Relays that Cooperate to Compute

Transcription

1 Reays that Cooperate to Compute Matthew Nokeby Rice University Bobak Nazer Boston University Behnaam Aazhang Rice University Natasha evroye University of Iinois - Chicago devroye@uic.edu Abstract This paper proposes a new coding scheme that combines the advantages of statistica cooperation and agebraic structure. Consider a mutipe-access reay channe where two transmitters attempt to send the moduo-sum of their finite fied messages to the receiver with the hep of the reay. The transmitters use nested attice codes to ensure that sums of codewords are protected against noise and to preserve the moduo operation of the finite fied. We deveop a bock Markov coding scheme where the reay recovers the rea sum of the codewords and retransmits it coherenty with the two transmitters. I. INTROUCTION Consider a wireess network comprised of severa users that wish to communicate with each other. In certain scenarios, it is beneficia to have some of these users act ike reays to hep other users recover their desired messages. This strategy is broady referred to as physica-ayer cooperation and incudes many powerfu schemes such as decode-and-forward, compress-and-forward, and ampify-and-forward [ [8. The key feature uniting these schemes is that they expoit statistica dependencies between the observations of the reays and the destinations to increase the achievabe rates. More recenty, a new famiy of schemes, dubbed physica-ayer network coding, has been proposed to harness the interference property of the wireess medium, through the use of codebooks with appropriate agebraic structure see [9 for a recent survey. For instance, compute-and-forward enabes reays to recover equations of the transmitted messages and pass them towards the destinations [0. Unti recenty, these two broad strategies have been studied in isoation: existing coding schemes expoit either statistica dependencies or agebraic structure, but not both. In this paper, we deveop a form of reay cooperation for compute-and-forward that can simutaneousy benefit from statistica cooperation and agebraic structure. The key is that the reay can directy recover the sum of the codewords and retransmit it for a coherent gain at the destination. Recent work has aso investigated user cooperation for compute-andforward [. There, each transmitter expoits the fu dupex nature of the channe to recover other transmitters codewords, which it can then send, aong with its own codeword, for a coherent gain. II. PROBLEM STATEMENT We wi deveop our scheme in the context of a Gaussian mutipe-access reay channe MARC where the destination wishes to decode the moduo-sum of the users messages. See Figure for an iustration. For ease of exposition, we ony consider rea-vaued channes and symmetric rates. w E x w 2 E 2 x 2 z R y R R x R z y u = w w 2 Fig.. Compute-and-forward over a Gaussian mutipe-access reay channe. The reay heps the receiver decode the moduo-sum of the transmitted messages. There are two transmitters indexed by, 2}, each with a ength-k message vector w, drawn independenty and uniformy over a finite fied F k p, where p is prime. An encoder, E : F k p R n, maps this message into a ength-n channe input vector x. As usua, the channe inputs must each satisfy a power constraint, x 2 np S. A reay observes the noisy sum of the the channe inputs, y R = x +x 2 +z R where z R is i.i.d. Gaussian noise with mean zero and variance N R. The reay produces its own channe input x R using its causa knowedge of y R. Specificay, et R i : R i R denote the mapping that produces the reay s channe input for time i, i.e., the ith component of x R. The reay s channe input must satisfy a power constraint, x R 2 np R. The destination observes the noisy sum of the channe inputs pus the signa from the reay, y = x +x 2 +x R +z 2 where z R is i.i.d. Gaussian noise with mean zero and variance N R. A decoder, : R n F k p, produces an estimate û = y of the moduo-p sum of the messages, u = w w 2. We say a computation rate R is achievabe if, for any ǫ > 0 and n arge enough, there exist encoding and decoding functions, such that the reay can reiaby decode the sum, k n og 2p > R ǫ 3 Pû u < ǫ. 4 û

2 The computation capacity is the supremum of a achievabe computation rates. III. NESTE LATTICE COES Our achievabe scheme reies on the use of nested attice codes. For competeness, we provide a brief set of definitions beow and refer interested readers to [0, [2 [4 for further detais as we as [7, which deveops a famiy of attice-based decode-and-forward and compress-and-forward schemes for cassica reay scenarios. A attice Λ is a discrete subgroup of R n with the property that if t,t 2 Λ then t +t 2 Λ. Any attice can be described using a rea-vaued generator matrix B R n, Λ = BZ n. 5 A pair of attices Λ,Λ FINE is nested if Λ Λ FINE. With each attice, we associate a quantizer, Q Λ : R n Λ, that maps vectors to the nearest attice point in Eucidean distance, Q Λ x = argmin x t. 6 t Λ The fundamenta Voronoi region is the subset of points in R n that quantize to the zero vector, V = x : Q Λ x = 0}. The moduo operation returns the quantization error with respect to the attice, [x mod Λ = x Q Λ x, 7 and satisfies the distributive aw, [ a[x mod Λ+b[y mod Λ mod Λ = [ ax+by mod Λ, for any integer-vaued coefficients a, b Z. A nested attice code L is created by taking the set of fine attice points that fa within the fundamenta Voronoi region of the coarse attice, L = Λ FINE V. The rate of such a code is R = n og L. Erez and Zamir have shown that there exist nested attice codes that can approach the capacity of a pointto-point Gaussian channe [3. This capacity-achieving attice ensembe is created using Construction A, which embeds a finite fied codebook into the reas. Specificay, et be the finite fied generator matrix for a inear code and B be the rea-vaued generator matrix for the coarse attice Λ. The fine attice is created as foows: Create a finite fied generator matrix G Fp n k with every eement drawn in an i.i.d. uniform fashion from F p. Let C denote the codebook induced by G, C = c = Gw : w F k p }. Embed this codebook into the unit cube and tie the resut over the integers, Λ FINE = p C +Z. Rotate by the generator matrix of the coarse attice to get the desired fine attice, Λ FINE = B Λ FINE. It can be shown [3, [5 that, for appropriate p and k a attice drawn from this random ensembe is good for Gaussian channe coding with probabiity that goes to with n. For our reaying strategy, we wi use a codebook that can be decoded in two stages by the destination. Foowing the framework set forth in [8, we wi spit our attice codebook into two parts, the termed the resoution codebook L r and vestigia codebook L v, respectivey. These codebooks are created by spitting the coumns of the finite fied generator matrix G = [G r G v and generating attices Λ r and Λ v from each part foowing the construction outined above. In effect, this means that the first k r symbos of a message are encoded onto Λ r and the ast k v = k k r symbos of a message are encoded onto Λ v. The rates of L r and L v are set to R r and R v with R r +R v = R. As shown in [0, Lemma 6, there exist mappings between finite fied vectors and nested attice codes that preserve inearity. Specificay, there is a one-to-one mapping φ : F k p L such that for any w,w 2 F k p, φ [φw +φw 2 mod Λ = w w 2. 8 This correspondence can be extended to the resoution and vestigia codebooks [8. That is, there exist mappings φ r : F k p L r and φ v : F k p L v such that for any w,w 2 F k p, φ r w w 2 = [φ r w +φ r w 2 mod Λ 9 φ v w w 2 = [φ v w +φ v w 2 mod Λ 0 [φ r w +φ v w mod Λ = φw. See Lemma 2 and 3 in [8 for a proof. IV. COMPUTE-AN-FORWAR Compute-and-forward is a framework for reiaby sending inear combinations of finite fied messages over muti-user networks [0. Here, we summarize some of the key resuts that wi be used as buiding bocks in our scheme. It was shown in [0 that the computation rate R = 2 + P S N 2 is achievabe for sending the sum over a two-user mutipeaccess channe, i.e., the network in Figure with the reay turned off. The basic scheme is described beow for competeness. Encoding: Each user maps its message to a attice codeword, t = φw. It then appies a dither 2 that is drawn independenty and uniformy over V and takes mod Λ to produce its channe input, x = [t d mod Λ. The second moment of Λ is chosen to meet the power constraint P S. This terminoogy is intended to convey that the vestigia component of a message is the component eftover after the destination has decoded the resoution codeword. 2 As in the standard random coding argument, these random dithers can be repaced with fixed ones after showing that the scheme works with high probabiity. That is, no shared randomness is necessary. See [0 for more detais.

3 ecoding: The receiver scaes its observation y by the minimum-mean squared error MMSE coefficient α = N +, removes the dithers, and takes mod Λ to get s = [ αy d d 2 mod Λ 3 = [ [t +t 2 mod Λ+ αx +x 2 +αz mod Λ. Thus, the receiver observes themod Λ sum of the attice codewords v = [t +t 2 mod Λ pus an independent noise term with variance N EFFEC = α 2 +α 2 N = 2PSN +N. It then quantizes onto the fine attices and takes mod Λ to get its estimate of the attice codeword sum, Using the resuts of [3, if ˆv = [ Q ΛFINE s mod Λ. 4 R < PS N EFFEC, 5 then, for anyǫ > 0 andnarge enough,pˆv v < ǫ. Finay, appying the inverse mapping, we obtain a reiabe estimate of the moduo-sum of the messages, û = φ ˆv. Recenty, it was shown that if a receiver can recover the mod Λ sum of the codewords, it can recover the rea sum as we [6. This wi be an essentia ingredient of our scheme, as it wi enabe the reay s transmission to coherenty combine with those from the source terminas. Beow, we state a specia case of [6, Lemma. Lemma : If a decoder can make an estimate ˆv of v = [t + t 2 mod Λ with vanishing probabiity of error, then it can aso make an estimate ˆq of the rea sum of the channe inputs q = x +x 2 with vanishing probabiity of error. V. RELAY COOPERATION FOR COMPUTE-AN-FORWAR We now propose a reay cooperation scheme for computeand-forward. The basic premise is that the reay has a better channe than the destination. To hep the destination decode, the reay recovers the rea sum of the codewords and retransmits them. To make this work under a causaity constraint, we empoy bock Markov coding in the same fashion as the cassica decode-and-forward scheme proposed by Cover and E Gama [. Our main resut is a new achievabe region for reiabe computation over the MARC. Theorem : The computation rate R = max maxr ρ,γ,r 2 ρ,γ} 6 ρ,γ [0, R 2 ρ,γ = min 2 + ρ 2 γ 2 P S R ρ,γ = min N R + ρ 2 γ 2 } 2 + ρ2 P S + PR 2 ρ 2 +N ρ2 γ 2 P S 7 N +min 2 + ρ2 γ 2 P S, N R } 2 + ρ2 P S + PR 2 ρ 2 +N 2 + ρ2 γ 2 P S N R + ρ 2 γ 2, 2 + ρ2 γ 2 P S N } 8 is achievabe for reiaby sending the moduo sum of messages over the mutipe-access reay channe. Proof: Using the construction in Section III, we draw a nested attice codebook L of rate R and scae it so the second moment of the coarse attice is equa to P S. This codebook can be decomposed into resoution and vestigia components, L r and L v, with rates R r and R v. Foowing standard bock Markov encoding, each transmitter encodes B messages w [,...,w[b each of rate R over B + bocks of n channe instances each. Assuming B correct decoding, this wi yied an overa rate of B+ R, meaning that for arge B the rate oss associated with the extra bock is negigibe. We now describe the encoding and decoding schemes at bock, bock 2 b B, and bock B+. Throughout, it is assumed that the dithers d [b,r and d[b,v are drawn independenty and uniformy over the fundamenta Voronoi region of the coarse attice V and made avaiabe to a terminas. As noted earier, these random dithers can be repaced with fixed dithers if desired. Let γ and ρ be constants in [0, chosen to maximize the rate expression in the theorem statement. Bock, Encoding: Each transmitter maps its message to both the resoution and vestigia attice codebooks, t [,r = φ rw [ It then appies dithers x [,r = [t[,r d[,r mod Λ t[,v = φ vw [. 9 x[,v = [t[,v d[,v mod Λ. and transmits the weighted sum of these codewords, x [ = ρ 2 γx [,r + γ 2 x [,v. 20 The reay sends nothing in this bock.,

4 Bock, ecoding: The reay observes y [ R = ρ γ 2 x [,r +x[ + γ 2 x [,v +x[ 2,v 2 +z [ R. 22 Since it ony needs to recover the sum of the resoution components, it has two options. First, it can decode the resoution sum whie treating the vestigia sum as noise. In this case, the effective signa power is ρ 2 γ 2 P S and the effective noise power is N R + ρ 2 γ 2. Thus, it can recover x [,r +x[ if R r < 2 + ρ 2 γ 2 P S N R + ρ 2 γ Second, the reay can initiay decode the vestigia sum x [,v + x[ 2,v, subtract it from the received signa, and then decode the resoution sum without any interference. Using this method, decoding is successfu provided R v < 2 + ρ2 γ 2 P S N R + ρ 2 γ 2 24 R r < 2 + ρ2 γ 2 P S N R. 25 The destination does not recover anything yet. Bock 2 b B, Encoding: Each transmitter maps its message to both the resoution and vestigia attice codebooks, t [b,r = φ rw [b It then appies dithers,r = [t[b,r d[b,r mod Λ t[,v = φ vw [b. 26 x[b,v = [t[b,v d[b,v mod Λ. and transmits the weighted sum of these codewords pus the resoution codeword from the previous bock, = ρ 2 γ,r + γ 2,v +ρx [b,r 27 The reay sends the sum of the resoution codewords from the previous bock, scaed to meet its power constraint, R = PR x [b,r +x [b. 28 Bock 2 b B, ecoding: The reay observes R = ρ γ 2,r +x[b + γ 2 + ρ x [b,r +x [b y [b,v +x[b 2,v +z [b R. 29 Since it aready know the resoution sum x [b,r +x [b from the previous bock, it can remove it and decode the new resoution sum using the scheme from bock.,r +x[b The destination observes y [b = ρ γ 2 + ρ+,r +x[b PR + γ 2,v +x[b 2,v x [b,r +x [b +z [b. 30 It first decodes the resoution sum x [b,r + x [b from y [b which is possibe if R r < 2 + ρ2 P S + PR 2 ρ N Assuming the destination decodes correcty, it subtracts the resoution sum from its observation to get ỹ [b = y[b P ρ+ R x [b,r +x [b. Next, it takes ỹ [b from the previous bock and removes the resoution sum to get ỹ [b γ ρ 2 x [b,r +x [b 32 = ρ 2 γ 2 x [b,v +x [b 2,v +z [b. 33 From here, it can subtract the vestigia sum x [b,v +x [b R v < 2 + ρ2 γ 2 P S N 2,v if. 34 Bock B +, Encoding: In the fina bock, the transmitters and reay coherenty send the sum of the resoution codewords from bock B, x [B+ x [B+ R = = ρx [B,r 35 PR x [B,r 2P +x[b. 36 S Bock B +, ecoding: The reay has nothing to decode in this bock. The destination can recover x [B,r +x[b from its observation under the same condition as in bocks 2 through B. It then subtracts this resoution sum from ỹ [B to expose the ast vestigia sum, x [B,v + x[b 2,v. Finay, it can recover this sum under the same condition as in bocks 2 through B. The destination now has a of the desired mod Λ sums of attice codewords as we as their rea sums. It appies the inverse mapping φ to each mod Λ sum to reiaby recover its desired message sums, w [ w[ 2,...,w[B w [B 2. Foowing standard union bound arguments, it can be shown that the average probabiity of error goes to zero as n increases. Therefore, there must exist a sequence of good fixed attice codebooks that achieve the desired rates. Remark : In the cassica decode-and-forward scheme [, there is no need to spit power between the resoution and vestigia messages. In fact, when the destination turns to decode the vestigia message, it can competey remove the effect of the resoution codeword and obtain the fu SNR of the resuting channe to itsef. In our considerations, it is not cear how to enabe the reay to decode the rea sum of the resoution codewords from the rea sum of the resoution and vestigia codewords. To overcome this issue, we have spit power between the two messages, which resuts in a sma rate oss. Future work wi focus on mitigating this effect. In Figure 2 we pot the achievabe rate from Theorem. We compare against two simiar schemes. First, we compare against the basic compute-and-forward scheme from [0, i.e., the reay is turned off. Second, we compare against a scheme

5 presented in [7 in which the destination decodes the messages individuay with the hep of the reay. We choosep S = 0dB, P R = 20dB, N = 0dB, and we vary N R in order to sweep out a range of vaues for the signa-to-noise ratio between the transmitters and the reay. Achievabe rate bits/channe use No reay Seperate decoding Proposed scheme SNR between Transmitters and Reay [9 B. Nazer and M. Gastpar, Reiabe physica ayer network coding, Proceedings of the IEEE, vo. 99, no. 3, pp , March 20. [0, Compute-and-forward: Harnessing interference through structured codes, IEEE Transactions on Information Theory, vo. 57, no. 0, pp , October 20. [ M. Nokeby and B. Aazhang, Cooperative compute-and-forward, IEEE Transactions on Information Theory, Submitted March 202, avaiabe onine: [2 R. Zamir, S. Shamai Shitz, and U. Erez, Nested inear/attice codes for structured mutitermina binning, IEEE Transactions on Information Theory, vo. 48, no. 6, pp , June [3 U. Erez and R. Zamir, Achieving og + SNR on the AWGN 2 channe with attice encoding and decoding, IEEE Transactions on Information Theory, vo. 50, no. 0, pp , October [4 R. Zamir, Lattices are everywhere, in Proceedings of the 4th Annua Workshop on Information Theory and its Appications ITA 2009, La Joa, CA, February [5 U. Erez, S. Litsyn, and R. Zamir, Lattices which are good for amost everything, IEEE Transactions on Information Theory, vo. 5, no. 0, pp , October [6 B. Nazer, Successive compute-and-forward, in Proceedings of the Internationa Zurich Seminar on Communications IZS 202, Zurich, Switzerand, March 202. Fig. 2. Achievabe rates vs. SNR For ow SNR vaues, the requirement that the reay must decode the sum constrains the achievabe rate, and the basic compute-and-forward scheme dominates. For higher SNR vaues, the reay can more easiy decode, and cooperation nets a rate gain. Both separate decoding and our proposed approach outperform non-cooperation. However, since our approach garners a coherence gain, and since the destination need ony decode the sum of messages rather than the messages individuay, it outperforms separate decoding for the parameter vaues shown. REFERENCES [ T. M. Cover and A. E Gama, Capacity theorems for the reay channe, IEEE Transactions on Information Theory, vo. 25, no. 5, pp , September 979. [2 G. Kramer, M. Gastpar, and P. Gupta, Cooperative strategies and capacity theorems for reay networks, IEEE Transactions on Information Theory, vo. 5, no. 9, pp , September [3 B. Schein and R. G. Gaager, The Gaussian parae reay network, in Proceedings of the IEEE Internationa Symposium on Information Theory ISIT 2000, Sorrento, Itay, June [4 S. Borade, L. Zheng, and R. Gaager, Ampify-and-forward in wireess reay networks: Rate, diversity, and network size, IEEE Transactions on Information Theory, vo. 53, no. 0, pp , October [5 I. Maric, A. Godsmith, and M. Médard, Anaog network coding in the high-snr regime, in Proceedings of the IEEE Wireess Network Coding Conference WiNC 200, Boston, MA, June 200. [6 S. H. Lim, Y.-H. Kim, A. E Gama, and S.-Y. Chung, Noisy network coding, IEEE Transactions on Information Theory, vo. 57, no. 5, pp , May 20. [7 Y. Song and N. evroye, Lattice codes for the Gaussian reay channe: ecode-and-forward and compress-and-forward, IEEE Transactions on Information Theory, Submitted October 20, avaiabe onine: [8 M. Nokeby and B. Aazhang, Lattice coding over the reay channe, in Proceedings of the IEEE Internationa Conference on Communications ICC 20, Kyoto, Japan, June 20.