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is (Shitz) Shamai S. and , 5 .I I. ,[ ], 6 [email protected] NTRODUCTION rrt pitn [ splitting rate or ] 2 ]–[ .Nzri ihteDepartment the with is Nazer B. , 4 ,wihrl nyo single- on only rely which ], 7 ebr IEEE Member, ]. otn MA, Boston, , nWireless in s h 15th the d 694630. . tthe at s . nteger- nteger- neering, orcing eword oa to s can g rithm kshop, itters rizon Inc., hile on- he H er th al h n g e d y e . n hooSaa (Shitz), Shamai Shlomo and , n qaiainmtie.T oiaetefis applicatio first [ the work motivate To prior matrices. integ beamform equalization optimizing downlink for and for algorithm iterative result an and optimality forcing constant-gap a ality: sum the of terms i in which duality channel, guarantee only rate. dual can the we in means ma turn users variances different noise One effective with matrices. the associated beamforming that is and obstacle equalization technical the of roles the iertasitracietrsfrteMM C(e,for (see, BC significant of MIMO MAC, performance the [ the 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H H T 10 sn h aesmpwrand power sum same the using .A ihlvl hsmeans this level, high a At ]. sas civbefrdecoding for achievable also is elw IEEE Fellow, 8 ]–[ H A 12 T yexchanging by .Specifically, ]. vraMIMO a over o and ion 16 .In ]. atrix be y gher sage ing the ds. er- ed ce ly n, le n n 1 r IEEE TRANS INFO THEORY, TO APPEAR 2 using only “digital” successive cancellation, assuming that the advantages of a random precoding matrix on the outage channel state information (CSI) is available at the transmitters. probability for integer-forcing [36]. Using duality, we demonstrate that integer-forcing can also Integer-forcing can also serve as a framework for distributed operate within a constant gap of the MIMO BC capacity source coding, and can be viewed as the “dual” of integer- using only “digital” dirty-paper coding, again assuming CSI forcing channel coding in a certain sense. See [37],[38] for is available at the transmitter. For the second application, it is further details. Very recent work has also established uplink- well-known that simultaneously optimizing beamforming and downlink duality for compression-based strategies for cloud equalization matrices corresponds to a non-convex problem. radio access networks [39]. However, for both the MIMO MAC and BC, finding the Finally, we note that there is a rich body of work on optimal equalization matrix for a fixed beamforming matrix lattice-aided reduction [40]–[46] for MIMO channels. For has a closed-form solution. Therefore, a natural algorithm is instance, in the uplink version of this strategy, each transmitter to iterate between a problem and its dual, updating the equal- employs a lattice-based constellation (such as QAM). The ization matrix at every iteration. For example, the maxSINR decoder steers the channel to a full-rank integer matrix using algorithm [21] relies on a variation of this idea to identify good equalization, makes hard estimates of the resulting integer- interference alignment solutions. Here, we propose an iterative linear combinations of lattice symbols, and then applies the algorithm for optimizing the beamforming and equalization inverse integer matrix to obtain estimates of the emitted matrices used in integer-forcing and show that it converges to symbols. Roughly speaking, integer-forcing can be viewed as a local optimum. Recent follow-up work has used a variation lattice-aided reduction that operates on the codeword, rather of our algorithm to identify good integer-forcing interference than symbol, level. This in turn allows us to write explicit alignment solutions [22]. achievable rate expressions for integer-forcing, whereas rates for lattice-aided reduction must be evaluated numerically. A. Related Work Prior work on integer-forcing [16]–[18],[23] has focused on the important special case where all codewords have the B. Paper Organization same effective power. This constraint is implicitly imposed by the original compute-and-forward framework [24]. In order The remainder of this paper is organized as follows. In to establish uplink-downlink duality, we need the flexibility Section II, we give problem statements for the uplink and to allocate power unequally across codewords. We will thus downlink. Next, in Section III, we give a high-level overview employ the expanded compute-and-forward framework [20], of our duality results. Section IV provides background results which can handle unequal powers. Our achievability results from nested lattice coding that will be needed for our achiev- draw upon capacity-achieving nested lattice codes, whose ability scheme. We present detailed achievability strategies for existence has been shown in a series of recent works [25]– the uplink and downlink in Sections V and VI, respectively. [30]. We refer interested readers to the textbook of Zamir for Section VII formally establishes an uplink-downlink duality a detailed history as well as a comprehensive treatment of relationship for integer-forcing, and handles the technical issue lattice codes [31]. associated with different effective noise variance associations For the sake of notational simplicity, we will state all across the MIMO MAC and BC. In Section VIII, we propose of our results for real-valued channels. Analogous results an iterative algorithm that uses uplink-downlink duality for can be obtained for complex-valued channels via real-valued optimizing the integer-forcing beamforming, equalization, and decompositions. Recent efforts have shown that compute-and- integer matrices. We provide simulations in Section IX and forward can also be realized for more general algebraic struc- Section X concludes the paper. tures [32]. For instance, building lattices from the Eisenstein integers yields better approximations for complex numbers on average, and can increase the average performance of compute-and-forward [33]. C. Notation Here, we will assume that full channel state information (CSI) is available to the transmitter and receiver, in order We will make use of the following notation. Column vectors to optimize the beamforming matrices and power allocations. will be denoted by boldface, lowercase font (e.g., a ∈ ZL) and However, CSI may not always be available, especially at the matrices with boldface, uppercase font (e.g., A ∈ ZL×L). Let transmitter. The original integer-forcing paper [16] numeri- a[i] denote the ith coordinate of the vector a. We will use kak cally demonstrated the performance gains over conventional to represent ℓ2-norm of a, Tr(A) to represent the trace of A, linear receivers in terms of outage rates. It also established and eig(A) to denote the set of eigenvalues (i.e., spectrum) of that, if each antenna encodes an independent data stream, A. We will also use diag(a) to denote the diagonal matrix then integer-forcing attains the optimal diversity-multiplexing formed by using the placing the elements of a along the tradeoff [34]. Subsequently, it was shown that if the transmitter diagonal. All logarithms are taken to base 2 and we define mixes the data streams using a space-time code with a non- log+(x) , max(0, log x). We denote the identity matrix by I, vanishing determinant, then integer-forcing operates within a the all-ones column vector of length k by 1k and the all-zeros constant gap of the capacity [35]. Recent work has also studied column vector of length k by 0k. IEEE TRANS INFO THEORY, TO APPEAR 3

We will work with both the real field R and prime-sized which lets us concisely write the channel output as finite fields Z = {0, 1,...,p − 1} where p is prime.1 We p Y H X Z will denote addition and summation over the reals by + and u = u u + u . N×n , respectively. For a prime-sized finite field, we will use ⊕ This channel output is sent through a decoder Du : R → and to denote addition and summation, respectively. Define {1, 2,..., 2nR1 }×···×{1, 2,..., 2nRL } that produces esti- P [a] mod p to be the modulo-p reduction of a. For vectors and mates of the messages, (ˆw ,..., wˆ )= D (Y ). L u,1 u,L u u matrices, the modulo-p operation is taken elementwise and Overall, we say that the uplink rates Ru,1,...,Ru,L are denoted by [a] mod p and [A] mod p, respectively. Taking a achievable if, for any ǫ > 0 and n large enough, there exist linear combination over a prime-sized finite field can be linked P L encoders and decoder such that ( ℓ=1{wˆu,ℓ 6= wu,ℓ}) < ǫ. to taking a linear combination over the reals as follows, The uplink capacity region is the closure of the set of all achievable rates. S q1w1 ⊕ q2w2 = [q1w1 + q2w2] mod p . Downlink Channel. The downlink channel model mirrors the Note that, on the left-hand side, q1, q2, w1, w2 are elements uplink channel model. There is a single N-antenna transmitter of the finite field whereas, on the right-hand side, they are and L receivers. Let Mℓ represent the number of antennas elements of the integers under the natural mapping. Finally, th at the ℓ receiver and let M = ℓ Mℓ be the total num- subscripts “u” and “d” will be used to denote variables ber of receive antennas. The transmitter has L messages: th P associated with the uplink and downlink, respectively. the ℓ message wd,ℓ is drawn independently and uniformly from {1, 2,..., 2nRd,ℓ } and is intended for the ℓth receiver. ROBLEM TATEMENT nRd,1 II. P S The transmitter uses an encoder Ed : {1, 2,..., 2 } × We now give problem statements for the uplink and down- {1, 2,..., 2nRd,L } → RN×n to map these messages into a link. See Figure 1 for a block diagram. We focus on real-valued channel input Xd = Ed(wd,1,...,wd,L) where n represents channels, and note that our results are directly applicable to the blocklength. This channel input must satisfy a total power E T complex-valued channels by using a real-valued decomposi- constraint [ Tr(XdXd )] ≤ nPtotal. tion as in [16]. Throughout the paper, we assume that full CSI For m =1,...,L, the channel output observed by the mth is available at the transmitters and receivers.2 receiver is Uplink Channel. The uplink channel (i.e., MIMO MAC) Y = H X + Z consists of L transmitters and a single N-antenna receiver. The d,m d,m d d,m th Mm×N ℓ transmitter is equipped with Mℓ transmit antennas. It has a where Hd,m ∈ R is the channel matrix from the th Mm×n message wu,ℓ that is drawn independently and uniformly from transmitter to the m receiver and the noise Zd,m ∈ R nRu,ℓ nRu,ℓ {1, 2,..., 2 } and an encoder Eu,ℓ : {1, 2,..., 2 } → is elementwise i.i.d. Gaussian with mean zero and variance RMℓ×n that maps this message into a channel input Xu,ℓ = one. The receiver passes its channel output through a decoder Mm×n nRd,m Eu,ℓ(wu,ℓ) of blocklength n. It will often be convenient to Dd,m : R → {1, 2,..., 2 } in order to get an work with the concatenation of the channel inputs estimate wˆd,m = Dd,m(Yd,m) of its desired message. Overall, we say that the downlink rates R ,...,R are Xu,1 d,1 d,L achievable if, for any ǫ> 0 and n large enough, there exist an , . Xu  .  , P L encoder and decoders such that ( ℓ=1{wˆd,ℓ 6= wd,ℓ}) < ǫ. X  u,L The downlink capacity region is the closure of the set of all   achievable rates. S which is of dimension M × n where M = ℓ Mℓ denotes the total number of transmit antennas. The transmitters must Finally, it will often be useful to work with the following E T P concatenated matrices, satisfy a total power constraint [ Tr(XuXu )] ≤ nPtotal. The receiver observes a noisy linear combination of the Yd,1 Hd,1 Zd,1 emitted signals, , . , . , . Yd  .  Hd  .  Zd  .  , L Yd,L Hd,L Zd,L Yu = Hu,ℓXu,ℓ + Zu       which enable us to compactly write the downlink channel Xℓ=1 N×Mℓ th output as where Hu,ℓ ∈ R is the channel matrix from the ℓ transmitter to the receiver and the additive noise Z ∈ RN×n u Yd = HdXd + Zd . is elementwise i.i.d. Gaussian with mean zero and variance one. We denote the concatenated channel matrices by Remark 1: Conventional MAC models impose a power constraint on each user individually. However, it is well-known , Hu [Hu,1 ··· Hu,L] , that uplink-downlink duality can be established only if we are free to reallocate the power across transmitters [9]–[11]. Note 1We note that some mathematicians prefer to use the notation Z/pZ or Z/(p) to avoid confusion with the p-adic integers, which are often denoted also that we use an expected power constraint rather than a Z T by p. However, since we do not invoke the p-adic integers and will often hard power constraint of the form Tr(XuX ) ≤ nPtotal. In Z u use superscripts to denote vector spaces, we prefer to use the notation p for order to impose a hard power constraint, we would first need the finite field. 2For the case of CSI at the receivers only, our expressions can be suitably to show that the nested lattice ensemble from [30], is also good modified to obtain outage rate expressions, along the lines of [16]. for covering in the sense of [29]. This is currently an open IEEE TRANS INFO THEORY, TO APPEAR 4

Zd,1 Xu,1 Yd,1 Tx 1 Hu,1 Hd,1 Rx 1 Zu Zd,2 Xu,2 Yu Xd Yd,2 Tx 2 Hu,2 Rx Tx Hd,2 Rx 2 . . . . . Uplink . Downlink Zd,L Xu,L Channel Channel Yd,L Tx L Hu,L Hd,L Rx L

Fig. 1. Block diagram of the uplink and downlink channel models. We say that the channels are duals of each other if their channel matrices satisfy T Hd,ℓ = Hu,ℓ. question and beyond the scope of this paper. Alternatively, receivers, and time sharing. See [12] or [47, §9.6.4] for more we could keep only a constant fraction of each codebook, details. throwing out the codewords with the highest powers. This Uplink-Downlink Duality. It can be argued that the uplink would result in codebooks that meet hard power constraints and downlink capacity regions described above are equal to and achieve the same rates, at the cost of disrupting the one another, Cu = Cd. This was first shown for the sum- symmetry of the encoding scheme. capacity [9]–[11] and then for the full capacity region [12].

III. OVERVIEW OF MAIN RESULTS We now give a high-level overview of our main results. B. Conventional Linear Architectures To put our results in context, we begin by stating the ca- We begin with a summary of classical linear uplink and pacity regions for the uplink and downlink. We then give a downlink architectures and their duality relationships. quick summary of the rates achievable via conventional linear Uplink Channel. The ℓth transmitter has a codeword s ∈ architectures and their uplink-downlink duality relationships. u,ℓ Rn with expected power 1 Eks k2= P . It uses a beam- Finally, we overview our integer-forcing architectures for the n u,ℓ u,ℓ forming vector c ∈ RMℓ to generate its channel input uplink and downlink and state our uplink-downlink duality, ℓ capacity approximation, and algorithmic results. T Xu,ℓ = cu,ℓsu,ℓ .

A. Capacity Regions Collecting the beamforming vectors into the matrix Uplink Channel. The uplink (i.e., MIMO MAC) capacity cu,1 0M1 ... 0M1 region Cu is the set of rate tuples (Ru,1,...,Ru,L) satisfying 0M2 cu,2 ... 0M2 1 C ,   T u . . .. . Rℓ ≤ log det I + Hu,ℓKℓHu,ℓ (1) . . . . 2   ℓ∈S  ℓ∈S  0 0 ... c  X X  ML ML u,L for all subsets S ⊆ {1, 2,...,L} and for some positive   and the codewords into the matrix semi-definite matrices K1,..., KL satisfying the sum power L T constraint Tr(Kℓ) ≤ Ptotal. It can be attained with ℓ=1 su,1 i.i.d. Gaussian encoding and simultaneous joint typicality S , . decoding. Alternatively,P it can be attained with i.i.d. Gaussian u  .  , sT encoding, successive interference cancellation decoding, and  u,L time sharing [5],[6] or rate splitting [7]. See [47, §9.2.1] for   more details. we can write the beamforming operation as Downlink Channel. As shown by [12], the downlink (i.e., Xu = CuSu . MIMO BC) capacity region Cd is the convex hull of the set of rate tuples (Rd,1,...,Rd,L) satisfying To recover the mth codeword, the receiver uses an equaliza- T N tion vector bu,m ∈ R to obtain the effective channel output det I + Hd,θ(m)Kθ(m)Hd,θ(m) 1 m≥ℓ R ≤ log   X (2) ˜yT θ(ℓ) 2 T u,m det I + Hd,θ(m)Kθ(m)Hd,θ(m) T   = bu,mYu  m>ℓ     T T T T T X b H c s b H c s b Z for some permutation θ of {1, 2,...,K} and positive semi- = u,m u,m u,m u,m + u,m u,ℓ u,ℓ u,ℓ + u,m u , ℓ=6 m definite matrices K1,..., KL satisfying the sum power con- signal X noise L interference straint ℓ=1 Tr(Kℓ) ≤ Ptotal. It can be attained using dirty- paper coding at the transmitter, joint typicality decoding at the | {z } | {z } P | {z } IEEE TRANS INFO THEORY, TO APPEAR 5

T which is fed into a single-user decoder. By employing using equalization matrix Cd = Cu and precoding matrix T i.i.d. Gaussian codewords, the transmitters can achieve the Bd = Bu . The same relationship can be established starting following rates from an achievable rate tuple for the downlink and going to T 2 the uplink. 1 Pu,m|b Hu,mcu,m| R = log 1+ u,m (3) In other words, any rates that are achievable on an uplink u,m 2 T 2 ℓ=6 m Pu,ℓ|bu,mHu,ℓcu,ℓ| ! channel can be achieved on a downlink channel with a trans- posed channel matrix by exchanging the roles of the equal- for m =1,...,L. P ization and beamforming matrices (and transposing them) as Downlink Channel. The transmitter has a codeword sd,ℓ ∈ Rn intended for the ℓth receiver with expected power well as reallocating the powers. 1 2 Remark 2: In some cases, it may be desirable to employ Eksd,ℓk = Pd,ℓ. It collects these codewords into a matrix n rate splitting [7] at the transmitter(s). This can be viewed as T sd,1 creating virtual transmitters (in the uplink) or virtual receivers , . (in the downlink). In this setting, uplink-downlink duality Sd  .  sT continues to hold so long as the uplink and downlink users  d,L are split into virtual users in the same fashion.   N×L and applies a beamforming matrix Bd ∈ R to create its channel input C. Integer-Forcing Linear Architectures Xd = BdSd . The linear architectures discussed above fall short of achiev- th RMm The m receiver uses an equalization vector cd,m ∈ ing the MIMO MAC or BC capacity, due to noise amplification to form an effective channel output from the equalization step (which worsens as the condition T number of the channel matrix increases). Integer-forcing linear ˜yd,m T architectures can substantially reduce this rate loss by allowing = c Y d,m d the single-user decoders to target integer-linear combinations, T T T T T = cd,mHd,mbd,msd,m + cd,mHd,ℓbd,ℓsd,ℓ + cd,mZd,m . rather than individual codewords. These linear combinations ℓ=6 m can then be solved for the desired codewords. By carefully signal X noise interference selecting the integer coefficients to match the interference pre- | {z } | {z } sented by the channel, we can reduce the noise amplification | {z } caused by the equalization step. Using i.i.d. Gaussian codewords, we can achieve the following rates for m =1,...,L: To streamline the overview, we will focus on the effective noise variances attained by the integer-forcing architecture T 2 1 Pd,m|c Hd,mbd,m| as well as the resulting achievable rates. Due to the fact R = log 1+ d,m . (4) d,m T 2 that the effective noise for a linear combination impacts all 2 Pd,ℓ|c Hd,ℓbd,ℓ| ! ℓ=6 m d,m participating codewords, care is needed to ensure that this Uplink-Downlink Duality.P We can now state the uplink- effective noise variance is only associated with the achievable downlink duality relationship for conventional linear architec- rate of a single user. The technical details will be given in tures. Define the uplink and downlink equalization matrices Sections V and VI for the uplink and downlink, respectively. T T T T c 0 ... 0 Uplink Channel. The operations at the transmitters mimic b d,1 M2 ML u,1 0T cT ... 0T those of a conventional linear architecture, except that we . M1 d,2 ML B , . C ,   , use a nested lattice codebook to ensure that integer-linear u  .  d . . .. . T . . . . combinations of codewords are themselves codewords. The b  T T T   u,L 0 0 ... c     M1 M2 d,L  goal is to recover L integer-linear combinations of the form   T T T respectively. Also, define the uplink and downlink power au,1Su,..., au,LSu where the au,m are the rows of a full-rank ZL×L matrices, integer matrix Au ∈ , i.e., T Pu = diag(Pu,1,...,Pu,L) Pd = diag(Pd,1,...,Pd,L) , au,1 . Au =  .  . respectively. The following theorem encapsulates the uplink- aT  u,L downlink result of [10] in our notation.   Theorem 1 ( [10]): H th T For a given uplink channel matrix u To recover the m linear combination au,mSu, the receiver Mm and (diagonal) power matrix Pu that meets the total power applies an equalization vector bu,m ∈ R to form the T constraint Tr(Cu CuPu) = Ptotal, let Ru,1,...,Ru,L be a effective channel output rate tuple that is achievable with equalization matrix Bu and T T precoding matrix Cu. Then, for the downlink channel matrix ˜yu,m = bu,mYu T T T Hd = Hu , there exists a unique (diagonal) power matrix T = au,mSu + zu,eff,m Pd with total power usage Tr(Bd BdPd) = Ptotal, such that T T T T z , b Zu + (b HuCu − a )Su . the rate tuple Rd,ℓ = Ru,ℓ for ℓ = 1,...,L is achievable u,eff,m u,m u,m u,m IEEE TRANS INFO THEORY, TO APPEAR 6

s SISO u,1 Xu,1 ˜yu,1 SISO ˆuu,1 wu,1 cu,1 Hu,1 bu,1 ˆwu,1 Encoder Zu Decoder s SISO u,2 Xu,2 Yu ˜yu,2 SISO ˆuu,2 wu,2 cu,2 H b −1 ˆw Encoder u,2 u,2 Decoder Qu u,2 ...... s SISO u,L Xu,L ˜yu,L SISO ˆuu,L wu,L cu,L H b ˆw Encoder u,L u,L Decoder u,L

Fig. 2. Block diagram of the integer-forcing uplink architecture. Each message vector wu,ℓ is encoded into a dithered lattice codeword su,ℓ and mapped T T to a channel input Xu,ℓ = cu,ℓsu,ℓ. For m = 1,...,L, the receiver uses an equalized channel output ˜yu,m = bu,mYu to make an estimate ˆuu,m of th the linear combination uu,m. At the m decoder, algebraic successive cancellation is used to (digitally) cancel out m − 1 codewords prior to applying a SISO decoder. These codewords are then restored to obtain an estimate of the mth linear combination. Finally, the linear combinations are inverted to recover estimates ˆwu,ℓ of the message vectors.

zu,eff,1 s ˜y ˆu SISO u,1 u,1 SISO u,1 wu,1 ˆw Encoder Decoder u,1 zu,eff,2 s ˜y ˆu SISO u,2 u,2 SISO u,2 wu,2 A −1 ˆw Encoder u Decoder Qu u,2 . . zu,eff,L s ˜y ˆu SISO u,L u,L SISO u,L wu,L ˆw Encoder Decoder u,L

Fig. 3. Block diagram of the effective channel induced by the integer-forcing uplink architecture. The mth decoder observes an integer-linear combination of the codewords plus effective noise, Pℓ au,m,ℓsu,ℓ + zu,eff,m from which it makes an estimate of the linear combination uu,ℓ with coefficients qu,m,ℓ = [au,m,ℓ] mod p. Finally, it applies the inverse of the matrix Qu = {qu,m,ℓ} over Zp to estimate the message.

We define the effective noise variance as employ a nested lattice codebook to ensure that the codebook 1 is closed under integer-linear combinations so that the users σ2 , Ekz k2 u,m n u,eff,m can decode linear combinations of the transmitted codewords. 2 Additionally, as first proposed by Hong and Caire [17],[18], 2 T T 1/2 = kbu,mk + (bu,mHuCu − au,m)Pu . (5) we apply a precoding step over the finite field in order to “pre-invert” the linear combinations before mapping the Assuming the receiver can successfully recover all L linear combinations, it can now apply the inverse of the integer messages to codewords. This step guarantees that the integer- matrix to obtain the transmitted messages. A block diagram il- linear combinations recovered by the users correspond to their lustrating these operations and the resulting effective channels desired messages. These operations and the resulting effective can be found in Figures 2 and 3, respectively. channel are illustrated in Figures 4 and 5, respectively. In Section V, we will provide a detailed description of The mth receiver attempts to recover the linear combination T T th the uplink achievability scheme proposed in [20]. Overall, it ad,mSd where ad,m is the m row of the full-rank, integer L×L establishes that the following rates are achievable matrix Ad ∈ Z , i.e.,

1 Pu,m + T Ru,m = log 2 m =1,...,L. 2 σ ad,1  u,π(m)  . Ad = . . for at least one permutation πu.  .  T Remark 3: Although it is not immediately obvious, any rate a  d,L tuple that is achievable via a conventional linear architecture   is also achievable via an integer-forcing linear architecture To do so, it uses an equalization vector c ∈ RMm to form by using the same beamforming matrix, setting the integer d,m the effective channel output matrix to be the identity matrix, and scaling the equalization vectors by the appropriate MMSE coefficient [16, Lemma 3]. T T While [16] only establishes this for the uplink, this follows ˜yd,m = cd,mYd,m naturally for the downlink as well. T T = a Sd + z (6) Downlink Channel. We use the same encoding operations d,m d,eff,m T , T T T at the transmitter as in a conventional linear architecture. We zd,eff,m (cd,mHd,mBd − ad,m)Sd + cd,mZd,m . (7) IEEE TRANS INFO THEORY, TO APPEAR 7

Zd,1 ¯wd,1 sd,1 Xd,1 Yd,1 ˜yd,1 wd,1 SISO b H cd,1 SISO ˆw Encoder d,1 d,1 Decoder d,1 Zd,2 ¯w s X Y ˜y −1 d,2 d,2 d,2 Xd d,2 d,2 wd,2 SISO b H cd,2 SISO ˆw Qd Encoder d,2 d,2 Decoder d,2 ...... Zd,L ¯wd,L sd,L Xd,L Yd,L ˜yd,L wd,L SISO b H cd,L SISO ˆw Encoder d,L d,L Decoder d,L

Fig. 4. Block diagram of the integer-forcing downlink architecture. The encoder applies the inverse of Qd =[Ad] mod p over Zp to the message vectors wd,1,..., wd,L and then maps the results to dithered lattice codewords sd,1,..., sd,L. (The arrows between the encoders indicate that, as each codeword is formed, an additional linear precoding step is needed to eliminate potential interference from codewords with smaller powers.) The channel input is formed T th T by beamforming these codewords, Xd = Pℓ bd,ℓsd,ℓ. The m decoder uses an equalized channel output ˜yd,m = ˜cd,mYd,m to make an estimate of an integer-linear combination of the lattice codewords, which, due to the inverse operation corresponds to an estimate of its desired message.

zd,eff,1 ¯w s ˜y d,1 SISO d,1 d,1 SISO wd,1 ˆw Encoder Decoder d,1 zd,eff,2 ¯wd,2 s ˜y −1 SISO d,2 d,2 SISO wd,2 A ˆw Qd Encoder d Decoder d,2 . . . . zd,eff,L ¯w s ˜y d,L SISO d,L d,L SISO wd,L ˆw Encoder Decoder d,L

Fig. 5. Block diagram of the effective channel induced by the integer-forcing downlink architecture. The mth decoder observes an integer-linear combination Z of the codewords plus effective noise, Pℓ ad,m,ℓsd,ℓ +zd,eff,m. Since the encoder applied the inverse of Qd =[Ad] mod p over p to the message vectors prior to mapping them to lattice codewords, then the mth integer-linear combination corresponds to the mth message.

We define the effective noise variance as always linked to the rates Ru,m and Rd,m. However, for uplink integer-forcing, the effective noise variance for the mth 2 , 1 E 2 σd,m kzd,eff,mk (8) linear combination may not correspond to Ru,m. Similarly, n th 2 for downlink integer-forcing, the m effective power may not c 2 cT H B aT P1/2 = k d,mk + ( d,m d,m d − d,m) d . correspond to Rd,m. While we can always find permutations that connect effective noise variances and power to rates, these Uplink-Downlink Duality. The following theorem establishes permutations may differ between the uplink and downlink, uplink-downlink duality for integer-forcing in terms of the sum which in turn limits the our approach in Section VII to sum- rate. rate duality. Theorem 2: For a given uplink channel matrix Hu, integer matrix Au, and power matrix Pu that meets the total power T constraint Tr(Cu CuPu)= Ptotal, let Ru,1,...,Ru,L be a rate Remark 5: As noted in Remark 2, uplink-downlink duality tuple that is achievable via integer-forcing with equalization continues to hold for conventional linear architectures under matrix Bu and precoding matrix Cu. Then, for the downlink rate splitting. The same is true for integer-forcing architectures, T T channel matrix Hd = Hu , integer matrix Ad = Au , there but in terms of the sum rate. exists a unique power matrix Pd with total power usage Tr(BTB P ) = P , such that the sum rate R ≥ d d d total ℓ d,ℓ Approximate Sum Capacity. For the uplink channel, it is ℓ Ru,ℓ is achievable via integer-forcing using equalization T T known that integer-forcing can operate with a constant gap of matrix C = C and precoding matrix B = B P. The same d u d u the sum capacity [19, Theorem 3], [20, Theorem 4]. Using relationshipP can be established starting from an achievable rate uplink-downlink duality, we can establish a matching result tuple for the downlink and going to the uplink. for the downlink channel. A full proof of this duality theorem will be given in Sec- tion VII. M×N Remark 4: Note that Theorem 2 only establishes duality for Theorem 3: For any channel matrix Hd ∈ R and the sum rate whereas, for conventional linear architectures, total power constraint Ptotal, there is a choice of the power Theorem 1 establishes duality for the individual rates. This allocation Pd, integer matrix Ad, beamforming matrix Bd, stems from the fact that, for a conventional linear architecture, and equalization vectors cd,m such that the integer-forcing the mth effective noise variance and mth effective power are beamforming architecture can operate within a constant gap IEEE TRANS INFO THEORY, TO APPEAR 8 of the downlink sum-capacity, B. Nested Lattice Codes and Properties L Our encoding strategies rely on the existence of good nested 1 T L Rd,m ≥ max log det I + Hd KHd − log L lattice codebooks. Below, we describe the nested lattice ensem- K0 2 2 m=1 K ble as well as properties that are central to our achievability X Tr( )≤Ptotal   proofs. Our notation closely follows that from [20, §IV], which contains a more detailed exposition. where K  0 means that the matrix K must be positive Recall that n denotes the blocklength of our coding scheme. semidefinite. Let p represent a prime number and Z the finite field of size The proof is deferred to Appendix A. p p. We will also need integer-valued parameters 0 ≤ k ≤ Iterative Optimization. In Section VIII, we present an ap- C,ℓ k , ℓ =1,...,L. Define k , min k , k , max k , plication of our uplink-downlink duality result for iteratively F,ℓ C ℓ C,ℓ F ℓ F,ℓ and k , k − k . optimizing the beamforming and equalization matrices used F C The construction begins with the generator matrix of a linear in either an uplink or downlink integer-forcing architecture. code G ∈ ZkF×n. For ℓ =1,...,L, define G and G to This problem is non-convex in general as is the corresponding p C,ℓ F,ℓ be the submatrices consisting of the first k and k rows problem for conventional linear architectures. Our algorithm C,ℓ F,ℓ of G, respectively. Let provably converges to a local optimum and performs well in simulations. T ZkC,ℓ CC,ℓ = GC,ℓw : w ∈ p

n T ZkF,ℓ o IV. NESTED LATTICE CODES CF,ℓ = GF,ℓw : w ∈ p n o Below, we review some basic lattice definitions as well as denote the resulting linear codebooks. For γ > 0 to be speci- nested lattice code constructions that we will need for our −1 fied later, define the mapping φ(w) , γp w from Zp to R. achievability results. See [31] for a thorough introduction to We also define the inverse mapping φ¯(κ) , [γ−1pκ] mod p, lattice codes. which is only defined on the domain γp−1Z. Both of these mappings are taken elementwise when applied to vectors and A. Lattice Definitions will be used to go back and forth between linear codebooks and lattices. A lattice Λ is a discrete additive subgroup of Rn such that, We now generate L coarse lattices and L fine lattices as if λ , λ ∈ Λ, then λ + λ ∈ Λ and −λ , −λ ∈ Λ. The 1 2 1 2 1 2 follows: nearest neighbor quantizer associated to Λ is defined as −1 n ΛC,ℓ = λ ∈ γp Z : φ¯(λ) ∈CC,ℓ QΛ(x) , arg minkx − λk λ∈Λ n −1 n o ΛF,ℓ = λ ∈ γp Z : φ¯(λ) ∈CF,ℓ . (with ties broken in a systematic fashion). The fundamental n n o Voronoi region V of Λ is the set of all points in R that By construction, these lattices are nested according to the quantize to 0. We define the second moment Λ as order for which the parameters kC,ℓ and kF,ℓ are increasing. 1 1 Define ΛC and ΛF to be the coarsest and finest lattices in the σ2(Λ) , kxk2 dx ensemble, respectively. Let VC,ℓ and VF,ℓ denote the Voronoi n V Vol(V) Z regions of ΛC,ℓ and ΛF,ℓ, respectively. Finally, we take the where denotes the volume of . Vol(V) V elements of the fine lattice ΛF,ℓ that fall in the Voronoi region modulo operation We also define the with respect to Λ as of the coarse lattice ΛC,ℓ to be the nested lattice codebook [x] mod Λ , x − Q (x) Λ Lℓ , ΛF,ℓ ∩VC,ℓ and note that it satisfies a distributive law, [a[x]modΛ + = [ΛF,ℓ] mod ΛC,ℓ y x y Z b[ ]modΛ]modΛ=[a + b ] mod Λ for all a,b ∈ and th x, y ∈ Rn. for the ℓ user. Lemma 1 (Crypto Lemma): Let x be a random vector over The theorem below summarizes results from [30] that Rn and d be an independent random vector drawn uniformly demonstrate that this nested lattice construction exhibits good over the Voronoi region V of the lattice Λ. The modulo sum shaping and noise tolerance properties. [x + d] mod Λ is independent of x and uniform over V. Theorem 4 ( [30, Theorem 2]): For ℓ = 1,...,L, select 2 See [31, Ch 4.1] for a full proof. parameters Pℓ > 0 and 0 < σeff,ℓ < Pℓ. Then, for any ǫ > 0 and n and p large enough, there are parameters γ, kC,ℓ, and The lattice ΛC is said to be nested in the lattice ΛF if ΛC ⊂ kF×n kF,ℓ and a generator matrix G ∈ Z such that, for ℓ = ΛF. In this case, ΛC is called the coarse lattice and ΛF the p 1,...,L fine lattice. A nested lattice codebook L =ΛF ∩VC consists of all fine lattice points that fall in the fundamental Voronoi (a) the submatrices GC,ℓ and GF,ℓ are full rank. region VC of the coarse lattice. Note that nested lattices satisfy (b) the coarse lattices ΛC,ℓ have second moments close to the following quantization property: their power constraints

2 [QΛF (x)] mod ΛC = [QΛF ([x] mod ΛC)] mod ΛC . (9) Pℓ − ǫ < σ (ΛC,ℓ) < Pℓ . IEEE TRANS INFO THEORY, TO APPEAR 9

(c) the lattices can tolerate the desired level of effective noise. Let z , z ,..., z be independent noise vectors ∗ Power 0 1 L Constraint where z0 ∼ N (0, I) and zℓ ∼ Unif(VC,ℓ). For any Linear ∗ R L 2 Labeling β0,β1,...,βL ∈ , let zeff = ℓ=0 zℓ. If β0 + L 2 2 ϕ Message ℓ=1 βℓ Pℓ ≤ σeff,m, any fine lattice point λ ∈ ΛF,m P Symbols can recover from zeff with high probability, P P λ z λ QΛF,m ( + eff) 6= < ǫ . Noise Tolerance 2 L 2  Nested Lattice Codebook Similarly, if β0 + ℓ=1 βℓ Pℓ ≤ Pm, any coarse lattice Lℓ =ΛF,ℓ ∩VC,ℓ point λ ∈ ΛC,m can recover from zeff with high proba- Vector Space over Finite Field P Zk bility, p P QΛC,m (λ + zeff) 6= λ < ǫ . th (d) the rates of the nested lattice codes satisfy Fig. 6. Illustration of the linear labeling of the ℓ nested lattice codebook. The first kC,ℓ − kC elements of the linear label are “don’t care” entries (denoted by the ∗ symbol) and correspond to the mod ΛC,ℓ operation. The 1 kF,ℓ − kC,ℓ 1 Pℓ next kF,ℓ − kC,ℓ elements are free to carry information symbols (denoted log|Lℓ|= log2 p> log 2 − ǫ . n n 2 σeff,ℓ by solid circles). The last kF − kF,ℓ elements are zero (denoted by dashed   lines). Finally, it can be argued that we can label lattice codewords so that integer-linear combinations of codewords correspond to linear combinations of the messages over Zp. We recall the the top kF − kF,ℓ elements of the vector. Therefore, the first definition of a linear labeling from [32]. kC,ℓ − kC elements are determined by the shaping operation Zk Definition 1: We say that a mapping ϕ :ΛF → p is a modLℓ and can be interpreted as enforcing the power con- linear labeling if straint Pℓ. The next kF,ℓ − kC,ℓ elements are occupied by

(a) λ ∈ ΛF,ℓ if and only if the last kF − kF,ℓ components of information symbols and the final kF − kF,ℓ elements are its label ϕ(λ) are zero. Similarly, λ ∈ ΛC,ℓ if and only if zero, which can be interpreted as enforcing the noise tolerance 2 the last kF − kC,ℓ components of its label ϕ(λ) are zero. threshold σeff,ℓ. (b) For all aℓ ∈ Z and λℓ ∈ ΛF, we have that Overall, we arrive at the following high-level intuitions: L L • Only the encoder with the largest power (i.e., the coarsest ϕ aℓλℓ = qℓϕ(λℓ) lattice) can control the very top signal levels. For an  ℓ=1  ℓ=1 encoder ℓ whose power is less than the maximum power, X M the top k − k signal levels are outside of its direct where q = [a ] mod p. C,ℓ C ℓ ℓ control. (In our strategy, these signal levels will be set Consider the mapping that sets ϕ(λ) to be the last k compo- ZkF ¯ T during the decoding process.) nents of the unique vector v ∈ p satisfying φ(λ)= G v. th • The lowest k − k are set to zero by the ℓ encoder From [20, Theorem 10], ϕ is a linear labeling. We also define F F,ℓ so that all of its information symbols lie above the noise the inverse map level (i.e, the corresponding fine lattice can tolerate the

T 0 desired effective noise variance). ϕ¯ , φ G kC , w   ! V. UPLINK INTEGER-FORCING ARCHITECTURE which satisfies ϕ(¯ϕ(w)) = w. Our uplink coding scheme is taken from [20, Section VI]. C. Intuition via Signal Levels Below, we summarize the encoding and decoding operations in order to highlight the similarities between the uplink and We now develop some intuition by describing the linear downlink integer-forcing schemes. labeling of our nested lattice construction in terms of “signal We begin by selecting a power allocation P = Zk u levels” over p. See Figure 6 for an illustration. Each code- diag(Pu,1,...,Pu,L) for the codewords and a beamforming word from the ℓth nested lattice codebook can be expressed as th matrix Cu. Note that, in order to meet the total power con- an element from the ℓ fine lattice, λF,ℓ ∈ ΛF,ℓ modulo the T th straint with equality, we require that Tr(Cu CuPu) = Ptotal. ℓ coarse lattice, [λF,ℓ] mod ΛC,ℓ ∈ Lℓ. We can thus write We also select a full-rank integer matrix A ∈ ZL×L and an the linear label of any codeword in Lℓ as T L×N equalization matrix Bu = [bu,1 ··· bu,L] ∈ R . These choices specify the effective noise variances σ2 from (5). ϕ ([λF,ℓ] mod ΛC,ℓ)= ϕ λF,ℓ − QΛC,ℓ (λF,ℓ) u,m The structure of the integer matrix A determines which = ϕ(λ ) ⊖ ϕ Q (λ ) u F,ℓ ΛC,ℓ F
,ℓ codewords can be cancelled out in each decoding step. In where ⊖ denotes the subtraction operation over Zp.
From order to keep our notation manageable, we assume that Au is th th Definition 1(a), we know that ϕ QΛC,ℓ (λF,ℓ) only occupies selected so that the m user can be associated with the m the top k − k elements of the vector corresponding to the effective noise variance. The following definition describes C,ℓ C
linear label. Similarly, we know that ϕ(λF,ℓ) only occupies when this is possible. IEEE TRANS INFO THEORY, TO APPEAR 10

Definition 2: We say that the identity permutation is admis- That is, the receiver attempts to recover L linear combinations sible for the uplink if of cosets of the messages. The top elements kC,ℓ −kC of ˜wu,ℓ (a) the effective noise variances are in increasing order, (denoted by e in (11)) can be thought of as “don’t care” entries. 2 2 σu,1 ≤···≤ σu,L and That is, they can take any values, but are not directly set by the th (b) the leading principal submatrices of Au are full rank, ℓ transmitter, since this would require it to exceed its power [1:m] rank(Au )= m for m =1,...,L. constraint. In our scheme, these entries are determined during These conditions can always be met by permuting the rows the decoding process and can be recovered by the receiver, and columns of the selected matrices. See the remark below although this is not required to recover the original messages for details. wu,ℓ. Remark 6: Assume that we have chosen uplink parameters We now state the encoding and decoding steps used in the Au, Bu, Cu, Hu, and Pu, and now we wish to satisfy uplink integer-forcing architecture. We select an ensemble of Definition 2 by permuting row and column indices. First, good nested lattices ΛC,1,..., ΛC,L, ΛF,1,..., ΛF,L with pa- 2 2 select a permutation π that places the effective noise variances rameters Pu,1,...,Pu,L and σu,1,...,σu,L using Theorem 4. 2 2 th in increasing order, σu,π(1) ≤ ··· ≤ σu,π(L). Next, let θ Encoding: The ℓ transmitter starts by taking the p-ary be a permutation such that, after row permutation of Au expansion of its message index wu,ℓ to obtain the message by π and column permutation by θ, we obtain a matrix vector w ∈ ZkF,ℓ−kC,ℓ . It then uses the inverse linear ˜ u,ℓ p A = {au,π(m),θ(ℓ)}m,ℓ whose leading principal submatrices labeling to map this to a lattice point [1:m] are full rank, rank(A˜ u ) = m for m = 1,...,L. Now, permute the rows of the equalization matrix Bu by π to obtain 0kC −kC ˜ ˜ ,ℓ Bu,m. Finally, reindex the users by θ by setting Cu to be the λu,ℓ = ϕ¯ wu,ℓ mod ΛC,ℓ ˜    beamforming matrix consisting of vectors ˜cu,m = cu,θ(ℓ), Hu 0kF−kF,ℓ ˜ to be the channel matrix consisting of submatrices Hu,ℓ =    ˜ Hu,θ(ℓ), and Pu to be the power matrix with diagonal entries and dithers it to produce the codeword ˜ Pu,ℓ = Pu,θ(ℓ). It can be verified that the effective noise 2 variances σ˜u,m,m =1,...,L that result from these permuted su,ℓ = [λu,ℓ + du,ℓ] mod ΛC,ℓ 2 2 parameters satisfies σ˜u,m = σu,π(m). Overall, we find that the 1 + 2 rates Ru,π(m) = 2 log (Pu,π(m)/σu,θ(m)), m =1,...,L are where the dither vector du,ℓ is drawn independently and achievable. uniformly over VC,ℓ. Thus, by the Crypto Lemma and Theo- For notational convenience, we assume going forward that rem 4(b), su,ℓ is independent of λu,ℓ and has expected power th the identity permutation is admissible. close to Pu,ℓ. Finally, the ℓ transmitter uses its beamforming In our decoding procedure, we will need to triangularize Au vector cu,ℓ to produce its channel input over Zp in the following sense. We need a lower unitriangular ¯ L×L ¯ ¯ T matrix L ∈ Z such that A = [LAu] mod p is upper p Xu,ℓ = cu,ℓsu,ℓ . triangular. First, note that the condition in Definition 2(b) is equivalent to the condition that there exists a lower unitriangu- Decoding: The receiver attempts to recover linear combina- lar matrix L ∈ RL×L such that LA is upper triangular. Given tions of the form (10) and then solve them to obtain estimates the existence of such an L, it follows from [19, Appendix A] of the message vectors. As an intermediate step, the receiver that, for p large enough, we can always find an appropriate L¯. will attempt to decode certain integer-linear combinations of It also follows that L¯ has a lower unitriangular inverse L¯(inv) the lattice codewords, i.e., over Zp. After our overview of the encoding and decoding steps, we provide an explicit example of such matrices in L Example 1. ˜ µu,m = au,m,ℓ λu,ℓ mod ΛC Using the linear labeling ϕ, we can show that each  ℓ=1  nested lattice codebook Lℓ is isomorphic to the vector space X kF,ℓ−kC,ℓ Zp . Each user will take the p-ary expansion of its mes- ˜ , where λu,ℓ λu,ℓ − QΛC,ℓ (λu,ℓ + du,ℓ). The linear labels ZkF,ℓ−kC,ℓ sage index wu,ℓ to obtain a message vector wu,ℓ ∈ p . of these integer-linear combinations correspond to the desired The intermediate goal of the receiver is to recover L linear linear combinations, ϕ(µu,m) = uu,m. (This decoding step combinations of the form sets the top kC,ℓ − kC “don’t care” entries of ˜wu,ℓ to the L values specified by the linear label ϕ( − QΛC,ℓ (λu,ℓ + du,ℓ)).) uu,m = qu,m,ℓ ˜wu,ℓ (10) The main obstacle is that, in order to decode the mth Mℓ=1 integer-linear combination, the receiver must first cancel out th where qu,m,ℓ = [au,m,ℓ] mod p, au,m,ℓ is the (m,ℓ) entry of the first m − 1 codewords using the prior m − 1 linear Au, and ˜wu,ℓ ∈ Jwu,ℓK with combinations. This is accomplished via the algebraic SIC e technique from [19], which takes advantage of the fact that , Zk ZkC,ℓ−kC Jwu,ℓK w ∈ p : w = wu,ℓ for some e ∈ . the nested lattice codebook is isomorphic to a vector space  0   over Z . Specifically, by adding an integer-linear combination  kF−kF,ℓ  p   (11) of µu,1,..., µu,m−1 it can “digitally” null out the lattice code-   IEEE TRANS INFO THEORY, TO APPEAR 11

˜ ˜ words λu,1,..., λu,m−1 without impacting effective noise. Theorem 5 ( [20, Lemma 13]): Choose a power allocation Define Pu = diag(Pu,1,...,Pu,L), beamforming matrix Cu ∈ M×L N×M m−1 R , channel matrix Hu ∈ R , full-rank integer matrix T ZL×L νu,m = µu,m + ¯lm,iµu,i mod ΛC (12) Au = [au,1 ··· au,L] ∈ , and equalization vectors N i=1 b ∈ R . Assume, without loss of generality, that the  X  u,m L identity permutation is admissible for the uplink according to ˜ = a¯m,ℓ λu,ℓ mod ΛC Definition 2. Then, the following rates are achievable  ℓ=1  X 1 Pu,m ¯ th ¯ th R = log+ , m =1,...,L, where lm,i is the (m,i) entry of L and a¯m,ℓ is the (m,ℓ) u,m 2 σ2 A¯  u,m  entry of the upper triangular matrix defined above. Note that 2 2 2 T T 1/2 νu,m ∈ ΛF,m and, given ν1,..., νm, we can recover µu,m: σu,m = kbu,mk + (bu,mHuCu − au,m)Pu . m (inv) For a full proof, see [20, §VI]. µu,m = l¯ νi mod ΛC m,i Example 1: To illustrate the algebraic SIC technique, con-  i=1  X sider the integer matrix ¯(inv) th ¯(inv) where lm,i is the (m,i) entry of L . th 2 1 For the m decoding step, we assume that the receiver A = 0 1 has already successfully recovered the previous m − 1   integer-linear combinations, i.e., µˆu,1 = µu,1,..., µˆu,m−1 = Z 2 and assume that the underlying finite field is 3 and σu,1 < µu,m−1. The receiver begins by equalizing its observation, 2 σu,2. Define the following lower unitriangular matrices T T ˜y = b Yu . u,m u,m ¯ 1 0 ¯ 1 0 L1 = L2 = . The receiver then removes the dithers3, nulls out the lattice 0 1 2 1     codewords corresponding to the first m−1 users, and quantizes Each matrix can be used to cancel out one of the codewords, th onto the m fine lattice, i.e., m−1 ¯ ¯ ¯ 2 1 νu,m = QΛF ˜yu,m + lm,iµˆu,i A = [L A]mod3= ,m 1 1 0 1 i=1   X   L 2 1 A¯ = [L¯ A]mod3= . − a d mod Λ 2 2 1 0 u,m,ℓ u,ℓ C   ℓ=1  ¯ ˜ L X Since A1 is upper triangular, the first lattice codeword λu,1 only needs to tolerate effective noise variance σ2 , and the = QΛF au,m,ℓ(su,ℓ − du,ℓ) u,1 ,m ˜   ℓ=1 second lattice codeword λu,2 must tolerate the larger noise X m−1 2 variance σu,2. In other words, the identity permutation is ¯ ¯ + lm,iµˆu,i + zu,eff,m mod ΛC admissible according to Definition 2. Alternatively, using L2, i=1 A¯ X  we obtain a matrix 2 that is upper triangular after permuting m−1 the two columns. In other words, the first lattice codeword ¯ = QΛF µu,m + lm,iµˆu,i + zu,eff,m mod ΛC 2 ,m must tolerate the larger noise variance σu,2 whereas the second   i=1  2 X lattice codeword only needs to tolerate the smaller one σu,1. In general, not all orderings are possible, as demonstrated by

= QΛF,m (νu,m + zu,eff,m) mod ΛC the following integer matrix, which admits only the identity h i permutation: where the last step follows from (12) and the distributive law. 1 0 A = . It then forms an estimate of its desired linear combination 1 1 m   µˆ = l¯(inv)νˆ mod Λ u,m m,i i C VI. DOWNLINK INTEGER-FORCING ARCHITECTURE  i=1  X The key idea underlying downlink integer-forcing is the ˆuu,m = ϕ(µˆm) . fact that the transmitter can pre-invert the linear combinations Finally, if all L linear combinations have been recovered prior to encoding. This technique, which was first proposed correctly, we can solve the linear combinations to recover by Hong and Caire [17],[18], allows each receiver to decode the original messages. This strategy leads to the following any integer-linear combination of the codewords in order to achievable rates. reduce the effective noise but still recover its desired message. 3For lattice coding proofs, it is usually assumed that the dithers are available These papers focused on the important special case where all at the transmitters and receivers. Note that it can be shown that fixed dither users employ the same fine and coarse lattices, and thus have vectors exist that attain the same performance [20, Appendix H]. In other equal powers and must tolerate the worst effective noise across words, the randomness dithers should not be viewed as common randomness, but rather as part of the usual probabilistic method used in random coding receivers. Specifically, as illustrated in Figure 4, in the equal −1 proofs. power case, the transmitter should apply the inverse Qd to IEEE TRANS INFO THEORY, TO APPEAR 12

˜ the p-ary expansions of the messages, prior to generating the get a permuted power matrix Pd. Finally, we reindex the users lattice codewords. As a result, each receiver’s integer-linear by θ, which in turns yields a permuted equalization matrix ˜ combination of codewords corresponds to its desired message. Cd consisting of equalization vectors ˜cd,m = ˜cd,θ(m) as well ˜ This basic strategy can be generalized to allow for unequal as permuted channel submatrices Hd,m = Hd,θ(m) to form ˜ powers and a unique effective noise variance associated to each overall channel matrix Hd. It can be verified that the effective receiver. However, if each lattice codeword is generated using 2 noise variances σ˜d,m,m = 1,...,L, that result from these a different coarse lattice, it does not suffice to apply the inverse permuted parameters satisfy σ˜2 = σ2 . Thus, the rates −1 d,m d,θ(m) Qd to the messages. As discussed in Section IV-C, the issue 1 + 2 Rd,θ(m) = 2 log (Pd,π(m)/σd,θ(m)) are achievable. is that the top kC,ℓ −kC elements (i.e., the “don’t care” entries) To keep our notation manageable, we assume below that th cannot be set directly by the ℓ encoder. Rather, in our coding the identity permutation is admissible. scheme, their values will be set as a function of the message We now describe the encoding and decoding steps and dither vectors, and will act as interference for encoders used in the integer-forcing beamforming architecture. whose higher power levels allows them access to these entries. Using the parameters Pd,1 ≥ ··· ≥ Pd,L and Thus, we will encode the messages in stages, starting with the σ2 ,...,σ2 , we pick a good ensemble of nested lattices signal levels accessible to all encoders and applying the inverse d,1 d,L ΛC,1,..., ΛC,L, ΛF,1,..., ΛF,L via Theorem 4. We will Q−1 d . For the next set of signal levels, the encoder with the assume that the prime p used in the lattice construction is lower power will not participate, other than to add interference [1:m] [1:m] large enough so that Q = [A ] mod p is full rank via its “don’t care” entries. The remaining encoder will pre- d d over Zp for m = 1,...,L. It is always possible to choose cancel this interference as well as apply the inverse of the such a prime, as argued in [20, Lemmas 3, 4]. submatrix of Qd with row and column indices corresponding Encoding: Take the p-ary expansion of each message wd,ℓ to to the active encoders. This process will continue, removing kF −kC obtain the message vector w ∈ Z ,ℓ ,ℓ for ℓ = 1,...,L. an additional encoder at each stage, until all signal levels have ℓ p These vectors are then zero-padded to obtain been filled.

0kC,ℓ−kC A. Integer-Forcing Beamforming ¯wd,ℓ = wd,ℓ . (13) 0  We begin by choosing a power allocation P = kF−kF,ℓ d   diag(Pd,1,...,Pd,L) for the codewords and a full-rank integer Recall from Section IV-C that the parameter kC,ℓ −kC denotes L×L matrix Ad ∈ Z . We also select a beamforming matrix how many of the top signal levels cannot be directly set by N×L Mm th Bd ∈ R and equalization vectors cd,m ∈ R , m = the ℓ encoder, due to its power constraint. Since the powers 1,...,L. To meet the total power constraint with equality, we are assumed to be in decreasing order, it follows that these T need that Tr(Bd BdPd)= Ptotal. Taken together, these choices parameters are in increasing order, kC,1 − kC ≤···≤ kC,L − 2 specify the effective noise variances σd,m from (8). kC. As in the uplink case, the structure of the integer matrix Ad We now proceed to pre-invert the linear combinations in L will determine the order in which interference cancellation stages. Recall that the notation w[i] refers to the ith entry of is possible via a digital variation on dirty-paper precoding the vector w. that occurs at the message level. To simplify our notation, Initialization Step, kC,L − kC +1 ≤ i ≤ k: These signal lev- th we will assume that Ad is selected so that the m user can els are accessible by every encoder, meaning that we can th be associated with the m power Pd,m. We specify when this simply apply the inverse, is possible below. v [i] ¯w [i] Definition 3: We say that the identity permutation is admis- d,1 d,1 . Q−1 . sible for the downlink if  .  = d  .  . v [i] ¯w [i] (a) the powers are in decreasing order, Pd,1 ≥ ··· ≥ Pd,L  d,L   d,L  and Note that this fully specifies all of the signal levels controlled (b) the leading principal submatrices of Ad are full rank, by the Lth encoder. Therefore, we set the remaining entries A[1:m] for . rank( d )= m m =1,...,L to zeros, vd,L[1],..., vd,L[kC,L − kC]=0, apply the inverse These conditions can be satisfied via reindexing the rows linear labeling to obtain a fine lattice point and columns of the chosen matrices, as demonstrated in the following remark. λd,L =ϕ ¯(vd,L) , Remark 7: Assume that we have selected downlink param- and then generate our dithered codeword eters Ad, Bd, Cd, Hd, and Pd, and now wish to satisfy Definition 3 by permuting row and column indices. We first sd,L = [λd,L + dd,L] mod ΛC,L choose a permutation π that puts the powers in decreasing where the dither vector dd,L is drawn independently and order Pd,π(1) ≥···≥ Pd,π(L). Next, we take a permutation uniformly over V . This process fixes the “don’t care” ˜ C,L θ such that, for Ad = {a }m,ℓ, the leading principal th d,θ(m),π(ℓ) entries of the L encoder (i.e., the top kC,L −kC signal levels) ˜ [1:m] submatrices are all full rank, rank(Ad ) = m for m = to 1,...,L. Then, we permute the columns of the beamforming ˜ ˜ ed,L = ϕ(QΛC,L (λd,L + dd,L)) , matrix Bd to get Bd as well as the powers Pd,ℓ = Pd,π(ℓ) to IEEE TRANS INFO THEORY, TO APPEAR 13 which will act as interference towards the remaining L − 1 removing the dither vectors, quantizing onto the mth fine encoders that send information in these levels. lattice, and taking the modulus with respect to the coarsest For the rest of the signal levels, we proceed by induction lattice, for m =1,...,L − 1, assuming that vd,ℓ, λd,ℓ, sd,ℓ, ed,ℓ have been set for ℓ = m +1,...,L. L

Induction Step, kC,m − kC +1 ≤ i ≤ kC,m+1: The first m µˆd,m = QΛF,m ˜yd,m − ad,m,ℓ dd,ℓ mod ΛC . encoders can directly set these signal levels and the remaining   Xℓ=1  L − m encoders contribute interference via their “don’t care entries.” Thus, the encoding task is to first cancel out the The linear label of this estimate can be viewed as an estimate interference from these “don’t care” entries and then apply of the desired message along with zero-padding, the the inverse of the mth leading principal submatrix,

L ˜ed,m vd,1[i] ¯wd,1[i] ⊕ ℓ=m+1 qd,1,ℓ ed,ℓ[i] −1 ϕ(µˆd,m)= ˆwd,m . . [1:m] . . = Q . . 0   .  d  L .  kF−kF,m L vd,m[i]   ¯w [i] ⊕ q e [i]      d,m ℓ=m+1 d,m,ℓ d,ℓ     (14) for some ˜e ∈ ZkC,m−kC . As we will argue below, if L d,m p µˆ = µ , then ˆw = w . Note that this fully specifies all of the signal lev- d,m d,m d,m d,m els controlled by the mth encoder. Therefore, we set Theorem 6: Choose a power allocation Pd = N×L diag(Pd,1,...,Pd,L), beamforming matrix Bd ∈ R , vd,m[1],..., vd,m[kC,m − kC]=0, apply the inverse linear RMm×N labeling to obtain a fine lattice point channel matrices Hd,m ∈ , full-rank integer matrix L×L Mm Ad ∈ Z , and equalization vectors cd,m ∈ R . Assume, λd,m =ϕ ¯(vd,m) , without loss of generality, that the identity permutation is admissible for the downlink according to Definition 3. Then, and then generate our dithered codeword the following rates are achievable

s = [λ + d ] mod Λ . d,m d,m d,m C,m 1 P R = log+ d,m , m =1,...,L, d,m 2 σ2 where the dither vector dd,m is drawn independently and  d,m  2 uniformly over VC,m. This sets the “don’t care” entries of 2 2 T T 1/2 th σd,m = kcd,mk + (cd,mHd,mBd − ad,m)Pd . the m encoder (i.e., the top kC,m − kC signal levels) to

e = ϕ(Q (λ + d )) , d,m ΛC,m d,m d,m Proof: By the Crypto Lemma, each dithered codeword s is uniformly distributed over V and independent of which will act as interference towards the remaining m − 1 d,ℓ C,ℓ the other dithered codewords. Thus, by Theorem 4(b), we encoders that send information in these levels. have that 1 Eks k2≤ P , which guarantees that the power After all signal levels have been set, we stack the dithered n d,ℓ d,ℓ constraint is met codewords T s 1 T 1 T T d,1 E[ Tr(X X )] = E[ Tr(S B B S )] . d d d d d d S = . n n d  .  1 T E T T s = [ Tr(Bd BdSd Sd)]  d,L n   1 T T and apply the beamforming matrix to create the channel input = Tr (B B E[S S ]) n d d d d 1 T Xd = BdSd . ≤ Tr (B B P )= P . n d d d total Decoding: The goal of each receiver is to decode its message vector wd,ℓ. As a first step, it will make an estimate of the At the receiver side, we need to argue that µˆd,m = µd,m following integer-linear combination of the lattice codewords, with high probability and, if so, ˆwd,m = wd,m. We begin by examining the linear label of µd,m, L ˜ µd,m = ad,m,ℓ λd,ℓ mod ΛC  ℓ=1  ud,m = ϕ(µd,m) X L where a is the (m,ℓ)th entry of A and λ˜ = λ − d,m,ℓ d d,ℓ d,ℓ = qd,m,ℓ ϕ(λd,ℓ) ⊖ ϕ(QΛC,ℓ (λd,ℓ + dd,ℓ)) QΛC (λd,ℓ + dd,ℓ). It forms its estimate by equalizing its ,ℓ Mℓ=1   observation L

T T = qd,m,ℓ ϕ(λd,ℓ) ⊖ ed,ℓ . ˜yd,m = cd,mYd,m , Mℓ=1   IEEE TRANS INFO THEORY, TO APPEAR 14

th [1:1] [1:1 Now, we examine the i symbol of this linear label for kC,m − able to invert Q1 = [A1 ] mod p, and this permutation is kC +1 ≤ i ≤ k, inadmissible. Finally, note that for A3, both permutations will be admissible. ud,m[i] L = qd,m,ℓ (vd,ℓ[i] ⊖ ed,ℓ[i]) ℓ=1 VII. UPLINK-DOWNLINK DUALITY Mm L (a) = q v [i] ⊖ q e [i] d,m,ℓ d,ℓ d,m,ℓ d,ℓ As discussed in Section III-A, the uplink and downlink ca- Mℓ=1 ℓ=Mm+1 pacity regions are duals of one another [9]–[12]. Furthermore, L L (b) for conventional linear architectures, we can achieve the same = ¯w [i] ⊕ q e [i] ⊖ q e [i] d,m d,m,ℓ d,ℓ d,m,ℓ d,ℓ rate tuple on dual uplink and downlink channels by exchanging ℓ=m+1 ℓ=m+1 M M the roles of the beamforming and equalization matrices (and = ¯w [i] d,m transposing them) [10]. (See Theorem 1 for a precise statement where (a) uses the fact that vd,ℓ[i]=0 for ℓ = m +1,...,L in our notation.) For integer-forcing architectures, we can by construction and ed,ℓ[i] = 0 for ℓ = 1,...,m via establish a similar form of uplink-downlink duality, but only Definition 1(a) since ed,ℓ is the linear label of a lattice point for the sum rate. from ΛC,ℓ and (b) follows from plugging in (14). From (13) it Let follows that, if µˆd,m = µd,m, then ˆwd,m = wd,m. Note that, since the last k − k entries of ¯w are zero, we know P F F,m d,m β , u,ℓ from Definition 1(a) that µ ∈ Λ . u,ℓ 2 d,m F,m σu,ℓ We need to argue that µˆd,m = µd,m with probability at least 1 − ǫ. Recall from (6) and (7) that ˜yT = aT S + zT . d,m d,m d d,eff,m denote the th effective SINR for the uplink and let Thus, ℓ L , Pd,ℓ ˜yd,m = ad,m,ℓ(λd + dd,m − QΛC,ℓ (λd,ℓ + dd,ℓ)) + zd,eff,m βd,ℓ 2 σd,ℓ Xℓ=1 L denote the ℓth effective SINR for the downlink. Our uplink- ˜ = ad,m,ℓ(λd,ℓ + dd,ℓ)+ zd,eff,m , downlink duality results stem from showing that if the effective Xℓ=1 SINRs βu,ℓ can be established on the uplink, then the effective and, using (9), SINRs βd,ℓ = βu,ℓ can be established on the downlink, and vice versa. Unfortunately, this does not immediately µˆ µ z d,m = [QΛF,m ( d,m + d,eff,m)] mod ΛC . translate to duality of the achievable rate tuples, since the rates R = 1 log+(β ) and R = 1 log+(β ) are only From Theorem 4(c), we know that, since µd,m ∈ ΛF,m, u,ℓ 2 u,ℓ d,ℓ 2 d,ℓ the quantization step can tolerate noise with effective vari- achievable within our integer-forcing framework if the identity 2 P permutation is admissible for both the uplink and downlink. ance σd,m, which implies that (µˆm 6= µm) < ǫ. From Theorem 4(d), we know that the rate satisfies Rd,m > In general, as discussed in Remark 4, the identity per- 1 + 2 mutation may not be admissible on both the uplink and 2 log (Pd,m/σd,m) − ǫ. Finally, following the steps in [20, Appendix H], we can show that good fixed dither vectors exist. downlink, even with the freedom to reindex the transmitters and receivers. However, we can always find permutations πu + 2 Example 2: The choice of the integer matrix places con- and πd such that the rates Ru,ℓ =1/2 log (Pu,ℓ/σu,πu(ℓ)) and + 2 straints on the power levels that can be associated with each Rd,ℓ = 1/2 log (Pd,πd(ℓ)/σd,ℓ) are achievable via integer- user. Consider the following integer matrices: forcing. Therefore, the duality of the effective SINRs allows us to establish sum-rate duality, as shown below. 2 1 0 1 2 1 A = A = A = 1 0 1 2 2 1 3 1 1 Lemma 2: Assume that the rates Ru,ℓ = 1 + 2       log (Pu,ℓ/σ ) > 0, ℓ = 1,...,L, are achievable on [1:m] 2 u,πu(ℓ) and assume that P1 > P2. Note that rank(A1 ) = m for the uplink for some permutation πu. Also, assume that the + 2 m = 1, 2, and thus the identity permutation is admissible, rates Rd,ℓ = 1/2 log (Pd,πd(ℓ)/σd,ℓ)), ℓ = 1,...,L, are meaning that power P1 can be associated with user 1 and achievable on the uplink for some permutation πd and that power P2 with user 2. Within our coding scheme, the key the effective SINRs are equal, βu,ℓ = βd,ℓ, ℓ = 1,...,L. [1:1] [1:1 Z step is that we can invert Q1 = [A1 ] mod p over p Then, the downlink sum rate is at least as large as the uplink since it is non-zero. In contrast, note that A2 corresponds sum rate, to exchanging the rows of A1. If we also exchange the beamforming vectors, then, according to Remark 7, this should L L correspond to associating power P1 with user 2 and power Rd,ℓ ≥ Ru,ℓ . [1:1] with user . However, since A , we will not be ℓ=1 ℓ=1 P2 1 2 = 0 X X IEEE TRANS INFO THEORY, TO APPEAR 15

Proof: We have that Proof: Our proof is inspired by the approach of [10]. We

L L begin by defining vector notation for the powers and effective 1 + Pu,ℓ SINRs, Ru,ℓ = log 2 2 σu,πu(ℓ) Xℓ=1 Xℓ=1   Pu,1 Pd,1 βu,1 βd,1 L . . . . (a) 1 Pu,ℓ ρ , . ρ , . β , . β , . . = log u  .  d  .  u  .  d  .  2 σ2 u,πu(ℓ) Pu,L Pd,L βu,L βd,L Xℓ=1           L         1 P = log u,ℓ 2 Let ¯c denote the ℓth column of C (with zero-padding 2 σu,πu(ℓ) u,ℓ u  ℓ=1  th YL included) and au,m,ℓ denote the (m,ℓ) entry of Au. De- 1 P th = log u,ℓ fine the L × L non-negative matrix Mu whose (m,ℓ) 2 σ2 T 2 ℓ=1 u,ℓ entry is [Mu]m,ℓ = (bu,mHu¯cu,ℓ − au,m,ℓ) . Let Ju =   2 2 YL diag ([kbu,1k ··· kbu,Lk ]). It can be verified that the (b) 1 Pd,ℓ relations = log 2 2 σd,ℓ 2  ℓ=1  2 2 T T 1/2 YL σu,m = kbu,mk + (bu,mHuCu − au,m)Pu 1 P d = log d,π (ℓ) 2 σ2 for m =1,...,L can be equivalently expressed as  ℓ=1 d,ℓ  L Y (I − diag(βu)Mu)ρu = Juβu . (15) 1 P d = log d,π (ℓ) 2 σ2 We now repeat this process for the downlink. Let b¯ ℓ=1  d,ℓ  d,ℓ X denote the th column of B and denote the th (c) L L ℓ d ad,m,ℓ (m,ℓ) 1 + Pd,πd(ℓ) ≤ log = Rd,ℓ entry of Ad. Define the L×L non-negative matrix Md whose 2 T 2 σd,ℓ th ¯ 2 ℓ=1   ℓ=1 (m,ℓ) entry is [Md]m,ℓ = (cd,mHd,mbd,ℓ − ad,m,ℓ) . Let X X 2 2 where (a) follows from the fact that all uplink rates are as- Gd = diag ([kcd,1k ··· kcd,Lk ]). It can be verified that the relations sumed to be positive, (b) from the assumption that βu,ℓ = βd,ℓ, + 2 and (c) from the fact that . T T 1/2 log(x) ≤ log (x) σ2 = kc k2+ (c H B − a )P We now recall the following basic results for non-negative d,m d,m d,m d,m d d,m d matrices. A vector or a matrix is non-negative (i.e., F ≥ 0) if for m =1,...,L can be equivalently expressed as all its entries are non-negative. A vector or a matrix is positive (i.e., F > 0) if all its entries are positive. A square matrix F (I − diag(βd)Md)ρd = Gdβd . is a Z-matrix if all its off-diagonal elements are non-positive. T T By assumption, we have that Ad = Au , Bd = Bu , An M-matrix is a Z-matrix with eigenvalues whose real parts T T T Cd = C , and Hd = H . It follows that M = Mu as well. are positive. The following lemma is a special case of [48, u u d Furthermore, since Mu and Md are non-negative, we have Theorem 1]. that (I−diag(βu)Mu) and (I−diag(βd)Md) are Z-matrices. Lemma 3 ( [48, Theorem 1]): Let F be a square Z-matrix. Since, by assumption, βu > 0, we have that Juβu > 0 and The following statements are equivalent: thus (I − diag(βu)Mu) satisfies condition (c) of Lemma 3. (a) F is a non-singular M-matrix. This implies that every real eigenvalue of (I − diag(βu)Mu) −1 (b) F has a non-negative inverse. That is, F exists and is positive. Setting βd = βu, we have that −1 F ≥ 0. T (c) There exists x ≥ 0 satisfying Fx > 0. eig(diag(βu)Mu)= eig(diag(βu)Md ) (d) Every real eigenvalue of F is positive. = eig(Md diag(βu))

The lemma below establishes uplink-downlink duality for = eig(Md diag(βd)) the effective SINRs. = eig(diag(βd)Md) , Lemma 4: Select a power matrix Pu and beamforming T which implies that all real eigenvalues of (I − diag(β )M ) matrix Cu, and let Ptotal = Tr(Cu CuPu) denote the total d d power consumption. Furthermore, select a channel matrix are also positive, satisfying condition (d) of Lemma 3. This −1 Hu, full-rank integer matrix Au, and equalization matrix implies that the inverse (I − diag(βd)Md) exists and is G Bu. Let βu,ℓ, ℓ = 1,...,L denote the effective uplink non-negative. Combining this with the fact that dβd ≥ 0, SINRs and assume that βu,ℓ > 0, ℓ = 1,...,L. Then, for we know that there exists a non-negative power vector T T beamforming matrix Bd = Bu , channel matrix Hd = Hu , −1 T T ρd = (I − diag(βd)Md) Gdβd (16) integer matrix Ad = Au , and equalization matrix Cd = Cu , there exists a unique power matrix Pd with total power usage that attains the desired effective downlink SINRs. T Tr(Bd BdPd) = Ptotal, that yields effective downlink SINRs It remains to show that the total downlink power consump- βd,ℓ = βu,ℓ, ℓ = 1,...,L. The same relationship can be tion is equal to the total uplink power consumption. Define 2 2 established starting from effective SINRs for the downlink and Gu = diag ([kcd,1k ··· kcd,Lk ]) and Jd as the L × L th 2 going to the uplink. matrix with (m,ℓ) entry [Jd]m,ℓ = bd,m,ℓ where bd,m,ℓ is IEEE TRANS INFO THEORY, TO APPEAR 16

th the (m,ℓ) entry of Bd. The total uplink power consumption Note that, even for a fixed integer matrix, simultaneously can be written as optimizing the remaining parameters is a non-convex problem. We now develop a suboptimal iterative algorithm based on CTC P Ptotal = Tr( u u u) uplink-downlink duality. Assume, for now, that the integer T T −1 = 1 Guρu = 1 Gu(I − diag(βu)Mu) Juβu matrix Au, beamforming matrix Cu, and power allocation P The total downlink power consumption can be written as matrix u are fixed. Consider the Cholesky decomposition T −1 T T −1 T T T −1 FF = (Pu + Cu Hu HuCu) (17) Tr(Bd BdPd)= 1 Jdρd = 1 Jd(I − diag(βd)Md) Gdβd . where F is a lower triangular matrix with strictly positive diagonal entries. It follows from [20, Lemma 2] that the We now demonstrate these quantities are equal: optimal equalization matrix is T −1 1 Jd(IL − diag(βd)Md) Gdβd T T T T −1 Bu = AuPuC H (I + HuCuPuC H ) , (18) −1 u u u u T −1 −1 −1 =1 Jd Gd diag(βd) − Gd Md 1 and, for this choice, the effective noises can be written as T T  −1 σ2 = ka Fk2 where a is the mth row of A . 1T β −1G−1 MTG−1 JT1 u,m u,m m u = diag( d) d − d d d We are now prepared to create the dual downlink channel. −1 T T T T T −1 −1 T −1 Set Ad = Au , Bd = Bu , Cd = Cu , and Hd = Hu . Use (16) =1 diag(βd) Gd − Md Gd Ju1 to solve for the downlink power vector ρd and set Pd = T  −1 diag(ρ ). By Theorem 2, the downlink sum rate is at least as =1 J−1 diag(β )−1G−1 − J−1M G−1 1 d u u d u u d large as the uplink sum rate. −1 th (a) T −1 −1 −1 −1 −1 The m downlink effective noise variance is = 1 J diag(βu) G − J MuG 1 u u u u 2 T T 1/2 T 2 2  −1  σd,m = kcd,mk + (cd,mHd,mBd − ad,m)P , =1 Gu(IL − diag(βu)Mu) Juβu = Ptotal . d where uses the fact that G G since CT C . which corresponds to a quadratic optimization problem in cd,m (a) d = u d = u Proof of Theorem 2: Without loss of generality, we assume (holding all other parameters fixed). The optimal equalization that the identity permutation is admissible for the uplink and vector is that all achievable rates are positive 1 + , T T T T T T T −1 Ru,ℓ = 2 log (βu,ℓ) > 0 cd,m = ad,mPdBd Hd,m(I + Hd,mBdPdBd Hd,m) (19). ℓ =1,...,L. From Lemma 4, we know that we can establish If we use these equalization vectors in place of the original effective downlink SINRs βd,ℓ = βu,ℓ with the same total power consumption. Finally, it follows from Lemma 2 that ones, we can only improve the effective noise variances, and hence the effective SINRs and sum rate. a sum rate satisfying ℓ Rd,ℓ ≥ ℓ Ru,ℓ is achievable on the downlink. The proof for starting from the downlink is Finally, we are ready to return to the uplink channel. We update the beamforming matrix C CT using the optimal identical. P P  u = d equalization vectors found above and use (15) to solve for the uplink power vector ρu and set Pu = diag(ρu). We begin a VIII.ITERATIVE OPTIMIZATION VIA DUALITY new iteration on the uplink and continue until we converge In this section, we present an iterative optimization algo- to a local optimum. (Note that, since each step has a unique rithm for the non-convex problem of optimizing the beam- minimizer, this two-block coordinate descent algorithm will forming and equalization matrices in order to maximize the always converge. See, for instance, [49, Corollary 2] for a sum rate. Our algorithm exploits uplink-downlink duality to convergence proof.) converge to a local optimum. We also explore algorithms for The overall process is succinctly summarized in Algo- optimizing the integer matrix. We first present our algorithm rithm 1. for the uplink channel, and afterwards state the modifications The choice of a good integer matrix Au is critical for the needed to use it on a downlink channel. performance of integer-forcing. Consider the lattice GTZL. Assuming Bu is chosen according to (18), respectively, the A. Uplink Optimization optimal integer matrix corresponds to finding the shortest set of L linearly independent basis vectors (i.e., the successive For a given uplink channel matrix Hu and total power minima) for the lattice GTZL. This problem, which is known constraint Ptotal, our task is to maximize the sum rate by as the Shortest Independent Vector Problem in the theoretical selecting a full-rank integer matrix Au, equalization matrix computer science literature, is conjectured to be NP-hard [50]. Bu, beamforming matrix Cu, and power allocation matrix However, it is possible to find approximately optimal solutions Pu. Assuming, without loss of generality, that the identity permutation is admissible, this corresponds to the following in polynomial time via the LLL algorithm [51]. See [52],[53] optimization problem: for more details. Remark 8: While it is possible to iteratively update the L 1 + Pu,ℓ integer matrix Au as part of the algorithm, we have observed max log 2 Au,Bu,Cu,Pu 2 σ (numerically) that good performance is available by choosing ℓ=1 u,ℓ ! X T a good basis at the beginning, and then refining the remaining subject to Tr(Cu CuPu) ≤ Ptotal. matrices. IEEE TRANS INFO THEORY, TO APPEAR 17

Algorithm 1 Iterative Uplink Optimization via Duality forcing as well as capacity bounds.4 Owing to uplink-downlink Given Hu and Ptotal. duality, we can simultaneously plot the sum rate for the Set initial parameters Au, Bu, Cu, and Pu. uplink and downlink channel. For simplicity, we will state our Calculate initial uplink SINRs βu. notation in terms of the uplink channel. We draw the channel while βu not converged do matrix Hd elementwise i.i.d. N (0, 1). Set Bu using (18). Create virtual dual downlink channel with A = AT, d u 7 B = BT, and C = CT. d u d u Capacity Solve for ρ using (16) and set P = diag(ρ ). d d d 6 Integer-Forcing Optimize Cd using (19). T Zero-Forcing Update Cu = Cd . 5 Solve for ρu using (15) and set Pu = diag(ρu). Update βu. end while 4 Output Au, Bu, Cu, Pu, and βu. 3

B. Downlink Optimization Average Sum Rate 2

For a given downlink channel matrix Hd and total power constraint Ptotal, our task is to maximize the sum rate by 1 selecting the power allocation matrix Pd, beamforming matrix B , full-rank integer matrix A , and equalization matrix 0 d d 0 2 4 6 8 10 12 14 16 18 20 Cd. Assuming, without loss of generality, that the identity permutation is admissible, we have the following optimization Total Power (in dB) problem: Fig. 7. Average sum rate under i.i.d. Gaussian fading for integer-forcing L and zero-forcing architectures with L = 4 single-antennas users and N = 2 1 + Pd,ℓ basestation antennas. max log 2 Ad,Bd,Cd,Pd 2 σ ℓ=1 d,ℓ ! X T subject to Tr(Bd BdPd) ≤ Ptotal. As in the uplink case, this is a non-convex optimization Capacity 10 problem, even if Ad is fixed. We will use iterative uplink- Integer-Forcing downlink optimization to converge to a local optimum. (As in Zero-Forcing the uplink, convergence follows from [49, Corollary 2].) 8 The overall process is summarized in Algorithm 2.

Algorithm 2 Iterative Downlink Optimization via Duality 6 Given Hd and Ptotal. Set initial parameters Ad, Bd, Cd, and Pd. Calculate initial downlink SINRs βd. Average Sum Rate 4 while βd not converged do Optimize Cd using (19). T T Create virtual uplink channel with Au = Ad , Bu = Bd , T 2 and Cu = Cd . Solve for ρ using (15) and set P = diag(ρ ). u u u 0 2 4 6 8 10 12 14 16 18 20 Optimize Bu using (18). T Total Power (in dB) Update Bd = Bu . Solve for ρd using (16) and set Pd = diag(ρd). Fig. 8. Average sum rate under i.i.d. Gaussian fading for integer-forcing Update βd. and zero-forcing architectures with L = 4 single-antennas users and N = 4 end while basestation antennas. Output Ad, Bd, Cd, Pd, and βd. In our plots, the “Capacity” curves correspond to the MIMO MAC sum capacity (under a total power constraint) from (1) IX. NUMERICAL RESULTS or to the MIMO BC sum capacity (2). These expressions

We now provide simulation results for our integer-forcing 4MATLAB code to generate these figures is available on the second author’s architecture and compare its performance to that of zero- website. IEEE TRANS INFO THEORY, TO APPEAR 18

8 25 Capacity Capacity 7 Integer-Forcing Integer-Forcing Zero-Forcing 20 Zero-Forcing 6

15 5

4 10 Average Sum Rate 3 Average Sum Rate 5 2

1 0 0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 Total Power (in dB) Number of Basestation Antennas

Fig. 9. Average sum rate under i.i.d. Gaussian fading for integer-forcing Fig. 10. Average sum rate under i.i.d. Gaussian fading for integer-forcing and and zero-forcing architectures with L = 2 single-antennas users and N = 4 zero-forcing architectures with L = 6 single-antennas users and N basestation basestation antennas. antennas at Ptotal = 20dB.

X. CONCLUSION are evaluated following the dual decomposition approach In this paper, we established an uplink-downlink duality from [54]. relationship for integer-forcing. In the process, we extended The “Integer-Forcing” curves correspond to the sum rate for prior work on downlink integer-forcing to allow for unequal uplink integer-forcing from Theorem 5 or downlink integer- powers and unequal rates. Using the duality relationship, we developed an iterative algorithm for the non-convex problem forcing from Theorem 6. The integer matrix Au is chosen using the LLL algorithm to approximate the successive minima of optimizing the beamforming and equalization matrices. We of the lattice FTZL where F is defined in (17) with the initial also demonstrated that downlink integer-forcing can operate within a constant gap of the MIMO BC sum capacity. choice of Cu = I. Afterwards, we iteratively optimize Bu, An interesting direction for future work is utilizing uplink- Cu, and Pu using Algorithm 1. downlink duality to optimize integer-forcing architectures for The “Zero-Forcing” curves correspond to the sum rate of up- more complicated Gaussian networks. For instance, recent link zero-forcing from (3) or downlink zero-forcing from (4). work [22] has utilized uplink-downlink duality as a building The matrices Bu, Cu, and Pu are iteratively optimized using block for optimizing the beamforming and equalization matri- Algorithm 1 while holding Au = I. ces used in integer-forcing interference alignment [55]. In Figure 7, we have plotted the average sum rate with Another direction is to establish uplink-downlink duality respect to Ptotal for L = 4 single-antenna users and a between uplink integer-forcing enhanced by successive in- basestation with N =2 antennas. In this scenario, there are not terference cancelation and downlink integer-forcing enhanced enough basestation antennas to invert the channel matrix, and by dirty-paper coding. We investigated this relationship in thus the performance of zero-forcing saturates, whereas both an earlier conference paper [56]. Unfortunately, this result the sum capacity and integer-forcing sum rate scale with Ptotal. requires the identity permutation to be admissible on both In Figure 8, we increase the number of basestation antennas the uplink and downlink without reindexing, which cannot to N =4 while holding the number of users fixed at L =4. be assumed without loss of generality. The key technical The zero-forcing sum rate now scales with Ptotal, but there is issue is that the SIC and DPC matrices must be lower and still a significant gap to the sum capacity and integer-forcing upper triangular, respectively, and this is not maintained under performance. This gap can be nearly closed by reducing the reindexing. number of users to L = 2 and keeping N = 4 basestation antennas. APPENDIX A PROOF OF THEOREM 3 Overall, we observe that the integer-forcing sum nearly matches the sum capacity. In contrast, zero-forcing operates It is well-known [9]–[11] that the sum capacity of the near the sum capacity only when the number of basestation MIMO BC is equal to that of the dual MIMO MAC, antennas N is at least as large as the number of (single- 1 T max log det I + HuKHu antenna) users L. This is demonstrated in Figure 10 by varying K0 2 Tr(K)≤P N from 1 to 12 for L = 12 and Ptotal = 20dB. total   IEEE TRANS INFO THEORY, TO APPEAR 19

T where Hu = Hd . Select a covariance matrix Kopt that attains [15] A. Wiesel, Y. C. Eldar, and S. Shamai, “Zero-forcing precoding and the MIMO MAC sum capacity. Next, select a power allocation generalized inverses,” IEEE Transactions on Signal Processing, vol. 56, T pp. 4409–4418, September 2008. Pu and a beamforming matrix Cu satisfying CuPuCu = [16] J. Zhan, B. Nazer, U. Erez, and M. Gastpar, “Integer-forcing linear Kopt. From [20, Theorem 4], there exists an integer matrix receivers,” IEEE Transactions on Information Theory, vol. 60, pp. 7661– 7685, December 2014. Au such that integer-forcing via Theorem 5 attains the sum [17] S.-N. Hong and G. Caire, “Reverse compute and forward: A low- rate complexity architecture for downlink distributed antenna systems,” in L Proceedings of the IEEE International Symposium on Information 1 T T L Theory (ISIT 2012), (Cambridge, MA), July 2012. Ru,ℓ = log det I + HuCuPuC H − log L 2 u u 2 [18] S. N. Hong and G. Caire, “Compute-and-forward strategies for coop- Xℓ=1   erative distributed antenna systems,” IEEE Transactions on Information 1 T L Theory, vol. 59, pp. 5227–5243, September 2013. = log det I + H K H − log L. 2 u opt u 2 [19] O. Ordentlich, U. Erez, and B. Nazer, “The approximate sum capacity   of the symmetric K-user Gaussian interference channel,” IEEE Trans- using the optimal equalization matrix Bu from (18). From actions on Information Theory, vol. 60, pp. 3450–3482, June 2014. Theorem 2, we can attain the same sum rate on the downlink [20] V. Ntranos, V. Cadambe, B. Nazer, and G. Caire, “Expanding the by using A = AT, B = BT, and C = CT as well compute-and-forward framework: Unequal powers, signal levels, and d u d u d u multiple linear combinations,” IEEE Transactions on Information The- as solving for the downlink power vector ρd using (16) and ory, vol. 62, pp. 4879 – 4909, September 2016. setting Pd = diag(ρd). [21] K. Gomadam, V. Cadambe, and S. 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[39] L. Liu, P. Patil, and W. Yu, “An uplink-downlink duality for cloud radio He is currently an Assistant Professor in the Department of Electrical access network,” in Proceedings of the IEEE International Symposium and Computer Engineering at Boston University, Boston, MA. From 2009 on Information Theory (ISIT 2016), (Barcelona, Spain), July 2016. to 2010, he was a postdoctoral associate in the Department of Electrical [40] H. Yao and G. W. Wornell, “Lattice-reduction-aided detectors for and Computer Engineering at the University of Wisconsin, Madison, WI. MIMO communication systems,” in Proceedings of the IEEE Global His research interests include information theory, communications, signal Communications Conference (GLOBECOM 2002), (Taipei, Taiwan), processing, and neuroscience. November 2002. Dr. Nazer received the Eli Jury Award from the EECS Department at UC [41] C. Windpassinger, R. F. Fischer, and J. B. 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Sciandrone, “On the convergence of the block nonlinear Gauss–Seidel method under convex constraints,” Operations Shlomo Shamai (Shitz) received the B.Sc., M.Sc., and Ph.D. degrees in Research Letters, vol. 26, pp. 127 – 136, April 2000. electrical engineering from the Technion—Israel Institute of Technology, in [50] D. Micciancio and S. Goldwasser, Complexity of Lattice Problems: 1975, 1981 and 1986 respectively. A Cryptographic Perspective, vol. 671 of The Kluwer International During 1975-1985 he was with the Communications Research Labs, in the International Series in Engineering and Computer Science. Cambridge, capacity of a Senior Research Engineer. Since 1986 he is with the Department UK: Kluwer Academic Publishers, 2002. of Electrical Engineering, Technion—Israel Institute of Technology, where he [51] A. K. Lenstra, H. W. Lenstra, and L. Lov´asz, “Factoring polynomials is now a Technion Distinguished Professor, and holds the William Fondiller with rational coefficients,” Mathematische Annalen, vol. 261, no. 4, Chair of Telecommunications. His research interests encompasses a wide pp. 515–534, 1982. spectrum of topics in information theory and statistical communications. [52] E. Agrell, T. Eriksson, A. Vardy, and K. Zeger, “Closest point search in Dr. Shamai (Shitz) is an IEEE Fellow, an URSI Fellow, a member of the lattices,” IEEE Transactions on Information Theory, vol. 48, pp. 2201– Israeli Academy of Sciences and Humanities and a foreign member of the 2214, August 2002. US National Academy of Engineering. He is the recipient of the 2011 Claude [53] M. R. Bremner, Lattice Basis Reduction. CRC Press, 2012. E. Shannon Award, the 2014 Rothschild Prize in Mathematics/Computer [54] W. Yu, “Sum-capacity computation for the Gaussian vector broadcast Sciences and Engineering and the 2017 IEEE Richard W. Hamming Medal. channel via dual decomposition,” IEEE Transactions on Information He has been awarded the 1999 van der Pol Gold Medal of the Union Theory, vol. 52, pp. 754–759, February 2006. Radio Scientifique Internationale (URSI), and is a co-recipient of the 2000 [55] V. Ntranos, V. Cadambe, B. Nazer, and G. Caire, “Integer-forcing inter- IEEE Donald G. Fink Prize Paper Award, the 2003, and the 2004 joint ference alignment,” in Proceedings of the IEEE International Symposium IT/COM societies paper award, the 2007 IEEE Information Theory Society on Information Theory (ISIT 2013), (Istanbul, Turkey), July 2013. Paper Award, the 2009 and 2015 European Commission FP7, Network of [56] W. He, B. Nazer, and S. Shamai (Shitz), “Dirty-paper integer-forcing,” Excellence in Wireless COMmunications (NEWCOM++, NEWCOM#) Best in Proceedings of the 53rd Annual Allerton Conference on Communi- Paper Awards, the 2010 Thomson Reuters Award for International Excellence cations, Control, and Computing, (Montcello, IL), September 2015. in Scientific Research, the 2014 EURASIP Best Paper Award (for the EURASIP Journal on Wireless Communications and Networking), and the 2015 IEEE Communications Society Best Tutorial Paper Award. He is also the recipient of 1985 Alon Grant for distinguished young scientists and the 2000 Technion Henry Taub Prize for Excellence in Research. He has served as Associate Editor for the Shannon Theory of the IEEE Transactions on Information Theory, and has also served twice on the Board of Governors of Wenbo He received the B.Sc. in electrical engineering from Polytechnic the Information Theory Society. He has also served on the Executive Editorial Institute of New York University and the M.Sc. and Ph.D. degrees in electrical Board of the IEEE Transactions on Information Theory and on the IEEE engineering from Boston University, in 2011, 2014, and 2016, respectively. Information Theory Society Nominations and Appointments Committee. In 2016, Wenbo He joined MathWorks as a senior software engineer working in the sampling and scheduling field for Simulink core group. During his time in Boston University, his research interests focus on topics in information theory and wireless communications, especially on multi- terminal coding for uplink and downlink channel.

Bobak Nazer (S ’02 – M ’09) received the B.S.E.E. degree from Rice University, Houston, TX, in 2003, the M.S. degree from the University of California, Berkeley, CA, in 2005, and the Ph.D degree from the University of California, Berkeley, CA, in 2009, all in electrical engineering.