Engineering, Technology & Applied Science Research Vol. 8, No. 4, 2018, 3141-3148 3141 www.etasr.com Samadi et al.: Achieving Optimal Degrees of Freedom for an Interference Network with General … Achieving Optimal Degrees of Freedom for an Interference Network with General Message Demand Zainalabedin Samadi School of Electrical Engineering Iran University of Science & Technology Tehran, Iran z.samadi@gmail.com Vahid Tabataba Vakili School of Electrical Engineering Iran University of Science & Technology Tehran, Iran vakily@iust.ac.ir Farzan Haddadi School of Electrical Engineering Iran University of Science & Technology Tehran, Iran haddadi@iust.ac.ir Abstract—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. Keywords-interference channel; interference alignment; degrees of freedom; channel state information I. 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 multiple‐ input 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 [7-10]. 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 [8-10], 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. Each transmitter em arb gen [6] sub net opt to alg Ma lea per eff int obt con ext cas sce mo wh ach cha pro me V, K tra the on req Le req and con tra gre and a r tim rel wh sig fad Th ass dis rec Engineerin www.etasr mits a unique bitrary subset neralization o ], to the case w bset of the twork is evalu timally alloca achieve the to As an alter gorithmic appr any of those akage at each rfectly be a fectiveness is terference leak tain the optim ncerned with tended the ite se of general enarios. The rest of t odel is introdu hy linear IA hieve the tota annel extensio oposed to ach essage demand and concludin Consider a K transmitters ansmitters and e system mod e and only o quest an arbitr et Sj, j ϵ[N] be quested by rec d is the s ntributing to is ansmitters are eater than zero d want to achi reliable commu me slot , t  lationship: here 1≤j≤N is gnal of transm ding factors o he channel fa sumed to be stribution. Z[j] ceiver j. The ng, Technology r.com message and t of the me of the multiple where each rec transmitted m uated in [11]. ate DoFs to th otal DoF of suc rnative to the roaches have methods aim receiver so t attained. The s that when kage will be mal solutions the feasibilit erative IA algo l IFC to eva the paper is o uced in Sectio scheme, over al number of ons. In Section hieve optimal ds. Simulation ng remarks are II. SY K×N user sing and N recei receivers hav del is shown in one independe rary set of mes the set of indi ceiver j, where set of indices the interfere s the set of all defined as the o. All transmi ieve the maxim unication. Cha  is characteriz the user inde mitter k, H[jk](t) f the channel ading factors e independen ](t) is the ad noise terms y & Applied Sci Sama each receiver ssages. This e unicasts scen ceiver is intere messages. Do In this work, e transmitters ch a network. e closed form been propose m to minimiz that ‐at the be e suggested n perfect IA zero and suc . In this pap ty of the per orithm develo luate IA feas organized as fo on II. In Secti r a single an DoF with a n IV, an iterat l DoF of an n results are p e presented in YSTEM MODEL gle‐hop interfe ivers. We as ve M antennas n Figure 1. E ent message. ssages from m ces of those tr e [N] is defined s of those tra ence at rece l active transm e transmitters itters share a c mum possible annel output at zed by the foll ex, X[k](t) is th t), 1≤k≤K is t from transmi at different ntly drawn fr dditive white are all assu ience Research adi et al.: Achiev r is interested construction nario consider ested in an arb oF region for we evaluate h of this archit m designs, s ed in the liter ze the interfe est case‐the IA insight for A is possible ch algorithms per, we are m rfect IA. We oped in [12], sibility of dif follows. The s ion III, it is a ntenna IFC, c limited numb tive IA algorit IFC with g provided in S Section VI. erence network ssume that a s. An illustrati Each transmitte Each receive multiple transm ransmitted mes d as [N]={1,… ansmitted mes eiver j. Obvi mitters, where with assigned common band sum rate along t receiver j and lowing input‐o (1) he M×1 transm the M×M mat itter k to recei time instant from a conti Gaussian no umed to be d h V ving Optimal D in an is a red in bitrary r this how to ecture everal rature. erence A can their e, the s may mostly have to the fferent system argued cannot ber of thm is eneral ection k with ll the ion of er has er can mitters. ssages …..,N} ssages ously, active d DoF dwidth g with d over output mitted trix of iver j. ts are inuous ise at drawn ind dist assu P: whe ind the belo exp diag tran be d whe sign bea ind is a rece info glob as a hav reg cor Vol. 8, No. 4, 20 Degrees of Free dependently f tribution with umed that all ere E is the dex is omitted time expansio ow, we use panded signals H[jk]=diag(H[ gonal matrix. nsmitted mess described as fo ere Y[j]=[Y[j](1 nal vector ov amforming ma dependent sign a d[k]×1 vector eived noise ormation (CSI bal CSI at tran Authors in [1 an interference ve derived Do ion of such a rresponding Do The total DoF 018, 3141-3148 edom for an Int from a Gau h zero mean transmitters a expectation ta for convenien on in number uppercase bo s. [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. Fig. 1. 11], have refer e network wit oF region of a system with oF region is de F number is de 8 terference Netw ussian indepe and unit va are subject to aken over tim nce. Let den of symbols. F old fonts to , , … H[jk](τ)), he beamformi the extended ., Y[j](τ)T]Τ is ded channel, V mitter k, and d[ ansmitted from mitted symbols ceiver j. Perf to be availab user IFC mode rred to the afo th general mes this setup. D h power cons efined as efined as: 3142 work with Gener endent identi ariance. It is a power cons (1) me. Hereafter, note the durati From this poin denote the is a Μτ×Μτ b ing matrix o channel mode (2) the Μτ×1 rec V[k] is the Μτ [k] is the numb m transmitter k and is a Μ fect channel ble at receivers el. orementioned ssage demand enote the cap straint P as (3) . ral … ically also straint time on of nt and time‐ block f the el can eived τ×d[k] ber of k. X[k] Μτ×1 state s and setup ds and pacity , Engineering, Technology & Applied Science Research Vol. 8, No. 4, 2018, 3141-3148 3143 www.etasr.com Samadi et al.: Achieving Optimal Degrees of Freedom for an Interference Network with General … III. LINEAR VECTOR IA LIMITATION 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 M M M β β β         d (4) Proof. Maximum total number of DoF for this network is obtained to be 1 KM β  , where β is the number of requested messages for each prime receiver [11]. Obviously, total DoF is achieved by [ ] 1 k Md β   . 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), we will have the following inequality: wh ach Do net  mo bro is n sat net ele con Do net me can a s the eac ach ext con me eve thi tot at usi me Engineerin www.etasr hich contradic hieves total D oF point that twork. Theorem 2. generic, the regular inter finite extensi Proof. It sho ore than 1 Do oadcast or mu not unique in t tisfies twork. For ex ements, receiv nsidered as a oF gain, and twork is 1. Th ethods like tim n achieve the special case o e channel stru ch receiver is s Fig. 2. The proof fo hieve the oute tensions of nsider a hypot essages achiev ery transmitte is is the only tal number of D Let message transmitter k ing zero‐forci essages ng, Technology r.com cts the assum oF number. T achieves the Assuming th total number rference netwo ion of the chan ould be noted oF, otherwise ultiple access c this case. In fa ac xample, if a s ver and its c a multiple acc therefore the here is no nee me division b total number o of 6×3 user IF ucture along w shown in Figu 6 3 user IFC or the general er bound of 6/ this channel. thetical achiev ves 1 DoF if er and zero‐for DoF point in DoF of this ne be bea k. Receiver j ing. At receiv , the y & Applied Sci Sama mption that thi herefore, e total numbe hat the chann of the DoF o ork cannot be nnel. that the regul , the network channel. Optim act, every K×1 chieves optim et, say Si, wo orresponding cess channel e total numb ed for IA in th based multiple of Dof of this FC with gener with requested ure 2. with general mes case is simila /3 DoF for thi Over this vable scheme w possible, usin rcing at every n achievable r etwork, accord mformed alon intends to d ver j, to dec e vectors cor ience Research adi et al.: Achiev (6) is DoF assign is the er of DoF o nel coefficient of a single an e achieved us ar IFC should k is equivalen mal DoF assign 1 vector mal DoF of ould have K o transmitters c without losin er of DoF o his case and s e access techn structure. Con ralized messag set of messa sage demands. ar. It is intend is setup. Cons extended ch where each of ng beamformi y receiver. Not egion that ach ding to Theorem ng 3×1 vector ecode ode 2 indepe rresponding t h V ving Optimal D nment e only of this ts are ntenna sing a d have nt to a nment that f this or K-1 can be g any of this simple niques nsider ge set, ges at ded to sider 3 hannel, f the 6 ing at te that hieves m 1. endent to the des dire the dim rece abo line rece Wh cor resp the unk que not are sho bili exp var Mo infe vec con ind whe Nυ as whe at r ante netw requ the acc whe usin eve Vol. 8, No. 4, 20 Degrees of Free sired message ections. Since 4 interferin mension. IA req Thus, the to eiver j can de ove criteria, th early indepen eiver. This req here D(S) is rresponding to pectively, and receivers. Fo Observe that known precod estion is raised t. Authors in [ selected rand ould have almo inear equations pected to be riables is less oreover, [8] p easible when ctor. It can be nstitutes an i dependent varia ere L is is less than th ere receiver j. The ennas at each work. In the f quirements is s channel. Consider a 3n cording to The ere each of th ng beamform ery receiver. T 018, 3141-3148 edom for an Int es should oc signals come ng vectors m quirements ca otal dimension ecode all its de he desired sig dent of the quirement imp defined as th are the se o the desired d 3 is the total or example, a . t (14) is a se ding and zer d as to whethe [7] argued tha domly and ind ost surely a per s is called pro feasible almo than or equ proved that im each transmit observed that mproper set ables in this sy the numb is the num he number of is th erefore, using h node, we c following, it i still infeasible n symbol exten orem 1, the on he 6 transmitt ming at every The 3n×n vec 8 terference Netw ccupy 2 lin e from a space must occupy an be written as n of the inte esired messag gnal vectors a interference d plies that, he dimension ets of the recei d and undesir subspace dim at user 1, et of bilinear ro‐forcing filt er a system ad at when the ch dependently, t rfect alignmen per, in other w ost surely wh al to the num mproper syste tter uses only the set of IA of equations ystem can be o ber of cha mber desired st equations, Νe, he number of 3 extensions o cannot achiev is proved that e using every nsion of the ch nly achievable ters achieves transmitter an ctors 3144 work with Gener early indepen e of 3 dimens the remainin s follows. (7) erference is 1 ges. Along wit are required t dimension at (8) of a subspac ived signal ve red signal vec mension availab is obtaine r equations in ters. A feasi dmits perfect I hannel coeffic the proper sys nt solution. A words, perfect hen the numb mber of equat em of equatio one beamfor requirements, s. The numbe obtained as (16) annel extens treams at recei , which is obt (17) interfering str of the channel ve 6 DoF for t this system o finite extensio hannel over w e scheme is the n DoF if pos nd zero‐forcin sh ral … ndent sions, ng 1 1 and th the to be each ce S, ectors ctors, ble at ed as n the ibility IA or cients stems set of IA is er of tions. ons is rming (14), er of ) sions. iver j. ained reams l or 3 r this of IA on of which, e case sible, ng at hould Engineering, Technology & Applied Science Research Vol. 8, No. 4, 2018, 3141-3148 3145 www.etasr.com Samadi et al.: Achieving Optimal Degrees of Freedom for an Interference Network with General … satisfy IA conditions along with the linear independence condition. IA requirements at 3 receivers can be summarized as follows, (9) Since diagonal channel matrices are almost surely full rank, after some algebraic manipulations, (18) becomes (10) where matrices are obtained as follows, (11) and CM 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 ei 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 zero‐ forcing 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) Engineering, Technology & Applied Science Research Vol. 8, No. 4, 2018, 3141-3148 3146 www.etasr.com Samadi et al.: Achieving Optimal Degrees of Freedom for an Interference Network with General … where (14) Let us define as the set of receivers that request message from transmitter k, i.e., . In the reciprocal network, the total interference leakage at receiver k due to all undesired transmitters is given by (26): (15) where (16) is the interference covariance matrix at receiver k. The iterative algorithm alternates between the original and reciprocal networks. In the original network, each receiver solves the following optimization problem. (17) In other words, receiver j chooses its interference suppression filter to minimize the leakage interference due to all undesired transmitters. The dimensional received signal subspace that contains the least interference is the space spanned by the eigenvectors corresponding to the smallest eigenvalues of the interference covariance matrix . Thus, the columns of are given by: (29) where is the eigenvector corresponding to the smallest eigenvalue of . For the second step, consider the reciprocal network obtained by reversing the roles of the transmitters and the receivers. The transmit precoding matrices in the reciprocal network, , are the receive interference suppression matrices from the original network. Each receiver in the reciprocal network solves the following optimization problem. (30) Similarly, the columns of are given by: (31) The receive zero‐forcing filters in the reciprocal network are then used as the transmit precoding matrices in the original network, and the algorithm iterates again. The iterations continue in this manner until the algorithm converges. The proof of convergence for this algorithm is similar to the one presented in [12]. It can be observed that each step in the algorithm reduces the value of the total leakage interference. The total leakage interference for general IFC is defines as follows, (32) where is defined in (24). The algorithm presented in this section seeks perfect IA. In particular it seeks to create an interference free subspace of the required number of dimensions that is designated as the desired signal subspace. However, note that perfect IA may not always be feasible. In such scenarios, the network structure should be reconfigured. Simulation results indicate that residual interference due to the imperfect IA significantly degrades the achievable sum rate performance in high signal to noise ratios. We can either reduce the number of transmitted streams, increase the number of receive antennas, use relays in the network to accommodate IA, or use channel extension to make IA feasible. In the following, we will consider the case of reduced number of transmitted streams (RNS) and increased number of receive antennas (INA) along with the case of optimal DoF assignment (ODA). V. SIMULATION ΡESULTS In this section we run simulations to illustrate practical feasibility of the optimal DoF assignment. Consider the 6×3 user regular IFC shown in Figure 2. It is proved that perfect IA is not feasible for this network, assuming single antenna nodes. We consider a MIMO interference channel where each transmitter/receiver is equipped with 3 antennas. Optimal DoF assignments for this structure is obtained as d[k]=1, k=[6]. The system of bilinear equations for IA requirements of this network constitutes an improper system because the number of the independent variables is 18 which is less than the number of equations which is 24. Therefore, we expect the perfect IA to be still infeasible. This fact is confirmed with simulation results. We allocate power 1/d to each column of the precoding matrices, and set the noise power level to σ2=10-P/10, where P=[0:10:60]dB. Final results are obtained by averaging over 200 channel realizations, where each channel element is i.i.d. zero mean unit variance circularly symmetric complex Gaussian. We run 104 iterations of the extended minimum leakage algorithm for each simulation. We plot the sum rate of each system. The sum rate is computed as , where is defined as the covariance matrix of desired signal for receiver j. It is observed in Figure 3 that the sum rate of ODA case is limited with residual interference at high signal to noise ratios. Sum rate performance is also plotted for two cases of RNS (d[1]=0, d[k]=1, k=2) and INA (4 receive antennas). We compare the performance of the iterative IA algorithms for this channel with orthogonal schemes and isotropic transmission. Th equ Th tra reg Fig reg hig the eac ent hig (d[ rec lea It i tha the sch fea use req S3= ass d[3] rec equ lik rec wit sol int ful sho fea sig des in Engineerin www.etasr he sum rate for ual time shari he isotropic tra ansmitter send gardless of the g. 3. Performa gular IFC. It is observed gh signal to n e upperbound ch receiver n tail channel e gh signal to n [1]=…=d[6]=1) ceiver nodes akage interfere is realized tha at, the system ese cases Nυ=N heme. Iterative algo asibility of IA er. We evalua quested messa ={3,4,5}. Solv signment for 3]=d[4]=d[5]=0 ceiver node f uations is prop kely to be feas ceived DoFs is th an idea of w lution or not. H terference is p ll‐rank, the slo ould be close asible. The slo gnal to noise sired signal sp this scenario ng, Technology r.com r the orthogon ing for the use ansmission ca ds d[k]=1, k e channel infor ance of the itera d that orthogo noise ratios in on the DoF fo node is 6, ap extensions. Th noise ratios sug ) can be achie without chan ence for each o at IA is feasibl of the bilinea Ne=16 for RN orithm can al A strategy for ate the iterativ age sets defi ving the linear this network .2. We have for the simul per in this cas sible for this s s 3 1 /j j n N   whether the alg Hence, if perfe perfectly suppr ope of the sum to 10/3. Figu ope of the sum e ratios. Eva pace confirms and there is y & Applied Sci Sama nal scheme is c ers, and with p ase refers to th k=[6] stream rmation. ative IA algorith onal schemes o this specific or this network pproaching the he slope of th ggests that 6 eved by using nnel extension of three receiv le in case of R ar equations is S scheme and lso be used to a given num ve algorithm f ned as S1={1 r programmin k is obtained e assumed 5 lations. The se, Nυ=40, Ne= scenario. The 10 / 3 . This m gorithms conv ect alignment ressed and the m rate at high s ure 4 suggests m rate is appro aluating leaka the fact that p no need for ience Research adi et al.: Achiev calculated assu power 6P per he case where m of equal p hm for the 6 outperform OD structure. Alth k with 3 anten e upperbound he INA sum r degrees of fre g 4 antennas n. For feasibl vers should be RNS and INA. s proper for bo Nυ=Ne=24 fo o check theor mber of stream for a 5×3 IFC 1,5}, S2={1,2} ng (6), optima d as d[1]=d[2] antennas at system of bi =36, hence, O average numb measure provid verge to a perfe is achieved, i. direct channe signal to noise s that perfect oximately 3 a age interferen perfect IA is fe the RNS and h V ving Optimal D uming node. e each power 3 user DA at hough nnas at d may rate at eedom at the le IA, e zero. . Note oth of or INA retical ms per C with } and l DoF ]=0.4, each ilinear DA is ber of des us fect IA e., the els are ratios IA is at high nce in easible d INA stra OD opt Fig. IFC as a equ enc that regu of netw min Thi resu gen real this and ben [1] [2] [3] [4] [5] [6] Vol. 8, No. 4, 20 Degrees of Free ategies in this DA in this ca timal DoF assi 4. Performan . The concept an interference ual number o countered in ne t perfect IA ular IFCs with the optimal D works is not nimum leakag is algorithm is ults suggest th neral IFC, iter lization of IA s network. Nu d simultaneous nefits of using R. H. Etkin, D capacity to with Vol. 54, No. 12 S. Jafar, M. interference cha 53, No. 7, pp. 2 A. Host-Madse networks”, IEE Adelaide, Austr M. A. Maddah over MIMO X Performance An 54, No. 8, pp. 3 S. A. Jafar, S. channel”, IEEE 151-170, 2008 V. R. Cadamb freedom of the Information Th 018, 3141-3148 edom for an Int network. In f ase, which em ignment. nce of the iterati VI. CO of regular int e network in w of optimal D etworks with f cannot be ac h generic chan DoF assignme easily mathe ge IA is exten s based on the hat, despite th rative IA algo and evaluation umerical comp s isotropic tran distributed IA REFER D. N. C. Tse, H. hin one bit”, IEE 2, pp. 5534-5562, Fakhereddin, “D annel,”, IEEE Tr 2637-2642, 2007 en, A. Nosratinia EE International ralia, pp. 2065-20 -Ali, A. S. Mota X channels: Interf nalysis”, IEEE Tr 457-3470, 2008 Shamai, “Degree E Transactions on e, S. A. Jafar, “ e K -user interfe eory,Vol. 54, No. 8 terference Netw fact, RNS has mphasizes the ive IA algorithm ONCLUSION terference netw which all activ DoF. 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Jafar, “A distributed numerical approach to interference alignment and applications to wireless interference networks”, IEEE Transactions on Information Theory, Vol. 57, No. 6, pp. 3309-3322, 2011 [13] Z. Samadi, V. T. Vakili, F. Haddadi, “Channel aided interference alignment”, IET Signal Processing, Vol. 11, No. 7, pp. 854-860, 2017 AUTHORS PROFILE Zainalabedin Samadi was born in Hashtroud, Iran, on April 26, 1985. He received the B.S. degree from the Ferdowsi University of Mashhad, Mashhad, Iran, in 2008, the M.S. degree from Iran University of Science and Technology, Tehran, Iran, in 2011. He is currently working toward the Ph.D. in the Department of Electrical Engineering, Iran University of Science and Technology. His research interests include multiuser information theory, joint source-channel coding, spread-spectrum systems and emerging interference alignment techniques. Vahid Tabataba Vakili received the B.S. degree from Sharif University of Technology, Tehran, Iran, in 1970, the M.S. degree from the University of Manchester, Manchester, U.K., in 1973, and Ph.D. from the University of Bradford, Bradford, U.K., in 1977, all in electrical engineering. In 1985, he joined the Department of Electrical Engineering, Iran University of Science and Technology, Tehran. In 1997, he was promoted to Associate Professor. He has served as the Head of the Communications Engineering Department and as the Head of postgraduate studies. His research interests are in the areas of mobile cellular systems, interference cancellation for CDMA systems, and space–time processing and coding. Farzan Haddadi was born in 1979. He received B.Sc., M.Sc., and Ph.D. degrees in communication systems from Sharif University of Technology, Tehran, Iran, in 2001, 2003, and 2010, respectively. In 2011, he joined Iran University of Science and Technology Faculty, Tehran, Iran. His research interests include array signal processing, statistical signal processing, nondestructive testing, sonar systems, and detection theory.