The DoF of the Asymmetric MIMO Interference Channel with Square Direct Link Channel Matrices
aa r X i v : . [ c s . I T ] O c t The DoF of the Asymmetric MIMO InterferenceChannel with Square Direct Link Channel Matrices
Tang Liu † , Daniela Tuninetti † , Syed A. Jafar ∗† University of Illinois at Chicago, Chicago, IL 60607, USA, Email: tliu44, [email protected] ∗ University of California at Irvine, Irvine, CA 92697, USA, Email: [email protected]
Abstract —This paper studies the sum Degrees of Freedom(DoF) of K -user asymmetric MIMO Interference Channel (IC)with square direct link channel matrices, that is, the u -thtransmitter and its intended receiver have M u ∈ N antennaseach, where M u need not be the same for all u ∈ [1 : K ] .Starting from a -user example, it is shown that existingcooperation-based outer bounds are insufficient to characterizethe DoF. Moreover, it is shown that two distinct operatingregimes exist. With a dominant user, i.e., a user that has moreantennas than the other two users combined, it is DoF optimalto let that user transmit alone on the IC. Otherwise, it isDoF optimal to decompose and operate the 3-user MIMO ICas an ( M + M + M ) -user SISO IC. This indicates thatMIMO operations are useless from a DoF perspective in systemswithout a dominant user.The main contribution of the paper is the derivation of anovel outer bound for the general K -user case that is tightin the regime where a dominant user is not present; this isdone by generalizing the insights from the 3-user example toan arbitrary number of users. I. I
NTRODUCTION
Interference channels (IC) have been extensively studied inthe past years due to their practical relevance. The capacityof even the simple two-user case is still open in general. Forthe Gaussian noise IC progress has been made by focusing onthe degrees-of-freedom (DoF), or scaling of the sum-capacitywith signal-noise-ratio (SNR) as SNR grows to infinity. Asignaling scheme, known as interference alignment [1], hasbeen shown to achieve 1/2 the interference-free capacity foreach user for almost all channel realizations, regardless ofthe number of users, in single antenna systems. This showedthe surprising result that ICs are not intrinsically interferencelimited.Multiple-input-multiple-output (MIMO) techniques arewidely used in practical wireless communication systems asa means to increase the spectral efficiency. The completecharacterization of the DoF of a general multiuser MIMOIC has been elusive so far. The case where every node hasthe same number of antennas was solved in [1], where it wasshown that MIMO operations are not needed to achieve theoptimal DoF. The question whether the same remains truein asymmetric
MIMO IC has been answered in some specialcases only.In [2] Jafar and Fakhereddin fully characterized the DoFof the 2-user MIMO IC with arbitrary number of antennasat each node. Their result has served as a fundamental outer bound for the K -user MIMO IC where each transmitter has M antennas, each receiver has N antennas, and M = N ,indicated as the ( M × N ) K IC [3], [4], [5], [6], [7], [8].The idea is to partition both the set of transmitters and theset of receivers into two groups, let the users in each groupperfectly cooperate and thus outer bound the performanceof the original IC by that of the so obtained 2-(super)userIC. For the ( M × N ) K IC, MIMO operations are neededin order to attain the optimal DoF; however it was observedthat, except for some values of
M/N , either M or N can bereduced without affecting the DoF [3], [5]. For this ( M × N ) K model, both the achievability and converse proofs reliedon the the symmetry of antennas across users and it is not apriori clear how to generalize them to settings that lack thissymmetry.The case where K MIMO users share the same channeland each node can have different number of antennas has notreceived so much attention as of yet, to the best of our knowl-edge. The reason may lie in the fact that known bounds for“almost symmetric” ICs do not seem to be tight in the generalcase. In this work we study the class of general asymmetricMIMO ICs with square direct link channel matrices , that is,each transmitter and its corresponding receiver have the samenumber of antennas, but different transmitter-receiver pairscan have different number of antennas. Although this settingis not fully general yet, it is a first step towards understandingthe impact of heterogeneous devices in ad-hoc networks.The main contribution of the paper is a full DoF char-acterization for the proposed setting. First we show thatexisting cooperation-based outer bounds are insufficient tocharacterize the DoF and derive a novel DoF outer bound.The novel bound reveals that two distinct operating regimesexist. With a dominant user, i.e., a user that has more antennasthan all the other users combined, it is optimal to let thatuser transmit alone on the IC. Otherwise, it is optimal to decompose and operate the MIMO IC as a multiuser single-input-single-output (SISO) IC where the number of users isgiven by the total number of transmit antennas. This rathersurprising result indicates that MIMO operations are uselessfrom a DoF perspective in systems without a dominant userif the direct link channel matrices are square.The paper is organized as follows. Section II presents thechannel model and summarizes known bounds. Section IIIhighlights the main ingredients in the converse proof byeans of a simple 3-user example. The rigorous proof forthe general K -user case is provided in Section IV. SectionV concludes the paper.II. C HANNEL M ODEL AND K NOWN B OUNDS
A. Channel model
We consider a specific multiuser asymmetric
MIMO ICthat consists of K transmitter-receiver pairs sharing the samewireless channel and thus interfering with one another. Welet M u be the number of antennas at Tx u and at Rx u , u ∈ [1 : K ] , where without loss of generality M ≥ M ≥ . . . ≥ M K ≥ . The channel input-output relationship is ¯ Y i = X k ∈ [1: K ] ¯ H ik ¯ X k + ¯ Z i ∈ C M i × , i ∈ [1 : K ] , (1a) ¯ X i ∈ C M i × : E [ k ¯ X i k ] ≤ P , (1b) ¯ Z i ∈ C M i × : ¯ Z i ∼ N ( , I ) , (1c)where ¯ H ij ∈ C M i × M j is the channel matrix from Tx j to Rx i , ( i, j ) ∈ [1 : K ] . Tx i has a message W i , of rate R i ( P ) , where P is the transmit power, for Rx i , i ∈ [1 : K ] .Achievable rates and capacity region are defined in the usualway [9].In this work we are interested in the high-SNR regime,i.e., P ≫ , and will use the DoF as performance metric.The (sum) DoF d Σ is defined as d Σ := sup X i ∈ [1: K ] d i , (2)where the supremum is over all achievable rate vectors ( R ( P ) , . . . , R K ( P )) and where d i is the DoF of i -th userdefined as d i := lim P → + ∞ R i ( P )log( P ) for i ∈ [1 : K ] . B. Inner bound
An achievable scheme is as follows. By ‘disabling’ MIMOoperations, i.e., treating each pair of antennas as a separateuser, we can transform the MIMO IC into a SISO IC with P i ∈ [1: K ] M i users; by interference alignment we can achieve / DoF per user [1], [10]. We shall refer to this simpleachievable scheme as the decomposition inner bound [11].Another simple achievable scheme is to let only Tx (the user with the largest number of antennas) transmit, andachieve d = M , d i = 0 , i ∈ [2 : K ] .By combining these two schemes, the DoF of our asym-metric MIMO IC satisfies max M , P i ∈ [1: K ] M i ! ≤ d Σ . (3) C. Outer bound
The DoF of 2-user MIMO IC with arbitrary number ofantennas at each node was derived in [2]. This result is widelyused in DoF converse proofs (see for example [3], [4]) wherethe main idea is to reduce a K -user MIMO IC to a 2-user oneby either ‘silencing’ all but two users, or by using cooperationto obtain a 2-user MIMO IC. Therefore, by partitioning the K users into two groups so as to form two ‘super users’ andby applying the result of [2], we immediately obtain that theDoF of our asymmetric MIMO IC satisfies d Σ ≤ min S⊆ [1: K ] max X i ∈S M i , X i ∈S c M i ! , (4)where S c is the complement of S in [1 : K ] . We shall referto this bound as the cooperation outer bound. D. Systems with a dominant user
When one user has more antennas than all the other userscombined, i.e., M ≥ X i ∈ [2: K ] M i , (5)we say that the IC has a dominant user (user 1 in our channelsetting). In this case the left hand side of (3) and the righthand side of (4) coincide, and thus the DoF is completelycharacterized. Therefore, for systems with a dominant user,the cooperation outer bound is tight and is achieved by lettingonly the dominant user transmit. E. Systems without a dominant user
When there is no dominant user, the inner bound in (3)and the outer bound in (4) do not coincide in general unlessthere exists a set
S ⊆ [1 : K ] such that X i ∈S M i = X i ∈S c M i , in which case the decomposition inner bound matches thecooperative outer bound. So in general either the cooperativeouter bound or the decomposition inner bound is not tight.In order to understand which bound might be loose, wenext consider a specific 3-user IC example. Through thisexample we will show that the decomposition inner boundin (3) is tight. This will provide the necessary intuition forthe extension of the proof to the general K -user case inSection IV.III. E XAMPLE : THE ( M , M , M ) = (2 , , CASE
For the case ( M , M , M ) = (2 , , , the outer boundin (4) gives d Σ ≤ while the inner bound in (3) gives d Σ ≥ / . In this section we aim to demonstrate that theouter bound is loose. Intuitively, 3 DoF appears to be toooptimistic since it is well known that the 3-user MIMO ICwith ( M , M , M ) = (2 , , has 3 DoF [4]. Therefore, ifthe outer bound were tight, it would indicate that removingone antenna at each terminal of the the third transmitter-receiver pair does not impact the DoF. Cases of ‘antennaredundancy’ are known in [3], [5], but we shall show thatthis is not the case for our asymmetric MIMO IC when nodominant user exists.In Section III-A we start by transforming the IC in (1)into an equivalent IC in which the channel matrices containzeros in carefully chosen positions. In Section III-B we givea ‘dimension counting argument’ to show that no more than / DoFs are achievable in the equivalent IC. Finally inSection III-C, we give an information theoretic proof of thisintuitive argument and show the outer bound d Σ ≤ / .With this, the tightness of the decomposition inner boundis proved. The example highlights the key steps for the proofof optimality of the lower bound in (3) for the general K -usercase without a dominant user. A. Channel transformation
In general, we can set ¯ X i = V i X i and construct Y i = U i ¯ Y i in the channel in (1), where the beamforming matrices V i and the shaping matrices U i are full-rank / invertiblesquare matrices of dimension M i , i ∈ [1 : K ] that donot depend on P . Since invertible transformations preserveDoF, the channel in (1) and the transformed one have thesame DoF. The input-output relationship of the transformedchannel reads Y i = X k ∈ [1: K ] H ik X k + Z i ∈ C M i × , (6a) H ik := U i ¯ H ik V k ∈ C M i × M k , ( i, k ) ∈ [1 : K ] , (6b)where in (6a) we neither specify the input power constraintson the inputs X k , k ∈ [1 : K ] , nor the covariance matrix ofthe noise terms, as they do not impact the DoF.In the following we assume that all channel coefficientsare generic , i.e., randomly chosen from a continuous distri-bution. Under this assumption, the goal is to show how tofind invertible beamforming and shaping matrices such thatthe transformed channel for our ( M , M , M ) = (2 , , example is Y = " h (11)11 h (11)12 h (11)21 h (11)22 X + " h (12)11 h (12)22 X + (cid:20) h (13)11 (cid:21) X ,Y = " h (21)11 h (21)22 X + " h (22)11 h (22)12 h (22)21 h (22)22 X + (cid:20) h (23)11 (cid:21) X ,Y = h h (31)11 i X + h h (32)11 i X + h (33)11 X , where h ( ij ) ab is the scalar channel gain from the b -th antenna of Tx j to the a -th antenna of Rx i , and where we no longer writethe noises for notation convenience. To show that indeed sucha transformed channel can be found, we proceed along anumber of steps. Step 1:
As a first step we neutralize at (the singleantenna of) Rx the signal from the second antenna of Tx and from the second antenna of Tx . We do so by carefullychoosing some columns of the matrices V and V . Let V := (cid:2) v v (cid:3) , V := (cid:2) v v (cid:3) , where v ki indicates the i -th column of the matrix V k . Wechoose ( v , v ) such that ¯ H v = 0 , ¯ H v = 0 . Since ¯ H and ¯ H are generic × matrices, v and v (which are × matrices) can be chosen from the (onedimensional) right null space of ¯ H and ¯ H , respectively. Step 2:
As a second step we neutralize the signal fromthe second antenna of Tx at the first antenna of Rx , andfrom the second antenna of Tx at the first antenna of Rx .We let U := (cid:20) u u (cid:21) , U := (cid:20) u u (cid:21) , where u ki indicates the i -th row of the matrix U k . In orderto achieve our goal, we impose u ¯ H v = 0 , u ¯ H v = 0 . Since v and v have been decided already based on ¯ H and ¯ H , we have that ¯ H v and ¯ H v are generic × matrices almost surely. Therefore, u and u (which are × matrices) can be chosen from the (one dimensional)left null space of ¯ H v and ¯ H v , respectively. Step 3:
As a third step, we neutralize the signal from(the single antenna of) Tx at the second antenna of Rx and at the second antenna of Rx . We thus impose u ¯ H V = 0 , u ¯ H V = 0 . Since V is a non-zero scalar, we choose u and u asrows in the (one dimensional) left null space of ¯ H and ¯ H , respectively. Step 4:
As a last step, we neutralize the signal receivedat the second antenna of Rx from the first antenna of Tx and the one received at the second antenna of Rx from thefirst antenna of Tx . For this we impose u ¯ H v = 0 , u ¯ H v = 0 . Since u and u have been decided already based on ¯ H and ¯ H , the vectors u ¯ H and u ¯ H have dimension × and are generic. Therefore, we choose v and v tobe columns in their respective (one dimensional) right nullspaces.By the above operations, V , V , U , U have been de-cided. V and U are scalars and can be set to one withoutloss of generality. Also all transform matrices were decidedbased on generic channel coefficients, so they do not havedependence or special structure. Thus all transform matricesare full rank and invertible almost surely, and the transfor-mation preserves the DoF. B. An intuitive dimension-counting argument
We start with a ‘dimension counting’ argument to give anintuitive reason as to why the decomposition inner bound d Σ ≥ / should be tight. Without loss of generality, we canassume d = d = d and d = d ′ .Since Rx has a single antenna, the total DoF of its ownand the interference signal cannot be larger than one. Thisimplies that the interference at Rx must have less than − d ′ DoF.Now consider Tx that must achieve d DoF. Since the partof its signal that causes interference at Rx must have lessthan − d ′ DoF, the part of its signal that does not interferet Rx must have at least d − (1 − d ′ ) DoF. This is to say, Tx controls d − (1 − d ′ ) dimensions to be neutralized at Rx and these dimensions are therefore decided.Now consider Rx , which has two antennas. By the genericsetting, the decided d − (1 − d ′ ) dimensions at Tx do notalign automatically with the interference from Tx ; thereforewe have the bound [ d ] + [ d − (1 − d ′ )] + [ d ′ ] ≤ ⇐⇒ d + d ′ ≤ . (7a)From the outer bound in (4) and from d i ≤ M i , i ∈ [1 : 3] ,we know d ≤ , d ′ ≤ . (7b)It is easy to see that the bounds in (7) define a pentagon withvertices ( d, d ′ ) = (1 , / and ( d, d ′ ) = (1 / , . Thereforethe largest DoF can be at most d + d ′ = 5 / . Since a DoFof is achievable by (3), we conclude that the cooperationouter bound in (4) might be loose. C. An information theoretic proof
We define the differential entropy of the noisy signal as in[12] ~ ( X n ) := h ( X n + Z n ) , (8)where h is standard differential entropy. X n is a signal vectorpower constrained to P and Z n ∼ N (0 , I ) is independentnoise vector. Joint and conditional differential entropies aredefined in the same manner [12].We next formalize the intuitive argument from Sec-tion III-B. In the transformed channel, let X k be the signalsent by the first antenna of Tx k , and X k be one sent by thesecond antenna of Tx k , k ∈ [1 : 2] . By Fano’s inequality, wehave n ( R − ǫ n ) (9a) ≤ I ( W ; Y n ) = h ( Y n ) − h ( Y n | W ) (9b) ≤ h ( Y n ) − h ( Y n | W , W ) (9c) ≤ n (1 · log( P ) + o (log( P ))) − h ( Y n | W , W ) (9d) = n (1 · log( P ) + o (log( P ))) − ~ ( X n ) , (9e)where the inequality in (9d) follows because Rx has onlyone antenna, and the one in (9e) since in the transformedchannel h (cid:16) Y n | W , W (cid:17) = h (cid:16) h (31)11 X n + h (32)11 X n + h (33)11 X n + Z n | W , W (cid:17) = h (cid:16) h (32)11 X n + Z n (cid:17) =: ~ ( X n ) , which implies that Rx can recover X n up to noise distortionof the order o (log( P )) . Hence, the bound in (9) implies ~ ( X n ) ≤ n (1 · log( P ) − R + ǫ n + o (log( P ))) . (10) Moreover, the bound in (10) together with n ( R − ǫ n ) ≤ I ( W ; Y n ) ≤ I ( X n ; Y n )= ~ ( X n , X n ) = ~ ( X n ) + ~ ( X n | X n ) , implies ~ ( X n | X n ) ≥ n ( R − · log( P ) + R − ǫ n + o (log( P ))) . (11)Now consider n ( R − ǫ n ) ≤ I ( Y n ; W ) (12a) ≤ h ( Y n ) − h ( Y n | W , X n ) (12b) ≤ n (2 · log( P ) + o (log( P ))) − ~ ( X n , X n | X n , X n ) (12c) = n (2 · log( P ) + o (log( P ))) − ~ ( X n ) − ~ ( X n | X n ) (12d) ≤ n (3 · log( P ) − R − R + 3 ǫ n + o (log( P ))) , (12e)where the inequality in (12c) follows since Rx has twoantennas, the one in (12d) since X n and X n = ( X n , X n ) are independent, and finally (12e) comes from (11) and n ( R − ǫ n ) ≤ I ( Y n ; X n ) ≤ ~ ( X n ) .Therefore, from (12) and for n ≫ , we conclude that R + R + 2 R log( P ) ≤ o (1) , (13)or equivalently that d + d ′ ≤ / (recall R = R = d log( P ) and R = d ′ log( P ) without loss of optimality for DoF).The argument at the end of Section III-B shows that thenovel bound d + d ′ ≤ / , together with known outer bounds,implies d Σ ≤ / . Since the outer bound is achievable bythe decomposition inner bound, we have d Σ = 5 / . Thiscompletes the proof for this specific example.IV. S UM D O F FOR THE GENERAL K - USER CASE
In the previous section, through suitable invertible trans-formations we could rewrite the original IC into a newone with a special structure in the channel matrices; thisstructure suggested how to provide genie side informationto the receivers in the outer bound proof. We extend herethe proof for the example in Section III in two ways. Firstwe give a DoF outer bound for the general 3-user IC withnumber of antennas specified by the vector ( M , M , M ) inSection IV-A. Then we generalize the result to the K -usercase in Section IV-B. A. The 3-user case
Without loss of generality let M ≥ M ≥ M . We assumethere is no dominant user, that is, M < M + M . By theinvertible transformations ¯ X i = V i X i , Y i = U i ¯ Y i , i ∈ [1 :3] , we aim to obtain an equivalent channel where the inputsare partitioned as X = ( X , X , X ) , X = ( X , X ) ,nd X = ( X , X ) , and similarly for the outputs. Let | X ij | indicate the size / number of antennas in X ij . We want | X | = | Y | = | X | = | Y | = M − M , | X | = | Y | = | X | = | Y | = M − M , | X | = | Y | = | X | = | Y | = | X | = | Y | = M + M − M . Let the channel matrix from ¯ X jb to ¯ Y ia in the originalchannel be denoted as ¯ h ( ij ) ab , with size | Y ia | × | X jb | .We now derive the channel input/output relationship of thetransformed channel. As before, the beamforming matrices in(6a) are denoted as V = (cid:2) v v v (cid:3) , V = (cid:2) v v (cid:3) , V = (cid:2) v v (cid:3) , where v ab has size M a × | X ab | , and the shaping matrices as U = u u u , U = (cid:20) u u (cid:21) , U = (cid:20) u u (cid:21) , where u ab has size | Y ab | × M a .We first choose the beamforming matrices by imposing ¯ H v = 0 , ¯ H v = 0 , h ¯ h (12)11 ¯ h (12)12 i v = 0 , h ¯ h (13)31 ¯ h (13)32 i v = 0 . Under the generic channel gain assumption, the matrices ¯ H , ¯ H , h ¯ h (12)11 ¯ h (12)12 i and h ¯ h (13)31 ¯ h (13)32 i have right nullspace of rank M − M , M − M , M + M − M ,and M + M − M , respectively, almost surely. Thus wecan pick columns from these right null spaces to formthe beamforming matrices v , v , v , v , which aretherefore of size M × ( M − M ) , M × ( M − M ) , M × ( M + M − M ) , and M × ( M + M − M ) ,respectively, and are still generic almost surely. The matrices v , v , and v are randomly chosen so that they are full-rank and with no specific relation with the previously chosenmatrices.We then choose the shaping matrices by imposing u ¯ H = 0 , u ¯ H = 0 ,u ¯ h (12)12 ¯ h (13)12 ¯ h (12)22 ¯ h (13)22 ¯ h (12)32 ¯ h (13)32 = 0 ,u " ¯ h (21)13 ¯ h (21)23 = 0 , u " ¯ h (23)11 ¯ h (23)21 = 0 ,u " ¯ h (31)11 ¯ h (31)21 = 0 , u " ¯ h (32)11 ¯ h (32)21 = 0 . Under the generic channel gain assumption, all channelmatrices are full rank almost surely; the shaping matricescan thus be chosen as rows is the respective right null spacesand are still generic almost surely. U is full rank matrix, since u and u are chosen from independent null spaces,thus are independent. Similarly, we claim U is full rank. Wethen show that U is also full rank. It is easy to see that u and u are independent. If U is not full rank, there mustexist non-zero row-vectors g of size × (2 M − M − M ) and g of size × ( M + M − M ) such that g (cid:20) u u (cid:21) = g u , that is = g u ¯ h (12)12 ¯ h (13)12 ¯ h (12)22 ¯ h (13)22 ¯ h (12)32 ¯ h (13)32 = g (cid:20) u u (cid:21) ¯ h (12)12 ¯ h (13)12 ¯ h (12)22 ¯ h (13)22 ¯ h (12)32 ¯ h (13)32 = g u ¯ h (12)12 ¯ h (12)22 ¯ h (12)32 u ¯ h (13)12 ¯ h (13)22 ¯ h (13)32 = g F . Since u is independent of ¯ H and u is independent of ¯ H , F is a full rank square matrix almost surely. Then G = , which contradicts our initial assumption. Therefore weclaim U is full rank almost surely.With the chosen beamforming and shaping matrices, thetransformed channel has input/output relationship Y = h (11)11 h (11)12 h (11)13 h (11)21 h (11)22 h (11)23 h (11)31 h (11)32 h (11)33 X + h (12)12 h (12)21
00 0 X + h (13)21 h (13)32 X Y = " h (21)12 h (21)22 h (21)23 X + " h (22)11 h (22)12 h (22)21 h (22)22 X + " h (23)11 h (23)12 h (23)22 X Y = " h (31)12 h (31)21 h (31)22 X + " h (32)11 h (32)12 h (32)22 X + " h (33)11 h (33)12 h (33)21 h (33)22 X , where h ( ij ) ab represents the transformed channel matrix from X jb to Y ia , which has size | Y ia | × | X jb | . Since the beam-forming and shaping matrices are full rank almost surely,we performed an invertible transformation that preserves theoF. Therefore we obtained a new channel that is not fullyconnected and whose structure suggests which genie sideinformation to provide to the receivers in the converse proof.We shall consider different choices of side information atthe various receivers. The idea is to start as usual by Fano’sinequality, by providing side information S nu to receiver u ,and by bounding the entropy of the output as a function ofthe number of antennas at receiver u , so as to obtain n ( R u − ε n ) ≤ n ( M u log( P ) + o (log( P ))) − h ( Y nu | W u , S nu ) . The entropy term h ( Y nu | W u , S u ) depends on the distributionof the interference at receiver u (since X nu can be cancelledthanks to the knowledge of W u ) conditioned on the sideinformation S nu ; if such an entropy term, which appears witha negative sign, cannot be single-letterized, then we proceedto provide side information to another receiver in such a waythat the same entropy term appears with positive sign; byadding the two bounds we ‘get rid’ of the entropy termsthat cannot be single-letterized. We continue in this fashionuntil we obtain a single-letter outer bound. For the generalasymmetric 3-user MIMO IC the steps are as follows. By providing Rx with side information W we have n ( R − ε n ) ≤ h ( Y n ) − h ( Y n | W , W ) ≤ n ( M log( P ) + o (log( P ))) − ~ ( X n , X n | X n , X n )= n ( M log( P ) + o (log( P ))) − ~ ( X n , X n ) , (14)where the inequality follows since Rx does not receive X .Similarly, by providing Rx with W we obtain n ( R − ε n ) ≤ n ( M log( P ) + o (log( P ))) − ~ ( X n , X n ) . (15)By adding (14) and (15) and since ~ ( X n , X n ) + ~ ( X n , X n ) ≥ ~ ( X n , X n ) + ~ ( X n | X n , X n ) + ~ ( X n )= ~ ( X n , X n , X n ) + ~ ( X n ) ≥ n ( R − ε n ) + ~ ( X n ) , we obtain n ( R + R + R − ε n ) ≤ n (( M + M ) log( P ) + o (log( P ))) − ~ ( X n ) . (16) Next, weprovide ( X n , X n ) as side information to Rx and obtain n ( R − ε n ) ≤ h ( Y n ) − h ( Y n | W , X n , X n ) ≤ n ( M log( P ) + o (log( P ))) − ~ ( X n , X n | W , X n , X n )= n ( M log( P ) + o (log( P ))) − ~ ( X n | X n ) − ~ ( X n | X n ) . (17)Similarly, we provide ( X n , X n ) as side information to Rx and obtain n ( R − ε n ) ≤ n ( M log( P ) + o (log( P ))) − ~ ( X n | X n ) − ~ ( X n | X n ) . (18) By adding (17) and (18) we obtain n ( R + R + R − ε n ) ≤ n (( M + M ) log( P )+ o (log( P ))) + ~ ( X n ) − ~ ( X n | X n ) − ~ ( X n | X n ) . (19) Now, we provide Rx withenough side information to enable the decoding of all mes-sages. After Rx has decoded its own message / removed X n from the received signal, it is left with M linearcombinations of M + M interfering symbols; if we weprovide Rx with M + M − M extra observations / antennaoutputs, it will be able to decode all interfering symbols. Nextwe derive two such ‘MAC-bounds’ by providing either X n or X n to Rx . We have n ( R + R + R − ε n ) ≤ I ( W , W , W ; Y n , X n )= h ( Y n , X n ) − o (log( P )) ≤ h ( Y n ) + ~ ( X n | X n ) − o (log( P ))= h ( Y n ) + ~ ( X n , X n ) − ~ ( X n ) − o (log( P )) ≤ n ( M log( P ) + o (log( P ))) + n ( R − ε n ) − ~ ( X n ) . (20)Similarly n ( R + R + R − ε n ) ≤ I ( W , W , W ; Y n , X n ) ≤ n ( M log( P ) + o (log( P ))) + n ( R − ε n ) − ~ ( X n ) . (21) Final bound:
By adding (16), (19), (20), (21), and bytaking n → ∞ , we obtain R + 4 R + 4 R ≤ M + M + M ) log( P ) + o (log( P )) , and therefore the DoF is outer bounded by d Σ ≤ lim P →∞ M + M + M ) log( P ) + o (log( P ))4 log ( P )= M + M + M . This concludes the proof for the general 3-user asymmetricIC in the case where there is no dominant user.
B. The general K -user case We are now ready to extend our 3-user result to the general K -user asymmetric MIMO IC. Our main result is Theorem 1.
For almost all channel realizations the asymmet-ric K -user MIMO IC, in which the i -th user has M i antennasat both the transmitter and the receiver, i ∈ [1 : K ] , the DoFis d Σ = max X i ∈ [1: K ] M i / , max i ∈ [1: K ] M i . (22) Proof:
As per our discussion in Section II-D, when thereis a dominant user (whose has more antennas than the restof the users combined) it is optional to let only that usertransmit. When there is no dominant user, we can alwaysartition the users into three groups such that no group hasmore antennas than the the other two groups combined. Thenwe allow the users in the same group to fully cooperate andapply our bound for 3-user IC, which shows that the DoFis half the sum of number of the total number of antennas.This concludes the proof.V. C
ONCLUSION
In this paper we studied a special class of K -user asym-metric MIMO interference channels in which a transmitterand its receiver are equipped with the same number ofantennas, while different users may have different number ofantennas. We showed that existing cooperation-based outerbounds are loose and gave a novel outer bound. Our resultindicates two operating regimes. For systems with a dominantuser (a user who has more antennas that the other userscombined), the optimal DoF is achieved by inactivating allbut the dominant user. For systems without a dominant user,the decomposition inner bound turns out to be tight, that is,the MIMO operations do not help in the DoF perspective. Thecharacterization of the DoF of arbitrary asymmetric K -userMIMO interference channels is part of ongoing investigation.A CKNOWLEDGEMENT
The work of T. Liu and D. Tuninetti was partially fundedby NSF under award number 1218635. S. Jafar is affiliatedwith the Center for Pervasive Communications and Comput-ing (CPCC) at UC Irvine. His work is supported in part byNSF grant CCF-1319104. The contents of this article aresolely the responsibility of the author and do not necessarilyrepresent the official views of the NSF.R
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