Achieving Optimal Degrees of Freedom for an Interference Network with General Message Demand

—The concept of degrees of freedom (DoF) has been adopted to resolve the difficulty of studying the multi-user wireless network capacity regions. Interference alignment (IA) is an important technique developed recently for quantifying the DoF of such networks. In the present study, a single-hop interference network with K transmitters and N receivers is taken into account. Each transmitter emits an independent message and each receiver requests an arbitrary subset of the messages. Using the linear IA techniques, the optimal DoF assignment has been analyzed. Assuming generic channel coefficients, it has been shown that the perfect IA cannot be achieved for a broad class of interference networks. Analytical evaluation of DoF feasibility for general interference channels (IFCs) is complicated and not available yet. Iterative algorithm designed to minimize the leakage interference at each receiver is extended to work with general IFCs. This algorithm provides numerical insights into the feasibility of IA, which is not yet available in theory.


INTRODUCTION
Despite the intensive research on multi-user wireless communication networks over the past three decades, the subject is still not well understood from information-theoretic perspective.Capacity limits of many multi-user networks are still unknown (even for a small number of users).For example, the capacity region of a two-user single-antenna interference channel (IFC) is still unknown, though the region can be bounded up to a constant value [1].Researchers have derived various approximations of the capacity region of the IFCs.For example, the maximum total degrees of freedom (DoF) correspond to the first order approximation of the sum-rate capacity in high SNR regime.DoF is not a new concept, it is widely known as the multiplexing gain in point-to-point communication scenarios.It was termed as spatial DoF in [2], referring to the maximum multiplexing gain.In the present study, the maximum multiplexing gain is presumed to be the total DoF.The study of DoF of interference networks was pioneered in [3].In [2], the total DoF of two-user multipleinput multiple-output IFC is provided.Interference alignment (IA) is a powerful scheme developed recently for quantifying the DoF region of multi-user networks.IA refers to the construction of signals in such a way that their mutual interference aligns at the receivers, facilitating simple interference cancellation techniques.The remaining dimensions are designated for communicating the desired signal, keeping it free from interference.IA was first introduced in [4], and clarified in [5].The analysis of DoF is shown to be very useful in revealing the capacity potential of the IFCs.The channel capacity of IFC is limited with large number of users.Authors in [3,6], showed that the total number of η=ΚΜ/2 DoF can be achieved asymptotically via infinite time (frequency) expansion under block fading channel for a K-user M antenna IFC.This result indicates that the capacity of each user is unbounded regardless of the user number K. The achieving method is based on IA.
The principal assumption enabling this surprising result is that the channel extensions are exponentially long in K and are generic (e.g., drawn from a continuous probability distribution).However, there is an important distinction between perfect IA schemes and partial IA schemes.Perfect IA schemes are able to exactly achieve the outer bound of the DoF with a finite symbol extension of the channel.In contrast, partial IA schemes pay a penalty in the form of the overflow room required to "almost" align interference.Study of the design and feasibility of linear beamforming for IA without symbol extensions has received significant attention [7][8][9][10].
In general, linear IA can be described by a set of bilinear equations which correspond to the zero-forcing conditions at each receiver.One can count the number of "independent unknowns" and the number of scalar equations in this quadratic system defining IA to verify if the total number of variables exceeds the total number of constraints in the system of equations.If a system has more variables than constraints, it is called a proper system.Otherwise, it is called an improper system [7].While it is known that almost all improper systems are infeasible [8][9], the feasibility of the proper systems is still an area of active research.In [8][9][10], a set of sufficient conditions for feasibility are established.In this paper, we consider the case of interference networks with general message demands.In this setup, there are K transmitters and N receivers, each equipped with M antennas. [jk] (1), H [jk] (2), .Denoting th age k as V [k] , ollows, 1) T , Y [j] (2) T ,… ver the extend atrix of transm nal symbols tra r of the transm vector at rec I) is assumed nsmitters.DoF region for the setup described in Section II has been derived by [11], as follows, (5) where is the set of message indices requested by receiver .Optimal DoF assignment in an IFC with general message demands is obtained by solving the following linear programming problem; (6) where z is defined as a G×1 vector consisted of scalar elements .implies that all elements of z should be smaller than M. The solution to the feasibility problem is not known in general.In other words, given a set of randomly generated channel matrices and a degree-of-freedom allocation , it is not known if one can almost surely find the transmit and receive filters that will satisfy the feasibility conditions.If we look at the problem of finding the precoding and zero-forcing matrices when the channel coefficients are arbitrary and given, the problem is computationally intractable.In particular, [8] established that for an arbitrary K-user MIMO IFC, checking the achievability of a certain DoF tuple is NP-hard when each user and transmitter has more than 2 antennas.The notion of regular IFC will be introduced below and it will be shown that perfect IA is not feasible for single antenna regular IFC.The distributed IA developed in this paper will be useful in numerically evaluating the feasibility problem for general IFC.Regular networks are defined as IFCs for which the only optimal DoF assignment is equal DoF assignment for all active transmitters.Theorem 1 implies that an IFC in which all prime receivers request the same number of transmitted messages and each transmitter sends message to an equal number of prime receivers is a regular network.
 Theorem 1.The only DoF point that achieves total number of DoF of an IFC where all receivers request the same number of transmitted messages and each transmitter sends message to equal number of receivers is , ,..., 1 1 1 Proof.Maximum total number of DoF for this network is obtained to be 1 , where β is the number of requested messages for each prime receiver [11].Obviously, total DoF is . It is now intended to show that this is the only DoF point that achieves total number of DoF.It is worth noting that as prime receivers are addressed those receivers whose requested message sets are not a subset of any other requested message set.If Theorem 1 is not true, there is at least one which is strictly greater than .We would also have the following lemma: In the specified channel structure, we should have (8) where G is the number of prime receivers.
Assume that there is a where , which implies that .Thus, using (5), we will have (4) The first equality, a , is derived from the fact that , is derived from DoF region inequalities in (5) and c is implied from the assumption that .Equation ( 9) contradicts the assumption that this DoF assignment achieves total DoF number.Therefore, the lemma is valid.
Based on (5), in order to characterize DoF region, we should consider G inequalities of the form (5) Since each message is requested by receivers, summing all G inequalities, we have (11) where is defined as .Using the fact that there is at least one strictly greater than , along with lemma 1, it is obtained that (12) substituting ( 12) in (11) Since diagonal channel matrices are almost surely full rank, after some algebraic manipulations, (18) becomes (10) where matrices are obtained as follows, (11) and C M is defined as (12) Assuming is of rank n, [13] has shown that (20) implies n eigenvectors of lie in .Since all channel matrices are diagonal, the set of eigenvectors of channel matrices, their inverse and product are column vectors of the identity matrix.Define and note that e i exists in , therefore, the set of equations in (19) implies that (13) Thus, at receiver 1, the desired signal is not linearly independent of the interference signal, , and hence, receiver 1 cannot fully decode W [1] and W [4] solely by zero-forcing the interference signal.Therefore, if the channel coefficients are completely random and generic, we cannot obtain 6/3 DoF for the 6×3 user single antenna IFC through linear IA schemes.
Regular IFCs are mostly encountered in networks with few users.We have evaluated optimal DoF assignment for random configurations of a 10×10 IFC.Roughly, 25% of the evaluated networks were regular IFCs.It is implied that there are many cases where perfect IA for a single antenna IFC is not feasible.

IV. DISTRIBUTED IA ALGORITHM FOR GENERAL IFC
In general, the optimal DoF solution for each specific configuration can be obtained by solving the linear programming problem (6), using methods like simplex algorithm.The closed form solution for each arbitrary requested message set structure is not straightforward.Moreover, analytical evaluation of DoF feasibility is not straightforward.Interference can be aligned in networks with single antenna nodes through the channel extension in frequency or time as long as the channel is varying across frequency or time.However, this approach requires long symbol extensions.For example, the achievable scheme presented in [11] uses at least 1536 extensions of the single 6×3 antenna introduced in Figure 2 to achieve d={1/3, 1/12, 1/24, 1/192, 1/1536} number of DoFs for the transmitters.Total DoF number of 0.47 is achieved for this scheme which is still far less than the optimal DoF of 2 for this channel.Due to these difficulties, it is preferred if IA can be accomplished without or with limited symbol extensions.Actually, IA schemes are most likely to be used in MIMO networks.Feasibility of perfect IA is not yet available in theory for most MIMO networks.Iterative algorithms provide numerical insights into the feasibility of IA in these cases.In this section, distributed IA algorithm presented in [12] is extended to be utilized for the case of general IFC with multiple antenna nodes and with no symbol extensions.
Based on the system model presented in Section II, IA feasibility conditions are derived as follows, (23) while the rank condition is typically assumed to be automatically satisfied, this is not true in the case of symbol extensions because channel matrices cannot be assumed to be generic.When no symbol extensions is applied and the MIMO channels are generic, the rank condition is satisfied with probability one.In other words, with generic channels there is no need to explicitly introduce the rank constraint into the optimization problem.Equation ( 23) requires that at each receiver, all interferences must be suppressed, leaving as many interference-free dimensions as the DoF allocated to that receiver.
The intention is to use alternating optimization procedure similar to [12], to find the transmit precoding and receive zeroforcing matrices.We start with arbitrary transmit and receive filters and iteratively update these filters to approach IA.Alternating optimization procedure is realized by switching the direction of communication.The total interference leakage at receiver j due to all undesired transmitters is given by: (24)

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, we will have the following inequality: