We characterize the degrees of freedom (DoF) of multiple-input and multiple-output (MIMO) interference channels with rank-deficient channel matrices. For the two-user rank-deficient MIMO interference channel, we provide a tight outer bound to show that the previously known achievable DoF in the symmetric case is optimal and generalize the result to fully asymmetric settings. For the K-user rank-deficient interference channel, we improve the previously known achievable DoF and provide a tight outer bound to establish optimality in symmetric settings. In particular, we show that for the K-user rank-deficient interference channel, when all nodes have M antennas, all direct channels have rank D0, all cross channels are of rank D, and the channels are otherwise generic, the optimal DoF value per user is min(D0, M - (min(M, (K - 1)D)/2)). Notably for interference channels, the rank-deficiency of direct channels does not help and the rank deficiency of cross-channels does not hurt. The main technical challenge is to account for the spatial dependences introduced by rank deficiencies in the interference alignment schemes that typically rely on the independence of channel coefficients.

We study the degrees of freedom (DoF) of the layered 2 × 2 × 2 multiple-input-multiple-output (MIMO) interference channel where each node is equipped with arbitrary number of antennas, the channels between the nodes have arbitrary rank constraints, and subject to the rank-constraints the channel coefficients can take arbitrary values. The DoF outer bounds reveal a fundamental rank-matching phenomenon, reminiscent of impedance matching in circuit theory. It is well known that the maximum power transfer in a circuit is achieved not for the maximum or minimum load impedance but for the load impedance that matches the source impedance. Similarly, the maximum DoF in the rank-constrained 2 × 2 × 2 MIMO interference network is achieved not for the maximum or minimum ranks of the destination hop, but when the ranks of the destination hop match the ranks of the source hop. In fact, for mismatched settings of interest, the outer bounds identify a DoF loss penalty that is precisely equal to the rank-mismatch between the two hops. For symmetric settings, we also provide achievability results to show that along with the min-cut max-flow bounds, the rank-mismatch bounds are the best possible, i.e., they hold for all channels that satisfy the rank-constraints and are tight for almost all channels that satisfy the rank-constraints. Limited extensions-from sum-DoF to DoF region, from 2 unicasts to X message sets, from 2 hops to more than 2 hops and from 2 nodes per layer to more than 2 nodes per layer-are considered to illustrate how the insights generalize beyond the elemental 2 × 2 × 2 channel model.

Seeking fundamental insights into multi-hop multi-flow networks we study the simplest non-trivial setting, a 2 × 2 × 2 MIMO interference network comprised of two sources, two relays and two destinations, wherein all nodes have M antennas, all first-hop channels are of rank D1, and all second hop channels are of rank D2. For this setting, we show that the optimal sum DoF is min(4D1, 4D2, 2M - |D1- D2|). While 4D1, 4D2are the obvious min-cut bottlenecks that are active when either hop is severely rank-deficient, what is remarkable is that under moderate rank-deficiencies the DoF are limited not by the higher or the lower of the two ranks D1, D2, but only by the difference of the two ranks |D1- D2|. This suggests an interesting "rank-matching" design principle for multi-hop networks, reminiscent of "impedance matching", wherein the goal is not necessarily to increase or decrease the rank of each hop, but rather to use linear processing at intermediate hops to create effectively a two-hop setting with matching ranks.

We consider the interference channel with K transmitters and K receivers all having a single antenna, wherein the K × K transfer matrix representing this channel has rank D (D <; K) . The degrees of freedom (DoF) of such channels are not known as the rank deficiency in the transfer matrix creates algebraic dependencies between the channel coefficients. We present a modified version of the [CJ08] alignment scheme, to handle these dependencies while aligning interference, and state the sufficient conditions for achieving half rate per user using this scheme. The difficulties in proving these sufficient conditions are shown for K = 4 and K = 5. We also show that these sufficient conditions are not satisfied for K ≥ 6.

This paper explores how degrees of freedom (DoF) results from wireless networks can be translated into capacity or linear capacity results for their finite field counterparts that arise in network coding applications. The main insight is that scalar (SISO) finite field channels over Fpn are analogous to n × n vector (MIMO) channels in the wireless setting, but with an important distinction-there is additional structure due to finite held arithmetic, which enforces commutativity of matrix multiplication and limits the channel diversity to n, making these channels similar to diagonal channels in the wireless setting. Within the limits imposed by the channel structure, the DoF optimal precoding solutions for wireless networks can be translated into capacity or linear capacity optimal solutions for their finite held counterparts. This is shown through the study of capacity of the 2-user X channel and linear capacity of the 3-user interference channel. Besides bringing the insights from wireless networks into network coding applications, the study of finite held networks over Fpn also touches upon important open problems in wireless networks (finite SNR, finite diversity scenarios) through interesting parallels between p and SNR, and n and diversity.

Precoding based network alignment (PBNA) is a network coding paradigm inspired by wireless networks where all the intelligence resides at the sources and destination nodes whereas intermediate relay nodes only perform arbitrary linear network coding operations, creating an effective one-hop finite field linear network between sources and destinations. The main question explored in this work is how degrees of freedom (DoF) results from wireless networks can be translated into capacity results for their finite field counterparts. A finite field X channel, i.e., a multiple unicast network comprised of 2 source nodes, 2 destination nodes and 4 independent messages (one for each source-destination pair) is considered in this work, where the channel outputs are arbitrary linear combinations of channel inputs over a finite field Fpn. Like its wireless counterpart, which has 4/3 sum DoF, this channel is shown to have a sum capacity of 4/3 symbols per channel use for most channel realizations. The main insight is that, with a few exceptions that are pointed out, scalar (SISO) finite field channels over Fpnare analogous to n×n complex vector (MIMO) channels in the wireless setting, so the DoF optimal precoding solutions for wireless networks can be translated into capacity optimal solutions for their finite field counterparts.

We study the degrees of freedom (DoF) of 2-user and 3-user multiple input multiple output (MIMO) interference channels with rank deficient channel matrices. Only achievable DoF results and trivial outer bounds were previously available for these problems, restricted to symmetric settings. For the 2-user rank deficient MIMO interference channel we prove the optimality of previously known achievable DoF in the symmetric case and generalize the result to fully asymmetric settings. For the 3-user rank deficient MIMO interference channel, we improve the achievable DoF and provide a tight outer bound to establish optimality. Linear precoding based achievable schemes are found to be DoF optimal in both cases.