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Aligned Image Sets: Instruments for Robust GDoF Bounds


As the first steps in the path towards progressively refined capacity approximations, degrees of freedom (DoF) and generalized degrees of freedom (GDoF) studies of wireless networks have turned out to be surprisingly useful. By exposing large gaps where they exist in our understanding of the capacity limits, these studies have been the catalysts for numerous discoveries over the past decade \cite{Jafar_FnT}. Some of the most interesting unresolved questions brought to light by recent DoF and GDoF studies have to do with channel uncertainty and the diversity of channel strengths. Consider the wireless network with $K$ transmitters and $K$ receivers, which could represent the $K$ user interference channel, the $K\times K$ $X$ channel, or the $K$ user MISO BC, i.e., the broadcast channel formed by allowing full cooperation among the transmitters in a $K$ user interference channel. Consider, first the issue of channel uncertainty. If the channel state information at the transmitter(s) (CSIT) is perfect then the $K$ user interference channel has $K/2$ DoF, the $K\times K$ $X$ channel has $K^2/(2K-1)$ DoF, and the MISO BC has $K$ DoF almost surely.\footnote{Channel state information at the receivers (CSIR) is assumed perfect throughout this work.} The optimal DoF are achieved by interference alignment for the interference and $X$ channel settings, and by transmit zero-forcing in the MISO BC. However, what happens if the CSIT is available only to finite precision?

Lapidoth et al. nearly a decade ago conjectured that the MISO BC has only $1$ DoF, i.e., the DoF collapse.

Since the MISO BC contains within it the $K$ user interference and $X$ channels, the collapse of DoF under finite precision CSIT implies that neither zero-forcing nor interference alignment is robust enough to provide a DoF advantage under finite precision CSIT, i.e., the DoF collapse for the interference and $X$ channels as well. Now consider the diversity of channel strengths which is explored through the studies of generalized degrees of freedom (GDoF). If all cross channels are much weaker relative to the direct channels then the DoF do not collapse even with finite precision CSIT. The collapse of DoF is avoided in the interference channel, simply by treating the weak interference as noise. Since the $X$ and BC settings include the interference channel, the collapse of DoF is avoided there as well. If some cross-channels are strong while others are so weak that they can be ignored entirely, as in the topological interference management problem, then even under finite precision CSIT, interference alignment plays a key role, albeit in a more robust form that does not depend on actual channel realizations.

In this thesis, we first prove the conjecture of Lapidoth, Shamai and Wigger as well as the ”PN” conjecture of Tandon et al., for all non-degenerate forms of finite precision CSIT. In the next step, we derive the GDoF of two user MISO BC problem, the sum GDoF of the K user symmetric MISO BC and the symmetric GDoF region of the K-user interference channel under finite precision CSIT. In the symmetric setting, we allow each cross channel to carry α DoF while each direct channel is capable of carrying 1 DoF. Finally, we derive sum-set inequalities specialized to the GDoF framework and derive DoF of MIMO BC under partial CSIT aiding from sum-set inequalities.

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