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Generalized degrees of freedom (GDoF) are characterized for the symmetric K-user multiple-input multiple-output interference channel under the assumption that the channel-state information at the transmitters (CSITs) is limited to finite precision. In this symmetric setting, each transmitter is equipped with M antennas, each receiver is equipped with N antennas, each desired channel (i.e., a channel between a transmit antenna and a receive antenna belonging to the same user) has strength ∼ P , while each undesired channel has strength ∼ Pα, where P is a nominal SNR parameter. The result generalizes a previous GDoF characterization for the SISO setting (M=N=1) and is enabled by a significant extension of the aligned image sets bound that is broadly useful. GDoF per user take the form of a W-curve with respect to α for fixed values of M and N. Under finite precision CSIT, in spite of the presence of multiple antennas, all the benefits of interference alignment are lost.

This paper obtains the first degrees of freedom (DoFs) bound that is provably sensitive to network coherence time, i.e., coherence time in an interference network, where all channels experience the same coherence patterns. This is accomplished by a novel adaptation of the aligned image sets bound and settles various open problems noted previously by Naderi and Avestimehr and by Gou et al. For example, a necessary and sufficient condition is obtained for the optimality of 1/2 DoF per user in a partially connected interference network, where the channel state information at the receivers (CSIRs) is perfect, the channel state information at the transmitters (CSITs) is instantaneous but limited to finite precision, and the network coherence time is Tc=1. The surprising insight that emerges is that even with perfect CSIR and instantaneous finite precision CSIT, the network coherence time matters, i.e., it has a DoF impact.

The main result of this paper is that an orthogonal access scheme, such as time division multiple access achieves the all-unicast degrees of freedom (DoF) region of the topological interference management problem if and only if the network topology graph is chordal bipartite, i.e., every cycle that can contain a chord, does contain a chord. The all-unicast DoF region includes the DoF region for any arbitrary choice of a unicast message set, so e.g., the results of Maleki and Jafar on the optimality of orthogonal access for the sum-DoF of one-dimensional convex networks are recovered as a special case. The result is also established for the corresponding topological representation of the index coding problem.

In order to gain new insights into multiple-input-multiple-output (MIMO) interference networks, the optimality ofσk=1 K Mk/2 (half the cake per user) degrees of freedom is explored for a K-user MIMO interference channel where the cross-channels have arbitrary rank constraints, and the k th transmitter and receiver are equipped with Mk antennas each. The result consolidates and significantly generalizes results from prior studies by Krishnamurthy et al., of rank-deficient interference channels where all users have M antennas; and by Tang et al., of full rank interference channels where the k th user pair has Mk antennas. The broader outcome of this paper is a novel class of replication-based outer bounds for arbitrary rank-constrained MIMO interference networks where replicas of existing users are added as auxiliary users and the network connectivity is chosen to ensure that any achievable scheme for the original network also works in the new network. The replicated network creates a new perspective of the problem, so that even simple arguments such as user cooperation become quite powerful when applied in the replicated network, giving rise to stronger outer bounds, than when applied directly in the original network. Remarkably, the replication-based bounds are broadly applicable not only to MIMO interference channels with arbitrary rank-constraints, but much more broadly, even beyond Gaussian settings.

In this paper, we propose a novel 'genie chains' approach to obtain information theoretic degrees of freedom (DoF) outer bounds for MIMO wireless interference networks. This new approach creates a chain of mappings from genie signals provided to a receiver to the exposed signal spaces at that receiver, and then the exposed signal spaces serve as the genie signals for the next receiver in the chain subject to certain linear independence requirements. Our approach essentially converts an information theoretic DoF outer bound problem into a linear algebra problem. Several applications of the genie chains approach are presented.

A conjecture made by Lapidoth et al. at Allerton 2005 (also an open problem presented at ITA 2006) states that the degrees of freedom (DoF) of a two user broadcast channel, where the transmitter is equipped with two antennas and each user is equipped with one antenna, must collapse under finite precision channel state information at the transmitter (CSIT). That this conjecture, which predates interference alignment, has remained unresolved, is emblematic of a pervasive lack of understanding of the DoF of wireless networks-including interference and X networks-under channel uncertainty at the transmitter(s). In this paper, we prove that the conjecture is true in all non-degenerate settings (e.g., where the probability density function of unknown channel coefficients exists and is bounded). The DoF collapse even when perfect channel knowledge for one user is available to the transmitter. This also settles a related recent conjecture by Tandon et al. The key to our proof is a bound on the number of codewords that can cast the same image (within noise distortion) at the undesired receiver whose channel is subject to finite precision CSIT, while remaining resolvable at the desired receiver whose channel is precisely known by the transmitter. We are also able to generalize the result along two directions. First, if the peak of the probability density function is allowed to scale as O(= Pα), representing the concentration of probability density (improving CSIT) due to, e.g., quantized feedback at rate (α/2)log (P), then the DoF is bounded above by 1+α, which is also achievable under quantized feedback. Second, we generalize the result to arbitrary number of antennas at the transmitter, arbitrary number of single-antenna users, and complex channels. The generalization directly implies a collapse of DoF to unity under non-degenerate channel uncertainty for the general K-user interference and M× N user X networks as well.

We characterize the degrees of freedom (DoF) of MIMO interference networks with rank-deficient channel matrices. For the 2-user rank deficient MIMO interference channel, we provethe optimality of previously known achievable DoF in the symmetric case and generalize the re-sult to fully asymmetric settings. For the K-user rank deficient interference channel, we improvethe previously known achievable DoF and provide a tight outer bound to establish optimalityin symmetric settings. In particular, we show that for the K-user rank deficient interferencechannel, when all nodes have M antennas, all direct channels have rank D0, all cross chan-nels are of rank D, and the channels are otherwise generic, the optimal DoF value per user ismin(D0, M −min(M,(K−1)D)). For 2-user and 3-user rank deficient channels, achievable schemes2are for both constant and time-varying channels, while for K-user rank deficient channels, we present schemes for time-varying channels and note that the insights would act as stepping stones for constant channels. 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 dependencies introduced by rank deficiencies in the interference alignment schemes that typically rely on the independence of channel coefficients.

© 2014 IEEE. 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, 4D2 are 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.

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 \(\mathbb pn are analogous to n × n vector (MIMO) channels in the wireless setting, but with an important distinction - there is additional structure due to finite field 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 field 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 field networks over pn 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. © 2014 IEEE.

We show that the three-user MT ×MR Multiple-Input Multiple-Output (MIMO) interference channel where each transmitter is equipped with MT antennas and each receiver is equipped with MR antennas has d(M, N) δ= min δ M 2?1/? , N 2+1/?degrees of freedom (DoF) normalized by time, frequency, and space dimensions, where M = min(MT , MR), N = max(MT , MR), κ δ = κ M N?M. While the DoF outer bound of d(M, N) is established for every MT , MR value, 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 MT ×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 δ = {1/2, 2/3, 3/4, . . .} and B δ = {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, 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×MR MIMO interference channel, completely for all values of MT , MR. In particular, we show that the linear interference alignment problem (MT×M R, 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/NA settings. © 1963-2012 IEEE.

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. © 1963-2012 IEEE.

It is shown that a greedy orthogonal access scheme achieves the sum degrees of freedom (DoF) of all one-dimensional (all nodes placed along a straight line) convex cellular networks (where cells are convex regions) when no channel knowledge is available at the transmitters except the knowledge of the network topology. In general, optimality of orthogonal access holds neither for two-dimensional convex cellular networks nor for one-dimensional non-convex cellular networks, thus revealing a fundamental limitation that arises when both one-dimensional and convex properties are simultaneously enforced. The result also establishes the sum capacity of the corresponding class of index coding problems. © 2013 IEEE.

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.

We explore similarities and differences in recent works on blind interference alignment under different models such as staggered block fading model and the delayed CSIT model. In particular we explore the possibility of achieving interference alignment with delayed CSIT when the transmitters are distributed. Our main contribution is an interference alignment scheme, called retrospective interference alignment in this work, that is specialized to settings with distributed transmitters. With this scheme we show that the 2 user X channel with only delayed channel state information at the transmitters can achieve 8/7 DoF, while the interference channel with 3 users is able to achieve 9/8 DoF. We also consider another setting where delayed channel output feedback is available to transmitters. In this setting the X channel and the 3 user interference channel are shown to achieve 4/3 and 6/5 DoF, respectively.

Aligned-Images Sum-set Inequalities are used in this work to study the Generalized Degrees of Freedom (GDoF) of the symmetric layered multi-hop interference channel under the robust assumption that the channel state information at the transmitters (CSIT) is limited to finite precision. First, the sum-GDoF value is characterized for the $2\times 2\times 2$ setting that is comprised of 2 sources, 2 relays, and 2 destinations. It is shown that the sum-GDoF does not improve even if perfect CSIT is allowed in the first hop, as long as the CSIT in the second hop is limited to finite precision. The sum GDoF characterization is then generalized to the $2\times 2\times \cdots \times 2$ 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, with finite precision CSIT. Previous studies of multi-hop interference networks either identified sophisticated GDoF optimal schemes under perfect CSIT, such as aligned interference neutralization and network diagonalization, that are powerful in theory but too fragile to be practical, or studied robust achievable schemes like classical amplify/decode/compress-and-forward without claims of information-theoretic optimality. In contrast, under finite precision CSIT, we show that the benefits of fragile schemes are lost, while 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. As such, this work represents another step towards bridging the gap between theory (optimality) and practice (robustness) with the aid of Aligned-Images Sum-set Inequalities. [ABSTRACT FROM AUTHOR]

We present a condition for 1/2 degree of freedom for each user in constant $K$ -user single-antenna interference channels. This condition is sufficient for all and necessary for almost all channel matrices. Moreover, it applies to all channel topologies, i.e., to fully-connected channels as well as channels that have individual links absent, reflected by corresponding zeros in the channel matrix. Moreover, it captures the essence of interference alignment by virtue of being expressed in terms of a generic injectivity condition that guarantees separability of signal and interference. Finally, we provide codebook constructions achieving 1/2 degree of freedom for each user for all channel matrices satisfying the condition we identified. [ABSTRACT FROM AUTHOR]

Blind Interference Alignment (BIA) schemes create and exploit channel coherence patterns without the knowledge of channel realizations at transmitters, while beamforming schemes rely primarily on channel knowledge available to the transmitters without regard to channel coherence patterns. In order to explore the compatibility of these disparate ideas and the possibility of synergistic gains, this work studies the Degrees of Freedom (DoF) of the 2-user ${(M_{1}\times N_{1})(M_{2}\times N_{2})}$ Multiple-Input Multiple-Output (MIMO) Interference Channel (IC) where Transmitter 1 is equipped with reconfigurable antennas and has no channel knowledge, while Transmitter 2 has partial channel knowledge but no reconfigurable antennas. Taking a fundamental dimensional analysis perspective, the main question is to identify which antenna configurations allow synergistic DoF gains. The main results of this work are two-fold. The first result identifies antenna configurations where both reconfigurable antennas and partial channel knowledge are individually beneficial, as those where $M_{1}< N_{1}< \min (M_{2},N_{2})$. The second result shows that synergistic gains exist in each of these settings, over the best known solutions that rely on either reconfigurable antennas or partial channel knowledge alone. Coding schemes that jointly exploit partial channel knowledge and reconfigurable antennas emerge as a byproduct of the analysis. [ABSTRACT FROM AUTHOR]

The Generalized Degrees of Freedom (GDoF) of the two user interference channel are characterized for all parameter regimes under the assumption of finite precision channel state information at the transmitters (CSIT), when a limited amount of (half-duplex or full-duplex) cooperation is allowed between the transmitters in the form of $\pi $ DoF of shared messages. 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. Indeed, the achievability and converse for the parameter regimes with $1/3$ slope are the most challenging aspects of this work. In particular, the converse relies on non-trivial applications of Aligned Images bounds. [ABSTRACT FROM AUTHOR]

The degrees of freedom (DoF) region is characterized for the 2-user multiple input multiple output (MIMO) broadcast channel (BC), where the transmitter is equipped with $M$ antennas, the two receivers are equipped with $N_{1}$ and $N_{2}$ antennas, and the levels of channel state information at the transmitter (CSIT) for the two users are parameterized by $\beta _{1}, \beta _{2}$ , respectively. The achievability of the DoF region was established by Hao, Rassouli and Clerckx, but no proof of optimality was heretofore available. The proof of optimality is provided in this work with the aid of sum-set inequalities based on the aligned image sets (AIS) approach. [ABSTRACT FROM AUTHOR]

We present sum-set inequalities specialized to the generalized degrees of freedom (GDoF) framework. These are information theoretic lower bounds on the entropy of bounded density linear combinations of discrete, power-limited dependent random variables in terms of the joint entropies of arbitrary linear combinations of new random variables that are obtained by power level partitioning of the original random variables. These bounds generalize the aligned image sets approach, and are useful instruments to obtain GDoF characterizations for wireless networks, especially with multiple antenna nodes, subject to arbitrary channel strength and channel uncertainty levels. To demonstrate the utility of these bounds, we consider various examples of interference and broadcast channels for which we obtain tight GDoF characterizations with the aid of sum-set inequalities. [ABSTRACT FROM AUTHOR]

We study the $K=3$ user multiple input single output (MISO) broadcast channel (BC) with $M=3$ antennas at the transmitter and 1 antenna at each receiver, from the generalized degrees of freedom (GDoF) perspective, under the assumption that the channel state information at the transmitter (CSIT) is limited to the finite precision. In particular, our goal is to identify a parameter regime where a simple layered superposition (SLS) coding scheme achieves the entire GDoF region. With $\alpha _{ij}$ representing the channel strength parameter for the link from the $j^{th}$ antenna of the transmitter to the $i^{th}$ receiver, we prove that the SLS is GDoF optimal without the need for time-sharing if $\max (\alpha _{ki},\alpha _{im})\leq \alpha _{ii}$ and $\alpha _{ki}+\alpha _{im}\le \alpha _{ii}+\alpha _{km}$ for all $i,k\in [{3}],m\in [M]$. The GDoF region under this condition is a convex polyhedron. The result generalizes to arbitrary $M\geq 3$. [ABSTRACT FROM AUTHOR]

The generalized degrees of freedom of the two-user symmetric multiple input multiple output interference channel are characterized as a function of the channel strength levels and the level of channel state information at the transmitters. In this symmetric setting, each transmitter is equipped with $M$ antennas, each receiver is equipped with $N$ antennas, and both cross links have the same strength parameter $\alpha $ and the same channel uncertainty parameter $\beta $. The main challenge resides in the proof of the outer bound which is accomplished by a generalization of the aligned image sets approach. [ABSTRACT FROM AUTHOR]

For the $K=2$ user MISO BC, i.e., the wireless broadcast channel where a transmitter equipped with $K=2$ antennas sends independent messages to $K=2$ receivers each of which is equipped with a single antenna, the generalized degrees of freedom (GDoFs) region are characterized for arbitrary channel strength and channel uncertainty levels for each of the channel coefficients. The result is extended to K>2 users under additional restrictions which include the assumption of symmetry. [ABSTRACT FROM AUTHOR]

The generalized degrees of freedom (GDoF) characterization of the symmetric $K$ user interference channel is obtained under finite precision channel state information at the transmitters (CSIT). The symmetric setting is where each cross channel is capable of carrying $\alpha $ DoF, while each direct channel is capable of carrying 1 DoF. Remarkably, under finite precision CSIT the symmetric $K$ user interference channel loses all the GDoF benefits of interference alignment. The GDoF per user diminish with the number of users everywhere except in the very strong (optimal for every receiver to decode all messages) and very weak (optimal to treat all interference as noise) interference regimes. The result stands in sharp contrast to prior work on the symmetric setting under perfect CSIT, where the GDoF per user remain undiminished due to interference alignment. The result also stands in contrast to prior work on a subclass of asymmetric settings under finite precision CSIT, i.e., the topological interference management problem, where interference alignment plays a crucial role and provides substantial GDoF benefits. [ABSTRACT FROM PUBLISHER]

We propose a cognitive blind interference alignment (BIA) scheme for macro-femto two-tier networks where the users are equipped with a single reconfigurable antenna. The proposed scheme fully cancels the intracell interference in both the upper and lower tiers. Moreover, it provides the optimal Degrees of Freedom (DoF) in the macro-tier while the lower tier subject to inter-tier interference achieves non-zero DoF. To do so, the proposed scheme relies on a topology management scheme where the users belong to the macro-tier if and only if they can optimally treat the interference caused by the transmission at the femto-tier as noise. The proposed scheme does not require any channel state information at the transmitter or data sharing between the macro- and femto-tiers. An outer bound of the DoF is derived for the considered macro-femto network. It is demonstrated that the proposed scheme can attain the outer bound subject to optimal sum DoF for the macro-tier. Furthermore, compared to previous approaches, cognitive BIA can achieve better sum rates for macro-femto networks for femtocells subject to different levels of inter-tier interference. [ABSTRACT FROM PUBLISHER]

The benefits of partial and full transmitter cooperation are evaluated for a two user interference channel under finite precision channel state information at the transmitters (CSITs), using the generalized degrees of freedom (GDoFs) metric. Under finite precision CSIT, the benefits of interference alignment are completely lost, so that the $X$ channel obtained by partial transmitter cooperation does no better than the underlying interference channels. Full transmitter cooperation produces a vector broadcast channel, which has a strict GDoF advantage over partial cooperation ( $X$ channel) and whose GDoF is fully achieved by interference enhancement. [ABSTRACT FROM AUTHOR]

A conjecture made by Lapidoth et al. at Allerton 2005 (also an open problem presented at ITA 2006) states that the degrees of freedom (DoF) of a two user broadcast channel, where the transmitter is equipped with two antennas and each user is equipped with one antenna, must collapse under finite precision channel state information at the transmitter (CSIT). That this conjecture, which predates interference alignment, has remained unresolved, is emblematic of a pervasive lack of understanding of the DoF of wireless networks—including interference and X networks—under channel uncertainty at the transmitter(s). In this paper, we prove that the conjecture is true in all non-degenerate settings (e.g., where the probability density function of unknown channel coefficients exists and is bounded). The DoF collapse even when perfect channel knowledge for one user is available to the transmitter. This also settles a related recent conjecture by Tandon et al. The key to our proof is a bound on the number of codewords that can cast the same image (within noise distortion) at the undesired receiver whose channel is subject to finite precision CSIT, while remaining resolvable at the desired receiver whose channel is precisely known by the transmitter. We are also able to generalize the result along two directions. First, if the peak of the probability density function is allowed to scale as O((\sqrt {P})^\alpha ) , representing the concentration of probability density (improving CSIT) due to, e.g., quantized feedback at rate ({\alpha }/{2})\log (P)$ , then the DoF is bounded above by $1+\alpha $ , which is also achievable under quantized feedback. Second, we generalize the result to arbitrary number of antennas at the transmitter, arbitrary number of single-antenna users, and complex channels. The generalization directly implies a collapse of DoF to unity under non-degenerate channel uncertainty for the general $K$ -user interference and $M\times N$ user $X$ networks as well. [ABSTRACT FROM AUTHOR]

In a $K$ -user Gaussian interference channel, it has been shown by Geng et al. that if for each user, the desired signal strength is no less than the sum of the strengths of the strongest interference from this user and the strongest interference to this user (all values in decibel scale), then power control and treating interference as noise (TIN) is optimal from the perspective of generalized degrees of freedom (GDoF) and achieves the entire channel capacity region to within a constant gap. In this paper, we generalize the optimality of TIN to compound networks. We show that for a $K$ -user compound Gaussian interference channel, if in every possible state for each receiver, the channel always satisfies the TIN-optimality condition identified by Geng et al., then the GDoF region of the compound channel is the intersection of the GDoF regions of all possible network realizations, which is achievable by power control and TIN. Furthermore, we demonstrate that for a general $K$ -user compound interference channel, regardless of the number of states of each receiver, we can always construct a counterpart $K$ -user regular interference channel that has the same TIN region as the original compound channel. The regular interference channel has only one state for each receiver, which may be different from all of the original states. Solving the GDoF-based power control problem for the compound channel is equivalent to solving the same problem in its regular counterpart. Exploring the power control problem further we develop a centralized power control scheme for $K$ -user compound interference channels, to achieve all the Pareto optimal GDoF tuples. Finally, based on this scheme, we devise an iterative power control algorithm which requires at most $K$ updates to obtain the globally optimal power allocation for any feasible GDoF tuple. [ABSTRACT FROM PUBLISHER]

We study the degrees of freedom (DoF) of the layered $2 \times 2 \times 2$ 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 $2 \times 2 \times 2$ 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 $2 \times 2 \times 2$ channel model. [ABSTRACT FROM AUTHOR]

In many wireless networks, link strengths are affected by many topological factors, such as different distances, shadowing, and intercell interference, thus resulting in some links being generally stronger than other links. From an information theoretic point of view, accounting for such topological aspects is still a novel approach, that has been recently fueled by strong indications that such aspects can crucially affect transceiver and feedback design, as well as the overall performance. This paper here takes a step in exploring this interplay between topology, feedback, and performance. This is done for the two user broadcast channel with random fading, in the presence of a simple two-state topological setting of statistically strong versus weaker links, and in the presence of a practical ternary feedback setting of alternating channel state information at the transmitter [alternating channel state information at the transmitter (CSIT)] where for each channel realization, this CSIT can be perfect, delayed, or not available. In this setting, the work derives generalized degrees-of-freedom bounds and exact expressions, that capture performance as a function of feedback statistics and topology statistics. The results are based on novel topological signal management schemes that account for topology in order to fully utilize feedback. This is achieved for different classes of feedback mechanisms of practical importance, from which we identify specific feedback mechanisms that are best suited for different topologies. This approach offers further insight on how to split the effort—of channel learning and feeding back CSIT—for the strong versus for the weaker link. Further intuition is provided on the possible gains from topological spatio-temporal diversity, where topology changes in time and across users [ABSTRACT FROM PUBLISHER]

It is shown that in the $K$ -user interference channel, if for each user the desired signal strength is no less than the sum of the strengths of the strongest interference from this user and the strongest interference to this user (all values in decibel scale), then the simple scheme of using point-to-point Gaussian codebooks with appropriate power levels at each transmitter and treating interference as noise (TIN) at every receiver (in short, TIN scheme) achieves all points in the capacity region to within a constant gap. The generalized degrees of freedom (GDoF) region under this condition is a polyhedron, which is shown to be fully achieved by the same scheme, without the need for time-sharing. The results are proved by first deriving a polyhedral relaxation of the GDoF region achieved by TIN, and then providing a dual characterization of this polyhedral region via the use of potential functions, and finally proving the optimality of this region in the desired regime. [ABSTRACT FROM AUTHOR]

We propose a blind interference alignment scheme for partially connected cellular networks. The scheme cancels both intracell and intercell interference by relying on receivers with one reconfigurable antenna and by allowing users at the cell edge to be served by all the base stations in their proximity. An outer bound for the degrees of freedom is derived for general partially connected networks with single-antenna receivers when knowledge of the channel state information at the transmitter is not available. It is demonstrated that for symmetric scenarios, this outer bound is achieved by the proposed scheme. On the other hand, for asymmetric scenarios, the achievable degrees of freedom are not always equal to the outer bound. However, the penalty is typically small, and the proposed scheme outperforms other blind interference alignment schemes. Moreover, significant reduction of the supersymbol length is achieved compared with a standard blind interference alignment strategy designed for fully connected networks. [ABSTRACT FROM PUBLISHER]

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 , all cross channels are of rank D$ , and the channels are otherwise generic, the optimal DoF value per user is \min (D_{0},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. [ABSTRACT FROM PUBLISHER]

Starting from the elemental $2\times 2\times 2$ 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 $2\times 2\times 2$ 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 $2\times 2\times 2\times 2$ 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. [ABSTRACT FROM PUBLISHER]

This paper studies linear interference networks, both wired and wireless, with no channel state information at the transmitters except a coarse knowledge of the end-to-end one-hop topology of the network that only allows a distinction between weak (zero) and significant (nonzero) channels and no further knowledge of the channel coefficients' realizations. The network capacity (wired) and degrees of freedom (DoF) (wireless) are found to be bounded above by the capacity of an index coding problem for which the antidote graph is the complement of the given interference graph. The problems are shown to be equivalent under linear solutions. An interference alignment perspective is then used to translate the existing index coding solutions into the wired network capacity and wireless network DoF solutions, as well as to find new and unified solutions to different classes of all three problems. [ABSTRACT FROM PUBLISHER]

We show that an M\times N user MIMO X network with A antennas at each node has A\left(MN\overM+N-1\right) 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\times 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. [ABSTRACT FROM PUBLISHER]

We explore the capacity and generalized degrees of freedom (GDOF) of the two-user Gaussian X channel, i.e., a generalization of the two-user interference channel where there is an independent message from each transmitter to each receiver. There are three main results in this paper. First, we characterize the sum capacity of the deterministic X channel under a symmetric setting. Second, we characterize the GDOF of the Gaussian X channel under a similar symmetric model. Third. we extend the sum capacity characterization previously obtained for the Gaussian interference channel in the noisy interference regime to the Gaussian X channel. Specifically, we show that the Gaussian X channel has the same sum capacity as the underlying Gaussian interference channel in this regime. [ABSTRACT FROM PUBLISHER]

We show that the 2\,\times \,2\,\times \,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. [ABSTRACT FROM PUBLISHER]

We study the effect of the absence of channel knowledge at the transmitters for multiple-input-multiple-output (MIMO) networks. Specifically, we assume perfect channel state information at the receivers, no channel state information at the transmitter(s), and independent identically distributed (i.i.d.) Rayleigh fading across antennas, users and time slots. We provide the characterization of the degrees of freedom (DoF) region for a 2-user MIMO broadcast channel. We then provide a DoF region outer bound for a 2-user MIMO interference channel. This bound is shown to be tight for all possible combinations of the number of antennas at each node except for one case. To analyze the unsolved case, we point out the potential of interference alignment in the 2-user MIMO interference channel with no channel state information at the transmitters. As a byproduct, we explore a special class of MIMO broadcast channels where the capacity region is established by using the outer bound developed in the DoF analysis. [ABSTRACT FROM AUTHOR]

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, Weingarten have 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 Weingarten is 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 Weingarten is 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. [ABSTRACT FROM AUTHOR]

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\over 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\times K~X network as well. Only finite symbol extensions are used, and no channel knowledge at the receivers is required to null the interference. [ABSTRACT FROM PUBLISHER]

Recent results establish the optimality of interference alignment to approach the Shannon capacity of interference networks at high SNR. However, the extent to which interference can be aligned over a finite number of signalling dimensions remains unknown. Another important concern for interference alignment schemes is the requirement of global channel knowledge. In this work, we provide examples of iterative algorithms that utilize the reciprocity of wireless networks to achieve interference alignment with only local channel knowledge at each node. These algorithms also provide numerical insights into the feasibility of interference alignment that are not yet available in theory. [ABSTRACT FROM AUTHOR]

We study a layered K-user M-hop Gaussian relay network consisting of Km nodes in the m^th layer, where M\geq 2 and K=K1=KM+1. We observe that the time-varying nature of wireless channels or fading can be exploited to mitigate the interuser interference. The proposed amplify-and-forward relaying scheme exploits such channel variations and works for a wide class of channel distributions including Rayleigh fading. We show a general achievable degrees of freedom (DoF) region for this class of Gaussian relay networks. Specifically, the set of all (d1,\ldots , dK) such that di\leq 1 for all i and \sum i=1^{K} di\leq K\Sigma is achievable, where di is the DoF of the i^th source-destination pair and K\Sigma is the maximum integer such that K\Sigma \leq \mathop min\limits m\{Km\} and M/K\Sigma is an integer. We show that surprisingly the achievable DoF region coincides with the cut-set outer bound if M/ \mathop min\limits m\{Km\} is an integer; thus, interference-free communication is possible in terms of DoF. We further characterize an achievable DoF region assuming multi-antenna nodes and general message set, which again coincides with the cut-set outer bound for a certain class of networks. [ABSTRACT FROM PUBLISHER]

The N+1 user, 1 \times 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 sum capacity within \cal 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. 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 the \cal O(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 the N+1 user, 1 \times N SIMO 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. [ABSTRACT FROM AUTHOR]

We provide inner bound and outer bound for the total number of degrees of freedom of the K user multiple-input multiple-output (MIMO) Gaussian interference channel with M antennas at each transmitter and N antennas 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) \over \min(M,N)=R is equal to an integer. For this case, we show that the total number of degrees of freedom is equal to \min(M,N)K if K \leq R and \min(M,N)R \over R+1K if K > 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. [ABSTRACT FROM PUBLISHER]

It has been conjectured by Hø-Madsen and Nosratinia that complex Gaussian interference channels with constant channel coefficients have only one degree-of-freedom regardless of the number of users. While several examples are known of constant channels that achieve more than 1 degree-of-freedom, these special cases only span a subset of measure zero. In other words, for almost all channel coefficient values, it is not known if more than 1 degree-of-freedom is achievable. In this paper, we settle the Høst-Madsen–Nosratinia conjecture in the negative. We show that at least 1.2 degrees-of-freedom are achievable for all values of complex channel coefficients except for a subset of measure zero. For the class of linear beamforming and interference alignment schemes considered in this paper, it is also shown that 1.2 is the maximum number of degrees-of-freedom achievable on the complex Gaussian 3 user interference channel with constant channel coefficients, for almost all values of channel coefficients. To establish the achievability of 1.2 degrees-of-freedom we use the novel idea of asymmetric complex signaling – i.e., the inputs are chosen to be complex but not circularly symmetric. It is shown that unlike Gaussian point-to-point, multiple-access and broadcast channels where circularly symmetric complex Gaussian inputs are optimal, for interference channels optimal inputs are in general asymmetric. With asymmetric complex signaling, we also show that the 2 user complex Gaussian X channel with constant channel coefficients achieves the outer bound of 4/3 degrees-of-freedom, i.e., the assumption of time-variations/frequency-selectivity used in prior work to establish the same result, is not needed. [ABSTRACT FROM PUBLISHER]

We characterize the generalized degrees of freedom of the K user symmetric Gaussian interference channel where all desired links have the same signal-to-noise ratio (SNR) and all undesired links carrying interference have the same interference-tonoise ratio, INR = SNRα. We find that the number of generalized degrees of freedom per user, d(α), does not depend on the number of users, so that the characterization is identical to the 2 user interference channel with the exception of a singularity at α = 1 where d(1) = 1/K. The achievable schemes use multilevel coding with a nested lattice structure that opens the possibility that the sum of interfering signals can be decoded at a receiver even though the messages carried by the interfering signals are not decodable. [ABSTRACT FROM AUTHOR]

In this paper, we explore the benefits, from the perspective of degrees of freedom (DOF), of user cooperation and cognitive message sharing for a two-user multiple-input multiple-output (MIMO) Gaussian interference channel with M1, M2 antennas at transmitters and N1, N2 antennas at receivers. For the case of user cooperation (including cooperation at transmitters only, at receivers only, and at transmitters as well as receivers), the sum DOF is min {M1 M1,N1 + N2), max(M1, N2), max(M2, N1)} which is the same as the sum DOF of the channel without cooperation. For the case of cognitive message sharing, the sum DOF is min{M1, M2 + N2, (1 - 1T2) ((1 - 1R2 max(M1, N2) + 1R2, (M1, N2)) + 1T2M1 + M2, (1 - 1T1)((1 - 1R1) ⋅ max(M2, N1) + 1R1M2 + N1) + 1T1M1 + M2)} where 1Ti = 1(0) when transmitter i is (is not) a cognitive transmitter and 1Ri is defined in the same fashion. Our results show that while both techniques may increase the sum capacity of the MIMO interference channel, only cognitive message sharing can increase the sum DOF. We also find that it may be more beneficial for a user to have a cognitive transmitter than to have a cognitive receiver. [ABSTRACT FROM AUTHOR]

It is known that the capacity of parallel (multicarrier) Gaussian point-to-point, multiple access and broadcast channels can be achieved by separate encoding for each subchannel (carrier) subject to a power allocation across carriers. In this paper we show that such a separation does not apply to parallel Gaussian interference channels in general. A counter example is provided in the form of a 3 user interference channel where separate encoding can only achieve a sum capacity of 2 log(1 + 3 SNR) while the actual capacity, achieved only by joint encoding across carriers, is 3 log(1 + 2 SNR). As a byproduct of our analysis, we propose a class of multiple-access-outer bounds on the capacity of the 3 user interference channel. [ABSTRACT FROM AUTHOR]

We explore the degrees of freedom of M x N user wireless X networks, i.e., networks of M transmitters and N receivers where every transmitter has an independent message for every receiver. We derive a general outer bound on the degrees of freedom region of these networks. When all nodes have a single antenna and all channel coefficients vary in time or frequency, we show that the total number of degrees of freedom of the X network is equal to MN/M+N-1 per orthogonal time and frequency dimension. Achievability is proved by constructing interference alignment schemes for X networks that can come arbitrarily close to the outer bound on degrees of freedom. For the case where either M = 2 or N = 2 we find that the degrees of freedom characterization also provides a capacity approximation that is accurate to within O(1). For these cases the degrees of freedom outer bound is exactly achievable. [ABSTRACT FROM AUTHOR]