interference alignment with finite extensions in partially connected networks in this paper we investigate the interference alignment ia for a class of partially connected multi input multi output heterogeneous networks which consist of l fully connected pico cells j partially connected pico cells and one macro cell in particular we investigate the feasibility of ia conditions with no channel extensions t independent channel extensions and t dependent channel extensions respectively for the single beam case we present three theorems to characterize some necessary conditions which l j t and the channel diversity order overline l must jointly satisfy when ia conditions are feasible these necessary conditions also implicitly impose the constraints on the upper bounds of achievable degrees of freedom with different channel extensions

interference alignment with asymmetric complex signaling settling the host madsen nosratinia conjecture it has been conjectured by ho 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 host 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

gaussian interference channel capacity to within one bit the capacity of the two user gaussian interference channel has been open for 30 years the understanding on this problem has been limited the best known achievable region is due to han and kobayashi but its characterization is very complicated it is also not known how tight the existing outer bounds are in this work we show that the existing outer bounds can in fact be arbitrarily loose in some parameter ranges and by deriving new outer bounds we show that a very simple and explicit han kobayashi type scheme can achieve to within a single bit per second per hertz bit s hz of the capacity for all values of the channel parameters we also show that the scheme is asymptotically optimal at certain high signal to noise ratio snr regimes using our results we provide a natural generalization of the point to point classical notion of degrees of freedom to interference limited scenarios

minimization of csi feedback dimension for interference alignment in mimo interference multicast networks it is well known that interference alignment ia can achieve substantial theoretical gains in multiple input multiple output mimo networks however the conventional works usually assume the full channel state information csi is available at the transmitter side which would create a overwhelming csi feedback overhead for practical frequency division duplexing fdd wireless systems to implement ia in practice it is highly desirable to reduce the amount of required csi feedback overhead in this paper we consider ia in mimo interference multicast networks under partial csi feedback and we attempt to minimize the csi feedback cost subject to ia feasibility constraints with a given degree of freedom dof requirements first we propose a csi feedback profile to embrace two important csi feedback reduction strategies and we use the metric of csi feedback dimension to quantify the associated csi feedback cost we then formulate the ia conditions under partial csi feedback in mimo interference multicast networks and we derive new ia feasibility conditions under the proposed partial csi feedback framework based on these results we consider the csi feedback dimension minimization subject to the ia feasibility constraints with a given dof requirements in mimo interference multicast networks which is formulated as a combinatorial optimization problem based on the specific problem structure we derive an asymptotically optimal solution for a category of network topologies and we further obtain closed form tradeoff results between dofs and the csi feedback cost for mimo interference multicast networks

on the capacity of the finite field counterparts of wireless interference networks 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 f p n are analogous to n times 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 mathbb f p n 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

replication based outer bounds on the optimality of half the cake for rank deficient mimo interference networks in order to gain new insights into multiple input multiple output mimo interference networks the optimality of sum k 1 k m k 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 m k 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 m k 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

degrees of freedom of full duplex cellular networks effect of self interference it was recently shown that full duplex fd operation at a base station bs can provide up to twice as many degrees of freedom dof as a conventional half duplex hd cellular network in the absence of self interference in practice however self interference in an fd bs may not be eliminated completely due to imperfect cancellation and it is not yet known whether fd operation can improve the sum dof of cellular networks in the presence of residual self interference in this paper we provide a complete characterization of the sum dof in an fd bs cellular network with self interference where the rank of the self interference matrix is arbitrary specifically we propose a self interference cancellation scheme that maximizes the sum dof and we prove its optimality by deriving the matching upper bound our results show that even in the presence of residual self interference an fd bs network can still outperform a conventional hd bs network and furthermore the sum dof coincides with that of an fd bs network with no self interference under certain conditions we also derive an achievable sum rate under ergodic phase fading showing that not only a sum dof gain but also a sum rate gain can be obtained over the entire signal to noise ratio range even if residual self interference exists

robust lattice alignment for k user mimo interference channels with imperfect channel knowledge in this paper we consider a robust lattice alignment design for k user quasi static multiple input multiple output mimo interference channels with imperfect channel knowledge with random gaussian inputs the conventional interference alignment ia method has the feasibility problem when the channel is quasi static on the other hand structured lattices can create structured interference as opposed to the random interference caused by random gaussian symbols the structured interference space can be exploited to transmit the desired signals over the gaps however the existing alignment methods on the lattice codes for quasi static channels either require infinite signal to noise ratio snr or symmetric interference channel coefficients furthermore perfect channel state information csi is required for these alignment methods which is difficult to achieve in practice in this paper we propose a robust lattice alignment method for quasi static mimo interference channels with imperfect csi at all snr regimes and a two stage decoding algorithm to decode the desired signal from the structured interference space we derive the achievable data rate based on the proposed robust lattice alignment method where the design of the precoders decorrelators scaling coefficients and interference quantization coefficients is jointly formulated as a mixed integer and continuous optimization problem the effect of imperfect csi is also accommodated in the optimization formulation and hence the derived solution is robust to imperfect csi we also design a low complex iterative optimization algorithm for our robust lattice alignment method by using the existing iterative ia algorithm that was designed for the conventional ia method numerical results verify the advantages of the proposed robust lattice alignment method compared with the time division multiple access tdma two stage maximum likelihood ml decoding generalized han kobayashi hk distributive ia and conventional ia methods in the literature

feasibility of interference alignment for the mimo interference channel we study vector space interference alignment for the multiple input multiple output interference channel with no time or frequency diversity and no symbol extensions we prove both necessary and sufficient conditions for alignment in particular we characterize the feasibility of alignment for the symmetric three user channel where all users transmit along d dimensions all transmitters have m antennas and all receivers have n antennas as well as feasibility of alignment for the fully symmetric m n channel with an arbitrary number of users an implication of our results is that the total degrees of freedom available in a k user interference channel using only spatial diversity from the multiple antennas is at most 2 this is in sharp contrast to the k 2 degrees of freedom shown to be possible by cadambe and jafar with arbitrarily large time or frequency diversity moving beyond the question of feasibility we additionally discuss computation of the number of solutions using schubert calculus in cases where there are a finite number of solutions

interference alignment algorithms for the k user constant mimo interference channel this paper considers the degrees of freedom dof for a k user multiple input multiple output mimo m n interference channel using interference alignment ia a new performance metric for evaluating the efficacy of ia algorithms is proposed which measures the extent to which the desired signal dimensionality is preserved after zero forcing the interference at the receiver inspired by the metric two algorithms are proposed for designing the linear precoders and receive filters for ia in the constant mimo interference channel with a finite number of symbol extensions the first algorithm uses an eigenbeamforming method to align sub streams of the interference to reduce the dimensionality of the interference at all the receivers the second algorithm is iterative and is based on minimizing the interference leakage power while preserving the dimensionality of the desired signal space at the intended receivers the improved performance of the algorithms is illustrated by comparing them with existing algorithms for ia using monte carlo simulations

degrees of freedom of mimo cellular networks decomposition and linear beamforming design this paper investigates the symmetric degrees of freedom dof of multiple input multiple output mimo cellular networks with g cells and k users per cell having n antennas at each base station and m antennas at each user in particular we investigate techniques for achievability that are based on either decomposition with asymptotic interference alignment or linear beamforming schemes and show that there are distinct regimes of g k m n where one outperforms the other we first note that both one sided and two sided decomposition with asymptotic interference alignment achieve the same dof we then establish specific antenna configurations under which the dof achieved using decomposition based schemes is optimal by deriving a set of outer bounds on the symmetric dof using these results we completely characterize the optimal dof of any g cell network with single antenna users for linear beamforming schemes we first focus on small networks and propose a structured approach to linear beamforming based on a notion called packing ratios packing ratio describes the interference footprint or shadow cast by a set of transmit beamformers and enables us to identify the underlying structures for aligning interference such a structured beamforming design can be shown to achieve the optimal spatially normalized dof sdof of two cell two user cell network and the two cell three user cell network for larger networks we develop an unstructured approach to linear interference alignment where transmit beamformers are designed to satisfy conditions for interference alignment without explicitly identifying the underlying structures for interference alignment the main numerical insight of this paper is that such an approach appears to be capable of achieving the optimal sdof for mimo cellular networks in regimes where linear beamforming dominates asymptotic decomposition and a significant portion of sdof elsewhere remarkably polynomial identity test appears to play a key role in identifying the boundary of the achievable sdof region in the former case

degrees of freedom for mutually interfering broadcast channels in this paper we study spatial degree of freedom dof for two mutually interfering broadcast channels ibc as the demand for space division multiple access sdma increases the ibc where each link has a single transmit node and multiple receive nodes becomes important this paper presents the results of the lower and upper bounds on the dof of the ibc from the derived results it is shown that for most cases zero forcing zf beamforming can achieve the optimal dof of the ibc except for some special cases also we identify a condition that disabling receive cooperation in the multiple input multiple output interference channels causes no dof loss it is confirmed that we cannot expect a dof improvement by enabling in cell receive cooperation if in at least one of two bss the number of antennas is greater than or equal to that of users per cell furthermore we observe a positive result that as the number of users goes to infinity the total dof of the ibc converges to the interference free dof which is the maximum achievable dof in the absence of the inter cell interference

decentralized delay optimal control for interference networks with limited renewable energy storage in this paper we consider delay minimization for interference networks with renewable energy source where the transmission power of a node comes from both the conventional utility power ac power and the renewable energy source we assume the transmission power of each node is a function of the local channel state local data queue state and local energy queue state only we consider two delay optimization formulations namely the decentralized partially observable markov decision process dec pomdp and noncooperative partially observable stochastic game posg in dec pomdp formulation we derive a decentralized online learning algorithm to determine the control actions and lagrangian multipliers lms simultaneously based on the policy gradient approach under some mild technical conditions the proposed decentralized policy gradient algorithm converges almost surely to a local optimal solution in the noncooperative posg formulation the transmitter nodes are noncooperative we extend the decentralized policy gradient solution and establish the technical proof for almost sure convergence of the learning algorithms in both cases the solutions are very robust to model variations

inner bound on the gdof of the k user mimo gaussian symmetric interference channel the k user multiple input multiple output mimo gaussian symmetric interference channel where each transmitter has m antennas and each receiver has n antennas is studied from a generalized degrees of freedom gdof perspective an inner bound on the gdof is derived using a combination of techniques such as treating interference as noise zero forcing zf at the receivers interference alignment ia and extending the han kobayashi hk scheme to k users as a function of the number of antennas and the log inr log snr level several interesting conclusions are drawn from the derived bounds it is shown that when k n m 1 a combination of the hk and ia schemes performs the best among the schemes considered when n m k n m 1 the hk scheme outperforms other schemes and is found to be gdof optimal in many cases in addition when the snr and inr are at the same level zf receiving and the hk scheme have the same gdof performance

outer bounds on the sum rate of the k user mimo gaussian interference channel this paper derives outer bounds on the sum rate of the k user mimo gaussian interference channel gic three outer bounds are derived under different assumptions of cooperation and providing side information to receivers the novelty in the derivation lies in the careful selection of side information which results in the cancellation of the negative differential entropy terms containing signal components leading to a tractable outer bound the overall outer bound is obtained by taking the minimum of the three outer bounds the derived bounds are simplified for the mimo gaussian symmetric ic to obtain outer bounds on the generalized degrees of freedom gdof the relative performance of the bounds yields insight into the performance limits of multiuser mimo gics and the relative merits of different schemes for interference management these insights are confirmed by establishing the optimality of the bounds in specific cases using an inner bound on the gdof derived by the authors in a previous work it is also shown that many of the existing results on the gdof of the gic can be obtained as special cases of the bounds e g by setting k 2 or the number of antennas at each user to 1

on feasibility of interference alignment in mimo interference networks we explore the feasibility of interference alignment in signal vector space based only on beamforming for k user mimo interference channels our main contribution is to relate the feasibility issue to the problem of determining the solvability of a multivariate polynomial system which is considered extensively in algebraic geometry it is well known e g from bezout s theorem that generic polynomial systems are solvable if and only if the number of equations does not exceed the number of variables following this intuition we classify signal space interference alignment problems as either proper or improper based on the number of equations and variables rigorous connections between feasible and proper systems are made through bernshtein s theorem for the case where each transmitter uses only one beamforming vector the multibeam case introduces dependencies among the coefficients of a polynomial system so that the system is no longer generic in the sense required by both theorems in this case we show that the connection between feasible and proper systems can be further strengthened since the equivalency between feasible and proper systems does not always hold by including standard information theoretic outer bounds in the feasibility analysis

maximum of the riemann zeta function on a short interval of the critical line we prove the leading order of a conjecture by fyodorov hiary and keating about the maximum of the riemann zeta function on random intervals along the critical line more precisely as t for a set of t t 2t of measure 1 o 1 t we have max t u 1log 12 iu 1 o 1 loglogt

degrees of freedom of interference channels with comp transmission and reception we study the degrees of freedom dof of the k user interference channel with coordinated multipoint comp transmission and reception each message is jointly transmitted by m t successive transmitters and is jointly received by m r successive receivers we refer to this channel as the comp channel with a transmit cooperation order of m t and receive cooperation order of m r since the channel has a total of k transmit antennas and k receive antennas the maximum possible dof is equal to k we show that the comp channel has k dof if and only if m t m r geq k 1 the key idea is that the zero forcing of the interference corresponding to the i rm th message at the decoder of the j rm th message where j ne i can be viewed as a shared responsibility between the m t transmitters carrying the i rm th message and the m r receivers decoding the j rm th message for the general case we derive an outer bound that states that the dof is bounded above by left lceil k m t m r 2 2 right rceil for the special case with only comp transmission i e m r 1 we propose a scheme that can achieve k m t 1 2 dof for all k and conjecture that the result holds true for all k in the proposed coding scheme the m t transmitters carrying each message are used to cancel the interference introduced by this message at the first m t 1 receivers thereby allowing each of these receivers to enjoy 1 dof and asymptotic interference alignment is used to align the interfering signals at each other receiver to occupy half the signal space the achievability proofs are based on the notion of algebraic independence from algebraic geometry

achievable sum dof of the k user mimo interference channel with delayed csit this paper considers a k user multiple input multiple output mimo interference channel ic where 1 the channel state information obtained by the transmitters csit is completely outdated and 2 the number of transmit antennas at each transmitter i e m is greater than the number of receive antennas at each user i e n the usefulness of the delayed csit was first identified in a k phase retrospective interference alignment ria scheme proposed by maddah ali and tse for the multiple input single output broadcast channel but the extension to the mimo ic is a non trivial step as each transmitter only has the message intended for the corresponding user recently abdoli et al focused on a single input single output ic and solved such bottleneck by inventing a k phase ria with distributed overheard interference retransmission in this paper we propose two k phase ria schemes suitable for the mimo ic by generalizing and integrating some key features of both abdoli s and maddah ali s works the two schemes jointly yield the best known sum degrees of freedom dof performance so far for the case m n geq k the achieved sum dof is asymptotically given by 64 15 n when k to infty

the approximate capacity region of the symmetric k user gaussian interference channel with strong interference the symmetric k user interference channel ic is studied with the goal of characterizing its capacity region in the strong interference regime within a constant gap the achievable rate region of a scheme combining rate splitting at the transmitters and interference alignment and successive decoding computation at the receivers is derived next it is shown that this scheme achieves the so called greedy max corner points of the capacity region within a constant gap by combining this result with previous results by ordentlich et al on the sum capacity of the symmetric ic a constant gap characterization of the capacity region for the strong interference regime is obtained this leads to the first approximate characterization of the capacity region of the symmetric k user ic furthermore a new scheme that achieves the sum capacity of the channel in the strong interference regime within a constant gap is also proposed and the corresponding gap is calculated the advantage of the new scheme is that it leads to a characterization within a constant gap without leaving an outage set contrary to the scheme by ordentlich et al

outer bounds on the capacity of interference channels corresp new outer bounds are demonstrated for the capacity regions of discrete memoryless interference channels and gaussian interference channels the bound for discrete channels coincides with the capacity region in special cases the bound for gaussian channels improves previous knowledge when the interference is of medium strength

subspace alignment chains and the degrees of freedom of the three user mimo interference channel 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 m d m n min m 2 1 n 2 1 degrees of freedom dof normalized by time frequency and space dimensions where min mt mr n max mt mr 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 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 mr mimo 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

on the interference management for k user partially connected fading interference channels the multiplexing gain muxg of the k user interference channel ic with partially connected interfering links is analyzed the motivation for the partially connected ic comes from the fact that not all interferences are equally strong in practice we introduce the idea of the graphical representation of interference alignment ia condition we show that for a k user partially connected mimo ic with m antennas at each transmitter and receiver the transmit beamforming matrices achieving the muxg of km 2 can be generated sequentially almost surely if the interference channel has a graphical representation which follows certain criteria we also present a novel scheme where we use the idea of partially connected ic to obtain improvements in conventional distributed interference alignment algorithm for fully connected ic the improvements are obtained in terms of greater average sum rate and faster convergence

on the degrees of freedom of finite state compound wireless networks 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

linear precoder design for mimo interference channels with finite alphabet signaling this paper investigates the linear precoder design for k user interference channels of multiple input multiple output mimo transceivers under finite alphabet inputs we first obtain general explicit expressions of the achievable rate for users in the mimo interference channel systems we study optimal transmission strategies in both low and high signal to noise ratio snr regions given finite alphabet inputs we show that a simple power allocation design achieves optimal performance at high snr whereas the well known interference alignment technique for gaussian inputs only utilizes a partial interference free signal space for transmission and leads to a constant rate loss when applied naively to finite alphabet inputs moreover we establish necessary conditions for the linear precoder design to achieve weighted sum rate maximization we also present an efficient iterative algorithm for determining precoding matrices of all the users our numerical results demonstrate that the proposed iterative algorithm achieves considerably higher sum rate under practical qam inputs than other known methods

a new achievable rate region for the interference channel a new achievable rate region for the general interference channel which extends previous results is presented and evaluated the technique used is a generalization of superposition coding to the multivariable case a detailed computation for the gaussian channel case clarifies to what extent the new region improves previous ones the capacity of a class of gaussian interference channels is also established

outer bounds on the capacity of gaussian interference channels two outer bounds on the capacity region of the two user gaussian interference channel ifc are derived the idea of the first bound is to let a genie give each receiver just enough information to decode both messages this bound unifies and improves the best known outer bounds of sato and carleial furthermore the bound extends to discrete memoryless ifcs and is shown to be equivalent to another bound of carleial the second bound follows directly from existing results of costa and sato and possesses certain optimality properties for weak interference

interference alignment with asymmetric complex signaling on mimo x channels it is known that with conventional spatial interference alignment i e linear interference alignment without symbol extension the achievable degrees of freedom dof of mimo x channels is no more than lfloor d outer rfloor where d outer denotes the outer bound of dof of mimo x channels in this paper we propose a spatial interference alignment and zero forcing framework in combination with asymmetric complex signaling which can achieve the dof of 1 2 lfloor 2d outer rfloor for arbitrary number of antennas on each terminal note that 1 2 lfloor 2d outer rfloor is equal to or larger than lfloor d outer rfloor the result shows that the technique of asymmetric complex signaling which was originally proposed for single antenna systems can be useful for mimo channels in particular when symbol extensions are not allowed

degrees of freedom of the k user m times n mimo interference channel 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 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 r and min m n r r 1 k 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

mimo interference alignment in random access networks in this paper we analyze a multiple input multiple output mimo interference channel where nodes are randomly distributed on a plane as a spatial poisson cluster point process a poisson cluster point process consists of clusters with fixed number of points randomly distributed as with the cluster centers distributed randomly on the plane the nodes in each cluster use interference alignment ia to suppress intra cluster interference but unlike most work on ia we do not neglect inter cluster interference we also connect the accuracy of channel state information to the distance between the nodes i e the quality of csi degrades with increasing distance accounting for the training and feedback overhead we derive the transmission capacity of this mimo ia ad hoc network and then compare it to open loop interference blind spatial multiplexing finally we present exemplary system setups where spatial multiplexing outperforms ia due to the imperfect channel state information or the non aligned inter cluster interference

multiple antenna interference channels with real interference alignment and receive antenna joint processing based on simultaneous diophantine approximation in this paper the degrees of freedom dof regions of constant coefficient multiple antenna interference channels are investigated first we consider a k user gaussian interference channel with mk antennas at transmitter k 1 k k and n j antennas at receiver j 1 j k denoted as a k m k n j channel relying on a result of simultaneous diophantine approximation a real interference alignment scheme with joint receive antenna processing is developed the scheme is used to obtain an achievable dof region the proposed dof region includes two previously known results as special cases namely 1 the total dof of k n n interference channel with k users and n antennas at each node is nk 2 and 2 the total dof of a k m n channel is at least kmn m n we next explore constant coefficient interference networks with k transmitters and j receivers all having n antennas each transmitter emits an independent message and each receiver requests an arbitrary subset of the messages employing the novel joint receive antenna processing the dof region for this setup is obtained we finally consider wireless x networks where each node is allowed to have an arbitrary number of antennas it is shown that the joint receive antenna processing can be used to establish an achievable dof region which is larger than what is possible with antenna splitting as a special case of the derived achievable dof region for constant coefficient x network the total dof of wireless x networks with the same number of antennas at all nodes and with joint antenna processing is tight while the best inner bound based on antenna splitting cannot meet the outer bound finally we obtain a dof region outer bound based on the technique of transmitter grouping

on the robustness of lattice interference alignment a static constant channel gains real k user interference channel is considered where all interference cross channel gains are integers for such channels previous results demonstrate that the number of degrees of freedom is very sensitive to slight variations in the direct channel gains in this paper we derive an achievable rate region for such channels that is valid for finite snr at moderate values of snr the derived rate region is robust to slight variations in the direct channel gains at asymptotic high snr conditions known results on the degrees of freedom are recovered the new rate region is based on lattice interference alignment the result is established via a new coding theorem for the two user gaussian multiple access channel where both users use a single linear code

complete characterization of the pareto boundary for the miso interference channel in this correspondence we study the achievable rate region of the multiple input single output miso interference channel under the assumption that all receivers treat the interference as additive gaussian noise our main result is an explicit parametrization of the pareto boundary for an arbitrary number of users and antennas the parametrization describes the boundary in terms of a low dimensional manifold for the two user case we show that a single real valued parameter per user is sufficient to achieve all points on the pareto boundary and that any point on the pareto boundary corresponds to beamforming vectors that are linear combinations of the zero forcing zf and maximum ratio transmission mrt beamformers we further specialize the results to the miso broadcast channel bc a numerical example illustrates the result

joint space decomposition and synthesis approach and achievable dof regions for k user mimo interference channels the paper proposes a new approach of joint space decomposition and synthesis jsds for deriving achievable degree of freedom dof region by interference alignment without symbol extension for k user mimo interference channel solving the interference alignment problem is to find the precoders that align desired signal and interference with disjoint subspaces at receivers a related property of these precoders is that the solution range for each one of them is a union of linear spaces termed transmit spaces based on this property the jsds approach is developed to derive transmit spaces that fulfill alignment conditions from this approach achievable dof regions for 2 and 3 user interference channels are obtained the region for the 2 user channel is shown to coincide with the existing result of the dof capacity of the channel for the k user k 3 case inner as well as outer bounds are given

degrees of freedom region of the mimo x channel we provide achievability as well as converse results for the degrees of freedom region of a multiple input multiple output mimo x channel i e a system with two transmitters two receivers each equipped with multiple antennas where independent messages need to be conveyed over fixed channels from each transmitter to each receiver the inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message with m 1 antennas at each node we find that the total sum rate degrees of freedom are bounded above and below as 1 les eta x les 4 3 if m 1 and channel matrices are nondegenerate then the precise degrees of freedom eta x 4 3 m thus the mimo x channel has noninteger degrees of freedom when m is not a multiple of 3 simple zero forcing without dirty paper encoding or successive decoding suffices to achieve the 4 3 m degrees of freedom if the channels vary with time frequency then the channel with single antennas m 1 at all nodes has exactly 4 3 degrees of freedom the key idea for the achievability of the degrees of freedom is interference alignment i e signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio

interference alignment under limited feedback for mimo interference channels this paper analyzes multiple input multiple output interference channels where each receiver knows its channels from all the transmitters and feeds back this information using a limited number of bits to all the other terminals it is shown that as long as the feedback bit rate scales sufficiently fast with the signal to noise ratio the transmitters can use an interference alignment strategy by treating the quantized channel estimates as being perfect to achieve the sum degrees of freedom of the interference channel attainable with perfect and global channel state information a tradeoff between the feedback rate and the achievable degrees of freedom is established by showing that a slower scaling of feedback rate for any one user leads to commensurately fewer degrees of freedom for that user alone it is then shown that under the same fixed transmission strategy but with random quantization the above mentioned sufficient condition on the feedback scaling rate to attain a given sum degrees of freedom up to the maximum attainable is also necessary in this setting

structured lattice codes for some two user gaussian networks with cognition coordination and two hops we study a number of two user interference networks with multiple antenna transmitters receivers mimo transmitter side information in the form of linear combinations over an appropriate finite field of the information messages and two hop relaying we start with a cognitive interference channel cic where one of the transmitters noncognitive has knowledge of a rank 1 linear combination of the two information messages while the other transmitter cognitive has access to a rank 2 linear combination of the same messages this is referred to as the network coded cic since such linear combination may be the consequence of some random linear network coding scheme implemented in the backbone wired network for such channel we develop an achievable region based on a few novel concepts precoded compute and forward pcof with channel integer alignment cia combined with standard dirty paper coding we also develop a capacity region outer bound and find the symmetric generalized degrees of freedom gdof region of the network coded cic through the gdof characterization we show that knowing mixed data linear combinations of the information messages provides a unbounded spectral efficiency gain over the classical cic counterpart if the ratio in decibel of signal to noise snr to interference to noise is larger than certain threshold then we consider a gaussian relay network having the two user mimo ic as the main building block we use pcof with cia to convert the mimo ic into a deterministic finite field ic then we use a linear precoding scheme over the finite field to eliminate interference in the finite field domain using this unified approach we derive the symmetric sum rate of the two user mimo ic with coordination cognition and two hops we also provide finite snr results not just dof which show that the proposed coding schemes are competitive against state of the art interference avoidance scheme based on orthogonal access for standard randomly generated rayleigh fading channels

the degrees of freedom of isotropic mimo interference channels without state information at the transmitters this paper fully determines the degree of freedom dof region of two user interference channels with arbitrary number of transmit and receive antennas in the case of isotropic and independent or block wise independent fading where the channel state information is available to the receivers but not to the transmitters thus the dof gap between previous upper and lower bounds due to the authors vaze and varanasi and huang jafar shamai and vishwanath is closed the dof result characterizes the capacity region to the first order of the logarithm of the signal to noise ratio snr in the high snr regime the dof region is achievable by using random gaussian codebooks independent of the channel states which implies that it is impossible to increase the dof using beamforming and interference alignment in the absence of channel state information at the transmitters

on the number of interference alignment solutions for the k user mimo channel with constant coefficients in this paper we study the number of different interference alignment ia solutions in a k user multiple input multiple output mimo interference channel when the alignment is performed via beamforming and no symbol extensions are allowed we focus on the case where the number of ia equations matches the number of variables in this situation the number of ia solutions is finite and constant for any channel realization out of a zero measure set and as we prove in this paper it is given by an integral formula that can be numerically approximated using monte carlo integration methods more precisely the number of alignment solutions is the scaled average of the determinant of a certain hermitian matrix related to the geometry of the problem interestingly while the value of this determinant at an arbitrary point can be used to check the feasibility of the ia problem its average properly scaled gives the number of solutions for single beam systems the asymptotic growth rate of the number of solutions is analyzed and some connections with classical combinatorial problems are presented nonetheless our results can be applied to arbitrary interference mimo networks with any number of users antennas and streams per user

approximate ergodic capacity of a class of fading two user two hop networks the fading awgn two user two hop network is considered where the channel coefficients are independent and identically distributed i i d according to a continuous distribution and vary over time for a broad class of channel distributions the ergodic sum capacity is characterized to within a constant number of bits second hertz independent of the signal to noise ratio the achievability follows from the analysis of an interference neutralization scheme where the relays are partitioned into m pairs and interference is neutralized separately by each pair of relays when m 1 the proposed ergodic interference neutralization characterizes the ergodic sum capacity to within 4 bits sec hz for i i d uniform phase fading and approximately 4 7 bits sec hz for i i d rayleigh fading it is further shown that this gap can be tightened to 4 log 4 bits sec hz approximately 2 6 for i i d uniform phase fading and 4 4 log 3 8 bits sec hz approximately 3 1 for i i d rayleigh fading in the limit of large m1

the degrees of freedom of the k user gaussian interference channel is discontinuous at rational channel coefficients the degrees of freedom of a k user gaussian interference channel gic has been defined to be the multiple of 1 2 log 2 p at which the maximum sum of achievable rates grows with increasing power p in this paper we establish that the degrees of freedom of three or more user real scalar gics viewed as a function of the channel coefficients is discontinuous at points where all of the coefficients are nonzero rational numbers more specifically for all k 2 we find a class of k user gics that is dense in the gic parameter space for which k 2 degrees of freedom are exactly achievable and we show that the degrees of freedom for any gic with nonzero rational coefficients is strictly smaller than k 2 these results are proved using new connections with number theory and additive combinatorics

the capacity of the gaussian interference channel under strong interference corresp the capacity region of a gaussian interference channel with two separate messages is obtained for the case of moderately strong interference it is shown that the region coincides with the one where both messages are required in both receiving terminals

linear degrees of freedom for the m times n mimo interference channels with constant channel coefficients this paper analyzes the linear degrees of freedom ldof for k user m n mimo interference channels with constant channel coefficients in this correspondence we interpret the interference alignment problem as a multivariate polynomial system in terms of properness the properness provides a ldof upperbound for interference alignment with this approach we propose a method to obtain the upperbound for total ldof achievable by linear interference alignment the ldof upperbound for properness condition results have two different cases which depend on the number of users the number of transmit antennas and the number of receive antennas a key contribution of this paper is that it provides a closed form ldof upperbound for m n mimo interference channels where the ldof of each user may be different

constrained interference alignment and the spatial degrees of freedom of mimo cognitive networks an interference alignment ia scheme is presented that allows multiple opportunistic transmitters secondary users to use the same frequency band of a pre existing primary link without generating any interference the primary and secondary transmit receive pairs are equipped with multiple antennas under power constraints on the primary transmitter the rate of the primary user is maximized by water filling on the singular values of its channel matrix leaving some eigen modes unused and hence the secondary users can align their transmitted signals to produce a number of interference free dimensions at each secondary receiver without causing any interference to the primary user an outer bound is developed on the degrees of freedom dof of the secondary users in the case of a symmetric secondary network with time varying channel coefficients having m antennas at each node and operating in the presence of a primary link with d0 active eigen modes a precoding scheme is presented for the secondary transmitters that can asymptotically achieve the available m d0 2 dof per secondary user an iterative algorithm is also presented that utilizes channel reciprocity to achieve the proposed cognitive ia scheme for a cognitive 3 user secondary network with constant channel coefficients a novel closed form solution is derived for the precoding matrices of the secondary users

beam tracking for interference alignment in slowly fading mimo interference channels a perturbations approach under a linear framework in this paper the beam design for signal space interference alignment in slowly fading multiuser multiple input multiple output mimo interference channels is considered based on a linear formulation for interference alignment a predictive beam tracking algorithm is proposed using matrix perturbation theory the proposed algorithm based on a mixture of iteration and update computes interference aligning beamforming vectors at the current time by updating the previous beam vectors based on the channel difference between the two time steps during the predictively updating phase and yields significant reduction in computational complexity compared with existing methods recalculating beams at each time step the tracking performance of the algorithm is analyzed in terms of mean square error and sum rate loss between the predictively updating approach and the recalculating approach and the impact of imperfect channel knowledge is also investigated under the state space channel model numerical results show that the proposed algorithm has almost the same performance as non predictive methods in sum rate thus the proposed algorithm provides a very efficient way to realize interference alignment in a realistic slowly fading mimo channel environment

degrees of freedom of mimo x networks spatial scale invariance and one sided decomposability 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

rank constrained schur convex optimization with multiple trace log det constraints rank constrained optimization problems have received an increasing intensity of interest recently because many optimization problems in communications and signal processing applications can be cast into a rank constrained optimization problem however due to the nonconvex nature of rank constraints a systematic solution to general rank constrained problems has remained open for a long time in this paper we focus on a rank constrained optimization problem with a schur convex concave objective function and multiple trace log determinant constraints we first derive a structural result on the optimal solution of the rank constrained problem using majorization theory based on the solution structure we transform the rank constrained problem into an equivalent problem with a unitary constraint after that we derive an iterative projected steepest descent algorithm which converges to a local optimal solution furthermore we shall show that under some special cases we can derive a closed form global optimal solution the numerical results show the superior performance of our proposed technique over the baseline schemes

hierarchical blind interference alignment over interference networks with finite coherence time we investigate a blind interference alignment bia scheme through staggered antenna switching over various interference networks e g broadcast channel interference channel and cellular networks with realistic channel assumptions in existing bia the coherence time of channel is assumed to be long enough but that may not always be true in realistic scenarios therefore we propose a dynamic supersymbol design method which can construct a supersymbol with limited symbol extension that is determined by the coherence time of channel it is demonstrated that the supersymbol block length can be reduced significantly by aligning interferences in a hierarchical manner referred to as hierarchical bia the key idea of hierarchical bia is to align interferences in groups and to use the same supersymbol structure between groups producing aligned inter group interferences without inner group interference consequently it is observed that with a given coherence time the proposed dynamic supersymbol design that exploits hierarchical bia achieves higher degrees of freedom than the conventional method

a feasibility test for linear interference alignment in mimo channels with constant coefficients in this paper we consider the feasibility of linear interference alignment ia for multiple input multiple output mimo channels with constant coefficients for any number of users antennas and streams per user and propose a polynomial time test for this problem combining algebraic geometry techniques with differential topology ones we first prove a result that generalizes those previously published on this topic in particular we consider the input set complex projective space of mimo interference channels the output set precoder and decoder grassmannians and the solution set channels decoders and precoders satisfying the ia polynomial equations not only as algebraic sets but also as smooth compact manifolds using this mathematical framework we prove that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is generically surjective if that mapping is not surjective then the solution variety projects into the input space in a singular way and the projection is a zero measure set this result naturally yields a simple feasibility test which amounts to checking the rank of a matrix we also provide an exact arithmetic version of the test which proves that testing the feasibility of ia for generic mimo channels belongs to the bounded error probabilistic polynomial complexity class

csi feedback reduction for mimo interference alignment interference alignment ia is a linear precoding strategy that can achieve optimal capacity scaling at high snr in interference networks most of the existing ia designs require full channel state information csi at the transmitters which induces a huge csi signaling cost hence it is desirable to improve the feedback efficiency for ia and in this paper we propose a novel ia scheme with a significantly reduced csi feedback to quantify the csi feedback cost we introduce a novel metric namely the feedback dimension this metric serves as a first order measurement of csi feedback overhead due to the partial csi feedback constraint conventional ia schemes can not be applied and hence we develop a novel ia precoder decorrelator design and establish new ia feasibility conditions via dynamic feedback profile design the proposed ia scheme can also achieve a flexible tradeoff between the degree of freedom dof requirements for data streams the antenna resources and the csi feedback cost we show by analysis and simulations that the proposed scheme achieves substantial reductions of csi feedback overhead under the same dof requirement in mimo interference networks

continuum of weakly coupled oscillatory mckean neurons the mckean model of a neuron possesses a one dimensional fast voltage like variable and a slow recovery variable a recent geometric analysis of the singularly perturbed system has allowed an explicit construction of its phase response curve s coombes 2001 phase locking in networks of synaptically coupled mckean relaxation oscillators physica d vol 160 173 188 here we use tools from coupled oscillator theory to study weakly coupled networks of mckean neurons using numerical techniques we show that the mckean system has traveling wave phase locked solutions consistent with that of a network of more biophysically detailed hodgkin huxley neurons

limited feedback design for interference alignment on mimo interference networks with heterogeneous path loss and spatial correlations interference alignment is degree of freedom optimal on k user mimo interference channels and many previous works have studied the transceiver designs however these works predominantly focus on networks with perfect channel state information at the transmitters and symmetrical interference topology in this paper we consider a limited feedback system with heterogeneous path loss and spatial correlations and investigate how the dynamics of the interference topology can be exploited to improve the feedback efficiency we propose a novel spatial codebook design and perform dynamic quantization via bit allocations to adapt to the asymmetry of the interference topology we bound the system throughput under the proposed dynamic scheme in terms of the transmit snr feedback bits and the interference topology parameters it is shown that when the number of feedback bits scales with snr as c s cdot log hbox snr cal o 1 the sum degrees of freedom of the network are preserved moreover the value of scaling coefficient c s can be significantly reduced in networks with asymmetric interference topology

on the achievable degrees of freedom of two cell multiuser mimo interference networks in this paper we study the sum degrees of freedom dof of an uplink two cell multiuser mimo interference network with asymmetric number of users in the cells the achievable dof is devised based on a two dimensional space time spreading code framework with linear precoding decoding design and finite channel extension the derivation of the achievable dof is shown related to a rank minimization problem which corresponds to the minimization of the dimension of the interference subspace the problem is solved by the proposed grouping algorithm ga based on aligning interfering signals into a low dimensional subspace as a group and attaining the minimum number of groups the achievable sum dof derived based on the proposed ga is shown to be greater than prior arts and achieves the theoretic upper bound in several cases we also give a closed form expression of the max imum achievable sum dof when there is the maximum number of admissible users in the considered finite diversity environment