Rateless Codes for the Gaussian Multiple Access Channel

Transcription

1 Rateess Codes for the Gaussian Mutipe Access Channe Urs Niesen Emai: Uri Erez Dept EE, Te Aviv University Te Aviv, Israe Emai: Devavrat Shah Emai: Gregory W Worne Emai: gww@mitedu Abstract We consider communication over the Gaussian mutipe access channe MAC with unnown set of active users The proposed mutipe access strategy is distributed and achieves a maximum sum rate point on the boundary of the capacity region for this channe for any set of active users S simutaneousy, as if S were nown at the transmitters The proposed coding scheme spits each user into a set of virtua users, each of which can be decoded using a singe-user decoder at the receiver instead of having to decode a users jointy We aso present a generaization of this scheme to the case where the channe gains differ between users and each user ony nows its own channe gain I INTRODUCTION AND MOTIVATION Consider a set of users, each having a queue of messages that it wants to transmit through a wireess channe to some centra base station This base station is aowed to occasionay broadcast a short beacon signa to eep the users synchronized Whenever a user detects the beacon signa, it starts transmitting a codeword corresponding to the first message in its queue, if it is not empty As soon as the base station is abe to decode a the messages currenty being transmitted, it emits again a beacon, indicating that the current round of communication is over and starting the next one Note that this setup is rather decentraized and there are various degrees of uncertainty the users face First, whie each user nows if it currenty has a message to transmit, it nows neither which nor how many other users are transmitting at the same time Moreover, the channe gains can differ across users Assuming the gain of the upin and downin channe are the same, as woud be the case with, eg, time division dupex, each user can infer the gain of its in from the strength of the beacon signa it received The users are, however, ignorant of the channe gains of the other users As the wireess medium is shared among a the users, these issues resut in an uncertainty about the amount of interference each user wi experience during any communication round We propose a mutipe access strategy for this setup using successive decoding of the different users That is, the decoder at the base station can use severa singe-user decoders in succession and does not need to decode a transmitting users jointy At the same time, this scheme is optima in the sense that it achieves the maximum possibe sum rate for the case This wor was supported in part by NSF under Grant No CCF-055, Draper aboratory, and Mitre Corporation where the set of transmitting users as we as a the in quaities were nown to a users That is, neither the ac of nowedge about the in quaities or number and identity of interfering other users nor the requirement of using successive decoding at the base station resut in a oss of achievabe rate Throughout most of this paper, we assume that the singeuser codes used in the successive decoding scheme above are rateess capacity achieving codes This simpifies the anaysis and aows to isoate the impact of successive decoding on the performance of the system We wi, however, briefy discuss how practica rateess singe-user codes can be used within this architecture Rate-spitting is a technique for the mutipe access channe MAC which is designed for use in conjunction with successive decoding [] In [] both the number as we as the identity of the users is fixed and nown to each user The case where the identity of the users is not nown, but their number is fixed and nown, has been investigated in [] and more recenty in [3] In there it is assumed that the channe gain is identica for each user As the number of transmitting users is nown, this impies that every user nows the amount of interference it wi experience This differs from the setup considered here, in which the unnown number as we as the unnown channe gains of interfering users resut in uncertainty about the amount of interference Queueing and scheduing aspects in a setup simiar to the one described in this paper are anayzed in [4] and [5] respectivey The remainder of this paper is organized as foows In Section II, the probem setup is formaized, and some notation is introduced Section III describes the proposed mutipe access scheme and estabishes the main resut of this paper Section IV contains concuding remars II PROBEM FORMUATION The scenario of interest can be modeed as communication over a Gaussian MAC We first consider the case where a the channe gains are equa to one As wi be shown in Section III- A, the genera case can be reduced to this setup There are K tota users out of which a subset S is active That is, ony the users in the set S are using the channe At time t assumed to be discrete here the received signa is Y t X s,t + Z t, X/06/$ IEEE This fu text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for pubication in the IEEE GOBECOM 006 proceedings

2 where {Z t } t is a sequence of iid Gaussian random variabes with zero mean and unit variance, independent of the channe inputs The set S is nown to the receiver; however, the transmitters now ony whether they are in S or not ie, transmitter s ony nows if s S, but do not now the set S in genera A code for this channe defines for every possibe user s an encoding function or codeboo f s : {,,M} R mapping the message m s of user s into an infinite sequence of channe symbos f s m s satisfying an average power constraint P The index m s of the codeword to be sent by transmitter s is chosen with uniform probabiity over the set {,,M} The receiver consists of K decoders, each of them for a specific set of active users S Each of these decoders is specified by a deterministic decoding time T S and a decoding function φ S : R TS {,,M} S mapping the sequence {Y t } TS into an estimate { ˆm s } φ S {Y t } TS t t of the messages sent by each active user The tripe {fs } K s, {T S } S, {φ S } S specifies a coding scheme The average probabiity of error ē of a coding scheme is defined as ē max ēs, S [ ] ēs Pr { ˆm s m s } S, and the rate of communication if S is the set of active users is R S og M T S Every coding scheme resuts in a K tupe of rates {R S } S A rate tupe {R S } S wi be caed achievabe if there exists a sequence of coding schemes indexed by M with rate converging to {R S } S and such that im M ē 0 From a purey information theoretic point of view, the communication probem described above can be soved using standard resuts Note first that the optima input distribution for each active user s is the same for a sets S namey, zero mean Gaussian with variance P As, moreover, the number of different sets S is finite, there exists a code and a joint decoder achieving any rate R S <C S S og + S P simutaneousy for a S as if S were nown at the transmitters In other words, there exists a coding scheme operating for every S at the equa rate point on the boundary of the capacity region of the ordinary Gaussian MAC with nown S at the transmitters as defined for exampe in [6] The coding scheme described in the ast paragraph has, however, the disadvantage of requiring a joint decoder That is, in the worst case K codes have to be decoded jointy, resuting in an unacceptaby arge decoding compexity In the next section, we wi describe a ower compexity coding scheme, buiding on the rate-spitting approach [] and its generaization in [], which aows the use of mutipe singeuser decoders at the receiver instead of a joint decoder III RATEESS CODES In this section, we construct a rateess code which uses severa singe-user decoders at the receiver instead of one joint decoder We require this coding scheme to achieve rates arbitrariy cose to C S simutaneousy for a sets S of active users In other words, we want the decoding times T S to satisfy T S og M C S ε for an arbitrary ε>0 and M arge enough Observe from and that the requirement on T S depends on S ony through S We spit each user s into virtua users For each s, virtua user uses a codeboo with M codewords We require that M M, ie, that the tota number of messages for each user s is M The codewords of virtua user of user s are created as infinite ength sequences of independent Gaussian random variabes with mean zero and variance P t for the t-th symbo in the sequence, subject to the constraint that P t P 3 for a times t A power aocation {P t},t wi be caed vaid if P t 0 for a t, and 3 is satisfied for a t The codewords of a virtua users of each user s are added to form the codeword of that user At time T S, the decoder φ S first decodes ayer of virtua users for a s S, regarding a other virtua users of the same and a other users in S as noise The decoded codewords of the virtua users in ayer are then subtracted from the received signa, and the decoder continues in the same manner with ayer of virtua users unti a virtua users have been decoded Hence the decoder φ S consists of S singe-user decoders, which are used successivey Note that for finite this decoding procedure is suboptima, as it regards a codewords of virtua users in the same ayer as noise, whereas some of them coud have been subtracted off the received signa after decoding It does aow, however, to choose the encoders {f s } K s to be identica for each user s, which is necessary due to the distributed nature of the probem Moreover, we show in the seque that this scheme can approach optimaity as the number of virtua users goes to infinity We first consider power aocations that are constant across time ie, P t P for a virtua users and times t emma shows that as M and apower aocation {P } can be found such that for every possibe set of users S there exists a decoding time T S and a spitting of messages {M } with the foowing two properties: A messages can be reiaby decoded at time T S, and the rate R S is cose to C S in the sense of Note that the choice of {M } is aowed to depend on the set of active users S As X/06/$ IEEE This 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3 in an actua system {M } woud have to be chosen before communication begins and without nowing S, the power aocation given by emma can not be used to guarantee optima communication for a possibe sets S simutaneousy This probem is addressed in Theorem It states that as M and a time varying power aocation {P t},t and a spitting of messages {M } can be found such that for every possibe set of users S there exists a decoding time T S with the foowing two properties: A messages can be reiaby decoded at time T S, and the rate R S is cose to C S in the sense of The first emma generaizes and strengthens a resut from []; part of the proof foows aong the ines of the one there emma For a vaid {P } and S there exists T S and {M } such that im M ēs 0and im im R S C S, M if and ony if im P 0for a N Moreover, for any finite C S im R S P M 4 sup P S + P Proof Ca R S the rate of the above scheme as M and Using genie aided decoder arguments as in [], the highest achievabe rate with successive decoding as described above is im im R S im M og + x, where P x + S P + S i P i Consider og + x + x x + For the second term, we get + x x 4 P 4 sup P P P 0 P P P P P, 5 which converges to zero if im P 0for a N For the first term in 4 we have P x + S i P i and hence if im P 0for a N im im R S im M im P x P + S i P i dy 6 y0 + S y og + S P S C S For finite, the approximation error in 6 is upper bounded by 4 P max d y [0,P ] dy + S y P S 4 4 P S 4 P max y [0,P ] sup P Together with 5 this impies that C S im M R S P S 4 S + S y sup P + P 4 sup P 4 sup P S + P P P Conversey, if there exists some such that P δ>0 then it is easiy seen that just by decoding this ayer suboptimay, we wi aways get a rate R S stricty beow C S for a As the codewords for each virtua user are a sequence of independent random variabes and as the channe is memoryess, the direct part of emma can be seen to appy aso to the case where the power aocation is time varying We use this ind of power aocation in the foowing More precisey, we choose powers P t constant on each time interva {,T }, {T +,T },,{T K +,T K } where T T S for any S such that S We denote the power for virtua user in time interva by P T The spitting of the messages is done uniformy, ie, we set M M / for a For the first time interva, choose P T such that og + P T + i P it R X/06/$ IEEE This fu text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for pubication in the IEEE GOBECOM 006 proceedings

4 for a and some constant R For subsequent time intervas, we aocate powers P T such that j T j P T j og + T + P T j + i P it j R 7 for a, constants R and with T j T j T j, T 0 0 From emma, it is cear that if a vaid power aocation of this type exists such that a powers P T j converge to zero as, then we can, for any, maer arbitrariy cose to C S for a S such that S by choosing M and arge enough The next theorem estabishes that a power aocation of this type exists and hence that the above scheme is asymptoticay in M and optima for a sets of active users S simutaneousy Even though the theorem ony caims existence of such a power aocation, its proof is constructive in the sense that it gives an agorithm to find such an aocation Theorem For M M / there exists a vaid {P T }, and {T S } S such that im M ē 0and for a S im im R S C S M Proof Identify R in 7 as R S for any S such that S Using emma, we then ony have to show that a vaid power aocation satisfying 7 exists and that im P T 0 for a and Note first that im R / 0for any power aocation satisfying 7 That is, the rate in each ayer of virtua users goes to zero Assume then that there exists a δ>0 such that for a we have P T δ for at east one {,,} and some {,,K} Then we get from 7 R T P T og + T + P T + i P it which is bounded away from zero Hence there exists some sequence such that im R / > 0, contradicting the fact that a ayers have the same rate Thus if a vaid power aocation satisfying 7 exists it must satisfy im P T 0for a N and {,,K} We wi show by induction that such a vaid power aocation exists Ceary, we can find a power aocation such that 7 is satisfied for with T ogm/c {s} ε for some ε>0 Assume then we have fixed decoding times {T j } j and that the resut hods up to that point That is, we have chosen vaid powers P T j for j {,, } such that for each ayer 7issatisfiedforsomeR Define δ T R + j T j og P T j + P T j + i P it j By the induction hypothesis δ 0 for a {,,} Using the fact that T R T R, we get from 7 that we have to find a power aocation such that T og P T + + P T + i P δ it Soving for P T, we find exp δ P T / T exp δ / T [ ] + P i T 8 Note that exp δ / T 0 Consider now P T as a function of T as defined through 8 et B be the greatest vaue of T such that exp δ / T 0, hods for a {,,} with equaity for at east one It is easiy checed that B>0 P T is a continuous function of T over B, with im T B P T and im T P T 0 Hence there exists T i such that P T P Moreover, for T B,, we have P T 0 for a {,,} Hence choosing T T resuts in a vaid power aocation for time sot Even though T is in genera not an integer, we can mae the rounding error as sma as desired by choosing M arge enough This concudes the induction step Figure shows the fraction of capacity C S achievabe with this scheme as a function of number of ayers used As expected the number of ayers needed to achieve a fixed fraction of capacity increases with the number of active users S It can, however, be observed from the same figure that, even for S as arge as 0, ony a moderate number of ayers is needed to operate at a rate cose to capacity For exampe, for S 0and 0 ayers, more than 90% of the sum capacity can be achieved Note aso that the rate oss incurred for a fixed finite number of ayers depends ony on the number of active users S, and not on the number of potentia active users K In other words, the system can be designed very conservativey assuming a arge number of potentia users K without having to pay a penaty at east in terms of achievabe rate A Arbitrary Channe Gains In this section, we remove the restriction that a channe gains are equa to one That is, each user has now an associated constant channe gain As each user nows its own channe gain, this is equivaent to imposing a possiby different average power constraint Ps for each user s S We assume that for a s Swe have P s s P for some s N Spit each user s into s virtua users This resuts in a set S of active virtua users each with identica power constraint P We can now use the coding scheme described in the ast section to X/06/$ IEEE This fu text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for pubication in the IEEE GOBECOM 006 proceedings

5 RS/CS S S 5 S 0 S 0 Fig Fraction of capacity C S achievabe as a function of number of ayers virtua users for different numbers of active users S The nomina SNR ie, without interference is 0 db achieve a rate R S arbitrariy cose to C S Hence each user s S can transmit at a rate arbitrariy cose to s C S s S og + S P We have s C S og + S P og + P s, and hence the above scheme achieves a maximum sum rate point on the boundary of the achievabe rate region of the Gaussian MAC Note that with this procedure not a users are communicating at the same rate As each user is spit into a number of virtua users proportiona to its average power constraint, this impies that the reative rate a user can communicate at is aso proportiona to its average power constraint For exampe, when two users are active both with equa average power constraint P they are both abe to communicate at the same rate, say R When two users are active, but this time the first user has an average power constraint of P and the second user has a constraint of P, the first user is abe to communicate at some rate R whereas the second user achieves a rate R Note that R R in genera in this situation There is aso a tradeoff between the accuracy P with which different power eves can be approximated and the number of virtua users s we need to spit each user into If P is chosen very sma, s wi be very arge for many users s, resuting in a higher decoding compexity As, however, in any practica system the actua channe gains or, equivaenty, the power constraints in our setup can ony be estimated within a certain precision, a nonzero P can aways be chosen such that the approximation error is negigibe compared to the estimation error B Practica Rateess Codes Up to this point, we have assumed that every virtua user uses a capacity achieving singe-user code More precisey, the codewords of virtua user of user s are created as infinite ength sequences of independent Gaussian random variabes with mean zero and variance P t for the t-th symbo in the sequence In a rea system, we have to repace this with a practica rateess code satisfying a time varying power constraint The design of such a code is simpified by the fact that the power aocated to each virtua user vanishes as the number of these users grows This maes it possibe to use simpe practica codes and efficient decoding techniques described in [7] These codes are constructed from a singe good base code of a fixed bocength say n for the standard additive white Gaussian noise channe with constant power constraint The codeword corresponding to message m in the rateess code is constructed by repeating a scaed and dithered version of the codeword corresponding to message m in the base code Decoding is performed by combining bocs of ength n of the received sequence into a singe vector of ength n, which is then decoded using the decoder for the base code For a detaied description of this construction and an anaysis of its performance, see [7] IV CONCUSION We have described a communication scheme for the Gaussian MAC, achieving a maximum sum rate point on the boundary of the capacity region for this channe for any set of active users S simutaneousy, even when S is unnown at the transmitters The proposed coding scheme spits each user into a set of virtua users, each of which can be decoded using a singe-user decoder at the receiver instead of having to decode a users jointy The presented soution aso generaizes easiy to the case where the channe gains differ between users and each user ony nows its own channe gain REFERENCES [] B Rimodi and R Urbane A rate spitting approach to the Gaussian mutipe-access channe IEEE Transactions on Information Theory, 4: , March 996 [] R S Cheng Stripping CDMA an asymptoticay optima coding scheme for -out-of-k white Gaussian channes IEEE GOBECOM, pages 4 46, November 996 [3] J Cao and E Yeh Distributed rate spitting in Gaussian and discrete memoryess mutipe-access channes IEEE ISIT, pages 6 66, September 005 [4] E Teatar and R G Gaager Combining queueing theory with information theory for mutiaccess IEEE Journa on Seected Areas in Communications, 36: , August 995 [5] S Raj, E Teatar, and D Tse Job scheduing and mutipe access DIMACS Series in Discrete Mathematics and Theoretica Computer Science, 003 [6] T M Cover and J A Thomas Eements of Information Theory Wiey, 99 [7] U Erez, G W Worne, and M D Trott Faster than Nyquist coding: The merits of a regime change Aerton Conference, pages , October X/06/$ IEEE This fu text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for pubication in the IEEE GOBECOM 006 proceedings