Robust Sum-GDoF of Symmetric 2 × 2 × 2 Weak Interference Channel With Heterogeneous Hops
Pages 1309 - 1319
Abstract
The symmetric <inline-formula> <tex-math notation="LaTeX">$2\times 2\times 2$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$\ell ^{th}$ </tex-math></inline-formula> hop, <inline-formula> <tex-math notation="LaTeX">$\ell \in \{1,2\}$ </tex-math></inline-formula>, both direct channels have strength <inline-formula> <tex-math notation="LaTeX">$\alpha _{[\ell]}$ </tex-math></inline-formula>, both cross channels have strength <inline-formula> <tex-math notation="LaTeX">$\beta _{[\ell]}$ </tex-math></inline-formula> (in logarithmic scale), and <inline-formula> <tex-math notation="LaTeX">$\beta _{[\ell]}\leq 0.5\alpha _{[\ell]}$ </tex-math></inline-formula>. Thus, while assuming symmetry within each hop, the model allows heterogeneity across hops <inline-formula> <tex-math notation="LaTeX">$(\alpha _{[{1}]},\beta _{[{1}]}) \neq (\alpha _{[{2}]},\beta _{[{2}]})$ </tex-math></inline-formula>. Because <inline-formula> <tex-math notation="LaTeX">$\beta _{[\ell]}\leq 0.5\alpha _{[\ell]}$ </tex-math></inline-formula>, 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 <inline-formula> <tex-math notation="LaTeX">$2\times 2\times 2$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$\beta _{[\ell]}\geq 2\alpha _{[\ell]}$ </tex-math></inline-formula>.
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