INTERFERENCE is a key bottleneck in wireless communication

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1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 60, NO 9, SEPTEMBER Feasibility of Intefeence Alignment fo the MIMO Intefeence Channel Guy Besle, Dustin Catwight, and David Tse, Fellow, IEEE Abstact We study vecto space intefeence alignment fo the multiple-input multiple-output intefeence channel with no time o fequency divesity, and no symbol extensions We pove both necessay and sufficient conditions fo alignment In paticula, we chaacteize the feasibility of alignment fo the symmetic thee-use channel whee all uses tansmit along d dimensions, all tansmittes have M antennas and all eceives have N antennas, as well as feasibility of alignment fo the fully symmetic (M = N) channel with an abitay numbe of uses An implication of ou esults is that the total degees of feedom available in a K-use intefeence channel, using only spatial divesity fom the multiple antennas, is at most 2 This is in shap contast to the K/2 degees of feedom shown to be possible by Cadambe and Jafa with abitaily lage time o fequency divesity Moving beyond the question of feasibility, we additionally discuss computation of the numbe of solutions using Schubet calculus in cases whee thee ae a finite numbe of solutions Index Tems Intefeence channel, intefeence alignment, feasibility of alignment, algebaic geomety I INTRODUCTION INTERFERENCE is a key bottleneck in wieless communication netwoks of all types: wheneve spectum is shaed between multiple uses, each use must deal with undesied signals Cellula netwoks in densely populated aeas, fo example, ae seveely limited by intefeence To addess this poblem, the eseach community as well as the wieless communications industy have invested a geat deal of effot in tying to develop efficient communication schemes to deal with intefeence Nevetheless, the cuent state-of-the-at systems ely on two basic appoaches: eithe othogonalizing the communication links acoss time o fequency, o shaing the same esouce while teating othe uses signals as noise If thee ae K co-located uses, these appoaches esult in Manuscipt eceived Mach 21, 2013; evised Febuay 25, 2014; accepted June 20, 2014 Date of publication July 29, 2014; date of cuent vesion August 14, 2014 G Besle and D Tse wee suppoted by the Cente fo Science of Infomation (CSoI), an NSF Science and Technology Cente, unde gant ageement CCF D Catwight was suppoted in pat by the Pogam on Algebaic Geomety with a View Towads Applications and in pat by NSF unde Gant DMS A potion of this wok was pesented at the Infomation Theoy Wokshop and the Alleton Confeence, both in 2011 G Besle is with the Laboatoy fo Infomation and Decision Systems, Depatment of Electical Engineeing and Compute Science, Massachusetts Institute of Technology, Cambidge, MA USA ( gbesle@mitedu) D Catwight is with the Depatment of Mathematics, Yale Univesity, New Haven, CT USA ( dustincatwight@yaleedu) D Tse is with the Infomation Systems Laboatoy, Depatment of Electical Engineeing, Stanfod Univesity, Stanfod, CA USA ( dtse@stanfodedu) Communicated by D Guo, Associate Edito fo Shannon Theoy Colo vesions of one o moe of the figues in this pape ae available online at Digital Object Identifie /TIT a faction 1/K of the total esouce being available to each use: pefomance seveely degades as the numbe of uses inceases Intefeence alignment is one ecent development (among othes, such as hieachical MIMO [3]) that has opened the possibility of significantly bette pefomance in intefeence-limited communications than taditionally thought possible The basic idea of intefeence alignment is to align, o ovelap, multiple intefeing signals at each eceive in ode to educe the effective intefeence Intefeence alignment has been used in the index coding liteatue since the late 90 s, seemingly fo the fist time by Bik and Kol [4] Fo wieless communication, intefeence alignment was used by Maddah-Ali et al [5] and made moe explicit by Jafa and Shamai [6], both fo the multiple-input multipleoutput (MIMO) X channel But the extent of the potential benefit of intefeence alignment was fist obseved by Cadambe and Jafa [7] in application to the K -use intefeence channel, when they showed that fo time-vaying o fequency selective K channels with unbounded divesity, 2 total degees of feedom ae achievable using a basic linea pecoding scheme In othe wods, somewhat amazingly, each use gets the same degees of feedom as with only two uses in the system, independent of the total numbe of uses K The numbe of degees of feedom in a system, defined late, is given by the total capacity nomalized by the capacity of a single point-topoint link, in the limit of high signal-to-noise atios (SNR) The K 2 esult of Cadambe and Jafa equies unbounded channel divesity, and it is unclea what the implication is fo eal systems with finite channel divesity Despite majo effot by eseaches ove the last six o so yeas, little is known about how divesity affects the ability to align intefeence Patial pogess in this diection includes [8] fo the thee-use channel, as well as [9] fo single-beam stategies Many pactical systems ae also equipped with multiple antennas and thus have spatial divesity Multiple antennas ae known to geatly incease the degees of feedom of point-topoint systems In this pape we focus on how spatial divesity helps to deal with intefeence by studying intefeence alignment fo the MIMO intefeence channel In ode to focus on the effect of spatial divesity, we assume thee is no time o fequency divesity, ie, the channel is constant ove time and fequency Fo technical easons we additionally estict attention to stategies making use of a single time-slot (no symbol extensions) Because [7] uses linea (vecto space) pecoding, we attempt to simplify mattes by esticting to the class of such vecto space schemes (defined caefully in Sec II) Ou fist esults ae fo the symmetic thee use channel We pove a necessay IEEE Pesonal use is pemitted, but epublication/edistibution equies IEEE pemission See fo moe infomation

2 5574 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 60, NO 9, SEPTEMBER 2014 condition fo alignment and give a constuctive achievable vecto space stategy These togethe chaacteize the feasibility of alignment fo d signaling dimensions pe use in a symmetic system with M antennas at the tansmittes and N antennas at the eceives The aguments equie only basic linea algeba Next we genealize to an abitay numbe of uses The aguments make use of tools fom algebaic geomety to analyze the bilinea equations aising fom the alignment conditions We fist pove a geneal necessay condition We subsequently show using Schubet calculus that if M = N and all tansmittes use d signalingdimensions, then the necessay condition is also sufficient Schubet calculus is a system fo computing the numbe of solutions to vaious enumeative poblems on algebaic vaieties such as Gassmannians The famewok gives an explicit (albeit complicated) combinatoial ule fo counting the numbe of alignment solutions, and in pinciple allows to check diectly if alignment is geneically feasible How to pefom this veification is not obvious in geneal; we ague that the numbe of solutions is positive in the afoementioned symmetic case The est of the pape is outlined as follows In Section II we intoduce the MIMO intefeence channel model and fomulate the alignment feasibility poblem Section III contains an oveview of the main esults and compaison with elated woks Section IV deives necessay conditions fo alignment, and Section V gives sufficient conditions Finally, Section VI discusses computing the numbe of alignment solutions as well as how to compute the solutions themselves, and Section VII gives some concluding emaks II THE MIMO INTERFERENCE CHANNEL The K -use MIMO intefeence channel has K tansmittes and K eceives, with tansmitte i having M i antennas and eceive i having N i antennas Fo i = 1,,K, eceive i wishes to obtain a message fom the coesponding tansmitte i The emaining signals fom tansmittes j = i ae undesied intefeence The channel is assumed to be constant ove time, and at each time-step the input-output elationship is given by y i = H [ii] x i + H [ij] x j + z i, 1 i K (1) 1 j K j =i Hee fo each use i we have x i C M i and y i, z i C N i, with x i the tansmitted signal, y i the eceived signal, and z i CN(0, I Ni ) is additive isotopic white Gaussian noise The channel matices ae given by H [ij] C N i M j fo 1 i, j K ; fo the est of the pape, we assume that the H [ij] ae geneic, meaning that thei enties lie outside of an algebaic hypesuface depending only on the paametes d i, M i,andn i If the enties ae andomly chosen fom some non-singula pobability distibution, this will be tue with pobability 1 Additionally, each use obeys an aveage powe constaint, 1 T E( xt i 2 ) P fo a block of length T We estict the class of coding stategies to (linea) vecto space stategies In this context, degees-of-feedom has a simple intepetation as the dimensions of the tansmit subspaces, descibed in the next paagaph Howeve, note that one can moe geneally define the degees-of-feedom egion in tems of an appopiate high tansmit-powe limit P of the Shannon capacity egion C(P) nomalized by log P ([5], [7]) In that geneal famewok, it is well-known and staightfowad that vecto space stategies give a concete non-optimal achievable stategy with ates R i (P) = d i log(p) + O(1), 1 i K Hee d i is the dimension of tansmitte i s subspace and P is the tansmit powe The tansmittes encode thei data using vecto space pecoding Suppose tansmitte j wishes to tansmit a vecto ˆx j C d j of d j data symbols These data symbols ae modulated on the subspace U j C M j of dimension d j, giving the input signal U j ˆx j,wheeu j is a M j d j matix whose columns give a basis of U j This signal is obseved by eceive i though the channel as H [ij] U j ˆx j The dimension of the tansmit space, d j, detemines the numbe of data steams, o degees-of-feedom, available to tansmitte j With this estiction to vecto space stategies, the output fo eceive i is given by y i = H [ij] U j ˆx j + z i, 1 i K (2) 1 j K The desied signal space at eceive i is thus H [ii] U i, while the intefeence space is the span of the undesied subspaces, ie, j =i H[ij] U j In the egime of asymptotically high tansmit powes, in ode that decoding can be accomplished we impose the constaint at each eceive i that the desied signal space H [ii] U i is complementay to the intefeence space j =i H[ij] U j Equivalently, eceive i must have a subspace V i (onto which it can poject the eceived signal) with dim V i = dim U i such that and H [ij] U j V i, 1 i, j K, i = j, (3) dim(poj Vi H [ii] U i ) = dim U i (4) Hee, H [ij] U j V i means that the two vecto spaces ae othogonal with espect to the standad Hemitian fom on C N i Equivalently, if we wite V i and U j fo matices whose columns fom bases fo V i and U j espectively, then othogonality means that all enties of the matix V i H[ij] U j ae zeo, whee V i denotes the Hemitian tanspose If each diect channel matix H [ii] has geneic (o iid continuously distibuted) enties, then the second condition (4) is satisfied assuming dim V i = d i fo each i This is because the set of channels fo which condition (4) is not satisfied obeys a deteminant equation and is theefoe contained in an algebaic hypesuface (algebaic set of codimension 1) This can be easily justified see [10] fo some bief emaks Hence we focus on condition (3) Ou goal in this pape is to detemine when intefeence alignment is feasible: given a numbe of uses K, numbes of

3 BRESLER et al: FEASIBILITY OF INTERFERENCE ALIGNMENT FOR THE MIMO INTERFERENCE CHANNEL 5575 antennas M 1,,M K and N 1,,N K, and desied tansmit subspace dimensions d 1,,d K, does thee exist a choice of subspaces U 1,,U K and V 1,,V K with dim U i = dim V i = d i satisfying (3)? III MAIN RESULTS As discussed in the pevious section, fo vecto space stategies the alignment poblem educes to finding vecto spaces U i C M i and V i C N i whee dim U i = dim V i is denoted d i, such that H [ij] U j V i, 1 i, j K, i = j, (5) whee the matix H [ij] C N i M j epesents the channel between tansmitte j and eceive i We again emphasize that the H [ij] ae assumed to be geneic Ou goal is to maximize the signal dimensions d i subject to the constaint that thee exist vecto spaces satisfying (5) Yetis et al [11] poposed compaing the numbe of vaiables and equations in the system of bilinea equations (5) in ode to detemine when it has solutions, and justified this using Benstein s Theoem in the case that d i = 1foalli Oufist esult makes this intuition pecise by showing that the feasible solutions ae an algebaic vaiety of the expected dimension, when the channel matices ae geneic Thus, we have the following necessay condition fo intefeence alignment: Theoem 1: Fix an intege K and integes d i,m i, and N i fo 1 i K and suppose the channel matices H [ij] ae geneic If, fo any subset A {1,,K }, the quantity t A = ( di (N i d i ) + d i (M i d i ) ) d i d j i A i, j A,i = j is negative, then thee ae no feasible stategies Moeove, if thee ae feasible stategies, then t {1,,K } is the dimension of the vaiety of solutions The constaint on t {1,,K } was obtained independently and simultaneously by Razaviyayn et al [12] The dimension of the vaiety of solutions is impotant, because when multiple stategies ae feasible, we may wish to optimize ove the feasible stategies accoding to some othe citeion, such as the obustness of the system The necessay condition fom Theoem 1 is not sufficient One, almost tivial, equiement fo thee to even exist vecto spaces is that d i M i and d i N i fo each i Similaly, by looking at the total capacity of two tansmittes and one eceive, we have the following constaint, which is also implied by combining the infomation-theoetic aguments of [13] and [14] Theoem 2: Fo any distinct indices i, j, and k, the feasibility of intefeence alignment equies d i + d j + d k max(n i, M j + M k ) Symmetically, Theoem 2 also holds with the oles of M and N evesed Moeove, this esult extends to chains of tansmittes and eceives longe than thee, at least when the tansmit and eceive dimensions ae identical To do so, athe than thee indices i, j, andk fom Theoem 2, we use a sequence of indices such that each consecutive tiple Fig 1 Fo a fixed value of d, the feasible egion in the M, N plane is white while the infeasible egion is shaded The labels 1, 2, 3, 4, indicate the maximum length of alignment paths fo M, N in the coesponding egion of indices is distinct Again, Theoem 3 also applies with M and N evesed Theoem 3: Fix a non-negative intege and let i 1,,i +2 be a sequence of indices, such that each consecutive tiple consists of thee distinct indices, ie, i j = i j+1 fo 1 j + 1 and i j = i j+2 fo 1 j Also, assume that if i j = i j then i j+1 = i j +2 Suppose that N i j is the same fo all 1 j, which we denote N, and similaly M i j = Mfo2 j + 2 Inode fo intefeence alignment to be feasible, we must have: +2 d i j + d i j max(n,( + 1)M) j=1 j=2 When is positive, we can ewite the left-hand side: d i1 + d i+1 + d i d i max(n,( + 1)M) j=2 The condition on the indices i j in Theoem 3 is somewhat technical, but the simplest choice is to have i 1,,i +2 cycle though thee distinct values Indeed, in the case of 3 uses such a cycle is the only possible sequence and we obtain the following simplification Coollay 4: Suppose that K = 3 and that N 1 = N 2 = N 3 = N and M 1 = M 2 = M 3 = M If intefeence alignment is feasible, then fo any positive intege, d 1 + d +1 + d d max(n,( + 1)M), j=2 whee fo i > 3, we define d i to be d i,wheei is the emainde of i when divided by 3 Of couse, Coollay 4 also holds fo any eodeing of the thee uses Moeove, if we assume that the tansmit dimensions d i ae all equal to some fixed d, then the infeasible paametes coespond to the shaded aeas of Figue 1 In fact,

4 5576 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 60, NO 9, SEPTEMBER 2014 as shown by ou next theoem, in this symmetic case we have completely chaacteized the feasible egion Theoem 5: Let K = 3 and without loss of geneality, let us assume that N M Then intefeence alignment is feasible if and only if fo each 0, (2 + 1)d max(n,( + 1)M) (6) Moeove, if, in addition, N + M = 4d, then thee is a unique solution if N > M and thee ae ( 2d) d solutions if N = M = 2d The sufficiency of these conditions has also been independently shown by Ami et al [15], but only in the zeo-dimensional case, when N +M = 4d Also independently and simultaneously, Wang et al [16] have obtained esults vey simila to Theoem 5 Thei necessay condition (which matches the one in Theoem 5) is infomation-theoetic, and thus, unlike ous, is not limited to linea stategies, constant channels, o no symbol-extensions The linea achievability esult of [16] also matches Theoem 5 Theoem 5 is constuctive in the sense that the solutions can be obtained fom basic opeations in linea algeba The eason that the K = 3 case is easy to analyze is that the constaints (5), which always link a pai of vecto space choices, fom a cycle in the case of K = 3 This cycle allows alignment stategies to be constucted fom alignment paths, as was also obseved by [16] (and peviously in [8] fo the thee-use intefeence channel with eithe time o fequency divesity) When thee ae moe uses, we nonetheless have a sufficiency condition in the case when N = M: Theoem 6: Suppose that K 3 and that d i = d and M i = N i = N fo all uses i Then, fo geneic channel matices, thee is a feasible stategy if and only if 2N (K + 1)d A vey simila theoem was also obtained by Razaviyayn et al [12] fo the case that d divides N They found a distinct, but ovelapping, set of paametes fo which they could chaacteize feasibility Theoem 7 ([12], Th 2): Suppose that d = d i fo all i and that M i and N i ae divisible by d fo all i Then intefeence alignment is possible if and only if the quantities t A fom Theoem 1 ae non-negative fo all subsets A Reaanging the inequality of Theoem 6, we have that the numbe of tansmit dimensions satisfies d K 2N +1 Coollay 8 (Fully Symmetic Achievable dof): The maximum nomalized dof is given by max dof = K 2N K 2 N K + 1 K In shap contast to the K 2 total nomalized dof achievable fo infinitely many paallel channels in [7], fo the MIMO case we see that at most two dof (nomalized by the single-use pefomance of N tansmit dimensions) ae achievable fo any numbe of uses K and antennas N The diffeence is due to the stuctue of the channel matices: fo the MIMO case with full geneic matices condition (3) is difficult to satisfy and (4) is easy, while fo the paallel case with diagonal matices the situation is evesed We note that this obsevation was peviously made in [11] based on thei conjectued necessay condition Theoem 6 suggests an engineeing intepetation fo the pefomance gain fom inceasing the numbe of antennas Depending on whethe N < d(k + 1)/2 o not, thee ae two diffeent egimes fo the pefomance benefits of inceasing N: (1) alignment gain o (2) MIMO gain To illustate these concepts, suppose that thee ae K = 5usesIfN = 1, ie, thee is only a single antenna at each node, then no alignment can be done and only one use can communicate on a single dimension, giving 1 total degee of feedom As the numbe of antennas inceases to 2 and 3, the numbe of degees of feedom becomes 3 and 5 espectively Because of alignment, the numbe of uses communicating and hence total degees of feedom inceases, which we call alignment gain The alignment gain has slope 2 Howeve, afte N = 3, inceasing N affods no additional possibilities fo intefeence alignment; the total numbe of degees of feedom inceases only because moe dimensions ae available The MIMO gain has slope 5/3, which is the asymptotic coefficient in Coollay 8 Unlike Theoem 5, the poof of Theoem 6 does not povide a way of computing the solutions Instead of linea algeba, it uses Schubet calculus to pove the existence of solutions In fact, in Section VI, we will see that thee cannot be a simple, exact desciption of the symmetic intefeence alignment poblem Nonetheless, as we will discuss, solutions may be found using numeical algebaic geomety softwae In addition to geneal algebaic methods of oot finding, othes have poposed heuistic algoithms, mainly iteative in natue (see [17] [21]) Some have poofs of convegence, but no pefomance guaantees ae known Schmidt et al [21], [22] study a efined vesion of the single-tansmit dimension poblem, whee fo the case that alignment is possible, they attempt to choose a good solution among the many possible solutions Papailiopoulos and Dimakis [20] elax the poblem of maximizing degees of feedom to that of a constained ank minimization and popose an iteative algoithm In tems of the feasibility poblem, González, Beltan, and Santamaía [23] have given a polynomial-time andomized algoithm fo deteming whethe given paametes ae feasible In a slightly diffeent diection, Razaviyayn et al [19] show that checking the feasibility of alignment fo specific system paametes, including the channel matices, is NP-had Note that thei esult does not contadict that of González et al, since the NP-hadness eduction equies special choices fo the channel matices and does not apply to geneic channels Finally, we emphasize that ou attention has been esticted to vecto space intefeence alignment, whee the effect of finite channel divesity can be moe easily obseved Intefeing signals can also be aligned on the signal scale using lattice codes, which was fist poposed in [24] with followup wok in [25] [27] Recent pogess in this diection includes [28] [31] IV NECESSARY CONDITIONS In this section, we pove the necessay conditions fo intefeence alignment, Theoems 1, 3, and 2 The poof of

5 BRESLER et al: FEASIBILITY OF INTERFERENCE ALIGNMENT FOR THE MIMO INTERFERENCE CHANNEL 5577 Theoem 1 is by counting equations, o, moe pecisely, by detemining the dimension of the elevant algebaic vaieties The poofs of Theoems 3 and 2 use only linea algeba Fo backgound on some of the concepts in algebaic geomety, see the texts by Hatshone [32] o Shafaevich [33] In pactice, intefeence alignment equies finding the feasible communications stategies given the fixed channel matices Howeve, it will be useful to think of the pocedue in evese: we fix the communication stategy and study the set of channels fo which the communication stategy is feasible As we will see in the poof of Lemma 10, the advantage hee is that fo a fixed stategy, the constaints on the channel matices ae linea To make this appoach pecise, we will epesent the space of stategies as a poduct of Gassmannians Recall (fo example fom [33]) that the Gassmannian G(d, N) is the vaiety whose points coespond to d-dimensional subspaces of an N-dimensional vecto space C N Thus, fo each i, the tansmit subspace U i coesponds to a point in the Gassmannian, which we also wite as U i G(d i, M i ), and similaly V i G(d i, N i ),wheev i is the complex conjugate of V i We choose to paametize by the complex conjugate because the elation (5) is defined by algebaic equations in the basis of V i, but not in V i The stategy space is thus the poduct of the Gassmannians, K K S = G(d i, M i ) G(d i, N i ) i=1 Likewise, the elevant channel matices ae a tuple of K (K 1) matices, which we can epesent as a point in the poduct H = i = j CN i M j In the poduct S H, wedefinethealignment vaiety to be the subvaiety I S H of those odeed pais (s, h) such that s is a feasible stategy fo h The dimensions of S and H ae the sums of the dimensions of thei factos, so and dim S = i=1 K ( di (M i d i ) + d i (N i d i ) ), (7) i=1 dim H = 1 i, j K i = j M i N j (8) The dimension of I will be computed in Lemma 10, using Theoem 9, which is a ough analogue of the ank-nullity theoem fom linea algeba Given a map f : X Y,thefibe of a point y Y is the invese image of y unde the map f : f 1 (y) ={x X : f (x) = y} A polynomial map is a function whose coodinates ae given polynomials Finally, an ieducible vaiety is one which cannot be witten as the union of two pope, closed subvaieties The following theoem in algebaic geomety can be found, fo example, as [33, Th 7, p 76] Theoem 9 (Dimension of Fibes): Let f : X Y be a polynomial map between ieducible vaieties Suppose that f is dominant, ie, its image is dense in Y Let n and m denote the dimensions of X and Y espectively Then m n and: 1) Fo any y f (X) Y and fo any component Z of the fibe f 1 (y) the dimension of Z is at least n m 2) Thee exists a nonempty open subset U Y such that dim f 1 (y) = n mfoy U In the poof of Theoem 1, we will apply Theoem 9 twice, fo each of the pojections fom I to the factos S and H The fist of these pojections computes the dimension of I Lemma 10: I is an ieducible vaiety of dimension K ( di (M i d i ) + d i (N i d i ) ) + (M i N j d i d j ) i=1 1 i, j K i = j Poof: We conside the pojection fom ou incidence vaiety on the space of stategies p : I S Fo any point s = (U 1,,U K, V 1,,V K ) S, we claim that the fibe p 1 (s) is a linea space of dimension dim p 1 (s) = (M i N j d i d j ) 1 i, j K i = j To see this claim, we give local coodinates to each of the subspaces compising the solution s S We wite u (i) a fo the ath basis element of subspace U i,wheeu (i) a has zeos in the fist d i enties except fo a 1 in the ath enty, and similaly fo v ( j) b (this is without loss of geneality) The othogonality condition V j H [ ji] U i can now be witten as the condition v ( j) b H [ ji] u (i) a fo each 1 a d i and 1 b d j Witing this out explicitly, we obtain 0 = ( j) v (k)h[ ji] (k, l)u (i) a (l) 1 k M i 1 l N j = v 1 k d i 1 l d j + b ( j) b (k)h[ ji] (k, l)u (i) a (l) v ( j) b k>d i o l>d j = H [ij] (a, b) + (k)h[ ji] (k, l)u (i) a (l) v ( j) b k>d i o l>d j (k)h[ ji] (k, l)u (i) a (l) Note that this equation is linea in the enties of H [ ji] Thee ae d i d j such linea equations, and each one has a unique vaiable H [ ji] (a, b), so the equations ae linealy independent and each equation educes the dimension by 1 The claim follows fom the fact that in total thee ae i = j d id j equations and we began with dim H = 1 i, j K M i N j dimensions (8) i = j We have shown that the fibes of I S ae vecto spaces, and, in paticula, ieducible vaieties of constant dimension Thus, since S is an ieducible vaiety, so is I [34, Example 143] Moeove, Theoem 9 gives the elation dim I = dim S + dim p 1 (s) Since the dimension of S is exactly the fist summation in the lemma statement, this poves the lemma

6 5578 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 60, NO 9, SEPTEMBER 2014 Poof of Theoem 1: We now conside the pojection onto the second facto q : I H This map is dominant if and only if the alignment poblem is geneically feasible In this case, Theoem 9 tells us that the fibe q 1 (h) fo a geneic h H has dimension dim q 1 (h) = dim I dim H (9) Using fomula (8) fo the dimension of H and Lemma 10 fo the dimension of I, we get that (9) is equal to the quantity t {1,,K } fom the theoem statement Theefoe, by Theoem 9, this quantity must be non-negative if thee ae to be feasible solutions, in which case t {1,,K } is the dimension of the set of solutions to the geneic alignment poblem Now we tun to the necessay conditions fo othe subsets A {1,,K } Any feasible stategy fo the full set of K tansmittes and eceives, will, in paticula be feasible fo any subset of tasmitte-ecieve pais Theefoe, a necessay condition fo a geneal set of channel matices to have a feasible stategy is that the same is tue fo any subset of the pais Since the numbe t A is the dimension of the vaiety of solutions when esticted just to the tansmittes and eceives indexed by i A, thent A must be non-negative in ode to have a feasible stategy We now focus on the second necessay condition, Theoem 3, and the closely elated Theoem 2 As we mentioned befoe, Theoem 3 is a genealization of the obvious constaint that d i M i fo each tansmitte, and the genealization is fomed by consideing + 1 tansmittes and eceives at the same time We fist handle the case of = 1 Poof of Theoem 2: We define A to be the N i (M j + M k ) block matix: (H [ij] H [ik] ) Fo geneic channel matices, A will have full ank, ie, ank equal to min(n i, M j +M k ) We conside the vecto space U = U j U k of dimension d j + d k in C M j +M k The othogonality condition (3) implies that V i AU IfN i M j + M k,thena will be injective, and so AU has dimension d j + d k Howeve, othogonal vecto spaces can have at most complementay dimensions, so we have that d i + d j + d k N i On the othe hand, if N i M j + M k, then the Hemitian tanspose A is injective, and fom the othogonality elation A V i U, we get that d i + d j + d k M j + M k Thus, we conclude that d i + d j + d k max(n i, M j + M k ) A key step in the poof of Theoem 2 was that the matix A had full ank Fo > 1, we again show that, unde appopiate hypotheses, the analogous matix has full ank Lemma 11: Let i 1,,i +2 be a sequence as in the statement of Theoem 3, and we assume that N i = N and M i = M fo all i Fo any 1 define the N ( + 1)M block matix A to be H [i 1i 2 ] H [i 1i 3 ] H [i 2i 3 ] H [i 2i 4 ] (10) H [i i +1 ] H [i i +2 ] Fo geneic channel matices H [ij], the matix A has full ank, min(n,( + 1)M) Lemma 11 is poved in the appendix Using the lemma, we now pove Theoem 3 Poof of Theoem 3: We fix the intege Define the poduct of tansmit spaces U = U i2 U i3 U i+2 (C M ) +1,and similaly let V = V i1 V i (C N ) Note that U and V have dimensions +2 dim U = d i j and dim V = d i j j=2 Fist, suppose that N ( +1)M Then Lemma 11 implies that the linea map A : (C M ) +1 (C N ) is injective By the othogonality condition (3), we have V A U, and thus dim(v) + dim(a U) = j=1 j=1 +2 d i j + is at most N Altenatively, if ( +1)M N, the Hemitian tanspose A is an injective linea map A : (C N ) (C M ) +1 Again, the othogonality conditions (3) imply that A V U so dim V + dim U ( + 1)M This poves the theoem j=2 V SUFFICIENT CONDITIONS In this section, we give citeia fo ensuing the achievability of intefeence alignment We have aleady seen the necessay diection of Theoem 5, and hee we pove the sufficient diection, fist when M = N, and second when M < N The fome case will also be coveed by Theoem 6 dealing with K > 3, but we give a specific K = 3 poof because it is constuctive and additionally it allows to compute the numbe of solutions in the bounday case A Thee Uses Poposition 12: If K = 3, and M = N 2d, then alignment is feasible Moeove, in the case of equality, the numbe of solutions is exactly ( 2d) d fo geneic channel matices We note that the constuction in the following poof has also appeaed in [7, Appendix IV] Poof: By fist esticting ou tansmit and eceive spaces to abitay subspaces, we can assume that M = N = 2d Since the channel matices ae squae, geneically, they ae invetible, so we can define the poduct B = H [1,2] (H [3,2] ) 1 H [3,1] (H [2,1] ) 1 H [2,3] (H [1,3] ) 1 Again, geneically, this matix will have 2d linealy independent eigenvectos, and we choose V 1 to be the span of any d of them Then we set U 3 = (H[1,3] ) 1 V 1 V 2 = H [2,3] U 3 U 1 = (H[2,1] ) 1 V 2 V 3 = H [3,1] U 1 U 2 = (H[3,2] ) 1 V 3 d i j

7 BRESLER et al: FEASIBILITY OF INTERFERENCE ALIGNMENT FOR THE MIMO INTERFERENCE CHANNEL 5579 Fig 2 Sub-egion 1: The figue indicates that no alignment is possible when 2M N, sinceim(h [12] ) and Im(H [13] ) ae complementay Since the thee subspaces V 1, H [12] U 2, H [13] U 3 ae each of dimension d, complementay, and lie in C N at eceive 1, we obtain the constaint 3d N These fom a feasible stategy, and thee ae ( 2d) d possible stategies Befoe we poceed to the poof in the case when M and N ae distinct, we infomally descibe the geomety undelying the constuction of solutions A given vecto u i in the signal space of tansmitte i is said to initiate an alignment path of length + 1 if thee exists a sequence of vectos u i+1, u i+2,,u i+ C M, such that H [i 1,i] u i = H [i 1,i+1] u i+1, H [i+ 2,i+ 1] u i+ 1 = H [i+ 2,i+] u i+ Hee channel indices ae intepeted modulo 3 Fo example, a vecto u 2 at tansmitte 2 initiating an alignment path of length 3 means that thee exist vectos u 3 and u 1 such that H [12] u 2 = H [13] u 3 and H [23] u 3 = H [21] u 1 The feasible egion of Figue 1 is divided up into subegions labeled with the maximum length of an alignment path; this numbe depends on M and N though the incidence geomety of the images of the channel matices Im(H [ij] ) We begin by examining sub-egion 1, and then look at how things genealize to the othe sub-egions The point of depatue is the obvious constaint d M in ode to have a d-dimensional subspace of an M dimensional vecto space Continuing, assuming M d, suppose 2M N, so(m, N) lies in sub-egion 1 of Figue 1 At eceive one, the images Im(H [12] ) and Im(H [13] ) of the channels fom tansmittes two and thee ae in geneal position and theefoe thei intesection has dimension [2M N] + = 0; in othe wods, alignment is impossible in sub-egion 1 Figue 2 shows pictoially that because alignment is not possible hee, we have the constaint 3d N Moving onwad to sub-egion 2, we have 2M > N and thus alignment is possible This means that alignment paths of length 2 ae possible (Fig 3), with up to 2M N intefeence dimensions ovelapping at each eceive Thus, the intefeence space H [12] U 2 + H [13] U 3 at eceive one occupies at least 2d (2M N) dimensions, and we have the constaint 3d 2M Howeve, because 3M 2N, no vecto at (say) tansmitte thee can be simultaneously aligned at both eceives one and two, as indicated in Figue 4 One can also see that no simultaneous alignment is possible by changing pespective to that of a combined eceive one and two Fig 3 Sub-egion 2: Alignment is possible hee The figue denotes an alignment path of length 2 Fig 4 Sub-egion 2: The stiped egions at eceives one and two each denote the dimension 2M N potion of the space in which alignment can occu Fom tansmitte thee s pespective, one sees that simultaneous alignment is not possible fo 2(2M N) M, o equivalently, 3M 2N By Lemma 11, the map ( H [12] H [13] H [23] H [21] ) (11) fom the thee tansmittes to C 2N is injective; analogously to the case in sub-egion 1, this is intepeted to mean that no alignment is possible in the combined eceive space C 2N Thus, five complementay d-dimensional subspaces lie in C 2N and we obtain the constaint 5d 2N As fa as achievability goes, the basic ule-of-thumb is to ceate alignment paths of maximum length Thus, in subegion 2, whee alignment is possible, the achievable stategy aligns (as pe Fig 3) as many vectos as possible and the emaining ones (if d > 2(2M N)) ae not aligned Fo example, in sub-egion 4, alignment paths of length fou ae used (Fig 5) Poposition 13: If K = 3 and M < N and (6) holds fo each 0, then alignment is feasible In the 0-dimensional case, when N + M = 4d, thee is a unique solution Poof: Let be the (unique) intege such that N <( + 1)M and ( + 1)N ( + 2)M (12) Note that this implies, fom (6), that (2 + 3)d ( + 1)N (13)

8 5580 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 60, NO 9, SEPTEMBER 2014 Fig 5 Sub-egion 4: Alignment paths of length fou ae denoted hee, initiated by vectos at tansmitte 1 and (2 + 1)d ( + 1)M (14) We pove achievability by examining two cases: fist d ( +1)[( +1)M N] and second, d >( +1)[( +1)M N] Case 1 means that all of the signal space U i can be obtained fom alignment paths of length + 1 (up to intege ounding), wheeas in case 2 we must use alignment paths of length as well in ode to attain the equied d dimensions We fist establish case 1 Let A i be the matix fom (10) fo the inceasing sequence of indices i, i + 1,,i + + 1, whee indices ae undestood modulo 3 Let W i beadimension d +1 subspace in the kenel of A i Letd := d (+1) d +1, and if d > 0letw i be a 1-dimensional subspace in ke A i \W i We define W i, j C M to be the pojection of W i onto its ( j i)th block of coodinates The spaces w i ae equied in ode to accommodate the emainde left when dividing d by +1, and will togethe contibute d dimensions to each signal space U j We put and V j = U j = j 1 i= j 1 W i, j + j d i= j 1 ( H [ j, j 1] W j, j+1 + H [ j, j 1] w j, j+1 j + H [ j, j+1] W i, j+1 + i= j w i, j (15) j d +1 i= j w i, j ), (16) whee again, the indices in H [ j, j 1] and H [ j, j+1] ae undestood to be taken modulo thee If all of U j s constituent subspaces ae complementay, then U j has dimension ( +1) d +1 +d = d We igoously justify this in Lemma 14 To see that V j has dimension (at least) d, we obseve that by subadditivity of dimension, dim V j N ( + 2) d + 1 d e, (17) whee e = 0if( + 1) d and e = 1 othewise Plugging in the inequality (13) we obtain dim V j d + 1 d d e + 1 = d + d d + 1 e d + 1 Suppose now that we ae in case 2: d >( + 1)[( + 1) M N] This means that not all of the signal space U i can be included in alignment paths of length + 1, so the emainde will be included in alignment paths of length Letd := d ( + 1)[( + 1)M N] and d = d d As befoe, denote by W i the kenel of the matix A i, having dimension ( + 1)M N Denote by π the pojection fom C (+1)M C M to the fist M coodinates The space π(ke A i ) is contained in ke A 1 i LetX i fo i = 1, 2, 3 each be a d dimensional subspace in ke A i 1 \ π(w i), andletw i be a 1-dimensional subspace in ke A i 1 \ (π(w i) + X i )Put and V j = U j = j 1 i= j 1 W i, j + j i= j 1 X i, j + j d i= j 1 ( H [ j, j 1] (W j, j+1 + X j, j+1 + w j, j+1 ) j + H [ j, j+1] W i, j i= j j d +1 i= j j +1 i= j w i, j (18) H [ j, j+1] X i, j+1 w i, j ) (19) As befoe, a naive count suggests that U j should have dimension d, and this will again be justified with Lemma 14 To see that V j has dimension at least d we again use subadditivity of dimension to get d dim V j N ( + 2)[( + 1)M N] ( + 1) d e 1 d = N ( + 2)[( + 1)M N] d e 1, whee e 1 is 0 if divides d and e 1 is 1 othewise Letting e 2 := d d,wehave dim V j N ( + 2)[( + 1)M N] d d + e 2 e 1 ( + 1)d = N + e 2 e 1 ( + 1) = d [( + 1)M N] M d + e 2 e 1 Substituting M fo d, the inequality (14) implies that dim V j d + e 2 e 1

9 BRESLER et al: FEASIBILITY OF INTERFERENCE ALIGNMENT FOR THE MIMO INTERFERENCE CHANNEL 5581 If e 1 is one then e 2 is stictly positive, so the fact that dim V j is an intege implies dim V j d Lemma 14: The subspaces U j and V j defined in (15), (16), (18), and (19) have dimension d Poof: We fist show that U 1 has dimension d; by symmety of the constuction, the dimensions of U 2 and U 3 will also be d The subspace U 1 = 0 i= W i,1 is the sum of + 1 subspaces W i, j, which we claim ae independent; suppose to the contay, that thee is some set of linealy dependent vectos w i1,w i2,,w is, with 0 i 1 i 2 i s, and w i W i,1, satisfying w is s 1 l=1 λ lw il = 0 Let s be the minimum such value, with all sets of subspaces W i1, j, W i2, j,,w is 1, j fo j = 1, 2, 3 being complementay Now, by the definition of the subspaces W i, j, fo each vecto w il W il,1 thee is a sequence u 2 i l,,u q+1 i l of length q := +1 i s 1 satisfying H [31] w il = H [32] u 2 i l,,h [q+2,q] u q i l = H [q+2,q+1] u q+1 The linea combination s 1 l=1 λ lw il thus i l gives ise to a sequence u 1,,u q+1 defined by u a = s 1 l=1 λ lu a i l satisfying H [31] w is ( s 1 ) = H [31] λ l w il = H [32] u 2, l=1 H [12] u 2 = H [13] u 3 H [q+2,q] u q = H [q+2,q+1] u q+1 (20) Note that by the minimality assumption of s, none of the u j vectos ae zeo By the definition of W is,1, thee is a length-(i s 1) sequence of vectos peceding w is satisfying alignment conditions simila to those in (20); togethe with w is and the vectos in (20), this sequence can be extended to a sequence of vectos of total length q + i s = (i s i s 1 )> + 1, none of which ae zeo Stacking the fist + 2 of these vectos poduces a nonzeo element in the kenel of A i s +1 Howeve, A i s +1 is full-ank by Lemma 11; the dimension of the kenel is [ ] + ( ) (+2)M (+1)N = M+d = M 2d < 0, ie, the kenel is tivial This is the desied contadiction We now check that V 1 has dimension d, and again by symmety, the dimensions of V 2 and V 3 will also be d Note that if V 1 had dimension geate than d, we could choose a d-dimensional subspace and this would still satisfy the alignment equations (3) But V 1 is the othogonal complement of the sum of + 2 subspaces W i, j of dimension d/( + 1), so by subadditivity of dimension, we have the lowe bound on dimension dim V 1 N ( + 2) dim W i, j = d B Moe Than Thee Uses We now pove achievability fo moe than 3 uses We do this unde the additional assumption that M = N Unlike ou techniques above, these existence esults will not be constuctive, a chaacteistic shaed by pevious existence poofs in [12] and [1] In [1], a poof of Theoem 6 was given using dimension theoy fo algebaic vaieties and linea algeba Hee, we give an intesection-theoetic poof, which involves moe advanced machiney, but that machiney allows fo a moe staightfowad computation Moeove, as we will see in Poposition 18, the Schubet calculus famewok will also allow us to go beyond the existence of solutions and count the numbe of solutions when that numbe is finite Ou poof will show that the expected numbe of solutions, as counted by Schubet calculus, is always positive Schubet calculus is a method fo computing the numbe of solutions to cetain enumeative poblems Fo algebaic vaieties, such as the poduct of Gassmannians that paametize alignment stategies, thee is a commutative ing whose elements coespond to conditions, on the paametes (such as the othogonality elation (5)), and whee the poduct coesponds to the simultaneous imposition of both conditions In algebaic geomety, this is known as the Chow ing and in the case of poducts of Gassmannians, it coincides with the cohomology ing fom algebaic topology The Chow ing of a Gassmannian has an explicit Z-basis indexed by patitions Specifically, the Chow ing of the Gassmannian G(d, m) has a basis coesponding to patitions with at most d pats of size at most m d Such a patition is a list of integes λ i with m d λ 1 λ d 0, and we wite λ fo its size, λ 1 + +λ d The poduct between two of these basis elements, known as Schubet classes, is given by an inticate combinatoial pocess known as the Littlewood-Richadson ule Thus, to detemine whethe a given alignment poblem is feasible, we poceed in two steps Fist, we detemine the elements in the Chow ing coesponding to each of the othogonality conditions (5) Second, we multiply these elements togethe and the esulting poduct is non-zeo if and only if alignment is feasible The fist step is done by Lemma 17 below Fo the second step, the cental difficulty is undestanding the esults of the Littlewood-Richadson ule fo poducts of Schubet classes In ode to establish sufficient conditions in the fully symmetic case, ou poof will use caefully chosen tems fom each Schubet class, which will be sufficient because the poducts of the othe tems will be non-negative: Theoem 15: In the Chow ing of the Gassmannian, the poduct of two patitions is a non-negative sum of othe patitions [35, p 146, eq (8)] Fo the chosen tems, it will be sufficient to compute thei poducts, not with the geneal Littlewood-Richadson ule, but by the following simplification: Poposition 16: The Schubet classes have the following popeties in the Chow ing of the Gassmannian: 1) If λ and μ ae two patitions with λ 1 + μ 1 d, then the poduct [λ][μ] has a coefficient of 1 in font of the tem [ν], wheeν i = λ i + μ i 2) Suppose that λ has l pats and μ has k pats and that l+k m d Then [λ][μ] has a coefficient of 1 in font of the tem [ν] whee ν is fomed by concatenating the pats of λ with the pats of μ, and then soting them in deceasing ode Both pats of this poposition can each be poved fom the Piei ule [35, p 146, eq (9)], and in fact they ae closely

10 5582 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 60, NO 9, SEPTEMBER 2014 Fig 6 The Young diagam of the patition (5, 4, 1) elated to each othe Patitions can be depicted as boxes in the uppe left cone, such as the depiction of (5, 4, 1) in Figue 6 Such diagams have an involution by eflecting them along a diagonal so that the conjugate of (5, 4, 1) is the patition (3, 2, 2, 2, 1) Fo any patition λ, we wite λ to denote the conjugate patition This conjugation coesponds to the isomophism between G(d, m) and G(m d, m), andit is compatible with the multiplication The last two items ae elated by this conjugation opeation The Chow ing of a poduct of Gassmannians has a basis indexed by tuples of patitions, one fo each Gassmannian Moeove, the poducts can be computed factowise Fomally, the Chow ing of the poduct of Gassmannians is the tenso poduct of the Chow ings of the Gassmannians We now compute the class in this Chow ing of the vaiety defined by a single othogonality condition (5) Ou method is simila to the elementay definition of the degee of a pojective vaiety as the numbe of points in the intesection with a geneic linea space of complementay dimension Howeve, instead of linea spaces, we look at the intesection of Schubet classes of complementay dimension with the set of pais of vecto spaces satisfying (5) Lemma 17: The alignment coespondence defined by H [ij] U j V i has class λ [λ] [dd λ ] in the Chow ing of G(d, M) G(d, N) The sum is taken to be ove all patitions with at most d pats of size at most d, and d d λ is the patition whose kth pat has size d λ d+1 k Poof: We compute the class by intesecting with dual Schubet classes to get a zeo-dimensional cycle In paticula, we let μ and ν be patitions into at most d pats of size at most M d and N d espectively, and such that the total size μ + ν is (M + N 3d)d, which is the dimension of the coespondence We fist ecall the definition of the Schubet vaiety in G(d, M) associated to a patition μ and a flag F Aflag in C M is a nested set of vecto spaces 0 = F 0 F 1 F M = C M such that F i has dimension i The Schubet vaiety is the closed subvaiety of those vecto spaces U such that dim(u F M d+i μi ) i fo 1 i d By symmety, we can assume that N is geate than o equal to M and thus a vecto space U G(d, M) uniquely defines an (N d)-dimensional subspace (H [ij] U) in C N Likewise, the othogonal complement (H [ij] F k ) C N is an (N k)-dimensional vecto space, which togethe fom a flag fom dimension N M though N We can choose a flag of additional vecto spaces contained within (H [ij] C M ) to get a full flag in C N, which we denote F ThenU is in the Schubet vaiety of μ if and only if (H [ij] U) is in the Schubet vaiety coesponding to F and the patition ((N M) d +μ),which we denote σ Note that σ = μ +(N M)d, soσ and ν togethe have total size (2M 3d)d We also fix a flag E in C N, which then defines a Schubet vaiety in G(d, N) indexed by ν We assume that this flag E is chosen geneically, by which we mean that the intesection F i E j is tivial if i + j N and has the expected dimension i + j N othewise Now we wish to find the points in the intesection of these two Schubet vaieties Passing fom U to Ũ = (H [ij] U) tuns the othogonality condition into containment, so we wish to find pais V Ũ satisfying the Schubet conditions dim(ũ F M d+i νi ) i, dim(v E d+ j σ j ) j, (21) whee 1 i d and 1 j N d Wenowleti and j be indices satisfying i + j = N d + 1 Since Ũ has dimension N d and contains V,thismeansthatŨ F M d+i νi and V E d+ j σ j must have a non-zeo element in common, and thus F M d+i νi and E d+ j+σ j must intesect non-tivially By the geneic choice of the flag E, this means that the sum of the dimensions of these vecto spaces must be at least the dimension of the ambient space, so that we have the inequality 2M d + 1 ν i σ j > M Reaanging, this means that ν i + σ j can be at most M d Because of thei degees, the only possibility is that ν i = M d σ M d+1 i fo 1 i d and that σ j = d fo 1 j M 2d Moeove, fo fixed patitions ν and σ of this type, thee is a unique pai of vecto spaces V Ũ satisfying (21) We set V to be the vecto space geneated by the one-dimensional vecto spaces F M d+i νi E d i+μi +1 fo 1 i d and take Ũ to be geneated by V and F M 2d What we have shown is that the only Schubet classes which occu in the coespondence class ae dual to the classes μ and σ above, and these occu with coefficient 1 Since the pats of σ have size at most d, this means that the pats of (N d) d ν have size at most d, and this is the patition λ fom the lemma statement Tacing backwads, we see that ν is (M d) d +λ, and thus the expession of the coespondence consists of the classes [λ] [d d λ ], as in the statement Poof of Theoem 6: It will be sufficient to pove that the poduct of the classes fom Lemma 17 ove all pais i = j esults in a positive multiple of some Schubet class Moeove, by Theoem 15, it is sufficient to find one combination of tems fom each incidence class whose poduct is non-zeo We shall exhibit such a combination in two sepaate cases Fist, we suppose that K is odd Fom the facto fo each incidence coespondence, we choose the tem based on the cyclic diffeence (i j) mod K {1,,K 1}, whee j is the index of the eceive and i is the index of the tansmitte In paticula, when this modula diffeence is between 1 and (K 1)/2 inclusive, we choose the tem whee the tansmitte patition is d d We wite a b to denote the patition consisting of b pats of size a By Lemma 17, the tem has the empty patition 0 fo the eceive s Gassmannian When this modula diffeence is at least (K + 1)/2, we choose the tem whee the eceive patition is d d and the tansmitte patition is 0 Thus, fo each Gassmannian, we have the poduct of (K 1)/2 copies of d d We have assumed that N d d

11 BRESLER et al: FEASIBILITY OF INTERFERENCE ALIGNMENT FOR THE MIMO INTERFERENCE CHANNEL 5583 TABLE I NUMBER OF SOLUTIONS TO SYMMETRIC ALIGNMENT PROBLEM Fig 7 Schematic diagam of the patitions whose poduct gives a non-zeo coefficient in the poof of Theoem 6 fo the case when K and d ae even On top ae the patitions fo the eceive s Gassmannian and on the bottom is the tansmitte s Gassmannian, when the index is odd Each block epesents a single copy of d d/2 o (d/2) d, and the aangement shows a patition with non-zeo coefficient in thei poduct (K 1)/2, so this poduct contains at least one copy the tuple with (d(k 1)/2) d in each spot Suppose now that K is even and d is also even We choose the tem of each incidence elation as follows (schematically depicted in Fig 7) The tansmitte s Gassmannian has patition (d/2) d, with the exception that when the tansmitte s index j is even and the tansmitte s index i is equal to j + 1 o j + 2, modulo K, the tansmitte has d d/2 In eithe case, the eceive s patitions ae the same Fo each eceive, we have the poduct of K 2 copies of d d/2 and one (d/2) d The poduct of d d/2 with itself has a non-zeo coefficient in font of d d Then, the poduct of (K 2)/2 copies of d d with (d/2) d has a non-zeo coefficient in font of (d(k 1)/2) d, using ou assumption that N d d(k 1)/2 At a tansmitte with odd index, we ae evaluating the poduct of K 1 copies of (d/2) d, yielding a non-zeo coefficient in font of (d(k 1)/2) d At an even index, it is simila except two of these copies ae eplaced by d d/2, which themselves multiply to d d,andwe get the same esult Finally, when K is even and d is odd, the poof is simila, except that since d/2 is not an intege, we have to ound it up o down each time it is used In paticula, fo the facto coesponding to the incidence elation between the ith eceive and jth tansmitte, we use the same patitions above except that we ound down d/2 in the tansmitte s patition when i j is even and ound up when i j is odd Of couse, this causes d/2 in the eceive s patition to ound in the opposite diection Oveall, fo each Gassmannian we have ounded up half the time and down half the time and the poduct woks out as above The assumption of a symmetic alignment poblem was cucial in being able to choose ectangula Schubet classes in the poof of Theoem 6 The difficulty of genealizing this poof to non-symmetic poblems is to find Schubet cycles whose poduct is povably positive, and unlike the symmetic case, ectangula patitions may not suffice in geneal VI COMPUTING FEASIBLE STRATEGIES In this section, we discuss the computation of stategies fom paticula channel matices In the case of K = 3, the poofs of sufficiency in Popositions 12 and 13 ae effective, in that the feasible stategies can be computed via eigenvectos o the kenel of a matix, espectively Fo K > 3, the feasibility poblem is not educed to linea algeba, and the method of Theoem 6 is not constuctive, but we can still solve the intefeence alignment poblem using geneal numeical methods fo polynomial equations The method used to pove Theoem 6 was to establish a lowe bound on the geneic numbe of solutions using Schubet calculus It was sufficient to find one poduct of classes which was positive Howeve, by computing all tems in the poduct of the incidence elations, we can compute the numbe of solutions fo a geneal system in small cases: Poposition 18: Suppose that eithe d is even o K is odd Then the numbe of solutions to the symmetic alignment poblem is as given in Table I Using vey diffeent methods, [36, Sec IV] counted the appoximate numbe of solutions to some intefeence alignment poblems In the common cases, the two methods agee up to thei stated magins of eo In paticula, we confim that fo M = N = 5, K = 4, and d = 2, thee ae 3700 solutions, which they could only claim with high confidence In addition, the fist two values in Table I fo K = 3 wee computed in [22] using Benstein s Theoem The solution counts given in Poposition 18 ae elevant fo computing solutions in two diffeent ways Fist, a lage numbe of solutions indicate the difficulties in enumeating all solutions o in using an iteative algoithm Second, the numbe of solutions also measues the algebaic complexity of finding a solution, since it is also the degee of the field extension of a solution fo andom ational paametes To illustate this pinciple, conside the case of Poposition 12, which showed that the solution to the intefeence alignment poblem with d = 3andN = M = 2d can be found by finding d eigenvectos of a 2d 2d matix Algebaically, finding an eigenvalue and eigenvecto of a geneic 2d 2d matix with ational enties equies solving an ieducible polynomial of degee 2d, and thus the solution lies in a degee 2d extension of the ationals Finding a second eigenvecto equies an extension of degee 2d 1, and so on, so that the solution lies in a degee (2d)!/d! extension A somewhat moe efined analysis would show that it lies in a smalle field of degee ( 2d) d, but in any case the pime divisos of the degee ae less than 2d, and likewise the polynomials that had to be factoed had degee less than 2d The stuctue