on the capacity of multiple access and broadcast fading channels with full channel state information...

a r X i v : 1 3 1 0 . 8 5 3 2 v 1 [ c s . I T ] 2 9 O c t 2 0 1 3 1 On the Capacity of Multiple Access and Broadcast Fading Channels with Full Channel State Information at Low SNR Zouheir Rezki Senior Member, IEEE, and Mohamed-Slim Alouini, Fellow, IEEE, Abstrac t—We study the thr oughput cap aci ty re gio n of the Gaussian multi-access (MAC) fading channel with perfect channel state information (CSI) at the receiver and at the transmitters, at low power re gime. We sho w that it has a mul tidime nsi ona l rectangle struc ture and thus is simpl y char acter ized by singl e user capacity points. More specifically, we show that at low power regime, the boundary surface of the capacity region shrinks to a single point cor re spondi ng to the sum rat e maximi zer and that the coo rdi nat es of thi s poi nt coi nci de wit h sin gle user capacity bounds . Inspired by this result, we prop ose an on-off scheme, compute its achievable rate, and show that this scheme achie ves single user capac ity bounds of the MAC channel for a wide cla ss of fading cha nne ls at asympt oti cal ly low power regime. We argue that this class of fading encompasses all known wireless cha nne ls for whi ch the capac ity re gio n of the MAC channel has even a simpler expression in terms of users’ average power constraints only. Using the duality of Gaussian MAC and broadcast channels (BC), we deduce a simple characterization of the BC capacity region at low power regime and show that for a class of fading channels (including Rayleigh fading), time-sharing is asymptotically optimal. Index Terms —Multi-access, broadcast, ergodic capacity, capacity region, low-SNR, low power, fading channel, on-off signaling. I. Introduction It is now widely accepted that energy e fficiency is a key parameter in designing wireless communication systems. This has catalyzed interests of many researchers inside the information / communication theory communities in order to better unders tand perfor mance limits of wireless commu nicat ion in the low power reg ime , and de vel op new tec hni que s to achieve / approa ch these limi ts, e.g., [1]–[5]. For inst ance, in wide band commu nicat ions, althou gh the signa l stren gth is generally very low, one can capitalize on the huge bandwidth and still achieve a high capacity [2], [6], [7]. The low-SNR framework is also useful to model cellular networks in some speci fic case s [4], [8], sensor networks where power sav ing is The authors are members of the KAUST Strategic Research Initiative (SRI) on Uncertainty Quantificatio n in Science and Engineering. Zouh eir Rezk i and Mohamed -Sli m Alo uini are wit h the Electri cal En- gine erin g Prog ram, Comp uter , Elec trical, and Math emat ical Scie nce and Engineerin g (CEMSE) Division, King Abdullah Universit y of Science and Te chno log y (KA UST), Thuwal, Makk ah Prov ince , Saud i Arab ia. Email : {zouheir.rezki,slim.alouini }@kaust.edu.sa. This work was funded by a Competitive Research Grant (CRG2) from the Office of Competitive Research Funding (OCRF) at KAUST. Part of this work has been accep ted for pres enta tion in the 201 3 IEEE International Conference on Communications (ICC’2013), Budapest, Hungary, and in the 2013 IEEE Inte rnat iona l Symp osiu m on Info rmat ion Theory (ISIT’2013), Istanbul, Turkey. 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 R1 (npcu) R 2 ( n p c u ) AWGN CSI−TR CSI−R D A B C Figu re 1. Cap acit y reg ion of a 2-us er MAC Rayl eigh fadi ng channe l wher e ¯ P 1 and ¯ P 2 denote the average transmit power at the transmitters, respectively. The acronyms CSI-TR in the legend corresponds to the case where the CSI is available at both transmitters and at the receiver as well. The acronym CSI-R correspond s to the case where CSI is avail able at the receiver only. detrimental [9], [10] and more generally any communication scenario where the bandwidth and the power are fixed, but the system degree of freed om is larg e enough such that the power per degre e of fre edom is ver y low [1] , [11]. Mai nly , the re are two tre nds in the lit era ture of communica tio ns at low- power regime. The first one focuses on studying fundamental limi ts in terms of channel capac ity , error probabil ity , error exponent, etc; e.g., [1]–[5], [7], [12]. The second one deals more with signaling design and practical schemes that achieve thes e performanc e limi ts asymptoti call y, e.g., [1], [13]–[16]. A comprehensive list of references regarding communications at low-power regime and energy e fficiency can be found in [11], [17], [18] In this paper, we aim at studying the throughput capacity re gion of fa ding multi-access cha nne ls (MAC) and fad ing broadcast channels (BC) with Gaussian noise, where perfect channel state information (CSI) at the receiver(s) (CSI-R) and at the transmitter(s) (CSI-T) is assumed, at low power regime. The throughput capacity of the Gaussian MAC fading channel has been derived in [19]. Therein, it has been shown that each point on the boundary surface of the capacity region can be obtained by successive decoding and that the optimal rate and power all oca tio ns can be see n as the genera li zat ion of the single-user water-filling construction to MAC channels. The bounda ry surf ace is defined as the set of use rs’ rate s suc h that no compon ent can be incre ased with other components remaining fixed, whi le st ayi ng in the capacity reg ion [19].

on the capacity of multiple access and broadcast fading channels with full channel state information...

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