Power-Bandwidth Tradeoff in Multiuser Relay Channels with Opportunistic Scheduling

Po wer -Bandwidth T radeof f in Multiuser Relay Channels with Opportunistic Scheduling ¨ Ozg ¨ ur Oyman Intel Research Santa Clara, CA 95 054 Email: ozgu r . oyman@intel.com Moe Z. W in Massachusetts Institute of T ech nology (MIT) Cambridge, MA 021 39 Email: moewin@mit.edu Abstract —The goal of this paper is to understand t he key merits of mult ihop relaying techniques jointly in terms of their ener gy efficiency and spectral efficiency advantages in the presence of multiuser diversity gains from opportunistic (i.e., channel-aware) scheduli ng and identify the regimes and conditions in which re lay-assisted multiuser c ommunication p ro vides a clear ad vantage over direct multiuser communication. For this pu rpose, we use Shannon-th eoretic tools to analyz e the tra deoff between energy efficiency and spectral efficiency (kn own as th e power -bandwidth tradeoff) over a fading multiuser relay channel wi th K users in the asymptotic regime of large (but finite) number o f users (i .e., dense networ k). Benefiting from the extreme-v alue theoretic results of [1 ], we char - acterize the p ower - bandwidth tradeoff and the associated energy and spectral efficiency measures of the bandwidth- limited high signal-to-noise ratio (SNR) and p ower -limited low SNR regimes, and u tilize them in inv estigating the large system b ehav i or of th e multiuser relay channel as a function of t he number of users and physical channel SNRs. Ou r analysis results in very accurate closed-fo rm fo rmulas in the large (but finite) K regime that quan tify energy and spectral efficiency performance, and provides insights on th e i mpact of multih op relaying and mu ltiuser divers ity techniques on the power -bandwidth tradeoff. Index T erms —Power -bandwidth tradeoff, relay-assisted multiuser communications, opp ortunistic sched uling, mul- tiuser diversity , multi hop relaying, energy efficiency , spec- tral efficien cy I . I N T R O D U C T I O N W e consider the u plink a nd downlink of a cellular multihop /mesh system (e.g., IEEE 8 02.1 6j systems), with one ba se station, one fixed r elay station and K users. Th e ro le of th e relay station is to enhan ce en d- to-end link quality in terms of capa city , coverage and reliability using multiho p rou ting techniques [2], and its presence allows the base station to choose b etween (i) sending/re ceiving d ata directly to/fr om a given user , (ii) commun icating over a two-h op route wher e the b ase station sends data to the r elay station an d the relay station forwards the data to the users in d ownlink, and vice versa for the uplink. W e refer to this communic ation model as the multiuser r elay channel ; wh ich inclu des both the m ultiaccess r elay channel (uplink ) [3], [4] and broadc ast relay chann el (downlink) [ 3], [5]. For fixed p ortable application s, where radio chann els are slowly varying, multiple acc ess method s based on oppor tunistic sched uling mechanisms take advantage of variations in users’ channel qua lity and alloc ate re- sources such th at the user with the best ch annel q uality is served at any given time or frequ ency (could be subject to certain fairness and delay constraints). It has been shown by the p ioneerin g works [6]-[8] that the su m capacity un der such oppo rtunistic schedulin g algorithm s inc reases with th e number o f users; yielding multiuser diversity gains b y exploiting the time an d frequen cy selectivity o f wireless chan nels as well as the indepen dent c hannel variations acro ss users. Relation to Pre viou s W ork: While multiuser diversity concepts over trad itional cellular systems is well und er- stood, th ere ar e o pen issues on the desig n and analy- sis of resour ce allocation and oppo rtunistic scheduling algorithm s in r elay-based cellular multiho p n etworks. Recently , low-complexity resourc e manag ement me th- ods for OFDMA- based cellular m ultihop networks were propo sed in [9]-[1 0], and were shown to simultan eously realize gains fr om both multiuser div er sity an d multihop relaying to en hance cap acity and coverage. More over , in [1], tools from extreme- value theory were used to char- acterize the spectral efficiency of oppor tunistic schedul- ing algor ithms over fading m ultiuser relay c hannels in the asymp totic regime o f large (but finite) number of users, and to provide insights on the role of multiu ser di- versity , mu ltihop ro uting and spectrum reuse techniqu es in leveraging the system-level perfo rmance. Finally , ap- plications of op portun istic co mmunicatio n principles to relay-assisted wireless networks were in vestigated in various other contexts in [11]-[14]. Contributions: Th e prio r art o n the analysis of op - portun istic scheduling algorithms over the mu ltiuser relay chan nel h as fo cused on spectra l efficiency , wh ich is clearly an impo rtant perf orman ce m etric given the scarcity of bandwidth resources in a cellular network and system operatio n in the band width-limited regime. Often howev er, ea ch user’ s mo bile termina l in a cellular network could additionally be se verely constrained by its computatio nal a nd transmission /receiving power and/o r could be subject to p oor signal quality du e to path loss and sha dow fading effects of the wireless chann el ( e.g., mobile ter minal in a coverag e h ole at the cell edge ), leading to system o peration in the power-limited regime, where the optimization of energy efficiency become s crucial in system design. The goal of this paper is to under stand th e key merits of m ultihop re laying techniqu es jointly in terms of the ir energy efficiency an d spec tral efficiency a dvantages in the p resence of multiuser diversity gains from opp or- tunistic (i.e., ch annel-aware) sch eduling and identif y the regimes and co nditions in wh ich relay- assisted multiuser commun ication provides a clear advantage over direct multiuser commu nication. For this pu rpose, we use Shannon -theore tic tools to analyze the tradeoff between energy efficiency and spectra l efficiency (kn own as th e power - bandwid th tradeo ff) over a fading multiuser r elay channel with K users in th e asym ptotic regime o f large (but finite) numb er of u sers (i.e., den se network ). Benefiting fr om the extreme-value th eoretic resu lts of [1], we char acterize the power-bandwidth tradeoff and the associated performanc e measures of the high and low signal-to-n oise ratio ( SNR) regimes, a nd utilize th em in in vestigating the large system beh avior of th e m ultiuser relay chan nel as a fu nction of the nu mber of u sers and physical chann el SNRs. Our analysis results in very accurate formulas in the large (but fin ite) K regime, an d provides insights on the im pact of multihop relay ing and multiuser diversity techniq ues o n the p ower -b andwidth tradeoff. I I . M U LT I U S E R R E L AY C H A N N E L M O D E L A N D P R OT O C O L A S S U M P T I O N S Consider the n etwork d epicted in Fig. 1 with K + 2 nodes, in which K users, indexed by k = 1 , ..., K , send/receive in formatio n to/from a ba se station. The relay station is designated to help users transmit/rec eiv e informa tion u tilizing its high capacity bac khaul link to the base station. In other words, m ultiple users share a single relay for u plink and downlink (i.e. multiaccess and broad cast). W e assume that all the links over the multiuser relay channel are corr upted by add itiv e white Gaussian n oise (A WGN). Furth ermore , the links b etween the b ase sta- tion and users are assumed to be un der frequen cy-flat multiplicative fading i.i.d . acr oss users, with com plex channel gains { h k } K k =1 , where | h k | 2 ∈ R is a real-valued User 1 User K Relay Station Base Station Fig. 1. Multiuse r relay channe l model (both uplink and downl ink). random variable rep resenting the channel p ower gain fo r the link between the b ase station and user k , dr awn from an arbitrary con tinuous distribution F h with E [ | h k | 2 ] = 1 , ∀ k . The average received SNR between the base station an d each user eq uals SNR ( b ) . Analogou sly , the links b etween th e relay station and users have average SNRs given by SNR ( r ) , 1 and ar e und er frequ ency-flat fading i.i.d . across users, with co mplex cha nnel gains { g k } K k =1 , wher e | g k | 2 ∈ R repr esents the ch annel power gain for th e link betwe en the rela y station an d user k , d rawn f rom an arbitrary con tinuous distribution F g with E [ | g k | 2 ] = 1 , ∀ k . The set of chan nels { h k } K k =1 and { g k } K k =1 are indepe ndent. It is assume d that th e cellular back haul link between th e b ase station and relay station is an A WGN line- of-sight (LOS) con nection with received SNR equ al to SNR B . Furthermo re, ou r ch annel mo del co ncentrates o n the quasi-static r egime, in which, on ce dr awn, the chann el gains { h k } K k =1 and { g k } K k =1 over th e mu ltiuser links remain fixed for th e entir e duration of a time slot allocated fo r co dew ord transmission i.e., eac h co de- word spans a single fadin g state, and that the ch annel coheren ce tim e is much larger than the coding block length, i. e., slow fadin g assum ption. Du e to slow fading , each terminal in the multiuser relay chann el is able to ob tain fu ll chan nel state informatio n regar ding its transmission/recep tion links. Because the tran smission rate of each codeword over each hop is adapti vely chosen so that reliab le decoding is always possible (the data r ate is ch anged o n a c odeword b y co dew ord basis to ad apt to the instantaneou s ch annel fading cond itions), system is never in o utage provided that th e c oding blocklen gth is arbitrarily large. For the fo llowing analysis, we a ssume a tim e-division 1 Superscript b stands for ”base” station and r stands for ”relay” station . based (half du plex) relay ing constraint for multi-h op routing proto cols, which is due to the p ractical limitation that terminals can o ften not transmit and rece iv e at the same time. I n p articular, we co nsider a two-phased decode- and-fo rward proto col; wh ere, fo r a g iv e n ro uting path between the base station and a given user, the relay station hea rs and fully deco des the transmitted data signal in the fir st ph ase and forwards its re-encod ed version in th e seco nd phase. Moreover , we consider the u se of p oint-to- point capacity achieving c odes o nly (without any kind of cooperatio n acr oss termina ls) over all transmissions over the multiuser r elay chan nel an d do not impose any d elay c onstraints on the multiho p system; in contrast, we allow each co ded tran smission to have an arbitrarily large blocklength, although we will be co ncerne d with the relative sizes of b lockleng ths over multiple hops. I I I . A S Y M P T O T I C M E A S U R E S O F E N E R G Y A N D S P E C T R A L E FFI C I E N C Y This section d escribes our me thodolo gy for e valuating power - bandwid th tr adeoff in multiuser relay chan nels and acco rdingly intro duces the key m easures of energy and spe ctral efficiency to be used in ou r per forman ce characterizatio n. As part of the pr ior ar t in this respect, the analytical tools to study the p ower -b andwidth trade- off in the p ower -lim ited regime have b een pr eviously developed in the c ontext of point-to -point single-u ser commun ications [ 15]-[16], and were extended to multi- user (point-to -multipo int and m ultipoint-to -point) set- tings [17]-[2 0], as well as to a dhoc wireless networkin g examples of single-relay chann els [ 21]-[22], A WGN multihop networks [23]-[25] and dense multi-anten na re- lay network s [2 6]. In the ba ndwidth- limited regime, the necessary too ls to p erform the power-bandwidth tradeoff analysis were d ev e loped by [17] in the context o f cod e- division m ultiple access (CDMA) systems and were later used by [27] an d [28] to character ize fun damental limits in mu lti-antenn a channels over poin t-to-poin t an d broadc ast communication, respectively , and by [24], [26] to study a variety of adh oc networking scen arios. W e assume that a ll term inals in the mu ltiuser relay channel are supplied with finite total average transmit power P (in W atts (W)) over un constrained ba ndwidth B (in Hertz (Hz)). The multiuser relay chan nel with desired end -to-end data rate R mu st respect th e fun - damental limit R/B ≤ C ( E b / N 0 ) , where C is the Shannon capacity (ergodic mutual info rmation 2 ) ( in bits/second/Hertz o r b/s/Hz), which w e will also r efer 2 W e emphasize that due to our assumptions on the channel statistics stated in Section II, a Shannon capacit y exists (this is obtaine d by av eraging the total mutual information ove r the statistics of the channel processes) for the multiuser relay channel. as th e spectral efficiency , and E b / N 0 is the ene rgy p er informa tion bit normalized by b ackgrou nd noise spectral lev el, expressed as E b / N 0 = SN R /C ( SNR ) , for SNR = P / ( N 0 B ) and C denotin g the spec tral efficiency as a function of SN R 3 (in n ats/second/Hertz or nats/s/Hz). There exists a trad eoff be tween the efficienc y measures E b /N 0 and C (k nown as the power-bandwidth tradeoff) in achieving a giv e n target d ata rate. When C ≪ 1 , the system oper ates in th e po wer - limited re gime ; i.e., the ban dwidth is large an d the main concern is the limitation on power . Sim ilarly , the case of C ≫ 1 correspo nds to the bandwidth -limited re g ime . Tightly framing achiev a ble per forman ce, particular empha sis in our power-bandwidth tradeoff a nalysis is placed o n the regions of low an d high E b /N 0 . Low E b /N 0 r egime: Defining ( E b /N 0 ) min as th e minimum system-wide E b /N 0 required to co n vey any positiv e rate reliably , we ha ve ( E b / N 0 ) min = min SNR /C ( SNR ) , over all SNR ≥ 0 . In most scenar- ios, E b /N 0 is minimized in the power-limited wideband regime when SNR is low a nd C is near zero. W e consider the first-ord er behavior of C as a func tion o f E b /N 0 when C → 0 by analyzin g th e affine f unction (in decibels) 4 10 log 10 E b N 0 ( C ) = 10 log 10 E b N 0 min + C S 0 10 log 10 2+ o ( C ) , where S 0 denotes the “wideband” slop e of spectral efficiency in b/s/Hz/(3 dB) a t the point ( E b /N 0 ) min , S 0 = lim E b N 0 ↓ E b N 0 min C ( E b N 0 ) 10 log 10 E b N 0 − 10 log 10 E b N 0 min 10 log 10 2 . It c an be shown tha t [1 5] E b N 0 min = lim SNR → 0 ln 2 ˙ C ( SNR ) , (1) and S 0 = lim SNR → 0 2 h ˙ C ( SNR ) i 2 − ¨ C ( SNR ) , (2) where ˙ C and ¨ C denote the first and secon d ord er deriv atives of C ( SNR ) (evaluated in nats/s/Hz). High E b /N 0 r egime: In the h igh SNR regime (i.e., SNR → ∞ ), the d epende nce b etween E b /N 0 and C ca n be characterized as [17] 10 log 10 E b N 0 ( C ) = C S ∞ 10 log 10 2 − 10 log 10 ( C ) + 10 lo g 10 E b N 0 imp + o (1) , 3 The use of C and C avoids assign ing the same sym bol to spectral ef ficiency functions of SNR and E b /N 0 . 4 u ( x ) = o ( v ( x )) , x → L stands for l im x → L u ( x ) v ( x ) = 0 . where S ∞ denotes the “high SNR” slope o f the spectral efficiency in b/s/Hz/(3 dB) S ∞ = lim E b N 0 →∞ C ( E b N 0 ) 10 log 10 E b N 0 10 log 10 2 = lim SNR →∞ SNR ˙ C ( SNR ) (3) and ( E b /N 0 ) imp is the E b /N 0 improvement factor with respect to a single-user single- antenna unfaded A WGN referenc e chann el 5 and it is expressed as E b N 0 imp = lim SNR →∞  SNR exp  − C ( SNR ) S ∞  . (4) I V . P O W E R - B A N D W I D T H T R A D E O FF C H A R AC T E R I Z AT I O N Consider th e sched uling p roblem such that K users in the multiuser relay ch annel ar e to be assigned time-slots for transmission/rece ption over a comm on ba ndwidth. This problem in volves transmission s over three ty pes of links: (i) L B : W ireless cellular b ackhaul link between the base statio n an d r elay station with re ceiv e d SNR equal to SNR B , (ii) L R : The link between th e relay station and u sers with average SNR equ al to SNR ( r ) and com plex channe l gain s { g k } K k =1 and (iii) L D : The direct link between the base station an d user s with av e rage SNR equ al to SNR ( b ) and complex chan nel gains { h k } K k =1 . In th is context, we employ the max i- mum sign al-to-inter ference- plus-noise ratio ( max-SINR) oppor tunistic scheduling algo rithm, wh ich always serves the user with the best instantaneou s ra te at any g iv en time/frequ ency resou rce, in con junction with two trans- mission protoco ls over the mu ltiuser relay channel: a) Dir ect transmission (n o r elay ing): Only link L D is activ e for all available time resou rces. The b ase station compare s the instanta neous r ates over the d irect links between itself and th e user s determ ined by the chan nel gains { h k } K k =1 and assigns link L D to th e be st u ser with the high est instantaneo us rate. b) T wo-hop r elayin g: Links L B and L R are a ctiv e for th is relay-assisted mu ltihop rou ting proto col. W e assign positive time-sharing coefficients β B ∈ [0 , 1 ] an d β R ∈ [0 , 1] to links L B and L R , respectively , to specif y the fr actional time d uring which these links are active, such that β B + β R = 1 . The relay station compare s the instantaneo us rates over the links between itself an d the users determined b y the channe l gains { g k } K k =1 and and assigns link L R to the best u ser with the hig hest instantaneou s r ate. The base station comm unicates with the selected user over a two-ho p route th roug h th e relay station. 5 For the A WGN channel; C ( SNR ) = ln(1 + SNR ) resulti ng in S 0 = 2 , ( E b /N 0 ) min = l n 2 , S ∞ = 1 and ( E b /N 0 ) imp = 1 . A. Opp ortunistic Mu ltiuser Scheduling with Dir ec t Communicatio n Assuming Gaussian inp uts, i.e., all input signals have the tempor ally i.i.d. zero-m ean circu larly sym metric complex Gaussian distribution, the maxim um suppo rt- able spectral efficiency over the multiu ser relay c hannel achieved by d irect comm unication in conjunc tion with oppor tunistic schedulin g is given by (in nats/s/Hz) C direct ( SNR ) = E  max k =1 ,..., K C ( SNR α ( b ) | h k | 2 )  (5) = c ( h ) K ( SNR ) κ + d ( h ) K ( SNR ) , (6) where the closed form expr ession in (6) follows from direct app lication of th e extreme- value theoretic an alysis in [1 ] f or the asym ptotic regime of large K and assumes T ype I convergence [2 9] on the max ima of the i.i. d. instantaneou s spe ctral efficiency r ealizations in (5), that depend on th e channel p ower gains { | h k | 2 } K k =1 , i.e., F h belongs to T ype I domain of attraction le ading to th e Gumbel limiting extreme-value distribution. I n (5), the spectral e fficiency fun ction C ( x ) is d efined as C ( x ) = ln(1 + x ) an d we expr ess SNR ( b ) as SNR ( b ) = α ( b ) SNR . Moreover , in (6), κ ≈ 0 . 5772 1566 is Euler’ s con stant and c ( h ) K ( SNR ) and d ( h ) K ( SNR ) are given b y c ( h ) K ( SNR ) = SNR α ( b ) a ( h ) K 1 + SNR α ( b ) b ( h ) K , d ( h ) K ( SNR ) = ln(1 + SNR α ( b ) b ( h ) K ) , where a ( h ) K and b ( h ) K are th e sequences of co nstants necessary to en sure the following conv ergence in d is- tribution as K → ∞ (cf. Lemma 1 in [1]) : P max k | h k | 2 − b ( h ) K a ( h ) K ≤ x ! → exp( − e xp( − x )) . (7) The following theo rem states o ur results on sp ectral efficiency vs. E b / N 0 characterizatio n for d irect commun ication in the presence of m ultiuser diversity gains fr om oppor tunistic schedulin g: Theorem 1: In the asymptotic re g ime of la r ge K , assuming that F h belongs to T ype I do main of attraction lead ing to th e Gumb el limiting extr eme-valu e distribution, th e power-bandwidth tradeoff for direct communica tion with op portunistic scheduling ca n be characterized thr oug h the following relationships: Low E b N 0 r egime: E b N 0 direct min = ln 2 α ( b )  κ a ( h ) K + b ( h ) K  , and S direct 0 = 2  κ a ( h ) K + b ( h ) K  2 b ( h ) K  2 κ a ( h ) K + b ( h ) K  . High E b N 0 r egime: E b N 0 direct imp = 1 α ( b ) b ( h ) K exp − a ( h ) K b ( h ) K κ ! , and S direct ∞ = 1 . W e clearly see fr om Theo rem 1 how multiuser diversity gains imp act the energy and spectral efficiency mea- sures o f th e b andwidth -limited h igh SNR an d power - limited low SNR regimes. In particular, th e scaling constants a ( h ) K and b ( h ) K impact both of ( E b / N 0 ) direct min and ( E b / N 0 ) direct imp , as well as S direct 0 , althoug h it sho uld be noted that S 0 conv e rges to 2 as K → ∞ . Assum ing Rayleigh fading d istribution on F h , we have a ( h ) K = 1 and b ( h ) K = log( K ) , which implies that ( E b / N 0 ) direct min and ( E b / N 0 ) direct imp decay at a rate of at least 1 / lo g( K ) in the regime of asymptotically large K . B. Opp ortunistic Mu ltiuser S cheduling with T wo-Hop Relaying Assuming Gaussian inputs, the m aximum supporta ble end-to- end spectral efficiency over the multiuser relay channel achieved b y two-hop relaying in con junction with opportu nistic scheduling is g iv en by (in nats/s/Hz) C relay = min [ β B C ( SNR ) , β R max k =1 ,..., K C  SNR α ( r ) | g k | 2   (8) = β B C ( SNR ) − β R c ( g ) K ( SNR ) Ei( z K ( SNR )) , (9) where the closed fo rm expression in (9) follows from the extreme-value theo retic analysis in [1] for the asym p- totic regime of large K (cf. T heorem 2 in [1]) and assumes T ype I conver gence o n the m axima o f the i.i.d. instantaneou s spe ctral efficienc y realizations in (8), that depend o n the channe l p ower gains {| g k | 2 } K k =1 , i. e., F g belongs to T ype I do main of attraction leading to th e Gumbel limiting extreme-value distribution. I n (8), we express SNR ( r ) and SNR B as SNR ( r ) = α ( r ) SNR and SNR B = SN R . Moreover, in (9), Ei( x ) is the expon en- tial integral function defin ed by Ei( x ) = R ∞ x e − y y dy , z K ( SNR ) is given by z K ( SNR ) = exp β R d ( g ) K ( SNR ) − β B C ( SNR B ) β R c ( g ) K ( SNR ) ! . and c ( g ) K ( SNR ) and d ( g ) K ( SNR ) are given by c ( g ) K ( SNR ) = SNR α ( r ) a ( g ) K 1 + SNR α ( r ) b ( g ) K , d ( g ) K ( SNR ) = ln(1 + SNR α ( r ) b ( g ) K ) , where a ( g ) K and b ( g ) K are the sequences of constants neces- sary to en sure the fo llowing conver g ence in distribution as K → ∞ (cf . L emma 1 in [1]): P max k | g k | 2 − b ( g ) K a ( g ) K ≤ x ! → exp( − e xp( − x )) . (10) The following theo rem states o ur results on sp ectral efficiency v s. E b / N 0 characterizatio n for two-hop relaying in the pre sence of multiuser div ersity gains from oppor tunistic scheduling : Theorem 2: In the asymptotic re g ime of la r ge K , assuming tha t F g belongs to T ype I domain of attraction leading to th e Gumb el limiting extr eme- value distribution, the power -ba ndwidth tradeoff fo r two-hop r ela ying with opportun istic scheduling can be characterized thr oug h the following relationships: Low E b N 0 r egime: E b N 0 relay min = ln 2 β B − β R α ( r ) a ( g ) K Ei ( ζ K ) , and S relay 0 = 2  β B − β R a ( g ) K α ( r ) Ei ( ζ K )  2 β B − 2 β R a ( g ) K b ( g ) K ( α ( r ) ) 2 Ei ( ζ K ) , wher e ζ K = e xp β R α ( r ) b ( g ) K − β B β R α ( r ) a ( g ) K ! . High E b N 0 r egime: E b N 0 relay imp =                             1 /  α ( r ) b ( g ) K  exp  − a ( g ) K κ / b ( g ) K  for β B < β R exp  a ( g ) K /b ( g ) K  Ei  α ( r ) b ( g ) K  b ( g ) K /a ( g ) K !! for β B = β R = 1 / 2 1 for β B > β R and S relay ∞ = min [ β B , β R ] . W e clearly see fr om Theo rem 2 how multiuser diversity gains imp act the energy and spectral efficiency mea- sures o f th e b andwidth -limited h igh SNR an d power - limited low SNR regimes. In particular, th e scaling constants a ( g ) K and b ( g ) K impact both of ( E b / N 0 ) relay min and ( E b / N 0 ) relay imp , as we ll as S relay 0 . Mor eover , ou r anal- ysis spe cifies th e dep endence of a ll p ower -b andwidth tradeoff measur es o n time-sharin g coefficients β B and β R . Finally , we note th at the behavior of the ene rgy efficiency measure of the ban dwidth- limited h igh SNR regime, ( E b / N 0 ) relay imp , varies in a p iecewise fashion as a function of β B and β R . V . N U M E R I C A L R E S U LT S For the following numerical study , we assume Rayleigh fadin g distribution on F h and F g that leads to a ( h ) K = a ( g ) K = 1 and b ( h ) K = b ( g ) K = lo g( K ) , and fur ther- more set K = 20 , α ( b ) = 0 . 01 , and α ( r ) = 1 . Selecting K = 2 0 is typica l as the num ber o f users per sector in a relay-based cellular network (e.g., wireless metropo litan area networks (WMANs) b ased on the IEEE 802.1 6j multihop relay standar d [30]), and more over the cho ices of α ( b ) = 0 . 01 , and α ( r ) = 1 a re realistic when th e users are located at the cell edge, when the use o f mu ltihop relaying is m ost b eneficial fo r leveraging the system- lev el per forman ce of cellular networks [10]. In Figs. 2-4, we plo t sp ectral efficiency vs. E b /N 0 for direct commu nication an d two-hop relay ing in th e presence of oppo rtunistic sched uling f or different values of β B and β R . Here , we compare empir ically gen- erated power - bandwid th trad eoff curves (solid curves) with their analytical cou nterpar ts (dashed cu rves) from Theorem s 1 and 2 for th e low and high E b / N 0 regimes. The emp irical results ar e ob tained by averaging th e expressions in (5) and (8) over a large number of random ly gene rated fading realizations ( based on Mo nte Carlo simu lations) for various SNRs and com puting the power-bandwidth tra deoffs from this set of average spectral e fficiencies based o n E b / N 0 = SNR /C ( SNR ) , where C ( SNR ) is determ ined empir ically . From Figs. 2-4, we validate the accuracy and tightness of th e closed- form expressions presented in The orems 1 and 2 for th e power - bandwid th tr adeoff in the scenarios of d irect comm unication and two-hop re laying. In par- ticular , we verify that our ana lytical results are well in agreemen t with the empirical results for all ran ges o f β B and β R , despite th e fact th at K is set a t the finite 0 5 10 15 20 25 30 35 0 1 2 3 4 5 6 7 8 9 E b /N 0 (dB) Spectral efficiency (b/s/Hz) Direct − Empirical Relay − Empirical Low SNR approximations High SNR approximations Fig. 2. Spectral ef ficiency vs. E b / N 0 for direct communicat ion vs. two-ho p relaying with opportunistic s cheduli ng for β B = 1 / 3 and β R = 2 / 3 . Solid curves represent empirical powe r-bandwidth trade- of fs while dashed curv es are analytical powe r-ban dwidth tradeof fs. 0 5 10 15 20 25 30 35 0 1 2 3 4 5 6 7 8 9 E b /N 0 (dB) Spectral efficiency (b/s/Hz) Direct − Empirical Relay − Empirical Low SNR approximations High SNR approximations Fig. 3. Spectral ef ficiency vs. E b / N 0 for direct communicat ion vs. two-ho p relaying with opportunistic s cheduli ng for β B = 1 / 2 and β R = 1 / 2 . Solid curves represent empirical powe r-bandwidth trade- of fs while dashed curv es are analytical powe r-ban dwidth tradeof fs. value of 20 . Mo reover , we ob serve that in the power- limited low SNR r egime, mu ltihop relay ing provides a superior p ower -b andwidth tradeoff (and a sign ificant power efficiency g ain) over d irect comm unication in the presence of multiuser d iv er sity gains from oppor tunistic scheduling . R E F E R E N C E S [1] ¨ O. Oyman, “Opportun ism in multiuser relay channels: Schedul- ing, routing and spectrum reuse, ” in Pr oc. IEEE Internat ional Symposium on Information Theory (ISIT’07) , Nice, France, June 2007, pp. 286 – 290. [2] ¨ O. Oyman, J. N. L aneman, and S. Sandhu, “Multih op relaying for broadband wireless mesh net works: F rom theory to practice , ” IEEE Communications Magazine , vol . 45, no. 11, pp. 116–122, Nov . 2007. 0 5 10 15 20 25 30 35 0 1 2 3 4 5 6 7 8 9 E b /N 0 (dB) Spectral efficiency (b/s/Hz) Direct − Empirical Relay − Empirical Low SNR approximations High SNR approximations Fig. 4. Spectral ef ficiency vs. E b / N 0 for direct communication vs. two-ho p relaying with opportunistic scheduling for β B = 2 / 3 and β R = 1 / 3 . Solid curve s represent empirical power -bandwidt h trade- of fs while dashed curv es are analytical power -bandwidth tradeof fs. [3] G. Kramer , M. Gastpar , and P . Gupta, “Coopera ti ve strate gies and capacity theorems for relay netwo rks, ” IEE E T rans. Inf. Theory , vol. 51, no. 9, pp. 3037–3063, Sep. 2005. [4] D. Chen and J. N. Laneman, “The di versity-mult iplexing tradeo f f for the m ultia ccess relay channe l, ” in Pr oc. Confer ence on Informatio n Science s and Systems (CISS) , Princeton , NJ, March 2006. [5] Y . Liang and V . V . V eerav alli, “Cooper ati ve relay broadcast channe ls, ” IE EE Tr ans. Inf. Theory , vol. 53, no. 3, pp. 900– 928, Mar . 2007. [6] R. Knopp and P . H umblet, “Information capacity and power control in single cell multiuse r communications, ” in Proc. IEE E Int. Compute r Conf. , Seattle, W A, Jun. 1995. [7] D. Tse and S. Hanly , “Multi-a ccess fading channels: Polymatroid structure , optimal resource all ocati on and throughput capaciti es, ” IEEE T rans. Inf. Theory , vol. 44, no. 7, pp. 2796–2815, Nov . 1998. [8] P . V iswanath, D. N. C. Ts e, and R. Laroia, “Opportunisti c beamforming using dumb antennas, ” IEEE T rans. Inf. Theory , vol. 48, no. 6, pp. 1277–1294, Jun. 2002. [9] ¨ O. Oyman, “ OFDM 2 A : A centra lized resource allocation polic y for cel lular multi-hop netw orks, ” in Pr oc. IEEE Asilomar Confer ence on Signals, Systems and Computers , Pacific Grove, CA, Oct. 2006. [10] ¨ O. Oyman, “Opportuni stic scheduling and spectrum reuse in relay-b ased cellular OFDMA networks, ” in Proc . IEEE Global T elecommuni cations Confer ence (GLOBECOM’07) , W ashington, D.C., Nov . 2007, pp. 3699 – 3703. [11] A. Bletsas, H. Shin, and M. Z. W in, “Cooperati ve communica- tions with outage-o ptimal opportunistic relaying, ” IEE E T rans. W ir eless Communications , vol. 6, no. 9, pp. 3450–3460, Sep. 2007. [12] D. G ¨ und ¨ uz and E. E rkip, “Opport unistic cooperat ion by dynamic resource allocati on, ” IEE E T rans. W irele ss Communicati ons , vol. 6, no. 4, pp. 1446–1454 , Apr . 2007. [13] C. K. L o, Jr . R. W . Heath, and S. V ishwanath , “Opportun istic relay selection with limited feedback, ” in P r oc. IEEE VTC , Dublin, Ireland , Apr . 2007, vol. 1, pp. 135–139. [14] M. Hu and J. Zhang, “Opportuni stic multi-acc ess: Multiuser di versity , relay-a ided opportunisti c scheduling, and traffic-a ided smooth admission control, ” J ournal on Mobile Networks and Applicat ions , vol. 9, no. 4, pp. 435–444, Aug. 2004. [15] S. V erd ´ u, “Spectra l effic iency in the wideband regi m e, ” IEEE T rans. Inf. Theory , vol. 48, no. 6, pp. 1319–1343, Jun. 2002. [16] A. Lozano, A. T ulino, and S. V erd ´ u, “Multiple -antenna capac ity in the low-po wer regime, ” IEEE T rans. Inf. Theory , vol. 49, no. 10, pp. 2527–2543, Oct. 2003. [17] S. Shamai (Shitz) and S. V erd ´ u, “The impact of flat-fading on the spect ral ef ficienc y of CDMA, ” IEEE T rans. Inf. Theory , vol. 47, no. 5, pp. 1302–1327, May 2001. [18] G. Caire, D. T uninetti, and S. V erd ´ u, “Suboptimality of T DMA in the low-po wer regime, ” IEEE T rans. Inf. Theory , vol. 50, no. 4, pp. 608–620, Apr . 2004. [19] A. L apidoth , I. E. T elatar , and R. Urbanke , “On wide-band broadca st channe ls, ” IEE E T rans. Inf. Theory , vol. 49, no. 12, pp. 3250–3258, Dec. 2003. [20] T . Muharemovi ´ c and B. Aazhang, “Rob ust slope region for wideband CDMA with m ultipl e antennas, ” in P r oc. 2003 IEEE Informatio n Theory W orkshop , Paris, France, March 2003, pp. 26–29. [21] A. E l Gamal and S. Zahedi, “Minimum energy communica- tion over a relay channel, ” in Proc. 2003 IEEE International Symposium on Information Theory (ISIT’03) , Y okohama, Japan, June-July 2003, p. 344. [22] X. Cai, Y . Y ao, and G. Giannak is, “ Achie vabl e rates in lo w-powe r relay links ov er fa ding channels, ” IE EE T rans. Communi cations , vol. 53, no. 1, pp. 184–194, Jan. 2005. [23] M. Sik ora, J. N. Laneman, M. Haenggi, D. J. Costello, and T . E. Fuja, “Bandwidt h and power efficien t routing in linear wireless netw orks, ” IEE E T rans. Inf. Theory , vol. 52, no. 6, pp. 2624– 2633, June 2006. [24] ¨ O. Oyman and S. Sandhu, “Non-ergodic powe r-bandwidth tra de- of f in linear multi-hop networks, ” in Proc. IEEE International Symposium on Information Theory (ISIT’06) , Seattl e, W A, July 2006, pp. 1514–1518. [25] ¨ O. Oyman and J. N. L aneman, “Multihop di versity in wideband OFDM systems: The impact of spatial reuse and frequency selec- ti vity , ” in Pr oc. 2008 IEEE International Symposium on Sprea d Spectrum T echnique s and A pplicat ions (ISSST A’08 ) , Bologna , Italy , Aug. 2008. [26] ¨ O. Oyman and A. J. Paulraj, “Po wer-bandwidt h tradeof f in dense multi-ant enna relay networks, ” IEEE T rans. W ire less Communicat ions , vol. 6, no. 6, pp. 2282–2293, June 2007. [27] A. Lozano, A. Tuli no, and S. V erd ´ u, “High-SNR power offset in multiant enna communicat ion, ” IE EE T rans. Inf. Theory , vol. 51, no. 12, pp. 4134–4151, Dec. 2005. [28] N. Jindal, “High SNR analysis of MIMO broadcast channels, ” in Pr oc. 2005 IE EE International Symposium on Informatio n Theory (ISIT’05) , Adelaide , Australia , Sep. 2005. [29] M. R. Leadbette r , G. Lindgren, and H. Rootze n, Extr emes and Related Pr opertie s of Random Sequences and Proce sses , Springer -V erlag, New Y ork, NY , 1983. [30] IEEE 802.16j: Relay T ask Gr oup F ixed B r oadband W irele ss Access , URL: http:// wirele ssman.org/re la y/index.html .