Lattice Coding for the Two-way Two-relay Channel

Transcription

1 01 IEEE International Symposium on Information Theory Lattice Coding for the Two-way Two-relay Channel Yiwei Song Natasha Devroye University of Illinois at Chicago Chicago IL ysong uicedu Huai-Rong Shao Chiu Ngo Samsung Electronics US R&D Center SISA) San Jose CA 951 hrshao Abstract We develop a novel lattice coding scheme for the Two-way Two-relay Channel: 1 where Node 1 and communicate with each other through two relay nodes and Each node only communicates with its neighboring nodes The key technical contribution is the lattice-based achievability strategy where each relay is able to remove the noise while decoding the sum of several signals in a Block Markov strategy and then re-encode the signal into another lattice codeword using the so-called Re-distribution Transform This allows nodes further down the line to again decode sums of lattice codewords The symmetric rate achieved by the proposed lattice coding scheme is within 1 log bit/hz/s of the symmetric rate capacity I INTRODUCTION Lattice codes may be viewed as linear codes in Euclidean space: the sum of two lattice codewords is again a codeword This group property is exploited in additive white Gaussian noise AWGN) relay networks such as [1] [] [] [] where it has been shown that lattice codes may sometimes outperform iid random codes particularly when interested in decoding a linear combination of the received codewords One such example is the AWGN Two-way Relay channel where two users communicate with each other through a relay node [] [] If the two users employ lattice codewords the relay node may decode the sum of the codewords from both users directly at higher rates than decoding them individually It is then sufficient for the relay to broadcast this sum of codewords to both users since each user knowing the sum and its own message may determine the other desired message ast work Beyond their use in the Two-way Relay Channel nested lattice codes have been shown to be capacity achieving in the point-to-point Gaussian channel [5] the Gaussian Multiple-access Channel [] Broadcast Channel [6] and to achieve the same rates as those achieved by iid Gaussian codes in the Decode-and-Forward rate and Compress-and- Forward rates [1] of the Relay Channel [7] Lattice codes have also been shown to be useful in the Compute-and-Forward framework for decoding linear equations of codewords of [] The Two-way Two-relay Channel: 1 where Nodes 1 and exchange information through the relay nodes and is related to [8] which considers the throughput of iid random code-based Amplify-and-Forward and Decodeand-Forward approaches for this channel model or the iid random coding based schemes of [9] where there are also links between all nodes This model is also different from that in ortions of this work were performed at Samsung Electronics US R&D Center during Yiwei Song s summer internship in 01 [10] where a two-way relay channel with two parallel rather than sequential as in this work) relays are considered Contributions The proposed scheme for the Two-way Two-relay Channel 1 may be seen as a generalization of the lattice based scheme of [] [] for the Two-way Relay Channel 1 However this generalization is not straightforward as the multiple relays need to repeatedly be able to decode the sum of codewords One may enable this by having the relays use lattice codewords as well something not required in the Two-way Relay Channel The scheme includes multiple Block Markov phases where the end users send new messages encoded by lattice codewords and the relays decode a combination of lattice codewords The relays then perform a Re-distribution Transform on the decoded lattice codeword combinations and broadcast the resulting lattice codewords The novelty of our scheme lies in this Redistribution Transform which enables both messages to fully utilize the relays power Furthermore all decoders are lattice decoders more computationally efficient than joint typicality decoders) and only a single nested lattice pair is needed II RELIMINARIES ON LATTICE CODES AND NOTATION Our notation for nested) lattice codes for transmission over AWGN channels follows that of [6] [11] An n-dimensional lattice Λ is a discrete subgroup of Euclidean space R n with Euclidean norm under vector addition We use bold x to denote column vectors x T to denote the transpose of x and 0 denote the all zeros vector All vectors lie in R n unless otherwise stated all logarithms are base and Nµ σ ) denotes a Gaussian random variable or vector) of mean µ and variance σ Further define or note that The nearest neighbor lattice quantizer of Λ as : Qx) = arg min λ Λ x λ ; The mod Λ operation as x mod Λ := x Qx); The Voronoi region of Λ as V := {x : Qx) = 0} which is of volume V := VolV) The second moment per dimension of a uniform distribution over V as σ Λ) := 1 V 1 n V x dx; For any s R n αs mod Λ)) mod Λ = αs) mod Λ α Z 1) βs mod Λ) = βs) mod βλ β R ) The definitions of Rogers and oltyrev good lattices are in [1]; we will not need these definitions explicitly Rather we will use the results derived from lattices with these properties /1/$ IEEE 11

2 01 IEEE International Symposium on Information Theory 1 Y 1 = X + Z 1 Y = X 1 + X + Z Y = X + X + Z Y = X + Z Fig 1 A Nested lattice codes The Gaussian Two-way Two-relay Channel Model Consider two lattices Λ and Λ c such that Λ Λ c with fundamental regions V V c of volumes V V c respectively; Λ Λ c ) is termed a nested lattice pair Here Λ is termed the coarse lattice which is a sublattice of Λ c the fine lattice and hence V V c When transmitting over the AWGN channel using the set C ΛcV = {Λ c VΛ)} as codebook the coding rate R is R = 1 n log C Λ cv = 1 n log V V c Nested lattice pairs satisfying certain properties were shown to be capacity achieving for the AWGN channel [5] In this work we only need one good nested lattice pair Λ Λ c in which Λ is both Rogers good and oltyrev good and Λ c is oltyrev good see definitions in [1]) The existence of such a pair may be guaranteed by [5]; and may be generated by Construction A [5] [] which maps the codebook of a linear block code over a finite field into real lattice points Then as described in [1] one may construct a one-to-one mapping 1:1) denoted by φ ) which maps an element in the finite field w F prime = {0 1 1} to a point in n-dimension real space t C ΛcV: t = φw) and w = φ 1 t) B Technical lemmas The following lemmas proven in [1] are needed in the proposed two-way lattice based scheme Let t ai and t bi C ΛcV be generated from w ai and w bi F prime as t ai = φw ai ) t bi = φw bi ) Furthermore let α α i β i Z such α α that i β i / Z and θ R We use and to denote modulo addition multiplication and subtraction over the finite field F prime Lemma 1 There exists a 1:1 mapping between v = i α iθt ai + i β iθt bi ) mod θλ and u = i α iw ai i β iw bi Lemma There exists a 1:1 mapping between α w and w Lemma If w ai and w bi are uniformly distributed over F prime then i α iθt ai + i β iθt bi ) mod θλ is uniformly distributed over {θλ c VθΛ)} III CHANNEL MODEL In the Gaussian Two-way Two-relay Channel Node 1 and exchange messages w a w b of respective rates R a R b through multiple full-duplex relays Node and ) and multiple hops as shown in Figure 1 Each node can only communicate with its neighboring nodes The channel model may be expressed as all bold symbols are n dimensional) Y 1 = X + Z 1 Y = X 1 + X + Z Y = X + X + Z Y = X + Z where Z i i {1 }) is an iid Gaussian noise vector with variance N i : Z i N0 N i I) and the input X i is subject to the transmit power constraint i : 1 n EXT i X i) i Standard definitions of achievable rate regions for the pairs R a R b ) are omitted due to space constraints; see [1] We first need the following tangential result which forms the basis for our Two-way Two-Relay Channel achievability scheme IV LATTICE CODES IN THE BC HASE OF THE TWO-WAY RELAY CHANNEL The work [] [] introduces a two-phase lattice scheme for the Gaussian Two-way Relay Channel 1 where nodes 1 and exchange information through node : the Multiple-access Channel MAC) phase and the Broadcast Channel BC) phase In the MAC phase if the codewords are from nested lattice codebooks the relay may decode the sum of the two codewords directly without decoding them individually This is sufficient as then in the BC phase the relay may broadcast the sum of the codewords to both users who may determine the other message using knowledge of their own transmitted message In the scheme of [] The relay re-encodes the decoded sum into a codeword from an iid random codebook in [] and a lattice codebook in [] In extending the schemes of [] [] to multiple relays we would want to use lattice codebooks in the BC phase as in [] This would for example allow the signal sent by Node to be aligned with Node s transmitted signal aligned is used to mean that the two codebooks are nested) in the Two-way Two-relay Channel: 1 and hence enable the decoding of the sum of codewords again at Node However the scheme of [] is only applicable to channels in which the SNR from the users to the relay are symmetric ie 1 = In this case the relay can simply broadcast the decoded and possibly scaled) sum of codewords sum without re-encoding it Thus before tackling the Two-way Two-relay channel we first devise a lattice-coding scheme for the BC phase in the Twoway Relay Channel with arbitrary uplink SNRs 1 In the Two-way Relay Channel [] Nodes 1 and exchange messages through the relay Node with channel model: Y 1 = X + Z 1 Y = X 1 + X + Z Y = X + Z where Z i i {1 }) is an iid Gaussian noise vector with variance N i : Z i N0 N i I) and the input X i is subject to the transmit power constraint i : 1 n EXT i X i) i Definitions of achievability are as in [1] We devise an achievability scheme which uses lattice codes in both the MAC phase and BC phase For simplicity to demonstrate the central idea of a lattice-based BC phase which is going to be used in the Two-way Two-relay Channel we do not use dithers nor MMSE scaling as in [5] [] [] We assume that 1 = p and = p where p R and N Z In the next section we incorporate arbitrary power Dithers and MMSE scaling allows one to go from achieving rates proportional to logsnr) to log1 + SNR) However we initially forgo the 1+ term for simplicity and so as not to clutter the main idea with additional dithers and MMSE scaling 11

3 01 IEEE International Symposium on Information Theory constraints by first truncating the powers to have the desired form; we show that even with this sub-optimal truncation constant gap-to-capacity results are possible We focus on the symmetric rate ie when the rates of the two messages are identical Codebook generation: Consider the messages w a w b F prime where = [ nrsym ] where R sym is the symmetric coding rate and [ ] denotes rounding to the nearest prime Nodes 1 and send the codewords X 1 = Npt a = Npφw a ) and X = pt b = pφw b ) where φ ) is defined in Section II-A with the nested lattices Λ Λ c Notice that their codebooks are scaled versions of the codebook C ΛcV The symmetric coding rate is then R sym := 1 n log V V c In the MAC phase the relay receives Y = X 1 + X + Z and decodes Npt a + pt b ) mod NpΛ with arbitrarily low probability of error as n with rate constraints 1 according to [1 Lemma ] We note that the first rate constant is redundant as 1 ; we are including it in the theorem statement for intuition so as to make it easier to understand the proofs of the main theorems in Section V In the BC phase if mimicking the steps of [] the relay simply broadcasts the scaled version of Npt a +pt b ) mod NpΛ: ) Np Npta + pt b) mod NpΛ) = t a + N t b mod [ )] 1 + [ Λ R a R b < min 1 1 we would achieve the rate log ] + [ )] for the 1 + [ direction and the rate log N ] + for the 1 1 direction While the rate constraint for the direction is as large as expected the rate constraint for the is achievable using lattice codes direction 1 does not fully utilize the power at the relay ie the codeword t b appears to use only the power /N rather than the full power One would thus want to somehow transform the decoded sum Npt a +pt b ) mod NpΛ such that in accordance with Theorem 6 both t a and t b of the transformed signal would be uniformly distributed over V roof: Λ) Notice that the relay can only nested lattice pair Λ Λ c operate on Npt a + pt b ) mod NpΛ rather than Npt a and pt b individually Re-distribution Transform: To alleviate this problem we propose the following Re-distribution Transform operation which consists of three steps: F prime 1) multiply the decoded signal by N to obtain NNpt a + pt b ) mod NpΛ) ) then perform mod Λ to obtain NNpt a + pt b ) mod NpΛ) mod Λ = pt a + Npt b ) mod NpΛ according to the operation rule in 1) and finally ) re-scale the signal to be of second moment as Np pt a + Npt b ) mod NpΛ) = N t a + t b ) mod in block i Node and send X i Λ according to ) Notice that N t a + t b ) mod Λ is uniformly distributed over { Λ c V Λ)} by Lemma The relay broadcasts X = N t a + t b ) mod Λ Notice that N t a + t b ) mod Λ is uniformly distributed over { Λ c V Λ)} and so its coding rate is R sym Node 1 and Node receive Y 1 = X + Z 1 and Y = X + Z respectively and according to [1 Lemma 5] may decode N t a + t b ) mod Λ at rate ] + ] + Nodes 1 and then map the decoded N R t a + R t b ) mod R Λ to Nw a w b by Lemma 1 With side information w a Node 1 may then determine w b ; likewise with side information w b Node obtains Nw a and w a by Lemma We note that for the single relay Two-way Relay Channel we achieve lower rates than those in [] [] We are describing it here to explain the intuition behind the scheme which will be used for multiple relays the goal of this work) Section V V TWO-WAY TWO-RELAY CHANNEL We first consider the full-duplex Two-way Two-relay channel where every node transmits and receives at the same time Theorem For the channel model described in Section III if 1 = p = M q = p and = q where p q R + and M N Z + the following rate region N )] ) + We again note that some terms are redundant but are included to allow for a simple easily generalizable expression Codebook generation: We consider the good with corresponding codebook C ΛcV = {Λ c V} and two messages w a w b F prime in which = [ nrsym ] R sym is the coding rate) The codewords associated with the messages w a and w b are t a = φw a ) and t b = φw b ) where the mapping φ ) from to C ΛcV R n is defined in Section II-A Encoding and decoding steps: We use a Block Markov Encoding/Decoding scheme where Node 1 and transmit a new message w ai and w bi respectively at the beginning of block i To satisfy the transmit power constraints Node 1 and send the scaled codewords X 1i = pt ai = pφw ai ) {pλ c VpΛ)} and X i = qt bi = qφw bi ) {qλ c VqΛ)} respectively and X i and Node j j {1 }) receives Y ji in block i The procedure of the first few blocks the initialization steps) are described and then a generalization is made We note that in general the coding rates R a for w a and R b for w b may be different as long as R sym = maxr a R b ) since we may always send dummy messages to make the two coding rates equal ) 11

4 01 IEEE International Symposium on Information Theory Block 1: Block : 1 = p = M q = p = q 1 pt a1 pt a1 qtb1 pt a Mqt a1 Nptb1 qt b pt a + Npt b1) mod NpΛ qt b + Mqt a1) mod MqΛ qt b1 Block : pt a Mqt a + NMqt b1) mod MqΛ Npt b + MNpt a1) mod NpΛ qt b w b1 pt a + Npt b + MNpt a1) mod NpΛ qt b + Mqt a + NMqt b1) mod MqΛ w a1 Block : pt Mqt a + NMqt a b+ NM qt a1) mod MqΛ Npt b + MNpt a+ M pt b1) mod NpΛ qt b Block 5: w b pt a + Npt b + MNpt a +M pt b1) mod NpΛ qt b + Mqt a + NMqt b+ NM qt a1) mod MqΛ w a pt a5 Mqt a + NMqt b + NM qt a + M qt b1) mod MqΛ Npt b + MNpt a + M pt b +M pt a1) mod NpΛ qt b5 w b pt a5 + Npt b + MNpt a + M pt b +M pt a1) mod NpΛ qt b5 + Mqt a + NMqt b + NM qt a + M qt b1) mod MqΛ w a Block i: pt ai Mqt ai 1) + NMqt bi ) + NM qt ai ) + + N i 1)/ M i 1)/ qt b1) mod MqΛ Npt bi 1) + MNpt ai ) + M pt bi ) + + M i 1)/ N i 1)/ pt a1) mod NpΛ qt bi w bi ) pt ai + Npt bi 1) + MNpt ai ) + +M i 1)/ N i 1)/ pt a1) mod NpΛ qt bi + Mqt ai 1) + NMqt bi ) + +N i 1)/ M i 1)/ qt b1) mod MqΛ w ai ) Fig Multi-phase Block Markov achievability strategy for Theorem Block 1: Codewords X 11 = pt a1 and X 1 = qt b1 sent from Nodes 1 and to Nodes and may be decoded if resp 1 ) 5) according to [1 Lemma 5] Block : Node 1 and send new codewords X 1 = pt a and X = qt b while Node and broadcast X = Mqt a1 and X = Npt b1 received in the last block Note they are scaled to fully utilize the transmit power = M q and = p Node receives Y = X 1 + X + Z and decodes pt a + Npt b1 ) mod NpΛ if ) and 6) Node decodes qt b + Mqt a1 ) mod MqΛ if 5) and 7) Block : Encoding: Node 1 and send new codewords as in the previous blocks Node further processes its decoded codewords combination according to the three steps of the Re-distribution Transform from previous block as Npt a + Npt b1 ) mod NpΛ)) mod NpΛ =Npt a + pt b1 ) mod NpΛ and scales this to utilize the full transmit power = M q as Mq Np Npt a + pt b1 ) mod NpΛ = Mqt a + NMqt b1 ) mod MqΛ It then broadcasts X = Mqt a + NMqt b1 ) mod MqΛ Notice that since Mqt a + NMqt b1 ) mod MqΛ {MqΛ c VMqΛ)} according to Lemma its coding rate is R sym Similarly Node broadcasts X = Npt b + MNpt a1 ) mod NpΛ again at coding rate R sym Decoding: At the end of this block Node is able to decode pt a + Npt b + MNpt a1 ) mod NpΛ with rate constraints ) and 6) according to [1 Lemma ] and Node decodes qt b + Mqt a + NMqt b1 ) mod MqΛ if 7) and 5) Node 1 decodes Mqt a +NMqt b1 ) mod MqΛ sent by Node as in the point-to-point channel with rate constraint 8) 115

5 01 IEEE International Symposium on Information Theory according to Lemma [1 Lemma 5] From the decoded Mqt a + NMqt b1 ) mod MqΛ it obtains w a Nw b1 Lemma 1) With its own information w a Node 1 can then obtain N w b1 = w a Nw b1 w a which may be mapped to w b1 since is a prime number Lemma ) Notice = [ nrsym ] as n so N = [ nr N ] and / Z Similarly Node can decode w a1 with rate constraint 9) Block and 5 proceed similarly as shown in Figure Block i: To generalize in Block i assume i is odd) Encoding: Node 1 and send new messages X 1i = pt ai and X i = qt bi resp Node and broadcast X i = Mqt ai 1) + NMqt bi ) + NM qt ai ) + M qt bi ) + + N i 1)/ M i 1)/ qt b1 ) X i = Npt bi 1) + MNpt ai ) + M pt bi ) + M pt ai ) + + M i 1)/ N i 1)/ pt a1 ) mod MqΛ mod NpΛ Decoding: Node 1 decodes the codeword from Node with rate constraint 8) [1 Lemma 5]) and maps it to w ai 1) Nw bi ) NMw ai ) Mw bi ) N i 1)/ M i 1)/ 1 w b1 Lemma 1) With its own messages w ai i) and the messages it decoded previously {w b1 w b w bi ) } Node 1 can obtain N w bi ) and determine w bi ) accordingly Lemma ) Similarly Node can decode w ai ) subject to rate constraint 9) Re-distribution Transform: In block i Node decodes pt ai + Npt bi 1) + MNpt ai ) + M pt bi ) + M pt ai ) + + M i 1)/ N i 1)/ pt a1 ) mod NpΛ from the received Y i = X 1i + X i + Z i subject to ) and 6) [1 Lemma ]) It then uses the Redistribution Transform to obtain Npt ai + Npt bi 1) + MNpt ai ) + + M i 1)/ N i 1)/ pt a1 mod NpΛ)) mod NpΛ = Npt ai + pt bi 1) + M pt ai ) + + M i 1)/ N i 1)/+1 pt a1 mod NpΛ and scales Mq it to utilize the full transmit power: Np Npt ai + pt bi 1) + M pt ai ) + + M i 1)/ N i 1)/+1 pt a1 mod NpΛ) = Mqt ai + NMqt bi 1) + NM qt ai ) + + N i 1)/ M i 1)/+1 qt a1 mod MqΛ This signal will be transmitted in the next block i+1 Node performs similar operations decoding qt bi + Mqt ai 1) + NMqt bi ) + + N i 1)/ M i 1)/ qt b1 mod MqΛ subject to constraints 7) and 5) and transforms it into Npt bi + MNpt ai 1) +M pt bi ) + +M i 1)/ N i 1)/+1 pt b1 mod MqΛ which is transmitted in the next block Combining all rate constraints we obtain [ 1 min 1 [ 1 N )] ) + For i even we have analogous steps with slightly different indices as may be extrapolated from the difference between Block and 5 in Fig Assuming there are I blocks in total the final achievable rate is I I R sym which as I approaches R sym We may achieve the same region for the permuted powers: Lemma 5 The rates of Theorem may also be achieved when 1 = p = p and/or = q = M q roof: The proof is shown in [1] Theorem and Lemma 5 both hold for powers for which 1 / and/or / are either the squares of integers or the reciprocal of the squares of integers However these scenarios do not cover general power constraints with arbitrary ratios We next present an achievable rate region for arbitrary powers: Theorem 6 For the Two-way Two-relay Channel with arbitrary transmit power constraints any rates satisfying [ 1 [ R a R b < max min i i 1 = or 1 =M or M N for some N M Z + and i {1 } are achievable This rate region for R a = R b is within 1 log bit/hz/s per user from the symmetric rate capacity roof: The proof is shown in [1] The half-duplex case and extensions to more than two relays are also discussed in [1] REFERENCES [1] Y Song and N Devroye Lattice codes for the gaussian relay channel: Decode-and-forward and compress-and-forward abs/ [] B Nazer and M Gastpar Compute-and-forward: Harnessing interference through structured codes IEEE Trans Inf Theory vol 57 no 10 pp [] W Nam S Y Chung and Y Lee Capacity of the Gaussian Two-Way Relay Channel to Within 1/ Bit IEEE Trans Inf Theory vol 56 no 11 pp Nov 010 [] M Wilson K Narayanan H fister and A Sprintson Joint physical layer coding and network coding for bi-directional relaying IEEE Trans Inf Theory vol 56 no 11 pp Nov 010 [5] U Erez and R Zamir Achieving 1 log1 + SNR) on the AWGN channel with lattice encoding and decoding IEEE Trans Inf Theory vol 50 no 10 pp 9 1 Oct 00 [6] R Zamir S Shamai and U Erez Nested Linear/Lattice codes for structured multiterminal binning IEEE Trans Inf Theory vol 8 no 6 pp [7] T M Cover and A El Gamal Capacity theorems for the relay channel IEEE Trans Inf Theory vol 5 no 5 pp Sep 1979 [8] opovski and H Yomo Bi-directional amplification of throughput in a wireless multi-hop network in roc IEEE Veh Technol Conf - Spring Melbourne May 006 pp [9] S Kim N Devroye and V Tarokh A class of bi-directional multi-relay protocols in roc IEEE Int Symp Inf Theory Seoul Jun 009 pp 9 5 [10] J ooniah and L-L Xie An achievable rate region for the two-way two-relay channel in roc IEEE Int Symp Inf Theory Jul 008 pp 89 9 [11] W Nam S-Y Chung and Y Lee Nested lattice codes for gaussian relay networks with interference IEEE Trans Inf Theory vol 57 no 1 pp Dec 011 [1] Y Song N Devroye H Shao and N Chiu Lattice coding for the two-way two-relay channel ) 116