UC Irvine

Capacity characterization of communication networks is the most fundamental problem in Information Theory, that underlies the design of various wireless and wired networks. The radical idea of "Interference alignment" has enabled Capacity or Degrees of Freedom characterization (DoF, a first order approximation) for many interference networks. Various alignment schemes developed have provided new and fundamental insights into the number of accessible signal dimensions in communication networks where the output signals are linear functions of the input signals. Most of the prior art deal with generic channels wherein the channel coefficients are assumed to be independent and drawn from a continuous distribution, continuous alphabet with infinite diversity, and the network is often single-hop. These assumptions are challenged due to the following reasons : 1) In MIMO systems, poor scattering environment and network topology lead to spatial dependencies that are manifested as rank deficient channels, 2) Multi-hop dependencies arise due to the presence of relays, and 3) Linear network coding applications (as in wired networks) act as finite field counterparts of wireless networks, with limited diversity.

In this thesis, Capacity / DoF of linear communication networks are characterized for "Non-generic channels". One of the significant problems considered is the DoF of the K-user MIMO rank deficient interference channel, with different ranks for the direct and the cross channels. For this rank deficient interference channel, it is shown that the rank deficiency of direct channels does not help DoF and the rank-deficiency of cross-channels does not hurt DoF. The main challenge is to account for the spatial dependencies introduced by rank deficiencies in the interference alignment schemes that typically rely on the independence of channel coefficients. Another interesting problem is the DoF of Two-hop MIMO rank deficient interference channel with different channel ranks in the first and the second hops, for which a rank-matching principle is identified reminiscent of impedance matching in circuit theory. For this channel, the DoF loss is shown to be the rank-mismatch between the two hops. Finally, capacity results for the finite field counterparts of wireless networks are presented, exploring the implications of channels being from a finite alphabet with limited

diversity. By characterizing the capacity of constant finite field channels over F_{p^n} for 2-user X channel and 3-user interference channel, interesting parallels are drawn between p and SNR, and n and Channel Diversity.