Starting from the elemental 2 × 2 × 2 interference channel, there has been much progress in the understanding of multihop multiflow wireless networks through degrees of freedom (DoF) studies that have produced important ideas such as (aligned) interference neutralization. However, much of this progress has been limited to layered connectivity models that are essentially motivated by the assumption that wireless networks can only operate in half-duplex mode. Motivated by recent breakthroughs in full-duplex radio technology, in this work, we expand the 2 × 2 × 2 interference channel model beyond layered connectivity in order to study the impact of full-duplex operation. In particular, we study the impact of intra-layer connectivity between relays that are in the same layer, the impact of direct inter-layer connectivity between sources and destinations, and the impact of intermediate inter-layer connectivity that connects sources or destinations, not directly to each other, but to relay nodes in non-adjacent layers in a 2 × 2 × 2 × 2 interference channel. We show that intra-layer links and intermediate inter-layer interference links do not cause a collapse of DoF, whereas direct interference links do cause a collapse of DoF.

We show that the three-user MT× MRMultiple-Input Multiple-Output (MIMO) interference channel where each transmitter is equipped with MTantennas and each receiver is equipped with MRantennas has (M) d(M, N) =Δmin (M/2-1/κ, N/2+1/κ) degrees of freedom (DoF) normalized by time, frequency, and space dimensions, where =Δmin(MT, MR), N =Δmax(MT, MR), κ =Δ⌈M/N-M⌉. While the DoF outer bound of d(M, N) is established for every MT, MRvalue, the achievability of d(M, N) DoF is established in general subject to a normalization with respect to spatial extensions, i.e., the scaling of the number of antennas at all nodes. In particular, we show that qd(M, N) DoF are achievable for the three-user qMT × qMR MIMO interference channel, for some positive integer q, which may be seen as a spatial extension factor. q is the scaling factor needed to make the value qd(M, N) an integer. Given spatial extensions, the achievability relies only on linear beamforming based interference alignment schemes and requires neither channel extensions nor channel variations in time or frequency. In the absence of spatial extensions, it is shown through examples how essentially the same interference alignment scheme may be applied over time-extensions over either constant or time-varying channels. The central new insight to emerge from this paper is the notion of subspace alignment chains as the DoF bottlenecks. The subspace alignment chains are instrumental both in identifying the extra dimensions to be provided by a genie to a receiver for the DoF outer bound, as well as in the construction of the optimal interference alignment schemes. The DoF value d(M, N) is a piecewise linear function of M, N, with either M or N being the bottleneck within each linear segment, whereas the other value contains some redundancy, i.e., it can be reduced without reducing the DoF. The corner points of these piecewise linear segments correspond to two sets, A={1/2, 2/3, 3/4, ...}and B={1/3, 3/5, 5/7, ...}. The set A contains all those values of M/N and only those values of M/N for which there is redundancy in both M and N, i.e., either can be reduced without reducing the DoF. The set B contains all those values of M/N and only those values of M/N for which there is no redundancy in either M or N, i.e., neither can be reduced without reducing the DoF. Because A and B represent settings with maximum and minimum redundancy, essentially they are the basis for the DoF outer bounds and inner bounds, respectively. Our results settle the question of feasibility of linear interference alignment, introduced previously by Cenk et al., for the three-user MT× MRMIMO interference channel, completely for all values of MT, MR. In particular, we show that the linear interference alignment problem (MT× MR, d)3(as defined in previous paper by Cenk et al.) is feasible if and only if d ≤ d(M, N). With the exception of the values M/N ∈ B, and only with that exception, we show that for every M/N value there are proper systems (as defined by Cenk et al.) that are not feasible. Evidently the redundancy contained in all other values of M/N manifests itself as superfluous variables that are not discounted in the definition of proper systems, thus creating a discrepancy between proper and feasible systems. Our results show that M/N ∈ A are the only values for which there is no DoF benefit of joint processing among co-located antennas at the transmitters or receivers. This may also be seen as a consequence of the maximum redundancy in the M/N ∈ A settings.

We show that an M×N user MIMO X network with A antennas at each node has A(MN/(M+N-1)) degrees of freedom (DoF), thus resolving in this case a discrepancy between the spatial scale invariance conjecture (scaling the number of antennas at each node by a constant factor will scale the total DoF by the same factor) and a decomposability property of overconstrained wireless networks. While the best previously known general DoF outer bound is consistent with the spatial invariance conjecture, the best previously known general DoF inner bound, inspired by the K user MIMO interference channel, was based on the decomposition of every transmitter and receiver into multiple single antenna nodes, transforming the network into an AM×AN user SISO X network. While such a decomposition is DoF optimal for the K user MIMO interference channel, a gap remained between the best inner and outer bounds for the MIMO X channel. Here we close this gap with the new insight that the MIMO X network is only one-sided decomposable, i.e., either all the transmitters or all the receivers (but not both) can be decomposed by splitting multiple antenna nodes into multiple single antenna nodes without loss of DoF. The result is extended to SIMO and MISO X networks as well and in each case the DoF results satisfy the spatial scale invariance property.

We show that an M × N user MIMO X network with A antennas at each node has A (MN/M+N-1) degrees of freedom (DoF), thus settling the spatial scale invariance conjecture (scaling the number of antennas at each node by a constant factor will scale the total DoF by the same factor) for this class of networks. The previously known best general DoF inner bound, inspired by the K user interference channel, was based on the decomposition of every transmitter and receiver into multiple single antenna nodes, transforming the network into an AM × AN user SISO X network. While such a decomposition is DoF optimal for the K user interference channel, a gap remained between the best inner and outer bound for the MIMO X channel. Here we close this gap with the new insight that the MIMO X network is only one-sided decomposable, i.e., either all the transmitters or all the receivers (but not both) can be decomposed by splitting multiple antenna nodes into multiple single antenna nodes without loss of DoF. The result is extended to SIMO and MISO X networks as well and in each case the DoF results satisfy the spatial scale invariance property. In addition, the feasibility of linear interference alignment is investigated based only on spatial beamforming without symbol extensions. Similar to MIMO interference networks, we show that when the problem is improper, it is infeasible.

We show that the 3 user MT× MRMIMO interference channel where each transmitter is equipped with MTand each receiver is equipped with MRantennas has min (M/2-1/k, N/2+1/k) degrees of freedom (DoF) per user normalized by time, frequency, and space dimensions, where N = max(MT, MR), M = min(MT, MR), k = [M/N-M]. While the information theoretic DoF outer bound is established for every M, N value, the achievability, relying only on linear interference alignment, is established in general subject to a normalization with respect to spatial-extensions, i.e., the scaling of the number of antennas at all nodes. In the absence of spatial extensions, we can also show through examples how essentially the same alignment scheme may be applied over time or frequency extensions. The central new insight to emerge from this work is the notion of subspace alignment chains as DoF bottlenecks. The subspace alignment chains are instrumental both in identifying the extra dimensions provided by a genie to a receiver for the DoF outer bound, as well as constructing the optimal interference alignment schemes. In addition, our results also settle the question of feasibility of linear interference alignment for the 3 user MT× MRMIMO interference channel, for all values of MT, MR.

We study precoder optimization gains and multiuser diversity gains with interference alignment in a two-cell wireless network. In this paper, we propose algorithms to improve the achievable sum rate by optimizing alignment directions. The proposed iterative algorithm can provide a substantial gain in the achievable rate while it requires only a few iterations. In addition, we investigate interference alignment to exploit multiuser diversity gains. A new criterion of user selection scheduling is proposed. This user selection can be done independently in different cells. Therefore, both the searching space and the information exchanged between base stations are significantly reduced compared to joint user scheduling over two cells.

We show that the 2 × 2 × 2 interference network, i.e., the multihop interference network formed by concatenation of two two-user interference channels achieves the min-cut outer bound value of 2 DoF, for almost all values of channel coefficients, for both time-varying or fixed-channel coefficients. The key to this result is a new idea, called aligned interference neutralization, that provides a way to align interference terms over each hop in a manner that allows them to be canceled over the air at the last hop.

We study the sum capacity of multiple unicasts in wireless multihop networks. With 2 source nodes and 2 sink 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). We explore the degrees of freedom (DoF) of wireless multihop X networks with a layered structure, allowing arbitrary number of hops and arbitrary connectivity within each hop. For the case when there are no more than two relay nodes in each layer, the DoF can only take values 1, 4/3, 3/2 or 2, based on the connectivity of the network, for almost all values of channel coefficients. When there are arbitrary number of relays in each layer, the DoF can also take the value 5/3. Achievability schemes incorporate linear forwarding, interference alignment and aligned interference neutralization principles. Information theoretic converse arguments specialized for the connectivity of the network are constructed based on the intuition from linear dimension counting arguments.

We show that the 2 × 2 × 2 interference network, i.e., the multihop interference network formed by concatenation of two 2-user interference channels achieves the min-cut outer bound value of 2 DoF, for almost all values of channel coefficients, for both time-varying or fixed channel coefficients. The key to this result is a new idea, called aligned interference neutralization, that provides a way to align interference terms over each hop in a manner that allows them to be cancelled over the air at the last hop.

TheN+1 user, 1 ×Nsingle-input multiple-output (SIMO) Gaussian interference channel where each transmitter has a single antenna and each receiver hasNantennas is studied. The sum capacity withinO(1) is characterized for the symmetric case where all direct links have the same signal-to-noise ratio (SNR) and all undesired links have the same interference-to-noise ratio (INR). The gap to the exact capacity is a constant which is independent of SNR and INR. To get this result, we first generalize the deterministic interference channel introduced by El Gamal and Costa to model interference channels with multiple antennas. We derive the capacity region of this deterministic interference channel. Based on the insights provided by the deterministic channel, we characterize the generalized degrees of freedom (GDOF) of Gaussian case, which directly leads to theO(1) capacity approximation. On the achievability side, an interesting conclusion is that the GDOF regime where treating interference as noise is found to be optimal in the two-user interference channel, does not appear in theN+ 1 user, 1 ×NSIMO case. On the converse side, new multiuser outer bounds emerge out of this work that do not follow directly from the two-user case. In addition to the GDOF region, the outer bounds identify a strong interference regime where the capacity region is established.

We explore the degrees of freedom (DoF) of three classes of finite state compound wireless networks in this paper. First, we study the multiple-input single-output (MISO) finite state compound broadcast channel (BC) with arbitrary number of users and antennas at the transmitter. In prior work, Weingartenhave found inner and outer bounds on the DoF with 2 users. The bounds have a different character. While the inner bound collapses to unity as the number of states increases, the outer bound does not diminish with the increasing number of states beyond a threshold value. It has been conjectured that the outer bound is loose and the inner bound represents the actual DoF. In the complex setting (all signals, noise, and channel coefficients are complex variables), we solve a few cases to find that the outer bound—and not the inner bound—of Weingartenis tight. For the real setting (all signals, noise, and channel coefficients are real variables), we completely characterize the DoF, once again proving that the outer bound of Weingartenis tight. We also extend the results to arbitrary number of users. Second, we characterize the DoF of finite state scalar (single antenna nodes) compound$X$networks with arbitrary number of users in the real setting. Third, we characterize the DoF of finite state scalar compound interference networks with arbitrary number of users in both the real and complex setting. The key finding is that scalar interference networks and (real)$X$networks do not lose any DoF due to channel uncertainty at the transmitter in the finite state compound setting. The finite state compound MISO BC does lose DoF relative to the perfect CSIT scenario. However, what is lost is only the DoF benefit of joint processing at transmit antennas, without which the MISO BC reduces to an$X$network.

We propose a blind interference alignment scheme for the vector broadcast channel where the transmitter is equipped with M antennas and there are K receivers, each equipped with a reconfigurable antenna capable of switching among M preset modes. Without any knowledge of the channel coefficient values at the transmitters and with only mild assumptions on the channel coherence structure we show that MK/(M+K-1) degrees of freedom are achievable. The key to the blind interference alignment scheme is the ability of the receivers to switch between reconfigurable antenna modes to create short term channel fluctuation patterns that are exploited by the transmitter. The achievable scheme does not require cooperation between transmit antennas and is therefore applicable to the M×K X network as well. Only finite symbol extensions are used, and no channel knowledge at the receivers is required to null the interference.

We provide inner bound and outer bound for the total number of degrees of freedom of theKuser multiple-input multiple-output (MIMO) Gaussian interference channel withMantennas at each transmitter andNantennas at each receiver if the channel coefficients are time-varying and drawn from a continuous distribution. The bounds are tight when the ratio [(max(M,N))/(min(M,N))]=Ris equal to an integer. For this case, we show that the total number of degrees of freedom is equal to min(M,N)KifK≤Rand min(M,N)[(R)/(R+1)]KifK>R. Achievability is based on interference alignment. We also provide examples where using interference alignment combined with zero forcing can achieve more degrees of freedom than merely zero forcing for some MIMO interference channels with constant channel coefficients.

The degrees of freedom (DoF) of K-user MIMO interference networks with constant channel coefficients are not known in general. Determining the feasibility of a linear interference alignment is a key step toward solving this open problem. Our approach in this paper is to view the alignment problem for interference networks as a multivariate polynomial system and determine its solvability by comparing the number of equations and the number of variables. Consequently, we divide the interference networks into two classes proper and improper, where interference alignment is and is not achievable, respectively. An interference network is called proper if the cardinality of every subset of equations in the corresponding polynomial system is less than or equal to the number of variables involved in that subset of equations. Otherwise, it is called improper. Our intuition in this paper is that for general channel matrices, proper systems are almost surely feasible and improper systems are almost surely infeasible. We prove the direct link between proper (improper) and feasible (infeasible) systems for some important cases, thus significantly strengthening our intuition. Numerical simulation results also support our intuition.

The N + 1 user, 1 × N single input multiple output (SIMO) Gaussian interference channel where each transmitter has a single antenna and each receiver has N antennas is studied. The symmetric capacity within O(1) is characterized for the symmetric case where all direct links have the same signal-to-noise ratio (SNR) and all undesired links have the same interference-to-noise ratio (INR). The gap to the exact capacity is a constant which is independent of SNR and INR. On the achievability side, an interesting conclusion is that the generalized degrees of freedom (GDOF) regime where treating interference as noise is found to be optimal in the 2 user interference channel, does not appear in the N + 1 user, 1 × N SIMO case. On the converse side, new multi-user outer bounds emerge out of this work that do not follow directly from the 2 user case. We also provide an outer bound on the capacity region of the 3 user SIMO Gaussian interference channel with 2 antennas at each receiver. This outer bound directly leads to the capacity region of this channel if the channel vectors satisfy certain conditions.

The 4 user single input multiple output (SIMO) Gaussian interference channel where each transmitter has a single antenna and each receiver has two antennas is studied. We show that if the channel coefficients are time-varying and drawn from a continuous distribution, the maximum number of spatial degrees of freedom per orthogonal time dimension for this channel is 8/3 almost surely. Achievability is based on the idea of interference alignment, i.e., signal spaces are aligned at receiver where they constitute interference while they are separable at receivers where they are desired.