Overlaid Cellular and Mobile Ad Hoc Networks

Overlaid Cellular a nd M obile Ad Hoc Netw orks Kaibin Huang ∗ , Y an Chen † ∗ , Bin Chen ‡ , Xia Y ang ‡ , and V incent K. N. Lau ∗ ∗ Departmen t of E lectronic & Com puter En gineering Hong Kong University of Science & T e chnolog y Clear W ater Bay , Hong Kong Email: k huang @ieee.org, eeknlau@ust.hk † Institute o f I nforma tion & Communicatio n Engineerin g Zhejiang University , Hangzhou , 31002 7, P .R. China Email: y anchen@ust.h k ‡ Huawei T ech nolog ies Co. Ltd . Bantian, Longgan g District, Shenzhen 518 129, P .R.China Email: binchen @huawei.com, yangxia@huawei.com Abstract — In cellular systems using frequency division du- plex, g rowing Internet services cause unbalance of uplink and downlink traffic, resulting in poor upli nk spectrum utilization. Addressing this issue, this paper considers overlaying an ad hoc network onto a cellu lar uplink network for impro ving spectrum utilization and spatial reuse efficiency . T ransmission capacities of the ov erlaid networks are analyzed, which are defined as the maximum densities of the ad h oc nodes and mobil e users under an outage constraint. Using tools from stochastic geometry , the capacity t radeoff curves for the overlaid networks are shown to be linear . Deploying overlaid networks based on frequency separation is prov ed to achiev e hi gher network capacities than that based on spatial separation. Furthermore, sp atial diversity is shown to enhance network capacities. I . I N T RO D U C T I O N In existing cellular systems based on frequen cy division duplex (FDD) such as FDD UMTS [ 1], equal b andwidths are usually allocated for uplin k an d downlink transmission. W ith rapidly growing wireless Intern et servic es, downlink tr affic load is increasingly h eavier than the uplink counterp art, which is a key chara cteristic of In ternet data [2], [3]. Consequently , uplink spectrum is underu tilized. T o addre ss this issue, this paper con siders overlaying an ad ho c network o nto a FDD cellular system f or imp roving the u tilization of up link spec- trum and also the spatial reu se efficiency . This paper focuses on chara cterizing th e network capacity trad eoff between the overlaid cellular and ad hoc networks. One measure of network capacity is transport capacity for multi-ho p network s that was in troduc ed in [4 ] and has attracted active r esearch (see e.g. [5 ], [6]). An other measur e of network cap acity is transmission ca pacity introd uced in [7] for sing le-hop networks, which is d efined as th e maxim um number of successful co mmunic ation links per unit area under the sig nal-to-inter ference- noise ratio (SINR) and o utage co n- straints. Transmission cap acity has been used fo r in vestigating the spatial reuse efficiency of different types of mob ile ad hoc networks ( see e.g. [8], [9]). Com pared with transport capacity , transmission cap acity allows more accurate analysis and easier computatio n [7], and h ence is ado pted in this paper . This p aper co nsiders overlaying the cellu lar uplink and ad hoc networks using two metho ds. Th e first is b lind transmis- sion where the transmission of ad hoc no des and mob ile users are indep endent; the second is fr equen cy mutual exclusion where ad h oc n odes transmit over frequ ency sub -chann els unoccu pied by mobile users. For both methods, the tr adeoff curves for the transmission capacities of two overlaid networks are shown to be linear . Th e r egion boun ded by each curve and positive axes is called the capacity region, which contain s all allowable co mbination s of ad hoc no de and mob ile user densities u nder the outage con straint. The capacity region for frequen cy mutual exclusion is p roved to b e larger th an th at for blind transmission. Thus, the former is a more efficient method for overlaying cellular an d ad hoc networks. Furth ermore, capacity regions fo r both overlay metho ds are shown to be enlarged by exploiting spatial diversity . I I . N E T W O R K A N D C H A N N E L M O D E L S A. Network Model The overlaid cellular and ad ho c networks is illu strated in Fig . 1. The locations of tr ansmitting nod es in the mobile ad hoc network are mod eled as a Poisson point pro cess (PPP) following the common ap proach in th e literatur e [7] , [10]. Specifically , the po sitions of the transmitters form a homog eneous PPP , denote d as Φ , on a 2 - dimension al plan e and with th e d ensity represen ted by λ a . Let T n denote the coordin ate of the n th tr ansmitting node. Each transmitting node is associated with a receiving no de located at a fixed distance denoted as d . Similarly , the positions of base stations in the cellular network are modeled as a homogeneo us PPP on the s ame plane as the ad hoc n etwork. Let Π b and λ b represent the process of base stations and its density , r espectiv ely . Each base station commun icates with M uplink mobile users, here after refe rred to simp ly as mobile users. Thus, the density o f m obile users is λ u = M λ b . The coor dinates of the n th base station and the m th associated mo bile user are represented by B n and U n,m = B n + Z n,m respectively . T o simplify an alysis, the distance between a b ase station and a mo bile user is fixed at r , thus | Z n,m | = r ∀ m, n . The case of random distances is considered in the fu ll-length paper . M o b i l e u s e r B a s e s ta ti o n A d h o c n o d e M o b i l e u s e r B a s e s ta ti o n A d h o c n o d e Fig. 1. Overl aid cel lular and ad hoc netwo rks The uplink spectrum is divided into K sub-ch annels. Each mobile u ser o ccupies one sub-ch annel. T o sim plify an alysis, different base stations assign the same sub set of M sub- channels to their corr espond sets of M mobile users. Let M denote th e ind ices of the M sub-cha nnels used b y m obile users. E ach tr ansmitter in the ad h oc n etwork either random ly selects a sub -chann el for transmission f rom the co mplete set of sub-chan nels or the subset not bein g u sed b y mobile users. These two m ethods are r eferred to as blind transmission and fr equency mutual exclusion , respectively . The n etworks are assumed to be interfer ence limited an d thus n oise is neglected for sim plicity . Consequently , the reli- ability of data p ackets receiv ed b y either a ba se station or a node is determine d by the signal-to- interferen ce ra tio ( SIR). Let S denote the r andom channe l p ower fo r an arbitrar y lin k, Ω the proc ess of in terferers 1 , an d I n the n th interf erer in Ω . Thus, assuming that d ata transmission p ower , deno ted as P D , is id entical for all tra nsmitters, the SIR at the receiv er en d of the link u nder co nsideration is g i ven as SIR = S P T n ∈ Ψ I n . (1) Since the SIR is independ ent o f P D , P D = 1 is assumed for simp licity . The corr ect decod ing of rec eiv ed data packets requires the SIR to exceed a threshold θ , w hich is id entical for all receiv ers in th e networks. In o ther words, the rate of informa tion sent fr om a transmitter to a r eceiv er is no less than log 2 (1 + θ ) assuming Ga ussian signaling . T o support this informa tion r ate with hig h probab ility , the outage pr obability that SIR is below θ m ust be smaller than o r equ al to a giv en threshold 0 < ǫ < 1 , i.e. P out ( λ ) = Pr(SIR < θ ) ≤ ǫ (2) where P out ( λ ) denotes the SIR ou tage p robab ility as a f unc- tion o f λ . 1 As discussed later , depending on the overla y m ethod , the interfe rers can include unintended mobile users, ad hoc transmitters in Φ , or both. λ u λ a α C 1 f o r b li n d t r a n s m i s s i o n α C 2 f o r f r e q u e n c y m u t u a l e x c l u s i o n C 1 f o r b li n d t r a n s m i s s i o n (1−α) C 2 f o r f r e q u e n c y m u t u a l e x c l u s i o n Fig. 2. The network capacity regi on for the overla id cellular ( λ u ) nd ad hoc ( λ a ) networks in the regime of small outage probabili ty . The parameter α is the percentage of sub-cha nnels used by the cellular networ ks; C 1 and C 2 are defined in (7) and (11), respecti vely . B. Cha nnel Mod el The mo del for a sing le sub- channel is described as follows, where the su b-chann el ind ex is omitted. Assume a narr ow- band sub- channel m odel with flat fading. Each su b-chann el consists of path-loss and small fading compon ents. Consider a typical receiv er wh ich can b e either a b ase station o r a ad h oc n ode. A sub-chann el between th e typical re ceiv er an d the associated transmitter is r − α Q 0 or d − α W 0 depend ing on whether the transmitter is a mobile u ser or a nod e, w here α and ( Q 0 and W 0 ) rep resent the p ath-loss expone nt an d fading gain s, respectively . Let Ω be th e po int proce ss groupin g all interferers to the typic al receiver , includ ing either m obile users, or ad hoc transmitters, or both depending on the network overlay method. Then a sub-chan nel between the n th interferer in Ω and the typical rec ei ver is D − α n G n , where D n is the distance and G n is the fading gain. The ran dom variables { D n } and { G n } ar e i.i.d. with in the same set. I I I . N E T W O RK C A P AC I T Y T R A D E O FF : A S Y M P T OT I C A N A L Y S I S Similar to the notion of transmission capacity in troduced in [7], [ 8], the network cap acity c onsidered in this paper is defined as the m aximum density of tran smitters supported by a network under th e outage co nstraint in (2 ). Let P b ( λ u , λ a ) and P a ( λ u , λ a ) d enote the outage pr obabilities at the ba se station and an ad hoc receiver , respecti vely . Then the capacities of th e cellular ( λ u ) and the ad ho c ( λ a ) networks can be written as ( λ u , λ a ) = min  arg P − 1 b ( ǫ ) , ar g P − 1 a ( ǫ )  . (3) Note that the thro ughp ut p er unit area for the cellular and the ad hoc network s ar e (1 − ǫ ) log(1 + θ ) λ u and (1 − ǫ ) log(1 + θ ) λ a , resp ectiv ely . Th e ne twork capacity tradeoff b etween the overlaid cellular and ad ho c networks is an alyzed fo r the regime of small outage probability regime ǫ → 0 in subsequent subsections. A. Ex isting Resu lts The cap acity of a mobile ad hoc network und er an o utage constraint is an alyzed in [ 8]. The key results in [8 ] are summarized in the following lemma and serve as the startin g point for our analy sis. Lemma 1 (W eber et al [8]): Consider a narrow-band ad hoc ne twork where the locations o f tr ansmitters are a hom o- geneou s PPP h aving the density λ . The ch annel between a typical pair of tran smitter and r eceiv er is d − α W , and that between the typical r eceiver and the n th interfe rer is D − α n G n . The SIR ou tage p robab ility fo r th e typical receiver is b ounded as P l out ≤ P out ≤ P u out where P l out = 1 − E  exp  − κλθ δ  P u out = 1 − E       1 − δ 2 − δ κλθ − δ  1 − δ 1 − δ κλθ − δ  2    + e − κλθ − δ    where δ = 2 α and κ = π E [ G δ ] W − δ d 2 . For ǫ → 0 , P out can be expa nded as 2 P out = E [ κ ] λθ δ + O  λ 2 θ 2 δ  . (4) B. Cap acity T radeoff: Blind T ransmission The outage prob ability for data received at a base station is derived f or ǫ → 0 . Consider the m th one of the M sub - channels assign ed to mo bile users. T he mob ile users using this sub-chan nel form a point proce ss denoted as ∆ m = { B + Z m | B ∈ Φ } = Φ + Z m , where Z m represents th e shift in location . Since the h omogen eous PPP Φ is tran slation inv ariant, ∆ m is also a homogen eous PPP with th e density λ b . Con sider a ty pical base station B and the assoc iated mob ile user U m using th e m th assigned su b-chann el, where the subscript n is o mitted for simplicity . W ithout loss o f g enerality , B is assumed to be located at the origin, and h ence U m = Z m . The data rece i ved at B from U m is interfered with b y other- cell mobile users and the transmitters in the ad hoc networks that use the same sub-chan nel, which are represented by the processes { ∆ m / { U m }} and Π m , respectively . Th e process of these interferers is { ∆ m / { U m }}∪ Π m . By Sli vnyak ’ s Theorem [11], cond itioned on U m , the process of other mob iles users { ∆ m / { U m }} remains as a hom ogeneo us PPP with the same density λ b . Next, consider a typic al pair of transm itter T and receiver R in the ad hoc ne twork. By blind tr ansmission, T and R ra ndomly selects one of K sub-c hannel for comm unica- tion. Let S T denote the index of the selected sub channel. Then S T follows the un iform distribution with the su pport { 1 , 2 , · · · , K } . By treating S T as a mark of T and using the Marking Th eorem [1 2], the m arked PPP Π k = { ( T , S T = k ) | T ∈ Π } is a homo geneou s PPP with density λ a /K . Combining ab ove r esults and the superpo sition proper ty of PPP , the process of interfer ers for U m , na mely { ∆ m / { U m }} ∪ Π m , is a hom ogeneo us PPP with the density λ b + λ a . Based o n the above re sults, the relationship λ u = λ a / M , and Lemma 1 , for a single- user d ata stream receiv ed at a 2 This result is generalize d from that in [8] where the asymptotic case θ → ∞ is considered. Note that ǫ → 0 corresponds to λθ δ → 0 . base station, the asym ptotically small outag e pr obability can be wr itten as P b = E [ κ b ]  λ u M + λ a K  θ δ + O  λ u M + λ a K  2 θ 2 δ ! . (5) where κ b = π E [ G δ ] Q − δ r 2 . For a rec ei ver in the ad h oc network, the outage pr obability is de riv ed for ǫ → 0 as fo llows. Consider a ty pical pair of transmitter T and receiver R commu nicating using the k th sub- channel. If k / ∈ M , the interf erers f or R are only uninten ded transmitters in the ad hoc network that also use the k th sub- channel, represented by Π k / { T } . I f k ∈ M , R is inter fered with b y both the transmitters in Π k / { T } and also the mob ile users u sing th e k th sub-chann el grou ped in the process ∆ m . Similar to earlier deriv ation, by u sing Lemm a 1, for the data received at a receiver in th e ad ho c network using th e k sub- channel, th e asymp totically small ou tage p robability c an be written a s P a ( k ) =                  E [ κ a ]  λ u M + λ a K  θ δ + O  λ u M + λ a K  2 θ 2 δ ! , k ∈ M E [ κ a ]  λ a K  θ δ + O  λ a K  2 θ 2 δ ! , k / ∈ M (6) where κ a = π E [ G δ ] W − δ d 2 . The tradeoff between the capacity of the cellular a nd ad hoc n etworks is obtain ed by using (3), (5) and (6). Denote asymptotic eq uiv alence as ∼ = and define it as the relation- ship between two fu nctions f 1 ( ǫ ) and f 2 ( ǫ ) that satisfies lim ǫ → 0 f 1 ( ǫ ) f 2 ( ǫ ) = 1 . Recall that th e capacities of the cellular and the ad h oc networks are d efined in (3) as the maximu m densities o f mobile users λ u and ad hoc tra nsmitters λ a under the outage constraint. The network capacity tradeoff curve thus obtained is given in the following proposition. Pr oposition 1: Using th e overlay meth od of b lind tr ansmis- sion, the capacity of the cellular network λ u and that of the ad h oc n etwork λ a satisfies the following r elationship λ u α + λ a ∼ = C 1 , ǫ → 0 (7) where α is the fraction of sub-ch annels assigned to mobile users, an d C 1 = K ǫθ − δ π E [ G δ ] min  1 d 2 E [ W − δ ] , 1 r 2 E [ Q − δ ]  . (8) The region bound ed by the cu rve in ( 7) and the con straints λ a ≥ 0 an d λ u ≥ 0 is the cap acity region for the overlaid cellular and ad hoc networks as illustrated in Fig. 2. Any combinatio n o f λ a and λ u within this region satisfies the outage co nstraint in (2). As observed fro m Fig. 2 (a), the area o f capacity region is increased by increasing α . This implies th at fo r blin d transmission , allocating a sub set of sub- channels r ather than the whole set to mo bile u sers results in inefficient use of the spectru m. Besides increasing the number of mobile users, the capacity region can b e also en larged by increasing C 1 in (8). This is equivalent to o ptimizing a set of parameters, including increasing the number of sub-cha nnels K o r the outage constraint ǫ , or reducing the distances between the transmitter and receiver in each link ( d and r ) or the function s of fading g ains ( E [ W − δ ] a nd E [ Q − δ ] ). Reducing these fu nctions can be achiev ed by exploiting spatial di versity as d iscussed in Section III- D. C. Capacity T radeoff: F requency Mutual Exclusion For frequen cy mutual exclusion, the c ellular and ad hoc networks u se different sub-sets of sub-cha nnels, namely M and K / M , and thus do not interf ere with each other . Specifi- cally , the data stream received a t a b ase statio n over the sub- channel m ∈ M co ntains inter ference fro m mob ile users in other cells using the same su b-chann els; a nod e rec eiving data using k th sub-cha nnel is interfered with on ly by unintend ed transmitting nodes also using the sub-ch annel k . Following a similar pr ocedure as in the preced ing sectio n, the ou tage probab ilities for the overlaid n etworks are obtained as P b = E [ κ b ] λ u θ δ M + O  λ 2 u θ 2 δ  (9) P a = E [ κ a ]  λ a K − M  θ δ + O  λ a K − M  2 θ 2 δ ! (10) where κ a = π E [ G δ ] W − δ d 2 . By apply ing the outage con- straint to (9 ) and (10), th e tradeoff curve f or the cap acities of the c ellular and ad h oc networks is derived a nd gi ven in th e following prop osition. Pr oposition 2: Using the overlay m ethod o f frequ ency m u- tual exclusion , the cap acity of the cellu lar network λ u and th at of th e ad h oc n etwork λ a satisfies the following r elationship λ u α + λ a 1 − α ∼ = C 2 , ǫ → 0 (11) where C 2 = K ǫθ − δ π E [ G δ ]  1 d 2 E [ W − δ ] + 1 r 2 E [ Q − δ ]  . ( 12) As shown in Fig. 2, the capacity region f or the case o f frequen cy m utual exclusion is bounded b y the curve defined in ( 11) an d th e first quad rant of the coord inate system. Sim ilar comments as for the c ase of b lind tran smission are also applicable for the present case. By co mparing (7) and (1 1), it can b e observed that C 2 > C 1 and the facto r (1 − α ) increases the cap acity λ a . In oth er words, the capa city region for frequen cy mu tual exclusion is larger than that fo r blind transmission. A com parison of capacity regions is given in Section IV. D. Effect o f Spatial Diversity The capacity region o f the overlaid cellular and ad hoc networks can be en larged by enhancing the channel g ains of data link s W and Q by exploiting spatial diversity . Spatial div ersity is created using multiple antennas an d provides an effecti ve way of counter acting fading . Amo ng many av ailable spatial diversity technique s, we consider the simp lest one: 1 2 3 4 5 6 7 8 9 10 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Diversity Order, L 1 Capacity Region Parameter, C 1 (or C 2 ) Blind transmission Frequency mutual exclusion Fig. 3. The capacity regi on parameter C 1 (or C 2 ) vs. div ersity order L 1 for the data link in the ad hoc network. The data link in the cellular network has the di versity order L 2 = L 1 + 2 . Other parameters are chosen as d = 5 meters, θ = 3 , and K = 1000 and ǫ = 0 . 01 . beamfor ming [13]. Assum ing i. i.d. Rayleig h fading a nd beam - forming at either the receiver o r transmitter (on e-sided), the channel g ains W and Q follow the chi- squared d istribution with the num ber of com plex degree s of freed om equal to L 1 and L 2 , resp ectiv ely , which are referred to as diversity o rders. The diversity order for the interferen ce chann el G is assumed to b e o ne. Based o n th e ab ove model, the n etwork capacity regions in Proposition 1 and 2 have clo se-form expressions a s giv en in th e fo llowing lemma. Lemma 2 : For i.i.d. Ray leigh fading, data link s using one- sided b eamform ing, and r = d , the capacity tradeoff curve is giv en a s: 1) Blind tr ansmission: λ u α + λ a = ˜ C Γ( L min ) Γ( L min − δ ) , ǫ → 0 (13) where ˜ C = K ǫθ − δ [ π d 2 Γ(1 + δ )] − 1 . 2) Frequen cy m utual exclusion: λ u α + λ a 1 − α = ˜ C  Γ( L 1 ) Γ( L 1 − δ ) + Γ( L 2 ) Γ( L 2 − δ )  , ǫ → 0 . The proof is given in Append ix. Let C 1 and C 2 denote the terms on the right- hand-size of (1 3) and (14), respectively . For illustration, c onsider the d iv ersity or ders L 2 = L 1 + 2 , the d istance r = d = 5 me ters, the SIR thresh old θ = 3 , the number of sub-ch annels K = 100 0 , and the outag e co nstraint ǫ = 0 . 01 . Fig 3 plots C 1 and C 2 versus the diversity ord er L 1 . As observed from Fig 3, both C 1 and C 2 increases as L 1 grows due to the diversity g ains. More over , C 2 is larger tha n C 1 and their difference is relati vely larger for a smaller diversity order, indicating that the cap acity for frequen cy mutu al exclusion is higher than tha t f or blind transmission . I V . S I M U L AT I O N A N D N U M E R I C A L R E S U LT S Fig. 4 compares the values o f the c apacity region p arameter C 1 computed u sing (7) a nd th ose o btained by simulation. The diversity order is cho sen as L 1 = L 2 = 4 and o ther 10 −4 10 −3 10 −2 10 −1 10 0 10 −2 10 −1 10 0 10 1 Capacity Region Parameter, C 1 Outage Constraint, ε Simulation Asymptotic Approximation Fig. 4. Comparison of the approximate d and simulated values of the paramete r C 1 that defines the capaci ty regi on for the over laid cellular and ad hoc networks using blind transmission. The approxi mated values for C 1 are computed using (7). parameters have identical values as used for Fig. 3. As observed fro m Fig. 4 , the theor etical values obtain ed b ased on the asymp totical assum ption ( ǫ → 0 ) m atch the simulation results very well. A similar observation is also mad e for C 2 giv en in (11), but the detailed comparison is o mitted in this paper . T hese ob servations lead to the conclusion that the asymptotical results obtain in Proposition 1 and 2 are also applicable f or the non-asymp totic r egime of ǫ e.g. ǫ > 0 . 1 . Fig. 5 co mpares the cap acity regions for two network overlay methods: b lind transm ission and fr equency mutual exclusion. The values of parameter s are ide ntical to those used f or g enerating Fig. 4. As observed f rom Fig. 5 , the network overlay based on frequency m utual exclu sion results in a mu ch larger capacity r egion than that using blind trans- mission. Therefore, the form er is a mor e efficient meth od for implementin g th e overlaid cellular and ad hoc n etworks. Nev- ertheless, the method of f requen cy mutua l exclu sion requires nodes in the ad h oc n etwork to detect sub-chan nels u nused by mobile u sers, b ut such an operation is unn ecessary for blind transmission. Detection o f av ailable sub -chann els can be based on ca rrier sending or side inf ormation broadcast by base stations. A P P E N D I X P R O O F F O R L E M M A 2 The resu lt fo r the c ase of frequ ency mu tual exclusion is obtained b y sub stituting E [ G δ ] = Γ(1 + δ ) , E [ Q − δ ] = Γ( L 2 − δ ) , and E [ W − δ ] = Γ( L 2 − δ ) into (11). T o ob tain the result for the case of blind transm ission, it is sufficient to use (7) and prove th at Γ( L 1 ) Γ( L 1 − δ ) > Γ( L 2 ) Γ( L 2 − δ ) if L 1 > L 2 and v ise versa. W ithout lo ss of generality , assume L 1 > L 2 . Using the prop erty Γ(1 + x ) = x Γ( x ) and Γ( n ) = 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Capacity of Ad Hoc Network, λ a Capacity of Cellular Network, λ u Frequency mutual exclusion Blind transmission Fig. 5. Comparison between the capacity region s for two netwo rk overlay methods, namely blind transmission and frequency mutual excl usion. ( n − 1)! Γ( L 1 ) Γ( L 1 − δ ) = ( L 1 − 1)( L 1 − 2) · · · L 2 ( L 1 − 1 − δ )( L 1 − 2 − δ ) · · · ( L 2 − δ ) Γ( L 2 ) Γ( L 2 − δ ) ( a ) > Γ( L 2 ) Γ( L 2 − δ ) . The inequality (a) holds since δ > 0 . This completes the proo f. R E F E R E N C E S [1] H. Holma and A. T oskala, WCDMA for UMTS: Radio Access for Third Genera tion Mobi le Communications . John Wil ey & Sons,, 2004. [2] P . Marques, J . Bastos, and A. Gameiro, “Opportunistic use of 3G uplink licensed bands, ” in Pr oc., IEEE Intl. Conf. on Communications , pp. 3588–3592, May 2008. [3] D. Kim and D. 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