Approximately achieving Gaussian relay network capacity with lattice codes

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1 Approxmately achevng Gaussan relay network capacty wth lattce codes Ayfer Özgür EFL, Lausanne, Swtzerland Suhas Dggav UCLA, USA and EFL, Swtzerland Abstract Recently, t has been shown that a quantze-mapand-forward scheme approxmately acheves wthn a constant number of bts the Gaussan relay network capacty for arbtrary topologes []. Ths was establshed usng Gaussan codebooks for transmsson and random mappngs at the relays. In ths paper, we show that the same approxmaton result can be establshed by usng lattces for transmsson and quantzaton along wth structured mappngs at the relays. I. INTRODUCTION Characterzng the capacty of relay networks has been a long-standng open queston n network nformaton theory. The semnal work of Cover and El-Gamal [] has establshed the basc achevablty schemes for relay channels. More recently there has been extenson of these technques to larger networks see [3] and references theren. In [], motvated by a determnstc model of wreless communcaton, t was shown that the quantze-map-and-forward scheme acheves wthn a constant number of bts from the nformatontheoretc cutset upper bound. Ths constant s unversal n the sense that t s ndependent of the channel gans and the operatng SNR, though t could depend on the network topology lke the number of nodes. In the quantze-map-and-forward scheme analyzed n [], each relay node frst quantzes ts receved sgnal at the nose level, then randomly maps t to a Gaussan codeword and transmts t. A natural queston that we address n ths paper s whether lattce codes retan the approxmate optmalty of the above scheme. Ths s motvated n part snce lattce codes along wth lattce decodng could enable computatonally tractable encodng and decodng methods. For example lattce codes were used for lnear functon computaton over multpleaccess networks [4] and for communcaton over multpleaccess relay networks wth orthogonal broadcast n [5]. The man result of ths paper s to show that the quantze-map-andforward scheme usng nested lattce codes for transmsson and quantzaton, stll acheves the Gaussan relay network capacty wthn a constant. Ths result s summarzed n Theorem.. The use of structured codes allows to specfy a structured mappng between the quantzaton and transmsson codebooks at each relay. The nested lattce codebooks consdered n ths paper are based on the random constructon n [6], where they are shown to acheve the capacty of the AWGN channel. Ths paper s organzed as follows: In Secton II, we state the network model and our man result. In Secton III, we summarze the constructon of the nested lattce ensemble. In Secton IV, we descrbe the network operaton. In partcular, we specfy how we use the nested lattce codes of Secton III for encodng at the source, quantzaton, mappng and transmsson at the relay nodes, and decodng at the destnaton node. In Secton V, we analyse the performance acheved by the scheme. The detaled proofs can be found n [0]. II. MAIN RESULT We consder a Gaussan relay network where a source node s wants to communcate to a destnaton node d, wth the help of N relay nodes, denoted N. The sgnal receved by node {s, d, N } s gven by y = H x + z where H s the N M channel matrx from node comprsng M transmt antennas to node comprsng N receve antennas. Each element of H represents the complex channel gan from a transmttng antenna of node to a recevng antenna of node. The nose z s complex crcularly-symmetrc Gaussan vector CN0, σ I and s..d. for dfferent nodes. The transmtted sgnals x are subect to an average power constrant. The followng theorem s the man result of ths paper. Theorem.: Usng nested lattce codes for transmsson and quantzaton along wth structured mappngs at the relays, we can acheve all rates R mn IX Ω ; Y Ω c X Ω c N Ω N between s and d, where Ω s a source-destnaton cut of the network and X Ω = {X, Ω} are..d. CN0, /M I. It has been shown n [] that the restrcton to..d. Gaussan nput dstrbutons s wthn N,d N bts/s/hz of the cut-set upper bound. Therefore the rate acheved usng lattce codes n the above theorem s wthn N,d N bts/s/hz to the cutset upper bound of the network. For smplcty of presentaton, n the rest of the paper we concentrate on a layered network where every node has a sngle transmt and receve antenna. More precsely, the sgnal receved by node n layer l, 0 l l d, denoted N l, s

2 gven by y = h x + z where h s the real scalar channel coeffcent from node n layer l, to node. s N 0, d N ld. The analyss can be extended to non-layered networks by followng the tmeexpanson argument of [], to multcast traffc wth multple destnaton nodes as well as to multple multcast where multple source nodes multcast to a group of destnaton nodes. III. CONSTRUCTION OF THE NESTED LATTICE ENSEMBLE Consder a lattce Λ or more precsely, a sequence of lattces Λ n ndexed by the lattce dmenson n wth V denotng the Vorono regon of Λ. Let us defne the second moment per dmenson of Λ as σ Λ = x dx n V where V denotes the volume of V and let the n n full-rank generator matrx of Λ be denoted by G Λ,.e., Λ = G Λ Z n. We assume that Λ or more precsely, the sequence of lattces Λ n s both Roger s and oltyrev good. The exstence of such lattces has been shown n [7], where the reader can also fnd the precse defntons of Roger s and oltyrev good. Ths fxed lattce Λ wll serve as the coarse lattce for all our nested lattce constructons. The fne lattce Λ s constructed usng Loelger s type-a constructon [8]. Let k, n, p be ntegers such that k n and p s prme. The fne lattce s constructed usng the followng steps. Draw an n k matrx G such that each of ts entres s..d accordng to the unform dstrbuton over Z p = {0,,..., p }. Form the lnear code V C = {c : c = G w,w Z k p}, where denotes modulo-p multplcaton. Lft C to R n to form Λ = p C + Z n. Λ = G Λ Λ s the desred fne lattce. Note that snce Z n Λ, we have Λ Λ. Draw v unformly over p Λ V and translate the lattce Λ by v. The nested lattce codebook conssts of all ponts of the translated fne lattce nsde the Vorono regon of the coarse lattce, Λ = v + Λ mod Λ = v + Λ V. In the above equaton, we defne x mod Λ as the quantzaton error of x R n wth respect to the lattce Λ,.e., x mod Λ = x Q Λ x, 3 where the lattce quantzer Q Λ x : R n Λ s gven by Q Λ x = argmn x λ. λ Λ Note that the quantzaton and mod operatons wth respect to a lattce can be defned n dfferent ways. The mod operaton n 3 maps x R n to the Vorono regon V of the lattce. More generally, t s possble to defne a mod or quantzaton operaton wth respect to any fundamental regon of the lattce. In partcular, when we consder the nteger lattce Z n n the sequel, or more generally ts multples p Z n where p s a postve nteger, we wll assume that x mod p Z n = x x p where x p denotes component-wse roundng to the nearest smaller nteger multple of p. In other words, the mod operaton wth respect to p Z n maps the pont x R n to the regon p [0, n. The above constructon yelds a random ensemble of nested lattce codes that has the followng desred propertes: There s a becton between Z n p p Z n [0, n p Λ G Λ [0, n p Λ V. The last observaton follows smply from the fact that both G Λ [0, n and V are fundamental regons of the lattce Λ,.e., they both tle R n. Snce C Z n p, the above becton restrcted to C yelds, C p C = Λ [0, n Λ G Λ [0, n Λ V Λ. 4 Note that Λ p Λ V. The bectons above can be explctly specfed n both drectons and we wll make use of ths fact n the next secton. The random codebook Λ has the followng statstcal propertes: Let λ p Λ V, Λ = λ = p Λ V = p n. Let λ, λ p Λ V,, Λ = λ, Λ = λ = p Λ V =. 5 pn In other words, the constructon n ths secton yelds an ensemble of nested lattce codes such that each codeword of the random codebook Λ s unformly dstrbuted over p Λ V and the codewords of Λ are parwse ndependent. These two propertes suffce to prove the random codng result of ths paper. IV. ENCODING, MAING AND DECODING The above constructon yelds a random ensemble of nested lattce pars Λ Λ wth codng rate, R = n log Λ whch can be tuned by choosng the precse magntudes of k and p. In ths ensemble, the coarse lattce Λ s fxed and the fne lattce Λ s randomzed. It has been shown n [9] that wth hgh probablty, a nested lattce Λ, Λ n ths ensemble s such that both Λ and Λ are Roger s and oltyrev-good. For quantzaton, we fx one such good member of the ensemble and use t at all the relay nodes. For transmsson, we draw a random nested lattce codebook from ths ensemble, ndependently at each relay. The mappng

3 between the quantzaton and transmsson codebooks at each relay s specfed below. Source: The source has p k messages, where p s prme and k n. The messages are represented as length-k vectors over the fnte feld Z p and mapped to a random nested lattce codebook Λ followng the constructon n Secton III. In the constructon, the coarse lattce Λ s scaled such that ts second moment σ Λ T = ǫ Λ, where Λ T now denotes the scaled verson of the lattce Λ to satsfy the power constrant. ǫ Λ 0 as n ncreases and choosng t carefully we can ensure that every codeword of Λ satsfes the power constrant. The nformaton transmsson rate s gven by R = n log pk. Let us denote by x w s, w {,..., e nr } the random transmt codewords correspondng to each message w of the source node. Relays: The relay node receves the sgnal y. The sgnal y s frst quantzed by usng a nested lattce codebook that has been generated by the constructon n Secton III. It s shown n [9] that ths constructon yelds nested lattces where the fne lattce s Roger s good wth hgh probablty f k log n. The coarse lattce s both Roger s and oltyrev good by constructon. We fx one such good nested lattce Λ Q, ΛQ and use the correspondng codebook Λ Q = ΛQ mod Λ Q at all the relay nodes for quantzaton. Therefore our quantzaton codebook s not random but fxed and moreover same for all relay nodes. We assume that the nested lattce Λ Q, ΛQ has been generated by usng the followng parameters: Let D s = max h. 6 The coarse lattce Λ Q s a scaled verson of the lattce Λ such that σ Λ Q = µd s + σ 7 for a constant µ > 0. Recall that σ s the nose varance. We denote the generator matrx of the scaled coarse lattce Λ Q by G Λ Q. The parameters k r and p r are chosen such that k r = log n and p r s the prme number such that p k r = e nrr, where R r = log σ Λ Q σ. 8 Note that snce R r s ndependent of n, p r = e nrr log n,.e, p r as n. It can be shown that wth the choce n 8 for R r, the second moment of Λ Q s such that σ Λ Q σ when n ncreases. Ths s a consequence of the fact that both Λ Q and ΛQ are Roger s good. Therefore, we are effectvely quantzng at the nose level. The quantzed sgnal s gven by ŷ = Q Λ Qy + u mod Λ Q where u s a random dther known at the destnaton node and unformly dstrbuted over the Vorono regon V Q of the More precsely, one should take p r to be the largest prme number such that p r e nrr/k. When n s large, the dfference becomes neglgble. fne lattce Λ Q. The dthers u are ndependent for dfferent nodes. Map and Forward: Let us scale the coarse lattce Λ such that ts second moment σ Λ T = ǫ Λ. Let G Λ T denote the generator matrx of the scaled coarse lattce. The quantzed sgnal ŷ at relay s mapped to the transmtted sgnal x by the followng mappng, x = G Λ T p r G p r G Λ Q ŷ mod Z n mod p r Z n + v mod Λ T, 9 where G s an n n random matrx wth ts entres unformly and ndependently dstrbuted n 0,,...,p r and v s a random vector unformly dstrbuted over p r Λ T V T, where V T s the Vorono regon of Λ T. G and v are ndependent for dfferent relay nodes. We ndex the e nrr codewords of Λ Q as, k {,...,e nrr }. The correspondng sequence that the codeword ŷ k s mapped to n 9 s denoted by x k. ŷ k roposton 4.: The above mappng has the followng propertes: At each relay, the transmtted sequences x Λ, where Λ s a nested lattce codebook. The mappng nduces a parwse ndependent and unform dstrbuton over p r ΛT V T. Formally, each quantzaton codeword ŷ k Λ Q s mapped unformly at random to the set p r Λ T V T. Two codewords ŷ k,ŷ k Λ Q such that k k are mapped ndependently. The mappng nduces an ndependent dstrbuton across the relays. The proposton says that the quantzaton codebooks at each relay are ndependently mapped to a random nested lattce codebook from the ensemble constructed n Secton III. The proof s based on the becton gven n 4: There s one-to-one correspondence between the codebook Λ Q and ts underlyng fnte feld codebook C Q. The mappng p r G Λ ŷ Q mod Z n takes the codeword ŷ Λ Q to ts correspondng codeword n C Q. Ths codeword n C Q s then mapped to a random fnte-feld codebook C = { c : c = G c, c C Q}. We then form the nested lattce codebook Λ correspondng to C followng agan the constructon of Secton III. The second property follows by observng that the random matrx G maps every nonzero vector c C Q unformly at random to another fnte feld vector n Z n p. The thrd property follows from the ndependence of the G s and v s for dfferent nodes. The mappng n 9 can be smplfed to the form, x = G Λ T G G Λ Q ŷ + v mod Λ T. Effectvely, t takes the quantzaton codebook Λ Q, expands t by multplyng wth a random matrx wth large entres of the order of p r and then folds t to the Vorono regon of Λ T. Snce the entres of G are potentally very large, even f two codewords are close n Λ Q, they are mapped ndependently to the codewords of the transmt codebook. Note that the complexty of the mappng s polynomal n n, whle random mappng of the form n [] has exponental complexty n n.

4 Destnaton: Gven ts receved sgnal y d, together wth the knowledge of all codebooks, mappngs, dthers and channel gans, the decoder performs a consstency check to recover the transmtted message. For each relay and quantzaton codeword ŷ k, t frst forms the sgnals ỹ k Note that for N l ỹ = ŷ u = ŷ k u mod Λ Q. 0 mod Λ Q = Q Λ Qy + u u mod Λ Q = y y + u mod Λ Q mod ΛQ = h x + z u mod Λ Q, where u = y +u mod Λ Q. u s ndependent of y and s unform over the Vorono regon of Λ Q Crypto Lemma, see [6]. The decoder then checks the set Ŵ of messages ŵ for whch there exsts some ndces k, such that x ŵ s,y d, {ỹ k,x k } N Ãǫ where Ãǫ denotes consstency and N denotes the set of relays. We defne consstency as follows: For a gven set of ndces {k } N, we say x ŵ s,y d, {ỹ k,x k } N Ãǫ f ỹ k h x k mod Λ Q n σc, for all N l, l l d where for convenence of notaton we have denoted x ŵ s = x k, N 0, and y d = ỹ k, N ld. We choose = + ǫσ σ c for a constant ǫ > 0 that can be taken arbtrarly small. We can nterpret the consstency check as follows. For each layer l =,...,l d the decoders pcks a set of potental quantzed receved sequences {ŷ k } Nl and the transmt sequences correspondng to them {x k } Nl. It checks for each layer l, whether the nputs and outputs are consstent,.e., whether the examned nputs {x k } Nl of the layer l could have generated the examned outputs {ŷ k } Nl. Note that the termnaton condtons are known,.e., x s s known for the message beng tested, and y d s the observed sequence at the destnaton. Therefore, effectvely the decoder checks whether there exsts a plausble set of nput and output sequences at each relay that under the message w yeld the observaton y d. Gven, note that the defnton of consstency n s closely related to weak typcalty. Indeed, t s a varant of the weak typcalty condton for Gaussan vectors. Therefore, effectvely our decoder s a typcalty decoder. V. ERROR ANALYSIS An error occurs f the transmtted message w s not n the lst,.e., w / Ŵ or when w w s also n the lst Ŵ. It s easy to show that the correct message w s n the lst wth hgh probablty. We concentrate on the probablty that there exst an error because w s not the unque message n Ŵ. Ths probablty can be upper bounded by concentratng on the par-wse error probabltes,.e., e e nr w w where w w s gven by {k } N s.t.x w s,y d, {ỹ k,x k } N Ãǫ x w s,y d, {ỹ k,x k } N Ãǫ k,...,k N We can condton on the event that the correct message produces ndces {k }, and snce ths s a generc ndex, we can carry out the entre calculaton condtoned on ths and then average over t. The summaton over the N ndces k,..., k N above can be rearranged to yeld Ω k, NΩ k k x w s,y d, {ỹ k,x k } Ãǫ s.t.k = k, NΩ c }{{} where Ω s a source-destnaton cut of the network,.e, Ω = {s, N Ω } where N Ω s a subset of the relayng nodes N. The frst summaton runs over all possble source-destnaton cuts Ω of the network, or equvalently over all subsets N Ω of the relayng nodes N. Followng [], the rearrangement of the summaton above can be nterpreted as ntroducng a noton of dstngushablty. The relay nodes n Ω are the ones that can dstngush between w and w because ỹ k ỹ k, when the relay nodes n Ω c cannot dstngush between w and w because ỹ k = ỹ k. The source node s naturally n the dstngushablty set Ω and the destnaton node s n Ω c. Thus, we sum over all possble cases for the dstngushablty set Ω. Now, let us examne the probablty denoted by. For a gven set of {k } N such that k = k, NΩ c and k k, N Ω, the consstency condton n takes two dfferent forms dependng on whether N Ω or Ω c. For nodes Ω c, the condton s equvalent to h x k x k + z u mod ΛQ n σc Ω l 3 where Ω l = Ω N l and we denote ths event by A. For nodes N Ω, the condton yelds ỹ k Ω c l h x k mod Λ Q n σc Ω l 4 where Ω c l = Ωc N l and we denote ths event by B. We have, = {A, Ω c }, {B, N Ω } h x k = A, Ω c B, N Ω A, Ω c. Note that due to roposton 4., x k,x k, {s, N } n expressons 3 and 4 are a set of ndependent random varables, unformly dstrbuted over p r Λ T V T. Due to the dtherng n 0, ỹ k n 4 s unformly dstrbuted over the Vorono regon V Q of the quantzaton lattce pont ŷk.

5 We wll frst bound the probablty A, Ω c by condtonng on the event defned n the followng lemma. Lemma 5.: Let us defne E to be the followng event, { {N, d}, {k, k } s.t. h x k x k +z u / V Q}. We have E 0. When E s true, we declare ths as an error. Ths adds a vanshng term to the decodng error probablty by the above lemma. Condtonng on the complement of E allows us to get rd of the mod operaton w.r.t Λ Q n 3. Gven E c, the condton A s equvalent to A = { h x k x k + z u n σc}. Ω l Therefore, we have A, Ω c E c = A, Ω c E c A, Ωc E c We upperbound the last probablty above n the followng lemma. Lemma 5.: h x k x k + z u n σc, Ω c Ω l e nixω;hxω+z Ω c Ωc +log+ǫ o n, where X, Ω are..d Gaussan random varables N0,, Z Ω c are..d Gaussan random varables N0, σ and H s the channel transfer matrx from nodes n Ω to nodes n Ω c. The proof of the lemma nvolves two man steps. Recall that x k,x k, Ω are dscrete random varables, ndependently and unformly dstrbuted over p r Λ T V T. We frst show that the probablty n the lemma s upper bounded by e nǫ h x x + z z n σc, Ω c Ω l 5 where x,x, Ω and z, Ωc are all ndependent Gaussan random varables such that x,x N0, σ xi n, z N0, σ z I n and as n ncreases, σx σ Λ T f Λ T s Roger s good, σz σ Λ Q σ f Λ Q s Roger s good whch s our case here. ǫ 0 when n, agan f Λ T and Λ Q are Roger s good. Gven ths translaton to Gaussan dstrbutons the problem becomes very smlar to the one for Gaussan codebooks n []. The second step s to bound the probablty n 5 by followng a smlar approach to []. We wll now upper bound the term B, N Ω A, Ω c 6 k, NΩ k k by frst removng the condton k k n the summaton above and then notng that ths term s equal to e NΩ nrr tmes the probablty B, N Ω A, Ω c evaluated for a randomly and ndependently chosen set of ndces {k } N Ω. When each of the ndces {k } N Ω s chosen unformly at random, ỹ k n 4 s a random varable unformly dstrbuted over V Q. Ths s due to the dtherng over the Vorono regon V Q of the fne lattce and the mod operaton wth respect to the coarse lattce Λ Q n 0. Moreover, by the Crypto Lemma, ν = ỹ k Ω c l Ω c l h x k h x k Ω l mod Λ Q s also unformly dstrbuted over V Q and s ndependent of h x k + h x k. Ω l Ths s due to the fact that ỹ k s ndependent of ths term, whch s due to the fact the ndex k and the dther u are chosen ndependently of everythng else. Therefore 6 s upper bounded by B, N Ω A, Ω c k, NΩ = e NΩ nrr ν nσ c N Ω e NΩ n log+ǫ++on 7 where the last nequalty follows from the below lemma. Lemma 5.3: Let ν be unformly dstrbuted over V Q. We have, ν n σc e n ««log + σ Λ Q σc o n. Combnng the results of Lemma 5. and 7, an consderng the summaton over all possble source-destnaton cuts proves the man result of ths paper stated n Theorem.. REFERENCES [] S. Avestmehr, S. Dggav and D. Tse, Wreless network nformaton flow: a determnstc approach, eprnt arxv: v - arxv.org. [] T. M. Cover and A. El Gamal, Capacty theorems for the relay channel, IEEE Trans. on Informaton Theory, vol.5, no.5, pp , September 979. [3] G. Kramer, I. Marc, and R. Yates, Cooperatve Communcatons. Foundatons and Trends n Networkng, 006. [4] B. Nazer and M. Gastpar, Computaton over Multple Access Channels,, IEEE Trans. on Informaton Theory, vol.53, no.0, pp , October 007. [5] W. Nam and S.-Y. Chung, Relay networks wth orthogonal components, n roc. 46th Annual Allerton Conference, Sept.008. [6] U. Erez and R. Zamr, Achevng log+snr on the AWGN channel wth lattce encodng and decodng, IEEE Trans. on Informaton Theory, vol.50, no.0, pp.93-54, Oct [7] U. Erez, S. Ltsyn and R. Zamr, Lattces whch are good for almost everythng, IEEE Trans. on Informaton Theory, vol.5, no.0, pp , Oct [8] H.-A. Loelger, Averagng bounds for lattces and lnear codes, IEEE Trans. on Informaton theory, vol.43, pp , Nov [9] D. Krthvasan and S. radhan, A proof of the exstence of good lattces, tech. rep., Unversty of Mchgan, July 007. See [0] A. Özgür and S. Dggav, Approxmately achevng Gaussan relay network capacty wth lattce codes, e-prnt - arxv.org, May 00. An alternatve way to upper bound 6 s to randomly choose the quantzaton lattces at each relay nstead of usng a fxed lattce.