# Fundamental Limits of Wireless Interference Networks Under Transmitter Channel Uncertainty

#### Abstract

The role of interference in a wireless communication network, consisting of a multitude of transmitters communicating with their respective receivers over a shared physical medium, can not be overstated. In such a network, a transmitter's signal often causes interference at an unintended receiver, thus making interference management of paramount importance in modern wireless networks. Moreover, the presence of multiple antennas modern wireless systems, which are known as multiple-input multiple-output (MIMO) systems, requires more intricate techniques for interference management. While the fundamental information capacity of all but the simplest such interference networks has remained elusive for decades, we are able to obtain and analyze first-order approximations of the information capacity of various networks, which lead to new insights about how to efficiently manage, and even utilize, the interference in such networks.

We study two such first-order approximation metrics for the information-theoretic capacity, known as degrees of freedom (DoF) and generalized degrees of freedom (GDoF). The DoF metric measures the growth of the capacity of a communication network with respect to the signal-to-noise ratio (SNR) in the dB scale when the SNR is asymptotically high, for which the interference in the network, rather than random noise, is the primary bottleneck for the capacity. While DoF analysis, for the sake of tractability, assumes all relevant links in a wireless network to be statistically equal in strength, the more refined GDoF metric goes a step further and allows for disparate link strengths in a network, and measures the high SNR growth of the capacity for such a network.

In wireless communications, the signal from a transmitter arrives at a receiver through multiple reflected paths, and this multi-path propagation channel is modeled, in a MIMO system, as a matrix with complex coefficients. All such channel matrices in a wireless network are collectively known as the channel state, and this channel state changes over time because of a wireless phenomenon known as fading. While the channel state information (CSI) at any given time is usually learnt at the receivers with great accuracy through pilot signals, it is much more difficult to obtain accurate channel state information at the transmitters (CSIT), which requires feedback from the receiver. In mobile communications where the channel state can change rapidly (on the order of a few milliseconds), this feedback can often be delayed, with the channel state having changed by the time the feedback reaches the transmitter. In the most adverse scenario, i.e., when the new channel is completely independent of the previous channel state, the transmitter gets completely outdated knowledge (about the previous channel states) through feedback, a situation known as delayed CSIT. Otherwise, when the current channel state is correlated with the previous channel states, the transmitters can also acquire an imperfect estimate of the current CSI after acquiring accurate delayed CSIT from feedback. This model is known as mixed CSIT. In this thesis, we study the fundamental communication limits of wireless interference MIMO networks under these two models of channel uncertainty, i.e., delayed CSIT and mixed CSIT.

We characterize the DoF region of the two-user MIMO Z-interference channel (Z-IC), a network where only one user causes interference at the other user, for the mixed CSIT model. When used over an interference channel (a channel where both users cause interference with each other) with one cross-link being statistically weaker than the other three links, the new communication schemes designed for the Z-IC, by explicitly recognizing the relative weakness of that link, can lead to a DoF improvement over the existing DoF-optimal scheme designed for the fully connected interference channel, while also requiring no feedback of the weak link's channel state information.

We also introduce the hybrid CSIT model for the two-user MIMO IC, where one transmitter obtains delayed CSI while the other has access to instantaneous CSI, and establish the DoF region under hybrid CSIT. These results are shown to be applicable to a large class of hybrid CSIT models with various combinations of no CSIT, delayed CSIT and instant CSIT for each link. Next, we characterize the DoF region, under mixed CSIT, of the K-user MIMO cyclic Z-interference channel (CZIC), wherein, except for one interference link at each transmitter and receiver, all other interference links are assumed to be weak. For the symmetric MIMO CZIC, we show that, even with outdated (delayed) CSIT, the maximum sum-DoF scales linearly with the number of users.

While the more refined GDoF of a communication network gives significant new insights beyond DoF, its characterization also brings formidable technical challenges with it. Consequently, very little is known about the GDoF of wireless networks under channel uncertainty. In this thesis, we provide the first characterization of the GDoF region under delayed CSIT of any MIMO network as well as any interference network (with distributed transmitters). To this end, we establish the GDoF region of the MIMO Z-IC under delayed CSIT. We also show that delayed CSIT is sufficient to achieve the GDoF region with instantaneous CSIT for many antenna configurations, and in such a scenario, the delay involved in feedback has no adverse effects on the GDoF region. We prove that both DoF-optimal achievability schemes, which do not take into account the knowledge about channel statistics, as well as a strategy of naively treating the interference as noise are both sub-optimal from the GDoF perspective.

Next, we obtain both inner and outer bounds on the symmetric GDoF (the GDoF that both users can achieve simultaneously) for the two-user symmetric MIMO IC with delayed CSIT, when the number of transmit antennas (equal at each transmitter) is larger than the number of receive antennas (equal at each receiver). We show that these bounds coincide over a range of alpha, where alpha is the ratio of the signal power and the interference power in the dB scale, and the symmetric GDoF is thus characterized in this range of alpha.

*This paper has been withdrawn.*