On the Sum Capacity of Multiaccess Block-Fading Channels with Individual Side Information

Transcription

1 On te Sum Capacity of Multiaccess Block-Fading Cannels wit Individual Side Information Yas Despande, Sibi Raj B Pillai, Bikas K Dey Department of Electrical Engineering Indian Institute of Tecnology, Bombay. {ykdespande,bsraj,bikas}@ee.iitb.ac.in Abstract We consider te problem of finding optimal, fair and distributed power-rate strategies to acieve te sum capacity of te Gaussian multiple-access block-fading cannel. Te transmitters ave access to only teir own fading coefficients, wile te receiver as access to all of te fading coefficients. We propose a distributed strategy called te midpoint strategy wic is optimal wen te system cannot tolerate outage. In addition, we demonstrate a successive decoding sceme tat can acieve tis maximal sum-rate. In presence of outage, we sow tat te strategies based on a single tresold are suboptimal. I. ITRODUCTIO Te multiple-access cannel is a widely used model to understand te fundamental limits on information transmission in a many-to-one communication scenario, suc as te uplink cannel of a cellular network. In te wireless regime, cannel fading due to multipat, sadowing and inerent cannel variability introduces interesting callenges in reliable communication. It is important to know weter te receiver (and transmitters ave access to measurements on te fading conditions, te delay and accuracy tereof. Trougout tis paper, we concentrate on te specific case of individual cannel state information (CSI at te transmitter, viz. eac transmitter as instantaneous access to its own fading state causally, but tat of no oter. Te receiver as complete cannel state information. More specifically, we consider te block-fading case: te fading coefficients are constant over a block of cannel uses, over wic te codeword lasts. Te transmitters, tus, are not allowed to take advantage of te ergodic nature of te fading process during coding, but may employ adaptive power and rates. Tis particular situation is motivated by systems involving occasional (opportunistic access to a sared medium, suc as in a cognitive radio or a sensor network wit a star topology. Here, multiple users wis to communicate teir data to te receiver over te awarded time slot in a fair but distributed fasion. If te slot duration is not fixed, as described in [], te receiver may employ a beacon signal for syncronizing te rounds of communication. Tere is considerable literature on multiaccess fading cannels wit instantaneous CSI. Te Sannon capacity of a Gaussian MAC wit CSI available only at te receiver is evaluated rigorously in [2]. Te optimal power control strategies to acieve capacity for te case of complete cannel state information at te transmitters (CSIT are given in [3] and [4]. Coming to partial side information at te transmitters, [5] gives te capacity region of a fading MAC under very general notions of CSI at te transmitters. Tese notions can be specialized to nearly all practical scenarios including individual transmitter CSI. However, our work differs from [5] due to te block-fading assumption. Te ergodic averaging inerently used in evaluating te Sannon capacity region in [5] turns out to be essential because of te absence of complete CSI. Alternate notions of capacity motivated by different practical scenarios ave also been investigated: delaylimited capacity for te fading MAC is dealt wit in [6] wile [7] defines te notions of expected capacity and capacity wit outage for information unstable single-user cannels. Our main results are summarized as follows: We introduce a fair, simple and distributed policy called te midpoint strategy for te Gaussian multiple-access block-fading cannel. Te midpoint strategy is sum trougput-optimal for symmetrical users wen outage cannot be tolerated. We also propose a low-complexity rate-splitting sceme tat allows te midpoint strategy trougput to be acieved troug successive decoding. Wen outage can be tolerated, we propose tresoldbased policies wic narrowly out-perform te midpoint strategy. We furter sow tat scemes based on a fixed tresold are suboptimal compared to variable ones. II. SYSTEM MODEL Consider M users communicating wit a single receiver. Tese users transmit real-valued signals X i, encountering realvalued fades H i. If Y is te value of te received signal at a (discrete time instant we ave Y = M H i X i + Z i were Z is an independent Gaussian noise process. Te fading space H i of te i-t user is te set of values taken by H i, and te joint fading space H is te set of values taken by te joint fading state H = (H, H 2,, H M. Similar vector quantities of user-wise parameters, like rate, power, cannel state realization, will be denoted wit a overbar symbol. We assume tat te (stationary and ergodic fading processes H i are independent, and teir distributions are known to all te transmitters and te receiver. In addition, we ave individual

2 CSIT, i.e. eac transmitter knows its own cannel fading coefficient H i but tat of no oter. Te receiver knows all te fading coefficients. Te transmitters ave individual average power constraints P avg i, and ave te freedom to adapt teir rate (and power according to teir own cannel conditions. Tis leads to te following notion of a power-rate strategy. Definition. A power-rate strategy is a collection of mappings (, R i : H i R + R + ; i =, 2,, M. Tus, in te fading state H i, te i t user expends power (H i and employs a codebook of rate R i (H i. Tis definition is reminiscent of power strategies in [4], but tere are two key differences. Firstly, we incorporate individual transmitter CSI in te definition. Secondly, te rate is allowed to be adaptive due to te block-fading restriction. Considered block-wise te cannel is a fixed-gain Gaussian multiaccess cannel (MAC. Consequently we assume tat te standard random Gaussian codebooks wit ML decoding are employed to acieve capacity tereof. Let C MAC (, P ( denote te capacity region of a Gaussian MAC wit fixed cannel gains =,, M and power allocations P ( = (P (,, P M ( M. We know tat, { C MAC (, P ( = R : S {, 2,, M} ( R i 2 log + } i 2 ( i Definition 2. We call a power-rate strategy as feasible if it satisfies te average power constraints for eac user i.e. i {, 2,, M}, E Hi (H i P avg i. Definition 3. A power-rate strategy is termed as outage-free if it never results in outage i.e. H, (R (,, R M ( M C MAC (, P ( Te trougput acieved by a given power-rate strategy is, ten: M R sum = E H R i (H i I {(R(H,,R M (H M C MAC ( H, P ( H} were I A is te indicator function for te condition A. Definition 4. Te sum capacity is te maximum (average trougput acievable, i.e. C sum = max R sum were te maximum is taken over all feasible power-rate strategies. III. OPTIMAL STRATEGIES WITHOUT OUTAGE Consider a situation werein it is required tat te system never suffer outage. Tis would be of importance wen te practical system under consideration involves occasional communication during arbitrarily allocated time slots, wic are small in comparison wit te cannel coerence time. Coordination being difficult in suc a setup, te callenge is to provide optimal, fair and distributed strategies for te system. We describe a distributed power-rate strategy called ( (2 R Fig.. Te users and 2 construct te innermost and outermost MAC capacity regions respectively. Te intermediate pentagon is te instantiated MAC. region and A denotes te operating point te midpoint strategy and a simple decoding sceme tereof. Prior to tat, let us consider te simple strategy of time saring among users, or plain time-division multiple-access (TDMA. A. Plain TDMA In plain TDMA te transmitters employ a simple taking turns policy. Eac block is divided into sub-blocks wit only one user transmitting in tat sub-block. Tis requires some extra coordination suc as agreeing on an ordering for te users. Te cannels for te users are now ortogonal and tey may water-fill over teir own sub-blocks to improve trougput. Tus, we obtain te power-rate strategy corresponding to plain TDMA as: ( ( i = + λ i i 2 R i ( i = 2M log ( + M i 2 were λ i is cosen suc tat E Hi (H i = P avg i. Te actual power employed by te user in its sub-block is M (H i and te full transmission rate supported tereby is cosen. B. Te Midpoint Rate Strategy For simplicity, assume tat te users ave different fixed powers P, P 2,, P M for te given round of communication (for instance, after deciding on an arbitrary feasible power strategy. Eac user assumes tat all oters are identical to itself and constructs te symmetrical MAC region based on tis assumption. It ten cooses te maximal equal-rates point for operation. Tus we ave A R 2 R mid i ( i = 2M log ( + M i 2. (3 Lemma 5. Te midpoint rate strategy is outage free, i.e. Rmid i C MAC (, P.

3 5 next section tat tis cost can be ameliorated troug ratesplitting and successive decoding. Rate Power mid-pt outer Fig. 2. Te midpoint strategy is only a constant off te full CSI bound [3] Proof: Te lemma follows directly from te concavity of te logaritm function, i.e. S {, 2,, M} : Ri mid ( i = 2M log ( + M i 2 M S + 2M log ( 2 log + i 2 As P,, P M are arbitrary, te users power strategies are now completely independent of eac oter. Te best power strategy for eac user would tus be to water-fill over its own cannel and we obtain ( ( i = + λ i i 2 (4 were λ i is cosen suc tat E Hi (H i = P avg i. Wen te users ave te same average power and identical fading distributions, we call tem a symmetric user set. We furter define a symmetric mid-point strategy, in wic eac user in a symmetric user set employs te same power allocation sceme and cooses te corresponding symmetric mid-point rate. For a symmetric user set, we ave te following result. Teorem 6. For a symmetric user set and any given outagefree strategy, tere is a symmetric midpoint strategy wic acieves at least as muc trougput as te given strategy. Proof: See appendix. ote ere tat te trougput acieved by te midpoint strategy is identical to tat acieved by plain TDMA. We compare tis in Figure 2 wit te opportunistic TDMA possible wit complete CSIT [3]. Te advantage of plain TDMA is its simplicity in decoding, since only M single-user decoders are needed. However, te price for tis is paid in te extra coordination required to set up an ordering for transmission between users. Te midpoint strategy avoids tis coordination, albeit at te cost of incurring joint decoding. We sow in te C. Rate Splitting We present an asymptotically optimal rate-splitting strategy tat replaces te joint decoder wit LM successive single-user decoders, were L is a parameter. Tis section is motivated by te work in [] and teir tecnique is useful in sowing te acievability. However, [] considers a rateless sceme wit variable coding block-lengts between rounds of communication. Te lengt of eac round is determined by a feedback beacon link from te receiver, block or time slot, and tere is no assumption of suc a feedback link. By a sligt abuse of te notation, we denote te received signal power for user i, i.e. i 2 as simply, trougout tis section. Assume tat te users ave different (received powers P, P 2... P M. For simplicity, we will assume tat te additive noise is of unit variance. Te values of may cange wit eac block of communication depending on te individual fading conditions. Eac user is unaware of te fade values and transmit powers of te rest of te users and, consequently, te interference tey may cause. Te encoding and decoding are done tus: eac user splits itself into L virtual users and splits its power, peraps unequally, among tese users. Eac user is to be visualized as a stack of virtual users. For decoding, we use a successive cancellation based single-user decoder, wic decodes one of te virtual users assuming all oter virtual users as yet undecoded as Gaussian noise, see [8] for te details. More specifically, transmitter i, aving power, splits its data stream in to L virtual users. Tis is done by allotting a power/rate pair (Pl i, ri l to te lt virtual user, suc tat l P l i =. Te transmitter i assumes tat all oter users are also at (received power and imagines identical power splitting strategies across all users. It ten cooses te rates rl i by considering all te oter virtual users in te same and lower layers as interference, i.e., r i l = 2 log ( + l + (M l + M l j. (5 However, in te actual setting, te interference encountered from te oter users are substantially different from tat accounted for in te denominator of (5 and a layer by layer decoding may fail. Surprisingly, it turns out tat tis can be compensated by not strictly adering to a layer by layer decoding. In particular, te receiver retains te freedom to decode te topmost iterto undecoded layer of any transmitter, irrespective of te number of layers wic were already decoded. We now sow tat tis is sufficient for complete decoding. Lemma 7. Assuming layer-wise rate allocation as per (5, it is always possible to find a virtual user wic can be decoded correctly, i.e. wit arbitrarily small error probability.

4 Proof: By induction: assume tat layers (virtual users above l k ave been decoded for te k t transmitter. Coose: k = arg max k l k Pj k Te actual interference for tis virtual user is given by: + j l k Pj k k k + = + M l k k= + M = + P k j j j + (M j Te inequality follows directly from te coice of k. Te RHS is te expected interference for te t virtual user of te k t transmitter. Tus, as te actual interference is less tan te expected interference, tis virtual user can be correctly decoded. In oter words, te user wit te best received SR can always be cosen for decoding. Teorem 8. As L and j, l, l 0, te rate acieved by all te users equals teir midpoint rate. Proof: Using 5, we ave ( L R i = 2 log Pl i + + (M Pl i + M l P j i Under te given conditions, we can use te same metod as in Lemma of [] to sow tat: lim R i = lim L L = 2 L Pi 0 P j l + (M l + M l j dy + My = 2M log ( + M Computational results in [] also sow tat only a nominal number of virtual users L suffice to yield good performance. IV. STRATEGIES WITH OUTAGE Tus far, we ave seen tat plain TDMA (or, equivalently, midpoint is trougput-optimal witout outage. However, a simple example demonstrates tat tis is not so wen we allow outage. By sacrificing on some blocks (or rounds of communication we may improve te overall trougput. Consider 2 symmetrical users transmit over a fading cannel wit two states: H (or ig and L (or low. Te fading coefficients are iid Bernoulli random variables for bot te users, wit P r(h = δ. Suppose te users do not employ power control. If δ is small enoug, any user wo as access to a fading level of H, sould not expect te oter user to be also at H and in turn try a pessimistic mid-point strategy. On te contrary, te better user sould expect te oter one to ave a value L, wic is more likely, and coose a rate of R (H, L = 2 log( + (H2 + L 2 P 4 log( + 2L2 P, were we adered to te mid-point strategy for te fading value L. Certainly, te (H, H fading states will result in outage, and te resulting trougput is (6 ( δ 2 log( + 2L2 P + 2( δδr (H, L, (7 wic can be greater tan tat of te outage-free strategies at low values of δ. Motivated by tis, we move to te general case werein te system can tolerate outage. We make some simplifying assumptions on te outage scenario. We consider only 2 completely symmetrical users (i.e. tey ave equal power constraints and fading marginals. Te fading distributions are assumed to be iid Rayleig. In addition, we ignore power control: te power is fixed to be P ( = P. In tis section, we detail two policies wic outperform te midpoint strategy in terms of long-term trougput. It is sown tat simple singletresold policies are strictly suboptimal. A. A Single-Tresold Strategy R ( C 2 t P ( C 2 P R ( 2, t A C( 2 2 P / Fig. 3. Here, C(x log( + x. User 2, wo is beyond te tresold, 2 constructs te outer two pentagons assuming to be at most as good as te intermediate region. User, wo is witin tresold constructs te innermost region, coosing its midpoint rate. A denotes te final operating point. Wit iid Rayleig fading, we can modify te above strategy to get good opportunistic access as follows: { ( R( = 4 log P for t (8 R (, t oterwise, were R (, is defined in (6. Here, beyond te tresold t, te transmitter assumes tat te oter transmitter is at most as good as t and operates on te boundary of te MAC constructed tereof (see Figure 3. Tus, te only time wen outage occurs is wen bot te transmitters are beyond t. Since muc of te probability mass is concentrated towards te bad cannel gains, te midpoint rates are retained in tat region, wile wit good cannels te transmitter takes a risk. R 2

5 Te trougput acieved by suc a strategy can be computed according to (2. Tey maximal trougput by using te best tresold is plotted in Figure 5, wic for te scales of our plot is indistinguisable from tat of te midpoint rate strategy, suggesting te utility of te midpoint rate sceme. B. A Single Tresold is Insufficient Trougput P = P = Tresold Fig (log scale Optimum variable tresold H(. On closer scrutiny, one can strictly improve te singletresold strategy. In an improved sceme, te transmitter is pessimistic wen below te tresold t, similar to te previous sceme. However, wen it experiences a better cannel, say, it attempts to coose te best tresold value H t ( wic maximizes te trougput wen averaged over te realizations of te oter user s fading coefficients. Tis involves maximization for eac value of t yielding a function H t ( wic depicts te optimal tresold assumption for tat cannel state. Te rate strategy ten gets modified as: { ( R( = 4 log P for t R (, H t ( oterwise were we ave H t ( = min [ { }] t, argmax ( e 2 R (,. Te function H t ( is plotted in te Figure 4. Te improvement of trougput by employing H t ( is sown in Figure 5 as te dased line, wic sows te difference in trougput, wen magnified 200 times to matc te scale of te plot. V. COCLUSIOS AD FUTURE WORK Te proposed midpoint strategy as straigtforward generalizations to MIMO MAC systems, since its basis is te concavity property of te logaritm. Coupled wit te proposed successive decoding, tis strategy is a viable alternative for many practical systems werein coordination is difficult to acieve due to large overead. Te notion of expected capacity matces te setup we consider ere. For te case of none or complete CSIT, te expected capacity matces te Sannon capacity of te cannel. However, wit partial CSIT and, in particular, our case of individual CSIT, te caracterization of te expected capacity is a line of work tat can be pursued Fig. 5. Power Trougput comparison furter. Similarly, te capacity wit outage can be considered for tis cannel, generalizing on te conclusions of section IV. APPEDIX A PROOF OF THEOREM 6 For a symmetric user set, te independent fading states H i ave te same distribution, say p(. Consider any outage-free power-rate strategy ( ( i, R i ( i. Let us define P ( = (/M M (. Since tere is no outage by te above coice for any of te users, te average sum-trougput is M M ER i (H i = R i (p(d = = ( M p( R i ( d ( ( M p( 0.5 log + 2 ( d p( ( 0.5 log ( + M 2 P ( d Te first inequality follows from te sum-rate bound of a MAC, wic sould be necessarily satisfied for no outage. Te rigt and side is te expected sum-rate of te symmetric mid-point strategy for power allocation P (. REFERECES [] U. iesen, U. Erez, D. Sa, and G. Wornell, Rateless codes for te gaussian multiple access cannel, in GLOBECOM 06, 2006, pp. 5. [2] S. Samai and A. D. Wyner, Information-teoretic considerations for symmetric, cellular, multiple-access fading cannels - part i, IEEE Transactions on Information Teory, vol. 43, no. 6, pp , 997. [3] R. Knopp and P. Humblet, Information capacity and power control in single-cell multiuser communications, in ICC 95 Seattle, pp [4] D. Tse and S. Hanly, Multiaccess fading cannels. i. polymatroid structure, optimal resource allocation and trougput capacities, Information Teory, IEEE Transactions on, vol. 44, no. 7, pp , ov [5] A. Das and P. arayan, Capacities of time-varying multiple-access cannels wit side information, Information Teory, IEEE Transactions on, vol. 48, no., pp. 4 25, Jan [6] S. V. Hanly and D.. C. Tse, Multiaccess fading cannels-part ii: Delaylimited capacities, IEEE Transactions on Information Teory, vol. 44, no. 7, pp , 998. [7] M. Effros, A. J. Goldsmit, and Y. Liang, Generalizing capacity: new definitions and capacity teorems for composite cannels, IEEE Transactions on Information Teory, vol. 56, no. 7, pp , 200.

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