interference as noise are known to be sum-GDoF optimal in a 1-hop setting. The main result of this work is the exact sum-GDoF of the symmetric 2 × 2 × 2 weak interference channel for heterogeneous hops under finite precision CSIT. Compared to prior work that assumes homogeneous hops, heterogeneous hops require not only more sophisticated optimal rate-splitting arguments, but also quantize-and-forward ideas which were not needed for homogeneous hops. The converse proof similarly involves generalizations to accommodate hop heterogeneity, as well as new bounds beyond the homogeneous case, based on sum-set inequalities and aligned images arguments. Additional results include sum-GDoF for perfect CSIT, and for a natural dual strong interference setting where β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[ℓ]</sub> ≥ 2α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[ℓ]</sub> .