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Structured Lattice Codes for 2 × 2 × 2 MIMO Interference Channel Song-Nam Hong Giuseppe Caire Dep. of Electrical Engineering Dep. of Electrical Engineering University of Southern California University of Southern California CA, USA CA, USA Email: [email protected] Email: [email protected] Abstract—We consider the 2×2×2 multiple-input multiple- output interference channel where two source-destination pairs S1 R1 D1 wish to communicate with the aid of two intermediate relays. In this paper, we propose a novel lattice strategy called Aligned Precoded Compute-and-Forward (PCoF). This scheme consists 3 of two phases: 1) Using the CoF framework based on signal 1 alignmentwetransformtheGaussiannetworkintoadeterministic 0 finitefieldnetwork.2)Usinglinearprecoding(overfinitefield)we S2 R2 D2 2 eliminate the end-to-end interference in the finite field domain. n Fthuertpheerrf,owrmeaexnpceloiattthfieniatlegeSbNraRic, ssturcuhcttuhraetobfelyaottnicdesatodeegnrheaencoef Fig.1. 2×2×2MIMOGaussianinterferencechannel. a freedom result (also achievable by other means). We can also J showthatAlignedPCoFoutperformstime-sharinginarangeof 8 reasonablymoderateSNR,withincreasinggainasSNRincreases. Also, there was the recent extension to the K × K × K 2 Gaussian IC in [10], achieving the optimal K DoF using aligned network diagonalization. T] I. INTRODUCTION We consider the 2×2×2 multiple-input multiple-output I In recent years, significant progress has been made on the (MIMO) IC as shown in Fig. 1 consisting of two sources, s. understanding of the theoretical limits of wireless commu- two relays, and two destinations. All nodes have M multiple c antennas. Each source k has a message for its intended nication networks. In [1], the capacity of multiple multicast [ network (where every destination desires all messages) is destination k, for k = 1,2. In the first hop, a block of 1 n channel uses of the discrete-time complex MIMO IC is approximated within a constant gap independent of SNR and v described by oftherealizationofthechannelcoefficients.Also,formultiple 3 5 flows over a single hop, new capacity approximations were (cid:20) Y (cid:21) (cid:20) F F (cid:21)(cid:20) X (cid:21) (cid:20) Z (cid:21) 4 obtainedintheformofdegreesoffreedom(DoF),generalized YR1 = F11 F12 X1 + ZR1 (1) 6 degreesoffreedom(GDoF),andO(1)approximations[2]–[4]. R2 21 22 2 R2 01. Ylaergt,eltyheunstsuodlvyedo.fTmhuelt2ip×le2fl×o2wsGaouvsesriamnuilnttieprlfeerheonpcsercehmananinesl wthheekre-ththseomuractericcehsanXnekl1inapnudt YseqRkuecnocnetsaixn, ar∈ranCg1e×dnbyforro(cid:96)w=s, k,(cid:96) 3 (IC) has received much attention recently, being one of the 1,...,M, the k-th relay channel output sequences y ∈ 1 Rk,(cid:96) : fundamental building blocks to characterize the DoFs of two- C1×n, and where Fjk ∈ CM×M denotes the channel matrix v flows networks [5] One natural approach is to consider this from source k to j. In the second hop, a block of n channel Xi model as a cascade of two ICs. In [6], the authors apply uses of the discrete-time complex MIMO IC is described by the Han-Kobayashi scheme [7] for the first hop to split each r (cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21) a messageintoprivateandcommonparts.Relayscancooperate Y1 = G11 G12 XR1 + Z1 (2) using the shared information (i.e., common messages) for Y2 G21 G22 XR2 Z2 the second hop, in order to improve the data rates. This wherethematricesX andY containthek-threlaychannel approach is known to be highly suboptimal at high signal- input sequences x Rk∈ C1×nk, the k-th destination channel to-noise ratios (SNRs), since two-user IC can only achieve Rk,(cid:96) output sequences y ∈ C1×n, and where G ∈ CM×M 1 DoF. In [8], Cadembe and Jafar show that 4 DoF is k,(cid:96) kj 3 denotes the channel matrix between relay j and destination k. achievable by viewing each hop as an X-channel. This is ThematrixZ (orZ )containsi.i.d.Gaussiannoisesamples accomplished using the interference alignment scheme for k Rk eachhop.Recently,theoptimalDoFwasobtainedin[9]using 1 We use “underline” to denote the matrices whose horizontal dimension aligned interference neutralization, which appropriately com- (columnindex)denotes“time”andverticaldimension(rowindex)runsacross bines interference alignment and interference neutralization. theantennas.ForanymatrixX,wedenotex bythe(cid:96)-throwofX. (cid:96) ∼ CN(0,1). We assume that the elements of F and G a natural labeling of the codewords of L by the information jk kj are drawn i.i.d. according to a continuous distribution (i.e., messages w ∈ Fr. Notice that the set p−1g(C)T is a system q Gaussian distribution). The channel matrices are assumed to of coset representatives of the cosets of Λ in Λ . Hence, the 1 be constant over the whole block of length n and known to natural labeling function f : Fr → L is defined by f(w) = q all nodes. Also, we consider a sum-power constraint equal to p−1g(wG)T mod Λ. MSNR at each transmitter. B. Compute-and-Forward In this paper, we propose a novel lattice strategy named Aligned Precoded Compute-and-Forward (PCoF). CoF makes We recall here the CoF scheme of [13]. Consider a 2-user useoflatticecodessothateachreceivercanreliablydecodea Gaussian multiple access channel (MAC) with M antennas at linear combination with integer coefficients of the interfering each transmitter and at the receiver, represented by codewords. Thanks to the fact that lattices are modules over Y =HCX+Z (4) the ringof integers, thelinear combinations translates directly into a linear combination of messages over a suitable finite where H ∈ CM×M, C ∈ Z[j]M×2M, and Z contains i.i.d. field.Inthisway,eachhopistransformedintoadeterministic Gaussian noise samples ∼ CN(0,1). This particular form of noiseless finite field IC. The end-to-end interferences in the channel matrix HC will be widely considered in this paper, finite field domain are eliminated by distributed precoding as a consequence of signal alignment. All users make use (over finite field) at relays. Using this framework, we char- of the same nested lattice codebook L = Λ ∩ V , where 1 Λ acterize a symmetric sum rate and prove that 2M −1 DoF Λ has second moment σ2 (cid:44) 1 (cid:82) (cid:107)r(cid:107)2dr = SNR . is achievable by lattice coding (this DoF result is proven in Eachuserk encodesM inΛformnaVtiooln(Vmλ)essVaλgesw ∈Fr inetfof k,(cid:96) q [9] without resorting to CoF and lattice coding). Further, we the corresponding codeword t =f(w ) and produces its k,(cid:96) k,(cid:96) use the lattice codes algebraic structure in order to obtain channel input according to also good performance at finite SNRs. We use integer-forcing receiver(IFR)of[11]inordertominimizetheimpactofnoise xk,(cid:96) =[tk,(cid:96)+dk,(cid:96)] mod Λ, (5) boosting at the receivers, and integer-forcing beamforming for (cid:96) = 1,...,M, where the dithering sequences d ’s (IFB),proposedbytheauthorsin[12],inordertominimizethe k,(cid:96) mutually independent across the users and the messages, powerpenaltyatthetransmitters.Weprovidenumericalresults uniformly distributed over V , and known to the receiver. Λ showing that Aligned PCoF outperforms time-sharing even at Notice that x is the (cid:96)-th row of X , k = 1,2. The reasonably moderate SNR, with increasing performance gain k,(cid:96) k decoder’s goal is to recover an integer linear combination as SNR increases. s=[bHCT] mod Λ,withsomeintegervectorb∈Z[j]M×1. Since Λ is a Z[j]-module (closed under liner combinations II. PRELIMINARIES 1 with Gaussian integer coefficients), then s ∈ L. Letting ˆs Inthissectionweprovidesomebasicdefinitionsandresults be decoded codeword, we say that a computation rate R is that will be extensively used in the sequel. achievable for this setting if there exists sequences of lattice codesLofrateRandincreasingblocklengthn,suchthatthe A. Nested Lattice Codes decoding error probability satisfies lim P(ˆs(cid:54)=s)=0. Let Z[j] be the ring of Gaussian integers and p be a prime. n→∞ In the scheme of [13], the receiver computes Let ⊕ denote the addition over F with q = p2, and let g : F → C be the natural mapping oqf F onto {a+jb : a,b ∈ yˆ = (cid:2)αHY−bHCD(cid:3) mod Λ q q Zp}⊂C.Werecallthenestedlatticecodeconstructiongiven = [s+z (HC,b,α)] mod Λ (6) in [13]. Let Λ = {λ = zT : z ∈ Zn[j]} be a lattice in Cn, eff with full-rank generator matrix T ∈ Cn×n. Let C = {c = where α ∈ CM×1 and zeff(HC,b,α) = (αHH−bH)CU+ wG:w∈Fr}denotealinearcodeoverF withblocklength αHZ denotes the effective noise, including the non-integer q q self-interference (due to the fact that αHH ∈/ Z[j]1×M in n and dimension r, with generator matrix G. The lattice Λ 1 general) and the additive Gaussian noise term. The scaling, is defined through “construction A” (see [14] and references dither and modulo-Λ operation in (6) is referred to as the therein) as Λ =p−1g(C)T+Λ, (3) CoF receiver mapping. Choosing αH =bHH−1, the variance 1 of the effective noise is given by σ2 = (cid:107)(H−1)Hb(cid:107)2. This exact where g(C) is the image of C under the mapping g (applied scheme is known as exact IFR [11]. In this way, the non- component-wise). It follows that Λ ⊆ Λ ⊆ p−1Λ is a chain integer penalty of CoF is completely eliminated. More in 1 of nested lattices, such that |Λ /Λ| = p2r and |p−1Λ/Λ | = general, the performance can be improved especially at low 1 1 p2(n−r). SNR by minimizing the variance of z (H,b,α) with respect For a lattice Λ and r∈Cn, we define the lattice quantizer to α. In this case, we obtain: eff Q (r) = argmin (cid:107)r − λ(cid:107)2, the Voronoi region V = Λ λ∈Λ Λ σ2(HC,b) = bHC(SNR−1I+CHHHHC)−1CHb.(7) {r ∈ Cn : Q (r) = 0} and [r] mod Λ = r−Q (r). For Λ eff Λ Λ and Λ given above, we define the lattice code L=Λ ∩V Since α is uniquely determined by HC and b, it will be 1 1 Λ with rate R = 1 log|L| = r logq. Construction A provides omitted in the following, for the sake of notation simplicity. n n Source 1 Relay1 Destination1 A. CoF framework based on signal alignment In this section we show how to turn any 2-user MIMO IC into a noiseless finite field IC using the CoF framework. We Precoding focus on the first hop of our 2×2×2 network since the same schemeisstraightforwardlyusedforthesecondhop.Consider "not used" the MIMO IC in (1). Let {w1,(cid:96) ∈Frq :(cid:96)=1,...,M} denote Source 2 "not used" Relay 2 Destination 2 themessagesofsource1and{w ∈Fr :(cid:96)=1,...,M−1} 2,(cid:96) q denote the messages of source 2. All transmitters make use of Fig.2. Adeterministicnoise-free2×2×2finitefieldinterferencechannel. the same nested lattice codebook L = Λ ∩ V , where Λ 1 Λ has the second moment σ2 = SNR . Also, we let V = Λ eff 1 From [13], we know that by applying lattice decoding to yˆ [v ···v ] ∈ CM×M and V = [v ···v ] ∈ 1,1 1,M 2 2,1 2,M−1 given in (6) the following computation rate is achievable: CM×M−1 denote the precoding matrices used at sources 1 R(H,b,SNR )=log+(SNR /σ2(HC,b)) (8) and 2, respectively. They are chosen to satisfy the alignment eff eff conditions, given by where log+(x) (cid:44) max{log(x),0}. Also, the receiver can F v = F v (12) reliably decode M linear combinations S = [BCT] mod Λ 11 1,k+1 12 2,k with integer coefficient vectors {bH :(cid:96)=1,...,M} (i.e., the F21v1,k = F22v2,k (13) (cid:96) (cid:96)-th row of B) if for k =1,...,M−1. The feasibility of the above conditions R≤min{R(H,b ,SNR )}(cid:44)R(H,B,SNR ). (9) was shown in [9] for any M. (cid:96) (cid:96) eff eff Let A ∈ Z[j]M×M and A ∈ Z[j]M−1×M−1 denote the 1 2 Using the lattice encoding linearity, the corresponding M fullrankintegermatrices.Theywillbeoptimizedtominimize F linear combinations over q for the messages are obtained the power penalty in Section IV. Each source k precodes its as messages over F as q U = g−1([B] mod pZ[j])g−1([C] mod pZ[j])W W(cid:48) =[A ]−1W , k =1,2. (14) k k q k (a) = [B] [C] W, (10) Then, the precoded messages are encoded using the nested q q lattice codes. Finally, the channel input sequences are given where we use the notation [B] (cid:44)g−1([B] mod pZ[j]). q by the rows of: III. ALIGNEDPCOF X =V A T =V T(cid:48), (15) k k k k k k In this section, we propose a novel lattice strategy called where t = [f(w(cid:48) ) + d ] mod Λ, and where T(cid:48) = “Aligned”PrecodedCoF(PCoF).Thisconsistsoftwophases: k,(cid:96) k,(cid:96) k,(cid:96) k A T .Duetothepowerconstraint,eachsourcekmustsatisfy 1) The CoF framework transforms a Gaussian network into k k a finite field network. 2) A linear precoding scheme is used SNR tr(V A AHVH)≤MSNR. (16) eff k k k k over finite field to eliminate the end-to-end interferences (see Since sources use the same nested lattice codes, we can Fig. 2). While using the CoF framework, the main perfor- choose: mance bottleneck consists of the non-integer penalty, which ultimately limits the performance of CoF scheme in the high SNR =min{SNR(V ,A ):k =1,2} (17) eff k k SNR regime [15]. To overcome this bottleneck, we employ where SNR(V ,A )=MSNR/tr(V A AHVH). signalalignmentinordertocreatean“aligned”channelmatrix k k k k k k The decoding procedure is as follows. We first consider the for which exact integer forcing is possible, as seen in Section aligned received signals. Relay 1 observes II-B. Namely, we use alignment precoding matrices V and 1 V2 at the two sources such that YR1 = F11X1+F12X2+ZR1 wprheecroediHngRkov∈er(cid:2)CCFMmk×1aMVyp1aronddFukCc2eRVka2p∈o(cid:3)wZ=e[rjHp]MeRnk×aC2lMtRyk.d,uHeotwoetvheer,n(1oth1ne)- (=a) (cid:124)F1(cid:123)1(cid:122)V(cid:125)1 t(cid:48)1,2t+(cid:48)1...,1t(cid:48)2,1 +ZR1 (18) unitary nature of the alignment matrices, and this can degrade (cid:44)HR1 t(cid:48)1,M +t(cid:48)2,M−1 (cid:20) (cid:21) the performance at finite SNR. In order to counter this effect, T = H C 1 +Z (19) we use the concept of IFB (see [12]). The main idea is that R1 R1 T R1 2 V canbepre-multiplied(fromtheoft)bysomeappropriately k where (a) follows from the fact that the precoding vec- chosen full-rank integer matrix A since its effect can be undone by precoding over F , usikng [A ] . Then, we can tors satisfy the alignment conditions in (13) and CR1 = q k q [ A C A ] with optimize the integer matrix in order to minimize the power 1 12 2 penalty. The detailed procedures of Aligned PCoF are given (cid:20) 01×M−1 (cid:21) C = . (20) in the following sections. 12 IM−1×M−1 Similarly, relay 2 observes the aligned signals: messages with a change of sign (multiplication by −1 in the  t(cid:48) +t(cid:48)  finitefield)atdestination2(seeFig.2).Noticethatthesystem 1,1 2,1 matrix is fixed and independent of the channel matrices, since .  .  Y = F V  . +Z (21) it is determined only by the alignment conditions. R2 (cid:124)2(cid:123)1(cid:122) (cid:125)1 t(cid:48)1,M−1+t(cid:48)2,M−1  R2 Lemma 1: Choosing precoding matrices M1 and M2 as (cid:44)HR2 t(cid:48)1,M M = (IM×M ⊕(−Q Q ))−1 (28) (cid:20) (cid:21) 1 12 21 T = H C 1 +Z (22) M = −(IM−1×M−1⊕(−Q Q ))−1 (29) R2 R2 T R2 2 21 12 2 the end-to-end system matrix becomes a diagonal matrix: where C =[ A C A ] with R2 1 22 2 (cid:20) M 0 (cid:21) (cid:20) IM×M 0 (cid:21) (cid:20) IM−1×M−1 (cid:21) Q 1 Q = . C22 = 01×M−1 . (23) 0 M2 0 −IM−1×M−1 Proof: See the long version of this paper [16]. The channel matrices in (19) and (22) follows the form Based on the above, we proved the following: considered in Section II-B. Following the CoF framework in Theorem 1: For the 2 × 2 × 2 MIMO IC defined in (1) (9) and (10), if R≤R(H C ,B ,SNR ), the relay k can Rk Rk Rk eff and (2), Aligned PCoF can achieve the symmetric sum rate of decode the M linear combinations with full-rank coefficients (2M −1)R with common message rate matrix B : Rk R= min{R(H C ,B ,SNR ),R(H C ,B ,SNR(cid:48) )} (cid:20) W(cid:48) (cid:21) k=1,2 Rk Rk Rk eff k k k eff U = [B ] [C ] 1 k = [BRRkk]qq(cid:2) [RAk q1]q W[C(cid:48)2k2]q[A2]q (cid:3)(cid:20) WW(cid:48)1(cid:48) (cid:21) wmfoharetrarinecyesfuVllkr,aVnkRkintotegseartismfyattrhiceesalAigkn,mAenRkt,cBonkd,iBtioRnk,sainnd(1a3n)y, 2 (cid:20) (cid:21) (=a) [BRk]q(cid:2) IM×M [Ck2]q (cid:3) WW1 HRk = Fk1V1, Hk =Gk1VR1 2 SNR = min{SNR(V ,A ):k =1,2} eff k k w[Bher]e−(1aU) isadnudeWtô thbeeptrheecofidrisntgMov−er1Fqroiwns(1o4f).[BLet]W−ˆ1U1 =. SNR(cid:48)eff = min{SNR(VRk,ARk):k =1,2}, R1 q 1 2 R2 q 2 and C = [A C A ] and C = [A C A ] with Then, we can define the deterministic noiseless finite field IC: Rk 1 k2 2 k R1 k2 R2 C ,C in (20) and (23). (cid:20) Wˆ1 (cid:21)=Q(cid:20) W1 (cid:21) (24) Sh1o2win2g2that R grows as logSNR yields: Wˆ2 W2 Corollary 1: Aligned PCoF achieves the 2M −1 DoF for the 2 × 2 × 2 MIMO IC when all nodes have M multiple where the so-called system matrix is obtained using C and 1 antennas. C as 2 (cid:20) IM×M Q (cid:21) Proof: See the long version of this paper [16]. Q= 12 (25) Q IM−1×M−1 21 IV. OPTIMIZATIONOFSYMMETRICSUMRATES where Suppose that precoding matrices V ,V are determined. (cid:20) 01×M−1 (cid:21) We need to optimize all integer matrkicesRkin Theorem 1 to Q = ,Q =[ IM−1×M−1 0M−1×1 ]. 12 IM−1×M−1 21 maximize the sum rates. First, the power-penalty optimization problems take on the form: B. Linear precoding over deterministic network M Eq.(24)definesthefirst hopofthenoiselessfinitefieldIC. argmin tr(cid:0)VAAHVH(cid:1)=(cid:88)(cid:107)Va (cid:107)2 Next,wefocusonthesecondhop.Therelaykusesaprecoded (cid:96) versionofdecodedlinearcombinationsW =M Wˆ asits (cid:96)=1 Rk k k subject to A is full rank (30) messages. Operating in a similar way as for the first hop, the second hop noiseless finite field IC is given by wherea denotesthe(cid:96)-thcolumnofA.Also,theminimization (cid:96) (cid:20) Wˆ (cid:21) (cid:20) W (cid:21) problem of variance of effective noise consists of finding an R1 = Q R1 . (26) integer matrix B solution of: Wˆ W R2 R2 argmin max{bH(SNR−1I+HHH)−1b } Concatenating (24) and (26), the end-to-end finite field noise- (cid:96) (cid:96) eff (cid:96) less network is described by subject to B is full rank (31) (cid:20) Wˆ (cid:21) (cid:20) M 0 (cid:21) (cid:20) W (cid:21) whereHdenotesanalignedchannelmatrixandbH isthe(cid:96)-th R1 =Q 1 Q 1 . (27) (cid:96) Wˆ R2 0 M2 W2 rtroiwangouflaBr.mBaytriCxhLolesuskcyh tdheactomposition, there exists a lower Lemma1showsthatthelinearcombinationsdecodedatdesti- nation1areequaltoitsdesiredmessagesandareequaltothe bH(SNR−1I+HHH)−1b =(cid:107)LHb (cid:107)2. (32) (cid:96) eff (cid:96) (cid:96) For the simulation, we used the following precoding matrices 70 to satisfy the above conditions: Time Sharing 60 AAlliiggnneedd PPCCooFF (w/o opt) 3 DoF V1 = (cid:2) F−211F121 F−111F221 (cid:3) and v2,1 =1. Also, the same construction method is used for the second Rates (bits per channel use)345000 2 DoF httprhreooalawpnany.semssrS-oiic1usnosrcnicaeosenntdsr2oapiu2o(nro)wtc,reiestrrhe.1elaIaln(ytoreoorr2lnre)rda,eetolitravhfyeetolsy1fooe)ufrrrfiemtcrvceaeeisnerrsns1ametlldwyiatnsaisnydaostnis2sruefecy(qmceuetqohisurreseieivsvaaesvhlteeirtgnreiahtmalgmeyeer, m Average Su20 sSIFlNoRtRs.agnInadinFIFiagBb.,o3cu,otmw5epdaoBrbinsbgeyrwvoeipthtthimsaitimzAipnllgiygntuhesediniPgnCtiedogeFenrtcimatynatmhriaacvteresictfehoser. Also, Aligned PCoF provides a higher sum rate than time- 10 sharing if SNR ≥ 15 dB, and its gain over time-sharing increases with SNR, showing that in this case the DoF result 0 0 10 20 30 40 50 60 70 matters also at finite SNR. SNR [dB] REFERENCES Fig. 3. Performance comparison of Aligned PCoF and time-sharing with [1] S. Avestimehr, S. Diggavi, and D. 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