We identify the role of equal strength interference links as bottlenecks on the ergodic sum capacity of a K user interference network with the fading process restricted primarily to inde- pendent and uniform phase variations while the channel magnitudes are held fixed across time. It is shown that even though there are K(K − 1) cross-links, only about K/2 disjoint and equal strength interference links suffice to determine the capacity of the network regardless of the strengths of the rest of the cross channels. This scenario is called a minimal bottleneck state. It is shown that ergodic interference alignment is capacity optimal for a network in a minimal bottleneck state. The results are applied to large networks. It is shown that large networks are close to bottleneck states with a high probability, so that ergodic interference alignment is close to optimal for large networks. Limitations of the notion of bottleneck states are also high- lighted for channels where both the phase and the magnitudes are allowed to vary with time. It is shown through an example that for these channels, joint coding across different bottleneck states makes it possible to circumvent the capacity bottlenecks.

In this paper we investigate the sum degrees of freedom (DoF) of multiple unicasts in a wireless network. With 2 source nodes, 2 destination nodes, there are a total of 4 independent unicast sessions (messages), one from each source to each sink node (this setting is also known as an X network), and also there is a delay-free relay working in full-duplex mode helping the transmissions from the source to destination nodes. For such a channel setting, we prove that 5/3 DoF is achievable almost surely for time-varying/frequency-selective channels, based on the ideas of aligned interference neutralization, linear forwarding and interference alignment. Also, the achievable scheme can be easily translated to the rational alignment scheme for the network with constant-values channel coefficients. In addition, we provide an intuition for the 5/3 DoF result from the perspective of counting the number of linear equations and variables.

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.

Generalized degrees of freedom (GDoF) analysis of wireless networks has contributed significantly to recent advances in the fundamental understanding of their capacity. A variety of schemes have been developed that are GDoF-optimal under the assumption that the Channel State Information at Transmitter(s) (CSIT) is perfect. However, these schemes become fragile in practice because channel uncertainty is unavoidable. Motivated by the need to understand the robust information-theoretic limits of wireless networks, we explore three settings under the assumption that the CSIT is limited to finite precision, and present their GDoF characterization.First, the optimality of Treating Interference as Noise (TIN) for K-user Interference Channel under finite precision CSIT is investigated. TIN is found to be GDoF optimal by Geng et al. in a parameter regime called the "TIN regime" under perfect CSIT. Our result shows that TIN is GDoF optimal in a much larger regime, called the CTIN regime, under finite precision CSIT. Next, the GDoF of the two-user interference channel are characterized for all parameter regimes under finite precision CSIT, when a limited amount of (half-duplex or full-duplex) cooperation is allowed between the transmitters. In all cases, the number of over-the-air bits that each cooperation bit buys is shown to be equal to either 0, 1, 1/2 or 1/3. The most interesting aspect of the result is the 1/3 slope, which appears only under finite precision CSIT and strong interference, and as such has not been encountered in previous studies that invariably assumed perfect CSIT. Finally, we explore the multi-hop interference channel. We consider the sum-GDoF of the symmetric multi-hop interference channel under finite precision CSIT. The sum-GDoF value is first characterized for the 2 × 2 × 2 setting that is comprised of 2 sources, 2 relays, and 2 destinations. The result is then generalized to the 2 × 2 × · · · × 2 setting that is comprised of L hops. Remarkably, for large L, the sum-GDoF value approaches that of the one-hop broadcast channel that is obtained by full cooperation among the two transmitters of the last hop, under finite precision CSIT. Under finite precision CSIT, a combination of classical random coding schemes that are simpler and much more robust, namely a rate-splitting between decode-and-forward and amplify-and-forward, is shown to be GDoF optimal.

Treating interference as noise (TIN) when it is sufficiently weak is one of the key principles of interference management for wireless networks. This dissertation revisits the optimality of TIN from an information theoretic perspective. It is shown that for K-user Gaussian interference channels, TIN achieves all points in the capacity region to within a constant gap, if for each user, the desired signal strength is no weaker than the sum of the strengths of the strongest interference caused by the user and the strongest interference suffered by the user (with all signal strengths measured in dB scale). We also extend the optimality of TIN to more general settings, including interference networks with general message sets, compound networks and MIMO interference channels, and characterize the secure capacity region within a constant gap for the identified TIN-optimal interference channels with secrecy constraints. Moreover, combining TIN with interference avoidance, we formulate a joint signal space and signal level optimization problem and propose a baseline decomposition approach.